Topological optimization via cost penalization

Transcription

1 Topological optimization via cost penalization arxiv:8.508v [math.oc] 8 Nov 08 Cornel Marius Murea, an Tiba épartement de Mathématiques, IRIMAS, Université de Haute Alsace, France, cornel.murea@uha.fr Institute of Mathematics Romanian Academy and Academy of Romanian Scientists, Bucharest, Romania, dan.tiba@imar.ro Abstract We consider general shape optimization problems governed by irichlet boundary value problems. The proposed approach may be extended to other boundary conditions as well. It is based on a recent representation result for implicitly defined manifolds, due to the authors, and it is formulated as an optimal control problem. The discretized approximating problem is introduced and we give an explicit construction of the associated discrete gradient. Some numerical examples are also indicated. Keywords: geometric optimization; optimal design; topological variations; optimal control methods; discrete gradient Introduction Shape optimization is a relatively young branch of mathematics, with important modern applications in engineering and design. Certain optimization problems in mechanics, thickness optimization for plate or rods, geometric optimization of shells, curved rods, drag minimization in fluid mechanics, etc are some examples. Many appear naturally in the form of control by coefficients problems, due to the formulation of the mechanical models, with the geometric characteristics entering the coefficients of the differential operators. See [5], Ch. 6, where such questions are discussed in details. It is the aim of this article to develop an optimal control approach, using penalization methods, to general shape optimization problems as investigated in [0], [], [5], [0], [8], etc. We underline that our methodology allows simultaneous topological and boundary variations.

2 Here, we fix our attention on the case of irichlet boundary conditions and we study the typical problem denoted by P: min j x, y Ω x dx, Ω E. y Ω = f in Ω,. y Ω = 0 on Ω,.3 where E R are given bounded domains, is of class C, and the minimization parameter, the unknown domain Ω, satisfies E Ω and other possible conditions defining a class of admissible domains. Notice that the case of dimension two is of interest in shape optimization. Moreover, f L, j : R R is some Carathéodory mapping. More assumptions or constraints will be imposed later. Other boundary conditions or differential operators may be handled as well via this control approach and we shall examine such questions in a subsequent paper. For fundamental properties and methods in optimal control theory, we quote [], [7], [], [6]. The problem.-.3 and its approximation are strongly non convex and challenging both from the numerical and theoretical points of view. The investigation from this paper continues the one in [5] and is essentially based on the recent implicit parametrization method as developed in [4], [8], [3], that provides an efficient analytic representation of the unknown domains. The Hamiltonian approach to implicitly defined manifolds will be briefly recalled together with other preliminaries in Section. The precise formulation of the problem and its approximation is analyzed in Section 3 together with its differentiability properties. In Section 4, we study the discretized version and find the general form of the discrete gradient. The last section is devoted to some numerical experiments, using this paper approach. The method studied in this paper has a certain complexity due to the use of Hamiltonian systems and its main advantage is the possibility to extend it to other boundary conditions or boundary observation problems. This will be performed in a subsequent article. Preliminaries Consider the Hamiltonian system x t = g x x t, x t, t I,. x t = g x x t, x t, t I,. x 0, x 0 = x 0, x 0,.3

3 where g : R is in C, x 0, x 0 and I is the local existence interval for..3, around the origin, obtained via the Peano theorem. The conservation property [3] of the Hamiltonian gives: Proposition. We have In the sequel, we assume that g x t, x t = g x 0, x 0, t I..4 g x 0, x 0 = 0, g x 0, x Under condition.5, i.e. in the noncritical case, it is know that the solution of..3 is also unique by applying the implicit functions theorem to.4, [4]. Remark. In higher dimension, iterated Hamiltonian systems were introduced in [4] and uniqueness and regularity properties are proved. Some relevant examples in dimension three are discussed in [8]. In the critical case, generalized solutions can be obtained [3], [4]. We define now the family F of admissible functions g C that satisfy the conditions: gx, x > 0, on,.6 gx, x > 0, on G = {x, x ; gx, x = 0},.7 gx, x < 0, on E..8 Condition.6 says that G = and condition.7 is an extension of.5. In fact, it is related to the hypothesis on the non existence of equilibrium points in the Poincare-Bendixson theorem, [], Ch. 0, and the same is valid for the next proposition. The family F defined by.6.8 is obviously very rich, but it is not closed we have strict inequalities. Our approach here, gives a descent algorithm for the shape optimization problem P and existence of optimal shapes is not discussed. Following [5], we have the following two propositions: Proposition. Under hypotheses.6,.7, G is a finite union of closed curves of class C, without self intersections, parametrized by..3, when some initial point x 0, x 0 is chosen on each component of G. If r F as well, we define the perturbed set G λ = {x, x ; g + λrx, x = 0, λ R}..9 We also introduce the neighborhood V, > 0 V = {x, x ; d[x, x, G] < },.0 where d[x, x, G] is the distance from a point to G. 3

4 Proposition.3 If > 0 is small enough, there is λ > 0 such that, for λ < λ we have G λ V and G λ is a finite union of class C closed curves. Remark. The inclusion G λ V shows that G λ G for λ 0, in the Hausdorff- Pompeiu metric [5]. 3 The optimization problem and its approximation Starting from the family F of admissible functions, we define the family O of admissible domains in the shape optimization problem.-.3 as the connected component of the open set Ω g, g F Ω g = {x, x ; gx, x < 0} 3. that contains E. Clearly E Ω g by.8. Notice as well that the domain Ω g defined by 3. we use this notation for the domain as well is not simply connected, in general. This is the reason why the approach to.-.3 that we discuss here is related to topological optimization in optimal design problems. But, it also combines topological and boundary variations. The penalized problem, > 0, is given by: { } subject to min g F, u L E j x, y x dx + I g y z g t z gt dt 3. y = f + g + +u, in, 3.3 y = 0, on, 3.4 where z g = z g, z g satisfies the Hamiltonian system.-.3 in I g with some x 0, x 0 \ E such that gx 0, x 0 = 0 z g g t = z g t, t I g, 3.5 x z g g t = z g t, t I g, 3.6 x z g 0 = x 0, x and I g = [0, T g ] is the period interval for , due to Proposition.. The problem is an optimal control problem with controls g F and u L distributed in. The state is given by [y, z g, z g] H C I g. We also have y H 0. In case the corresponding domain Ω g is not simply connected, in 4

5 3.7 one has to choose initial conditions on each component of Ω g and the penalization term becomes a finite sum due to Proposition.. The method enters the class of fixed domain methods in shape optimization and can be compared with [3], [4], [7]. It is essentially different from the level set method of Osher and Sethian [9], Allaire [] or the SIMP approach of Bendsoe and Sigmund [3]. From the computational point of view, it is easy to find initial condition 3.7 on each component of G and the corresponding period intervals I g associated to We have the following general suboptimality property: Proposition 3. Let j, be a Carathéodory function on R, bounded from below by a constant. enote by [yn, gn, u n] a minimizing sequence in Then, on a subsequence nm the not necessary admissible pairs [Ω g nm, ynm ] give a minimizing sequence in., ynm satisfies. in {x, x ; gx, x < } and.3 is fulfilled with a perturbation of order /. Proof. Let [y gm, g m ] H Ω gm F be a minimizing sequence in the problem.-.3, 3.. By the trace theorem, since Ω gm and are at least C, under our assumptions, there is ỹ gm H H0, not unique, such that ỹ gm = y gm in Ω gm. We define the control u gm L as following: u gm = 0, in Ω gm, u gm = ỹ g m + f, in \ Ω g m + gm, + where Ω gm is the open set defined in 3.. Notice that on the second line in the above formula, we have no singularity. It is clear that the triple [ỹ gm, g m, u gm ] is admissible for the problem with the same cost as in the original problem.-.3 since the penalization term in 3. is null due to the boundary condition.3 satisfied by ỹ gm. Consequently, there is nm sufficient big, such that j x, ynmx dx + y E nmz g nm t z g 3.8 nmt dt I g nm j x, ỹ gm x dx = j x, y gm x dx infp. E Since j is bounded from below, we get from 3.8: E Ω gm y nm dσ C 3.9 with C a constant independent of > 0. Then, 3.9 shows that.3 is fulfilled with a perturbation of order /. 5

6 Moreover, again by 3.8, we see the minimizing property of {ynm } in the original problem P. We notice that in the state equation 3.3, the right-hand side coincides with f in the set {x, x ; gx, x < }, which is an approximation of Ω g nm. Namely, we notice that for any g F, the open sets {x, x ; gx, x < } form a nondecreasing sequence contained in Ω g, when 0. Take x, x such that gx, x = 0 and take some sequence x n, x n x, x, x n, x n Ω g. We have gx n, x n < 0 by 3. and gx n, x n 0. Moreover, x n, x n Ω n = Ω g+n, for n > 0 sufficiently small. Consequently, we have the desired convergence property by [5], p. 46. This ends the proof. Remark 3. A detailed study of the approximation properties in the penalized problem is performed in [5], in a slightly different case. We consider now variations u + λv, g + λr, where λ R, u, v L, g, r F, gx 0, x 0 = rx 0, x 0 = 0. Notice that u + λv L and g + λr F for λ sufficiently small. The conditions.6,.7,.8 from the definition of F are satisfied for λ sufficiently small depending on g due to the Weierstrass theorem and the fact that E, and G are compacts. Here, we also use Proposition.3. Consequently, we assume λ small. We study first the differentiability properties of the state system : Proposition 3. The system of variations corresponding to is q = g + +v + g + + u r, in, 3.0 q = 0, on, 3. w = gz g w rz g, in I g, 3. w = gz g w + rz g, in I g, 3.3 w 0 = 0, w 0 = 0, 3.4 y where q = lim λ y z λ 0, w = [w λ, w ] = lim g+λr z g λ 0 with y λ λ H H0 being the solution of corresponding to g + λr, u + λv, and z g+λr C I g is the solution of corresponding to g + λr. The limits exist in the above spaces. We denote by the scalar product on R. Proof. We subtract the equations corresponding to y λ and y and divide by λ 0, small: yλ y = [ g + λr + λ λ + u + λv g + +u ], in, 3.5 with 0 boundary conditions on. The regularity conditions on F and u, v L give the convergence of the right-hand side in 3.5 to the right-hand side in 3.0 6

7 strongly in L via some calculations. Then, by elliptic regularity, we have yλ y λ q strongly in H H0 and 3.0, 3. follows. For , the argument is the same as in Proposition 6, [3]. The convergence of the ratio z g+λr z g is in C I λ g on the whole sequence λ 0, due to the uniqueness property of the linear system Here, we also use Remark., on the convergence G λ G and the continuity with respect to the perturbations of g in the Hamiltonian system.-.3, according to [3]. Remark 3. We have as well imposed the condition gx 0, x 0 = 0, g F, 3.6 where x 0, x 0 \ E is some given point. Similarly, constraints like 3.6 may be imposed on a finite number of points or on some curves in \ E and their geometric meaning is that the boundary Ω g of the admissible unknown domains should contain these points, curves, etc. Proposition 3.3 Assume that f L p, jx, is of class C R and bounded, g F and u L p, p >. Then, for any direction [r, v] F L p, the derivative of the penalized cost 3. is given by: j x, y x q xdx + y z g tq z g t z E gt dt 3.7 I g + y z g t y z g t wt z gt dt + y z g t I g Ig z gt w t dt z gt where q W,p W,p 0, w C I g, z g C I g satisfy and.-.3 respectively, and I g = [0, T g ] is the period interval for z g. Proof. In the notations of Proposition 3., we compute { [ lim j x, y λ λ 0 λ x j x, y x ] dx 3.8 E + [ y λ z g+λr t } ] z λ g+λh t y z g t z gt dt. I g In 3.8, λ > 0 is small and Proposition.3 ensures that g + λr F see [4] as well. By Proposition. we know that the trajectories associated to g + λh are periodic, that is the functions in the second integrals are defined on I g. Moreover, since f, u L p, then y λ, y defined as in 3.3, 3.4 are in W,p C, by the Sobolev theorem and elliptic regularity. Consequently, all the integrals appearing in 3.7, 3.8 make sense. 7

8 We also have y λ y in C, for λ 0, by elliptic regularity. Then, under the assumptions on j,, we get [ j x, y λ λ x j x, y x ] dx j x, y x q xdx. 3.9 E For the second integral in 3.8, we intercalate certain intermediary terms and we compute their limits for λ 0: [ y λ lim z g+λr t ] z λ 0 λ g+λr t y z g+λr t z g+λrt dt 3.0 I g = y z g tq z g t z gt dt I g due to the convergence z g+λr z g in C I g by g, r C and the continuity properties in.-.3; [ lim y z g+λr t z λ 0 λ g+λrt y z g t z g+λrt ] dt 3. I g = y z g t y z g t wt z gt dt, I g where w = w, w satisfies and again we use the regularity and the convergence properties in C, respectively C I g. [ lim y z g t z λ 0 λ g+λrt y z g t z gt ] dt 3. I g = y z g t Ig z gt w t dt, z gt where we recall that z gt = zg t + zg t is non zero by.7 and the Hamiltonian system, and standard derivation rules may be applied under our regularity conditions. By summing up , we end the proof of 3.7. Remark 3.3 In the case that Ω g is not simply connected, the penalization integral in 3. is in fact a finite sum and each of these terms can be handled separately, in the same way as above, due to Proposition.3 and Remark.. Now, we denote by A : C L p W,p W,p 0 the linear continuous operator given by r, v q, defined in 3.0, 3.. We also denote by B : C C I g the linear continuous operator given by , Br = [w, w ]. In these definitions, g C and u L p are fixed. We have: 8 E

9 Corollary 3. The relation 3.7 can be rewritten as: j x, y x Ar, vxdx + y z g tar, vz g t z E gt dt 3.3 I g + y z g t y z g t Brt z gt dt I g + y z g t z I g z gt [ r, r]z g tdt gt + y z g t Ct wtdt, I g z gt where the vector Ct is explained below. Proof. In the last integral in 3.7, we replace w t by the right-hand side in 3., 3.3. We compute: z gt w t 3.4 = z gt [ gz g t wt rz g t, gz g t wt + rz g t] = z gt [ rz g t, rz g t] + z gt [,gz g tw t,,gz g tw t] + z gt [,gz g tw t,,gz g tw t]. We denote by Ct the known vector Ct = [ z g t,gz g t + z g t,gz g t, z g t,gz g t + z g t,gz g t] and together with 3.4, we get 3.3. This ends the proof. 4 Finite element discretization We assume that and E are polygonal. Let T h be a triangulation of with vertices A i, i I = {,..., n}. We consider that T h is compatible with E, i.e. T T h, T E or T \ E where T designs a triangle of T h and h is the size of T h. We consider a triangle as a closed set. For simplicity, we employ piecewise linear finite element and we denote W h = {ϕ h C; ϕ h T P T, T T h }. 9

10 We use a standard basis of W h, {φ i } i I, where φ i is the hat function associated to the vertex A i, see for example [6], []. The finite element approximations of g and u are g h x = i I G iφ i x, u h x = i I U iφ i x, for all x. We set the vectors G = G i T i I Rn, U = U i T i I Rn and g h can be identified by G, etc. The function u is in L p, as in Proposition 3.3. Alternatively, for u h, we can use discontinuous piecewise constant finite element P 0. In order to approach g C, we can use high order finite elements. 4. iscretization of the optimization problem We introduce V h = {ϕ h W h ; ϕ h = 0 on }, I 0 = {i I; A i / } and n 0 = cardi 0. The finite element weak formulation of is: find y h V h such that y h ϕ h dx = f + gh + +u h ϕh dx, ϕ h V h. 4. As before, for y h x = j I 0 Y j φ j x, we set Y = Y j T j I 0 R n 0. In order to obtain the linear system, we take the basis functions ϕ h = φ i in 4. for i I 0. Let us consider the vector T F = fφ i dx R n 0, i I 0 the n 0 n 0 matrix K defined by K = K ij i I0,j I 0, K ij = and the n 0 n matrix B G, defined by B G, = Bij i I0,j I, Bij = φ j φ i dx g h + +φ j φ i dx. The matrix K is symmetric, positive definite. The finite element approximation of the state system is the linear system: KY = F + B G, U. 4. Now, we shall discretize the objective function 3.. We denote I E = {i I; A i E} and n E = cardi E. For the first term of 3., we introduce J Y = jx, y h xdx. E 0

11 We shall study the second term of 3.. In order to solve numerically the OE system , we use a partition [t 0,..., t k,..., t m ] of [0, T g ], with t 0 = 0 and t m = T g. We can use the forward Euler scheme: Zk+ = Zk t k+ t k g h Z x k, Zk, 4.3 Zk+ = Zk + t k+ t k g h Z x k, Zk, 4.4 Z0, Z0 = x 0, x 0, 4.5 for k = 0,..., m. We set Z k = Zk, Z k and we impose Z m = Z 0. In fact, Z k is an approximation of z g t k. We do not need to stock Z 0 and we set Z = Z, Z R m R m, with Z = Zk T k m and Z = Zk T k m. In the applications, one can use more performant numerical methods for the OE s, like explicit Runge-Kutta or backward Euler, but here we want to avoid a too tedious exposition. Without risk of confusion, we introduce the function Z : [0, T g ] R defined by Zt = t k+ t t k+ t k Z k + t t k t k+ t k Z k+, t k t < t k+ for k = 0,,..., m. We have Zt k = Z k and we can identify the function Z by the vector Z R m R m. We remark that Z is derivable on each interval t k, t k+ and Z t = t k+ t k Z k+ Z k, Z k+ Z k for t k t < t k+. We introduce the n 0 n 0 matrix NZ defined by Tg NZ = φ j Ztφ i Zt Z t dt 0 i I 0,j I 0 and the second term of 3. is approached by Y T NZY, then the discrete form of the optimization problem is min JG, U = J Y + G,U R n Y T NZY 4.6 subject to 4.. We point out that Y depends on G and U by 4. and Z depends on G by iscretization of the derivative of the objective function Let r h, v h be in W h and R, V R n be the associated vectors. The finite element weak formulation of is: find q h V h such that q h ϕ h dx = gh + +v h + g h + + u h r h ϕh dx, ϕ h V h. 4.7

12 Let Q R n 0 be the associated vector to q h and we construct the n 0 n matrix C G,, U defined by C G,, U = g h + + u h φ j φ i dx. i I 0,j I The linear system of 4.7 is KQ = B G, V + C G,, UR. 4.8 In order to approximate jx, y x, y given by , we consider the nonlinear application Y R n 0 LY R n E such that jx, y h x = i I E LY i φ i Ex where φ i E is the restriction of φ i to E. We define the n E n 0 matrix M E defined by M E = φ i φ j dx The first term of 3.7 is approached by and the second term of 3.7 is approached by i I E,j I 0. LY T M E Q 4.9 Y T NZQ 4.0 where the matrix NZ was introduced in the previous subsection. Next, we introduce the partial derivative for a piecewise linear function. Let g h W h and G R n its associated vector, i.e. g h x = i I G iφ i x. Let Π h G Rn defined by Π h G = areat i j g h T j J i areat j j j J i where J i is the set of index j such that the triangle T j has the vertex A i. Since g h is a linear function in each triangle T j, then g h T j is constant. Similarly, we construct Π h G Rn for. In fact, Π h and Π h are two n n matrices depending on T h. Then, we set h g h x = Π h G φ i ix i I and similarly for h g h. Finally, we put h g h = h g h, h g h. Since y h V h W h, we can define h y h and h y h.

13 Example 4. We shall give a simple example to understand the discrete derivative of W h functions. We consider the square [A A A 4 A 3 ] of vertices A = 0, 0, A =, 0, A 4 =,, A 3 = 0, and the triangulation of two triangles T = [A A A 4 ] and T = [A A 4 A 3 ]. We shall present the discrete derivative of the hat function { x in T φ 4 x, x = x in T. We have J = {, } and Π h φ 4 = = areat φ 4 T areat + areat + areat φ 4 T / 0 + / = /. / + / Similarly, J = {}, J 3 = {}, J 4 = {, }, then Π h φ 4 = / 0 = 0, / Π h φ 4 = 3 / = /, / Π h φ 4 = / 0 + / = / 4 / + / h φ 4 x, x = / φ x, x + 0 φ x, x + φ 3 x, x + / φ 4 x, x. In order to solve the OE system , we use the forward Euler scheme on the same partition as for : W k+ = W k t k+ t k h h g h Z k W k, W k 4. t k+ t k h r h Z k, W k+ = W k + t k+ t k h h g h Z k W k, W k 4. +t k+ t k h r h Z k, W 0 = 0, W 0 = 0, 4.3 for k = 0,..., m. We set W k = Wk, W k and now we have W m W 0 generally. In fact, W k is an approximation of wt k. We do not need to stock W 0 and we set W = W, W R m R m, with W = Wk T k m and W = Wk T k m. As mentioned before, we can use more performant numerical methods for the OE, like explicit Runge-Kutta or backward Euler. 3

14 We construct W : [0, T g ] R in the same way as for Zt W t = t k+ t t k+ t k W k + t t k t k+ t k W k+, t k t < t k+ for k = 0,,..., m. We have W t k = W k and W t = W t k+ t k k+ W k, W k+ Wk for t k t < t k+. If ψ k is the one-dimensional piecewise linear hat function associated to the point t k of the partition [t 0,..., t k,..., t m ], we can write equivalently W t = m k=0 W kψ k t for t [0, T g ]. The third term of 3.7 is approached by = = + m tk+ k=0 m tk+ k=0 t k m tk+ k=0 t k m tk+ k=0 t k y h Zt h y h Zt W t Z t dt 4.4 t k y h Zt h y h Zt W k ψ k t + W k+ ψ k+ t Z t dt y h Zt h y h Zt W k ψ k t + W k+ψ k+ t Z k Z k+ t k+ t k dt y h Zt h y h Zt W k ψ k t + W k+ψ k+ t Z k Z k+ t k+ t k dt where Z k Z k+ is the length of the segment in R with ends Z k and Z k+. We have tk+ y h Zt h y h Zt Wk ψ k t + Wk+ψ k+ t Z k Z k+ dt 4.5 t k t k+ t k tk+ = Wk Y i φ i Zt Π hy j φ j Zt ψ k t Z kz k+ t k t k+ t k dt i I 0 j I + W k+ tk+ t k Y i φ i Zt Π hy j φ j Zt i I 0 j I We introduce the n 0 n matrices N [k,k+] k N [k,k+] k Z = N [k,k+] k+ Z = tk+ t k tk+ then 4.5 can be rewritten as Y T t k W k N [k,k+] k ψ k+ t Z kz k+ t k+ t k dt. Z and N [k,k+] k+ Z defined by φ i Ztφ j Ztψ k t Z kz k+ t k+ t k dt φ i Ztφ j Ztψ k+ t Z kz k+ t k+ t k dt Z + Wk+N [k,k+] k+ Z Π hy 4 i I 0,j I i I 0,j I

15 and finally the third term of 3.7 is approached by Y T m k=0 + m Y T k=0 W k N [k,k+] k W k N [k,k+] k Z + Wk+N [k,k+] k+ Z Π hy 4.6 Z + Wk+N [k,k+] k+ Z Π hy. We can introduce the linear operators T Z and T Z by W R m T ZW = W R m T ZW = then 4.6 can be rewritten as m k=0 m k=0 W k N [k,k+] k W k N [k,k+] k Z + Wk+N [k,k+] k+ Z Z + Wk+N [k,k+] k+ Z 4.7 Y T T ZW Π hy + Y T T ZW Π hy. 4.8 The fourth term of term of 3.7 is approached by m tk+ Y i φ i Zt Y j φ j Zt t k i I 0 j I 0 k=0 But Z t and W t are constants for t k t < t k+, then Z t W t dt. 4.9 Z t Z t W t Z t = Z k+ Z k, Z k+ Z k W k+ W k, W k+ W k t k+ t k Z k Z k+ where Z k Z k+ is the length of the segment in R with ends Z k and Z k+. We introduce the n 0 n 0 matrix R k Z defined by tk+ R k Z = φ i Ztφ j Zt Z t dt and the linear operators T 3 Z t k i I 0,j I 0 W R m R m T 3 ZW 4.0 T 3 ZW = m k=0 Z k+ Z k, Z k+ Z k W k+ W k, W k+ W k Z k Z k+ R k Z. 5

16 The 4.9 can be rewritten as Y T T 3 ZW Y. 4. The study of this subsection can be resumed as following: Proposition 4. The discret version of 3.7 is dj G,U R, V = LY T M E Q + Y T NZQ 4. + Y T T ZW Π hy + Y T T ZW Π hy + Y T T 3 ZW Y which represents the derivative of J at G, U in the direction R, V. Proof. We get 4. just by assembling 4.9, 4.0, 4.8 and iscretization of the formula 3.3 From 4.8, we get Q = K B G, V + K C G,, UR and the discrete version of the operator A in the Corollary 3. is R, V R n R n A R, V = K B G, V + K C G,, UR. Replacing Q in the first two terms of 4., we get LY T M E + Y T NZ K B G, V LY T M E + Y T NZ K C G,, UR. We denote Λ t = y h Zt y h Zt Z t Λ t = y hzt Z t Z t Λ 3 t = y hzt Ct. Z t 6

17 The third term of 3.3 is approached by Tg Λ t W t dt 0 and using the trapezoidal quadrature formula on each sub-interval [t k, t k+ ], we get m t k+ t k [ Λ t k Wk, Wk + Λ t k+ W k+, Wk+ ]. 4.4 k=0 Similarly, for the 4th and 5th terms of 3.3, we get m t k+ t k [ Λ t k [ h r h, h r h ]Z k + Λ t k+ [ h r h, h r h ]Z k+ ] 4.5 and k=0 m t k+ t k [ Λ 3 t k Wk, Wk + Λ 3 t k+ W k+, Wk+ ]. 4.6 k=0 In order to write shorter, we introduce the vectors: Λ R m with first components t k+ t k Λ t k, k m and the last component t m t m Λ t m, Λ R m with first components t k+ t k Λ t k, k m and the last component t m t m Λ t m, Λ 3 R m with first components t k+ t k Λ 3t k, k m and the last component t m t m Λ 3t m, Λ 3 R m with first components t k+ t k Λ 3t k, k m and the last component t m t m Λ 3t m. Also, we introduce the vectors in R n : Λ = Λ = 0 k m 0 k m where ΦZ k = φ i Z k T i I Rn. t k+ t k Λ t k ΦZ k + Λ t k+ ΦZ k+ t k+ t k Λ t k ΦZ k + Λ t k+ ΦZ k+ 7

18 Proposition 4. The discrete version of the 3.3 is dj G,U R, V = LY T M E + Y T NZ K B G, V LY T M E + Y T NZ K C G,, UR + Λ T W + Λ T W + Λ T Π hr + Λ T Π hr + Λ 3 T W + Λ 3 T W. Proof. We obtain 4.7 by summing Next, we give more details about the relationship between W and R. Let us introduce the matrices tk+ t M k = k h h g h Z k t k+ t k h h g h Z k t k+ t k h h g h Z k + t k+ t k h h, g h Z k 0 I = 0 and the n matrice N k = tk+ t k Φ T Z k Π h t k+ t k Φ T Z k Π h We remark that M depends on G and Z and N on Z. The system can be written as W k+ W Wk+ = M k k Wk + N kr. Proposition 4.3 We have the following equality W W. = M m Wm Wm N 0 N. N m. R 4.8 where at the right-hand side, M m is a m m matrix defined by I M I M m M, M m M, M m, I 8

19 and the size of the second matrix, which contains N, is m n. Proof. From 4.3 and the recurrent relation, we have W W = N 0R W W W = M W + N R = M N 0R + N R W m W m W m W m. = M m M N 0R + M m M N R + + M m N m 3R + N m R W = M m m Wm + N m R = M m M m M N 0R +M m M m M N R + + M m M m N m 3R +M m N m R + N m R which gives 4.8. Since W depends on R by 4.8, we can introduce the linear operator approximation of B in the Corollary 3. R R n W = W, W = B G, ZR, B 3 G, ZR R m R m. If we denote by l i the i-th line of the matrix M m at the right-hand side of 4.8, for i m, then l N 0 B l 3 N G, Z =.., N m B 3 G, Z = l m l l 4. l m N 0 N. N m and B G, Z, B 3 G, Z are m n matrices. The size of the matrix containing N is m n. 9

20 4.4 Gradient type algorithm We start by presenting the algorithm. Step Start with k = 0, > 0 some given small parameter and select some initial G k, U k. Step Compute Y k the solution of 4. and Z k solution of Step 3 Find R k, V k such that dj G k,u k R k, V k < 0. We say that R k, V k is a descent direction. Step 4 efine G k+, U k+ = G k, U k + λ k R k, V k, where λ k > 0 is obtained via some line search λ k arg min λ>0 J G k, U k + λr k, V k. Step 5 If JG k+, U k+ JG k, U k is below some prescribed tolerance parameter, then Stop. If not, update k := k + and go to Step 3. In the Step 3, we have to provide a descent direction. We present in the following a partial result. Let us introduce a simplified adjoint system: find p h V h such that ϕ h p h dx = jx, y h xϕ h dx + Tg y h Ztϕ h Zt Z t dt 4.9 E 0 for all ϕ h V h and with Zt given by We have p h x = i I 0 P i φ i x and P = P i T i I 0 R n 0. The linear system associated to 4.9 is KP = M T ELY + NZY. We recall that K and NZ are symmetric matrices. Proposition 4.4 Given g h, u h W h, let y h V h the solution of 4.. If r h = p h u h and v h = p h, where p h V h is the solution of 4.9, then jx, y h xq h dx + Tg y h Ztq h Zt Z t dt 0, 4.30 E 0 where q h V h is the solution of 4.7 depending on r h and v h. Proof. Putting ϕ h = p h in 4.7 and ϕ h = q h in 4.9, we get gh + +v h + g h + + u h r h ph dx = q h p h dx 4.3 = jx, y h xq h dx + Tg y h Ztq h Zt Z t dt. E 0 0

21 For v h = p h, we have g h + +v h p h dx = and for r h = p h u h, we have g h + + u h r h p h dx = since g h in. This ends the proof. g h + +p hdx 0 g h + + u h p h dx 0 Remark 4. The left-hand side of 4.30 represents the first two terms of 4.. We can obtain a similar result as in Proposition 4.4, without using the adjoint system, by taking V T = LY T M E + Y T NZ K B G, R T = LY T M E + Y T NZ K C G,, U, in place of V and R in 4.3. In this case, 4.3 becames V R n R R n 0. We point out that V T = P T B G, and R T = P T C G,, U, so V, R is different from the direction given by Proposition 4.4. Now, we present a descent direction, obtained from the complete gradient of the discrete cost 4.. Proposition 4.5 For R, V R n R n given by V T = LY T M E + Y T NZ K B G, R T = LY T M E + Y T NZ K C G,, U Λ T B G, Z + Λ T B 3 G, Z Λ T Π h + Λ T Π h Λ 3 T B G, Z + Λ 3 T B 3 G, Z. we obtain a descent direction for J at G, U. Proof. In 4.7, we replace W by B G, ZR and W by B 3 G, ZR, we obtain that dj G,U R, V = V T V R T R, then dj G,U R, V = V R n R R n 0.

22 5 Numerical tests Shape optimization problems and their penalization are strongly nonconvex. The computed optimal domain depends on the starting domain, but also on the penalization or other numerical parameters. It may be just a local optimal solution. Moreover, the final computed value of the penalization integral is small, but not null. This allows differences between the optimal computed domain Ω g and the zero level curves of the computed optimal state y. Consequently, we compare the obtained optimal cost in the penalized problem with the costs in the original problem. -.3 corresponding to the optimal computed domain Ω g and the zero level curves of y. This is a standard procedure, to inject the approximating optimal solution in the original problem. Notice that in all the experiments, the cost corresponding to Ω g is the best one, but the differences with respect to the other computed cost values are small. This shows that the rather complex approximation/penalization that we use is reasonable. Its advantage is that it may be used as well in the case of boundary observation or for Neumann boundary conditions and this will be performed in a subsequent paper. In the examples, we have employed the software FreeFem++, [9]. Example. The computational domain is =] 3, 3[ ] 3, 3[ and the observation zone E is the disk of center 0, 0 and radius 0.5. The load is f =, jg = y y d, where y d x, x = x 0.5 x 0.5 +, then the cost function 3. becomes 6 { min Jg, u = y d g F, u L Ey dx + } y z g t z gt dt. 5. I g The mesh of has triangles and 3754 vertices. The penalization parameter is = 0 3 and the tolerance parameter for the stopping test at the Step 5 of the algorithm is tol = 0 6. The initial domain is the disk of center 0, 0 and radius.5 with a circular hole of center, and radius 0.5. At the Step 3 of the Algorithm, we use R k, V k given by Proposition 4.4. At the Step 4, in order to have E Ω k, we use a projection P at the line search λ k arg min λ>0 J PG k + λr k, U k + λv k and G k+ = PG k + λ k R k. If the value of gh k + λrk h at a vertex from E is positive, then we set this value to 0.. We recall that the left-hand side of 4.30 represents only the first two terms of 4., not the whole gradient. If r h, v h are given by Proposition 4.4 and γ > 0 is a scaling parameter, then γr h and v h verify 4.30, that is they also give a descent direction. We take the scaling parameter for r h given by γ =, that is a normalization of r maxr h h. In this way we

23 Figure : Example. The solution of the elliptic problem.-.3 in the domain Ω g left, in the domain bounded by the zero level sets of y right and the final computed state y in bottom. avoid the appearance of very high values of the objective function, that may stop the algorithm even in the first iteration. For the line search at the Step 4, we use λ = ρ i λ 0, with λ 0 =, ρ = 0.5 for i = 0,,..., 30. The stopping test is obtained for k = 94, some of values of the objective function are: JG 0, U 0 = 330.5, JG 30, U 30 = 54.75, JG 94, U 94 = At the final iteration, the first term of the optimal objective function is and Ω g y sds = We point out that the optimal Ω g has a hole and the penalization term is a sum of two integrals Ω g y sds = j= I j y z g t z gt dt where the integral over I corresponding to the exterior boundary of Ω g and I to the boundary of the hole. In Figure in the bottom, we can see the computed optimal state y in iteration 94. We also compute the costs E y y d dx = where y is the solution of the initial elliptic problem.-.3 in the domain Ω g with g obtained in iteration 94 and E y y d dx =.0403 where y is the solution of the elliptic problem.-.3 in the domain bounded by the zero level sets of y in iteration 94, see Figure. 3

24 Example. The domains, E and the mesh of are the same as in Example. For f = 4 and y d x, x = x x +, we have the exact optimal state y = y d defined in the disk of center 0, 0 and radius, that gives an optimal domain of the problem.-.3. We have used = 0 and the starting configuration: the disk of center 0, 0 and radius.5 with the circular hole of center, and radius 0.5. We use R k, V k given by Proposition 4.4. The parameters for the line search and γ are the same as in the precedent example. The stopping test is obtained for k = 64. The initial and the final computed values of the objective function are and.3. We obtain a local minimum that is different from the above global solution. The first term of the final computed objective function is The term Ω g y sds is and it was computed over the exterior boundary as well as over the boundaries of two holes. The length of the total boundary of the optimal domain is and of the initial domain is π = Figure : Example. The numerical solution of the elliptic problem.-.3 in the optimal domain Ω g left, in the domain bounded by the zero level sets of y right and the computed optimal state y bottom. The domain changes its topology. The computed optimal state y is presented in Figure in the bottom. At the left, we show y the solution of the elliptic problem.-.3 in the domain Ω g which gives E y y d dx = , at the right we show y the solution of the elliptic problem.-.3 in the domain bounded by the zero level sets of y, which gives E y y d dx =

25 Example 3. We have also used the descent direction given by Proposition 4.5, for the starting configuration the disk of center 0, 0 and radius.5, = 0, γ = r h and a mesh of of 3446 triangles and 6464 vertices. For solving the OE systems and we use m = 30. At the initial iteration, we have y E y d dx = , Ω g y sds = and the value of the objective function is J 0 = The algorithm stops after iterations and we have at the final iteration y E y d dx =.86, Ω g y sds = and the value of the penalized objective function is J = The final domain is a perturbation of the initial one, the circular non-smooth curve in the top, left image of Figure 3. We have y E y d dx =.0398 for y the solution of the elliptic problem.-.3 in the final domain Ω g and y E y d dx =.767 for y the solution of the elliptic problem.-.3 in the domain bounded by the zero level sets of y, Figure 3 at the bottom, right. Figure 3: Example 3. The zero level sets of the computed optimal g, y top, left, the final state y top, right, the solution of the elliptic problem.-.3 in the domain Ω g bottom, left and in the domain bounded by the zero level sets of y bottom, right. 5

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