Modi ed xed-grid nite element method in shape optimization...

Scientia Iranica B (2014) 21(1), 147{161

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Modi�ed �xed-grid �nite element method in shapeoptimization problems based on the gradientlessmethod

M. Heshmatia,*, F. Daneshmandb,c and Y. Aminic

a. Department of Mechanical Engineering, Kermanshah University of Technology, Kermanshah, Iran.b. Department of Mechanical Engineering, McGill University, Canada, H3A 2K6; Department of Bioresource Engineering, McGill

University, Canada, H9X 3V9.c. School of Mechanical Engineering, Shiraz University, Shiraz, Iran.

Received 13 February 2012; received in revised form 6 January 2013; accepted 4 November 2013

KEYWORDSShape optimization;Fixed-grid �niteelement method;Gradientless method;Stress concentration.

Abstract. This paper presents a methodology for solving shape optimization problemswhere the unknown is the shape of the problem domain. The proposed algorithm is basedon minimization of the stress along the design boundary calculated by the Modi�ed FixedGrid Finite Element Method (MFGFEM). Using MFGFEM eliminates mesh adaptationand re-meshing processes, as needed in the standard �nite element method, and reducesthe analysis cost signi�cantly. In this study, a new approach for computing the sti�nessmatrix of boundary intersecting elements is also presented and the optimal shape of theproblem domain is obtained via a simple optimization algorithm. The performance of theproposed approach is investigated for shape optimization problems. It is concluded thatthe results of the present method are in good agreement with other analytical and �niteelement solutions.© 2014 Sharif University of Technology. All rights reserved.

1. Introduction

Shape optimization problems are associated with �nd-ing the optimum pro�le of a component to improvethe behavior of a mechanical system and minimizesome properties, e.g. the weight of the body or highstress concentrations near the corners. Many investi-gations have been carried out and various structuraloptimization methods have been proposed for di�erentoptimization problems [1].

Conventional shape optimization methods arebased on using the Finite Element Method (FEM) anda set of elements formed from the boundary of theproblem domain. A direct relationship is then created

*. Corresponding author. Tel.: +98 937 3154288;Fax: +98 831 7267530E-mail address: [email protected] (M. Heshmati)

between the problem boundary and the �nite elementmeshes and any changes in the domain boundarywill also be produced in the computational meshes.The drawback of this strategy is clear: Signi�cantboundary changes can necessitate the adaptation orregeneration of the mesh, as it no longer provides acorrect representation of the medium. Therefore, in thestandard shape optimization methods based on FEM,re-meshing cannot be avoided during the optimiza-tion process if accurate analysis is to be guaranteed,especially for design problems with large changes inshape [2]. These factors incorporate a considerableamount of ine�ciency in the shape optimization meth-ods based on FEM.

An interesting approach to decrease FEM de-pendency on conventional mesh in shape optimizationproblems is to use the Fixed Grid Finite ElementMethod (FGFEM) [3-10] and eXtended Finite Element

148 M. Heshmati et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 147{161

Method (XFEM) [11,12]. Both FGFEM and XFEMuse a �xed non-boundary-�tted grid to perform a�nite element analysis. In FGFEM, the problemdomain is covered with uniform �nite elements, anda homogenization technique [3] is used to computethe sti�ness matrix for boundary intersecting elements.The problem domain is discretized with uniform ornon-uniform �nite elements in XFEM, and the stepfunction is used to carry out integration over boundaryintersecting elements [12]. It has been shown thatthe FGFEM and XFEM are e�ective in approximatingthe strain and stress �eld by their low requirementsof time and computational resources. Though theabove mentioned methods are equipped with excel-lent remesh-free properties and avoid cumbersomeremeshing processes, the homogenization procedureand step function used in FGFEM and XFEM re-sult in an inaccurate representation of the problemdomain and decreases the accuracy of numerical so-lution [4,5,9].

The integration over discontinuous elements hasalready been treated by several authors and has beenapplied in multiphase problems [13]. In the presentstudy, basic modi�cations are introduced in the originalFGFEM and a new approach, called Modi�ed FGFEM(MFGFEM), is introduced to encounter the boundaryintersecting elements. The modi�cation deals withsti�ness matrix computation of boundary intersectingelements by introducing material coordinates (r; s) thatmap the internal part of the boundary intersectingelement to the rectangular or triangular master elementwhen the internal part of the boundary element is arectangle or triangle. A new formulation is also pre-sented for mapping the internal part of the boundarywhen the internal part of the boundary intersectingelement is a pentagon.

It should be noted that the stress gradient infor-mation must also be determined through a sensitivityanalysis in the structural optimization problems tominimize stress concentrations [14-19]. Since the sensi-tivity calculation time is usually high when the numberof design variables is large, a gradientless approach isused in this paper. The gradientless approaches [20-27]do not use stress derivatives for determining optimalgeometries. It is well-known that the convergencerate is independent of the number of design variablesin these approaches, and they are particularly appro-priate for minimizing stress concentrations in shapeoptimization problems with a large number of designvariables. The underlying strategy for these methodsis to achieve a constant or nearly constant stresseddesign boundary by adding material where stresses are`high' and removing it where stresses are `low'. Thisstrategy has been shown to be consistent with the aimof minimizing the peak stress [22-27].

In the present study, the modi�ed �xed-grid

method, based on a gradientless approach for shapeoptimization of components and for minimizing stressconcentration, is developed to overcome drawbacksin solving shape optimization problems with FEMgradient-based methods. Cubic splines are also used inthe proposed method to model the shape of the designboundary and the optimal shape of the design bound-ary with constant stress is achieved iteratively. Thevalidity and performance of the method are demon-strated by solving numerical examples and comparingthe results of the present analysis with those reportedin the literature.

The paper is organized as follows: First, the de-sign parameterization is described and then the notionof non-boundary �tted grids is presented. The originalFGFEM is described followed by an explanation ofMFGFEM. After that, the application of boundaryconditions is discussed and the numerical integrationof a Galerkin weak form is presented. Finally, the opti-mization approach is described and numerical examplesare conducted. The results obtained are also comparedwith analytical or numerical solutions.

2. Design parameterization

Several methods are available in representing theboundary geometry of the structure. Most researchersuse the nodal coordinates of the discrete �nite elementmodel as design variables [25,27]. However, it is notsuitable for the �xed-grid solver, since node locationsfor the analysis remain unchanged throughout theoptimization process.

Various types of parametric spline are commonlyused for representing the local shape of geometric mod-eling. The cubic spline curves and cubic B-spline curvesare as certain varieties of cubic spline function. In thecubic spline representation, the cubic spline curve isexpressed in terms of the coordinates of control pointsand their corresponding tangents, whereas, in the B-spline [9-10] representations, the curve is representedin terms of control points of a polygon. However, whenthe curve passes through a set of given points, the cubicspline function, which has two continuous derivativeseverywhere and possesses minimum mean curvature, isa useful representation of the design boundary. TheB-spline curve is particularly useful where the curve is�tted by interactive manipulation.

In this paper, cubic spline curves are selectedto represent the design boundary. Because a cubicspline is a piece-wise cubic polynomial curve that hasa smooth shape, the movement of a point on the linecan be controlled precisely and the control points arepart of the generated curve. Waldman et al. [27]tested the suitability of using a typical cubic splineinterpolation on optimal shoulder �llet shapes andfound good results.

M. Heshmati et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 147{161 149

3. Non-boundary-�tted meshes

In the shape optimization problems based on the stan-dard �nite element method, the problem domain mustbe divided into a set of boundary �tted elements withprede�ned shapes. The elements are used for shapefunction construction and numerical integration. Themain drawbacks of these boundary �tted elements arethe di�culties that arise during the mesh generationperiod. Although there are many automatic meshgeneration techniques, the generation of an acceptableboundary �tted mesh for bodies with a moving bound-ary is not a simple task.

Another strategy for the numerical solution ofshape optimization problems is the use of non-boundary �tted meshes. In this strategy, it is notnecessary for the elements to coincide with domainboundaries and, therefore, the boundaries can intersectthe elements. The use of non-boundary-�tted meshesin the analysis of shape optimization problems reducesthe mesh generation costs, because the mesh generationprocess is performed without considering the boundarymovement. In the same way, using non-boundary-�ttedmeshes in the shape optimization problems reducesanalysis costs signi�cantly. A typical non-boundary-�tted mesh is shown in Figure 1. As can be seenfrom this �gure, the intersection of elements withboundaries causes the production of elements, someportions of which are located outside the problemdomain. These elements, in this paper, are calledBoundary Intersecting Elements (BIE). The elementslocated completely inside the problem domain arenamed Internal Elements (IE) and the others arenamed External Elements (EE). The collection of IEand BIE is considered an active element set. As shownin Figure 1, the nodes are also divided into threecategories: Internal Nodes (IN), External Nodes (EN),and External Boundary Nodes (EBN) which locateoutside the domain boundary; this type of nodes lieon the corners of boundary intersecting elements. Thecollection of IN and EBN is considered an active nodeset.

Figure 1. A non-boundary-�tted mesh and classi�cationof nodes and elements.

4. Fixed grid �nite element method

The original FGFEM uses the idea of the application ofnon-boundary-�tted grids in the �nite element solutionof the problems. In this method the homogenizationtechnique is used for the formulation of boundary inter-secting elements [3]. In the homogenization technique,it is assumed that the problem is not restricted bythe original domain boundaries, and the material isdistributed over the entire space with the assumptionthat the density and sti�ness of the media is a functionof space. In other words, it is assumed that the sti�nessof IE is equal to the original material, but that thesti�ness of EE is very low compared to the sti�ness ofIE. With these assumptions, the sti�ness of BIE is avalue between the sti�ness of IE and EE correspondingto the area fraction of the internal part of BIE. In thisapproach, if Ci is the elasticity tensor of IE, Ce is theelasticity tensor of EE; the elasticity tensor of BIE iswritten as:

Cb = rCi + (1� r)Ce; r =AiA; (1)

where A and Ai are the total area and area of theinternal part of the BIE, respectively, and, therefore,r fraction. This representation of BIE causes someinaccuracy in the formulation of BIE because onlythe area fraction is used for the formulation of BIEand the orientation of the boundary, with respect tothe element and the shape of the internal part of theelement, is not introduced in the formulation of BIE.Figure 2 shows examples of 6 di�erent BIE whichintersect the boundary in 6 di�erent con�gurations.The area fraction, r, for all these elements is equal and,from the view point of the homogenization technique,the sti�ness matrices of all these elements are the same.This causes an inaccurate representation of boundaryintersecting elements.

5. Modi�ed �xed grid �nite element method

In this paper, some modi�cations on the formulationof BIE are implemented to improve the accuracyof the sti�ness matrix and application of boundaryconditions. The following sections explain these modi-�cations.

5.1. Governing equationsThepagebreak[3] governing equations for the staticdeformations of body, , with boundary �, are as

Figure 2. Examples of di�erent BIEs with same areafraction.


follows:

r:� + b = 0 in ; (2)

u = �u on �u; (3)

t = �:n = �t on �t: (4)

In these equations, u is the displacement vector andb is the body force. The domain boundary, �, isdivided into two parts, �u and �t, which representthe boundaries with essential and natural boundaryconditions, respectively, where �u \ �t = 0 and�u [ �t = �. Vectors �u and �t are the prescribeddisplacements and traction forces, respectively, whichare applied on the domain boundaries. n is the unitoutward normal vector on the boundary. The totalpotential energy of a continuous system can be writtenas follows [28]:

� = �S ��L; (5)

in which �S is the strain energy and �L is the workdone by external forces. We have:

�S =12

Z

�T "d; (6)

�L =Z

uT bd +Z�t

uT �td� +mlXi=1

uT (xi)Ft: (7)

Fi is the external concentrated force applied at point xi,m1 is the total number of point forces, u is displacementvector, � is stress vector and " is strain vector. For 2Dcase, these values are de�ned as:

u = [ux uy]T ; (8)

� = [�x �y �xy]T ; (9)

" = ["x "y xy]T : (10)

The stress strain relation for a linear elastic materialwithout residual stress and strains can be de�ned usingelasticity matrix C as follows:

� = C": (11)

5.2. Essential and natural boundary conditionsIn the presented method, the penalty function methodis used only for essential boundary conditions, and thenatural boundary conditions are applied via numericalintegration of traction forces on the boundary. Inthis approach, for application of essential boundaryconditions, the total potential energy of the system is

modi�ed as:

�� =�S ��L +12�1

Z�u

(u� �u)T (u� �u)d�

+12�2

m2Xi=1

(u(xi)� �ui)T (u(xi)� �ui); (12)

where, �ui is the prescribed displacement in the pointconstraint at point xi, m2 is the total number of pointconstraints and �1 and �2 are the penalty parameters.The use of a penalty function in the application ofessential boundary conditions means that the rigidsupports are actually replaced by a set of deformablesupports, which are much sti�er than the materialsused in the problem.

5.3. Discretization processIn this paper, a new approach for the construction ofsti�ness is presented. In this approach, in contrast totraditional FGFEM, the degrees of freedom of all ENsare deleted and only the degrees of freedom of INs andEBNs remain as the unknown degrees of freedom.

In this method, IEs are treated as standard �niteelements with no di�culty. The strain energy ofBIEs is computed using the integration of the strainenergy density over the internal parts of these elements.Displacement, u(x), is approximated as:

u(x) = N(x)U; (13)

where N(x) is the shape function matrix and U is theglobal displacement vector. The strain vector is de�nedusing di�erentiation operator B as:

� = Bu = (BN)U: (14)

Using Eq. (14), the stress vector can be written as:

� = C" = C(BN)U: (15)

�� is then obtained by substituting Eqs. (6), (7), (13),(14) and (15) into Eq. (12) as follows:

�� =Z

UT (BN)TC(BN)UdZ

UTNT bd

�Z�t

UTNT �td��m1Xi=1

UTNTi Fi

+12�lZ�u

(NU � �u)T (NU � �u)d�

+12�2

m2Xi=1

(NiU � �ui)T (NiU � �ui): (16)


It should be noted that Ni = N(xi). Using theminimum potential energy principle and di�erentiatingEq. (16), with respect to displacement vector U , wehave:� Z

(BN)TC(BN)d + �lZ�u

NTNd�

+ �2

m2Xi=1

NTi Ni

�U =

� Z

NT bd +Z�t

NT �td�

+miXi=1

NTi Fi + �l

Z�u

NT �ud� + �2

m2Xi=1

NTi �u�:(17)

The above equation is a set of algebraic equations thatcan be solved simultaneously for the unknown vector,U . This equation can be written as:

(K +K�)U = R+R�; (18)

K =Z

(BN)TC(BN)d;

K� = �1

Z�

NTNd� + �2

m2Xi=1

NTi Ni; (19)

R =Z

NT bd +Z�t

NT �td� +mlXi=1

NTi Fi;

R� = �lZ�u

NT �ud� + �2

m2Xi=1

NTi �u: (20)

These integrals must be numerically evaluated usingnonboundary-�tted mesh. The intersection of the meshwith the domain boundary will divide the domainboundary into some boundary segments. These bound-ary segments are approximated by straight lines. Theboundary segment of a typical boundary intersectingelement (e), in Figure 1, is shown in Figure 3.

To improve the solution accuracy substantially,we develop a new scheme to calculate accurately thesti�ness matrix of the corresponding element. To thisend, we �rst determine the intersection points of thedomain boundary and the element boundaries, e.g.points 3 and 4 in Figure 3, and connect these pointsto form the approximate boundary line. Note that theoblique boundary usually crosses the element domainwithout passing through the analysis nodes.

Now, we propose a new strategy to evaluate thesti�ness. Here, we simply work with one elementdenoted by e in Figure 1. First, we express the sti�ness

Figure 3. Typical boundary intersecting element withlocal coordinates (�; �) and material coordinate (r; s).

matrix, ke, as:

ke =1Z�1

lZ�1

(BN)T (�; �)CbBN(�; �)jJ jd�d�: (21)

where (�; �) are the element local coordinates, jJ jis the Jacobian relating local coordinates (�; �) andglobal coordinates, and BN denotes the matrix relatingstrains to nodal displacements. To integrate Eq. (21),we introduce the material coordinates (r; s), whichmaps a normalized rectangular domain, [-1,1]�[-1,1],to the region bounded by 1-2-3-4 in Figure 3. InFigure 3, wei and ei j!ei are the approximations ofinternal and external parts of the BIE, respectively.As in the standard bilinear �nite element, the elementlocal coordinates are expressed as:�

��

�=

4Xi=1

Ni(r; s)��i�i

�; (22)

where (�i; �i) are the coordinates of points 1, 2, 3 and4 in Figure 3 and Ni(r; s) represent standard bilinearfunctions. If !ei becomes a triangular region, it can bealso treated by Eq. (22) as a degenerate case. Using thetransformation (22), the element sti�ness in Eq. (21)can be written as:

ke =1Z�1

1Z�1

(BN)T (�; �)CbBN(�; �)jJ jd�d�

�Zwi

(BN)T (�; �)CBN(�; �)jJ jd�d�

=1Z�1

1Z�1

(BN)T (�(r; s);

�(r; s))CBN(�(r; s); �(r; s))jJ jj bJ jdrds; (23)


where j bJ j is the Jacobian relating (r; s) and (�; �):

bJ =�@�@r

@�@r

@�@s

@�@s

�: (24)

When ei j!ei is a triangle, !ei becomes a pentagon. Inthis case, Eq. (23) is not applicable and the sti�nessmatrix for the element is obtained by subtracting thesti�ness of ei j!ei from the sti�ness of an internalelement:

ke =1Z�1

1Z�1

(BN)T (�; �)CBN(�; �)jJ jd�d�

�Z

ij!i(BN)T (�; �)CBN(�; �)jJ jd�d�

=1Z�1

1Z�1

(BN)T (�; �)CBN(�; �)jJ jd�d�

�1Z�1

1Z�1

(BN)T (�(r; s); �(r; s))CBN)T (�(r; s);

�(r; s))jJ jjJ�jdrds;(25)

where jJ�j results from the mapping between a normal-ized rectangle and ei j!ei .

As mentioned above, the domain integrals havebeen divided to integrations over IEs and BIEs. Weemphasize that the sti�ness evaluation above is appliedonly to elements lying on the boundary. For IEs, theelement sti�ness evaluation is exact in the proposedmethod. Thus, the advantages of �xed grid-basedmethods are the fast meshing and fast formulation ofthe sti�ness matrix. It only requires small additionalexpenses to calculate the intersection points andelement sti�ness matrices along the boundary of thedomain.

6. Optimization approach using splines

Figure 4 shows a plate with a �llet with a free boundary,�, on which there are a number of control points,p1; p2; � � � ; pk. To achieve a boundary shape with a con-stant tangential stress distribution around the bound-ary, a number of �nite element based gradientless meth-ods have been proposed in the literature [25,29]. Theunderlying optimization algorithms for these methodsare based on biological growth analogies, as elucidatedby Heller et al. [25]. We add material where stressesare high and remove it where stresses are low. Theamount of material added or removed at any point on

Figure 4. Geometry of plate with a �llet-shapedboundary.

the boundary is taken to be directly proportional tothe di�erence between the local tangential stress and asuitable reference value. This process is then repeatediteratively until the boundary hoop stress is constant ornearly constant, within a prescribed tolerance. Hence,the amount to move a given point, pi, on the designboundary (in the normal direction) is speci�ed by thefollowing equation [25,27,29]:

�i = K�i � �th�th

; i = 1; 2; :::; n; (26)

where positive �i indicates material addition, �i is thetangential stress at pi on the boundary, �th is thenonzero threshold boundary hoop stress, and K isan arbitrary to accelerate convergence and should bedetermined by trial.

The selection of the threshold stress will de�ne thetype of optimization process and the initial boundaryshape [25,27]. For example, if �th is selected to beequal to the maximum (minimum) stress, only materialremoval (addition) will occur. However, parameter�th is initially unknown and, hence, an arbitrarymagnitude of the threshold stress has to be selected.

In this paper, for the optimization problem ofminimizing the stress concentration with the prescribeddesign domain, a �xed point on the design boundary,p1, in Figure 4 is selected. This point is �xedthroughout the optimization process and its stressshould converge to the uniform solution stress, �nally.It should be noted that the optimization logic willbecome much simpler if the stress at this point isselected as �th [29]. Here, the cubic splines, whichare determined by a set of control points, p1; p2; :::; pk,are used to de�ne the design boundary, as shown inFigure 4. The coordinates of control points are changedby simulating biological adaptive growth and the designboundary is updated accordingly. The coordinates of a


control point are updated in the following way:

Y j+1i (Pi) = Y ji (Pi) + �ji � dji ; i = 1; 2; :::;m; (27)

where Y j+1i (Pi) and Y ji (Pi) are the coordinates of con-

trol point, Pi, at the (j+1)th and jth iterations, respec-tively. dji is the direction of movement for Pi (normaldirection, n, in Figure 4), m is the total number ofcontrol points, �ji is the step of Pi along the movementdirection, which can be obtained by Eq. (26).

All the control points are updated sequentiallyusing Eqs. (26) and (27), and the splines are updatedaccordingly. With the new shape, the stress distri-bution along the new design boundary will becomemore uniform and another modi�ed �xed-grid �niteelement analysis is then undertaken. This processis repeated until the boundary tangential stress isconstant or nearly constant. For such an iterativeprocedure, an appropriate function, RE, to monitorthe solution convergence, can be written as clearly asRE approaches zero; the stresses become more uniformaround the boundary:

RE =��max � �min

�max + �min

��: (28)

7. Illustrative examples

Three numerical examples are presented and discussedhere to demonstrate the accuracy and power of theboundary curve approximation and proposed shapeoptimization method. It must be mentioned that inthe �rst example, only the e�ciency of the MFGFEMis considered and no optimization is done.

7.1. In�nite plate with a circular hole subjectto a unidirectional load

In this example, the problem of 2D stress analysisaround a circular hole in an in�nite plate underunidirectional tension was examined. The schematicdiagram of the problem is presented in Figure 5.This problem is essentially a two-dimensional elasticityproblem and has an analytic solution. The problemwas solved in plane strain condition with a circular holein its center. Due to symmetry, only a quarter of theproblem is considered. In the cutting lines, the properboundary conditions must be applied to simulate an in-�nite media. Therefore, traction boundary conditionsfrom the exact solution were applied on these lines.The analytic solution of this problem can be written inthe following form [30]:

u =1 + �v

�E��

11 + �v

r cos � +2

1 + �va2

rcos �

+12a2

rcos 3� � 1

2a4

r3 cos 3��; (29)

Figure 5. Geometry of the plate with circular hole.

v =1 + �v

�E�� v

1 + �vr sin � +

1� �v1 + �v

a2

rsin �

+12a2

rsin 3� � 1

2a4

r3 sin 3��; (30)

�x = ��1� a2

r2

�32

cos 2� + cos 4��

+3a4

2r4 cos 4��;(31)

�y = �� a2

r2

�12

cos 2� + cos 4��

+3a4

2r4 cos 4��;(32)

�xy = �� a2

r2

�12

sin 2� + sin 4��

+3a4

2r4 sin 4��:(33)

The material properties are E = 2 GPa and v = 2:26.The problem is solved for 3 di�erent grid resolutions.The �rst grid density consists of 16 square elements,the second grid density consists of 64 square elementsand the �nal mesh density consists of 256 squareelements. A schematic diagram of the problem domainand boundary conditions are shown in Figure 5. Therelative error in energy and displacement norms is usedas a measure to compare the results. The relativeerror in the displacement norm is calculated using thefollowing expression:

d =

R

(uex � unu)T (uex � unu)dRuexT uexd

! 12

; (34)

and the relative error in strain energy norm is calcu-


lated using:

e =

R

("ex � "nu)TC("ex � "nu)dR"exTC"exd

! 12

: (35)

In these equations, superscripts ex and nu denote theexact and numeric values of the parameters, respec-tively. To calculate the relative errors in displacementand energy norm (Eqs. (33) and (34)), the domainintegrals should be divided into integrations over IEsand internal parts of BIEs using the Gauss quadratureformula, where uex and unu are written as:

uex =�uv

�; unu =

26644Pi=1

ueiNi4Pi=1

veiNi

3775 ; (36)

where u and v are substituted from Eqs. (29) and (30).uei ; vei are nodal displacements of the element obtainedfrom the present method.

The relative error norms in strain energy anddisplacement are presented in Figures 6 and 7 fordi�erent three type grid densities. It can be seenthat the error in displacement and energy norms inMFGFEM for grid density #1 are 0.5% and 8.8%, forgrid density #2 are 0.12% and 4.5%, and for the thirdmesh density are 0.04% and 2.3%, respectively. It is

Figure 6. Relative error norm in displacement fordi�erent numbers of element.

Figure 7. Relative error norm in strain energy fordi�erent numbers of element.

also obvious that the error norms in the present methodare less than those obtained by FEM and FGFEMsolvers.

7.2. Hole in a biaxial stress �eldIn this section, the benchmark structural optimizationproblem of a biaxial loaded plate of �nite width witha hole at its center is studied. The objective ofthis problem is to �nd out the optimal hole pro�leto produce the minimum Von-Mises stress on theboundary of the hole. The geometry, dimensions,boundary conditions and loading of the plate are shownin Figure 8. Only a quarter of the plate has beenanalyzed due to symmetry. On the symmetry lines,the proper boundary conditions must be applied tosimulate the complete media. A circular arc is used tode�ne the initial design boundary and six control pointsare selected to parameterize the design boundary. Afterassuming the initial geometry, a �xed non-boundary-�tted mesh is used to solve the shape optimization ofthe stress concentration problem. The stress �eld isobtained using the MFGFEM and the parameters arethen updated via a conjugate gradientless algorithm.This process is continued until achieving convergency.A typical mesh and an initial guess for the hole pro�leare shown in Figure 9.

To display the results, a polar coordinate systemis used, in which � = 0 and � = 90 show points Band A, respectively (Figures 8 and 9). Point A is the�xed point and the stress at this point is selected asthe threshold stress.

The stress far away from the concentration zoneis 38.971 MPa. It is the value of stress which wouldexist in the plate without a hole. Therefore, thisvalue has been taken for normalization of the results.

Figure 8. Dimensions, loads and boundary conditions ofthe problem.


Figure 9. Fixed-grid mesh of one quarter of the platewith a hole.

Figure 10(a)-(c) shows the normalized Von-Mises stresscontours for the initial and other iterations aroundthe hole boundary as a function of �. The stressconcentration around the initial hole is 3.2637 at � = 0and the minimum stress is 0.51 at � = 90. It can beseen from Figure 10(a) and (b) that the convergencespeed during the �rst iterations is considerably higherthan the last iterations, because the driving force inEq. (26) diminishes with increasing iteration. Notethat the uniform von-Mises stresses appear along thehole boundary in Figure 10(c). The ratio of the majorto minor axes of the resulting ellipse is 1.951, whichis close to the analytical solution for the in�nite platewith an ellipse, with the aspect ratio of 2:1.

The minimum possible uniform stress level alongthe boundary of an elliptical hole in an in�nite plateis 67.5 MPa [20]. In the present results, the Von-Misesstress along the design boundary is an essentially con-stant level in the range 62.87-67.502 MPa, which hasgood agreement with the result for the in�nite plate.Tekkaya and Guneri [24] investigated the same problemwith the same plate geometry and boundary conditionsusing the biological growth method. They reported theVon-Mises stress along the design boundary is withinthe range 62.3- 77.9 MPa. Zhixue [29] also reportedthat the Von-Mises stress along the design boundaryis within the range 68.98{70.30 MPa. Comparing theabove results, it can be concluded that the results ofthe present study have extremely high accuracy.

The normalized distributions of Von-Mises stressalong both the initial and �nal hole boundaries areshown in Figure 10(c) as a function of �. It can be notedthat the stress concentration is initially 3.2637 and isreduced to 1.7404 for the optimal shape of the hole.

A summary of di�erent results that are available

Figure 10. Normalized distribution of Von-Mises stressalong the hole boundary.

in the literature for the same problem is presentedin Table 1. The analytical solution for an in�niteplate and the result of our optimization for normalizedmaximum Von-Mises stress around the optimal holeare given. It is shown that the present normalizedmaximum Von-Mises stress is lower, and there is goodagreement between the present result and analyticalsolution.

Figure 11 shows the variation of the normalizedmaximum Von-Mises stress with iteration number.This result clearly shows the stability and high ef-�ciency of the proposed optimization method. InFigure 12, the variation of boundary hole with iterationnumber is also shown.


Table 1. Comparison of published results for normalized maximum Von-Mises stress around optimal hole with the presentmethod.

Research Analysis method andoptimization approach

Normalized maximumVon-Mises stress around

optimal hole

Present studyMFGFEM based ongradientless method

1.7405

Tekkaya and Guneri [20]FEM based on biologicalgrowth method

1.998

Zhixue Wu [25]FEM based ongradientless method

1.804

Timoshinko [26]Analytic solutionfor in�nite plate

1.732

Figure 11. Variation of normalized maximum Von-Misesstress with iteration.

Figure 12. Variation of hole pro�le with iteration

In the following, the e�ect of an initial guess forthe hole boundary on convergency of the uniform stressdistribution along the optimal shape is considered.Thus, we assume the following curves for this purpose:�

xa

��+�ya

��= 1: (37)

The above equation is plotted for di�erent values of� in Figure 13. It is obvious that the above equationgives a circle arc for � = 2.

The normalized maximum Von-Mises stress alongthe initial hole boundary for di�erent values of � isshown in Table 2. It is obvious that the normalizedmaximum Von-Mises stress decreases with increasingthe value of � .

In Figure 14, the normalized distributions of theVon-Mises stress along the optimal hole boundary, as

Figure 13. The assumed curves for initial hole pro�le.

Figure 14. Normalized distribution of Von-Mises stressalong the optimal hole boundary for di�erent values of �.

a function of �, for di�erent initial guesses for the holeboundary, are shown. It can be seen that for all initialguesses, the normalized stress distributions along theoptimal hole boundary are the same. These resultsclearly show the stability and high e�ciency of theoptimization method.


Table 2. Normalized maximum Von-Mises stress around di�erent initial hole boundaries.

� 1 1.5 1.75 2 2.5 3Normalized maximum

Von-Mises stress around initial hole4.2377 3.665 3.299 3.2637 2.833 2.686

Figure 15. Geometry, dimension and boundaryconditions of the shoulder �llet.

7.3. Fillet plate under tensionThe general geometry and notation for a symmetric at plate with a shoulder �llet subject to uniaxialtension is shown in Figure 15. The objective of thisoptimization problem is to �nd out the optimal �lletshape to produce the minimum stress concentrationfactor. Due to symmetry, only one-half of the completeplate is modeled in the MFGFE analysis. Values ofla = lb = 5l are used to keep the e�ect of plate lengthon the state of stress in the vicinity of the �llets at anegligible level. The load is assumed to be the uniformtension of 1 MPa. A wide range of �llet geometrieswas optimized in order to produce useful design data.The values of a=b = 1:33, 1.50, 1.67, 2, and the valuesof l=h = 1, 2, 3, 4, 5 are considered in this example.A straight line is used to de�ne the initial �llet shape.Point A is the �xed point and the stress at this point isselected as the threshold stress. The number of controlpoints is 20.

The iteration history of the maximum principlestress is shown in Figure 16(a)-(d). These results showthat the maximum principle stress converges after afew iterations.

The optimum normalized maximum principalstress values (Kt) obtained for all combinations of a=band l=h are presented in Table 3. It can be seen that fora given a=b, the Kt value decreases with increasing l=h,

Table 3. The optimum normalized maximum �rstprincipal stress values for di�erent geometries.

l=h a=b =1.33 a=b =1.5 a=b =1.67 a=b =21 1.5501 1.4157 1.3481 1.23652 1.2545 1.1856 1.13 1.05913 1.1625 1.1023 1.0567 1.01424 1.1066 1.0543 1.0158 1.00475 1.0696 1.0238 1.0028 1

Figure 16. Variations of maximum �rst principle stresswith iteration number for l=h = 1 anda=b = 1:33; 1:5; 1:67; 2.


as expected. Also, for a given l=h, Kt decreases withincreasing a=b, which demonstrates the �nite widthe�ects. Clearly, the lowest values of Kt are, therefore,obtained for cases where both a=b and l=h are large.It is interesting to note that in many instances, it ispossible to reduce the Kt values very close to unity, anda signi�cant reduction for Kt values is achieved. Thenormalized maximum principle stress along both theinitial and optimal shoulder �llet boundaries, as a func-tion of x=l, for di�erent values of a=b and l=h = 1, areshown in Figure 17(a)-(d). It is seen that the stressesare uniform along the full length of the optimal �lletboundary and the reduction of the maximum principalstress along the optimal �llet boundary for a=b = 1:33,1.5, 1.67, 2 (l=h = 1) are 33.1%, 40.5%, 34.7% and39.9%, respectively. For instance, the stresses alongthe optimal �llet boundary for a=b = 1:67 and l=h = 1are in the range of 1.313-1.348, while the initial line�llet gives maximum stress, which is 34.7% higher. heoptimal stress concentration factor with the present ge-ometry is reported in the study by Waldman et al. [27]to be 1.351, which shows excellent agreement with thatachieved using the present method. This comparisondemonstrates that the present method can achievegood result quality in terms of accuracy and e�ciency.

Non-dimensional optimal tension �llet shapes fordi�erent values of a=b and l=h are also shown in

Figure 18. Optimal non-dimensionalised tension �lletshape for a=b = 1:33 and l=h = 1, 2, 3, 4, 5.

Figures 18-21. It is obvious that, as the �llet getslonger (i.e. l=h gets larger), the optimal �llet geometrybecomes atter.

8. Conclusion

In this paper, a new method is developed for shapeoptimization problems using a modi�ed �xed grid FEM

Figure 17. Distributions of normalized �rst principle stress along initial and optimal �llet boundaries as a function of x=lfor l=h = 1 and a=b = 1:33; 1:5; 1:67; 2.




and a gradientless approach. The objective of theshape optimization problem is to minimize the stressconcentration. A new formulation for calculating theBIEs sti�ness matrix is also presented. The shape ofthe design boundary is modeled using cubic splines.The optimal shape of a design boundary with constantstress is achieved iteratively by adjusting the designboundary shape, based on a simple algorithm. Accu-rate pro�le shapes and associated stress concentrationfactors have been determined for optimal �llets inshoulder plates subjected to remote tension. A signi�-cant range of �llet geometries has been considered. Themain advantage of using non-boundary-�tted meshes isto reduce the computational costs of the analysis viarelaxation of the boundary conforming requirementsof an acceptable mesh. This approach simpli�es thepreprocessing stage signi�cantly and it is clear thatpreprocessing makes up a noticeable part of analysis

Figure 21. Optimal non-dimensionalised tension �lletshape for a=b = 2 and l=h = 1, 2, 3, 4, 5.

cost. The only cost that should be paid for this gainis the slight increase of e�ort required for evaluation ofthe sti�ness matrix of boundary intersecting elements.Obviously, this additional e�ort is negligible when com-pared to the cost of boundary-�tted mesh generation,as in standard FEM. The accuracy and convergenceof the proposed method are analyzed via numericalexamples and results are compared with analytic ornumerical solutions. The results show good agreementwith these analytic or numerical solutions.

References

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Biographies

Mahmood Heshmati received his BS degree inMechanical Engineering from Razi University of Ker-manshah, Iran, in 2005, and his MS degree in Me-chanical Engineering from Shiraz University, Iran,in 2008. He is currently an assistant professor inKermanshah University of Technology. His research�elds include composite material, functionally gradedmaterial, nanomechanics, MEMS, NEMS, nonlocalelasticity, dynamic contact, vibrational analysis, platesand beam structures, and �nite element analysis.

Farhang Daneshmand is Associate Adjunct Profes-sor in the Department of Bioresource Engineering in


the Faculty of Agricultural and Environmental Sci-ences, and the Department of Mechanical Engineeringin the Faculty of Engineering at McGill University,USA. He is also Associate Professor in the Depart-ment of Solid Mechanics in the Faculty of MechanicalEngineering at Shiraz University, Iran. His inter-disciplinary experience is related to solid mechanics,including developing a fundamental understanding ofnanotubes and nanoscale biological beam and shellstructures, functions and behaviors including small-scale e�ects using nonlocal elasticity and gradientelasticity theories, linear and non-linear vibrations andthe dynamic behavior of structures (in atable struc-

tures, hydraulic structures, shell containers) with andwithout uid-structure interaction, and advanced two-and three-dimensional computational solid mechanics(Isogeometric Analysis (IA), meshless methods, �xed-grid FEM).

Yasser Amini received BS and MS degrees in Me-chanical Engineering, in 2006 and 2009, respectively,from Shiraz University, Iran, where he is currently aPhD degree student. His research interests includeCFD, mesh-less methods, SPH method, uid structureinteraction, rare�ed gas ow dynamics and DSMCmethod.

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