Two generalized conforming quadrilateral Mindlin–Reissner plateelements based on the displacement function

Yan Shang a,c, Song Cen a,c,d,n, Chen-Feng Li b, Xiang-Rong Fu e

a Department of Engineering Mechanics, School of Aerospace Engineering, Tsinghua University, Beijing 100084, Chinab Zienkiewicz Centre for Computational Engineering & Energy Safety Research Institute, College of Engineering, Swansea University, Swansea SA2 8PP, UKc High Performance Computing Center, School of Aerospace Engineering, Tsinghua University, Beijing 100084, Chinad Key Laboratory of Applied Mechanics, School of Aerospace Engineering, Tsinghua University, Beijing 100084, Chinae College of Water Conservancy & Civil Engineering, China Agricultural University, Beijing 100083, China

a r t i c l e i n f o

Article history:Received 26 June 2014Received in revised form7 January 2015Accepted 16 January 2015

Keywords:Finite elementMindlin–Reissner plateDisplacement functionGeneralized conformingMesh distortion

a b s t r a c t

This work presents two 4-node, 12-DOF quadrilateral displacement-based finite elements for analysis ofthe Mindlin–Reissner plate. Derived from the fundamental analytical solutions of the displacementfunction F, the deflection and rotation fields of the proposed elements satisfy a priori all relatedgoverning equations. The unknown coefficients are determined through the generalized conformingelement method, a relaxed and rational conforming approach. The resulting elements perform likenonconforming elements on a coarse mesh, and with mesh refinement they converge as conformingelements. Numerical benchmarks demonstrate that the new elements are insensitive to mesh distortionand free of shear locking, and can provide satisfactory results for most cases.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

As a major class of finite element models, Mindlin–Reissner platebending elements have been extensively studied over the past fewdecades and numerous successful models were proposed [1–15].Playing a central role in practical engineering analysis, the Mindlin–Reissner plate element is continuously being improved for betterperformance and to meet specific requirement. Among others, someof the latest developments include the NIPE formulations derivedfrom the assumed-strain technique [16], the models with continuousdisplacement and discontinuous rotations [17,18], the element basedon the high-order linked interpolation [19], the alternative alphafinite element method with discrete shear gap technique [20], and soon. However, two major challenges remain outstanding: (1) theperformance of the element cannot be guaranteed when the mesh isseverely distorted; and (2) the precisions for stress/resultant resultsare relatively lower than those for displacements.

A series of new developments have been reported recently, whichmake it possible to address the aforementioned challenges. Liu et al.[21] proposed the smoothed FEM (SFEM), which integrates the strainsmoothing technique into the conventional FEM and formulatesdifferent smoothed FEM models, including the cell-based SFEM

[22,23], the edge-based SFEM [24] and the node-based SFEM [25].The plate elements based on these SFEM models [26,27] are insensi-tive to mesh distortion and free of shear locking. Ribatić et al. [28]constructed two 9-node distortion-immune displacement-based qua-drilateral thick plate elements by using bubble parameters. Cen et al.[29] proposed a hybrid displacement function (HDF) method forformulating Mindlin–Reissner plate elements, where they took thedisplacement function F [30] to derive the stress trial functions thatsatisfy all governing equations and employed the Timoshenko’s beamformulae to determine boundary displacement modes. The HDF elem-ent enables superior precision for both displacements and resultants,even in cases where a severely distorted mesh containing concavequadrilateral or degenerated triangular elements is used.

By combining the displacement function F [29,30] with the generalized conforming element method [1], this work develops two high-performance displacement-based Mindlin–Reissner plate elements.The concept of the generalized conforming element was first proposedby Long et al. [31], and since then has been successfully applied toplane problems [32,33], plate problems [34–37] and shell problems[38] as well. In this concept, the displacement fields are usuallydetermined by the relaxed compatibility requirements, i.e., the gen-eralized conforming conditions.

The new generalized conforming elements are derived via threesteps. First, the deflection and rotation fields within the element areassumed according to the fundamental analytical solutions of thedisplacement function F, and the corresponding unknown coefficients

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Finite Elements in Analysis and Design

http://dx.doi.org/10.1016/j.finel.2015.01.0120168-874X/& 2015 Elsevier B.V. All rights reserved.

n Corresponding author at: Department of Engineering Mechanics, School ofAerospace Engineering, Tsinghua University, Beijing 100084, China.

E-mail address: censong@tsinghua.edu.cn (S. Cen).

Finite Elements in Analysis and Design 99 (2015) 24–38

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are introduced. Next, the relations between these unknown coeffi-cients and the element’s nodal displacement DOFs are determined byemploying a set of generalized compatibility conditions, from whichthe deflection and rotation fields can be expressed as functions of theelement’s nodal displacement DOFs. Finally, the new elements can beconstructed following the principle of minimum potential energy.These two quadrilateral elements are denoted as GCP4-12α andGCP4-14α, respectively, indicating that they are “Generalized Con-forming Plate element with 4 nodes and 12 or 14 displacementcoefficients”. Since the displacement components derived from thedisplacement function F can satisfy all related governing equations,these new elements also possess the advantages from analyticalmethods, as the HDF elements [29] do.

The performance of the new elements is assessed with standardbenchmark examples. Numerical results show that these two newelements are free of shear locking, insensitive to mesh distortions,and can provide satisfactory results for most cases. Only a smalldeviation exists for thick plates under soft simply supported bou-ndary conditions due to interelement incompatibility.

2. The formulations of new elements

2.1. The displacement function F

For a Mindlin–Reissner plate (see Fig. 1), the transverse deflectionw and the rotations ψx, ψy can be expressed by the displacement

function F [30]:

w¼ F�DC∇2F; ψx ¼

∂F∂x

; ψy ¼∂F∂y

; ð1Þ

in which F should satisfy the equation:

D∇2∇2F ¼ q: ð2ÞSubstitution of Eq. (1) into the strain–displacement relations

yields

κxx ¼ �∂2F∂x2

; κyy ¼ �∂2F∂y2

; κxy ¼ �2∂2F∂x∂y

; ð3Þ

γxz ¼ �DC

∂∂x

∇2F� �

; γyz ¼ �DC

∂∂y

∇2F� �

; ð4Þ

where D and C denote the bending and shear stiffness of the plate.The solutions of the deflection, rotations and strains can be

readily derived from the fundamental analytical solutions ofdisplacement function F [29]. For the general (homogeneous) partof F, the displacement components and strains are listed in Table 1.For the particular part corresponding to a plate subjected to auniformly distributed transverse load q, the displacement compo-nents are

un ¼wn

ψnxψny

8><>:

9>=>;¼

q48D x

4þy4� �� q4C x2þy2� �q

12Dx3

q12Dy

3

8>><>>:

9>>=>>;; ð5Þ

x

w/ y

y

w/ x

z,w z,w

x,u y,v

Mxy

Mxy

Mxy

Mx

My

My

Ty

Ty

TxMid-surface (xoy plane)

z

x

y h

TxMx

Mxy∂ ∂ ∂ ∂

ψψ

Fig. 1. The definitions of the displacements and resultants for a Mindlin–Reissner plate.

Table 1Fourteen fundamental analytical solutions for the general part F0 of the displacement function and the corresponding deflections, rotations and strains.

i 1 2 3 4 5 6 7 8 9

F0i 1 x y x2 xy y2 x3 x2y xy2

w0i 1 x y x2�2D/C xy y2�2D/C x3�6xD/C x2y�2yD/C xy2�2xD/C

ψ0xi 0 1 0 2x y 0 3x2 2xy y2

ψ0yi 0 0 1 0 x 2y 0 x2 2xy

κ0xxi 0 0 0 �2 0 0 �6x �2y 0κ0yyi 0 0 0 0 0 �2 0 0 �2xκ0xyi 0 0 0 0 �2 0 0 �4x �4yγ0xzi 0 0 0 0 0 0 �6D/C 0 �2D/Cγ0yzi 0 0 0 0 0 0 0 �2D/C 0

i 10 11 12 13 14

F0i y3 x3y xy3 x4�y4 6x2y2�x4�y4

w0i y3�6yD/C x3y�6xyD/C xy3�6xyD/C x4�y4�(12x2�12y2)D/C 6x2y2�x4�y4

ψ0xi 0 3x2y y3 4x3 12xy2�4x3

ψ0yi 3y2 x3 3xy2 �4y3 12x2y�4y3

κ0xxi 0 �6xy 0 �12x2 �12y2þ12x2

κ0yyi �6y 0 �6xy 12y2 �12x2þ12y2

κ0xyi 0 �6x2 �6y2 0 �48xy

γ0xzi 0 �6yD/C �6yD/C �24xD/C 0γ0yzi �6D/C �6xD/C �6xD/C 24yD/C 0

43

5

6

7

nodes

Gauss points

Mid-side points

12

8

A1 B1

B2

B3

B4 A2

A3

A4

Fig. 2. The generalizing conforming quadrilateral plate element with 4 nodes.

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–38 25

and the strains are

En ¼

κnxxκnyyκnxyγnxzγnyz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

� q4Dx2� q4Dy2

0� q2Cx� q2Cy

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;: ð6Þ

2.2. The procedure for constructing new generalized conformingelements

Unlike the traditional generalized conforming elements or non-conforming elements, the deflection and rotation fields of the newelements are assumed according to the fundamental analytical solu-tions of the displacement function F. For a generalized conforming

4-node quadrilateral element, as shown in Fig. 2, the displacementfunction F is set as the linear combination of its fundamentalanalytical solutions:

F ¼ F0þFn ¼Xki ¼ 1

F0i αiþFn; ð7Þ

where k is the number of the fundamental analytical solutions for thegeneral part of F0, and αi (i¼1�k) are the corresponding unknowncoefficients. Substituting Eq. (7) into Eq. (1), the deflection androtation fields within the element are obtained as

w¼Xki ¼ 1

w0i αiþwn; ψx ¼Xki ¼ 1

ψ0xiαiþψnx ; ψy ¼Xki ¼ 1

ψ0yiαiþψny: ð8Þ

Fig. 3. The model and mesh for the patch test.

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–3826

The displacements can be expressed in the following matrixform:

u¼w01 w

02 w

03 ⋯ w

0k

ψ0x1 ψ0x2 ψ

0x3 ⋯ ψ

0xk

ψ0y1 ψ0y2 ψ

0y3 ⋯ ψ

0yk

2664

3775

α1

α2

α3

⋮αk

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;þ

wn

ψnxψny

8><>:

9>=>;¼Uαþun: ð9Þ

Then, the corresponding strain field can be obtained as

E¼

κxx1 κxx2 κxx3 ⋯ κxxkκyy1 κyy2 κyy3 ⋯ κyykκxy1 κxy2 κxy3 ⋯ κxykγxz1 γxz2 γxz3 ⋯ γxzkγyz1 γyz2 γyz3 ⋯ γyzk

26666664

37777775

α1

α2

α3

⋮αk

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;þ

κnxxκnyyκnxyγnxzγnyz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼ EαþEn: ð10Þ

The detailed components of matrices U and E in the aboveequations are listed in Table 1.

In order to solve these unknown coefficients αi (i¼1�k), it isnecessary to introduce k generalized conforming conditions. Afteremploying an appropriate set of generalized conforming condi-tions, the relations between the coefficients αi (i¼1�k) and theelement’s nodal displacement DOFs are obtained as

λαþUn ¼ Γqe; ð11Þwhere

qe ¼ w1 ψx1 ψy1 w2 ψx2 ψy2 w3 ψx3 ψy3 w4 ψx4 ψy4h iT

:

ð12ÞThe matrices λ, Un and Γ have different expressions for

different sets of generalized conforming conditions. The detailsfor this matter will be discussed in Section 2.3.

Following Eq. (11), α can be expressed in terms of qe:

α¼ Lλqe�Lu; ð13Þwhere

Lλ ¼ λ�1Γ; Lu ¼ λ�1Un: ð14ÞThen, substitution of Eq. (13) into Eqs. (9) and (10) yields

u¼U Lλqe�Lu� �þun ¼NqeþNn; ð15Þ

E¼ E Lλqe�Lu� �þEn ¼ BqeþBn; ð16Þ

in which

N¼ULλ; Nn ¼ un�ULu; ð17Þ

B¼ ELλ; Bn ¼ En�ELu: ð18Þ

The potential energy functional of a Mindlin–Reissner plateelement can be expressed in the following matrix form [1]:

ΠeP ¼∬Ae12ETAEdxdy�∬Ae fTudxdyþ

ZSeσ

RTdds; ð19Þ

where A is the elasticity matrix as defined in Eq. (20), f is thedistributed transverse load vector within the plate as defined inEq. (21), R is the boundary load vector at the element’s edge as

y = 0c

x

y

L /3

L /3

L /3

L /3x = 0ψ

ψ

Fig. 4. A quarter of square plate (c is the central point of plate). (a) 4×4 regular mesh and (b) 4×4 distorted mesh.

Table 2The dimensionless results of central deflection wc/(qL4/100D) and moment Mc/(qL2/10) of the clamped square plate, thin plate case (h/L¼0.001).

Mesh N�N 2�2 4�4 8�8 16�16 32�32 Reference

Central deflection wc/(qL4/100D)GCP4-12α 0.1227 0.1259 0.1264 0.1265 0.1265 0.1265GCP4-14α 0.1227 0.1259 0.1264 0.1265 0.1265MITC4 [2,27] 0.1211 0.1251 0.1262 0.1264 0.1265Q4BL [10] 0.1116 0.1238 0.1259 0.1265 0.1265ARS-Q12 [14] 0.1245 0.1263 0.1265 0.1265 0.1265AC-MQ4 [37] 0.1245 0.1263 0.1265 0.1265 0.1265MISC2 [27] 0.1266 0.1264 0.1265 0.1265 0.1265

Central moment Mc/(qL2/10)GCP4-12α 0.2234 0.2283 0.2290 0.2290 0.2291 0.2291GCP4-14α 0.2234 0.2283 0.2290 0.2290 0.2291MITC4 [2,27] 0.1890 0.2196 0.2267 0.2285 0.2289Q4BL [10] 0.2179 0.2261 0.2283 0.2289 0.2290ARS-Q12 [14] 0.2869 0.2433 0.2326 0.2299 0.2293AC-MQ4 [37] 0.2712 0.2407 0.2321 0.2298 0.2292MISC2 [27] 0.1976 0.2218 0.2273 0.2286 0.2289

Table 3The dimensionless results of central deflection wc/(qL4/100D) and moment Mc/(qL2/10) of the clamped square plate, thick plate case (h/L¼0.1).

Mesh N�N 2�2 4�4 8�8 16�16 32�32 Reference

Central deflection wc/(qL4/100D)GCP4-12α 0.1494 0.1499 0.1501 0.1502 0.1502 0.1499GCP4-14α 0.1494 0.1499 0.1501 0.1502 0.1502MITC4 [2,27] 0.1431 0.1488 0.1500 0.1504 0.1504Q4BL [10] 0.1427 0.1491 0.1501 0.1504 0.1504ARS-Q12 [14] 0.1678 0.1548 0.1515 0.1507 0.1505AC-MQ4 [37] 0.1474 0.1477 0.1494 0.1502 0.1504MISC2 [27] 0.1483 0.1500 0.1503 0.1504 0.1505

Central moment Mc/(qL2/10)GCP4-12α 0.2350 0.2306 0.2310 0.2315 0.2317 0.231GCP4-14α 0.2350 0.2306 0.2310 0.2315 0.2317MITC4 [2,27] 0.1898 0.2219 0.2295 0.2314 0.2318Q4BL [10] 0.2186 0.2288 0.2313 0.2318 0.2320ARS-Q12 [14] 0.3061 0.2527 0.2376 0.2334 0.2324AC-MQ4 [37] 0.2813 0.2494 0.2373 0.2334 0.2324MISC2 [27] 0.1982 0.2241 0.2300 0.2315 0.2319

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–38 27

defined in Eq. (22), and d is the boundary displacement along theboundary Seσ as defined in Eq. (23).

A¼

D μD 0 0 0μD D 0 0 00 0 1�μ2 D 0 00 0 0 C 00 0 0 0 C

26666664

37777775; ð20Þ

f ¼ q 0 0� �T ; ð21ÞR¼ �T Mn Mns

h iT; ð22Þ

d¼w

ψnψ s

8><>:

9>=>;�������Seσ

¼1

l m�m l

264

375

w

ψxψy

8><>:

9>=>;�������Seσ

¼ Luj Seσ ; ð23Þ

where l,m denote the direction cosines of the element boundaries’outer normal n.

By substituting Eqs. (15) and (16) into Eq. (19) and applying theprinciple of minimum potential energy,

∂ΠeP∂qe

¼∬AeBTABdxdyqeþ∬AeBTABndxdy�∬AeNT fdxdy

þZSσ

eNjTSeσL

TRds¼ 0; ð24Þ

the following relation can be obtained:

Kqe ¼ pe; ð25Þwhere K is the element stiffness matrix

K¼∬AeBTABdxdy; ð26Þand pe is the element nodal equivalent load vector

pe ¼∬AeNT fdxdy�∬AeBTABndxdy�ZSeσ

NjTSeσLTRds; ð27Þ

where Nj Seσ is the value of N along the boundary Seσ .

The unknown coefficient α can be derived from Eq. (13) aftersolving for qe. Then the displacements and the strains at any pointof the element can be determined by substituting its Cartesiancoordinates into Eqs. (9) and (10).

The above derivation shows the general formulations for con-structing new generalized conforming elements. If different gen-eralized conforming conditions are employed for deriving theelements, the detailed expressions of related matrices (see fromEqs. (14)–(18)) will also be different.

2.3. The formulations of the new generalized conforming plate (GCP)elements

In this section, two new generalized conforming plate (GCP)elements are constructed following the procedure described inSection 2.2. According to the number of the fundamental analytical

Table 4The dimensionless results of central deflection wc/(qL4/100D) and moment Mc/(qL2/10) of the SS1 square plate, thin plate case (h/L¼0.001).

Mesh N�N 2�2 4�4 8�8 16�16 32�32 Reference

Central deflection wc/(qL4/100D)GCP4-12α 0.4063 0.4062 0.4062 0.4062 0.4062 0.4062GCP4-14α 0.4063 0.4062 0.4062 0.4062 0.4062Q4BL [10] 0.4593 0.4292 0.4164 0.4110 0.4086ARS-Q12 [14] 0.4045 0.4060 0.4062 0.4062 0.4062AC-MQ4 [37] 0.4033 0.4061 0.4062 0.4062 0.4062

Central moment Mc/(qL2/10)GCP4-12α 0.4774 0.4788 0.4789 0.4789 0.4789 0.4789GCP4-14α 0.4774 0.4788 0.4789 0.4789 0.4789Q4BL [10] 0.5649 0.5010 0.4876 0.4830 0.4809ARS-Q12 [14] 0.5005 0.4839 0.4800 0.4792 0.4789AC-MQ4 [37] 0.5897 0.5050 0.4850 0.4799 0.4790

Table 5The dimensionless results of central deflection wc/(qL4/100D) and moment Mc/(qL2/10) of the SS1 square plate, thick plate case (h/L¼0.1).

Mesh N�N 2�2 4�4 8�8 16�16 32�32 Reference

Central deflection wc/(qL4/100D)GCP4-12α 0.4430 0.4500 0.4532 0.4542 0.4544 0.4617GCP4-14α 0.4430 0.4500 0.4532 0.4542 0.4544Q4BL [10] 0.4957 0.4727 0.4644 0.4624 0.4618ARS-Q12 [14] 0.4280 0.4419 0.4544 0.4596 0.4612AC-MQ4 [37] 0.4358 0.4437 0.4543 0.4595 0.4611

Central moment Mc/(qL2/10)GCP4-12α 0.4895 0.4976 0.5010 0.5025 0.5029 0.5096GCP4-14α 0.4895 0.4976 0.5010 0.5025 0.5029Q4BL [10] 0.5694 0.5169 0.5112 0.5100 0.5100ARS-Q12 [14] 0.5206 0.5087 0.5081 0.5091 0.5094AC-MQ4 [37] 0.6115 0.5236 0.5122 0.5101 0.5097

Table 6The dimensionless results of central deflection wc/(qL4/100D) and moment Mc/(qL2/10) of the SS2 square plate, thin plate case (h/L¼0.001).

Mesh N�N 2�2 4�4 8�8 16�16 32�32 Reference

Central deflection wc/(qL4/100D)GCP4-12α 0.4051 0.4062 0.4062 0.4062 0.4062 0.4062GCP4-14α 0.4051 0.4062 0.4062 0.4062 0.4062MITC4 [2,27] 0.3969 0.4041 0.4057 0.4061 0.4062Q4BL [10] 0.4305 0.4058 0.4062 0.4062 0.4062ARS-Q12 [14] 0.4045 0.4060 0.4062 0.4062 0.4062AC-MQ4 [37] 0.4052 0.4062 0.4062 0.4062 0.4062MISC2 [27] 0.4123 0.4077 0.4066 0.4063 0.4063

Central moment Mc/(qL2/10)GCP4-12α 0.4786 0.4788 0.4789 0.4789 0.4789 0.4789GCP4-14α 0.4786 0.4788 0.4789 0.4789 0.4789MITC4 [2,27] 0.4075 0.4612 0.4745 0.4778 0.4786Q4BL [10] 0.4712 0.4773 0.4784 0.4788 0.4788ARS-Q12 [14] 0.5009 0.4839 0.4801 0.4792 0.4789AC-MQ4 [37] 0.5106 0.4872 0.4810 0.4794 0.4790MISC2 [27] 0.4171 0.4637 0.4751 0.4779 0.4786

Table 7The dimensionless results of central deflection wc/(qL4/100D) and moment Mc/(qL2/10) of the SS2 square plate, thick plate case (h/L¼0.1).

Mesh N�N 2�2 4�4 8�8 16�16 32�32 Reference

Central deflection wc/(qL4/100D)GCP4-12α 0.4225 0.4256 0.4268 0.4272 0.4273 0.4273GCP4-14α 0.4225 0.4256 0.4268 0.4272 0.4273MITC4 [2,27] 0.4190 0.4255 0.4268 0.4272 0.4273Q4BL [10] 0.4269 0.4274 0.4273 0.4273 0.4273ARS-Q12 [14] 0.4228 0.4255 0.4267 0.4271 0.4272AC-MQ4 [37] 0.4139 0.4206 0.4251 0.4267 0.4271MISC2 [27] 0.4285 0.4277 0.4274 0.4273 0.4273

Central moment Mc/(qL2/10)GCP4-12α 0.4810 0.4767 0.4778 0.4785 0.4788 0.4789GCP4-14α 0.4810 0.4767 0.4778 0.4785 0.4788MITC4 [2,27] 0.4075 0.4612 0.4745 0.4778 0.4786Q4BL [10] 0.4716 0.4773 0.4785 0.4788 0.4788ARS-Q12 [14] 0.5223 0.4941 0.4834 0.4801 0.4792AC-MQ4 [37] 0.5419 0.5028 0.4862 0.4808 0.4794MISC2 [27] 0.4172 0.4637 0.4751 0.4779 0.4786

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–3828

solutions of the displacement function F being used, these twoelements are denoted by GCP4-12α and GCP4-14α, respectively.

2.3.1. The element GCP4-12αAs shown in Fig. 2, for the element GCP4-12α, the displacement

function F (see Eq. (7)) is assumed to be the linear combination of itsfirst 12 terms of fundamental analytical solutions (see Table 1). Then, kin Eqs. (7–10) equals 12. To determine these coefficients αi (i¼1–12),12 generalized conforming conditions must be employed.

First, the following nodal compatibility conditions for deflec-tion are considered:

w xj; yj� �

¼wj; j¼ 1�4ð Þ; ð28Þ

where (xj, yj), (j¼1–4), are the Cartesian coordinates of the node j,w(xj, yj) is calculated by using Eq. (9), and as one of the element’sDOFs, wj is the deflection at the node j.

Next, the point compatibility conditions are considered fornormal rotations at two Gauss points Ai and Bi (the local para-metric coordinates are 71=

ffiffiffi3

p, respectively) along each edge:

ψn xk; yk� �¼ ψnk; k¼ A1;B1;A2;B2;A3;B3;A4;B4ð Þ; ð29Þ

where ψn(xk, yk) are also calculated by substituting the Cartesiancoordinates into Eq. (9), and ψnk are determined by the boundarynormal rotations ψn varying linearly along the edge ij:

ψnij ¼ 1�sð Þψniþsψnj; s¼ 0�1ð Þ: ð30Þ

where ψni and ψnj are the normal rotations at the node i, j alongthe edge ij.

Substituting Eqs. (7–10) into Eqs. (28–30), the matrices inEq. (11) can be obtained for the element GCP4-12α. Specifically,the matrix λ can be written as:

λ¼Wφn

" #; φn ¼ TnΦ; ð31Þ

in which

W¼

w01 x1; y1� �

w02 x1; y1� �

w03 x1; y1� �

⋯ w012 x1; y1� �

w01 x2; y2� �

w02 x2; y2� �

w03 x2; y2� �

⋯ w012 x2; y2� �

w01 x3; y3� �

w02 x3; y3� �

w03 x3; y3� �

⋯ w012 x3; y3� �

w01 x4; y4� �

w02 x4; y4� �

w03 x4; y4� �

⋯ w012 x4; y4� �

2666664

3777775;

ð32Þ

Φ¼

ψ0x1 xA1; yA1� �

ψ0x2 xA1; yA1� �

⋯ ψ0x12 xA1; yA1� �

ψ0y1 xA1; yA1� �

ψ0y2 xA1; yA1� �

⋯ ψ0y12 xA1; yA1� �

⋮ ⋮ ⋱ ⋮ψ0x1 xB4; yB4

� �ψ0x2 xB4; yB4

� �⋯ ψ0x12 xB4; yB4

� �ψ0y1 xB4; yB4

� �ψ0y2 xB4; yB4

� �⋯ ψ0y12 xB4; yB4

� �

266666664

377777775; ð33Þ

Fig. 5. Errors of the central deflections and moments for clamped square plates in regular mesh. (a) h/L=0.001 (thin plate case) and (b) h/L=0.1 (thick plate case).

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–38 29

Tn ¼

T1T2

T3T4

26664

37775; Ti ¼ lij mij lij mij

" #; ð34Þ

where i¼ 1;2;3;4

!; j¼ 2;3;4;1

!, and lij and mij denote the direc-tion cosines of the outer normal of the element edge ij.

The matrix Un can be written as

Un ¼Wn

φnn

" #; φnn ¼ TnΦn; ð35Þ

Wn ¼

wn x1; y1� �

wn x2; y2� �

wn x3; y3� �

wn x4; y4� �

266664

377775; ð36Þ

Φn ¼

ψnx xA1; yA1� �

ψny xA1; yA1� �

⋮ψnx xB4; yB4

� �ψny xB4; yB4

� �

266666664

377777775: ð37Þ

The matrix Γ is given by

Γ¼ΓwΓφ

" #; ð38Þ

Γw ¼

I1�3I1�3

I1�3I1�3

26664

37775; I1�3 ¼ 1 0 0� �; ð39Þ

Fig. 6. Errors of the central deflections and moments for SS1 square plates in regular mesh. (a) h/L=0.001 (thin plate case) and (b) h/L=0.1 (thick plate case).

Γφ ¼

0 � 1l12 1�sA1ð Þy121l12

1�sA1ð Þx12 0 � 1l12sA1y121l12sA1x12 0 0 0 0 0 0

0 � 1l12 1�sB1ð Þy121l12

1�sB1ð Þx12 0 � 1l12sB1y121l12sB1x12 0 0 0 0 0 0

0 0 0 0 � 1l23 1�sA2ð Þy231l23

1�sA2ð Þx23 0 � 1l23sA2y231l23sA2x23 0 0 0

0 0 0 0 � 1l23 1�sB2ð Þy231l23

1�sB2ð Þx23 0 � 1l23sB2y231l23sB2x23 0 0 0

0 0 0 0 0 0 0 � 1l34 1�sA3ð Þy341l34

1�sA3ð Þx34 0 � 1l34sA3y341l34sA3x34

0 0 0 0 0 0 0 � 1l34 1�sB3ð Þy341l34

1�sB3ð Þx34 0 � 1l34sB3y341l34sB3x34

0 � 1l41sA4y411l41sA4x41 0 0 0 0 0 0 0 � 1l41 1�sA4ð Þy41

1l41

1�sA4ð Þx410 � 1l41sB4y41

1l41sB4x41 0 0 0 0 0 0 0 � 1l41 1�sA4ð Þy41

1l41

1�sA4ð Þx41

26666666666666666664

37777777777777777775

; ð40Þ

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–3830

where xij ¼ xi�xj; yij ¼ yi�yj, and lij is the length of edge ij.Then, by substituting the matrices λ, Un and Γ (see Eqs. (31–40))

into the equations discussed in Section 2.2, the expressions of thestiffness matrix and the nodal equivalent load vector for the elementGCP4-12α can be obtained.

2.3.2. The element GCP4-14αFor the element GCP4-14α (see Fig. 2), the displacement

function F is formed by the first 14 terms of its fundamentalanalytical solutions (see Table 1).

To determine the unknown coefficients αi (i¼1–14), fourteengeneralized conforming conditions are needed. Besides thoseconditions used for the element GCP4-12α, two additional com-patibility conditions are considered:

wðx5; y5Þþwðx7; y7Þ ¼w5þw7wðx6; y6Þþwðx8; y8Þ ¼w6þw8; ð41Þ

where (xj, yj) (j¼5–8) are the Cartesian coordinates of the mid-sidepoints 5–8 at each element edge, and wj (j¼5�8) are the deflec-tions at these mid-side points. The deflections wj (j¼5–8) aredetermined by using the locking-free formulae of Timoshenko’s

beam [29]:

wij ¼ ½1�sþð1�2δijÞZ3�wiþ½s�ð1�2δijÞZ3�wj

þ lij2½Z2þð1�2δijÞZ3�ψ si�

lij2½Z2�ð1�2δijÞZ3�ψ sj; ð42Þ

in which

Z2 ¼ sð1�sÞ; Z3 ¼ sð1�sÞð1�2sÞ; δij ¼6λij

1þ12λij; λij ¼

D

Cl2ij; ð43Þ

where D and C denote the bending and shear stiffness, respectively; and lij is the edge length. For these mid-side points 5–8,s equals 1/2.

Then, by substituting Eqs. (7–10) into these fourteen general-ized conforming conditions (see Eqs. (28)–(30) and (41)–(43)), thedetails of matrices in Eq. (11) can be obtained. Specifically, thematrix λ is given by

λ¼Wφn

Wmid

264

375; ð44Þ

where W and φn share the same for as in Eq. (31), and Wmid isobtained as

Fig. 7. Errors of the central deflections and moments for SS2 square plates in regular mesh. (a) h/L=0.001 (thin plate case) and (b) h/L=0.1 (thick plate case).

Wmid ¼w01 x5; y5

� �þw01 x7; y7� � w02 x5; y5� �þw02 x7; y7� � ⋯ w014 x5; y5� �þw014 x7; y7� �w01 x6; y6

� �þw01 x8; y8� � w02 x6; y6� �þw02 x8; y8� � ⋯ w014 x6; y6� �þw014 x8; y8� �" #

: ð45Þ

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–38 31

The matrix Un is

Un ¼Wn

φnnWnmid

264

375; ð46Þ

where Wn and φnn are the same as Eq. (35), and Wn

mid is obtainedas:

Wnmid ¼wn x5; y5

� �þwn x7; y7� �wn x6; y6

� �þwn x8; y8� �" #

: ð47Þ

The matrix Γ is

Γ¼ΓwΓφΓnw

264

375; ð48Þ

where Γw and Γφ are the same as Eq. (38), and Γnw is obtained as

Γnw ¼12 �18x12 �18y12 12 18x12 18y12 12 �18x34 �18y34 12 18x34 18y3412

18x41

18y41

12 �18x23 �18y23 12 18x23 18y23 12 �18x41 �18y41

" #;

ð49Þwhere xij ¼ xi�xj; yij ¼ yi�yj.

Fig. 8. Errors of the central deflections and moments for square plates in distorted mesh. (a) Clamped case; (b) SS1 case and (c) SS2 case.

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–3832

Thus, the stiffness matrix and the nodal equivalent load vectorfor the element GCP4-14α can be readily obtained. Since theelement GCP4-14α uses 14 terms of fundamental analytical solu-tions, which guarantees the completeness of the displacementfunction up to the fourth order, it performs better than theelement GCP4-12α in distorted meshes.

3. Numerical tests

Several benchmark examples are employed to assess the newelements GCP4-12α and GCP4-14α. In order to clearly illustrate theperformance, the results from some well-known Mindlin–Reissnerplate elements are also provided for comparison, including MITC4[2,27], Q4BL [10], ARS-Q12 [14], RDKMQ [15], MISC2 [27], AC-MQ4[37], S4R [39], CRB1 [40], CRB2 [40], S1 [40], DKQ [41], 9βQ4 [42],MiSP4 [43], MMiSP4 [43], MiSP4þ [44] and DKMQ [45].

3.1. The patch test

The model and the mesh used for the patch test are shown inFig. 3a together with the parameter settings. Three cases withdifferent span–thickness ratios (2a/h¼1000, 100, 20) are consid-ered, where 2a is the length of the longer edge and h is the plate’sthickness.

(a) Displacement loading caseIn this case, the deflections and rotations at the outer nodes1–4 are applied to the patch as the boundary conditions, andtheir values at the inner nodes 5–8 will be evaluated.For the constant bending state, the exact displacement fieldsare given by [28]

w¼ 1þxþ2yþx2þy2� �

2; ψx ¼

1þ2xð Þ2

; ψy ¼2þ2yð Þ

2: ð50Þ

Both GCP4-12α and GCP4-14α can provide exact solutions inall span–thickness ratio cases (2a/h¼1000, 100, 20).For the constant twisting state, the exact displacement fieldsare given by [28]

w¼ 1þxþ2yþxyð Þ2

; ψx ¼1þyð Þ2

; ψy ¼2þxð Þ2

: ð51Þ

In case 2a/h¼1000, both new elements can pass the testexactly. In case 2a/h¼100, the maximum relative percentageerrors of elements GCP4-12α and GCP4-14α are 0.078% and0.074%, respectively, and increase to 1.73% and 2.05% in case2a/h¼20. These errors are mainly caused by the interelementincompatibilities in a coarse mesh. With the refinement of themesh, these errors will vanish rapidly.

For the constant shear deformation state, the exact displace-ment fields are [46]

w¼ xþyð Þ2

; ψx ¼�12

; ψy ¼�12

: ð52Þ

This test is significant only for the thick plate with a very largethickness in which only shear energy works. The element

L /2=5 L /2=5

L/2=5

L/2=5

C C

Δ

Δ

Δ

Fig. 9. The clamped thin square plate for sensitivity test to mesh distortion.

Fig. 10. Results for the sensitivity test to mesh distortion: top: symmetric case;bottom: asymmetric case.

P

C

P

L

LL

Δ

Fig. 11. The clamped cantilever plate for sensitivity test to mesh distortion.

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–38 33

GCP4-14α can exactly pass this test, while the maximumrelative percentage error of GCP4-12α is 6.0%.

(b) Constant bending moment loading case (Mn¼1)Fig. 3b shows this loading case in which the plate is subjectedto moments along its all edges. The equivalent nodal forces arealso given in Fig. 3b. In this case, the bending moments Mx¼1and My¼1, while other resultants are equal to zero. For alldifferent span–thickness ratios (2a/h¼1000, 100, 20), bothGCP4-12α and GCP4-14α can exactly pass this patch test.

(c) Constant twisting moment loading case (Mxy¼1)As shown in Fig. 3c, the plate is subjected to twisting momentalong its all edges. The equivalent nodal forces, derived fromthe assumption that the plate’s edge behaves like a locking-free Timoshenko’s beam [29], are also given in Fig. 3c. In thiscase, the twisting moments Mxy¼1, while other resultants areequal to zero. When 2a/h¼1000, both two new elements canpass the test exactly. For the ratio 2a/h¼100, the maximumrelative percentage errors of the elements GCP4-12α andGCP4-14α are 2.12% and 1.70%, respectively. And for the ratio2a/h¼20, they increase to 3.24% and 2.80%.

(d) Constant shear resultant loading case (Tx¼1)As stated in Section 3.1(a), this test is significant only for thethick plate limit [45]. Fig. 3d shows the cantilever plate, with avery large value of thickness h, is subjected to a distributedline shear force at the free edge (replaced by two concentratednodal forces). The value of shear force Tx should equal 1. Theelement GCP4-14α can exactly pass this test, while the max-imum relative percentage error of GCP4-12α is 2.62%.

Fig. 14. Errors of the central deflection wc and moment My of Razzaque’s skew plate.

y=0 y=0

x=0

A

x=0

x

y

Cx

y

C

Radius R=5E μ=10.92; =0.3;Uniform loading: q=1Displacement BCs:w=0 (SS1) along AB w= n= s=0 (Clamped) along AB

B

A

Bψ

ψψ

ψ

ψ ψ

Fig. 15. A quarter of circular plate.

Fig. 12. Results for deviation of the transverse tip displacement wc.

DEE=10.92

=0.3hμ

=0.1L=100

A B

60°

L

FreeL

x

yC

Free

Fig. 13. The Razzaque’s 601 skew plate.

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–3834

(e) Some discussionsFrom above tests, it can be observed that, for the thin plate (2a/h¼1000), both new elements can provide exact solutions.However, small errors may appear when the thickness incr-eases for constant twisting moment cases. These errors aremainly caused by the interelement incompatibilities, and can

be eliminated by using a refiner mesh. Since the element GCP4-14α employs two more trial functions, it indeed exhibits betterperformance. As described in previous sections, these twoelements belong to the nonconforming model in a coarse mesh,and will converge into conforming models when the mesh isrefined. Thus, the convergence can still be guaranteed.

Table 9Normalized center deflection wc/wref and moments Mc/Mref of clamped circular plates.

Mesh N 3 12 48 192 Analytical

(a) h/R¼0.02 (h¼0.1) wc/wrefMITC4 [2,27] 0.9269 0.9914 0.9983 — 1.0000 (the reference value is 9783.48)DKMQ [45] 1.1009 1.0316 1.0083 —ARS-Q12 [14] 1.1007 1.0301 1.0077 1.0018AC-MQ4 [37] 0.7604 0.9455 0.9859 0.9959GCP4-12α 1.0223 0.9520 0.9874 0.9968GCP4-14α 0.8225 0.9510 0.9873 0.9970

Mc/MrefMITC4 [2,27] 0.9255 1.0092 1.0043 — 1.0000 (the reference value is 2.03125)DKMQ [45] 1.2505 1.0585 1.0142 —ARS-Q12 [14] 1.2533 1.0584 1.0166 1.0045AC-MQ4 [37] 0.9607 1.0152 1.0044 1.0012GCP4-12α 0.7639 0.9602 0.9936 0.9985GCP4-14α 0.8884 0.9743 0.9940 0.9986

(b) h/R¼0.2 (h¼1) wc/wrefMITC4 [2,27] 0.9311 0.9897 0.9978 — 1.0000 (the reference value is 11.5513)DKMQ [45] 1.0651 1.0182 1.0043 —ARS-Q12 [14] 1.0686 1.0179 1.0041 1.0010AC-MQ4 [37] 0.7359 0.9205 0.9773 0.9941GCP4-12α 1.0217 0.9665 0.9907 0.9977GCP4-14α 0.8345 0.9537 0.9883 0.9971

Mc/MrefMITC4 [2,27] 0.9502 1.0043 1.0043 — 1.0000 (the reference value is 2.03125)DKMQ [45] 1.2603 1.0732 1.0191 —ARS-Q12 [14] 1.2932 1.0738 1.0192 1.0049AC-MQ4 [37] 0.9756 1.0237 1.0083 1.0022GCP4-12α 0.7972 0.9762 0.9925 0.9983GCP4-14α 0.9261 0.9738 0.9928 0.9988

Table 8Normalized center deflection wc/wref and moments Mc/Mref of SS1 circular plates.

Mesh N 3 12 48 192 Analytical

(a) h/R¼0.02 (h¼0.1) wc/wrefMITC4 [2,27] 0.9146 0.9801 0.9951 — 1.0000 (the reference value is 39831.5)DKMQ [45] 0.9573 0.9900 0.9976 —ARS-Q12 [14] 0.9573 0.9897 0.9974 0.9993AC-MQ4 [37] 1.0145 1.0058 1.0014 1.0002GCP4-12α 1.0790 1.0075 1.0018 1.0043GCP4-14α 1.0301 1.0071 1.0017 1.0004

Mc/MrefMITC4 [2,27] 0.9173 0.9891 0.9969 — 1.0000 (the reference value is 5.15625)DKMQ [45] 1.0453 1.0085 1.0027 —ARS-Q12 [14] 1.0454 1.0088 1.0030 1.0008AC-MQ4 [37] 1.0413 1.0211 1.0056 1.0014GCP4-12α 0.9646 0.9995 1.0014 1.0004GCP4-14α 1.0158 1.0051 1.0015 1.0004

(b) h/R¼0.2 (h¼1) wc/wrefMITC4 [2,27] 0.9166 0.9801 0.9951 — 1.0000 (the reference value is 41.5994)DKMQ [45] 0.9535 0.9880 0.9969 —ARS-Q12 [14] 0.9544 0.9881 0.9969 0.9992AC-MQ4 [37] 0.9972 0.9963 0.9983 0.9995GCP4-12α 1.0766 1.0096 1.0021 1.0005GCP4-14α 1.0246 1.0055 1.0014 1.0004

Mc/MrefMITC4 [2,27] 0.9270 0.9872 0.9969 — 1.0000 (the reference value is 5.15625)DKMQ [45] 1.0473 1.0143 1.0046 —ARS-Q12 [14] 1.0613 1.0149 1.0040 1.0010AC-MQ4 [37] 1.0495 1.0246 1.0072 1.0019GCP4-12α 0.9787 1.0062 1.0010 1.0003GCP4-14α 1.0301 1.0051 1.0011 1.0005

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–38 35

3.2. The square plate

The square plate is subjected to a uniformly distributed load, asshown in Fig. 4. Due to symmetry, only a quarter of the plate ismodeled. The edge length and thickness are denoted by L and h,respectively. Poisson’s ratio μ is set as 0.3. Three cases with differentboundary conditions are considered: the clamped case (w¼0, ψn¼0,ψs¼0), the soft simply supported (SS1) case (w¼0), and the hardsimply supported (SS2) case (w¼0, ψs¼0).

Tables 2–7 give the dimensionless central deflections and mom-ents of the plate for two ratios (h/L¼0.001, 0.1), which are calculatedby using the regular mesh (see Fig. 4a). For comparison, resultsobtained by some other quadrilateral elements [2,10,14,27,37] arealso presented. To visually show the convergences, the correspondingplots using log scale are given in Figs. 5–7. Furthermore, the resultscalculated by using the distorted meshes (see Fig. 4b) are also plottedin Fig. 8. In these figures, the relative errors are defined as follows:

ew ¼w�wref�� ��

wref�� �� ; eM ¼ M�Mref

�� ��Mref�� �� : ð53Þ

Note, because the number of the significant digits for the ref-erence solutions is only four, the low bound of the relative error isabout 10�5.

It can be seen that, in most cases, these two elements canprovide good results, especially for thin plate cases. But for the SS1thick plate case, they may converge to a slightly stiffer solution.This problem is mainly caused by the interelement incompatibility.Since the errors are only 1%, the results are still in an acceptablerange for engineering applications. An interesting feature is thatthe results obtained by these two new elements are entirely inagreement with each other in the regular meshes.

3.3. The test for sensitivity to mesh distortions

3.3.1. The clamped thin square plateFig. 9 shows a clamped thin square plate (h/L¼0.001) subjected

to a uniformly distributed load. Due to symmetry, only a quarter ofthe plate is modeled by a coarse mesh (2�2). Two distortion casesare considered: the symmetric case and the asymmetric case. Inthe symmetric case, the central mesh node is moved along themain diagonal of the plate to the corner node. In the asymmetriccase, the central mesh node is moved to a plate edge along thedirection parallel to another plate edge.

The normalized results obtained by the new elements andsome other elements [2,27,39–41] are plotted in Fig. 10. The resultsshow that the proposed elements are quite insensitive to the meshdistortion, and the element GCP4-14α performs more stably. It

Fig. 16. Distributions of resultants along the radius of clamped circular plate (48 elements).

Y. Shang et al. / Finite Elements in Analysis and Design 99 (2015) 24–3836

should be noted that, in the symmetric case, when the absolutevalue of the distortion parameter Δ reaches or exceeds 1.25, oneelement will degenerate into a triangle or a concave quadrangle.Under these situations, the new elements can still perform well,while most other elements fail to provide a solution.

3.3.2. The clamped cantilever plateAs shown in Fig. 11, a clamed cantilever plate [42], which is

subjected to a tip load P¼0.5, is modeled by two elements. Thematerial parameters are E¼10.92 and μ¼0.3. The thickness andwidth are h¼0.01 and L¼1, respectively. The mesh distortion ismeasured by the parameter Δ. Fig. 12 shows the percentage devi-ations of the transverse tip displacement at node C obtained bydifferent elements [14,40,42]. This test proves again that the twonew elements are more insensitive to the mesh distortion thanothers.

These two tests sufficiently show the robustness of the newelements in distorted meshes. It can also be found that the elem-ent GCP4-14α exhibits a more outstanding performance, since itstrial functions are derived from the displacement function with upto the fourth order completeness.

3.4. The 601 skew plate

The Razzaque 601 skew plate [47] (h/L¼0.001) is subjected to auniformly distributed load, as shown in Fig. 13. The transversedeflection w and the bending moment My at the central node C arecalculated. Results obtained by the new elements and others[2,14,15,27,43–45] are plotted in Fig. 14. The results show thatthe new elements can produce good results for both deflectionsand bending moments.

3.5. The circle plate

The circular plate subjected to a uniformly distributed load q¼1is shown in Fig. 15. Due to symmetry, only a quarter of the plate ismodeled. The geometrical parameters and material parameters arealso given in Fig. 15. Two different thickness–radius ratio cases(h/R¼0.02, 0.2) and two different boundary condition cases (the SSlBC case and the clamped BC case) are analyzed. The normalizedcenter deflection and moments are listed in Tables 8 and 9. Fig. 16shows the distributions of bending moments Mr,Mθ and shear forceTr along the radius of the clamped circular plate calculated by using48 elements. It can be observed in Fig. 16 that the two newelements can provide resultants which are in good agreement withthe exact solutions [43,44], while most conventional displacement-based models fail.

4. Conclusions

Two generalized conforming plate elements GCP4-12α and GCP4-14α, whose trial functions are derived from the analytical solutions ofthe displacement function [29,30,48], are proposed in this paper. Thegeneralized conforming technique relaxes the requirements of com-patibility while improving the element performances. Since thedisplacement and strain fields of the new elements are assumedaccording to the fundamental analytical solutions of the displace-ment function F, which satisfy a priori all related governing equa-tions, these new elements possess the advantages from analyticalmethods. Some benchmarks are employed to assess the performanceof these new elements. In most cases, these two new elements canboth provide rather good results, especially for thin plate cases. Butfor the thick plate under SS1 boundary condition, the result mayconverge to a slightly stiffer solution. This phenomenon is mainlycaused by the interelement incompatibility. Since the error is only

about 1%, those results are still in an acceptable range for engineeringapplications.

Acknowledgement

This work is financially supported by the National NaturalScience Foundation of China (11272181), the Specialized ResearchFund for the Doctoral Program of Higher Education of China(20120002110080), and the Tsinghua University Initiative Scien-tific Research Program. The authors would also like to thank thesupport from British Council and the Chinese Scholarship Councilthrough the Sino-UK Higher Education Research Partnership forPhD Studies Scheme, and the support from European Commissionthrough the International Research Staff Exchange Scheme (IRSES).

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Two generalized conforming quadrilateral Mindlin–Reissner plate elements based on the displacement functionIntroductionThe formulations of new elementsThe displacement function FThe procedure for constructing new generalized conforming elementsThe formulations of the new generalized conforming plate (GCP) elementsThe element GCP4-12αThe element GCP4-14α

Numerical testsThe patch testThe square plateThe test for sensitivity to mesh distortionsThe clamped thin square plateThe clamped cantilever plate

The 60deg skew plateThe circle plate

ConclusionsAcknowledgementReferences