Structural and Multidisciplinary Optimization manuscrip t No.(will be inserted by the editor)

Laurent Van Miegroet · Pierre Duysinx

Stress Concentration Minimization of 2D filets using X-FEM andLevel Set Description

Received: date / Revised: date

Abstract This paper presents and applies a novel shape op-timization approach based on the Level Set description ofthe geometry and the eXtended Finite Element Method (X-FEM). The method benefits from the fixed mesh work us-ing X-FEM and from the curves smoothness of the LevelSet description. Design variables are shape parameters ofbasic geometric features which are described with a LevelSet representation. The number of design variables of thisformulation remains small whereas global (i.e. compliance)and local constraints (i.e. stress) can be considered. To il-lustrate the capability of the method to handle stress con-straints, numerical applications revisit the minimization ofstress concentration in a 2D filet in tension which has beenstudied previously by Pedersen in (27). Our results illustratethe great interest of using X-FEM and Level Set descriptiontogether. A special attention is also paid to stress computa-tion and accuracy with the X-FEM.

Keywords Shape optimization· Topology optimization·X-FEM · Level set

1 Introduction

Minimizing stress concentrations has been the topics ofmany works in structural optimization. Reducing stress con-centrations and stress intensity factors or controlling thestress failure criteria is essential in mechanical engineer-ing design (see for instance Ref. (29)). Very early structural

P. DuysinxAutomotive EngineeringAerospace and Mechanics Department - University of LiegeChemin des Chevreuils, 1, building B52, 4000 Liege, BelgiumTel.: +32-43669194Fax: +32-43669159E-mail: P.Duysinx@ulg.ac.be

L. Van MiegroetAutomotive EngineeringAerospace and Mechanics Department - University of LiegeChemin des Chevreuils, 1, building B52, 4000 Liege, BelgiumTel.: +32-43669270Fax: +32-43669159E-mail: L.VanMiegroet@ulg.ac.be

optimization techniques have been identified as a powerfuland rationale tool to support the engineering task in findingdesigns with high stiffness and strength. While sizing vari-ables bring a rather limited improvement of stress concen-tration factors, shape optimization can substantially increasethe strength of the design by modifying the boundaries ofthe structures. The reader can refer to the two famous re-view papers by Ding (12) or Haftka and Grandhi (17) and thereferences cited therein. Among many papers one can men-tion the contributions by Bennet and Botkin (7), Braibantand Fleury (8), Pedersen and Laursen (24), which proposedseveral applications of shape optimization to reduce stressconcentrations and failure criteria.

Following the impetus initiated by the work of Bendsøeand Kikuchi (3), a very important amount of research has fo-cused on topology optimization. As topology optimizationis formulated as an optimal material distribution, topologyoptimization introduces a very large number of design vari-ables so that most of the works consider only global or inte-gral objective function with a single volume constraint. Dueto the large number of restrictions to be considered, worksincluding stress constraints are rather limited (see for in-stance Duysinx and Bendsøe, (13) and Pereira et al. (28),Allaire et al. (2)). Fortunately for a single load case, Peder-sen in Ref. (26) showed that compliance minimization leadsto a good improvement of stress concentration. However formultiple load case, unequal stress constraints or multi ma-terial problems it is known (see for instance Rozvany (30))that there is no guarantee that optimum designs for stiffnessand strength are the same and the stress constraints have tobe introduced in the design problem formulation.

While topology was experiencing a very spectacular suc-cess in industrial applications, shape optimization has beenmore difficult to apply in industrial environment. Practically,one major advantage of the optimal material distribution for-mulation is to be able to work on a fixed regular mesh, whileshape optimization is forced to struggle with difficult prob-lems related to mesh distorsion, automatic mesh regenera-tion problems, finite element errors, etc. Moreover a majortechnical problem also stems from the sensitivity analysisthat requires the calculation of the so-called velocity field

2

because of the moving mesh and it affects the complexityand cost of the sensitivity analysis. It turns out that shapeoptimization remains generally quite fragile and delicatetouse in industrial context.

In order to circumvent the technical difficulties of themoving mesh problems, a couple of researches have tried toapply the fixed mesh principle to shape optimization. Let’smention the use of fictitious domain approach by Haslingerand co-authors as in Ref. (10), the fixed grid finite elementsby Kim et al. in Ref. (18) or more recently the projectionmethods as in Norato et al. (20). The present work relieson the noveleXtended Finite Element Method(X-FEM) thathas been proposed as an alternative to remeshing methods(see Ref. (19) or (5) for instance). The X-FEM method isnaturally associated with theLevel Setproposed by Osherand Sethian (Ref. (23)) description of the geometry to pro-vide a very efficient treatment of difficult problems involv-ing discontinuities and propagations. Up to now the X-FEMmethod has been mostly developed for crack propagationproblems (19), but the potential interest of the X-FEM andthe Level Set description for other problems like topologyoptimization was identified very early in Belytschko et al.(6), while the advantages of the Level Set method in struc-tural optimization was clearly demonstrated by Wang et al.(34) or Allaire et al. (1).

In Ref. (14), the authors see the X-FEM and the LevelSet description as an elegant way to fill the gap betweentopology and shape optimization and the method can bequalified asgeneralized shape optimizationas it is basedon smooth boundary descriptions while allowing topologymodifications as holes can merge and disappear. X-FEM en-ables working on a fixed mesh, as in topology optimization,removing the technical difficulties related to the mesh andvelocity field management from the formulation. The op-timization formulation has the advantage of using a smallnumber of design variables related to the shape parametersof geometric model of the structure described in Level Setsentities, while constraints can be either global (compliance,volume) or local (stresses) responses as in shape optimiza-tion.

The work wants to demonstrate that stress constraintscan be considered efficiently when using X-FEM and notonly compliance or energy functions. To this end the paperillustrates the capabilities of the methodology to minimizestress concentration factors in a well known family of bench-marks coming from shape optimization. The example of 2Dfilets has been selected because of the numerous publishedresults for this test case.

The layout of the paper is organized as follows. At first insection 2 the Extended Finite Element Method is presentedfor cracks, bi material and void. A special attention is alsopaid to stress computation and accuracy with the X-FEM.Then, section 3 summarizes the Level Set representation andits combination with X-FEM. In section 4 the formulationof optimization problem is presented whereas the sensitivityanalysis is addressed in section 5. Finally in section 6, 2Dfilet stress concentration minimization problems (inspired

by Ref. (27)) are used to illustrate optimization strategy us-ing X-FEM and Level Set when stress constraints are con-sidered.

2 The eXtended Finite Element Method

The eXtended Finite Element Method (X-FEM) (19; 5) hasbeen firstly introduced to study propagating cracks in me-chanical structures. The method has obtained so promisingresults in fracture mechanics that some authors have im-mediately foreseen the opportunities of applying X-FEM tomany kinds of problems in which discontinuities and mov-ing boundaries have to be modeled. As exemple publicationswe can mention Belytschkoet al. (5) who applied the X-FEM to the modelization of composite fibre orientation ina micro-structure, Chessaet al. (9) who studied the case oftwo phase fluid problems, and recently Guetariet al. (16)who simulated material removal in machine tooling prob-lems using X-FEM formalism.

2.1 Principle of the method

In the classical Finite Element Method, modeling disconti-nuities inside an element is not possible because the shapefunctions are required to be at leastC1. Therefore, if discon-tinuities are present in the model, these discontinuity bound-aries have to coincide with the element mesh boundaries andthey require a remeshing process each time the singularityevolves in the structure.

The X-FEM overcomes this restriction by adding to theclassical FEM approximation some particular discontinuousor singular shape functions. Hence, these additional shapefunctions, directly related to the nature of the discontinu-ity, allow to include geometric boundaries, cracks, materialor phase changes that are not coincident with the mesh andavoid the expensive and delicate mesh generation.

2.2 Modeling cracks

In presence of cracked structures, the displacement is dis-continuous. Hence, modeling the displacement field calls fordiscontinuous shape functions. The classical finite elementapproximation used is then extended to embed the discon-tinuous shape function as in the following equation:

u(x) = ∑i

uiNi(x)+∑j

ajNj (x)H(x) (1)

whereNi(x) are the classical shape functions related to nodaldegrees of freedomui . The Nj (x)H(x) are the discontinu-ous shape functions constructed by multiplying a classicalshape functionNj (x) with a Heaviside functionH(x), whichis equal to+1 on one side of the crack and−1 on the otherside (see Fig. 1) .

3

3

1

2

Boundary

Fig. 1 Representation of the extended shape functionN1(x)H(x) ofnode 1 for a cracked element

Note that this set of extended shape functions are onlysupported by the enrichment degrees of freedomaj . More-over, only the elements near the discontinuity usually sup-port extended shape functions whereas the other elementsremain unchanged (i.e. classical FE). The modification ofthe displacement field approximation does not introduce anew form of the discretized finite element equilibrium equa-tion but rather leads to an enlarged problem (see Ref. (19)for details):

K ·q = g⇔

[

Kuu KuaKau Kaa

][

ua

]

=

[

f extu

f exta

]

(2)

As the elements can now present discontinuous shape func-tions, the numerical integration scheme has to be modified inorder to take care of the discontinuity. The implementationused here is similar to the one described in Ref. (4). In or-der to use the classical Gauss numerical integration schemes,the elements embedding a singularity are divided into sub-triangular mesh coincident with the discontinuity boundary.Over this working mesh, a quadrature integration rule canbe applied (see Fig. 9). The procedure is described in moredetails in section 2.5.

2.3 Modeling material-material interfaces

When an interface between two different materials is present,the displacement field is continuous but the strain field isdiscontinuous. Therefore, in order to model this kind of dis-continuity inside an element, one have to add special shapefunctions that contain a discontinuity of the derivative ofthedisplacement field. According to Ref. (9) and (31), one canuse the following expression to discretize the displacementfield:

u(x) = ∑i

uiNi(x)+∑j

ajNj (x)(∣

∣φ j∣

∣−|φ(x)|) (3)

whereφ j is the distance from the nodej to the interface andφ(x) is the distance of the current pointx to the interface.

In order to illustrate this approximation, let’s considerthe simple problem of a bimaterial beam of Young modulusE1 = 1 andE2 = 2 subject to a traction forceF = 1 along the

1 10,8 0,8

eta 0,6 xi0,60,4 0,4

0,2 0,200 0

0,05

0,1

0,15

0,2

0,25

Fig. 2 Representation of the shape function of node 1 for a bi-materialtriangular element

x axis (Fig. 3). With classical finite elements approximation,this very simple test case requires a minimum of two finiteelements to be modeled. Within the X-FEM framework, theanalysis of the problem can be carried out with a single X-FEM. Using the enrichment of the shape functions proposedin eq. (3), the axial displacement field of the X-FEM rod is:

u(x) = u1(1−xL

)+u2(xL

)+a1(1−xL

)x+a2(xL

)x, (4)

x ∈ [0,L/2]

u(x) = u1(1−xL

)+u2(xL

)+a1(1−xL

)(L−x) (5)

+ a2(xL

)(L−x),x ∈ [L/2,L]

E2E1F

L

U1

a1

U2

a2

Fig. 3 Geometrical model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Dis

plac

emen

t

X

Analytical solution

a1 contribution

a2 contribution

u2 contribution

Numerical solution

Fig. 4 Degrees of freeedom contributions, analytical and numericalsolutions

4

Different enrichment shape functions can be used tomodel a bimaterial interface (see Ref. (31) for more details),however, the displacement field of eqn. (4) and (5) has theparticular and attractive advantage of zeroing on the two endnodes, which prevent from connecting the degree of freedomaj to other elements.

2.4 Modeling material-void interfaces

The modeling of material-void interfaces with X-FEM (31)differs only marginally from the cracked structure case (i.e.the discontinuous field is the displacement). For void inclu-sions and holes, the displacement field is now approximatedby:

u(x) = ∑i

uiNi(x)V(x) (6)

where

V(x) =

{

0 if x ∈ material zone1 if x ∈ void (7)

One can note that this functionV(x) models perfectly thesingularity in the displacement field as the functionV(x)cancels the displacement field in the void (see Fig. 5).

Boundary

3

2

1

Fig. 5 Representation of the shape function of node 1 on a cut element

Modeling holes with the X-FEM is a very appealingmethod for the shape optimization but also for the topologyoptimization as no remeshing is needed and no approxima-tion is made on the nature of the voids conversely to nu-merous methods as the power penalization of intermediatedensities method used (SIMP) in which void is replaced bya very weak material. However, as we will note in section2.5, the X-FEM method requires more complicated integra-tion procedure and a careful attention has to be paid to stressand sensitivity computation.

2.5 Implementation of X-FEM

In the case of material-void interfaces, elements are of onlythree types: solid ones, void ones and cut ones. The void el-ements have all their nodes lying fully in the void. These

elements (red ones in Fig. 6) are then consequently removedfrom the problem and have no contribution to the stiffnessmatrix. Their nodal degrees of freedom are eliminated fromthe system of equations when assembling the global stiff-ness matrix. The filled elements (green ones in Fig. 6) withall nodes lying in the solid domain are treated as normal ele-ments whereas the partially filled elements (blue ones in Fig.6), which have a mix of void and solid nodes require purelyan X-FEM.

Fig. 6 Representation of the different element type

Fig. 7 Representation of the model with cut and filled element only

Fig. 8 Representation of the model with the sub-triangles

5

2.5.1 Numerical integration of the stiffness matrix

First of all when an X-FEM element is to be integrated, wefirst have to detect which part of the element is filled withmaterial in order to determine if the solid part is a triangleor a quadrangle. As one can see in section 3 devoted to theLevel Set description of the geometry, this is easily donewhen using a Level Set Description.

For X-FEM elements a major issue is the computationof the stiffness matrix. Because of the strong discontinuityof the shape functions that takes place at the void solid inter-face, one can observe that using a quadrature rule (e.g. Gausspoints) would lead to a totally unsatisfactory results evenwhen increasing dramatically the number of Gauss points.At first one has to notice that because of the zero displace-ment field in the void domain (see eqn. 6), the void partof the element does not contribute to the stiffness matrix.Thus the computation is restricted to an integration proce-dure solely over the solid sub-domain of the element(see Fig.8). To integrate over the solid part of the element domain, wedivide it into a partition of triangles in 2D (or tetrahedronsin 3D) and use a classical Gauss quadrature over each of thesub-triangles. As the field is linear, Gauss quadrature ruleisexact on each sub-triangles. The procedure can easily be de-veloped either for triangular or quadrangular sub-domains.The main idea is to use the center of gravity to support anew integration mesh. The Fig. 9 illustrates the domain sub-division of a cut element in order to integrate it. One hasto notice that the sub-triangles are only a background meshonly used for the integration and that the size of the prob-lem is not augmented regarding to a classical finite elementmodel.

Attention has still to be paid to two kinds of numeri-cal problems that can occur with the X-FEM modeling ofvoid. The first one happens when the remaining solid partof the element is a quadrangle that is nearly degeneratedinto a triangle. The case is illustrated in Fig. 10. Hence, thisquadrangle could be divided into four triangles, but one ofthese triangles would be very ill conditioned and would in-troduce a rank deficiency. Thus, it is better to consider thissolid domain of the element as a triangle and divide it intothree triangles. The error is very limited and the precisionis good, which would never be achieved with a degeneratedsub-triangle .

The second kind of numerical error occurs when the areaof the solid part becomes very small regarding to the area ofthe supporting element (Fig. 11). One can observe that inthis case the element becomes very stiff and the structuralstiffness matrix becomes somewhat not so well conditioned.This explains the reason why some authors (for example seeRef. (11)) do not include these elements and eliminate themfrom the model when building the stiffness matrix. In thisstudy we could circumvent the problem thanks to a scalingprocedure of the stiffness matrix and we could not noticeany degradation of the precision in the displacement field.

void

solid Boundary

(a) Sub-division of a quadrangular ele-ment and Gauss points

boundary

void

solid

(b) Sub-division of a quadran-gular element and Gausspoints

Fig. 9 Sub division of elements

Especially compliance value has always been in very goodagreement with FEM that is taken as a reference solution asshown in Ref. (32). However we could observe that thesevery small solid elements are the source of a more importantdifficulty resulting in a very high overestimation of the localstress. This is a crucial issue in this study in which we wantto minimize the stress level. The problem is explained in thenext section.

Fig. 10 Quadrangle degenerated into a triangle

2.5.2 Estimation of stresses

A test case illustrates the problem of the overestimation ofthe stress when the area of the solid part becomes very small

6

Fig. 11 Small solid triangle

regarding to the area of the supporting element. The prob-lem is a classical quarter plate with a hole under an uniaxialload as illustrated in Fig. 12 and Fig. 13. The Fig. 12 showsthe stress component along thex (computed at Gauss point)obtained with a FEM model, whereas the second Fig. 13.illustrates the same result given by an X-FEM analysis.

OOfelie GraphOOfelie Graphsigma_xsigma_x

-0.0805-0.0805

0.497 0.497

1.08 1.08

1.65 1.65

2.23 2.23

2.81 2.81

3.39 3.39

Fig. 12 FEM stress distribution

As we can remark, a large error occurs on the maxi-mum stress for the X-FEM model. This overestimation ofthe stress level is due to a small solid element as pointedout in the zoomed part of Fig. 13. This is even more clearwhen we consider the following experiment. If the bound-ary of the circle is slightly modified (perturbation of 10−4 ofthe radius circle) in order to remove the small cut element, itis seen that the agreement between the results obtained withthe two methods become excellent again.

To prevent the phenomenon, one can resort to the fol-lowing strategy:

– Eliminate elements that have a too small solid part fromthe model when building the stiffness matrix (see (11));

– Move the node on the boundary in order to remove thesmall elements;

– Post process the stress results and eliminate stresseswhen the ratio of the solid area to the supporting areais too low;

OOfelie GraphOOfelie Graphsigma_xsigma_x

−0.0856−0.0856

0.603 0.603

1.29 1.29

1.98 1.98

2.67 2.67

3.36 3.36

4.05 4.05

Fig. 13 X-FEM stress over estimation with a small solid element

OOfelie GraphOOfelie Graphsigma_xsigma_x

-0.0910-0.0910

0.479 0.479

1.05 1.05

1.62 1.62

2.19 2.19

2.76 2.76

3.33 3.33

Fig. 14 X-FEM stress estimation without small solid elements

– Compute nodal stresses using a smoothing procedure asin Finite Element error estimation.

The first and second ones sound a bit tricky, while thefourth solution is obviously the best one. Because this solu-tion was not available in our computer code, we preferredhere the third solution because of it simplicity since it is justa matter of post-processing the results.

3 The Level Set Description

The low performances generally obtained with the shape op-timization is directly related to the fact that no deep bound-ary or topological changes such as creation or fusion of holesis allowed with an explicit parametric CAD representation.This restriction is overcome when using an implicit repre-sentation based on the level Set description.

The Level Set method, introduced by Osher and Sethianin Ref. (23), is a numerical technique first developed to trackmoving interfaces in physics problems. It is based upon the

7

idea of implicitly representing the interfaces as a Level Setcurve of a higher dimension functionψ(x, t). The boundariesof the structure are then conventionally represented by thezero level i.e.ψ(x, t)=0 of this functionψ, whereas the filledregion is attached to the positive part of theψ function (seeFig. 15).

If one knows also the parametric expression of the curveγ , a corresponding Level Set can be easily constructed usingthe signed distance function to the curveγ :

ψ(x, t) = ± minxγ∈γ(t)

∥

∥x−xγ∥

∥ (8)

Level Sets can be easily combined using Boolean oper-ations via min or max operators. This allows to create newentities and to model complex geometries.

3.1 Level Set and X-FEM

Applied to the X-FEM framework, the Level Set is definedon the structural mesh and a geometrical degree of freedomrepresenting its Level Set function value is associated at eachelement node. Inside the elements, the Level Set is interpo-lated with the shape function used for a finite elements ap-proximation:

ψ(x, t) = ∑i

ψiNi(x) (9)

The sign ofψ is then used to know whether or not a nodelies in the material domain. It is also used to give a directaccess to the type of the elements (solid, void or cut) and tothe cut edges of the finite elements:

– a triangle with 3 positive (negative) values of the LevelSet if filled with material (void);

– a triangle which has both positive and negative values iscut.

ψ(x, t) > 0⇔ Solidψ(x, t) < 0⇔Void (10)

Besides these advantages, the Level Set can also presentsome drawbacks. For instance in optimization, the dis-cretization approximation of eqn. (9) interpolates linearlythe Level Set when first order finite element are used. As aconsequence, the representation of the boundaries can over-estimate or underestimate the surface (volume) of the struc-ture. Hence, the estimation of the surface depends on thenumber of cut elements, on the position of the inteface in-side the element and of course on the mesh refinement. Thiserror can causes some ”zigzagging” problems during opti-mization if one uses a very coarse mesh and a constraint orobjective function related to the area (volume) as pointed outin Ref. (33).

4 Formulation of optimization problem

The formulation of the considered optimization problem issimilar to a shape optimization problem, but its solution is

OOfelie Graph

0.755

OOfelie GraphLevel Set

-0.274

-0.103

0.0688

0.240

0.412

0.584

(a) Level Set representation of NURBS curve

OOfelie GraphOOfelie Graphuu

0.000 0.000

5.30e-0065.30e-006

1.06e-0051.06e-005

1.59e-0051.59e-005

2.12e-0052.12e-005

2.65e-0052.65e-005

3.18e-0053.18e-005

(b) Displacement of the X-FEM model

Fig. 15 Geometric representation and X-FEM displacement result

greatly simplified thanks to the use of the X-FEM and LevelSet description as no velocity field and mesh perturbationare needed.

The geometry and the material domains are specified us-ing Level Sets description of the boundaries. The positivepart of the Level Set represents material domain while thenegative part is the void. To describe the structural geometrythe user has a library of basic geometric features (in LevelSets) that can be combined to create almost any structuralgeometry. The library geometric features are circles, ellipsesand all polygons. The design variables are chosen among thegeometric parameters of these features.

The optimization problem aims at finding the best shapefor minimizing a given objective function while satisfyingmechanical and geometrical design restrictions. The me-chanical constraints can either be global responses (e.g.compliance, volume) or local ones such as displacements orstress constraints.

The number of design variables is generally small as inshape optimization. However the number of constraints maybe large if many local stress restrictions (e.g. stress con-

8

straints) are considered. Nonetheless, large scale problemsas in topology optimization are avoided.

The design problem is cast as a general constrained op-timization problem:

minx

g0(x)

s.t.: gj (x) ≤ gmaxj j = 1. . .m

xi ≤ xi ≤ xi i = 1. . .n

(11)

The solution to this problem is obtained using the so-calledsequential convex programming. At each iteration, the X-FEM analysis problem is solved and a sensitivity analysisis performed. The solution of the optimization problem isthen found by using a CONvex LINearization approxima-tion scheme of each constraint functions (see (15)). The so-lution becomes the new design and the procedure is repeateduntil convergence.

Because of the X-FEM characteristics, the geometry hasnot to coincide with the mesh and the shape optimizationproblem is carried out on afixed mesh. One works here in aneulerian approach and not in a lagrangian approach. This cir-cumvents the mesh perturbation problems of classical shapeoptimization. Sensitivity analysis can be performed withoutconsidering a velocity field. The present formulation is then,up to a certain point, simpler. However, some technical dif-ficulties can be encountered if a finite difference or a semi-analytical scheme is used for sensitivity analysis. Basically,the problem is that the perturbation must not change thenumber of degrees of freedom of the X-FEM stiffness matrixwhile a finite perturbation of the Level Set could introducenew cut element and new nodes into the formulation.

The Level Set approach is very convenient to modify thegeometry because the Level Sets (and so the holes) can pen-etrate each other or disappear. Creation of new holes is moreproblematic since it leads to a non smooth problem. Topo-logical derivatives have then to be used for a rigorous treat-ment of the problem (see ref. (21) for instance). This capa-bility has not been implemented in this study.

5 The sensitivity analysis method

Similarly to classical shape optimization, the sensitivityanalysis of the various responses (such as compliance, dis-placements, stresses,. . .) is carried out using a semi-analyticapproach. The derivatives of stiffness matrix (K ) and loadvectors (f) are calculated by finite differences with respectto a small perturbationδx of Level Set parameters:

∂K∂x

≃K(x+δx)−K(x)

δx(12)

∂ f∂x

≃f(x+δx)− f(x)

δx(13)

These derivatives are then used to compute the sensitivity ofthe various objective functions. For the complianceC, thesensitivity (in the case of invariant loading forces) is then

given by:

∂C∂x

= −12

uT ∂K∂x

u (14)

Now, if the objective function or constraint involves thestresses of the problem, the sensitivity of this response isneeded. Two basic methods are available to get the deriva-tive of the stresses. The first one, which has the drawbackof forcing us to know explicitly the tension matrix, consistsin deriving the expression of the stresses (σ ) in all the ele-ments:

σ = T j uj = HB j u (15)

whereH is the Hooke’s matrix,Bj the matrix of the derivedshape functions of the elementj andT j the tension matrix ofelementj . The second method is based on the computationof the stresses related to the perturbed statex+δx by usingthe expression of the displacement sensitivities:

σ(x) = HB j u(x) (16)

u(x+δx) ≃ u(x)+∂u∂x

δx (17)

σ(x+δx) ≃ HB j u(x+δx) (18)

∂ σ∂x

≃σ(x+dx)−σ(x)

∂x(19)

This procedure reduces the sensitivity of the stresses as afunction of the displacement derivative. In the present pa-per, the implementation is performed in an industrial codeOOFELIE for which the knowledge of tension matrixT j isnot available for any element. Moreover for the maintenanceof industrial code it is often preferred to assume that sensi-tivity analysis is made less dependent upon element imple-mentation. Therefore, the second method has been chosenand implemented.

OOfelie GraphOOfelie Graph

Fig. 16 X-FEM Mesh for sensitivity analysis validation

9

The quarter plate with an elliptical hole and non uniformbi-axial loading (Fig. 16) serves as a tool for the valida-tion of the approximated semi-analytical sensitivity analy-sis that has been used. Table 1 gives the sensitivities of themaximum stress calculated by finite differences and semi-analytical approach for different combinations of the designvariablesa andb, the major and minor axes of the ellipticalhole. The results were obtained with a relative perturbationof the design variables ofδ = 10−4. The table presents thequality of the proposed semi-analytical approximation.

Table 1 Validation of semi-analytical sensitivity analysis approxima-tion.

Variable Finite difference Semi-analytical Relative error

a = 0.41 -213.2172401 -213.2279160 0.005%b = 0.41 2075.6012932 2074.0930198 0.073%

a = 0.55 -51.9288917 -51.888882944695 0.077%b = 0.55 5019.8392594 5003.5389826985 0.32%

In the classical shape optimization, the computing com-plexity of the stiffness matrix sensitivity is due to the mod-ifications of the mesh associated to the perturbationδx andto the velocity field calculation. In the present X-FEM basedapproach, one has not to deal with the mesh perturbationsas one works on a fixed grid. However, this method exhibitsa different drawback with respect to the general shape op-timization as, with X-FEM, the number of elements maychange. Therefore, when Level Set is perturbed in order tocompute the sensitivity, there is a possibility that previouslyempty element become partly filled with material. As a con-sequence, the number of degrees of freedom change and thedimension of the stiffness matrix is modified between theLevel Set perturbation which forbid to compute the stiffnessmatrix derivative by finite difference (eqn. 12). The strategythat is implemented presently (see Ref. (33) and (14) for de-tails) to circumvent the difficulty is the following. As one hasonly the displacement (ui) for the elements that are presentin the reference configuration, only these elements are takeninto account while the contributions coming from the newpartly filled elements are ignored. Hence, no new elementsare introduced and the size of the stiffness matrix remainsunchanged. Of course, the ultimate solution to the problemshould resort to a fully analytical sensitivity of the stiffnessmatrix. However, this would be rather restrictive for indus-trial applications.

6 Numerical applications

The X-FEM method for the modeling of material-void dis-continuity and its Level Set description have been imple-mented in an object oriented (C++) multiphysics finite ele-ment code, OOFELIE that is commercialized byOpen En-gineering(information available on the web site (22)).

In this software, various mechanical responses can bechosen as objective functions and design restrictions thatis:compliance and potential energy, all stress components andVon Mises equivalent stress, displacements and geometricresults. Presently, the implementation of the X-FEM methodis only available in 2-D with a library of both first degreequadrangle and triangle elements. The Level Set descriptioncan be defined in different ways. They can be constructedclassically from functions (circle, quadrangles, ellipses, ...)or from a set of points which are interpolated by a NURBScurve. The CONLIN optimizer by C. Fleury described inRef. (15) has been coupled in the OOFELIE environment torealize the numerical applications.

6.1 The 2D-filet in tension

The numerical applications are inspired by the famous prob-lems of stress concentration factors evaluation as in (29).Pedersen in Ref.(27) has stated this problem as an optimiza-tion problem. The problem consist in studying the effect ofthe connection zones in the following 2D structure and re-ducing the stress concentration factor.

l2

l3

l4

l1

lf

Fig. 17 2D filet model in tension

The following geometry parameters are fixed to the val-ues:

– l1=30mm,l4=60mm,l2=30mm,l3=90mm;– l f =30mm for section 6.1.1 and 6.1.2 and is a variable for

the section 6.1.3.

The plane stress state is assumed and a uni-axial stressfield is appliedσx=1. The material properties associated are:Young’s modulusE = 1N/m2, Poisson’s rationν=0.3. Dueto symmetries, only a quarter part is studied.

6.1.1 Super circular connection

As P. Pedersen pointed out in Ref.(27), a circular connectionto a straight domain is not the best solution and it is better toreplace it by a super-circular connection as defined by:

xη1 +xη

2 = r (20)

10

(a) Mesh used for all the test cases

OOfelie Graphsigma_x

-0.0216

0.242

0.506

0.770

1.03

1.30

1.56

(b) Initial configuration:η=2, σmax=1.563

Fig. 18 Mesh and X-FEM analysis

Since a parameter is introduced, the goal of the optimizationproblem is to find the configuration presenting the minimumstress concentration:minmax

iσ2

ix (21)

wherei is the index of the finite elements. In practice, a multiobjective approach is used by taking into account only themost critical elements, i.e. the top 10 % of the most stressedfinite elements. No other constraint is prescribed in the prob-lem except the side-constraint over the variableη which re-stricts η to the domain 2≤ η ≤ 4. The initial configura-tion, with η=2, presents a stress concentrationσmax/σ∞ =1.56253 with the mesh given in Fig. 18 (8000 degrees offreedom). The optimum, obtained after ten iterations, pro-vide a rather limited improvement as the stress concentra-tion is reduced to the value ofσmax/σ∞ = 1.52612 (gain of2.75%) and an optimum value ofη=2.0935 (Fig. 19).

6.1.2 Generalized super circular connection

The previous test case shows that only a minor improvementcan be obtained by using a super circle in comparison toa normal circle. Therefore, we replace the super circle filetwith a generalized super circle defined as:

xη1 +xξ

2 = r (22)

OOfelie GraphOOfelie Graph

Iteration nbIteration nb0.000 0.000 3.00 3.00 6.00 6.00 9.00 9.00 12.0 12.0 15.0 15.0

Obj FctObj Fct

1.57 1.57

1.56 1.56

1.55 1.55

1.54 1.54

1.53 1.53

(a) Evolution of the maximum stress

OOfelie GraphOOfelie Graphsigma_xsigma_x

-0.0233-0.0233

0.235 0.235

0.493 0.493

0.751 0.751

1.01 1.01

1.27 1.27

1.53 1.53

(b) Final configuration:η=2.09,σmax=1.526

Fig. 19 Optimization with a super circle

The problem is defined as the previous one, however ithandles now two variables,ξ andη which are constrainedbetween the values 2 and 4. As it can be remarked, the ad-ditional exponent allows a deepest modification of the filetcurvature which leads to a significant reduce of the stressconcentration of 6.7 % (Fig. 20).

6.1.3 Generalized super ellipse connection

This last test case shows the important effect of the filetlength on the stress concentration. A generalized super el-lipse is used to define the boundary of the filet:∣

∣

∣

x1

a

∣

∣

∣

η+

∣

∣

∣

x2

b

∣

∣

∣

ξ= r (23)

With an unlimited filet length this stress concentration couldbe avoided. Hence, the optimization variablea is constrained

11

OOfelie GraphOOfelie Graph

Iteration nbIteration nb0.000 0.000 5.00 5.00 10.0 10.0 15.0 15.0 20.0 20.0 25.0 25.0

Obj FctObj Fct

1.57 1.57

1.54 1.54

1.51 1.51

1.48 1.48

1.45 1.45

1.42 1.42

(a) Evolution of the maximum stress

OOfelie GraphOOfelie Graphsigma_xsigma_x

-0.0285-0.0285

0.214 0.214

0.458 0.458

0.701 0.701

0.944 0.944

1.19 1.19

1.43 1.43

(b) Final configuration:ξ=3.259,η=2.09,σmax=1.42938

Fig. 20 Optimization with a generalized super circle

between the values 20 and 60 (initial value at 30). The othervariables are constrained between 2≤ ξ ≤4 and 2≤ η ≤4.

As expected, the optimization reaches the sameξ andη obtained with the previous application and maximizes thevalue of the parametera representing the filet length. Thestress concentration has been reduced by 22% with respectto the previous case and nearly 30% to the initial circularfilet (Fig. 21).

7 CONCLUSIONS

This work has presented an novel approach for shape opti-mization based on the Level Set description and the X-FEMfor the structural optimization. This new approach takesplace between shape and topology optimization as it pos-

OOfelie GraphOOfelie Graph

Iteration nbIteration nb0.000 0.000 3.00 3.00 6.00 6.00 9.00 9.00 12.0 12.0 15.0 15.0

Obj FctObj Fct

1.61 1.61

1.51 1.51

1.41 1.41.

1.31 1.31

1.21. 1.21

1.11 1.11

(a) Evolution of the maximum stress

OOfelie GraphOOfelie Graphsigma_xsigma_x

-0.0275-0.0275

0.163 0.163

0.354 0.354

0.545 0.545

0.736 0.736

0.927 0.927

1.12 1.12

(b) Final configuration: ξ=3.259, η=2.09, a=60,σmax=1.11809

Fig. 21 Optimization with a generalized super ellipse

sesses the same characteristics of both ones. The X-FEMmethod has shown to be very useful as it easily takes ad-vantage of the fixed mesh work approach of topology op-timization whereas the smooth curve description from theshape optimization is preserved. Moreover, void is not ap-proximated as a smooth material in opposition to the SIMPmethod. Contrary to shape optimization, no remeshing pro-cess is needed in our applications, and only one mesh hasbeen created and used for all the test cases.

A semi-analytic sensitivity analysis with X-FEM andLevel Set for various responses have been developed andvalidated. One issue that has been investigated is the qualityof the X-FEM approximation for stresses. The source of theproblems are the small elements that have only a very lowfraction of their surface filled with solids. These elements

12

leads to large errors in the stress estimation. The estimationfor these elements should not be considered in a rough way,but rather require special post processing.

The novel shape optimization approach based on the X-FEM and Level Set description has been illustrated with theminimization of the stress concentration of a 2D mechanicalstructure. It gives encouraging results to study large scale in-dustrial problems, especially 3-D problems. Other analysesproblems will also be investigated in the future as dynamicproblems and multiphysics (electro-mechanical) problems.

8 Acknowledgments

The authors gratefully acknowledge the support of projectARC MEMS, Action de recherche concertee 03/08-298funded by the Communaute Francaise de Belgique.

References

1. Allaire G. Jouve F. and Toader A.M. (2004) Structural optimizationusing sensitivity analysis and a level-set method.Journal of Compu-tational Physics. Vol. 194, Issue 1, pp 363-393.

2. Allaire G., Jouve F. and Maillot H. (2004) Topology optimizationfor minimum stress design with the homogenization method.Struc-tural and Multidisciplinary OptimizationVol. 28, No 2-3, 87–98.

3. Bendsøe M. P. and Kikuchi N. (1988). Generating Optimal Topolo-gies in Structural Design Using a Homogenization Method,Com-puter Methods in Applied Mechanics and Engineering, 1988, 71,197-224.

4. Belytschko T., Moes N., Usui S., Parimi C.(2001) Arbitrary dis-continuities in finite elements. Int. J. Num. Meth. in Engng 50: 993-1013.

5. Belytschko. T. Parimi C. , Moes N. , Sukumar N. and Usui. S.(2003). Structured extended finite element methods for solids de-fined by implicit surfaces.Int. J. Numer. Meth. in Engng2003; 56 :609-635.

6. Belytschko T., Xiao S. and Parimi C. (2003). Topology optimiza-tion with implicit functions and regularization.Int. J. Num. Meth. inEngng2003; 57 : 1177-1196.

7. Bennet J.A. and Botkin E. (1985) Structural Shape Optimizationwith Geometric Description and Adaptive Mesh Refinement. AIAAJournal Vol 23 No 3, 458–464.

8. Braibant V. and Fleury C. (1984) Shape optimal design using B-splines.Comput. Meths Appl. Mech Engrg.Vol 44 No 3, 247–267.

9. Chessa J. and Belytschko T.(2003) The Extended Finite ElementMethod for Two-Phase Fluids. ASME Journal of Applied Mechan-ics, 70(1): 10–17.

10. Dankova . and Haslinger J. (1996) Numerical realizationof a ficti-tious domain approach used in shape optimization. Part I Distributedcontrols.Applications of Mathematics1996, 41(2): 123-147.

11. Daux C., Moes N., Dolbow J., Sukumar N., Belytschko T.(2000)Arbitrary discontinuities in finite elements. Int. J. Num. Meth. in En-gng 48: 1741-1760.

12. Ding Y. (1986) Shape optimization of structures: a literature sur-vey.Computers and Structures. Vol 24 No 4, 985 – 1004.

13. Duysinx P. and Bendsøe M.P. (1988) Control of local stresses intopology optimization of continuum structures,International Jour-nal for Numerical Methods in Engineering, 1998, 43, 1453-1478.

14. Duysinx P., Van Miegroet L., Jacobs T. and Fleury C. (2006) Gen-eralized Shape Optimization Using X-FEM and Level Set Methods,IUTAM Symposium on Topological Desgin, Optimization of Struc-tures, Machines and Materials, Bendsøe M. P., Olhoff N. and Sig-mund O. Eds. 2006, 23-32.

15. Fleury C. (1989) CONLIN: an efficient dual optimizer based onconvex approximation concepts.Structural Optimization. Vol. 1, pp81-89.

16. Guetari Y., Le Corre S., Moes N. (2005)Etude des possibilitesde la methode X-FEM pour la simulation numerique de la coupe.Mecanique & Industries 6: 315-319.

17. Haftka R.T. and Grandhi R.V. (1986). Structural shape optimiza-tion - A survey.Computer Methods in Applied Mchanics and Engi-neeringVol 57, 91–106.

18. Kim H., Querin O.M., Steven G.P., Xie Y.M. (2002). Improvingefficiency of evolutionary structural optimization by implementingfixed grid mesh.Structural and Multidisciplinary OptimizationVol24, No 6, 441–448.

19. Moes N., Dolbow J., Sukumar N. and Belytschko T. (1999) A fi-nite element method for crack growth without remeshing.Interna-tional journal for numerical methods in engineering1999, no. 46,131-150.

20. Norato J., Haber R., Tortorelli D. and Bendsøe M.P. (2004) A ge-ometry projection method for shape optimization.Int. J. Num. Meth.in Engng1999, 60(14): 2289-2312.

21. Novotny A.A., Feijoo R.A., Taroco E. and Padra C. (2003) Topo-logical Sensitivity Analysis.Comput. Methods Appl. Mech. Engrg.2003, 192, 803–829

22. OOFELIE, an Object Oriented Finite Element Code Led by Inter-active Execution. Internet resources: www.open-engineering.com.

23. Osher, S. and Sethian, J. A (1988) Fronts Propagating withCurvature-Dependent Speed: Algorithms Based on Hamilton-JacobiFormulations.Journal of Computational Physics1988, Vol. 79, 12–49.

24. Pedersen P. and Laursen C.L. (1983) Design for Minimum StressConcentration by Finite Element and Linear Programming.J. struc-tural Mech.Vol 10 No 4, 375–391.

25. Pedersen P. (1991). Optimal shape design with anisotropic mate-rials. NATO/DFG ASI Proceeding of vol. 1Optimization of LargeStructural Systems(Dir. G. Rozvany), Berchtesgaden, Germany,Sept. 23 - Oct.4 1991.

26. Pedersen P. (2000) On optimal shapes in materials and structures.Structural OptimizationVol 19, No 3, 169–182.

27. Pedersen P. (2003)Optimal Designs - Structures and Materials -Problems and Tools -, May 28, 2003.ISBN 87-90416-06-6

28. Pereira J., Fancello E. and Barcellos D.S. (2004) Topology opti-mization of continuum structures with material failure constraints.Structural and Multidisciplinary Optimization. Vol. 26, No 1-2, 50–66.

29. Peterson (1997)Stress Concentration factors, Second edition,John Wiley and Sons, 1997.

30. Rozvany G.I.N. (1996). Some Shortcomings in Michell’s TrussTheory.Structural Optimization, Vol.12, 244–250.

31. Sukumar N., Chopp D. L., Moes N. and Belytschko T. (2001)Modelling holes and inclusions by Level Set in the extended finiteelement method.Comput. Methods Appl. Mech. Engrg. 2001, 190,pp 6183-6200

32. Van Miegroet L., Moes N., Fleury C. and Duysinx P. (2005).Gen-eralized shape optimization based on the Level Set method. Proceed-ings of the 6th World Congress of Structural and MultidisciplinaryOptimization (J. Herskowitz ed.), Rio de Janeiro, Brazil, May 30 -June 3 2005.

33. Van Miegroet L., Lemaire E., Jacobs T., Dusyinx P. (2006). Stressconstrained optimization using X-FEM and Level Set description.Proceedings of the 3rd European Conference on Computational Me-chanics. Shape and Topological Sensitivity Analysis: Theory andApplications, Lisbon, Portugal, June 5 - 9 2006.

34. Wang M., Wang X. and Guo D. (2003). A Level Set method struc-tural topology optimization.Comp. Methods in Appl. Mech Engng.2003; 191:227-246.