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TOPOLOGICAL OPTIMIZATION OF STRUCTURES SUBJECT TO VON

MISES STRESS CONSTRAINTS

S. AMSTUTZ AND A.A. NOVOTNY

Abstract. The topological asymptotic analysis provides the sensitivity of a given shape func-tional with respect to an infinitesimal domain perturbation. Therefore, this sensitivity canbe naturally used as a descent direction in a structural topology design problem. However,according to the literature concerning the topological derivative, only the classical approachbased on flexibility minimization for a given amount of material, without control on the stresslevel supported by the structural device, has been considered. Yet, one of the most importantrequirements in mechanical design is to find the lightest topology satisfying a material fail-ure criterion. In this paper, therefore, we introduce a class of penalty functionals that mimic

a pointwise constraint on the Von Mises stress field. The associated topological derivative isobtained for plane stress linear elasticity. Only the formal asymptotic expansion procedure ispresented, but full justifications can be easily deduced from existing works. Then, a topologyoptimization algorithm based on these concepts is proposed, that allows for treating local stresscriteria. Finally, this feature is shown through some numerical examples.

1. Introduction

Structural topology optimization is an expanding research field of computational mechanicswhich has been growing very rapidly in the last years. For a survey on topology optimizationmethods, the reader may refer to the review paper [17], or to the monographs [2, 11, 23]. A

relatively new approach for this kind of problem is based on the concept of topological derivative[15, 18, 33]. This derivative allows to quantify the sensitivity of a given shape functional withrespect to an infinitesimal topological domain perturbation, like typically the nucleation of ahole. Thus, the topological derivative has been successfully applied in the context of topologyoptimization [7, 13, 28], inverse problems [5, 12, 20, 29] and image processing [9, 25, 24, 26].Concerning the theoretical development of the topological asymptotic analysis, the reader mayrefer to [30], for instance. However, in the context of structural topology design, the topolog-ical derivative has been used as a descent direction only for the classical approach based onminimizing flexibility for a given amount of material. Although widely adopted, through thisformulation the stress level supported by the structural device cannot be controlled. This limi-tation is not admissible in several applications, because one of the most important requirementsin mechanical design is to find the lightest topology satisfying a material failure criterion. Even

the methods based on relaxed formulations [1, 10, 11] have been traditionally applied to min-imum compliance problems. In fact, only a few works dealing with local stress control can befound in the literature (see, for instance, [3, 4, 14, 16, 19, 31]). This can be explained by themathematical and numerical difficulties introduced by the large number of highly non-linearconstraints associated to local stress criteria.

Following the original ideas presented in [8] for the Laplace equation, in this paper we intro-duce a class of penalty functionals in order to approximate a pointwise constraint on the VonMises stress field. The associated topological derivative is then obtained for plane stress linearelasticity. We show that the obtained topological asymptotic expansion can be used within atopology optimization algorithm, which allows for treating local stress criteria. Finally, the effi-ciency of this algorithm is verified through some numerical examples. In particular, the obtained

structures are free of geometrical singularity, unlike what occurs by the compliance minimizationKey words and phrases. topological sensitivity, topological derivative, topology optimization, local stress cri-

teria, von Mises stress.

1

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approach. We recall that such singularities lead to stress concentrations which are highly unde-sirable in structural design. For a detailed description of the stress concentration phenomenon,the reader may refer to [22, 32].

The paper is organized as follows. The plane stress linear elasticity problem and the proposed

class of Von Mises stress penalty functionals are presented in Section 2. The topological deriv-ative associated to these functionals is calculated in Section 3. The proposed topology designalgorithm is described in Section 4. Finally, Section 5 is dedicated to the numerical experiments.

2. Problem statement

In this Section we introduce a class of Von Mises stress penalty functionals under plane stresslinear elasticity assumptions.

2.1. The constrained topology optimization problem. LetD be a bounded domain ofR2

with Lipschitz boundary . We assume that is split into three disjoint parts D, Nand 0,where

D is of nonzero measure, and

N is of class C1. We consider the topology optimization

problem:

MinimizeD

I(u) (2.1)

subject to the state equations

div((u)) = 0 in D,u = 0 on D,

(u)n = g on N,(u)n = 0 on 0,

(2.2)

and the constraint

M(u) M a.e. in D. (2.3)

The notations used above are the following. The system (2.2) is understood in the weak sense,as this will be the case throughout all the paper, and admits an unique solution

u V={u H1(D)2, u|D = 0}.

The material density is a piecewise constant function which takes two positive values:

=

in in ,out in D\ .

In the applications,D \ is occupied by a weak phase that approximates an empty region, thuswe assume that

out in. (2.4)The stress tensor (u), normalized to a unitary Young modulus, is related to the displacementfield u through the Hooke law:

(u) =C e(u),

where

e(u) =u + uT

2is the strain tensor, and

C= 2II+ (I I)

is the elasticity tensor. Here, I and IIare the second and fourth order identity tensors, respec-

tively, and the Lame coefficients and are given in plane stress by

= 1

2(1 + ), and =

1 2,

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where is the Poisson ratio. The Neumann data g is assumed to belong to L2(N)2. The Von

Mises stress M(u) is given by

M(u) = 12

B(u).(u) = 12

B(u).e(u),

whereB = 3II (I I) (2.5)

andB = CB = 6II+ ( 2)(I I). (2.6)

The set Dis an open subset ofD and M is a prescribed positive number. Finally, the objectivefunctional I :V R is assumed to admit a known topological derivative DTI as defined inSection 2.3.

2.2. Penalization of the constraint. Problem (2.1)-(2.3) is very difficult to address directlybecause of the pointwise constraint. Therefore we propose an approximation based on the

introduction of a penalty functional. Let : R+ R+ be a nondecreasing function of class C2

.To enable proper justifications of the subsequent analysis, we assume further that the derivatives and are bounded. We consider the penalty functional:

J(u) =

D

(1

2B(u).e(u))dx. (2.7)

Then, given a penalty coefficient >0, we define the penalized objective functional:

I(u) =I(u) + J(u).

Henceforth we shall solve the problem:

MinimizeD

I(u) subject to (2.2). (2.8)

We will see that solving (2.8) instead of (2.1)-(2.3) leads to feasible domains provided that and are appropriately chosen, namely that the two following conditions are fulfilled:

is large enough, admits a sharp variation around .

2.3. Topology perturbations. Given a pointx0 D \ and a radius >0, we consider acircular inclusion = B(x0, ), and we define the perturbed domain (see Fig. 1):

=

\ if x0 ,( ) D if x0 D\ .

We denote for simplicity (u , ) by (u, ) and (u, ) by (u0, 0). Then, for all [0, 1], can be expressed as:

=

0 in D\ ,1 in .

We note that 0 and 1 are two positive functions defined in D and constant in a neighborhoodofx0. For all 0, the state equations can be rewritten:

div ((u)) = 0 in D,

u = 0 on D,(u)n = g on N,

(u)n = 0 on 0.

(2.9)

In order to solve (2.8), we are looking for an asymptotic expansion, named as topological as-ymptotic expansion, of the form

I(u) I

(u0) =f()DTI

(x0) + o(f()),

where f : R+ R+ is a function that goes to zero with , and DTI : D R is the so-called

topological derivative of the functional I. Since such an expansion is assumed to be known for

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the objective functional I, we subsequently focus on the penalty functional J. We adopt thesimplified notation:

J(u) :=J(u) = D (1

2B(u).e(u))dx. (2.10)

N

D

0

D

D\D~

0

out=

in

0 =

1

out=

1 in

=

Figure 1. Sketch of the working domain.

3. Topological sensitivity analysis of the Von Mises stress penalty functional

In this section, the topological sensitivity analysis of the penalty functionalJ is carried out.We follow the approach described in [8] for the Laplace problem. Here, the calculations aremore technical, but the estimates of the remainders detached from the topological asymptoticexpansion are analogous. Hence we do not repeat these estimates. The reader interested in thecomplete proofs may refer to [8].

Possibly shifting the origin of the coordinate system, we assume henceforth for simplicity thatx0= 0.

3.1. A preliminary result. The reader interested in the proof of the proposition below mayrefer to [6].

Proposition 3.1. LetV be a Hilbert space and 0 > 0. For all [0, 0), consider a vectoru Vsolution of a variational problem of the form

a(u, v) =(v) v V, (3.1)

where a and are a bilinear form onV and a linear form onV, respectively. Consider also,for all [0, 0), a functional J : V R and a linear form L(u0) V. Suppose that thefollowing hypotheses hold.

(1) There exist two numbersa and and a function R+ f() R such that, when

goes to zero,(a a0)(u0, v) = f()a+ o(f()), (3.2)

( 0)(v) = f()+ o(f()), (3.3)

lim0

f() = 0, (3.4)

wherev V is an adjoint state satisfying

a(, v) = L(u0), V. (3.5)

(2) There exist two numbersJ1 andJ2 such that

J(u) = J(u0) + L(u0), u u0 + f()J1+ o(f()), (3.6)

J

(u0

) = J0

(u0

) + f()J2

+ o(f()). (3.7)

Then we have

J(u) J0(u0) =f()(a + J1+ J2) + o(f()).

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3.2. Adjoint state. The bilinear and linear forms associated with Problem (2.9) are classicallydefined in the space V={u H1(D)2, u|D = 0} by:

a(u, v) = D

(u).e(v)dx u, v V, (3.8)

(v) =

N

g.v ds v V. (3.9)

At the point u0 (unperturbed solution), the penalty functional admits the tangent linear ap-proximationL(u0) given by:

L(u0), =

D

(

1

2B(u0).e(u0))B(u0).e()dx V.

We define the function

k1= (

1

2B(u0).e(u0))D, (3.10)

where D

is the characteristic function ofD. Then the adjoint state is (a weak) solution of theboundary value problem:

div ((v)) = +div (k1B(u0)) in D,

v = 0 on D,(v)n = k1B(u0)n on N 0.

(3.11)

3.3. Regularity assumptions. We make the following assumptions.

(1) For any r1 > 0 there exists r2 (0, r1) such that every function u H1(D\ B(x0, r2))

2

satisfying

div (0(u)) = 0 in D\ B(x0, r2),u = 0 on D,

0(u)n = 0 on N 0

belongs to W1,4(D\ B(x0, r1))2.(2) The load g is such thatu0 W1,4(D)2.

Note that, by elliptic regularity, u0 and v0 are automatically of class C1,, >0, in the vicinityofx0 provided that x0 D\ \ D.

Remark 3.2. The above assumption is satisfied in many situations, including nonsmooth do-mains, like for instance in the following case:

D is a Lipschitz polygon, N D= and D D= , the interface \ D is the disjoint union of smooth simple arcs, if a junction point between the interface and Dbelongs toD, then the Young modulus

distribution around this point is quasi-monotone (see the definition in [27]); in particular,if only one arc touches D at this point, it is sufficient that the angle defined by thesecurves in D \ is less than .

We refer to [27] and the references therein for justifications and extensions.

3.4. Variation of the bilinear form. In order to apply Proposition 3.1, we need to obtain aclosed form for the leading term of the quantity:

(a a0)(u0, v) =

(1 0)(u0).e(v)dx. (3.12)

In the course of the analysis, the remainders detached from this expression will be denoted byEi(), i= 1, 2,... By setting v =v v0 and assuming that is sufficiently small so that is

constant in , we obtain:

(a a0)(u0, v) = (1 0)(x0)

(u0).e(v0)dx+

(u0).e(v)dx

.

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For the readers convenience, the values of0(x0) and 1(x0) are reported in Table 1 (see alsoFig. 1). Since u0 and v0 are smooth in the vicinity ofx0, we approximate (u0) and e(v0) in

x0 0(x0) 1(x0)

in outD\ out in

Table 1. Coefficients 0(x0) and 1(x0) according to the location ofx0.

the first integral by their values at the point x0, and write:

(a a0)(u0, v) = (1 0)(x0)

2(u0)(x0).e(v0)(x0) +

(u0).e(v)dx + E1()

.

As v is solution of the adjoint equation (3.11), then the function v solves

div((v)) = 0 in (D\ ),[(v)n] = (1 0) (k1B(u0)n + (v0)n) on ,v = 0 on D,

(v)n = 0 on N 0,

(3.13)

where [(v)n] H1/2()2 denotes the jump of the normal stress through the interface. We recall that, as before, the boundary value problem (3.13) is to be understood in theweak sense for v H

1(D)2. We set S= S1+ S2, with

S1= k1(x0)B(u0)(x0) and S2 = (v0)(x0).

We approximate (v) by(hS) solution of the auxiliary problem:

div (hS

) = 0 in (R

2 \ ),[(hS)n] = (1 0)(x0)Sn on ,

(hS ) 0 at .(3.14)

In the present case of a circular inclusion, the tensor (hS) admits the following expression in apolar coordinate system (r, ):

for r

r(r, ) = (1+ 2) 1

1 +

2

r2

1

1 +

4

2

r2 3

4

r4

(1cos 2+ 2cos 2(+ )) ,(3.15)

(r, ) = (1+ 2) 1

1 +

2

r2 3

1

1 +

4

r4(1cos 2+ 2cos 2(+ )) , (3.16)

r(r, ) = 1 1 +

2 2

r2 3 4

r4

(1sin 2+ 2sin 2(+ )) , (3.17)

for 0< r <

r(r, ) = (1+ 2) 1

1 + +

1

1 + (1cos 2+ 2cos 2(+ )) , (3.18)

(r, ) = (1+ 2) 1

1 +

1

1 + (1cos 2+ 2cos 2(+ )) , (3.19)

r(r, ) = 1

1 + (1sin 2+ 2sin 2(+ )) , (3.20)

Some terms in the above formulas require explanation. The parameter denotes the angle

between the eigenvectors of tensors S1 and S2,

i=1

2(siI+ s

iII) and i =

1

2(siI s

iII), i= 1, 2,

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where siI and siIIare the eigenvalues of tensors Si for i= 1, 2. In addition, the constants and

are respectively given by

=1 +

1 , =

3

1 + ,

and is the contrast, that is, = 1(x0)/0(x0).From these elements, we obtain successively:

(u0).e(v)dx=

(v).e(u0)dx=

(hS).e(u0)dx+ E2().

Then approximating e(u0) in by its value at x0 and calculating the resulting integral withthe help of the expressions (3.18)-(3.20) yields:

(u0).e(v)dx =

(hS).e(u0)(x0)dx+ E2() + E3()

= 2 (k1T B(u0).e(u0) + T (u0).e(v0)) (x0) + E2() + E3(),

with

= 1 0

1+ 0(x0) and T =II+

12

1 +

I I. (3.21)

Finally, the variation of the bilinear form can be written in the form:

(a a0)(u0, v) =2(1 0)(x0)

k1B(u0).e(u0) +

1

2k1

1 + trB(u0)tre(u0)

+ 1

1(u0).e(v0) +

1

2

1 + tr(u0)tre(v0)

(x0) + (1 0)(x0)

3i=1

Ei(). (3.22)

3.5. Variation of the linear form. Since here is independent of, it follows trivially that

( 0)(v) = 0. (3.23)

3.6. Partial variation of the penalty functional with respect to the state. We nowstudy the variation:

VJ1() := J(u) J(u0) L(u0), u u0

=

D

(

1

2B(u).e(u)) (

1

2B(u0).e(u0))

(1

2B(u0).e(u0))B(u0).e(u u0)

dx.

By setting u= u u0, we can write:

VJ1() =

D

(

1

2B(u0).e(u0) + B(u0).e(u) +

1

2B(u).e(u)) (

1

2B(u0).e(u0))

(

1

2 B(u0).e(u0))B(u0).e(u)

dx. (3.24)

Since u is solution of the state equation (2.9), then by difference we find that u solves:

div((u)) = 0 in (D\ ),[(u)n] = (1 0)(u0)n on ,

u = 0 on D,(u)n = 0 on N 0.

(3.25)

By setting nowS= (u0)(x0), we approximate ubyhS solution of the auxiliary problem (3.14).

It comes:

VJ1() = D

(1

2

B(u0).e(u0) + B(u0).e(hS) +

1

2

B(hS ).e(hS)) (

1

2

B(u0).e(u0))

(1

2B(u0).e(u0))B(u0).e(h

S)

dx+ E4(). (3.26)

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Ifx0 D\D, we obtain easily, using a Taylor expansion of and the estimate |(hS)(x)|= O(

2)which holds uniformly with respect to x a fixed distance away from x0, that VJ1() = o(

2).

Thus we assume that x0 D (the special case where x0 D is not treated). In view of the

decay of(h

S

) at infinity and the regularity ofu0 near x0, we write

VJ1() =

R2

(

1

2B(u0)(x0).e(u0)(x0) + B(u0)(x0).e(h

S) +

1

2B(hS ).e(h

S))

(1

2B(u0)(x0).e(u0)(x0))

(1

2B(u0)(x0).e(u0)(x0))B(u0)(x0).e(h

S)

dx+E4()+E5(),

with (x) =1(x0) ifx , (x) =0(x0) otherwise. The above expression can be rewritten

as

VJ1() =

R2

(

1

2BS.S+ BS.(hS) +

1

2B(hS).(h

S)) (

1

2BS.S)

(

1

2

BS.S)BS.(h

S

)

dx + E4() + E5().

We denote byVJ11() andVJ12() the parts of the above integral computed over and R2 \ ,

respectively. Using the expressions (3.18)-(3.20), we find

VJ11() =21(x0)

(

1

2BS.S TBS.S+ 2

1

2TBTS.S) (

1

2BS.S) + (

1

2BS.S)TBS.S

.

Next, we define the function independent of

S (x) =(hS)(x). (3.27)

A change of variable yields

VJ12() =

2 R2\ 0(x0)

(

1

2

BS.S+BS.

S

+

1

2

B

S

.

S

)(

1

2

BS.S)

(

1

2

BS.S)BS.

S

dx.

We set

(S) =

R2\

(

1

2BS.S+BS.S +

1

2BS .

S )(

1

2BS.S)(

1

2BS.S)(BS.S+

1

2BS .

S )

dx.

(3.28)

The extra term 12BS .

S has been added so that (S) vanishes whenever is linear. Thus

we have

VJ12() =20(x0)

(S) +

1

2(

1

2BS.S)

R2\

BS .Sdx

.

Using the expressions (3.15)-(3.17), a symbolic calculation of the above integral provides

VJ12() =20(x0)

(S) +

14

2k1(x0)

5(2S.S tr2S) + 3

1 + 1 +

2

tr2S

.

Besides, after a change of variable and rearrangements, (S) reduces to

(S) =

10

0

1

t2[(s2M+ (t, )) (s

2M)

(s2M)(t, )]ddt, (3.29)

where

(t, ) =t

2

(s2I s

2II)(2 + 3

1 +

1 + )cos + 3(sI sII)

2(2 3t)cos2

+2t2

43(s

I+ s

II)2(

1 +

1 + )2 + (s

I s

II)2(3(2 3t)2 + 4 cos2 ) + 6

1 +

1 + (s2I s2

II)(2 3t)cos ,

s2M =1

2BS.S.

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Finally we obtain:

V J1() =1(x0)

(

1

2BS.S TBS.S+ 2

1

2TBTS.S) (

1

2BS.S) + (

1

2BS.S)TBS.S

+ 0(x0)

(S) +

1

42k1(x0)

5(2S.S tr2S) + 3

1 +

1 +

2

tr2S

+ E4() + E5().

(3.30)

3.7. Partial variation of the penalty functional with respect to the domain. The lastterm is treated as follows:

V J2() := J(u0) J0(u0)

=

D

(1 0)(1

2B(u0).e(u0))dx

= 2D(x0)(1 0)(x0)(1

2

B(u0)(x0).e(u0)(x0)) + E6().

3.8. Topological derivative. Like in [8] for the Laplace equation, we can prove that the re-minders Ei(), i = 1, ..., 6 behave like o(

2). Therefore, after summation of the different termsaccording to Proposition 3.1 and a few simplifications, we arrive at the final formula for thetopological asymptotic expansion of the penalty functional. It is given by

J(u) J0(u0) =2DTJ(x0) + o(

2)

with the topological derivative

DTJ = (1 0) [k1TBS.E+ (T II)S.Ea]

+ 1D(1

2BS.S TBS.S+ 2

1

2TBTS.S) + k1TBS.S

+ 0D

(S) +

1

42k1

5(2S.S tr2S) + 3

1 +

1 +

2tr2S

D0(

1

2BS.S). (3.31)

Formula (3.31) is valid for all x0 D \ D \ . We recall that and Tare given by (3.21); B,B, k1 and (S) are respectively given by (2.5), (2.6), (3.10), (3.29) and

S= (u0), E= e(u0), Ea= e(v0).

The coefficients 0 and 1 are given by Table 1. Moreover, u0 =u is the solution of the stateequation (2.2) and v0= v is the solution of the adjoint equation (3.11) for = 0, that is,

div (0(v0)) = +div (0k1B(u0)) in D,

v0 = 0 on D,

0(v0)n = 0k1B(u0)n on N 0.

(3.32)

Remark 3.3. For the particular case in which = 1/3, Formula (3.31) admits the simplerexpression:

DTJ = (1 0)

4k1s2M (1 2)S.Ea

+ 1D

((1 2)2s2M) + 4k1s

2M

+ 0D

(S) +

1

22k1(5S.S tr

2S)

0D(s

2M),

where (S) is given by (3.29) with

(t, ) =t

25(s2

I

s2

II

)cos + 3(sI sII)2(2 3t)cos2

+ 2t2

4

3(sI+ sII)

2 + (sI sII)2(3(2 3t)2 + 4 cos2 ) + 6(s2I s

2II)(2 3t)cos

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and

= 1 021+ 0

(x0).

Remark 3.4. Formula (3.31) has some similarity with results proved in [30], where a theoryis developed for a broad class of elliptic state equations and shape functionals in three spacedimensions, then it is applied to the linear elasticity case. However, the shape functionalsaddressed in this context are linear or quadratic in (u), and there is no background material(the inclusions are Neumann holes).

4. A topology design algorithm

Given a real parameter p 1, we consider the penalty function (see Fig. 2):

p(t) = p( t

2M)

withp : R+ R+,

t (1 + tp)1/p 1.

It is clear that this function satisfies the required assumptions (see Section 2.2). The penalized

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 2. Function pn for pn= 2n, n = 0, ..., 6.

problem that we shall solve reads:

Minimize

D

I(u) =I(u) + D

p(M(u)2)dx subject to (2.2). (4.1)

In practice, p must be chosen as large as possible, provided that the resolution of (4.1) canaccommodate with the sharp variation of p around 1. In all the numerical examples, we takethe valuep = 32 which, after several trials, proved to be a good compromise. In order to reducethe computer time, the function has been tabulated (see Fig. 3).

The unconstrained minimization problem (4.1) is solved by using the algorithm devised in [7].We briefly describe this algorithm here. It relies on a level-set domain representation and theapproximation of topological optimality conditions by a fixed point method. Thus, the currentdomain is characterized by a function L2(D) such that ={x D, (x)< 0}andD\ ={x D, (x) > 0}. We compute the topological derivative DTI

() = DTI() +DTJ()where DTJ() is given by formula (3.31) with 0 and 1 chosen according to Table 1. Then

we set G(x) = DTI

()(x) if x D\ and G(x) = DTI

()(x) if x . We define theequivalence relation on L2(D):

> 0, = .

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0

0.5

1

1.5

2

2.5

3 0

0.5

1

1.5

2

2.5

3

2

0

2

4

6

8

Figure 3. Function for = 1 (hole creation), p = 8 and = 0.3, with

sI+ sIIin abscissa and sI sIIin ordinate.

Clearly, the relation G is a sufficient optimality condition for the class of perturbationsunder consideration. We construct successive approximations of this condition by means of asequence (n)nN verifying

0 L2(D),

n+1 co(n, Gn)n N.

Above, the convex hull co(n, Gn) applies to the equivalence classes, namely half-lines. Choosingrepresentatives of unitary norm for n, n+1 and Gn, we obtain the algorithm:

0 S,n+1=

1

sin n[sin((1 n)n)n+ sin(nn)Gn] n N.

The notations are the following: Sis the unit sphere ofL2(D),n= arccos Gn,nGnn

is the angle

between the vectors Gn and n, and n [0, 1] is a step which is determined by a line searchin order to decrease the penalized objective functional. The iterations are stopped when thisdecrease becomes too small. At this stage, if the optimality condition is not approximated ina satisfactory manner (namely the angle n is too large), an adaptive mesh refinement using aresidual based a posteriori error estimate on the solution un is performed and the algorithm iscontinued.

5. Numerical Experiments

Given a fixed multiplier >0, we consider the objective functional

I(u) =|| + K(u),

with || the area of and the compliance

K(u) =

N

g.uds.

Unless otherwise specified, the domain D is equal to the whole computational domain D. The

material densities are in = 1 and out = 103. The Poisson ratio is = 0.3. The topologicalderivative of the area is obvious, and that of the compliance is known (see [6, 21]). In each case,the initial guess is the full domain 0= D.

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5.1. Bar. Our first example is a bar under traction (see Fig. 4). The computational domainis the unit square and the load is uniformly distributed along two line segments of length 0 .4.For comparison, we first address the unconstrained situation (i.e. = 0). We choose = 1,so that the theoretical optimal domain is known as the horizontal central band of width 0 .4.

Next, we want to retrieve this domain with the local stress constraint M(u) M= 1, and comprised between 0 and 1. We choose = 0.2 and = 1, then = 10. We observe thatthe obtained domains satisfy the constraint, but they are slightly bigger than the theoreticaloptimum. This is a consequence of the fact that p(s) is slightly positive for s

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Figure 6. Michells structure: optimal domain for = 100 (left) and Von Misesstress distribution, optimal domain for = 500 (right).

Area Compliance I(u) maxM(u)M

CPU time (s) Mesh

0 0.078 1007 11453 1901 1.68 188 13599

100 0.04 1007 11979 1486 1.02 319 17124500 0.04 1062 11616 1527 0.97 176 13599Table 2. Michells structure.

5.3. Eyebar. In this example, the working domain is a rectangle of size 16 8 deprived of acircular hole of radius 1.5. On the border of this hole, a horizontal load of density g(x, y) =((y2 1.52)x0, 0) is applied, where (x, y) denotes a local coordinate system whose origin is atthe center of the hole. On the right side, the structure is clamped along a segment of length 2 (see

Fig. 7). Here and in the subsequent examples, the subdomain D is represented in gray. Again,we show results obtained in the unconstrained ( = 0) and constrained ( = 500, M = 5)cases, with adjusted in order to obtain similar areas (see Fig. 7 and Table 3).

Figure 7. Eyebar: boundary conditions and obtained domains for = 0 and= 500.

Area Compliance maxM(u)M

CPU time (s) Mesh

0 0.25 46.3 187 1.31 169 13520500 0.2 46.4 193 0.99 206 13520

Table 3. Eyebar.

5.4. L-beam. We now turn to a classical problem containing a geometrical singularity, namelythe L-shaped beam (see Fig. 8 and 9 and Table 4). The length of the two branches is 2.5. Thestructure is clamped at the top, and a unitary pointwise force is applied at the middle of the

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right tip. We first show a result obtained in the unconstrained case ( = 0, = 0.01). Thenwe take the parameters = 104, = 0.01 and three different values for M: 40, 30 and 25. Weobserve that, in these last three cases, the reentrant corner is rounded, unlike what occurs inthe first case, when minimizing the compliance without stress constraint.

Figure 8. L-beam: boundary conditions and obtained design in the uncon-

strained case (top), obtained designs with the penalization for M= 40, M= 30and M= 25 (bottom, from left to right).

0 10 20 30 40 50 60 701.5

2

2.5

3

3.5

4

4.5

0 10 20 30 40 50 60 70100

120

140

160

180

200

220

0 10 20 30 40 50 60 700

1

2

3

x 104

Figure 9. L-beam with M= 30: convergence history for the area, the compli-

ance and the area of the region in which the constraint is violated (from left toright).

M Area Compliance maxM(u)M

CPU time (s) Mesh

104 102 40 1.74 186 1.01 891 26693104 102 30 1.93 203 0.99 672 26708104 102 25 2.05 181 1.01 536 26678

Table 4. L-beam.

5.5. U-beam. This last example consists in an U-shaped structure included in a box of size3 2.5 (see Fig. 10). We show a result obtained with the parameters = 0.3, M = 4 and

= 104. We get maxM(u)M

= 1.02 on a mesh of 28385 nodes, in 363s of CPU time.

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Figure 10. U-beam: boundary conditions, obtained design and zoom near areentrant corner.

5.6. Conclusion. In the above examples, we have minimized a linear combination of the area

and the compliance of an elastic structure while prescribing an upper bound on the Von Misesstress at each point. The two basic ingredients of our approach are the use of the topologicalderivative as a descent direction and of a penalty method for the constraint imposition. This isin contrast with the existing literature on the topic, where either dual methods are implemented[14, 16, 19, 31], with the well-known difficulties related to the irregularity of the Lagrangemultiplier, or a simple power law penalization is considered [3, 4], generally leading to unfeasibledomains. Furthermore, the topological derivative does not rely on any relaxation, which isa quite delicate issue for local criteria. Finally, the computational cost of this algorithm isremarkably low.

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Laboratoire danalyse non lineaire et geometrie, Faculte des Sciences, 33 rue Louis Pasteur,84000 Avignon, France.

E-mail address: samuel.amstutz@univ-avignon.fr

Laboratorio Nacional de Computacao Cientfica LNCC/MCT, Coordenacao de MatematicaAplicada e Computacional, Av. Getulio Vargas 333, 25651-075 Petropolis - RJ, Brasil.

E-mail address: novotny@lncc.br