Topological Sensitivity Analysis for Three-dimensional Linear Elasticity Problem

Transcription

1 Topological Sensitivity Analysis for Three-imensional Linear Elasticity Problem A.A. Novotny, R.A. Feijóo, E. Taroco Laboratório Nacional e Computação Científica LNCC/MCT, Av. Getúlio Vargas 333, Petrópolis - RJ, Brasil C. Para Centro Atómico Bariloche, 8400 Bariloche, Argentina September 7, 2005 Abstract In this work we use the Topological-Shape Sensitivity Metho to obtain the topological erivative for three-imensional linear elasticity problems, aopting the total potential energy as the cost function an the equilibrium equation as the constraint. This metho, base on classical shape sensitivity analysis, leas to a systematic proceure to compute the topological erivative. In particular, firstly we present the mechanical moel, later we perform the shape erivative of the corresponing cost function an, finally, we compute the final expression for the topological erivative using the Topological-Shape Sensitivity Metho an results from classical asymptotic analysis aroun spherical cavities. 1 Introuction The topological erivative has been recognize as a promising tool to solve topology optimization problems (see [3, where 425 references concerning topology optimization of continuum structures are inclue. See also [4, 8, 24 an references therein. Nevertheless, this concept is wier. In fact, the topological erivative may also be applie to solve inverse problems an to simulate physical phenomena with changes on the configuration of the omain of the problem. In aition, extension of the topological erivative in orer to inclue arbitrary shape holes an its applications to Laplace, Poisson, Helmoltz, Navier, Stokes an Navier-Stokes equations were evelope by Masmoui an his co-workers an by Sokolowsky an his co-workers (see, for instance, [18 for applications of the topological erivative in the context of topology esign an inverse problems. On the other han, although the topological erivative is extremely general, this concept may become restrictive ue to mathematical ifficulties involve in its calculation. However, several approaches to compute the topological erivative may be foun in the literature. In particular, we propose an alternative metho base on classical shape sensitivity analysis (see [1, 13, 14, 22, 25, 26, 27 an references therein. This approach, calle Topological-Shape Sensitivity Metho, was alreay applie in the following two-imensional problems: Poisson: steay-state heat conuction problem taking into account both homogeneous an nonhomogeneous Neumann an Dirichlet an also Robin bounary conitions on the hole [6, 20; Navier: plane stress an plane strain linear elasticity [7; Kirchhoff: thin plate bening problem [21; 1

2 More specifically, we have respectively consiere scalar secon-orer, vector secon-orer an scalar forth-orer PDE two-imensional problems. As a natural sequence of our work, therefore, in the present paper we apply the Topological-Shape Sensitivity Metho to compute the topological erivative in a vector secon-orer PDE three-imensional problem. In particular, we consier the three-imensional linear elasticity problem taking the total potential energy as the cost function an the state equation as the constraint. Therefore, for the sake of completeness, in Section 2 we present a short escription of the Topological-Shape Sensitivity Metho. In Section 3 we use this approach to compute the topological erivative for the problem uner consieration: in Section 3.1 we present the mechanical moel associate to three-imensional linear elasticity; in Section 3.2 we compute the shape erivative for this problem aopting the total potential energy as the cost function an the weak form of the state equation as the constraint an in Section 3.3, we compute the final expression for the topological erivative using classical asymptotic analysis aroun spherical cavities. Finally, it is important to mention that the obtaine result can be applie in several engineering problems such as topology optimization of three-imensional linear elastic structures. 2 Topological-Shape Sensitivity Metho Let us consier an open boune omain Ω R 3 with a smooth bounary Ω. If the omain Ω is perturbe by introucing a small hole at an arbitrary point ˆx Ω, we have a new omain Ω = Ω B, whose bounary is enote by Ω = Ω B, where B = B B is a ball of raius centere at point ˆx Ω. Therefore, we have the original omain without hole Ω an the new one Ω with a small hole B as shown in fig. (1. Thus, consiering a cost function ψ efine in both omains, its topological erivative is given in [8 as ψ (Ω ψ (Ω D T (ˆx = lim, (1 0 f ( where f ( is a negative function that ecreases monotonically so that f ( 0 with 0 +. n x^ n x^ B Figure 1: topological erivative concept Recently an alternative proceure to compute the topological erivative, calle Topological-Shape Sensitivity Metho, have been introuce by the authors [20. This approach makes use of the whole mathematical framework (an results evelope for shape sensitivity analysis (see, for instance, the pioneer work of Murat & Simon [17. The main result obtaine in [20 may be briefly summarize in the following Theorem (see also [6, 19: Theorem 1 Let f ( be a function chosen in orer to 0 < D T (ˆx <, then the topological erivative given by eq. (1 can be written as 1 D T (ˆx = lim 0 f ( τ ψ (Ω τ, (2 2

3 where τ R + is use to parameterize the omain. That is, for τ small enough, we have Ω τ := { x τ R 3 : x τ = x + τv, x Ω }. (3 Therefore, x τ = x an Ω τ = Ω. In aition, consiering that n is the outwar normal unit vector (see fig. 1, then we can efine the shape change velocity v, which is a smooth vector fiel in Ω assuming the following values on the bounary Ω { v = n on B (4 v = 0 on Ω an the shape sensitivity of the cost function in relation to the omain perturbation characterize by v is given by τ ψ (Ω τ ψ (Ω τ ψ(ω = lim. (5 τ 0 τ Proof. Re-writing eq. (1 like a Taylor series expansion we have ψ (Ω = ψ (Ω + f ( D T (ˆx + R(f(, (6 where R(f( contains all higher orer terms than f(, that is, it satisfies R(f( R(f( : lim = 0. (7 0 f( Let us take the erivative in relation to in both sies of eq. (6 to obtain ψ (Ω = f ( D T (ˆx + R (f( f (, (8 where, from eq. (5, we observe, for τ R + small enough, that ψ (Ω ψ (Ω τ ψ(ω = lim = τ 0 τ τ ψ (Ω τ. (9 Consiering the shape erivative of the cost function given by above expression (eq. 9 an rearranging eq. (8 we obtain 1 f ( τ ψ (Ω τ = D T (ˆx + R (f(. (10 Finally, taking the limit 0 in eq. (10 an consiering the efinition of R(f( given by eq. (7, we observe that 1 lim 0 R (f( = 0 D T (ˆx = lim 0 f ( τ ψ (Ω τ (11 an we get the proof of the Theorem Observe that the topological erivative given by eq. (1 can be seen as an extension of classical shape erivative, but with the mathematical ifficulty concerning the lack of homeomorphism between Ω an Ω. On the other han, the above Theorem highlights that the topological erivative may be obtaine by means of shape sensitivity analysis. Consequently, Topological-Shape Sensitivity Metho leas to a systematic approach to compute the topological erivative of the cost function ψ consiering eq. (2. In fact, the omains Ω an Ω τ have the same topology, that allow us to buil an homeomorphic map between them. In aition, Ω an Ω τ can be respectively seen as the material an the spatial configurations. Therefore, in orer to compute the shape erivative of the cost function (see eq. 5 we can use classical results from Continuum Mechanics like the Reynols transport theorem an the concept of material erivatives of spatial fiels [11. Finally, in this work we will show these features in the context of three-imensional elasticity. 3

4 3 The topological erivative in three-imensional linear elasticity Now, to highlight the potentialities of the Topological-Shape Sensitivity Metho, it will be applie to three-imensional linear elasticity problems consiering the total potential energy as the cost function an the equilibrium equation in its weak form as the constraint. Therefore, consiering the above problem, firstly we introuce the mechanical moel, later we perform the shape sensitivity of the aopte cost function with respect to the shape change of the hole an finally we compute the associate topological erivative. 3.1 Mechanical moel In this work, we consier a mechanical moel restricte to small eformation an isplacement an for the constitutive relation we aopt an isotropic linear elastic material. These assumptions lea to the classical three-imensional linear elasticity theory [10. In orer to compute the topological erivative associate to this problem, we nee to state the equilibrium equations in the original omain Ω (without hole an in the new one Ω (with hole Problem formulation in the original omain without hole The mechanical moel associate to the three-imensional linear elasticity problem can be state in its variational formulation as following: fin the isplacement vector fiel u U, such that T(u E(η = q η η V, (12 Ω Γ N where Ω represents a eformable boy with bounary Ω = Γ N Γ D, such that Γ N Γ D =, submitte to a set of surface forces q on the Neumann bounary Γ N an isplacement constraints ū on the Dirichlet bounary Γ D. Therefore, assuming that q L 2 (Γ N, the amissible functions set U an the amissible variations space V are given, respectively, by U = { u H 1 (Ω : u = u on Γ D }, V = { η H 1 (Ω : η = 0 on Γ D }. (13 In aition, the linearize Green eformation tensor E(u an the Cauchy stress tensor T(u are efine as E(u = 1 ( u + u T := s u an T(u = CE(u = C s u, (14 2 where C = C T is the elasticity tensor, that is, since I an II respectively are the secon an forth orer ientity tensors, E is the Young s moulus an ν is the Poisson s ratio, we have C = E (1 + ν (1 2ν [(1 2ν II + ν (I I C 1 = 1 [(1 + ν II ν (I I. (15 E The Euler-Lagrange equation associate to the above variational problem, eq. (12, is given by the following bounary value problem: fin u such that iv T(u = 0 in Ω u = ū on Γ D. (16 T(un = q on Γ N 4

5 3.1.2 Problem formulation in the new omain with hole The problem state in the original omain Ω can also be written in the omain Ω with a hole B. Therefore, assuming null forces on the hole, we have the following variational problem: fin the isplacement vector fiel u U, such that T (u E (η = q η η V. (17 Ω Γ N where the set U an the space V are respectively efine as U = { u H 1 (Ω : u = u on Γ D }, V = { η H 1 (Ω : η = 0 on Γ D }. (18 As seen before, the tensors E (u an T (u are respectively given as E (u = s u an T (u = C s u, (19 where the elasticity tensor C is efine in eq. (15. In accorance with the variational problem given by eq. (17, the natural bounary conition on B is T (u n = 0 (homogeneous Neumann conition. Therefore, the Euler-Lagrange equation associate to this new variational problem is given by the following bounary value problem: fin u such that 3.2 Shape sensitivity analysis iv T (u = 0 in Ω u = ū on Γ D T (u n = q on Γ N T (u n = 0 on B. (20 Let us choose the total potential energy store in the elastic soli uner analysis as the cost function. For simplicity, we assume that the external loa remains fixe uring the shape change. As it is wellknown, ifferent approaches can be use to obtain the shape erivative of the cost function. However, in our particular case, as the cost function is associate with the potential of the state equation, the irect ifferentiation metho will be aopte to compute its shape erivative. Therefore, consiering the total potential energy alreay written in the configuration Ω τ, efine through eq. (3, then ψ(ω τ := J τ (u τ can be expresse by J τ (u τ = 1 T τ (u τ E τ (u τ q u τ, (21 2 Ω τ Γ N where the tensors E τ (u τ an T τ (u τ are respectively given by with τ ( use to enote E τ (u τ = s τ u τ an T τ (u τ = C s τu τ, (22 τ ( := x τ (. (23 In aition, u τ is the solution of the variational problem efine in the configuration Ω τ, that is: fin the isplacement vector fiel u τ U τ such that T τ (u τ E τ (η τ = q η τ η τ V τ, (24 Ω τ Γ N where the set U τ an the space V τ are efine as U τ = { u τ H 1 (Ω τ : u τ = u on Γ D }, Vτ = { η τ H 1 (Ω τ : η τ = 0 on Γ D }. (25 5

6 Observe that from the well-known terminology of Continuum Mechanics, the omains Ω τ = Ω an Ω τ can be interprete as the material an the spatial configurations, respectively. Therefore, in orer to compute the shape erivative of the cost function J τ (u τ, at τ = 0, we may use the Reynols transport theorem an the concept of material erivatives of spatial fiels, that is [11 τ Ω τ ϕ τ where ϕ τ is a spatial scalar fiel an ( is use to enote = ( ϕ τ + ϕ τ iv v, (26 Ω ( := ( τ. (27 Taking into account the cost function efine through eq. (21 an assuming that the parameters E, ν, ū, an q are constants in relation to the perturbation represente by τ, we have, from eq. (26 an following Theorem 1, eqs. (3,4, that τ J τ (u τ = 1 [ 2 τ (T τ (u τ E τ (u τ + T (u E (u iv v q u, (28 Γ N Ω where, accoring to the material erivatives of spatial fiels [11, we have τ (T τ (u τ E τ (u τ = 2 (T (u E ( u T (u ( u v s. (29 Substituting eq. (29 in eq. (28 we obtain [ τ J τ (u τ 1 = Ω 2 T (u E (u iv v T (u ( u v s + T (u E ( u q u. (30 Ω Γ N Since u is the solution of the variational problem given by eq. (17 an consiering that u V, the eq. (30 becomes τ J τ (u τ = Σ v, (31 Ω where Σ is the Eshelby energy-momentum tensor (see, for instance, [5, 26 given in this particular case by Σ = 1 2 (T (u E (u I ( u T T (u. (32 Remark 2 It is interesting to observe that the Eshelby tensor Σ appears as a uality pair with respect to v, as can be seen in eq. (31. This fact allow us to interpret Σ as the set of configurational forces [12 associate to the change in the configuration of Ω characterize by v. Let us compute again the shape erivative of the cost function J τ (u τ efine through eq. (21, at τ = 0, using another version for the Reynols transport theorem [11, that is, ϕ τ τ = ϕ τ + ϕ τ (v n, (33 Ω τ Ω Ω where ϕ τ is a spatial scalar fiel an ( is use to enote ( := ( τ = ( τ xτ fixe. (34 6

7 Which results in τ J τ(u τ = 1 (T (u E (u (v n+ 1 2 Ω 2 where u can be written as [11 Ω τ (T τ (u τ E τ (u τ q u, (35 Γ N u = u + ( u v u = u ( u v. (36 Taking into account the notation introuce through eq. (34 an from eq. (36, we have τ (T τ (u τ E τ (u τ = 2T (u E (u where = 2 (T (u E ( u T (u E (ϕ, (37 ϕ = ( u v E (ϕ = s ϕ. (38 Substituting eq. (37 in eq. (35 we obtain τ J τ (u τ = 1 (T (u E (u (v n T (u E (ϕ 2 Ω Ω + T (u E ( u q u Ω Γ N = 1 (T (u E (u (n v T (u E (ϕ 2, (39 Ω Ω since u V an u is the solution of eq. (17. In aition, we observe that T (u E (ϕ = T (u ϕ n iv(t (u ϕ. (40 Ω Ω Ω Consiering this last result (eq. 40 in eq. (39 an taking into account again that u is the solution of eq. (20, we have τ J τ(u τ = 1 (T (u E (u (v n T (u ϕ 2 n Ω Ω [ 1 = Ω 2 (T (u E (u I ( u T T (u n v = Σ n v, (41 Ω remembering that Σ an ϕ are respectively given by eq. (32 an eq. (38. On the other han, taking into account eq. (31 an consiering the tensorial relation iv(σ T v = Σ v+ivσ v, (42 we can apply the ivergence theorem to obtain τ J τ (u τ = Σ n v ivσ v. (43 Ω Ω Thus, from eqs. (43,41 we observe that the Eshelby tensor has null ivergence. In fact, since v is an arbitrary velocity fiel, then from the funamental theorem of the calculus of variations it is straightforwar to verify that Ω ivσ v = 0 v ivσ = 0 (44 7

8 an the shape erivative of the cost function J τ (u τ efine through eq. (21, at τ = 0, becomes an integral efine on the bounary Ω, that is, τ J τ(u τ = Σ n v. (45 Ω In other wors, if the velocity fiel v is smooth enough in the omain Ω, then the shape sensitivity of the problem only epens on the efinition of this fiel on the bounary Ω. 3.3 Topological sensitivity analysis In orer to compute the topological erivative using the Topological-Shape Sensitivity Metho, we nee to substitute eq. (45 in the result of Theorem 1 (eq. 2. Therefore, from the efinition of the velocity fiel (eq. 4 an consiering the shape erivative of the cost function (eq. 45, we have that τ J τ (u τ = Σ n n, (46 B where Σ n n = 1 2 T (u E (u T (u n ( u n. (47 In aition, taking into account homogeneous Neumann bounary conition on the hole, we have, from eq. (20, that T (u n = 0 on B, therefore τ J τ (u τ = 1 T (u E (u. (48 2 B Finally, substituting eq. (48 in the result of the Theorem 1 (eq. 2, the topological erivative becomes D T (ˆx = 1 2 lim 1 0f T (u E (u. (49 ( B Consiering the inverse of the constitutive relation E (u = C 1 T (u (see eq. 15, then the integran of eq. (49 may be expresse as a function of the stress tensor as following T (u E (u = 1 E [ (1 + ν T (u T (u ν (trt (u 2. (50 Let us introuce a spherical coorinate system (r, θ, ϕ centere in ˆx (see fig. 2, then the stress tensor T (u = (T (u T, when efine on the bounary B, can be ecompose as T (u B = T rr + T rθ + T rϕ (e r e r + T rθ (e r e θ + T rϕ (e r e ϕ (e θ e r + T θθ (e θ e θ + T θϕ (e θ e ϕ (e ϕ e r + T θϕ (e ϕ e θ + T ϕϕ (e ϕ e ϕ, (51 where e r, e θ an e ϕ are the basis of the spherical coorinate system such that e r e r = e θ e θ = e ϕ e ϕ = 1 an e r e θ = e r e ϕ = e θ e ϕ = 0. (52 Since we have homogeneous Neumann bounary conition on B, then T (u n = 0 T (u e r = 0 on B. (53 From the ecomposition of the stress tensor shown in eq. (51 an taking into account eqs. (52,53, we observe that T (u e r = T rr e r + T rθ e θ + T rϕ e ϕ = 0 T rr = T rθ = T rϕ = 0. (54 8

9 Substituting eqs. (51,54 into eq. (50, the topological erivative given by eq. (49 may be written in terms of the components of the stress tensor in spherical coorinate, as following D T (ˆx = 1 2E lim 1 0f T (T θθ, T θϕ, T ϕϕ ( B = 1 2E lim 1 2π ( π 0f T (T θθ (, T θϕ, T ϕϕ 2 sin θθ ϕ, (55 where T (T θθ, T θϕ, T ϕϕ = (T θθ (T ϕϕ 2 2νT θθ T ϕϕ + 2(1 + ν(t θϕ 2. (56 Now, it is enough to calculate the limit 0 in the eq. (55 to obtain the final expression of the topological erivative. Thus, an asymptotic analysis [15 shall be performe in orer to know the behavior of the solution T θθ, T θϕ an T ϕϕ when 0. This behavior may be obtaine from the analytical solution for a stress istribution aroun a spherical voi in a three-imensional elastic boy [23, which is given, for any δ > 0 an at r =, by (see Appenix A T θθ = 3 1 { σ1 (u [ 3 5(1 2ν cos 2ϕ + 10 cos 2θ sin 2 ϕ B 4 7 5ν + σ 2 (u [ 3 + 5(1 2ν cos 2ϕ + 10 cos 2θ cos 2 ϕ +σ 3 (u [2(4 5ν 10 cos 2θ} + O( 1 δ, (57 = 15 1 ν B 2 7 5ν (σ 1 (u σ 2 (u cos θ sin 2ϕ + O( 1 δ, (58 B = 3 1 { σ1 (u [ 8 5ν + 5(2 ν cos 2ϕ + 10ν cos 2θ sin 2 ϕ 4 7 5ν + σ 2 (u [ 8 5ν 5(2 ν cos 2ϕ + 10ν cos 2θ cos 2 ϕ T θϕ T ϕϕ 2σ 3 (u (1 + 5ν cos 2θ} + O( 1 δ, (59 where σ 1 (u, σ 2 (u an σ 3 (u are the principal stress values of the tensor T (u, associate to the original omain without hole Ω (see eq. 12, evaluate in the point ˆx Ω, that is T (u ˆx. Substituting the asymptotic expansion given by eqs. (57,58,59 in eq. (55 we observe that function f ( must be chosen such that f ( = B = 4π 2 f ( = B = 4 3 π3 (60 in orer to take the limit 0 in eq. (55. Therefore, from this choice of function f ( shown in eq. (60, the final expression for the topological erivative becomes a scalar function that epens on the solution u associate to the original omain Ω (without hole, that is (see also [9, 16: in terms of the principal stress values σ 1 (u, σ 2 (u an σ 3 (u of tensor T (u D T (ˆx = 3 1 ν 4E 7 5ν [10(1 + νs 1(u (1 + 5νS 2 (u, (61 where S 1 (u an S 2 (u are respectively given by S 1 (u = σ 1 (u 2 + σ 2 (u 2 + σ 3 (u 2 an S 2 (u = (σ 1 (u + σ 2 (u + σ 3 (u 2 ; (62 in terms of the stress tensor T (u D T (ˆx = 3 1 ν [ 10(1 + νt (u T (u (1 + 5ν(trT (u 2 ; (63 4E 7 5ν 9

10 in terms of the stress T (u an strain E (u tensors D T (ˆx = 3 [ 1 ν 10T (u E (u 1 5ν trt (u tre (u 4 7 5ν 1 2ν ; (64 which was obtaine from a simple manipulation consiering the constitutive relation given by eq. (14. See also eq. (15. Remark 3 It is interesting to observe that if we take ν = 1/5 in eq. (64, the final expression for the topological erivative in terms of T (u an E (u becomes 4 Conclusions D T (ˆx = T (u E (u. (65 In this work, we have compute the topological erivative in three-imensional linear elasticity taking the total potential energy as the cost function an the state equation in its weak form as the constraint. The relationship between shape an topological erivatives was formally establishe in Theorem 1, leaing to the Topological-Shape Sensitivity Metho. Therefore, results from classical shape sensitivity analysis coul be use to compute the topological erivative in a systematic way. In particular, we have obtaine the explicit formula for the topological erivative for the problem uner consieration given by eqs. (61,63,64, whose result can be applie in several engineering problems such as topology optimization of three-imensional linear elastic structures (see, for instance, [2. Acknowlegments This research was partly supporte by CONICET (Argentina an the brazilian agencies CNPq/FAPERJ- PRONEX (E-26/ /2003. The support from these Institutions is greatly appreciate. References [1 J. Céa. Problems of Shape Optimal Design, in Haug & Céa [14. [2 J. Céa, S. Garreau, Ph. Guillaume & M. Masmoui. The Shape an Topological Optimizations Connection. Computer Methos in Applie Mechanics an Engineering, 188: , [3 H.A. Eschenauer & N. Olhoff. Topology Optimization of Continuum Structures: A Review. Applie Mechanics Review, 54: , [4 H.A. Eschenauer, V.V. Kobelev & A. Schumacher. Bubble Metho for Topology an Shape Optimization of Structures. Structural Optimization, 8:42-51, [5 J.D. Eshelby. The Elastic Energy-Momentum Tensor. Journal of Elasticity, 5: , [6 R.A. Feijóo, A.A. Novotny, E. Taroco & C. Para. The Topological Derivative for the Poisson s Problem. Mathematical Moels an Methos in Applie Sciences, 13-12:1-20, [7 R.A. Feijóo, A.A. Novotny, C. Para & E. Taroco. The Topological-Shape Sensitivity Metho an its Application in 2D Elasticity. To appear on Journal of Computational Methos in Sciences an Engineering. [8 S. Garreau, Ph. Guillaume & M. Masmoui. The Topological Graient. Research Report, UFR MIG, Université Paul Sabatier, Toulouse 3, France,

11 [9 S. Garreau, Ph. Guillaume & M. Masmoui. The Topological Asymptotic for PDE Systems: The Elasticity Case. SIAM Journal on Control an Optimization, 39: , [10 P. Germain & P. Muller. Introuction à la Mécanique es Milieux Continus. Masson, [11 M.E. Gurtin. An Introuction to Continuum Mechanics. Mathematics in Science an Engineering vol Acaemic Press, [12 M.E. Gurtin. Configurational Forces as Basic Concept of Continuum Physics. Applie Mathematical Sciences vol Springer-Verlag, [13 E.J. Haug, K.K. Choi & V. Komkov. Design Sensitivity Analysis of Structural Systems. Acaemic Press, [14 E.J. Haug & J. Céa. Proceeings: Optimization of Distribute Parameters Structures, Iowa, EUA, [15 A.M. Il in. Matching of Asymptotic Expansions of Solutions of Bounary Value Problems. Translations of Mathematical Monographs vol AMS, Provience, [16 T. Lewiński & J. Sokolowski. Energy change ue to the appearance of cavities in elastic solis. International Journal of Solis an Structures, 40: , [17 F. Murat & J. Simon. Sur le Contrôle par un Domaine Géométrique. Thesis, Université Pierre et Marie Curie, Paris VI, France, [18 P. Neittaanmäki et alli (es.. European Congress on Computational Methos in Applie Sciences an Engineering. Mini-symposium on Topological Sensitivity Analysis: Theory an Applications, Jyväskylä, Finlan, ECCOMAS [19 A.A. Novotny. Análise e Sensibiliae Topológica. Ph. D. Thesis, LNCC/MCT, Petrópolis - RJ, Brasil, 2003 ( novotny/principal.htm. [20 A.A. Novotny, R.A. Feijóo, C. Para & E. Taroco. Topological Sensitivity Analysis. Computer Methos in Applie Mechanics an Engineering, 192: , [21 A.A. Novotny, R.A. Feijóo, C. Para & E. Taroco. Topological Derivative for Linear Elastic Plate Bening Problems. Control & Cybernetics, 34-1: , [22 O. Pironneau. Optimal Shape Design for Elliptic Systems. Springer-Verlag, [23 M.A. Saowsky & E. Sternberg. Stress Concentration Aroun a Triaxial Ellipsoial Cavity. Journal of Applie Mechanics, , June [24 J. Sokolowski & A. Żochowski. On the Topological Derivative in Shape Optimization. SIAM Journal on Control an Optimization, 37: , [25 J. Sokolowski & J.-P. Zolésio. Introuction to Shape Optimization - Shape Sensitivity Analysis. Springer-Verlag, [26 E. Taroco. G.C. Buscaglia & R.A. Feijóo. Secon-Orer Shape Sensitivity Analysis for Nonlinear Problems. Structural Optimization, 15: , [27 J.-P. Zolézio. The Material Derivative (or Spee Metho for Shape Optimization. In Haug & Céa [14. 11

12 A Asymptotic Analysis In this appenix we present the analytical solution for the stress istribution aroun a spherical cavity in a three-imensional linear elastic boy, whose result was use to perform the asymptotic analysis in relation to the parameter in Section 3.3. Therefore, let us introuce a spherical coorinate system (r, θ, ϕ centere in ˆx, as shown in fig. 2. e r e e r n e 3 e 2 e1 x^ Figure 2: spherical coorinate system (r, θ, ϕ positione in the center ˆx of the ball B. Then, the stress istribution aroun the spherical cavity B is given, for any δ > 0, by T rr T rθ T rϕ T θθ T θϕ T ϕϕ = T1 rr + T2 rr + T3 rr + O(1 δ, = T1 rθ + T2 rθ + T3 rθ + O( 1 δ, = T rϕ 1 + T rϕ 2 + T rϕ 3 + O( 1 δ, = T1 θθ + T2 θθ + T3 θθ + O( 1 δ, = T θϕ 1 + T θϕ 2 + T θϕ 3 + O( 1 δ, = T ϕϕ 1 + T ϕϕ 2 + T ϕϕ 3 + O( 1 δ, (66 where T rr i, T rθ i, T rϕ i, T θθ i, T θϕ i an T ϕϕ i, for i = 1, 2, 3, are written, as: 12

13 for i = 1 T rr 1 = T rθ 1 = T rϕ 1 = T θθ 1 = T θϕ 1 = T ϕϕ 1 = for i = 2 T rr 2 = T rθ 2 = σ 1 σ 1 [ ( 3 12 r 3 5 r 5 [7 5ν + 5(1 + ν 3 + ( 10(5 ν 3 r r 5 σ 1 [7 5ν + 5(1 + ν 3 r r 5 σ 1 [ + (1 + 10ν ν r (28 20ν 10(1 2ν 3 r r 5 σ 1 [7 5ν + 5(1 2ν 3 r r 5 σ 1 [28 20ν + (11 10ν ν r σ 2 σ 2 ((1 2ν 3 r 3 5 r 5 cos 2θ sin 2 ϕ [ ( 3 12 r 3 5 r 5 [7 5ν + 5(1 + ν 3 r r 5 sin 2 θ sin 2 ϕ, (67 sin 2θ sin 2 ϕ, (68 sin θ sin 2ϕ, (69 (14 r 5 10ν + 25(1 2ν 3 cos 2θ sin 2 ϕ + ( 10(5 ν 3 r r 5 T rϕ 2 = σ 2 [7 5ν + 5(1 + ν 3 r r 5 T2 θθ σ 2 = [ + (1 + 10ν ν r (28 20ν 10(1 2ν 3 r r 5 T θϕ 2 = σ 2 [7 5ν + 5(1 2ν 3 r r 5 [28 20ν + (11 10ν 3 T ϕϕ 2 = σ ν 30 r ((1 2ν 3 r 3 5 r 5 cos 2θ cos 2 ϕ r 3 95 r 5 cos 2ϕ, (70 cos θ sin 2ϕ, (71 (28 r ν + 5(1 2ν 3 r r 5 cos 2ϕ, (72 r r 5 sin 2 θ cos 2 ϕ, (73 cos 2 ϕ sin 2θ, (74 sin θ sin 2ϕ, (75 (14 r ν + 25(1 2ν 3 cos 2θ cos 2 ϕ r 3 95 r 5 cos 2ϕ, (76 cos θ sin 2ϕ, (77 (28 r 5 20ν + 5(1 2ν 3 r r 5 cos 2ϕ, (78 13

14 for i = 3 T3 rr σ 3 = [ (38 10ν 3 r r 5 ( 10(5 ν 3 r r 5 T3 rθ = σ 3 [ + 10(1 + ν 3 r r 5 T rϕ 3 = 0, T θθ 3 = T θϕ 3 = 0, T ϕϕ 3 = σ 3 σ 3 sin 2 θ, (79 cos θ sin θ, (80 [(9 15ν (14 3 r r ν 5(1 2ν 3 r r 5 sin 2 θ [(9 15ν ((1 3 r r ν 3 r 3 5 r 5 sin 2 θ (81, (82 (83, (84 where σ 1, σ 2 an σ 3 are the principal stress values of the tensor T (u, associate to the original omain without hole Ω, evaluate in the point ˆx Ω, that is T (u ˆx. In other wors, the tensor T (u was iagonalize in the following way T (u ˆx = 3 σ i (e i e i, (85 where σ i is the eigen-value associate to the e i eigen-vector of the tensor T (u ˆx. i=1 Remark 4 It is important to mention that the stress istribution for i = 1, 2 was obtaine from a rotation of the stress istribution for i = 3. In aition, the erivation of this last result (for i = 3 can be foun in [23, for instance. 14