Review Article Hybrid Fundamental Solution Based Finite...

Review ArticleHybrid Fundamental Solution Based Finite Element MethodTheory and Applications

Changyong Cao and Qing-Hua Qin

Research School of Engineering The Australian National University Acton ACT 2601 Australia

Correspondence should be addressed to Qing-Hua Qin qinghuaqinanueduau

Received 13 October 2014 Revised 23 December 2014 Accepted 24 December 2014

Academic Editor Luigi C Berselli

Copyright copy 2015 C Cao and Q-H Qin This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An overview on the development of hybrid fundamental solution based finite element method (HFS-FEM) and its applicationin engineering problems is presented in this paper The framework and formulations of HFS-FEM for potential problem planeelasticity three-dimensional elasticity thermoelasticity anisotropic elasticity and plane piezoelectricity are presented In thismethod two independent assumed fields (intraelement filed and auxiliary frame field) are employedThe formulations for all casesare derived from the modified variational functionals and the fundamental solutions to a given problem Generation of elementalstiffness equations from the modified variational principle is also described Typical numerical examples are given to demonstratethe validity and performance of the HFS-FEM Finally a brief summary of the approach is provided and future trends in this fieldare identified

1 Introduction

A novel hybrid finite element formulation called the hybridfundamental solution based FEM (HFS-FEM) was recentlydeveloped based on the framework of hybrid Trefftz finiteelement method (HT-FEM) and the idea of the methodof fundamental solution (MFS) [1ndash5] In this method twoindependent assumed fields (intraelement filed and auxiliaryframe field) are employed and the domain integrals in thevariational functional can be directly converted to boundaryintegrals without any appreciable increase in computationaleffort as in HT-FEM [6ndash8] It should be mentioned that theintraelement field of HFS-FEM is approximated by the linearcombination of fundamental solutions analytically satisfyingthe related governing equation instead of 119879-complete func-tions as in HT-FEM The resulting system of equations fromthe modified variational functional is expressed in terms ofsymmetric stiffness matrix and nodal displacements onlywhich is easy to implement into the standard FEM It is notedthat no singular integrals are involved in the HFS-FEM bylocating the source point outside the element of interest anddo not overlap with field point during the computation [9]

The HFS-FEM mentioned above inherits all the advan-tages of HT-FEM over the traditional FEM and the boundaryelement method (BEM) namely domain decomposition andboundary integral expressions while avoiding the majorweaknesses of BEM [10ndash12] that is singular element bound-ary integral and loss of symmetry and sparsity [13] Theemployment of two independent fields also makes the HFS-FEM easier to generate arbitrary polygonal or even curve-sided elements It also obviates the difficulties that occur inHT-FEM [14 15] in deriving119879-complete functions for certaincomplex or new physical problems [16] The HFS-FEM hassimpler expressions of interpolation functions for intraele-ment fields (fundamental solutions) and avoids the coordi-nate transformation procedure required in the HT-FEM tokeep the matrix inversion stable Moreover this approachalso has the potential to achieve high accuracy using coarsemeshes of high-degree elements to enhance insensitivity tomesh distortion to give great liberty in element shape and toaccurately represent various local effects (such as hole crackand inclusions) without troublesome mesh adjustment [17ndash20] Additionally HFS-FEM makes it possible for a more

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 916029 38 pageshttpdxdoiorg1011552015916029

2 Advances in Mathematical Physics

flexible element material definition which is important indealingwithmultimaterial problems rather than thematerialdefinition being the same in the entire domain in BEMHowever we noticed that there are also some limitationsof HFS-FEM for example determining the positions ofsource points used for approximation interpolations It is alsoknown that fundamental solution based approximations canperform remarkably well in smooth problems but tend todeteriorate when high-gradient stress fields are presented

This paper is organized as follows in Section 2 thebasic idea and formulations of the HFS-FEM are presentedthrough a simple potential problem Then plane elasticityproblems are described in Section 3 Section 4 extends the 2Dformulations of the HFS-FEM to general three-dimensional(3D) elasticity problems The method of particular solutionand radial basis function approximation are shown to dealwith body force in this Section In Section 5 we extend theHFS-FEM to thermoelastic problems with arbitrary bodyforce and temperature change In Section 6 the HFS-FEMfor 2D anisotropic elastic materials is described based onthe powerful Stroh formalism Plane piezoelectric problemis discussed in Section 7 Finally typical numerical examplesare presented in Section 8 to illustrate applications and per-formance of the HFS-FEM Concluding remarks and futuredevelopment are discussed at the end of this paper

2 Potential Problems

21 Basic Equations of Potential Problems The Laplace equa-tion of a well-posed potential problem (eg heat conduction)in a general plane domainΩ can be expressed as [21 22]

nabla2119906 (x) = 0 forallx isin Ω (1)

with the boundary conditions

119906 = 119906 on Γ119906 (2)

119902 = 119906119894119899119894= 119902 on Γ

119902 (3)

where 119906 is the unknown field variable and 119902 represents theboundary flux 119899

119894is the 119894th component of outward normal

vector to the boundary Γ = Γ119906cup Γ119902 and 119906 and 119902 are specified

functions on the related boundaries respectively The spacederivatives are indicated by a subscript comma that is 119906

119894=

120597119906120597119909119894 and the subscript index 119894 takes values (1 2) for two-

dimensional and (1 2 3) for three-dimensional problemsAdditionally the repeated subscript indices imply summationconvention

For convenience (3) can be rewritten in matrix form as

119902 = A[

1199061

1199062

] = 119902 (4)

with A = [11989911198992]

22 Assumed Independent Fields In this section the pro-cedure for developing a hybrid finite element model withfundamental solution as interior trial function is described

based on the boundary value problem defined by (1)ndash(3)Similar to the conventional FEM and HT-FEM the domainunder consideration is divided into a series of elements [1516 21 23ndash30] In each element two independent fields areassumed in the way as described in [31] and are given inSection 22

221 Intraelement Field Similar to themethod of fundamen-tal solution (MFS) in removing singularities of fundamentalsolution for a particular element 119890 occupying subdomainΩ119890 we assume that the field variable defined in the element

domain is extracted from a linear combination of funda-mental solutions centered at different source points locatedoutside the element (see Figure 1)

119906119890(x) =

119899119904

sum

119895=1

119873119890(x y

119895) 119888119890119895= N

119890(x) c

119890

forallx isin Ω119890 y

119895notin Ω

119890

(5)

where 119888119890119895is undetermined coefficients 119899

119904is the number of

virtual sources and 119873119890(x y

119895) is the fundamental solution to

the partial differential equation

nabla2119873119890(x y) + 120575 (x y) = 0 forallx y isin R

2 (6)

as

119873119890(x y) = minus

1

2120587ln 119903 (x y) (7)

Evidently (5) analytically satisfies (1) due to the solutionproperty of119873

119890(x y

119895)

In implementation the number of source points is takento be the same as the number of element nodes which isfree of spurious energy modes and can keep the stiffnessequations in full rank as indicated in [21] The source pointy119904119895(119895 = 1 2 119899

119904) can be generated bymeans of themethod

employed in the MFS [32ndash35]

y119904= x0+ 120574 (x

0minus x119888) (8)

where 120574 is a dimensionless coefficient x0is the point on

the element boundary (the nodal point in this work) andx119888is the geometrical centroid of the element (see Figure 1)

Determination of 120574 was discussed in [31 36] and 120574 = 5ndash10 isusually used in practice

The corresponding outward normal derivative of 119906119890on Γ

119890

is

119902119890=120597119906119890

120597119899= Q

119890c119890 (9)

where

Q119890=120597N119890

120597119899= AT

119890(10)

with

T119890= [

120597N119890

1205971199091

120597N119890

1205971199092

]

119879

(11)

Advances in Mathematical Physics 3

1 2

34

Centroid

SourceNode

X2

X1

Intraelement field

Ωe

cx

0xΓe

u = Nece

Frame field u(x) = edeNsx

(a) 4-node 2D element

1 3

57

2

4

6

8

Centroid

SourceNode

X2

X1

Intraelement field

Ωe

Γe

Frame field u(x) = ede

u = Nece

N

cx

0x

sx

(b) 8-node 2D element

Figure 1 Intraelement field and frame field of a HFS-FEM element for 2D potential problems

222 Auxiliary Frame Field In order to enforce the con-formity on the field variable 119906 for instance 119906

119890= 119906

119891on

Γ119890cap Γ119891of any two neighboring elements 119890 and 119891 an auxiliary

interelement frame field is used and expressed in terms ofthe same degrees of freedom (DOF) d as those used in theconventional finite elements In this case is confined to thewhole element boundary as

119890(x) = N

119890(x) d

119890(12)

which is independently assumed along the element boundaryin terms of nodal DOF d

119890 where N

119890(x) represents the con-

ventional finite element interpolating functions For examplea simple interpolation of the frame field on a side with threenodes of a particular element can be given in the form

= 11199061+

21199062+

31199063 (13)

where 119894(119894 = 1 2 3) stands for shape functions in terms of

natural coordinate 120585 defined in Figure 2

23 Modified Variational Principle For the boundary valueproblem defined in (1)ndash(3) and (5) since the stationaryconditions of the traditional potential or complementaryvariational functional cannot guarantee the interelementcontinuity condition required in the proposedHFS FEmodelas in the HT FEM [21 26] a variational functional corre-sponding to the new trial functions should be constructedto assure the additional continuity across the common

N1

N2

N3

120585 = minus1 120585 = 0 120585 = +1

minus120585(1 minus 120585)

2

1 minus 1205852

120585(1 + 120585)

2

1 2 3

Figure 2 Typical quadratic interpolation for frame field

boundariesΓIef between intraelement fields of element ldquo119890rdquo andelement ldquo119891rdquo (see Figure 3) [36 37]

119906119890= 119906

119891(conformity)

119902119890+ 119902119891= 0 (reciprocity)

on ΓIef = Γ119890cap Γ119891

(14)


e f

ΓIef

Figure 3 Illustration of continuity between two adjacent elementsldquo119890rdquo and ldquo119891rdquo

Amodified variational functional is developed as follows

Π119898= sum

119890

Π119898119890

= sum

119890

Π119890+ intΓ119890

( minus 119906) 119902dΓ (15)

where

Π119890=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ (16)

in which the governing equation (1) is assumed to be satisfieda priori for deriving the HFS FE model The boundary Γ

119890of

a particular element consists of the following parts

Γ119890= Γ119906119890cup Γ119902119890cup ΓIe (17)

where ΓIe represents the interelement boundary of the ele-ment ldquo119890rdquo shown in Figure 3

To show that the stationary condition of the functional(15) leads to the governing equation (Euler equation) bound-ary conditions and continuity conditions invoking (16) and(15) gives the following functional for the problem domain

Π119898119890

=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ + intΓ119890

119902 ( minus 119906) dΓ (18)

from which the first-order variational yields

120575Π119898119890

= intΩ119890

119906119894120575119906119894dΩ minus int

Γ119902119890

119902120575dΓ + intΓ119890

(120575 minus 120575119906) 119902dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ(19)

Using divergence theorem

intΩ

119891119894ℎ119894dΩ = int

Γ

ℎ119891119894119899119894dΓ minus int

Ω

ℎnabla2119891dΩ (20)

we can obtain

120575Π119898119890

= intΩ119890

119906119894119894120575119906dΩ minus int

Γ119902119890

(119902 minus 119902) 120575dΓ + intΓ119906119890

119902120575dΓ

+ intΓIe

119902120575dΓ + intΓ119890

( minus 119906) 120575119902dΓ(21)

For the displacement-based method the potential confor-mity should be satisfied in advance

120575 = 0 on Γ119906119890

(∵ = 119906)

120575119890= 120575

119891 on ΓIef (∵ 119890=

119891)

(22)

then (21) can be rewritten as

120575Π119898119890

= intΩ119890

119906119894119894120575119906dΩ minus int

Γ119902119890

(119902 minus 119902) 120575dΓ + intΓIe

119902120575dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ(23)

The Euler equation and boundary conditions can be obtainedas

119906119894119894= 0 in Ω

119890

119902 = 119902 on Γ119902119890

= 119906 on Γ119890

(24)

using the stationary condition 120575Π119898119890

= 0As for the continuous requirement between two adjacent

elements ldquo119890rdquo and ldquo119891rdquo given in (14) we can obtain it in thefollowing way When assembling elements ldquo119890rdquo and ldquo119891rdquo wehave

120575Π119898(119890+119891)

= intΩ119890+119891

119906119894119894120575119906dΩ minus int

Γ119902119890+Γ119902119891

(119902 minus 119902) 120575dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ + intΓ119891

( minus 119906) 120575119902dΓ

+ intΓIef

(119902119890+ 119902119891) 120575

119890119891dΓ + sdot sdot sdot

(25)

from which the vanishing variation of Π119898(119890+119891)

leads to thereciprocity condition 119902

119890+ 119902

119891= 0 on the interelement

boundary ΓIefIf the following expression

intΓ119902

120575119902120575d119904

minussum

119890

[intΓIe

120575119902119890120575119890d119904 + int

Γ119890

120575119902119890120575 (

119890minus 119906119890) d119904]

(26)

is uniformly positive (or negative) in the neighborhood of1199060 where the displacement 119906

0has such a value that

Π119898(119906

0) = (Π

119898)0and where (Π

119898)0stands for the stationary

value of Π119898 we have

Π119898ge (Π

119898)0

or Π119898le (Π

119898)0

(27)

inwhich the relation that 119890=

119891is identical on Γ

119890capΓ119891has

been used This is due to the definition in (14) in Section 23


Proof For the proof of the theorem on the existence ofextremum we may complete it by the so-called ldquosecond vari-ational approachrdquo [38] In doing this performing variation of120575Π119898and using the constrained conditions (26) we find

1205752Π119898= intΓ119902

120575119902120575d119904

minussum

119890

[intΓIe

120575119902119890120575119890d119904 + int

Γ119890

120575119902119890120575 (

119890minus 119906119890) d119904]

(28)

Therefore the theorem has been proved from the sufficientcondition of the existence of a local extreme of a functional[38]

24 Element Stiffness Equation With the intraelement fieldand frame field independently defined in a particular element(see Figure 1) we can generate element stiffness equation bythe variational functional derived in Section 23 Followingthe approach described in [21] the variational functionalΠ119890corresponding to a particular element 119890 of the present

problem can be written as

Π119890=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ + intΓ119890

119902 ( minus 119906) dΓ (29)

Appling the divergence theorem (20) to the functional (29)we have the final functional for the HFS-FE model

Π119890=1

2[intΓ119890

119902119906dΓ + intΩ119890

119906119896nabla2119906dΩ] minus int

Γ119902119890

119902dΓ

+ intΓ119890

119902 ( minus 119906) dΓ

= minus1

2intΓ119890

119902119906dΓ minus intΓ119902119890

119902dΓ + intΓ119890

119902dΓ

(30)

Then substituting (5) (9) and (12) into the functional (30)produces

Π119890= minus

1

2c119879119890H119890c119890minus d119879119890g119890+ c119879119890G119890d119890

(31)

in which

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

N119879119890Q119890dΓ

G119890= intΓ119890

Q119879119890N119890dΓ g

119890= intΓ119902119890

N119879119890119902dΓ

(32)

The symmetry ofH119890is obvious from the scalar definition (31)

of variational functional Π119890

To enforce interelement continuity on the common ele-ment boundary the unknown vector c

119890should be expressed

in terms of nodal DOF d119890Theminimization of the functional

Π119890with respect to c

119890and d

119890 respectively yields

120597Π119890

120597c119890

119879= minusH

119890c119890+ G

119890d119890= 0

120597Π119890

120597d119890

119879= G119879

119890c119890minus g119890= 0

(33)

from which the optional relationship between c119890and d

119890and

the stiffness equation can be produced

c119890= Hminus1

119890G119890d119890 K

119890d119890= g119890 (34)

whereK119890= G119879

119890Hminus1119890G119890stands for the element stiffness matrix

25 Numerical Integral for H and G Matrix Generally it isdifficult to obtain the analytical expression of the integral in(32) and numerical integration along the element boundaryis required Herein the widely used Gaussian integration isemployed [22]

For theH119890matrix one can express it as

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x) dΓ (35)

by introducing the matrix function

F (x) = [119865119894119895(x)]

119898times119898= Q119879

119890N119890 (36)

Equation (36) can be further rewritten as

119867119894119895= intΓ119890

119865119894119895(x) dΓ =

119899119890

sum

119897=1

intΓ119890119897

119865119894119895(x) dΓ (37)

where

dΓ = radic(d1199091)2

+ (d1199092)2

= radic(d1199091

d120585)

2

+ (d1199092

d120585)

2

d120585 = 119869d120585

(38)

and 119869 is the Jacobean expressed as

119869 = radic(d1199091

d120585)

2

+ (d1199092

d120585)

2

(39)

where

[d1199091

d120585d1199092

d120585]

119879

=

119899119900

sum

119894=1

d119873119894(120585)

d120585

1199091119894

1199092119894

(40)

Thus the Gaussian numerical integration forHmatrix can becalculated by

119867119894119895=

119899119890

sum

119897=1

[int

+1

minus1

119865119894119895(x (120585)) 119869 (120585) d120585]

asymp

119899119890

sum

119897=1

[

119899119901

sum

119896=1

119908119896119865119894119895(x (120585

119896)) 119869 (120585

119896)]

(41)

where 119899119890is the number of edges of the element and 119899

119901

is the Gaussian sampling points employed in the Gaussiannumerical integration Similarly we can calculate the G

119890

matrix using

119866119894119895=

119899119890

sum

119897=1

[int

1

minus1

119865119894119895[x (120585)] 119869 (120585) d120585]

asymp

119899119890

sum

119897=1

119899119901

sum

119896=1

119908119896119865119894119895[x (120585

119896)] 119869 (120585

119896)

(42)


The calculation of vector g119890in (32) is the same as that

in the conventional FEM so it is convenient to incorporatethe proposed HFS-FEM into the standard FEM programBesides the flux is directly computed from (9)The boundaryDOF can be directly computed from (12) while the unknownvariable at interior points of the element can be determinedfrom (5) plus the recovered rigid-body modes in eachelement which is discussed in the following section

26 Recovery of Rigid-BodyMotion Considering the physicaldefinition of the fundamental solution it is necessary torecover themissing rigid-bodymotionmodes from the aboveresults Following the method presented in [21] the missingrigid-body motion can be recovered by writing the internalpotential field of a particular element 119890 as

119906119890= N

119890c119890+ 1198880 (43)

where the undetermined rigid-bodymotion parameter 1198880can

be calculated using the least square matching of 119906119890and

119890at

element nodes119899

sum

119894=1

(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894

= min (44)

which finally gives

1198880=1

119899

119899

sum

119894=1

Δ119906119890119894 (45)

in which Δ119906119890119894

= (119890minus N

119890c119890)|node119894 and 119899 is the number

of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c

119890

can be evaluated from (34) and then 1198880is evaluated from (45)

Finally the potential field119906 at any internal point in an elementcan be obtained by means of (43)

3 Plane Elasticity Problems

31 Linear Theory of Plane Elasticity In linear elastic theorythe strain displacement relations can be used and equilibriumequations refer to the undeformed geometry [39] In therectangular Cartesian coordinates (119883

1 1198832) the governing

equations of a plane elastic body can be expressed as

120590119894119895119895

= 119887119894 119894 119895 = 1 2 (46)

If written as matrix form it can be presented as

L120590 = b (47)

where 120590 = [12059011

12059022

12059012]119879 is a stress vector b = [119887

1 1198872]119879 is

a body force vector and the differential operator matrix L isgiven as

L =[[[

[

120597

1205971199091

0120597

1205971199092

0120597

1205971199092

120597

1205971199091

]]]

]

(48)

120576 = LTu (49)

where 120576 = [12057611

12057622

12057612]119879 is a strain vector and u = [119906

1 1199062]119879

is a displacement vectorThe constitutive equations for the linear elasticity are

given in matrix form as

120590 = D120576 (50)

where D is the material coefficient matrix with constantcomponents for isotropic materials which can be expressedas follows

D =[[[

[

+ 2119866 0

+ 2119866 0

0 0 119866

]]]

]

(51)

where

=2]

1 minus 2]119866 119866 =

119864

2 (1 + ])

] =

] for plane strain]

1 + ]for plane stress

(52)

The two different kinds of boundary conditions can beexpressed as

u = u on Γ119906

t = A120590 = t on Γ119905

(53)

where t = [11990511199052]119879 denotes the traction vector and A is a

transformation matrix related to the direction cosine of theoutward normal

A = [

1198991

0 1198992

0 11989921198991

] (54)

Substituting (49) and (50) into (47) yields the well-knownNavier partial differential equations in terms of displace-ments

LDL119879u = b (55)

32 Assumed Independent Fields For elasticity problem twodifferent assumed fields are employed as in potential prob-lems intraelement and frame field [1 31 36] The intraele-ment continuity is enforced on nonconforming internaldisplacement field chosen as the fundamental solution ofthe problem [36] The intraelement displacement fields areapproximated in terms of a linear combination of fundamen-tal solutions of the problem of interest

u (x) =

1199061(x)

1199062(x)

=

119899119904

sum

119895=1

[

[

119906lowast

11(x y

119904119895) 119906

lowast

12(x y

119904119895)

119906lowast

21(x y

119904119895) 119906

lowast

22(x y

119904119895)

]

]

1198881119895

1198882119895

= Nece (x isin Ω119890 y119904119895notin Ω

119890)

(56)


where 119899119904is again the number of source points outside the

element domain which is equal to the number of nodes ofan element in the present research based on the generationapproach of the source points [31] The vector ce and thefundamental solution matrix Ne are now in the form

Ne

= [

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) sdot sdot sdot 119906

lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

)

119906lowast

21(x y

1199041) 119906

lowast

22(x y


lowast

21(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

)

]

]

ce = [11988811 11988821

sdot sdot sdot 1198881119899

1198882119899]119879

(57)

in which x and y119904119895

are respectively the field point andsource point in the local coordinate system (119883

1 1198832) The

components 119906lowast119894119895(x y

119904119895) are the fundamental solution that is

induced displacement component in 119894-direction at the fieldpoint x due to a unit point load applied in 119895-direction at thesource point y

119904119895 which are given by [40 41]

119906lowast

119894119895(x y

119904119895) =

minus1

8120587 (1 minus ]) 119866(3 minus 4]) 120575

119894119895ln 119903 minus 119903

119894119903119895 (58)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032

2 The virtual source points

for elasticity problems are generated in the same manner asthat in potential problems described in Section 2

With the assumption of intraelement field in (56) thecorresponding stress fields can be obtained by the constitutiveequation (50)

120590 (x) = [12059011 12059022

12059012]119879

= Tece (59)

whereTe

=[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

)

]]]

]

(60)

As a consequence the traction is written as

1199051

1199052

= n120590 = Qece (61)

in which

Qe = nTe n = [

1198991

0 1198992

0 11989921198991

] (62)

The components 120590lowast119894119895119896(x y) for plane strain problems are given

as

120590lowast

119894119895119896(x y) = minus1

4120587 (1 minus ]) 119903

sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903

119894119903119895119903119896]

(63)

The unknown ce in (56) is calculated by a hybrid tech-nique [31] in which the elements are linked through anauxiliary conforming displacement framewhich has the sameform as that in conventional FEM (see Figure 1) This meansthat in the HFS-FEM a conforming displacement fieldshould be independently defined on the element boundary toenforce the field continuity between elements and also to linkthe unknown c with nodal displacement d

119890 Thus the frame

is defined as

u (x) =

1

2

=

N1

N2

d119890= N

119890d119890 (x isin Γ

119890) (64)

where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N

119890is the matrix of shape

functions and d119890is the nodal displacements of elements

Taking the side 3-4-5 of a particular 8-node quadrilateralelement (see Figure 1) as an example N

119890and d

119890can be

expressed as

N119890= [

[

0 sdot sdot sdot 0 1

0 2

0 3

0 0 sdot sdot sdot 0

0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

4

0 1

0 2

0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

6

]

]

de = [11990611 11990621

11990612

11990622


11990628]119879

(65)

where 1 2 and

3can be expressed by natural coordinate

as

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(66)

33 Modified Functional for the Hybrid FEM As in Section 2HFS-FE formulation for a plane elastic problem can also beestablished by the variational approach [36] In the absenceof body forces the hybrid functional Π

119898119890used for deriving

the present HFS-FEM can be constructed as [22]

Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ (67)

where 119894and 119906

119894are the intraelement displacement field

defined within the element and the frame displacementfield defined on the element boundary respectively Ω

119890

and Γ119890are the element domain and element boundary

respectively Γ119905 Γ119906 and Γ

119868stand respectively for the specified

traction boundary specified displacement boundary andinterelement boundary (Γ

119890= Γ

119905+ Γ

119906+ Γ

119868) Compared

to the functional employed in the conventional FEM thepresent variational functional is constructed by adding ahybrid integral term related to the intraelement and elementframe displacement fields to guarantee the satisfaction ofdisplacement and traction continuity conditions on the com-mon boundary of two adjacent elements


By applying the Gaussian theorem (67) can be simplifiedto

Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ) minus int

Γ119905

119905119894119894dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ

(68)

Due to the satisfaction of the equilibrium equation withthe constructed intraelement field we have the followingexpression for HFS finite element model

Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(69)

The variational functional in (69) contains boundary inte-grals only and will be used to derive HFS-FEM formulationfor the plane isotropic elastic problem

34 Element Stiffness Matrix As in Section 2 the elementstiffness equation can be generated by setting 120575Π

119898119890= 0

Substituting (56) (64) and (61) into the functional of (69)we have

Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (70)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(71)

To enforce interelement continuity on the common elementboundary the unknown vector c

119890should be expressed in

terms of nodal DOF d119890 The stationary condition of the

functional Π119898119890

with respect to c119890and d

119890yields respectively

(33) and (34)

35 Recovery of Rigid-Body Motion For the same reasonstated in Section 26 it is necessary to reintroduce thediscarded rigid-body motion terms after we have obtainedthe internal field of an elementThe least squares method canbe employed for this purpose and the missing terms can berecovered easily by setting for the augmented internal field[22]

u119890= N

119890c119890+ [

1 0 1199092

0 1 minus1199091

] c0 (72)



119890at

element nodes119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

] = min (73)

which finally gives

c0 = Rminus1119890re (74)

where

Re =119899

sum

119894=1

[[[

[

1 0 1199092119894

0 1 minus1199091119894

1199092119894

minus1199091119894

1199092

1119894+ 1199092

2119894

]]]

]

re =119899

sum

119894=1

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

(75)

in which Δ119906119890119895119894

= (119890119895119894minus 119890119895119894) (119895 = 1 2) and 119899 is the number


119890


Finally the displacement field u119890at any internal point in an

element can be obtained by (72)

4 Three-Dimensional Elastic Problems

In this section the HFS-FEM approach is extended to three-dimensional (3D) elastic problem withwithout body forceThe detailed 3D formulations of HFS-FEM are firstly derivedfor elastic problems by ignoring body forces and then aprocedure based on the method of particular solution andradial basis function approximation are introduced to dealwith the body force [42] As a consequence the homogeneoussolution is obtained by using theHFS-FEMand the particularsolution associated with body force is approximated by usingthe strong form of basis function interpolation

41 Governing Equations and Boundary Conditions Let(1199091 1199092 1199093) denote the coordinates in a Cartesian coordinate

system and consider a finite isotropic body occupying thedomain Ω as shown in Figure 4 The equilibrium equationfor this finite body with body force can be expressed as

120590119894119895119895

= minus119887119894

119894 119895 = 1 2 3 (76)

The constitutive equations for linear elasticity and thekinematical relation are given as

120590119894119895=

2119866V1 minus 2V

120575119894119895119890119896119896+ 2119866119890

119894119895

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(77)

where 120590119894119895is the stress tensor 119890

119894119895is the strain tensor and

120575119894119895is the Kronecker delta Substituting (77) into (76) the

equilibrium equation is rewritten as

119866119906119894119895119895

+119866

1 minus 2V119906119895119895119894

= minus119887119894 (78)


Γux1

x2

x3

b1

b2

b3

Γt

O

Figure 4 Geometrical definitions and boundary conditions for ageneral 3D solid

For a well-posed boundary value problem the boundaryconditions are prescribed as follows

119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(79)

where Γ119906cupΓ119905= Γ is the boundary of the solution domainΩ 119906

119894

and 119905119894are the prescribed boundary values In the following

parts we will present the procedure for handling the bodyforce appearing in (78)

42 The Method of Particular Solution The inhomogeneousterm 119887

119894associated with the body force in (78) can be

effectively handled by means of the method of particularsolution presented in [22] In this approach the displacement119906119894is decomposed into two parts a homogeneous solution 119906ℎ

119894

and a particular solution 119906119901119894

119906119894= 119906

ℎ

119894+ 119906119901

119894 (80)

where the particular solution 119906119901119894should satisfy the governing

equation

119866119906119901

119894119895119895+

119866

1 minus 2V119906119901

119895119895119894= minus119887

119894(81)

without any restriction of boundary condition However thehomogeneous solution should satisfy

119866119906ℎ

119894119895119895+

119866

1 minus 2V119906ℎ

119895119895119894= 0 (82)

with the modified boundary conditions

119906ℎ

119894= 119906

119894minus 119906119901

119894on Γ

119906

119905ℎ

119894= 119905119894minus 119905119901

119894on Γ

119905

(83)

From the above equations it can be seen that once theparticular solution 119906

119901

119894is known the homogeneous solution

119906ℎ

119894in (82) and (83) can be obtained using HFS-FEM The

final solution can then be given by (80) In the next sectionradial basis function approximation is introduced to obtainthe particular solution and the HFS-FEM is presented forsolving (82) and (83)

43 Radial Basis Function Approximation For body force 119887119894

it is generally impossible to find an analytical solution whichenables us to convert the domain integral into a boundaryone So we must approximate it by a combination of basis(trial) functions or other methods with the HFS-FEM Herewe use radial basis function (RBF) [33 43] to interpolate thebody force We assume

119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (84)

where 119873 is the number of interpolation points 120593119895 arethe RBFs and 120572

119895

119894are the coefficients to be determined

Subsequently the particular solution can be approximated by

119906119901

119894=

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (85)

where Φ119895

119894119896is the approximated particular solution kernel

of displacement satisfying (86) below Once the RBFs areselected the problem of finding a particular solution isreduced to solve the following equation

119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (86)

To solve this equation the displacement is expressed interms of the Galerkin-Papkovich vectors

Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(87)

Substituting (87) into (86) we can obtain the followingbiharmonic equation

nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (88)

Taking the Spline Type RBF 120593 = 1199032119899minus1 as an example we get

the following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (90)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(91)


and 119903119895represents the Euclidean distance between a field point

(1199091 1199092 1199093) and a given point (119909

1119895 1199092119895 1199093119895) in the domain of

interestThe corresponding particular solution of stresses canbe obtained as

119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(92)

where 120582 = (2V(1minus2V))119866 Substituting (90) into (92) we have

119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897 (93)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(94)

44 HFS-FEM for Homogeneous Solution After obtainingthe particular solution in Sections 42 and 43 we can

determine the modified boundary conditions (83) Finallywe can treat the 3D problem as a homogeneous problemgoverned by (82) and (83) by using the HFS-FEM presentedbelow It is clear that once the particular and homogeneoussolutions for displacement and stress components at nodalpoints are determined the distribution of displacement andstress fields at any point in the domain can be calculated usingthe element interpolation functionHowever for 3D elasticityproblems in the absent of body force that is 119887

119894= 0 the

procedures in Sections 42 and 43 will become unnecessaryand we can employ the following procedures to find thesolution directly

441 Assumed Intraelement and Auxiliary Frame Fields Theintraelement displacement fields are approximated in termsof a linear combination of fundamental solutions of theproblem as

u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (95)

where the matrix Ne and unknown vector ce can be furtherwritten as

Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(96)



1 1199092) The

fundamental solution 119906lowast119894119895(x y

119904119895) is given by [40]

119906lowast

119894119895(x y

119904119895) =

1

16120587 (1 minus ]) 119866119903(3 minus 4]) 120575

119894119895+ 119903119894119903119895 (97)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032

2+ 1199032

3 119899119904is the number of

source points The source point y119904119895(119895 = 1 2 119899

119904) can also

be generated by means of the following method [36] as intwo-dimensional cases

y119904= x0+ 120574 (x

0minus x119888) (98)

where 120574 is a dimensionless coefficient x0is the point on the

element boundary (the nodal point in this work) and x119888is

the geometrical centroid of the element (see Figure 5)According to (50) and (49) the corresponding stress

fields can be expressed as

120590 (x) = [12059011 12059022

12059033

12059023

12059031

12059012]119879

= Tece (99)

where

Te =

[[[[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

133(x y

1) 120590

lowast

233(x y

1) 120590

lowast

333(x y


lowast

133(x y

119899119904

) 120590lowast

233(x y

119899119904

) 120590lowast

333(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]]]]

]

(100)


1

4 3

2

Source pointsNodeCentroid

5 6

8 7

8-node 3D element

X2

X1

X3

Intraelement field

Ωe

Γe

Frame field

u = Nece

u(x) = edeN

cx

0x

sx

(a)

1

7 5

2

13 15

1917

3

4

6

89 10

1112

1416

1820

20-node 3D element

X2

X1

X3

Ωe

(b)

Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)

The components 120590lowast119894119895119896(x y) are given by

120590lowast

119894119895119896(x y) = minus1

8120587 (1 minus ]) 1199032

sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903

119894119903119895119903119896

(101)

As a consequence the traction can be written in the form

1199051

1199052

1199053

= n120590 = Qece (102)

in which

Qe = nTe n =[[

[

1198991

0 0 0 11989931198992

0 1198992

0 1198993

0 1198991

0 0 119899311989921198991

0

]]

]

(103)

To link the unknown c119890and the nodal displacement d

119890

the frame is defined as

u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (104)

where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N

119890is the matrix

of shape functions and d119890is the nodal displacements of

elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N

119890and

vector d119890can be expressed as

N119890= [0 N

1N20 0 N

4N30]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(105)

where the shape functions are expressed as

N119894=[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(106)

where 119894(119894 = 1ndash4) can be expressed by natural coordinate

120585 120578 isin [minus1 1]

1=(1 + 120585) (1 + 120578)

4

2=(1 minus 120585) (1 + 120578)

4

3=(1 minus 120585) (1 minus 120578)

4

4=(1 + 120585) (1 minus 120578)

4

(107)

and (120585119894 120578119894) is the natural coordinate of the 119894-node of the

element (Figure 6)


1 3

57

2

4

6

8

(0minus1)(minus1minus1) (1minus1)

(10)

(11)(01)

(minus10)

(minus11)

120578

120585

Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements

442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ

119898119890used

for deriving the present HFS-FEM is written as [22]

Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)

By applying the Gaussian theorem (108) can be simplified as

Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)

Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model

Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)

The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection

443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have

Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again

respectively (33) and (34)

444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by

V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V

120578are the tangential vectors in the 120585-direction

and 120578-direction respectively calculated by

V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)

where 119899119889is the number of nodes of the surface and (119909

119894 119910119894 119911119894)

are the nodal coordinates Thus the unit normal vector isgiven by

119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)

is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)


For the119867matrix we introduce the matrix function

F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)

and we rewrite it to the component form as

119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)

Using the relationship

d119878 = 119869 (120585 120578) d120585 d120578 (120)

and the Gaussian numerical integration we can obtain

119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899

119901are respectively the number of surface of

the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G

119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)

It should be mentioned that the calculation of vector g119890

in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section

445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field

u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)

and using a least-square procedure tomatch 119906119890ℎand

119890ℎat the

nodes of the element boundary

min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)

The above equation finally yields

c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)

5 Thermoelasticity Problems

Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4

51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909

1 1199092 1199093)

denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as

120590119894119895119895

= minus119887119894 (127)



119894is the body force vector

and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as

120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)

where 119890119894119895is the strain tensor 119906

119894is the displacement vector 119879

is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575

119894119895is the Kronecker delta and

119898 =2119866120572 (1 + V)(1 minus 2V)

(129)

is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as

119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)

For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as

119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)

where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω

119906119894and 119905

119894are the prescribed boundary values and

119905119894= 120590119894119895119899119895 (132)

is the boundary traction in which 119899119895denotes the boundary

outward normal

52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879

119894minus 119887

119894can be eliminated by employing

the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906

119894into

two parts the homogeneous solution 119906ℎ

119894and the particular

solution 119906119901119894as follows

119906119894= 119906

ℎ

119894+ 119906119901

119894(133)

in which the particular solution 119906119901119894should satisfy the govern-

ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)

but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy

119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)

with modified boundary conditions

119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)

From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906

ℎ

119894in (135)ndash

(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)

53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887

119894and the temperature field 119879

in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887

119894and the temperature field 119879 separately as

in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]

Here we demonstrated that the performance of the latter oneis usually better than the former one

531 Interpolating Temperature and Body Force SeparatelyThe body force 119887

119894and temperature 119879 are assumed to be by

the following two equations

119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)

where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572

119895

119894and 120573

119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows

119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895

119894are the approximated particular solution

kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations

119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)

To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]

Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)

Substituting (142) into (140) one can obtain the followingbiharmonic equation

nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)


If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the

following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)

for two-dimensional problem and

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)

for three-dimensional problem where 119903119895represents the

Euclidean distance of the given point (1199091 1199092) from a fixed

point (1199091119895 1199092119895) in the domain of interest The corresponding

stress particular solution can be obtained by

119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)

where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain

119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)

for two-dimensional problem and substituting (147) into(149) one can obtain

119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)

for three-dimensional problemTo solve (141) one can treat Ψ

119894as the gradient of a scalar

function

Ψ119894= 119880

119894 (154)

Substituting (154) into (141) obtains the Poissonrsquos equation

nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)

Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained

[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)

Then from (154) we can get Ψ119894as follows

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)

The corresponding stress particular solution can be obtainedby substituting (147) into

119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903

119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)


532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879

119894minus 119887119894together by the following

equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)

Thus the approximate particular solution can be written as

119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)

Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895

119894119896and 119878

119897119894119895 which are the same as those for

body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution

Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions

6 Anisotropic Composite Materials

In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]

In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical

methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]

61 Linear Anisotropic Elasticity

611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909

1 1199092 1199093) if we neglect the body force

119887119894 the equilibrium equations stress-strain laws and strain-

displacement equations for anisotropic elasticity are [61]

120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897

is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as

119862119894119895119896119897

119906119896119895119897

= 0 (165)

The boundary conditions of the boundary value problem(163)ndash(165) are

119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905

119894are the prescribed boundary displacement

and traction vector respectively In addition 119899119894is the unit

outward normal to the boundary and Γ = Γ119906+ Γ

119905is the

boundary of the solution domainΩFor the generalized two-dimensional deformation of

anisotropic elasticity 119906119894is assumed to depend on 119909

1and 119909

2

only Based on this assumption the general solution to (165)can be written as [61 62]

u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)

where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =

(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =

[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of

three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3

which is an arbitrary function with argument 119911120572

= 1199091+

1199011205721199092and will be determined by satisfying the boundary and

loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901

120572are the material

eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901

120572 which can be

obtained by the following Eigen relations [61]

N120585 = 119901120585 (168)


where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by

N = [

N1 N2

N3 NT1] 120585 =

ab (169)

where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT

minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862

119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)

The stresses can be obtained from the derivation of stressfunctions 120593 as follows

1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where

119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)

612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901

1 1199012 1199013) applied

at an internal point x = (1199091 1199092) far from the boundary The

boundary conditions of this problem can be written as

int119862

d120601 = p for any closed curve 119862 enclosing x

int119862

du = p for any closed curve 119862

limxrarrinfin

120590119894119895= 0

(173)

Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]

119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)

Therefore fundamental solutions of the problem can beexpressed as

u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)

The corresponding stress components can be obtained fromstress function 120601 as

120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)

where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879

respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose

y

120593 x1

x2

x

Figure 7 Schematic of the relationship between global coordinatesystem (119909

1 1199092) and local material coordinate system (119909 119910)

613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909

1 1199092 and 119909

3of the structural element rather than to

coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame

For the two coordinate systems mentioned in Figure 7the angle between the axis-119909

1and axis-119909 is denoted by 120593

which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by

[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)

where the transformation matrix T and its inverse matrix aredefined as

T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)


with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by

[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1C (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(179)

62 Formulations of HFS-FEM

621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated

in terms of a linear combination of fundamental solutions ofthe problem as

u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)

in which 119899119904is the number of source points x and y

119904119895are

respectively the field point and source point in the coordinatesystem (119909

1 1199092) local to the element under considerationThe

fundamental solution 119906lowast

119894119895(x y

119904119895) is given by (175) for general

elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections

The corresponding stress fields can be expressed as

120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)

The components 120590lowast

119894119895119896(x y) are given by (176) when p

119894is

selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively

As a consequence the traction can be written as

1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)

The unknown c119890in (180) and (182) may be calculated using

a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as

u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)




Taking the side 3-4-5 of a particular 8-node quadrilateral


element (see Figure 1) as an example N119890and d

119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and

3are expressed by natural coordinate 120585 isin

[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)

622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π

119898119890for a particular

element 119890 is constructed as

Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)

where the boundary Γ119890of the element 119890 is

Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)

and Γ119890119868is the interelement boundary of element 119890 Performing

a variation of Π119898 one obtains

120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)

Applying Gaussian theorem

∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)

and the definitions of traction force

119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)

Then substituting (195) into (192) gives

120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)

Considering the fact that

intΓ119890119906

119905119894120575119894dΓ = 0 (197)

we can finally obtain the following form

120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)

Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906

119894 120575119905119894 and 120575

119894may be arbitrary

As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891

120575Π119898(119890cup119891)

= minus∬Ω119890cupΩ119891

120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)

This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations

623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows

Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)

Substituting (180) (184) and (186) into the functional (200)yields the formulation as

Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)

The stationary condition of the functional Π119898119890

with respectto c

119890and d

119890 respectively yields (33) and (34)


7 Piezoelectric Materials

Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections

71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909

119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)


119894is the electric displacement

vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as

120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897

is the elasticity tensormeasured under zero electricfield 119890

119894119896119897and 120581

119894119895are respectively the piezoelectric tensor

and dielectric tensor measured under zero strain 120576119894119895and

119864119894are the elastic strain tensor and the electric field vector

respectivelyThe relation between the strain tensor 120576119894119895and the

displacement 119906119894is given by

120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)

and the electric field component 119864119894is related to the electric

potential 120601 by119864119894= minus120601

119894 (206)

The boundary conditions of the boundary value problem(203)ndash(206) can be defined by

119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)

where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-

ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899

119894is the unit outward normal to the boundary

and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)

is the boundary of the solution domainΩ

For the transversely isotropic material if 1199091-1199092is taken

as the isotropic plane one can employ either 1199091-1199093or 119909

2-1199093

plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576

22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)

For the plane stress piezoelectric problem (12059022= 120590

32= 120590

12=

0 and 1198632= 0) the constitutive equations can be obtained by

replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and

12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)

72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]

721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain

ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)

where the fundamental solution matrix Ne is now given by


Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)


are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909

1 1199092) The component

119906lowast

119894119895(x y

119904119895) is the induced displacement component (119894 = 1 2)

or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y

119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three

different roots of the characteristic equation 1198861199046119894minus 119887119904

4

119894+ 119888119904

3

119894minus

119889 = 0 The source point y119904119895(119895 = 1 2 119899

119904) can be generated

by the following method [36]

y119904= x0+ 120574 (x

0minus x119888) (217)

Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as

120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)

in which 120590lowast

119894119895(x y

119904119895) denotes the corresponding stress

components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point

load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting


the fundamental solutions into constitutive equations[85]

120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)

in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872

13are

defined as in literature [84]From (204) (208) and (209) the generalized traction

forces and electric displacement are given as

1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)

722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as

u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)

where N119890is a matrix of the corresponding shape functions

For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N

119890and nodal vector de

can be given in the form

N119890

=

[[[[

[


0 0 2

0 0 3

0 0 0 sdot sdot sdot 0

0 sdot sdot sdot 0 0 1

0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621

1206011sdot sdot sdot 119906

1411990624


1811990628

1206018]119879

(224)

where the shape functions N119890are expressed by natural

coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)

73 HFS-FEM Formulations

731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]

119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ

119890cap Γ119891 conformity) (226)

119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ

119890cap Γ119891 reciprocity)

(227)

where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational

principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]

Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π



Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)

and the boundary Γ119890of the element 119890 is

Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863

cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)

and Γ119890119868is the interelement boundary of element 119890 Compared

to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements

It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π

119898 one obtains [86]

120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)

in which the first term is

120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

minus intΓ119890119863

119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)


and the definitions of traction force and electrical displace-ment

119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)

we finally get the following form

120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)

Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906

119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601

may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891

120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)

which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme

732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π

119898119890= 0 To simplify the

derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as

Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)

Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model

Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)

Substituting (214) (223) and (221) into the above functional(241) yields the formulation as

Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10

Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading

0 1000 2000 3000 400030

32

34

36

38

40

42

44

46

Number of degrees of freedomHFS-FEMC3D8Hybrid EAS

Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125

Number of degrees of freedom

Reference

Stre

ss12059011

(MPa

)

HFS-FEMC3D8Hybrid EAS

(b)

Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590

11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)

Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)


u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)

Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590

11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)

Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)


with respectto c

119890and d

119890 respectively yields (33) and (34) Following

the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as

u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)

where the undetermined rigid-bodymotion parameter c0can

be calculated using the least square matching of ue and ue atelement nodes

min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u

119890minus u119890)|nodei and 119899 is the number of element

nodes As a consequence c0can be calculated by (246) once


50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05

Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution

120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)

Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius

120590r

0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22

20

15

10

5

020151050

y

x

(a)

120590t

4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9

20

15

10

5

020151050

y

x

(b)

Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)

the nodal DEP fields d119890and the interpolation coefficients c

119890

are respectively determined by (214) and (223) Then thecomplete DEP fields u

119890can be obtained from (244)

74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This

will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888

0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10

minus9(CmV) and 120576

0= 10

minus3(Vm)

respectively The reference values of other quantities as


1

1

1

T = 30x3

b3 = minus2000[(x minus 05)2 + (y minus 05)2]

x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)

Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)

shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10

0(m)of the problem so that the normalized governing

equations remain in exactly the same form as the originalequations

8 Numerical Examples

Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples

81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590

11at point 119860 of the block

which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for


z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)

Figure 16 (a) Displacement 1199063and (b) stress 120590

33along one cube edge when subjected to arbitrary temperature and body force

Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and

1199090

Displacement 1199060= 119909

01205760= 10

minus3(m) Electric Potential 120601

0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10

8(Nm2) Electric induction 119863

0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10

minus11(m2N) Impermeability 120573

0= 119864

01198630= 10

9(mVC)

Electric field 1198640= 120590

01198900= 10

7(Vm) Piezoelectric strain constant 119892

0= 119864

01205900= 10

minus1(mVN)

comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906

1

and 12059011obtained by HFS-FEM on Mesh 3 are also presented

in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]

82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879

0= 10 The two approaches listed in

Section 4 to approximate the body force and temperature arediscussed and analyzed in this example

Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy

Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable

83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =

5000 Poissonrsquos ratio ] = 03 and linear thermal expansion


L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)

Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements

coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be

119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)

Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15

Figure 16 presents the displacement 1199063and stress 120590

33

along one edge of the cube which is coinciding with 1199093axis

It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes

84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform


0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7

120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)

120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)

Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)

Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593

tension of 1205900

= 1GPa is applied in 1199092direction The

material parameters of the orthotropic plate are taken as 1198641=

113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045

The length and width of the plate are 119871 = 20mm and 119882 =

20mm the major and minor lengths of the ellipses 119886 and 119887

are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17

Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590

11 12059022 and 120590

12around the elliptic hole

in the composite plate for several different fiber angles areshown in Figure 19

Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90

∘ whereas the smallest appears at120593 = 0

∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well

85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21

an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =

3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864

2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=

0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements

In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901

120572= 119894 (120572 = 1 2 3) [59] However

a small perturbation of the material constants such as 1199011=

(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to

make the eigenvalues distinct and the results can be appliedconveniently

Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909

2= 0) and are 07

and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906

1and

1199062along the right edge (119909 = 8) by HFS-FEM are shown in

Figure 22

86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin

119909119909= 1205900= 10 parallel to the 119909

axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis

Figure 24 shows the distribution of hoop stress 120590120579and

radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin

119909119909

and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively

Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863

1205791205900times10

10

along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]

It can be seen from Figure 24 that hoop stress 120590120579has

maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590

120579tends to be equal to the remote

applied load 1205900when 119903 increases toward infinity Compared

with the hoop stress 120590120579 it is obvious that the radial stress 120590

119903

is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect


S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)

Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45

∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)

Figure 20 Variation of SCF with the lamina angle 120593

W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6

minus886420minus2minus4minus6minus8Z

(b)

Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions

Table 2 Comparison of displacement and stress at points A and B

Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )

Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)

x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8

(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)

Figure 22 The variation of displacement component (a) 1199061and

(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and

ABAQUS

Table 3 Properties of the material PZT-4 used for the model

Parameters Values Parameters Values11988811

139 times 1010 Nmminus2 119890

151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm

The maximum values of 120590120579appear at 120579 = 90

∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180

∘ for case of loading 120590infin

119911119911 both

of which agree well with the analytical solution from Sosa[74]The electric displacement119863

1205791205900times10

10 produced by 120590infin119909119909

and 120590infin119911119911

is nearly the same and is symmetrical with respect to

x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z

(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS


0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32

120590xx (HFS-FEM)120590zz (HFS-FEM)

120590xx (ABAQUS)120590zz (ABAQUS)

120590xx (Sosa)120590zz (Sosa)

Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)

Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590

119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin

119909119909and

along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911

000 025 050 075 100minus2

minus1

0

1

2

3

4

120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)

120590infinxx (ABAQUS)120590infinzz (ABAQUS)

120590infinxx (Sosa)120590infinzz (Sosa)

Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)

Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863

1205791205900times 10

10 along the hole boundary under remotemechanical loading

the 119909 axis It is found that the maximum values of 119863120579appear

at 120579 = 65∘ and 120579 = 115

∘ which also agrees well with theanalytical solution

9 Conclusions

In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily

adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric


problems and so on However there are still many possibleextensions and areas in need of further development in thefuture

(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement

(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics

(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities

(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010

[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012

[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014

[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014

[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013

[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005

[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014

[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014

[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011

[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993

[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015

[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015

[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005

[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996

[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995

[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007

[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005

[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006

[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996

[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014

[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000

[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008

[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009

[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996


[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010

[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013

[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014

[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994

[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003

[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005

[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009

[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006

[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007

[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005

[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999

[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010

[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012

[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987

[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951

[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010

[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990

[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012

[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001

[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988

[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999

[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010

[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993

[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987

[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010

[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012

[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004

[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999

[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012

[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009

[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000

[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012

[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011

[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007

[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981

[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998

[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996


[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958

[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999

[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997

[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000

[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998

[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005

[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003

[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010

[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002

[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013

[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001

[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003

[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991

[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997

[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007

[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996

[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998

[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000

[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000

[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996

[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012

[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013

[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996

[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005

[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013

[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005

[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005

[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014

[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


flexible element material definition which is important indealingwithmultimaterial problems rather than thematerialdefinition being the same in the entire domain in BEMHowever we noticed that there are also some limitationsof HFS-FEM for example determining the positions ofsource points used for approximation interpolations It is alsoknown that fundamental solution based approximations canperform remarkably well in smooth problems but tend todeteriorate when high-gradient stress fields are presented

This paper is organized as follows in Section 2 thebasic idea and formulations of the HFS-FEM are presentedthrough a simple potential problem Then plane elasticityproblems are described in Section 3 Section 4 extends the 2Dformulations of the HFS-FEM to general three-dimensional(3D) elasticity problems The method of particular solutionand radial basis function approximation are shown to dealwith body force in this Section In Section 5 we extend theHFS-FEM to thermoelastic problems with arbitrary bodyforce and temperature change In Section 6 the HFS-FEMfor 2D anisotropic elastic materials is described based onthe powerful Stroh formalism Plane piezoelectric problemis discussed in Section 7 Finally typical numerical examplesare presented in Section 8 to illustrate applications and per-formance of the HFS-FEM Concluding remarks and futuredevelopment are discussed at the end of this paper

2 Potential Problems

21 Basic Equations of Potential Problems The Laplace equa-tion of a well-posed potential problem (eg heat conduction)in a general plane domainΩ can be expressed as [21 22]

nabla2119906 (x) = 0 forallx isin Ω (1)

with the boundary conditions

119906 = 119906 on Γ119906 (2)

119902 = 119906119894119899119894= 119902 on Γ

119902 (3)

where 119906 is the unknown field variable and 119902 represents theboundary flux 119899

119894is the 119894th component of outward normal

vector to the boundary Γ = Γ119906cup Γ119902 and 119906 and 119902 are specified

functions on the related boundaries respectively The spacederivatives are indicated by a subscript comma that is 119906

119894=

120597119906120597119909119894 and the subscript index 119894 takes values (1 2) for two-

dimensional and (1 2 3) for three-dimensional problemsAdditionally the repeated subscript indices imply summationconvention

For convenience (3) can be rewritten in matrix form as

119902 = A[

1199061

1199062

] = 119902 (4)

with A = [11989911198992]

22 Assumed Independent Fields In this section the pro-cedure for developing a hybrid finite element model withfundamental solution as interior trial function is described

based on the boundary value problem defined by (1)ndash(3)Similar to the conventional FEM and HT-FEM the domainunder consideration is divided into a series of elements [1516 21 23ndash30] In each element two independent fields areassumed in the way as described in [31] and are given inSection 22

221 Intraelement Field Similar to themethod of fundamen-tal solution (MFS) in removing singularities of fundamentalsolution for a particular element 119890 occupying subdomainΩ119890 we assume that the field variable defined in the element

domain is extracted from a linear combination of funda-mental solutions centered at different source points locatedoutside the element (see Figure 1)

119906119890(x) =

119899119904

sum

119895=1

119873119890(x y

119895) 119888119890119895= N

119890(x) c

119890

forallx isin Ω119890 y

119895notin Ω

119890

(5)

where 119888119890119895is undetermined coefficients 119899

119904is the number of

virtual sources and 119873119890(x y

119895) is the fundamental solution to

the partial differential equation

nabla2119873119890(x y) + 120575 (x y) = 0 forallx y isin R

2 (6)

as

119873119890(x y) = minus

1

2120587ln 119903 (x y) (7)

Evidently (5) analytically satisfies (1) due to the solutionproperty of119873

119890(x y

119895)

In implementation the number of source points is takento be the same as the number of element nodes which isfree of spurious energy modes and can keep the stiffnessequations in full rank as indicated in [21] The source pointy119904119895(119895 = 1 2 119899

119904) can be generated bymeans of themethod

employed in the MFS [32ndash35]

y119904= x0+ 120574 (x

0minus x119888) (8)

where 120574 is a dimensionless coefficient x0is the point on

the element boundary (the nodal point in this work) andx119888is the geometrical centroid of the element (see Figure 1)

Determination of 120574 was discussed in [31 36] and 120574 = 5ndash10 isusually used in practice

The corresponding outward normal derivative of 119906119890on Γ

119890

is

119902119890=120597119906119890

120597119899= Q

119890c119890 (9)

where

Q119890=120597N119890

120597119899= AT

119890(10)

with

T119890= [

120597N119890

1205971199091

120597N119890

1205971199092

]

119879

(11)


1 2

34

Centroid

SourceNode

X2

X1

Intraelement field

Ωe

cx

0xΓe

u = Nece



1 3

57

2

4

6

8

Centroid

SourceNode

X2

X1

Intraelement field

Ωe

Γe


u = Nece

N

cx

0x

sx




119890= 119906

119891on



119890(x) = N

119890(x) d

119890(12)


119890 where N



= 11199061+

21199062+

31199063 (13)




N1

N2

N3

120585 = minus1 120585 = 0 120585 = +1

minus120585(1 minus 120585)

2

1 minus 1205852

120585(1 + 120585)

2

1 2 3



119906119890= 119906

119891(conformity)

119902119890+ 119902119891= 0 (reciprocity)

on ΓIef = Γ119890cap Γ119891

(14)


e f

ΓIef



Π119898= sum

119890

Π119898119890

= sum

119890

Π119890+ intΓ119890

( minus 119906) 119902dΓ (15)

where

Π119890=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ (16)


119890of


Γ119890= Γ119906119890cup Γ119902119890cup ΓIe (17)



Π119898119890

=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ + intΓ119890

119902 ( minus 119906) dΓ (18)


120575Π119898119890

= intΩ119890

119906119894120575119906119894dΩ minus int

Γ119902119890

119902120575dΓ + intΓ119890

(120575 minus 120575119906) 119902dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ(19)


intΩ

119891119894ℎ119894dΩ = int

Γ

ℎ119891119894119899119894dΓ minus int

Ω


we can obtain

120575Π119898119890

= intΩ119890

119906119894119894120575119906dΩ minus int

Γ119902119890

(119902 minus 119902) 120575dΓ + intΓ119906119890

119902120575dΓ

+ intΓIe

119902120575dΓ + intΓ119890

( minus 119906) 120575119902dΓ(21)


120575 = 0 on Γ119906119890

(∵ = 119906)

120575119890= 120575

119891 on ΓIef (∵ 119890=

119891)

(22)


120575Π119898119890

= intΩ119890

119906119894119894120575119906dΩ minus int

Γ119902119890

(119902 minus 119902) 120575dΓ + intΓIe

119902120575dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ(23)


119906119894119894= 0 in Ω

119890

119902 = 119902 on Γ119902119890

= 119906 on Γ119890

(24)




120575Π119898(119890+119891)

= intΩ119890+119891

119906119894119894120575119906dΩ minus int

Γ119902119890+Γ119902119891

(119902 minus 119902) 120575dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ + intΓ119891

( minus 119906) 120575119902dΓ

+ intΓIef

(119902119890+ 119902119891) 120575


(25)



119890+ 119902



intΓ119902

120575119902120575d119904

minussum

119890

[intΓIe

120575119902119890120575119890d119904 + int

Γ119890

120575119902119890120575 (

119890minus 119906119890) d119904]

(26)



Π119898(119906

0) = (Π




Π119898ge (Π

119898)0

or Π119898le (Π

119898)0

(27)



119890capΓ119891has




1205752Π119898= intΓ119902

120575119902120575d119904

minussum

119890

[intΓIe

120575119902119890120575119890d119904 + int

Γ119890

120575119902119890120575 (

119890minus 119906119890) d119904]

(28)




Π119890=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ + intΓ119890

119902 ( minus 119906) dΓ (29)


Π119890=1

2[intΓ119890

119902119906dΓ + intΩ119890


Γ119902119890

119902dΓ

+ intΓ119890

119902 ( minus 119906) dΓ

= minus1

2intΓ119890

119902119906dΓ minus intΓ119902119890

119902dΓ + intΓ119890

119902dΓ

(30)


Π119890= minus

1

2c119879119890H119890c119890minus d119879119890g119890+ c119879119890G119890d119890

(31)

in which

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

N119879119890Q119890dΓ

G119890= intΓ119890

Q119879119890N119890dΓ g

119890= intΓ119902119890

N119879119890119902dΓ

(32)







119890and d


120597Π119890

120597c119890

119879= minusH

119890c119890+ G

119890d119890= 0

120597Π119890

120597d119890

119879= G119879

119890c119890minus g119890= 0

(33)


119890and


c119890= Hminus1

119890G119890d119890 K

119890d119890= g119890 (34)

whereK119890= G119879




H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x) dΓ (35)


F (x) = [119865119894119895(x)]

119898times119898= Q119879

119890N119890 (36)


119867119894119895= intΓ119890

119865119894119895(x) dΓ =

119899119890

sum

119897=1

intΓ119890119897

119865119894119895(x) dΓ (37)

where

dΓ = radic(d1199091)2

+ (d1199092)2

= radic(d1199091

d120585)

2

+ (d1199092

d120585)

2

d120585 = 119869d120585

(38)


119869 = radic(d1199091

d120585)

2

+ (d1199092

d120585)

2

(39)

where

[d1199091

d120585d1199092

d120585]

119879

=

119899119900

sum

119894=1

d119873119894(120585)

d120585

1199091119894

1199092119894

(40)


119867119894119895=

119899119890

sum

119897=1

[int

+1

minus1

119865119894119895(x (120585)) 119869 (120585) d120585]

asymp

119899119890

sum

119897=1

[

119899119901

sum

119896=1

119908119896119865119894119895(x (120585

119896)) 119869 (120585

119896)]

(41)


119901


119890

matrix using

119866119894119895=

119899119890

sum

119897=1

[int

1

minus1

119865119894119895[x (120585)] 119869 (120585) d120585]

asymp

119899119890

sum

119897=1

119899119901

sum

119896=1

119908119896119865119894119895[x (120585

119896)] 119869 (120585

119896)

(42)





119906119890= N

119890c119890+ 1198880 (43)



119890at

element nodes119899

sum

119894=1

(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894

= min (44)

which finally gives

1198880=1

119899

119899

sum

119894=1

Δ119906119890119894 (45)

in which Δ119906119890119894

= (119890minus N



119890







120590119894119895119895

= 119887119894 119894 119895 = 1 2 (46)


L120590 = b (47)

where 120590 = [12059011

12059022


1 1198872]119879 is


L =[[[

[

120597

1205971199091

0120597

1205971199092

0120597

1205971199092

120597

1205971199091

]]]

]

(48)

120576 = LTu (49)

where 120576 = [12057611

12057622


1 1199062]119879



120590 = D120576 (50)


D =[[[

[

+ 2119866 0

+ 2119866 0

0 0 119866

]]]

]

(51)

where

=2]

1 minus 2]119866 119866 =

119864

2 (1 + ])

] =

] for plane strain]


(52)


u = u on Γ119906

t = A120590 = t on Γ119905

(53)



A = [

1198991

0 1198992

0 11989921198991

] (54)


LDL119879u = b (55)


u (x) =

1199061(x)

1199062(x)

=

119899119904

sum

119895=1

[

[

119906lowast

11(x y

119904119895) 119906

lowast

12(x y

119904119895)

119906lowast

21(x y

119904119895) 119906

lowast

22(x y

119904119895)

]

]

1198881119895

1198882119895


119890)

(56)




Ne

= [

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

)

119906lowast

21(x y

1199041) 119906

lowast

22(x y


lowast

21(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

)

]

]

ce = [11988811 11988821


1198882119899]119879

(57)



1 1198832) The





119906lowast

119894119895(x y

119904119895) =

minus1

8120587 (1 minus ]) 119866(3 minus 4]) 120575

119894119895ln 119903 minus 119903

119894119903119895 (58)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032




120590 (x) = [12059011 12059022

12059012]119879

= Tece (59)

whereTe

=[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

)

]]]

]

(60)


1199051

1199052

= n120590 = Qece (61)

in which

Qe = nTe n = [

1198991

0 1198992

0 11989921198991

] (62)


as

120590lowast

119894119895119896(x y) = minus1

4120587 (1 minus ]) 119903

sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903

119894119903119895119903119896]

(63)



is defined as

u (x) =

1

2

=

N1

N2

d119890= N

119890d119890 (x isin Γ

119890) (64)





119890and d

119890can be

expressed as

N119890= [

[


0 2

0 3



4

0 1

0 2


6

]

]

de = [11990611 11990621

11990612

11990622


11990628]119879

(65)

where 1 2 and


as

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(66)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ (67)

where 119894and 119906



119890





119890= Γ

119905+ Γ

119906+ Γ

119868) Compared




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ) minus int

Γ119905

119905119894119894dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ

(68)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(69)



119898119890= 0


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (70)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(71)







(33) and (34)


u119890= N

119890c119890+ [

1 0 1199092

0 1 minus1199091

] c0 (72)



119890at

element nodes119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

] = min (73)

which finally gives

c0 = Rminus1119890re (74)

where

Re =119899

sum

119894=1

[[[

[

1 0 1199092119894

0 1 minus1199091119894

1199092119894

minus1199091119894

1199092

1119894+ 1199092

2119894

]]]

]

re =119899

sum

119894=1

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

(75)

in which Δ119906119890119895119894



119890








120590119894119895119895

= minus119887119894

119894 119895 = 1 2 3 (76)


120590119894119895=

2119866V1 minus 2V

120575119894119895119890119896119896+ 2119866119890

119894119895

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(77)





119866119906119894119895119895

+119866

1 minus 2V119906119895119895119894

= minus119887119894 (78)


Γux1

x2

x3

b1

b2

b3

Γt

O



119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(79)


119894






119894


119906119894= 119906

ℎ

119894+ 119906119901

119894 (80)


equation

119866119906119901

119894119895119895+

119866

1 minus 2V119906119901

119895119895119894= minus119887

119894(81)


119866119906ℎ

119894119895119895+

119866

1 minus 2V119906ℎ

119895119895119894= 0 (82)


119906ℎ

119894= 119906

119894minus 119906119901

119894on Γ

119906

119905ℎ

119894= 119905119894minus 119905119901

119894on Γ

119905

(83)


119901


119906ℎ





119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (84)


119895



119906119901

119894=

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (85)

where Φ119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (86)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(87)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (88)



119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (90)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(91)



(1199091 1199092 1199093) and a given point (119909

1119895 1199092119895 1199093119895) in the domain of


119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(92)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897 (93)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(94)



119894= 0 the



u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (95)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(96)



1 1199092) The


119904119895) is given by [40]

119906lowast

119894119895(x y

119904119895) =

1

16120587 (1 minus ]) 119866119903(3 minus 4]) 120575

119894119895+ 119903119894119903119895 (97)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032

2+ 1199032



119904) can also


y119904= x0+ 120574 (x

0minus x119888) (98)





120590 (x) = [12059011 12059022

12059033

12059023

12059031

12059012]119879

= Tece (99)

where

Te =

[[[[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

133(x y

1) 120590

lowast

233(x y

1) 120590

lowast

333(x y


lowast

133(x y

119899119904

) 120590lowast

233(x y

119899119904

) 120590lowast

333(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]]]]

]

(100)


1

4 3

2


5 6

8 7

8-node 3D element

X2

X1

X3

Intraelement field

Ωe

Γe

Frame field

u = Nece

u(x) = edeN

cx

0x

sx

(a)

1

7 5

2

13 15

1917

3

4

6

89 10

1112

1416

1820

20-node 3D element

X2

X1

X3

Ωe

(b)



120590lowast

119894119895119896(x y) = minus1

8120587 (1 minus ]) 1199032

sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903

119894119903119895119903119896

(101)


1199051

1199052

1199053

= n120590 = Qece (102)

in which

Qe = nTe n =[[

[

1198991

0 0 0 11989931198992

0 1198992

0 1198993

0 1198991

0 0 119899311989921198991

0

]]

]

(103)


119890


u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (104)


119890is the matrix



119890and


N119890= [0 N

1N20 0 N

4N30]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(105)


N119894=[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(106)


120585 120578 isin [minus1 1]

1=(1 + 120585) (1 + 120578)

4

2=(1 minus 120585) (1 + 120578)

4

3=(1 minus 120585) (1 minus 120578)

4

4=(1 + 120585) (1 minus 120578)

4

(107)


element (Figure 6)


1 3

57

2

4

6

8


(10)

(11)(01)

(minus10)

(minus11)

120578

120585



119898119890used


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)



Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again



V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V



V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)


119894 119910119894 119911119894)


119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)




F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2

34

Centroid

SourceNode

X2

X1

Intraelement field

Ωe

cx

0xΓe

u = Nece



1 3

57

2

4

6

8

Centroid

SourceNode

X2

X1

Intraelement field

Ωe

Γe


u = Nece

N

cx

0x

sx




119890= 119906

119891on



119890(x) = N

119890(x) d

119890(12)


119890 where N



= 11199061+

21199062+

31199063 (13)




N1

N2

N3

120585 = minus1 120585 = 0 120585 = +1

minus120585(1 minus 120585)

2

1 minus 1205852

120585(1 + 120585)

2

1 2 3



119906119890= 119906

119891(conformity)

119902119890+ 119902119891= 0 (reciprocity)

on ΓIef = Γ119890cap Γ119891

(14)


e f

ΓIef



Π119898= sum

119890

Π119898119890

= sum

119890

Π119890+ intΓ119890

( minus 119906) 119902dΓ (15)

where

Π119890=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ (16)


119890of


Γ119890= Γ119906119890cup Γ119902119890cup ΓIe (17)



Π119898119890

=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ + intΓ119890

119902 ( minus 119906) dΓ (18)


120575Π119898119890

= intΩ119890

119906119894120575119906119894dΩ minus int

Γ119902119890

119902120575dΓ + intΓ119890

(120575 minus 120575119906) 119902dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ(19)


intΩ

119891119894ℎ119894dΩ = int

Γ

ℎ119891119894119899119894dΓ minus int

Ω


we can obtain

120575Π119898119890

= intΩ119890

119906119894119894120575119906dΩ minus int

Γ119902119890

(119902 minus 119902) 120575dΓ + intΓ119906119890

119902120575dΓ

+ intΓIe

119902120575dΓ + intΓ119890

( minus 119906) 120575119902dΓ(21)


120575 = 0 on Γ119906119890

(∵ = 119906)

120575119890= 120575

119891 on ΓIef (∵ 119890=

119891)

(22)


120575Π119898119890

= intΩ119890

119906119894119894120575119906dΩ minus int

Γ119902119890

(119902 minus 119902) 120575dΓ + intΓIe

119902120575dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ(23)


119906119894119894= 0 in Ω

119890

119902 = 119902 on Γ119902119890

= 119906 on Γ119890

(24)




120575Π119898(119890+119891)

= intΩ119890+119891

119906119894119894120575119906dΩ minus int

Γ119902119890+Γ119902119891

(119902 minus 119902) 120575dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ + intΓ119891

( minus 119906) 120575119902dΓ

+ intΓIef

(119902119890+ 119902119891) 120575


(25)



119890+ 119902



intΓ119902

120575119902120575d119904

minussum

119890

[intΓIe

120575119902119890120575119890d119904 + int

Γ119890

120575119902119890120575 (

119890minus 119906119890) d119904]

(26)



Π119898(119906

0) = (Π




Π119898ge (Π

119898)0

or Π119898le (Π

119898)0

(27)



119890capΓ119891has




1205752Π119898= intΓ119902

120575119902120575d119904

minussum

119890

[intΓIe

120575119902119890120575119890d119904 + int

Γ119890

120575119902119890120575 (

119890minus 119906119890) d119904]

(28)




Π119890=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ + intΓ119890

119902 ( minus 119906) dΓ (29)


Π119890=1

2[intΓ119890

119902119906dΓ + intΩ119890


Γ119902119890

119902dΓ

+ intΓ119890

119902 ( minus 119906) dΓ

= minus1

2intΓ119890

119902119906dΓ minus intΓ119902119890

119902dΓ + intΓ119890

119902dΓ

(30)


Π119890= minus

1

2c119879119890H119890c119890minus d119879119890g119890+ c119879119890G119890d119890

(31)

in which

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

N119879119890Q119890dΓ

G119890= intΓ119890

Q119879119890N119890dΓ g

119890= intΓ119902119890

N119879119890119902dΓ

(32)







119890and d


120597Π119890

120597c119890

119879= minusH

119890c119890+ G

119890d119890= 0

120597Π119890

120597d119890

119879= G119879

119890c119890minus g119890= 0

(33)


119890and


c119890= Hminus1

119890G119890d119890 K

119890d119890= g119890 (34)

whereK119890= G119879




H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x) dΓ (35)


F (x) = [119865119894119895(x)]

119898times119898= Q119879

119890N119890 (36)


119867119894119895= intΓ119890

119865119894119895(x) dΓ =

119899119890

sum

119897=1

intΓ119890119897

119865119894119895(x) dΓ (37)

where

dΓ = radic(d1199091)2

+ (d1199092)2

= radic(d1199091

d120585)

2

+ (d1199092

d120585)

2

d120585 = 119869d120585

(38)


119869 = radic(d1199091

d120585)

2

+ (d1199092

d120585)

2

(39)

where

[d1199091

d120585d1199092

d120585]

119879

=

119899119900

sum

119894=1

d119873119894(120585)

d120585

1199091119894

1199092119894

(40)


119867119894119895=

119899119890

sum

119897=1

[int

+1

minus1

119865119894119895(x (120585)) 119869 (120585) d120585]

asymp

119899119890

sum

119897=1

[

119899119901

sum

119896=1

119908119896119865119894119895(x (120585

119896)) 119869 (120585

119896)]

(41)


119901


119890

matrix using

119866119894119895=

119899119890

sum

119897=1

[int

1

minus1

119865119894119895[x (120585)] 119869 (120585) d120585]

asymp

119899119890

sum

119897=1

119899119901

sum

119896=1

119908119896119865119894119895[x (120585

119896)] 119869 (120585

119896)

(42)





119906119890= N

119890c119890+ 1198880 (43)



119890at

element nodes119899

sum

119894=1

(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894

= min (44)

which finally gives

1198880=1

119899

119899

sum

119894=1

Δ119906119890119894 (45)

in which Δ119906119890119894

= (119890minus N



119890







120590119894119895119895

= 119887119894 119894 119895 = 1 2 (46)


L120590 = b (47)

where 120590 = [12059011

12059022


1 1198872]119879 is


L =[[[

[

120597

1205971199091

0120597

1205971199092

0120597

1205971199092

120597

1205971199091

]]]

]

(48)

120576 = LTu (49)

where 120576 = [12057611

12057622


1 1199062]119879



120590 = D120576 (50)


D =[[[

[

+ 2119866 0

+ 2119866 0

0 0 119866

]]]

]

(51)

where

=2]

1 minus 2]119866 119866 =

119864

2 (1 + ])

] =

] for plane strain]


(52)


u = u on Γ119906

t = A120590 = t on Γ119905

(53)



A = [

1198991

0 1198992

0 11989921198991

] (54)


LDL119879u = b (55)


u (x) =

1199061(x)

1199062(x)

=

119899119904

sum

119895=1

[

[

119906lowast

11(x y

119904119895) 119906

lowast

12(x y

119904119895)

119906lowast

21(x y

119904119895) 119906

lowast

22(x y

119904119895)

]

]

1198881119895

1198882119895


119890)

(56)




Ne

= [

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

)

119906lowast

21(x y

1199041) 119906

lowast

22(x y


lowast

21(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

)

]

]

ce = [11988811 11988821


1198882119899]119879

(57)



1 1198832) The





119906lowast

119894119895(x y

119904119895) =

minus1

8120587 (1 minus ]) 119866(3 minus 4]) 120575

119894119895ln 119903 minus 119903

119894119903119895 (58)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032




120590 (x) = [12059011 12059022

12059012]119879

= Tece (59)

whereTe

=[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

)

]]]

]

(60)


1199051

1199052

= n120590 = Qece (61)

in which

Qe = nTe n = [

1198991

0 1198992

0 11989921198991

] (62)


as

120590lowast

119894119895119896(x y) = minus1

4120587 (1 minus ]) 119903

sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903

119894119903119895119903119896]

(63)



is defined as

u (x) =

1

2

=

N1

N2

d119890= N

119890d119890 (x isin Γ

119890) (64)





119890and d

119890can be

expressed as

N119890= [

[


0 2

0 3



4

0 1

0 2


6

]

]

de = [11990611 11990621

11990612

11990622


11990628]119879

(65)

where 1 2 and


as

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(66)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ (67)

where 119894and 119906



119890





119890= Γ

119905+ Γ

119906+ Γ

119868) Compared




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ) minus int

Γ119905

119905119894119894dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ

(68)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(69)



119898119890= 0


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (70)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(71)







(33) and (34)


u119890= N

119890c119890+ [

1 0 1199092

0 1 minus1199091

] c0 (72)



119890at

element nodes119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

] = min (73)

which finally gives

c0 = Rminus1119890re (74)

where

Re =119899

sum

119894=1

[[[

[

1 0 1199092119894

0 1 minus1199091119894

1199092119894

minus1199091119894

1199092

1119894+ 1199092

2119894

]]]

]

re =119899

sum

119894=1

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

(75)

in which Δ119906119890119895119894



119890








120590119894119895119895

= minus119887119894

119894 119895 = 1 2 3 (76)


120590119894119895=

2119866V1 minus 2V

120575119894119895119890119896119896+ 2119866119890

119894119895

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(77)





119866119906119894119895119895

+119866

1 minus 2V119906119895119895119894

= minus119887119894 (78)


Γux1

x2

x3

b1

b2

b3

Γt

O



119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(79)


119894






119894


119906119894= 119906

ℎ

119894+ 119906119901

119894 (80)


equation

119866119906119901

119894119895119895+

119866

1 minus 2V119906119901

119895119895119894= minus119887

119894(81)


119866119906ℎ

119894119895119895+

119866

1 minus 2V119906ℎ

119895119895119894= 0 (82)


119906ℎ

119894= 119906

119894minus 119906119901

119894on Γ

119906

119905ℎ

119894= 119905119894minus 119905119901

119894on Γ

119905

(83)


119901


119906ℎ





119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (84)


119895



119906119901

119894=

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (85)

where Φ119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (86)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(87)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (88)



119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (90)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(91)



(1199091 1199092 1199093) and a given point (119909

1119895 1199092119895 1199093119895) in the domain of


119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(92)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897 (93)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(94)



119894= 0 the



u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (95)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(96)



1 1199092) The


119904119895) is given by [40]

119906lowast

119894119895(x y

119904119895) =

1

16120587 (1 minus ]) 119866119903(3 minus 4]) 120575

119894119895+ 119903119894119903119895 (97)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032

2+ 1199032



119904) can also


y119904= x0+ 120574 (x

0minus x119888) (98)





120590 (x) = [12059011 12059022

12059033

12059023

12059031

12059012]119879

= Tece (99)

where

Te =

[[[[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

133(x y

1) 120590

lowast

233(x y

1) 120590

lowast

333(x y


lowast

133(x y

119899119904

) 120590lowast

233(x y

119899119904

) 120590lowast

333(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]]]]

]

(100)


1

4 3

2


5 6

8 7

8-node 3D element

X2

X1

X3

Intraelement field

Ωe

Γe

Frame field

u = Nece

u(x) = edeN

cx

0x

sx

(a)

1

7 5

2

13 15

1917

3

4

6

89 10

1112

1416

1820

20-node 3D element

X2

X1

X3

Ωe

(b)



120590lowast

119894119895119896(x y) = minus1

8120587 (1 minus ]) 1199032

sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903

119894119903119895119903119896

(101)


1199051

1199052

1199053

= n120590 = Qece (102)

in which

Qe = nTe n =[[

[

1198991

0 0 0 11989931198992

0 1198992

0 1198993

0 1198991

0 0 119899311989921198991

0

]]

]

(103)


119890


u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (104)


119890is the matrix



119890and


N119890= [0 N

1N20 0 N

4N30]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(105)


N119894=[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(106)


120585 120578 isin [minus1 1]

1=(1 + 120585) (1 + 120578)

4

2=(1 minus 120585) (1 + 120578)

4

3=(1 minus 120585) (1 minus 120578)

4

4=(1 + 120585) (1 minus 120578)

4

(107)


element (Figure 6)


1 3

57

2

4

6

8


(10)

(11)(01)

(minus10)

(minus11)

120578

120585



119898119890used


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)



Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again



V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V



V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)


119894 119910119894 119911119894)


119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)




F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










e f

ΓIef



Π119898= sum

119890

Π119898119890

= sum

119890

Π119890+ intΓ119890

( minus 119906) 119902dΓ (15)

where

Π119890=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ (16)


119890of


Γ119890= Γ119906119890cup Γ119902119890cup ΓIe (17)



Π119898119890

=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ + intΓ119890

119902 ( minus 119906) dΓ (18)


120575Π119898119890

= intΩ119890

119906119894120575119906119894dΩ minus int

Γ119902119890

119902120575dΓ + intΓ119890

(120575 minus 120575119906) 119902dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ(19)


intΩ

119891119894ℎ119894dΩ = int

Γ

ℎ119891119894119899119894dΓ minus int

Ω


we can obtain

120575Π119898119890

= intΩ119890

119906119894119894120575119906dΩ minus int

Γ119902119890

(119902 minus 119902) 120575dΓ + intΓ119906119890

119902120575dΓ

+ intΓIe

119902120575dΓ + intΓ119890

( minus 119906) 120575119902dΓ(21)


120575 = 0 on Γ119906119890

(∵ = 119906)

120575119890= 120575

119891 on ΓIef (∵ 119890=

119891)

(22)


120575Π119898119890

= intΩ119890

119906119894119894120575119906dΩ minus int

Γ119902119890

(119902 minus 119902) 120575dΓ + intΓIe

119902120575dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ(23)


119906119894119894= 0 in Ω

119890

119902 = 119902 on Γ119902119890

= 119906 on Γ119890

(24)




120575Π119898(119890+119891)

= intΩ119890+119891

119906119894119894120575119906dΩ minus int

Γ119902119890+Γ119902119891

(119902 minus 119902) 120575dΓ

+ intΓ119890

( minus 119906) 120575119902dΓ + intΓ119891

( minus 119906) 120575119902dΓ

+ intΓIef

(119902119890+ 119902119891) 120575


(25)



119890+ 119902



intΓ119902

120575119902120575d119904

minussum

119890

[intΓIe

120575119902119890120575119890d119904 + int

Γ119890

120575119902119890120575 (

119890minus 119906119890) d119904]

(26)



Π119898(119906

0) = (Π




Π119898ge (Π

119898)0

or Π119898le (Π

119898)0

(27)



119890capΓ119891has




1205752Π119898= intΓ119902

120575119902120575d119904

minussum

119890

[intΓIe

120575119902119890120575119890d119904 + int

Γ119890

120575119902119890120575 (

119890minus 119906119890) d119904]

(28)




Π119890=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ + intΓ119890

119902 ( minus 119906) dΓ (29)


Π119890=1

2[intΓ119890

119902119906dΓ + intΩ119890


Γ119902119890

119902dΓ

+ intΓ119890

119902 ( minus 119906) dΓ

= minus1

2intΓ119890

119902119906dΓ minus intΓ119902119890

119902dΓ + intΓ119890

119902dΓ

(30)


Π119890= minus

1

2c119879119890H119890c119890minus d119879119890g119890+ c119879119890G119890d119890

(31)

in which

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

N119879119890Q119890dΓ

G119890= intΓ119890

Q119879119890N119890dΓ g

119890= intΓ119902119890

N119879119890119902dΓ

(32)







119890and d


120597Π119890

120597c119890

119879= minusH

119890c119890+ G

119890d119890= 0

120597Π119890

120597d119890

119879= G119879

119890c119890minus g119890= 0

(33)


119890and


c119890= Hminus1

119890G119890d119890 K

119890d119890= g119890 (34)

whereK119890= G119879




H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x) dΓ (35)


F (x) = [119865119894119895(x)]

119898times119898= Q119879

119890N119890 (36)


119867119894119895= intΓ119890

119865119894119895(x) dΓ =

119899119890

sum

119897=1

intΓ119890119897

119865119894119895(x) dΓ (37)

where

dΓ = radic(d1199091)2

+ (d1199092)2

= radic(d1199091

d120585)

2

+ (d1199092

d120585)

2

d120585 = 119869d120585

(38)


119869 = radic(d1199091

d120585)

2

+ (d1199092

d120585)

2

(39)

where

[d1199091

d120585d1199092

d120585]

119879

=

119899119900

sum

119894=1

d119873119894(120585)

d120585

1199091119894

1199092119894

(40)


119867119894119895=

119899119890

sum

119897=1

[int

+1

minus1

119865119894119895(x (120585)) 119869 (120585) d120585]

asymp

119899119890

sum

119897=1

[

119899119901

sum

119896=1

119908119896119865119894119895(x (120585

119896)) 119869 (120585

119896)]

(41)


119901


119890

matrix using

119866119894119895=

119899119890

sum

119897=1

[int

1

minus1

119865119894119895[x (120585)] 119869 (120585) d120585]

asymp

119899119890

sum

119897=1

119899119901

sum

119896=1

119908119896119865119894119895[x (120585

119896)] 119869 (120585

119896)

(42)





119906119890= N

119890c119890+ 1198880 (43)



119890at

element nodes119899

sum

119894=1

(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894

= min (44)

which finally gives

1198880=1

119899

119899

sum

119894=1

Δ119906119890119894 (45)

in which Δ119906119890119894

= (119890minus N



119890







120590119894119895119895

= 119887119894 119894 119895 = 1 2 (46)


L120590 = b (47)

where 120590 = [12059011

12059022


1 1198872]119879 is


L =[[[

[

120597

1205971199091

0120597

1205971199092

0120597

1205971199092

120597

1205971199091

]]]

]

(48)

120576 = LTu (49)

where 120576 = [12057611

12057622


1 1199062]119879



120590 = D120576 (50)


D =[[[

[

+ 2119866 0

+ 2119866 0

0 0 119866

]]]

]

(51)

where

=2]

1 minus 2]119866 119866 =

119864

2 (1 + ])

] =

] for plane strain]


(52)


u = u on Γ119906

t = A120590 = t on Γ119905

(53)



A = [

1198991

0 1198992

0 11989921198991

] (54)


LDL119879u = b (55)


u (x) =

1199061(x)

1199062(x)

=

119899119904

sum

119895=1

[

[

119906lowast

11(x y

119904119895) 119906

lowast

12(x y

119904119895)

119906lowast

21(x y

119904119895) 119906

lowast

22(x y

119904119895)

]

]

1198881119895

1198882119895


119890)

(56)




Ne

= [

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

)

119906lowast

21(x y

1199041) 119906

lowast

22(x y


lowast

21(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

)

]

]

ce = [11988811 11988821


1198882119899]119879

(57)



1 1198832) The





119906lowast

119894119895(x y

119904119895) =

minus1

8120587 (1 minus ]) 119866(3 minus 4]) 120575

119894119895ln 119903 minus 119903

119894119903119895 (58)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032




120590 (x) = [12059011 12059022

12059012]119879

= Tece (59)

whereTe

=[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

)

]]]

]

(60)


1199051

1199052

= n120590 = Qece (61)

in which

Qe = nTe n = [

1198991

0 1198992

0 11989921198991

] (62)


as

120590lowast

119894119895119896(x y) = minus1

4120587 (1 minus ]) 119903

sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903

119894119903119895119903119896]

(63)



is defined as

u (x) =

1

2

=

N1

N2

d119890= N

119890d119890 (x isin Γ

119890) (64)





119890and d

119890can be

expressed as

N119890= [

[


0 2

0 3



4

0 1

0 2


6

]

]

de = [11990611 11990621

11990612

11990622


11990628]119879

(65)

where 1 2 and


as

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(66)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ (67)

where 119894and 119906



119890





119890= Γ

119905+ Γ

119906+ Γ

119868) Compared




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ) minus int

Γ119905

119905119894119894dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ

(68)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(69)



119898119890= 0


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (70)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(71)







(33) and (34)


u119890= N

119890c119890+ [

1 0 1199092

0 1 minus1199091

] c0 (72)



119890at

element nodes119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

] = min (73)

which finally gives

c0 = Rminus1119890re (74)

where

Re =119899

sum

119894=1

[[[

[

1 0 1199092119894

0 1 minus1199091119894

1199092119894

minus1199091119894

1199092

1119894+ 1199092

2119894

]]]

]

re =119899

sum

119894=1

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

(75)

in which Δ119906119890119895119894



119890








120590119894119895119895

= minus119887119894

119894 119895 = 1 2 3 (76)


120590119894119895=

2119866V1 minus 2V

120575119894119895119890119896119896+ 2119866119890

119894119895

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(77)





119866119906119894119895119895

+119866

1 minus 2V119906119895119895119894

= minus119887119894 (78)


Γux1

x2

x3

b1

b2

b3

Γt

O



119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(79)


119894






119894


119906119894= 119906

ℎ

119894+ 119906119901

119894 (80)


equation

119866119906119901

119894119895119895+

119866

1 minus 2V119906119901

119895119895119894= minus119887

119894(81)


119866119906ℎ

119894119895119895+

119866

1 minus 2V119906ℎ

119895119895119894= 0 (82)


119906ℎ

119894= 119906

119894minus 119906119901

119894on Γ

119906

119905ℎ

119894= 119905119894minus 119905119901

119894on Γ

119905

(83)


119901


119906ℎ





119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (84)


119895



119906119901

119894=

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (85)

where Φ119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (86)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(87)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (88)



119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (90)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(91)



(1199091 1199092 1199093) and a given point (119909

1119895 1199092119895 1199093119895) in the domain of


119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(92)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897 (93)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(94)



119894= 0 the



u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (95)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(96)



1 1199092) The


119904119895) is given by [40]

119906lowast

119894119895(x y

119904119895) =

1

16120587 (1 minus ]) 119866119903(3 minus 4]) 120575

119894119895+ 119903119894119903119895 (97)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032

2+ 1199032



119904) can also


y119904= x0+ 120574 (x

0minus x119888) (98)





120590 (x) = [12059011 12059022

12059033

12059023

12059031

12059012]119879

= Tece (99)

where

Te =

[[[[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

133(x y

1) 120590

lowast

233(x y

1) 120590

lowast

333(x y


lowast

133(x y

119899119904

) 120590lowast

233(x y

119899119904

) 120590lowast

333(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]]]]

]

(100)


1

4 3

2


5 6

8 7

8-node 3D element

X2

X1

X3

Intraelement field

Ωe

Γe

Frame field

u = Nece

u(x) = edeN

cx

0x

sx

(a)

1

7 5

2

13 15

1917

3

4

6

89 10

1112

1416

1820

20-node 3D element

X2

X1

X3

Ωe

(b)



120590lowast

119894119895119896(x y) = minus1

8120587 (1 minus ]) 1199032

sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903

119894119903119895119903119896

(101)


1199051

1199052

1199053

= n120590 = Qece (102)

in which

Qe = nTe n =[[

[

1198991

0 0 0 11989931198992

0 1198992

0 1198993

0 1198991

0 0 119899311989921198991

0

]]

]

(103)


119890


u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (104)


119890is the matrix



119890and


N119890= [0 N

1N20 0 N

4N30]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(105)


N119894=[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(106)


120585 120578 isin [minus1 1]

1=(1 + 120585) (1 + 120578)

4

2=(1 minus 120585) (1 + 120578)

4

3=(1 minus 120585) (1 minus 120578)

4

4=(1 + 120585) (1 minus 120578)

4

(107)


element (Figure 6)


1 3

57

2

4

6

8


(10)

(11)(01)

(minus10)

(minus11)

120578

120585



119898119890used


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)



Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again



V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V



V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)


119894 119910119894 119911119894)


119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)




F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205752Π119898= intΓ119902

120575119902120575d119904

minussum

119890

[intΓIe

120575119902119890120575119890d119904 + int

Γ119890

120575119902119890120575 (

119890minus 119906119890) d119904]

(28)




Π119890=1

2intΩ119890

119906119894119906119894dΩ minus int

Γ119902119890

119902dΓ + intΓ119890

119902 ( minus 119906) dΓ (29)


Π119890=1

2[intΓ119890

119902119906dΓ + intΩ119890


Γ119902119890

119902dΓ

+ intΓ119890

119902 ( minus 119906) dΓ

= minus1

2intΓ119890

119902119906dΓ minus intΓ119902119890

119902dΓ + intΓ119890

119902dΓ

(30)


Π119890= minus

1

2c119879119890H119890c119890minus d119879119890g119890+ c119879119890G119890d119890

(31)

in which

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

N119879119890Q119890dΓ

G119890= intΓ119890

Q119879119890N119890dΓ g

119890= intΓ119902119890

N119879119890119902dΓ

(32)







119890and d


120597Π119890

120597c119890

119879= minusH

119890c119890+ G

119890d119890= 0

120597Π119890

120597d119890

119879= G119879

119890c119890minus g119890= 0

(33)


119890and


c119890= Hminus1

119890G119890d119890 K

119890d119890= g119890 (34)

whereK119890= G119879




H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x) dΓ (35)


F (x) = [119865119894119895(x)]

119898times119898= Q119879

119890N119890 (36)


119867119894119895= intΓ119890

119865119894119895(x) dΓ =

119899119890

sum

119897=1

intΓ119890119897

119865119894119895(x) dΓ (37)

where

dΓ = radic(d1199091)2

+ (d1199092)2

= radic(d1199091

d120585)

2

+ (d1199092

d120585)

2

d120585 = 119869d120585

(38)


119869 = radic(d1199091

d120585)

2

+ (d1199092

d120585)

2

(39)

where

[d1199091

d120585d1199092

d120585]

119879

=

119899119900

sum

119894=1

d119873119894(120585)

d120585

1199091119894

1199092119894

(40)


119867119894119895=

119899119890

sum

119897=1

[int

+1

minus1

119865119894119895(x (120585)) 119869 (120585) d120585]

asymp

119899119890

sum

119897=1

[

119899119901

sum

119896=1

119908119896119865119894119895(x (120585

119896)) 119869 (120585

119896)]

(41)


119901


119890

matrix using

119866119894119895=

119899119890

sum

119897=1

[int

1

minus1

119865119894119895[x (120585)] 119869 (120585) d120585]

asymp

119899119890

sum

119897=1

119899119901

sum

119896=1

119908119896119865119894119895[x (120585

119896)] 119869 (120585

119896)

(42)





119906119890= N

119890c119890+ 1198880 (43)



119890at

element nodes119899

sum

119894=1

(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894

= min (44)

which finally gives

1198880=1

119899

119899

sum

119894=1

Δ119906119890119894 (45)

in which Δ119906119890119894

= (119890minus N



119890







120590119894119895119895

= 119887119894 119894 119895 = 1 2 (46)


L120590 = b (47)

where 120590 = [12059011

12059022


1 1198872]119879 is


L =[[[

[

120597

1205971199091

0120597

1205971199092

0120597

1205971199092

120597

1205971199091

]]]

]

(48)

120576 = LTu (49)

where 120576 = [12057611

12057622


1 1199062]119879



120590 = D120576 (50)


D =[[[

[

+ 2119866 0

+ 2119866 0

0 0 119866

]]]

]

(51)

where

=2]

1 minus 2]119866 119866 =

119864

2 (1 + ])

] =

] for plane strain]


(52)


u = u on Γ119906

t = A120590 = t on Γ119905

(53)



A = [

1198991

0 1198992

0 11989921198991

] (54)


LDL119879u = b (55)


u (x) =

1199061(x)

1199062(x)

=

119899119904

sum

119895=1

[

[

119906lowast

11(x y

119904119895) 119906

lowast

12(x y

119904119895)

119906lowast

21(x y

119904119895) 119906

lowast

22(x y

119904119895)

]

]

1198881119895

1198882119895


119890)

(56)




Ne

= [

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

)

119906lowast

21(x y

1199041) 119906

lowast

22(x y


lowast

21(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

)

]

]

ce = [11988811 11988821


1198882119899]119879

(57)



1 1198832) The





119906lowast

119894119895(x y

119904119895) =

minus1

8120587 (1 minus ]) 119866(3 minus 4]) 120575

119894119895ln 119903 minus 119903

119894119903119895 (58)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032




120590 (x) = [12059011 12059022

12059012]119879

= Tece (59)

whereTe

=[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

)

]]]

]

(60)


1199051

1199052

= n120590 = Qece (61)

in which

Qe = nTe n = [

1198991

0 1198992

0 11989921198991

] (62)


as

120590lowast

119894119895119896(x y) = minus1

4120587 (1 minus ]) 119903

sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903

119894119903119895119903119896]

(63)



is defined as

u (x) =

1

2

=

N1

N2

d119890= N

119890d119890 (x isin Γ

119890) (64)





119890and d

119890can be

expressed as

N119890= [

[


0 2

0 3



4

0 1

0 2


6

]

]

de = [11990611 11990621

11990612

11990622


11990628]119879

(65)

where 1 2 and


as

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(66)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ (67)

where 119894and 119906



119890





119890= Γ

119905+ Γ

119906+ Γ

119868) Compared




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ) minus int

Γ119905

119905119894119894dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ

(68)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(69)



119898119890= 0


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (70)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(71)







(33) and (34)


u119890= N

119890c119890+ [

1 0 1199092

0 1 minus1199091

] c0 (72)



119890at

element nodes119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

] = min (73)

which finally gives

c0 = Rminus1119890re (74)

where

Re =119899

sum

119894=1

[[[

[

1 0 1199092119894

0 1 minus1199091119894

1199092119894

minus1199091119894

1199092

1119894+ 1199092

2119894

]]]

]

re =119899

sum

119894=1

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

(75)

in which Δ119906119890119895119894



119890








120590119894119895119895

= minus119887119894

119894 119895 = 1 2 3 (76)


120590119894119895=

2119866V1 minus 2V

120575119894119895119890119896119896+ 2119866119890

119894119895

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(77)





119866119906119894119895119895

+119866

1 minus 2V119906119895119895119894

= minus119887119894 (78)


Γux1

x2

x3

b1

b2

b3

Γt

O



119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(79)


119894






119894


119906119894= 119906

ℎ

119894+ 119906119901

119894 (80)


equation

119866119906119901

119894119895119895+

119866

1 minus 2V119906119901

119895119895119894= minus119887

119894(81)


119866119906ℎ

119894119895119895+

119866

1 minus 2V119906ℎ

119895119895119894= 0 (82)


119906ℎ

119894= 119906

119894minus 119906119901

119894on Γ

119906

119905ℎ

119894= 119905119894minus 119905119901

119894on Γ

119905

(83)


119901


119906ℎ





119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (84)


119895



119906119901

119894=

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (85)

where Φ119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (86)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(87)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (88)



119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (90)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(91)



(1199091 1199092 1199093) and a given point (119909

1119895 1199092119895 1199093119895) in the domain of


119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(92)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897 (93)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(94)



119894= 0 the



u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (95)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(96)



1 1199092) The


119904119895) is given by [40]

119906lowast

119894119895(x y

119904119895) =

1

16120587 (1 minus ]) 119866119903(3 minus 4]) 120575

119894119895+ 119903119894119903119895 (97)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032

2+ 1199032



119904) can also


y119904= x0+ 120574 (x

0minus x119888) (98)





120590 (x) = [12059011 12059022

12059033

12059023

12059031

12059012]119879

= Tece (99)

where

Te =

[[[[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

133(x y

1) 120590

lowast

233(x y

1) 120590

lowast

333(x y


lowast

133(x y

119899119904

) 120590lowast

233(x y

119899119904

) 120590lowast

333(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]]]]

]

(100)


1

4 3

2


5 6

8 7

8-node 3D element

X2

X1

X3

Intraelement field

Ωe

Γe

Frame field

u = Nece

u(x) = edeN

cx

0x

sx

(a)

1

7 5

2

13 15

1917

3

4

6

89 10

1112

1416

1820

20-node 3D element

X2

X1

X3

Ωe

(b)



120590lowast

119894119895119896(x y) = minus1

8120587 (1 minus ]) 1199032

sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903

119894119903119895119903119896

(101)


1199051

1199052

1199053

= n120590 = Qece (102)

in which

Qe = nTe n =[[

[

1198991

0 0 0 11989931198992

0 1198992

0 1198993

0 1198991

0 0 119899311989921198991

0

]]

]

(103)


119890


u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (104)


119890is the matrix



119890and


N119890= [0 N

1N20 0 N

4N30]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(105)


N119894=[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(106)


120585 120578 isin [minus1 1]

1=(1 + 120585) (1 + 120578)

4

2=(1 minus 120585) (1 + 120578)

4

3=(1 minus 120585) (1 minus 120578)

4

4=(1 + 120585) (1 minus 120578)

4

(107)


element (Figure 6)


1 3

57

2

4

6

8


(10)

(11)(01)

(minus10)

(minus11)

120578

120585



119898119890used


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)



Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again



V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V



V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)


119894 119910119894 119911119894)


119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)




F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119906119890= N

119890c119890+ 1198880 (43)



119890at

element nodes119899

sum

119894=1

(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894

= min (44)

which finally gives

1198880=1

119899

119899

sum

119894=1

Δ119906119890119894 (45)

in which Δ119906119890119894

= (119890minus N



119890







120590119894119895119895

= 119887119894 119894 119895 = 1 2 (46)


L120590 = b (47)

where 120590 = [12059011

12059022


1 1198872]119879 is


L =[[[

[

120597

1205971199091

0120597

1205971199092

0120597

1205971199092

120597

1205971199091

]]]

]

(48)

120576 = LTu (49)

where 120576 = [12057611

12057622


1 1199062]119879



120590 = D120576 (50)


D =[[[

[

+ 2119866 0

+ 2119866 0

0 0 119866

]]]

]

(51)

where

=2]

1 minus 2]119866 119866 =

119864

2 (1 + ])

] =

] for plane strain]


(52)


u = u on Γ119906

t = A120590 = t on Γ119905

(53)



A = [

1198991

0 1198992

0 11989921198991

] (54)


LDL119879u = b (55)


u (x) =

1199061(x)

1199062(x)

=

119899119904

sum

119895=1

[

[

119906lowast

11(x y

119904119895) 119906

lowast

12(x y

119904119895)

119906lowast

21(x y

119904119895) 119906

lowast

22(x y

119904119895)

]

]

1198881119895

1198882119895


119890)

(56)




Ne

= [

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

)

119906lowast

21(x y

1199041) 119906

lowast

22(x y


lowast

21(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

)

]

]

ce = [11988811 11988821


1198882119899]119879

(57)



1 1198832) The





119906lowast

119894119895(x y

119904119895) =

minus1

8120587 (1 minus ]) 119866(3 minus 4]) 120575

119894119895ln 119903 minus 119903

119894119903119895 (58)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032




120590 (x) = [12059011 12059022

12059012]119879

= Tece (59)

whereTe

=[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

)

]]]

]

(60)


1199051

1199052

= n120590 = Qece (61)

in which

Qe = nTe n = [

1198991

0 1198992

0 11989921198991

] (62)


as

120590lowast

119894119895119896(x y) = minus1

4120587 (1 minus ]) 119903

sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903

119894119903119895119903119896]

(63)



is defined as

u (x) =

1

2

=

N1

N2

d119890= N

119890d119890 (x isin Γ

119890) (64)





119890and d

119890can be

expressed as

N119890= [

[


0 2

0 3



4

0 1

0 2


6

]

]

de = [11990611 11990621

11990612

11990622


11990628]119879

(65)

where 1 2 and


as

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(66)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ (67)

where 119894and 119906



119890





119890= Γ

119905+ Γ

119906+ Γ

119868) Compared




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ) minus int

Γ119905

119905119894119894dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ

(68)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(69)



119898119890= 0


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (70)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(71)







(33) and (34)


u119890= N

119890c119890+ [

1 0 1199092

0 1 minus1199091

] c0 (72)



119890at

element nodes119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

] = min (73)

which finally gives

c0 = Rminus1119890re (74)

where

Re =119899

sum

119894=1

[[[

[

1 0 1199092119894

0 1 minus1199091119894

1199092119894

minus1199091119894

1199092

1119894+ 1199092

2119894

]]]

]

re =119899

sum

119894=1

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

(75)

in which Δ119906119890119895119894



119890








120590119894119895119895

= minus119887119894

119894 119895 = 1 2 3 (76)


120590119894119895=

2119866V1 minus 2V

120575119894119895119890119896119896+ 2119866119890

119894119895

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(77)





119866119906119894119895119895

+119866

1 minus 2V119906119895119895119894

= minus119887119894 (78)


Γux1

x2

x3

b1

b2

b3

Γt

O



119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(79)


119894






119894


119906119894= 119906

ℎ

119894+ 119906119901

119894 (80)


equation

119866119906119901

119894119895119895+

119866

1 minus 2V119906119901

119895119895119894= minus119887

119894(81)


119866119906ℎ

119894119895119895+

119866

1 minus 2V119906ℎ

119895119895119894= 0 (82)


119906ℎ

119894= 119906

119894minus 119906119901

119894on Γ

119906

119905ℎ

119894= 119905119894minus 119905119901

119894on Γ

119905

(83)


119901


119906ℎ





119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (84)


119895



119906119901

119894=

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (85)

where Φ119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (86)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(87)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (88)



119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (90)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(91)



(1199091 1199092 1199093) and a given point (119909

1119895 1199092119895 1199093119895) in the domain of


119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(92)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897 (93)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(94)



119894= 0 the



u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (95)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(96)



1 1199092) The


119904119895) is given by [40]

119906lowast

119894119895(x y

119904119895) =

1

16120587 (1 minus ]) 119866119903(3 minus 4]) 120575

119894119895+ 119903119894119903119895 (97)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032

2+ 1199032



119904) can also


y119904= x0+ 120574 (x

0minus x119888) (98)





120590 (x) = [12059011 12059022

12059033

12059023

12059031

12059012]119879

= Tece (99)

where

Te =

[[[[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

133(x y

1) 120590

lowast

233(x y

1) 120590

lowast

333(x y


lowast

133(x y

119899119904

) 120590lowast

233(x y

119899119904

) 120590lowast

333(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]]]]

]

(100)


1

4 3

2


5 6

8 7

8-node 3D element

X2

X1

X3

Intraelement field

Ωe

Γe

Frame field

u = Nece

u(x) = edeN

cx

0x

sx

(a)

1

7 5

2

13 15

1917

3

4

6

89 10

1112

1416

1820

20-node 3D element

X2

X1

X3

Ωe

(b)



120590lowast

119894119895119896(x y) = minus1

8120587 (1 minus ]) 1199032

sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903

119894119903119895119903119896

(101)


1199051

1199052

1199053

= n120590 = Qece (102)

in which

Qe = nTe n =[[

[

1198991

0 0 0 11989931198992

0 1198992

0 1198993

0 1198991

0 0 119899311989921198991

0

]]

]

(103)


119890


u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (104)


119890is the matrix



119890and


N119890= [0 N

1N20 0 N

4N30]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(105)


N119894=[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(106)


120585 120578 isin [minus1 1]

1=(1 + 120585) (1 + 120578)

4

2=(1 minus 120585) (1 + 120578)

4

3=(1 minus 120585) (1 minus 120578)

4

4=(1 + 120585) (1 minus 120578)

4

(107)


element (Figure 6)


1 3

57

2

4

6

8


(10)

(11)(01)

(minus10)

(minus11)

120578

120585



119898119890used


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)



Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again



V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V



V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)


119894 119910119894 119911119894)


119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)




F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Ne

= [

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

)

119906lowast

21(x y

1199041) 119906

lowast

22(x y


lowast

21(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

)

]

]

ce = [11988811 11988821


1198882119899]119879

(57)



1 1198832) The





119906lowast

119894119895(x y

119904119895) =

minus1

8120587 (1 minus ]) 119866(3 minus 4]) 120575

119894119895ln 119903 minus 119903

119894119903119895 (58)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032




120590 (x) = [12059011 12059022

12059012]119879

= Tece (59)

whereTe

=[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

)

]]]

]

(60)


1199051

1199052

= n120590 = Qece (61)

in which

Qe = nTe n = [

1198991

0 1198992

0 11989921198991

] (62)


as

120590lowast

119894119895119896(x y) = minus1

4120587 (1 minus ]) 119903

sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903

119894119903119895119903119896]

(63)



is defined as

u (x) =

1

2

=

N1

N2

d119890= N

119890d119890 (x isin Γ

119890) (64)





119890and d

119890can be

expressed as

N119890= [

[


0 2

0 3



4

0 1

0 2


6

]

]

de = [11990611 11990621

11990612

11990622


11990628]119879

(65)

where 1 2 and


as

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(66)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ (67)

where 119894and 119906



119890





119890= Γ

119905+ Γ

119906+ Γ

119868) Compared




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ) minus int

Γ119905

119905119894119894dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ

(68)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(69)



119898119890= 0


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (70)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(71)







(33) and (34)


u119890= N

119890c119890+ [

1 0 1199092

0 1 minus1199091

] c0 (72)



119890at

element nodes119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

] = min (73)

which finally gives

c0 = Rminus1119890re (74)

where

Re =119899

sum

119894=1

[[[

[

1 0 1199092119894

0 1 minus1199091119894

1199092119894

minus1199091119894

1199092

1119894+ 1199092

2119894

]]]

]

re =119899

sum

119894=1

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

(75)

in which Δ119906119890119895119894



119890








120590119894119895119895

= minus119887119894

119894 119895 = 1 2 3 (76)


120590119894119895=

2119866V1 minus 2V

120575119894119895119890119896119896+ 2119866119890

119894119895

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(77)





119866119906119894119895119895

+119866

1 minus 2V119906119895119895119894

= minus119887119894 (78)


Γux1

x2

x3

b1

b2

b3

Γt

O



119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(79)


119894






119894


119906119894= 119906

ℎ

119894+ 119906119901

119894 (80)


equation

119866119906119901

119894119895119895+

119866

1 minus 2V119906119901

119895119895119894= minus119887

119894(81)


119866119906ℎ

119894119895119895+

119866

1 minus 2V119906ℎ

119895119895119894= 0 (82)


119906ℎ

119894= 119906

119894minus 119906119901

119894on Γ

119906

119905ℎ

119894= 119905119894minus 119905119901

119894on Γ

119905

(83)


119901


119906ℎ





119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (84)


119895



119906119901

119894=

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (85)

where Φ119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (86)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(87)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (88)



119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (90)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(91)



(1199091 1199092 1199093) and a given point (119909

1119895 1199092119895 1199093119895) in the domain of


119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(92)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897 (93)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(94)



119894= 0 the



u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (95)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(96)



1 1199092) The


119904119895) is given by [40]

119906lowast

119894119895(x y

119904119895) =

1

16120587 (1 minus ]) 119866119903(3 minus 4]) 120575

119894119895+ 119903119894119903119895 (97)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032

2+ 1199032



119904) can also


y119904= x0+ 120574 (x

0minus x119888) (98)





120590 (x) = [12059011 12059022

12059033

12059023

12059031

12059012]119879

= Tece (99)

where

Te =

[[[[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

133(x y

1) 120590

lowast

233(x y

1) 120590

lowast

333(x y


lowast

133(x y

119899119904

) 120590lowast

233(x y

119899119904

) 120590lowast

333(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]]]]

]

(100)


1

4 3

2


5 6

8 7

8-node 3D element

X2

X1

X3

Intraelement field

Ωe

Γe

Frame field

u = Nece

u(x) = edeN

cx

0x

sx

(a)

1

7 5

2

13 15

1917

3

4

6

89 10

1112

1416

1820

20-node 3D element

X2

X1

X3

Ωe

(b)



120590lowast

119894119895119896(x y) = minus1

8120587 (1 minus ]) 1199032

sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903

119894119903119895119903119896

(101)


1199051

1199052

1199053

= n120590 = Qece (102)

in which

Qe = nTe n =[[

[

1198991

0 0 0 11989931198992

0 1198992

0 1198993

0 1198991

0 0 119899311989921198991

0

]]

]

(103)


119890


u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (104)


119890is the matrix



119890and


N119890= [0 N

1N20 0 N

4N30]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(105)


N119894=[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(106)


120585 120578 isin [minus1 1]

1=(1 + 120585) (1 + 120578)

4

2=(1 minus 120585) (1 + 120578)

4

3=(1 minus 120585) (1 minus 120578)

4

4=(1 + 120585) (1 minus 120578)

4

(107)


element (Figure 6)


1 3

57

2

4

6

8


(10)

(11)(01)

(minus10)

(minus11)

120578

120585



119898119890used


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)



Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again



V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V



V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)


119894 119910119894 119911119894)


119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)




F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ) minus int

Γ119905

119905119894119894dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ

(68)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(69)



119898119890= 0


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (70)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(71)







(33) and (34)


u119890= N

119890c119890+ [

1 0 1199092

0 1 minus1199091

] c0 (72)



119890at

element nodes119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

] = min (73)

which finally gives

c0 = Rminus1119890re (74)

where

Re =119899

sum

119894=1

[[[

[

1 0 1199092119894

0 1 minus1199091119894

1199092119894

minus1199091119894

1199092

1119894+ 1199092

2119894

]]]

]

re =119899

sum

119894=1

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

(75)

in which Δ119906119890119895119894



119890








120590119894119895119895

= minus119887119894

119894 119895 = 1 2 3 (76)


120590119894119895=

2119866V1 minus 2V

120575119894119895119890119896119896+ 2119866119890

119894119895

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(77)





119866119906119894119895119895

+119866

1 minus 2V119906119895119895119894

= minus119887119894 (78)


Γux1

x2

x3

b1

b2

b3

Γt

O



119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(79)


119894






119894


119906119894= 119906

ℎ

119894+ 119906119901

119894 (80)


equation

119866119906119901

119894119895119895+

119866

1 minus 2V119906119901

119895119895119894= minus119887

119894(81)


119866119906ℎ

119894119895119895+

119866

1 minus 2V119906ℎ

119895119895119894= 0 (82)


119906ℎ

119894= 119906

119894minus 119906119901

119894on Γ

119906

119905ℎ

119894= 119905119894minus 119905119901

119894on Γ

119905

(83)


119901


119906ℎ





119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (84)


119895



119906119901

119894=

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (85)

where Φ119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (86)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(87)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (88)



119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (90)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(91)



(1199091 1199092 1199093) and a given point (119909

1119895 1199092119895 1199093119895) in the domain of


119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(92)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897 (93)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(94)



119894= 0 the



u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (95)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(96)



1 1199092) The


119904119895) is given by [40]

119906lowast

119894119895(x y

119904119895) =

1

16120587 (1 minus ]) 119866119903(3 minus 4]) 120575

119894119895+ 119903119894119903119895 (97)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032

2+ 1199032



119904) can also


y119904= x0+ 120574 (x

0minus x119888) (98)





120590 (x) = [12059011 12059022

12059033

12059023

12059031

12059012]119879

= Tece (99)

where

Te =

[[[[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

133(x y

1) 120590

lowast

233(x y

1) 120590

lowast

333(x y


lowast

133(x y

119899119904

) 120590lowast

233(x y

119899119904

) 120590lowast

333(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]]]]

]

(100)


1

4 3

2


5 6

8 7

8-node 3D element

X2

X1

X3

Intraelement field

Ωe

Γe

Frame field

u = Nece

u(x) = edeN

cx

0x

sx

(a)

1

7 5

2

13 15

1917

3

4

6

89 10

1112

1416

1820

20-node 3D element

X2

X1

X3

Ωe

(b)



120590lowast

119894119895119896(x y) = minus1

8120587 (1 minus ]) 1199032

sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903

119894119903119895119903119896

(101)


1199051

1199052

1199053

= n120590 = Qece (102)

in which

Qe = nTe n =[[

[

1198991

0 0 0 11989931198992

0 1198992

0 1198993

0 1198991

0 0 119899311989921198991

0

]]

]

(103)


119890


u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (104)


119890is the matrix



119890and


N119890= [0 N

1N20 0 N

4N30]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(105)


N119894=[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(106)


120585 120578 isin [minus1 1]

1=(1 + 120585) (1 + 120578)

4

2=(1 minus 120585) (1 + 120578)

4

3=(1 minus 120585) (1 minus 120578)

4

4=(1 + 120585) (1 minus 120578)

4

(107)


element (Figure 6)


1 3

57

2

4

6

8


(10)

(11)(01)

(minus10)

(minus11)

120578

120585



119898119890used


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)



Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again



V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V



V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)


119894 119910119894 119911119894)


119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)




F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Γux1

x2

x3

b1

b2

b3

Γt

O



119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(79)


119894






119894


119906119894= 119906

ℎ

119894+ 119906119901

119894 (80)


equation

119866119906119901

119894119895119895+

119866

1 minus 2V119906119901

119895119895119894= minus119887

119894(81)


119866119906ℎ

119894119895119895+

119866

1 minus 2V119906ℎ

119895119895119894= 0 (82)


119906ℎ

119894= 119906

119894minus 119906119901

119894on Γ

119906

119905ℎ

119894= 119905119894minus 119905119901

119894on Γ

119905

(83)


119901


119906ℎ





119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (84)


119895



119906119901

119894=

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (85)

where Φ119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (86)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(87)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (88)



119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (90)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(91)



(1199091 1199092 1199093) and a given point (119909

1119895 1199092119895 1199093119895) in the domain of


119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(92)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897 (93)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(94)



119894= 0 the



u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (95)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(96)



1 1199092) The


119904119895) is given by [40]

119906lowast

119894119895(x y

119904119895) =

1

16120587 (1 minus ]) 119866119903(3 minus 4]) 120575

119894119895+ 119903119894119903119895 (97)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032

2+ 1199032



119904) can also


y119904= x0+ 120574 (x

0minus x119888) (98)





120590 (x) = [12059011 12059022

12059033

12059023

12059031

12059012]119879

= Tece (99)

where

Te =

[[[[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

133(x y

1) 120590

lowast

233(x y

1) 120590

lowast

333(x y


lowast

133(x y

119899119904

) 120590lowast

233(x y

119899119904

) 120590lowast

333(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]]]]

]

(100)


1

4 3

2


5 6

8 7

8-node 3D element

X2

X1

X3

Intraelement field

Ωe

Γe

Frame field

u = Nece

u(x) = edeN

cx

0x

sx

(a)

1

7 5

2

13 15

1917

3

4

6

89 10

1112

1416

1820

20-node 3D element

X2

X1

X3

Ωe

(b)



120590lowast

119894119895119896(x y) = minus1

8120587 (1 minus ]) 1199032

sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903

119894119903119895119903119896

(101)


1199051

1199052

1199053

= n120590 = Qece (102)

in which

Qe = nTe n =[[

[

1198991

0 0 0 11989931198992

0 1198992

0 1198993

0 1198991

0 0 119899311989921198991

0

]]

]

(103)


119890


u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (104)


119890is the matrix



119890and


N119890= [0 N

1N20 0 N

4N30]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(105)


N119894=[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(106)


120585 120578 isin [minus1 1]

1=(1 + 120585) (1 + 120578)

4

2=(1 minus 120585) (1 + 120578)

4

3=(1 minus 120585) (1 minus 120578)

4

4=(1 + 120585) (1 minus 120578)

4

(107)


element (Figure 6)


1 3

57

2

4

6

8


(10)

(11)(01)

(minus10)

(minus11)

120578

120585



119898119890used


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)



Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again



V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V



V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)


119894 119910119894 119911119894)


119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)




F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1199091 1199092 1199093) and a given point (119909

1119895 1199092119895 1199093119895) in the domain of


119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(92)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897 (93)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(94)



119894= 0 the



u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (95)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(96)



1 1199092) The


119904119895) is given by [40]

119906lowast

119894119895(x y

119904119895) =

1

16120587 (1 minus ]) 119866119903(3 minus 4]) 120575

119894119895+ 119903119894119903119895 (97)

where 119903119894= 119909

119894minus 119909

119894119904 119903 = radic119903

2

1+ 1199032

2+ 1199032



119904) can also


y119904= x0+ 120574 (x

0minus x119888) (98)





120590 (x) = [12059011 12059022

12059033

12059023

12059031

12059012]119879

= Tece (99)

where

Te =

[[[[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

133(x y

1) 120590

lowast

233(x y

1) 120590

lowast

333(x y


lowast

133(x y

119899119904

) 120590lowast

233(x y

119899119904

) 120590lowast

333(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

212(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]]]]

]

(100)


1

4 3

2


5 6

8 7

8-node 3D element

X2

X1

X3

Intraelement field

Ωe

Γe

Frame field

u = Nece

u(x) = edeN

cx

0x

sx

(a)

1

7 5

2

13 15

1917

3

4

6

89 10

1112

1416

1820

20-node 3D element

X2

X1

X3

Ωe

(b)



120590lowast

119894119895119896(x y) = minus1

8120587 (1 minus ]) 1199032

sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903

119894119903119895119903119896

(101)


1199051

1199052

1199053

= n120590 = Qece (102)

in which

Qe = nTe n =[[

[

1198991

0 0 0 11989931198992

0 1198992

0 1198993

0 1198991

0 0 119899311989921198991

0

]]

]

(103)


119890


u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (104)


119890is the matrix



119890and


N119890= [0 N

1N20 0 N

4N30]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(105)


N119894=[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(106)


120585 120578 isin [minus1 1]

1=(1 + 120585) (1 + 120578)

4

2=(1 minus 120585) (1 + 120578)

4

3=(1 minus 120585) (1 minus 120578)

4

4=(1 + 120585) (1 minus 120578)

4

(107)


element (Figure 6)


1 3

57

2

4

6

8


(10)

(11)(01)

(minus10)

(minus11)

120578

120585



119898119890used


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)



Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again



V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V



V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)


119894 119910119894 119911119894)


119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)




F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

4 3

2


5 6

8 7

8-node 3D element

X2

X1

X3

Intraelement field

Ωe

Γe

Frame field

u = Nece

u(x) = edeN

cx

0x

sx

(a)

1

7 5

2

13 15

1917

3

4

6

89 10

1112

1416

1820

20-node 3D element

X2

X1

X3

Ωe

(b)



120590lowast

119894119895119896(x y) = minus1

8120587 (1 minus ]) 1199032

sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903

119894119903119895119903119896

(101)


1199051

1199052

1199053

= n120590 = Qece (102)

in which

Qe = nTe n =[[

[

1198991

0 0 0 11989931198992

0 1198992

0 1198993

0 1198991

0 0 119899311989921198991

0

]]

]

(103)


119890


u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (104)


119890is the matrix



119890and


N119890= [0 N

1N20 0 N

4N30]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(105)


N119894=[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(106)


120585 120578 isin [minus1 1]

1=(1 + 120585) (1 + 120578)

4

2=(1 minus 120585) (1 + 120578)

4

3=(1 minus 120585) (1 minus 120578)

4

4=(1 + 120585) (1 minus 120578)

4

(107)


element (Figure 6)


1 3

57

2

4

6

8


(10)

(11)(01)

(minus10)

(minus11)

120578

120585



119898119890used


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)



Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again



V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V



V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)


119894 119910119894 119911119894)


119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)




F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 3

57

2

4

6

8


(10)

(11)(01)

(minus10)

(minus11)

120578

120585



119898119890used


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(108)


Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

minus intΓ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(109)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ

(110)



Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (111)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(112)






119890yields again



V119899= V120585times V120578=

V119899119909

V119899119910

V119899119911

=

d119909d120585d119910d120585d119911d120585

times

d119909d120578d119910d120578d119911d120578

=

d119910d120585

d119911d120578

minusd119910d120578

d119911d120585

d119911d120585

d119909d120578

minusd119911d120578

d119909d120585

d119909d120585

d119910d120578

minusd119909d120578

d119910d120585

(113)

where V120585and V



V120585=

d119909d120585d119910d120585d119911d120585

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120585

119909119894

119910119894

119911119894

V120578=

d119909d120578d119910d120578d119911d120578

=

119899119889

sum

119894=1

120597119873119894(120585 120578)

120597120578

119909119894

119910119894

119911119894

(114)


119894 119910119894 119911119894)


119899 =V119899

1003816100381610038161003816V1198991003816100381610038161003816

(115)

where

119869 (120585 120578) =1003816100381610038161003816V119899

1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911

(116)




F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











F (x y) = [119865119894119895(119909 119910)]

119898times119898= Q119879

119890N119890 (117)

Then we can get

H119890= intΓ119890

Q119879119890N119890dΓ = int

Γ119890

F (x y) dΓ (118)


119867119894119895= intΓ119890

119865119894119895(119909 119910) d119878 =

119899119891

sum

119897=1

intΓ119890119897

119865119894119895(119909 119910) d119878 (119)


d119878 = 119869 (120585 120578) d120585 d120578 (120)


119867119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(121)

where 119899119891and 119899



119890matrix by

119866119894119895=

119899119891

sum

119897=1

int

1

minus1

119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578

asymp

119899119891

sum

119897=1

119899119901

sum

119904=1

119899119901

sum

119905=1

119908119904119908119905119865119894119895[119909 (120585

119904 120578119905) 119910 (120585

119904 120578119905)] 119869 (120585

119904 120578119905)

(122)




u119890= N

119890c119890+[[

[

1 0 0 0 1199093

minus1199092

0 1 0 minus1199093

0 1199091

0 0 1 1199092

minus1199091

0

]]

]

c0

(123)


119890ℎat the


min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (1199063119894minus 3119894)2

] (124)


c0 = Rminus1119890re (125)

where

Re

=

119899

sum

119894=1

[[[[[[[[[[[

[

1 0 0 0 1199093119894

minus1199092119894

0 1 0 minus1199093119894

0 1199091119894

0 0 1 1199092119894

minus1199091119894

0

0 minus1199093119894

1199092119894

1199092

2119894+ 1199092

3119894minus11990911198941199092119894

minus11990911198941199093119894

1199093119894

0 minus1199091119894

minus11990911198941199092119894

1199092

1119894+ 1199092

3119894minus11990921198941199093119894

minus1199092119894

1199091119894

0 minus11990911198941199093119894

minus11990921198941199093119894

1199092

1119894+ 1199092

2119894

]]]]]]]]]]]

]

re =119899

sum

119894=1

[[[[[[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ1199061198903119894

Δ11990611989031198941199092119894minus Δ119906

11989021198941199093119894

Δ11990611989011198941199093119894minus Δ119906

11989031198941199091119894

Δ11990611989021198941199091119894minus Δ119906

11989011198941199092119894

]]]]]]]]]]

]

(126)




1 1199092 1199093)


120590119894119895119895

= minus119887119894 (127)





120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120590119894119895119895

=2119866]1 minus 2]

120575119894119895119890 + 2119866119890

119894119895minus 119898120575

119894119895119879

119890119894119895=1

2(119906119894119895+ 119906119895119894)

(128)





119898 =2119866120572 (1 + V)(1 minus 2V)

(129)


119866119906119894119895119895

+119866

1 minus 2]119906119895119895119894

= 119898119879119894minus 119887119894 (130)


119906119894= 119906

119894on Γ

119906

119905119894= 119905119894

on Γ119905

(131)


119906119894and 119905


119905119894= 120590119894119895119899119895 (132)


outward normal


119894minus 119887



119894into




119906119894= 119906

ℎ

119894+ 119906119901

119894(133)


ing equation

119866119906119901

119894119895119895+

119866

1 minus 2]119906119901

119895119895119894= 119898119879

119894minus 119887119894

(134)


119866119906ℎ

119894119895119895+

119866

1 minus 2]119906ℎ

119895119895119894= 0 (135)


119906ℎ

119894= 119906

119894minus 119906119901

119894 on Γ

119906 (136)

119905ℎ

119894= 119905119894+ 119898119879119899

119894minus 119905119901

119894 on Γ

119905 (137)


ℎ

119894in (135)ndash











119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895

(119894 = 1 2 in R2 119894 = 1 2 3 in R

3)

119879 asymp

119873

sum

119895=1

120573119895120593119895

(138)


119895

119894and 120573


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896+

119873

sum

119895=1

120573119895Ψ119895

119894 (139)

where Φ119895119894119896and Ψ

119895



119866Φ119894119897119896119896

+119866

1 minus 2]Φ119896119897119896119894

= minus120575119894119897120593 (140)

119866Ψ119894119896119896

+119866

1 minus 2]Ψ119896119896119894

= 119898120593119894 (141)


Φ119894119896=1 minus ]119866

119865119894119896119898119898

minus1

2119866119865119898119896119898119894

(142)


nabla4119865119894119897= minus

1

1 minus ]120575119894119897120593 (143)



following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











following solutions

119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1)2(2119899 + 3)

2(R2) for 119899 = 1 2 3

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

2) for 119899 = 1 2 3

(144)

where

1198600= minus

1

2119866 (1 minus V)1199032119899+1

(2119899 + 1)2(2119899 + 3)

1198601= 5 + 4119899 minus 2V (2119899 + 3)

1198602= minus (2119899 + 1)

(145)


119865119897119894= minus

120575119897119894

1 minus V1199032119899+3

(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)

(R3) for 119899 = 1 2 3

(146)

Φ119897119894= 119860

0(1198601120575119897119894+ 119860

2119903119894119903119897) (R

3) for 119899 = 1 2 3

(147)

where

1198600= minus

1

8119866 (1 minus V)1199032119899+1

(119899 + 1) (119899 + 2) (2119899 + 1)

1198601= 7 + 4119899 minus 4V (119899 + 2)

1198602= minus (2119899 + 1)

(148)





119878119897119894119895= 119866 (Φ

119897119894119895+ Φ

119897119895119894) + 120582120575

119894119895Φ119897119896119896

(149)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R2) for 119899 = 1 2 3

(150)

where

1198610= minus

1

(1 minus V)1199032119899

(2119899 + 1) (2119899 + 3)

1198611= (2119899 + 2) minus V (2119899 + 3)

1198612= V (2119899 + 3) minus 1

1198613= 1 minus 2119899

(151)


119878119897119894119895= 119861

01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861

2120575119894119895119903119897+ 119861

3119903119894119903119895119903119897

(R3) for 119899 = 1 2 3

(152)

where

1198610= minus

1

4 (1 minus V)1199032119899

(119899 + 1) (119899 + 2)

1198611= 3 + 2119899 minus 2V (119899 + 2)

1198612= 2V (119899 + 2) minus 1

1198613= 1 minus 2119899

(153)



function

Ψ119894= 119880

119894 (154)


nabla2119880 =

119898 (1 minus 2])2119866 (1 minus ])

120593 (155)


[55] as follows

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1)2

(R2) for 119899 = 1 2 3

119880 =119898 (1 minus 2])2119866 (1 minus ])

1199032119899+1

(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3

(156)


Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

2119899 + 1(R2) for 119899 = 1 2 3

Ψ119894=119898 (1 minus 2])2119866 (1 minus ])

1199031198941199032119899

(2119899 + 2)(R3) for 119899 = 1 2 3

(157)


119878119894119895= 119866 (Ψ

119894119895+ Ψ

119895119894) + 120582120575

119894119895Ψ119896119896 (158)

Then we have

119878119894119895=

1198981199032119899minus1

(1 minus V) (2119899 + 1)(1 + 2119899V) 120575

119894119895+ (1 minus 2V) (2119899 minus 1) 119903

119894119903119895

(R2) for 119899 = 1 2 3

119878119894119895=

1198981199032119899minus1


119894119903119895+ 120575119894119895(1 + 2119899V)

(R3) for 119899 = 1 2 3

(159)




equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












equation

119898119879119894minus 119887119894asymp

119873

sum

119895=1

120572119895

119894120593119895 (160)


119906119901

119894=

3

sum

119896=1

119873

sum

119895=1

120572119895

119894Φ119895

119894119896 (161)


119894119896and 119878













120590119894119895119895

= 0 (162)

120590119894119895= 119862

119894119895119896119897119890119896119897 (163)

119890119894119895=1

2(119906119894119895+ 119906119895119894) (164)

where 119894 119895 = 1 2 3 119862119894119895119896119897


119862119894119895119896119897

119906119896119895119897

= 0 (165)


119906119894= 119906

119894on Γ

119906

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905

(166)

where 119906119894and 119905




119905is the



1and 119909

2


u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)






= 1199091+





120572 which can be


N120585 = 119901120585 (168)



N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











N = [

N1 N2

N3 NT1] 120585 =

ab (169)



119894119895119896119897as

follows

119876119894119896= 119862

11989411198961 119877

119894119896= 119862

11989411198962 119879

119894119896= 119862

11989421198962 (170)


1205901198941 = 2Re Lf1015840 (119911) 120590

1198942 = 2Re Bf1015840 (119911) (171)

where



1 1199012 1199013) applied



int119862


int119862


limxrarrinfin

120590119894119895= 0

(173)


119891 (119911) =1

2120587119894⟨ln (119911

120572minus 120572)⟩A119879p (174)


u =1

120587Im A ⟨ln (119911

120572minus 120572)⟩AT

p

120601 =1

120587Im B ⟨ln (119911

120572minus 120572)⟩AT

p(175)


120590lowast

1198941= minus120601

2= minus

1

120587ImB⟨

119901120572

(119911120572minus 120572)⟩AT

p

120590lowast

1198942= 120601

1=1

120587ImB⟨

1

(119911120572minus 120572)⟩AT

p

(176)



y

120593 x1

x2

x




1 1199092 and 119909






[12059011 12059022 12059023 12059031 12059012]119879

= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879

[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879

= (Tminus1)119879

[12057611 12057622 12057623 12057631 12057612]119879

(177)


T =

[[[[[[[

[

1198882

11990420 0 2119888119904

1199042

11988820 0 minus2119888119904

0 0 119888 minus119904 0

0 0 119904 119888 0

minus119888119904 119888119904 0 0 1198882minus 1199042

]]]]]]]

]

Tminus1 =

[[[[[[[

[

1198882

1199042

0 0 minus2119888119904

1199042

1198882

0 0 2119888119904

0 0 119888 119904 0

0 0 minus119904 119888 0

119888119904 minus119888119904 0 0 1198882minus 1199042

]]]]]]]

]

(178)



[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[12059011 12059022 12059023 12059031 12059012]119879


[12057611 12057622 12057623 12057631 12057612]119879

(179)




u (x) =

1199061(x)

1199062(x)

1199063(x)


119890) (180)


Ne =[[[

[

119906lowast

11(x y

1199041) 119906

lowast

12(x y

1199041) 119906

lowast

13(x y


lowast

11(x y

119904119899119904

) 119906lowast

12(x y

119904119899119904

) 119906lowast

13(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

23(x y


lowast

12(x y

119904119899119904

) 119906lowast

22(x y

119904119899119904

) 119906lowast

23(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

32(x y

1199041) 119906

lowast

33(x y


lowast

13(x y

119904119899119904

) 119906lowast

32(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]

]

ce = [11988811 11988821

11988831


1198882119899

1198883119899]119879

(181)


119904119895are




119894119895(x y




120590 (x) = [12059011 12059022

12059023

12059031

12059012]119879

= Tece (182)

where

Te =

[[[[[[[[[[

[

120590lowast

111(x y

1) 120590

lowast

211(x y

1) 120590

lowast

311(x y


lowast

111(x y

119899119904

) 120590lowast

211(x y

119899119904

) 120590lowast

311(x y

119899119904

)

120590lowast

122(x y

1) 120590

lowast

222(x y

1) 120590

lowast

322(x y


lowast

122(x y

119899119904

) 120590lowast

222(x y

119899119904

) 120590lowast

322(x y

119899119904

)

120590lowast

123(x y

1) 120590

lowast

223(x y

1) 120590

lowast

323(x y


lowast

123(x y

119899119904

) 120590lowast

223(x y

119899119904

) 120590lowast

323(x y

119899119904

)

120590lowast

131(x y

1) 120590

lowast

231(x y

1) 120590

lowast

331(x y


lowast

131(x y

119899119904

) 120590lowast

231(x y

119899119904

) 120590lowast

331(x y

119899119904

)

120590lowast

112(x y

1) 120590

lowast

231(x y

1) 120590

lowast

312(x y


lowast

112(x y

119899119904

) 120590lowast

212(x y

119899119904

) 120590lowast

312(x y

119899119904

)

]]]]]]]]]]

]

(183)



119894is



1199051

1199052

1199053

= n120590 = Qece (184)

in which

Qe = nTe

n =[[

[

1198991

0 0 11989931198992

0 11989921198993

0 1198991

0 0 11989921198991

0

]]

]

(185)



u (x) =

1

2

3

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (186)







119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119890can be

expressed as

N119890= [0 0 N

1N2N30 0 0]

d119890= [11990611 119906

2111990631

11990612

11990622

11990632


11990628

11990638]119879

(187)


Ni =[[[

[

119894

0 0

0 119894

0

0 0 119894

]]]

]

0 = [[

[

0 0 0

0 0 0

0 0 0

]]

]

(188)

and 1 2 and


[minus1 1]

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(189)




Π119898119890

=1

2∬Ω119890

120590119894119895120576119894119895dΩ minus int

Γ119905

119905119894119894dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ

(190)


Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

(191)



120575Π119898119890

= ∬Ω119890

120590119894119895120575119906119894119895dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

(192)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (193)


119905119894= 120590119894119895119899119895 (194)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890

119905119894120575119906119894dΓ minus int

Γ119890119905

119905119894120575119894dΓ (195)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ

+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

(196)


intΓ119890119906

119905119894120575119894dΓ = 0 (197)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ + int

Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119868

119905119894120575119894dΓ

(198)


119894 120575119905119894 and 120575



120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ + int

Γ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894

(199)



Π119898119890

= minus1

2intΓ119890

119905119894119906119894dΓ + int

Γ119890

119905119894119894dΓ minus int

Γ119905

119905119894119894dΓ (200)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (201)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ

(202)


with respectto c

119890and d






119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119894(119894 = 1 2 3) are given by

120590119894119895119895

= 0 119863119894119894= 0 in Ω (203)




120590119894119895= 119888119894119895119896119897

120576119896119897minus 119890119896119894119895119864119896 119863

119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)

where 119888119894119895119896119897


119894119896119897and 120581






120576119894119895=1

2(119906119894119895+ 119906119895119894) (205)



119894 (206)


119906119894= 119906

119894on Γ

119906 (207)

119905119894= 120590119894119895119899119895= 119905119894

on Γ119905 (208)

119863119899= 119863

119894119899119894= minus119902

119899= 119863

119899on Γ

119863 (209)

120601 = 120601 on Γ120601 (210)




and

Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)




2-1199093


22= 120576

32= 120576

12= 0 and 119864

2= 0) (204) can be

reduced to

12059011

12059033

12059013

=[[

[

11988811

11988813

0

11988813

11988833

0

0 0 11988844

]]

]

12057611

12057633

212057613

minus[[

[

0 11989031

0 11989033

11989015

0

]]

]

1198641

1198643

1198631

1198633

= [

0 0 11989015

11989031

11989033

0]

12057611

12057633

212057613

+ [

12058111

0

0 12058133

]

1198641

1198643

(212)


32= 120590

12=



12058133in (212) as

119888lowast

11= 11988811minus1198882

12

11988811

119888lowast

13= 11988813minus1198881211988813

11988811

119888lowast

33= 11988833minus1198882

13

11988811

119888lowast

44= 11988844 119890

lowast

15= 11989015

119890lowast

31= 11989031minus1198881211989031

11988811

119890lowast

33= 11989033minus1198881311989031

11988811

120581lowast

11= 12058111 120581

lowast

33= 12058133+1198902

31

11988811

(213)



ue =

1199061

1199062

120601

=

119899119904

sum

119895=1

[[[

[

119906lowast

11(x y

119904119895) 119906

lowast

21(x y

119904119895) 119906

lowast

31(x y

119904119895)

119906lowast

12(x y

119904119895) 119906

lowast

22(x y

119904119895) 119906

lowast

32(x y

119904119895)

119906lowast

13(x y

119904119895) 119906

lowast

23(x y

119904119895) 119906

lowast

33(x y

119904119895)

]]]

]

sdot

1198881119895

1198882119895

1198883119895


119890)

(214)



Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ne =

[[[[[

[

119906lowast

11(x y

1199041) 119906

lowast

21(x y

1199041) 119906

lowast

31(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

12(x y

1199041) 119906

lowast

22(x y

1199041) 119906

lowast

32(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

119906lowast

13(x y

1199041) 119906

lowast

23(x y

1199041) 119906

lowast

33(x y


lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

) 119906lowast

33(x y

119904119899119904

)

]]]]]

]

ce = [11988811 11988821

11988831

sdot sdot sdot 1198881119899119904

1198882119899119904

1198883119899119904

]119879

(215)




119906lowast

119894119895(x y



119904119895 The


119904119895) is given as [76 82 84]

119906lowast

11=

1

12058711987211

3

sum

119895=1

11990411989511199051

(2119895)1ln 119903119895

119906lowast

12=

1

12058711987212

3

sum

119895=1

11990411989521199051

(2119895)2arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

13=

1

12058711987213

3

sum

119895=1

11990411989531199051

(2119895)3arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

21=

1

12058711987211

3

sum

119895=1

11988911989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

22=

1

12058711987212

3

sum

119895=1

11988911989521199051

(2119895)2ln 119903119895

119906lowast

23=

1

12058711987213

3

sum

119895=1

11988911989531199051

(2119895)3ln 119903119895

119906lowast

31=

1

12058711987211

3

sum

119895=1

11989211989511199051

(2119895)1arc 119905119892

1199091minus 1199091199041

119904119895(1199093minus 1199091199043)

119906lowast

32=

1

12058711987212

3

sum

119895=1

11989211989521199051

(2119895)2ln 119903119895

119906lowast

33=

1

12058711987213

3

sum

119895=1

11989211989531199051

(2119895)3ln 119903119895

(216)

where 119903119895= radic(119909

1minus 1199091199041)2+ 1199042

119895(1199093minus 1199091199043)2 and 119904

119895is the three


4

119894+ 119888119904

3

119894minus




y119904= x0+ 120574 (x

0minus x119888) (217)


120590 = Tece (218)

where 120590 = [12059011

12059022

12059012

11986311198632]119879 and

Te =

[[[[[[[[[[[[[

[

120590lowast

11(x y

1199041) 120590

lowast

12(x y

1199041) 120590

lowast

13(x y


lowast

11(x y

119904119899119904

) 120590lowast

12(x y

119904119899119904

) 120590lowast

13(x y

119904119899119904

)

120590lowast

21(x y

1199041) 120590

lowast

22(x y

1199041) 120590

lowast

23(x y


lowast

21(x y

119904119899119904

) 120590lowast

22(x y

119904119899119904

) 120590lowast

23(x y

119904119899119904

)

120590lowast

31(x y

1199041) 120590

lowast

32(x y

1199041) 120590

lowast

33(x y


lowast

31(x y

119904119899119904

) 120590lowast

32(x y

119904119899119904

) 120590lowast

33(x y

119904119899119904

)

120590lowast

41(x y

1199041) 120590

lowast

42(x y

1199041) 120590

lowast

43(x y


lowast

41(x y

119904119899119904

) 120590lowast

42(x y

119904119899119904

) 120590lowast

43(x y

119904119899119904

)

120590lowast

51(x y

1199041) 120590

lowast

52(x y

1199041) 120590

lowast

53(x y


lowast

51(x y

119904119899119904

) 120590lowast

52(x y

119904119899119904

) 120590lowast

53(x y

119904119899119904

)

]]]]]]]]]]]]]

]

(219)


119894119895(x y






120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120590lowast

11=

1

12058711987211

3

sum

119895=1

[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

12=

1

12058711987212

3

sum

119895=1

[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

13=

1

12058711987213

3

sum

119895=1

[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

21=

1

12058711987211

3

sum

119895=1

[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

22=

1

12058711987212

3

sum

119895=1

[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

23= minus

1

12058711987213

3

sum

119895=1

[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

31=

1

12058711987211

3

sum

119895=1

[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

32=

1

12058711987212

3

sum

119895=1

[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

33= minus

1

12058711987213

3

sum

119895=1

[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

41=

1

12058711987211

3

sum

119895=1

[(119890151199041198951119904119895+ 119890151198891198951minus 120582

111198921198951)

sdot 1199051

(2119895)1

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

42=

1

12058711987212

3

sum

119895=1

[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582

111198921198952)

sdot 1199051

(2119895)2

1199091minus 1199091199041

1199032

119895

]

120590lowast

43= minus

1

12058711987213

3

sum

119895=1

[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582

111198921198953)

sdot 1199051

(2119895)3

1199091minus 1199091199041

1199032

119895

]

120590lowast

51=

1

12058711987211

3

sum

119895=1

[(119890311199041198951minus 119890331198891198951119904119895+ 120582

331198921198951119904119895)

sdot 1199051

(2119895)1

1199091minus 1199091199041

1199032

119895

]

120590lowast

52=

1

12058711987212

3

sum

119895=1

[(119890311199041198952+ 119890331198891198952119904119895minus 120582

331198921198952119904119895)

sdot 1199051

(2119895)2

119904119895(1199093minus 1199091199043)

1199032

119895

]

120590lowast

53= minus

1

12058711987213

3

sum

119895=1

[(119890311199041198953+ 119890331198891198953119904119895minus 120582

331198921198953119904119895)

sdot 1199051

(2119895)3

119904119895(1199093minus 1199091199043)

1199032

119895

]

(220)


13are



1199051

1199052

119863119899

=

Q1

Q2

Q3

ce = Qece (221)


where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

Qe = nTe

n =[[

[

1198991

0 1198992

0 0

0 11989921198991

0 0

0 0 0 11989911198992

]]

]

(222)


u (x) =

1

2

120601

=

N1

N2

N3

d119890= N

119890d119890 (x isin Γ

119890) (223)





N119890

=

[[[[

[


0 0 2

0 0 3



0 0 2

0 0 3



6

0 0 1

0 0 2

0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

9

]]]]

]

de = [11990611 11990621


1411990624


1811990628

1206018]119879

(224)


coordinate 120585

1= minus

120585 (1 minus 120585)

2

2= 1 minus 120585

2

3=120585 (1 + 120585)

2

(120585 isin [minus1 1])

(225)



119906119894119890= 119906

119894119891120601119890= 120601

119891(on Γ


119905119894119890+ 119905119894119891= 0 119863

119899119890+ 119863

119899119891= 0 (on Γ


(227)






Π119898119890

= Π119890+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ (228)

where

Π119890=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(229)



cup Γ119890119868

Γ119890119906= Γ119890cap Γ119906 Γ

119890119905= Γ119890cap Γ119905

Γ119890120601= Γ119890cap Γ120601 Γ

119890119863= Γ119890cap Γ119863

(230)





120575Π119898119890

= 120575Π119890+ intΓ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894(120575

119894minus 120575119906

119894)] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899(120575120601 minus 120575120601)] dΓ

(231)


120575Π119890= ∬

Ω119890

120590119894119895120575120576119894119895dΩ +∬

Ω119890

119863119894120575119864119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

= ∬Ω119890

120590119894119895120575119906119894119895dΩ +∬

Ω119890

119863119894120575120601119894dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ

(232)


∬Ω119890

119891119894dΩ = int

Γ119890

119891 sdot 119899119894dΓ (233)



119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905119894= 120590119894119895119899119895 119863

119899= 119863

119894119899119894 (234)

we obtain

120575Π119890= minus∬

Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ + int

Γ119890

119905119894120575119906119894dΓ

+ intΓ119890

119863119899120575120601dΓ minus int

Γ119890119905

119905119894120575119894dΓ minus int

Γ119890119863

119863119899120575120601dΓ

(235)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ minus int

Γ119890119905

119905119894120575119894dΓ


119863119899120575120601dΓ + int

Γ119890

[(119894minus 119906119894) 120575119905

119894+ 119905119894120575119894] dΓ

+ intΓ119890

[(120601 minus 120601) 120575119863119899+ 119863

119899120575120601] dΓ

(236)


intΓ119890119906

119905119894120575119894dΓ = 0 int

Γ119890120601

119863119899120575120601dΓ = 0 (237)


120575Π119898119890

= minus∬Ω119890

120590119894119895119895120575119906119894dΩ minus∬

Ω119890

119863119894119894120575120601dΩ

+ intΓ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119868

119905119894120575119894dΓ + int

Γ119868

119863119899120575120601dΓ

(238)


119894 120575119905119894 120575120601 120575119863

119899 120575

119894 and 120575120601


120575Π119898(119890cup119891)


120590119894119895119895120575119906119894dΩ minus∬

Ω119890cupΩ119891

119863119894119894120575120601dΩ

+ intΓ119890119905+Γ119890119905

(119905119894minus 119905119894) 120575

119894dΓ + int

Γ119890119863+Γ119891119863

(119863119899minus 119863

119899) 120575120601dΓ

+ intΓ119890+Γ119891

(119894minus 119906119894) 120575119905

119894dΓ + int

Γ119890+Γ119891

(120601 minus 120601) 120575119863119899dΓ

+ intΓ119890119891119868

(119905119894119890+ 119905119894119891) 120575

119894+ intΓ119890119891119868

(119863119899119890+ 119863

119899119891) 120575120601dΓ

(239)





Π119898119890

=1

2∬Ω119890

(120590119894119895120576119894119895+ 119863

119894119864119894) dΩ minus int

Γ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

=1

2(intΓ119890

119905119894119906119894dΓ minus∬

Ω119890

120590119894119895119895119906119894dΩ)

+1

2(intΓ119890

119863119899120601dΓ minus∬

Ω119890

119863119894119894120601dΩ)

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

+ intΓ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

(240)


Π119898119890

=1

2intΓ119890

119905119894119906119894dΓ + 1

2intΓ119890

119863119899120601dΓ minus int

Γ119905

119905119894119894dΓ

minus intΓ119863

119863119899120601dΓ + int

Γ119890

119905119894(119894minus 119906119894) dΓ + int

Γ119890

119863119899(120601 minus 120601) dΓ

= minus1

2intΓ119890

(119905119894119906119894+ 119863

119899120601) dΓ + int

Γ119890

(119905119894119894+ 119863

119899120601) dΓ

minus intΓ119905

119905119894119894dΓ minus int

Γ119863

119863119899120601dΓ

(241)


Π119898119890

= minus1

2c119890

119879H119890c119890+ c119890

119879G119890d119890minus d119890

119879g119890 (242)

where

H119890= intΓ119890

Q119879119890N119890dΓ G

119890= intΓ119890

Q119879119890N119890dΓ

g119890= intΓ119905

N119879119890tdΓ + int

Γ119863

N119879119890DdΓ

(243)


x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x1

x2

x3

L = 10

W = 10

A

H = 10

P = 10


0 1000 2000 3000 400030

32

34

36

38

40

42

44

46


Reference

Disp

lace

men

tu1

(10minus4

m)

(a)

0 1000 2000 3000 400090

95

100

105

110

115

120

125


Reference

Stre

ss12059011

(MPa

)


(b)


11

X Y

ZMesh 1

(a)

Mesh 2

X Y

Z

(b)

Mesh 3

X Y

Z

(c)



u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










u124222181614121080604020

(a)

230220210200190180170160150140130120110100908070605040

(b)


11of the cube

a

bT =ln(rb)

ln(ab)T0

(a)

X

Y

Z

(b)



with respectto c

119890and d



u119890= N

119890c119890+[[

[

1 0 1199092

0

0 1 minus11990910

0 0 0 1

]]

]

c0 (244)



min =

119899

sum

119894=1

[(1199061119894minus 1119894)2

+ (1199062119894minus 2119894)2

+ (120601119894minus 120601119894)2

] (245)

which finally gives

c0 = Rminus1119890re (246)

Re =119899

sum

119894=1

[[[[[

[

1 0 1199092119894

0

0 1 minus1199091119894

0

1199092119894

minus1199091119894

1199092

1119894+ 1199092

21198940

0 0 0 1

]]]]]

]

(247)

re =119899

sum

119894=1

[[[[[

[

Δ1199061198901119894

Δ1199061198902119894

Δ11990611989011198941199092119894minus Δ119906

11989021198941199091119894

Δ120601119890119894

]]]]]

]

(248)

in whichΔu119890119894= (u




50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










50 75 100 125 150 175 200minus25

minus20

minus15

minus10

minus05

00

05


120590r

r (m)

(a)


50 75 100 125 150 175 200minus12

minus9

minus6

minus3

0

3

6

1205900

r (m)

(b)


120590r


20

15

10

5

020151050

y

x

(a)

120590t


20

15

10

5

020151050

y

x

(b)



119890





0= 10

11(Nm2) 119890

0=

101(NmV) 119896

0= 10


0= 10

minus3(Vm)



1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

1

1

T = 30x3


x2

x3

x1

(a)

X

Y

Z

(b)

X

Y

Z

(c)











z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

u3

ABAQUSHFS-FEM

00 02 04 06 08 10minus0035

minus0030

minus0025

minus0020

minus0015

minus0010

minus0005

0000

(a)

minus1000

minus500

0

12059033

ABAQUSHFS-FEM

z

00 02 04 06 08 10

(b)




1199090


01205760= 10


0= 119909

01198640= 10

7(V)

Stress 1205900= 11988801205760= 10


0= 119896

01198640= 10

minus2(Cm2)

Compliance 1199040= 12057601205900= 10


0= 119864

01198630= 10

9(mVC)


01198900= 10


0= 119864

01205900= 10

minus1(mVN)


1











L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










L

B

A

W

2b

2a

x2

x1

1205900

1205900

(a)

Z

Y

X

X

Y

Z

(b)

Z

Y

X

X

Y

Z

(c)



119879 = 301199093 119887

1= 0 119887

2= 0

1198873= minus2000 [(119909 minus 05)

2+ (119910 minus 05)

2

]

(249)



33





0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90minus2

minus1

0

1

2

3

4

5

6

7



Hoo

p str

ess120590

120579120579

(GPa

)

Angle 120579 (deg)


tension of 1205900



113GPa 1198642= 527GPa 119866

12= 285GPa and V

12= 045





11 12059022 and 120590









2= 119864

3= 1103GPa 119866

23= 298GPa

11986631

= 11986612

= 285GPa and V23

= 049 V31

= V12

=



120572= 119894 (120572 = 1 2 3) [59] However


(1 minus 0004)119894 1199012= 119894 and 119901

3= (1 + 0004)119894 can be applied to



2= 0) and are 07


1and


Figure 22


119909119909= 1205900= 10 parallel to the 119909




119909119909



1205791205900times10

10







119903



S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

1312

1064

0816

0567

0319

0071

minus0178

minus0426

minus0674

minus0923

minus1171

minus1419

minus1668

Y

Z Z ZX

Y

X

Y

X

4624

4234

3844

3454

3064

2673

2283

1893

1503

1113

0723

0333

minus0057

1646

1372

1098

0824

0551

0277

0003

minus0271

minus0545

minus0819

minus1093

minus1366

minus1640

(a)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

4475

4029

3584

3138

2693

2247

1802

1356

0911

0465

0020

minus0425

minus0871

3241

2925

2610

2294

1979

1663

1348

1032

0716

0401

0085

minus0230

minus0546

2679

2489

2199

1959

1719

1480

1240

1000

0760

0520

0280

0041

minus0199

(b)

Y

Z Z ZX

Y

X

Y

X

S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)

6497

5952

5407

4862

4316

3771

3226

2681

2136

1590

1045

0500

minus0045

0910

0767

0624

0480

0337

0194

0051

minus0092

minus0235

minus0379

minus0522

minus0655

minus0808

1583

1319

1055

0791

0526

0262

minus0002

minus0266

minus0531

minus0795

1059

minus minus1323

minus1588

(c)


∘ and (c) 120593 = 90∘


40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










40

45

50

55

60

65

70

ABAQUS HFS-FEM

0 10 20 30 40 50 60 70 80 90

SCF

(1205901205791205791205900)

Angle 120593 (deg)


W

L

bb

ab

aaa aa

a

1205900

A

x2

x1

BC

(a)

X

Y

8

6

4

2

0

minus2

minus4

minus6


(b)




Disp 1199061

A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091

Stress 12059011

A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625


ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ABAQUSHFS-FEM

0034

0036

0038

0040

0042

0044

0046

Disp

lace

men

tu1

(mm

)


(a)

ABAQUSHFS-FEM

minus0015

minus0010

minus0005

0000

0005

0010

0015

Disp

lace

men

tu2

(mm

)


(b)



ABAQUS




151344Cmminus2

11988812


31minus698Cmminus2

11988813


331384Cmminus2

11988833


1160 times 10

minus9 CNm11988844


33547 times 10

minus9 CNm


∘ for case of 120590infin119909119909

and at 120579 = 0∘ and 120579 = 180


119911119911 both


1205791205900times10


and 120590infin119911119911


x

z

rA

L

120590infin xx=1205900

120590infin xx=1205900

120579

(a)

X

Y

Z

(b)

X

Y

Z



0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16 18 20

8

12

16

20

24

28

32




Stre

ss120590120579

(MPa

)

r (m)

(a)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

3

4

5

Stre

ss120590r

(MPa

)

r (m)




(b)



119909119909and


000 025 050 075 100minus2

minus1

0

1

2

3

4




Stre

ss1205901205791205900

120579 (120587 rad)

(a)

000 025 050 075 100

minus3

minus2

minus1

0

1

2

3




120579 (120587 rad)

Elec

tric

al d

ispla

cem

entD

1205791205900

(1010)

(b)


1205791205900times 10



at 120579 = 65∘ and 120579 = 115


9 Conclusions











References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















References




































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Review Article Hybrid Fundamental Solution Based Finite...

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Transcript of Review Article Hybrid Fundamental Solution Based Finite...