Removing holes in topological shape optimization
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 160-191.

The gradient based topological optimization tools introduced during the last ten years tend naturally to modify the topology of a domain by creating small holes inside the domain. Once these holes have been created, they usually remain unchanged, at least during the topological phase of the optimization algorithm. In this paper, a new asymptotic expansion is introduced which allows to decide whether an existing hole must be removed or not for improving the cost function. Then, two numerical examples are presented: the first one compares topological optimization with standard shape optimization, and the second one, issued from a lake oxygenation problem, illustrates the use of the new asymptotic expansion.

DOI : https://doi.org/10.1051/cocv:2007045
Classification : 49Q10,  49Q12,  74P05,  74P10,  74P15
Mots clés : topological optimization, topological sensitivity, topological gradient, shape optimization, Stokes equations
@article{COCV_2008__14_1_160_0,
     author = {Hassine, Maatoug and Guillaume, Philippe},
     title = {Removing holes in topological shape optimization},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {160--191},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {1},
     year = {2008},
     doi = {10.1051/cocv:2007045},
     zbl = {1140.49029},
     mrnumber = {2375755},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2007045/}
}
TY  - JOUR
AU  - Hassine, Maatoug
AU  - Guillaume, Philippe
TI  - Removing holes in topological shape optimization
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2008
DA  - 2008///
SP  - 160
EP  - 191
VL  - 14
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2007045/
UR  - https://zbmath.org/?q=an%3A1140.49029
UR  - https://www.ams.org/mathscinet-getitem?mr=2375755
UR  - https://doi.org/10.1051/cocv:2007045
DO  - 10.1051/cocv:2007045
LA  - en
ID  - COCV_2008__14_1_160_0
ER  - 
Hassine, Maatoug; Guillaume, Philippe. Removing holes in topological shape optimization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 160-191. doi : 10.1051/cocv:2007045. http://www.numdam.org/articles/10.1051/cocv:2007045/

[1] M. Abdelwahed, M. Amara, F. El Dabaghi and M. Hassine, A numerical modelling of a two phase flow for water eutrophication problems. ECCOMAS 2000, European Congress on Computational Methods in Applied Sciences and Engineering, Barcelone, 11-14 September (2000).

[2] G. Allaire and A. Henrot, On some recent advances in shape optimization. C. R. Acad. Sci. Paris, Ser. II B 329 (2001) 383-396. | Zbl 0986.49023

[3] G. Allaire and R. Kohn, Optimal bounds on the effective behavior of a mixture of two well-orded elastic materials. Quart. Appl. Math. 51 (1996) 643-674. | MR 1247433 | Zbl 0805.73043

[4] G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004) 363-393. | MR 2033390 | Zbl 1136.74368

[5] S. Amstutz, M. Masmoudi and B. Samet, The topological asymptotic for the Helmholtz equation. SIAM J. Contr. Optim. 42 (2003) 1523-1544. | MR 2046373 | Zbl 1051.49029

[6] J.A. Bello, E. Fernández-Cara, J. Lemoine and J. Simon, The differentiability of the drag with respect to the variations of a lipschitz domain in a Navier-Stokes flow. SIAM J. Control Optim. 35 (1997) 626-640. | MR 1436642 | Zbl 0873.76019

[7] M. Bendsoe, Optimal topology design of continuum structure: an introduction. Technical report, Departement of mathematics, Technical University of Denmark, DK2800 Lyngby, Denmark (1996).

[8] M. Bendsoe, N. Olhoff and O. Sigmund, IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials. Springer (2006). | Zbl 1099.74005

[9] F. Brezzi and M. Fortin, Mixed and hybrid finite element method, Springer Series in Computational Mathematics 15. Springer Verlag- New York (1991). | MR 1115205 | Zbl 0788.73002

[10] G. Buttazzo and G. Dal Maso, Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991) 17-49. | MR 1076053 | Zbl 0762.49017

[11] J. Céa, Conception optimale ou identification de forme, calcul rapide de la dérivée directionnelle de la fonction coút. RAIRO Math. Modél. Anal. Numér. 20 (1986) 371-402. | Numdam | MR 862783 | Zbl 0604.49003

[12] J. Céa, A. Gioan and J. Michel, Quelques résultats sur l'identification de domains. Calcolo (1973). | Zbl 0303.93023

[13] J. Céa, S. Garreau, P. Guillaume and M. Masmoudi, The shape and topological optimizations connection. Comput. Methods Appl. Mech. Engrg. 188 (2000) 713-726. | MR 1784106 | Zbl 0972.74057

[14] M. Chipot and G. Dal Maso, Relaxed shape optimization: the case of nonnegative data for the Dirichlet problems. Adv. Math. Sci. Appl. 1 (1992) 47-81. | MR 1161483 | Zbl 0769.35013

[15] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR 520174 | Zbl 0383.65058

[16] R. Dautray and J. Lions, Analyse mathémathique et calcul numérique pour les sciences et les techniques. Masson, collection CEA (1987). | Zbl 0642.35001

[17] S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778. | MR 1825864 | Zbl 0990.49028

[18] V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations, Theory and Algorithms. Springer Verlag (1986). | MR 851383 | Zbl 0585.65077

[19] R. Glowinski and O. Pironneau, Toward the computational of minimun drag profile in viscous laminar flow. Appl. Math. Model. 1 (1976) 58-66. | MR 455851 | Zbl 0361.76035

[20] P. Guillaume, Dérivées d'ordre supérieur en conception optimale de forme. Ph.D. thesis, Université Paul Sabatier, Toulouse, France (1994).

[21] P. Guillaume and M. Masmoudi, Computation of high order derivatives in optimal shape design. Numer. Math. 67 (1994) 231-250. | MR 1262782 | Zbl 0792.65044

[22] P. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet Problem. SIAM J. Control. Optim. 41 (2002) 1052-1072. | MR 1972502 | Zbl 1053.49031

[23] P. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations. SIAM J. Control Optim. 43 (2004) 1-31. | MR 2081970 | Zbl 1093.49029

[24] M.D. Gunzburger and H. Kim, Existence of an optimal solution of a shape control problem for the stationary Navier-Stokes equations. SIAM J. Control Optim. 36 (1998) 895-909. | MR 1613877 | Zbl 0917.49004

[25] M. Hassine and M. Masmoudi, The topological sensitivity analysis for the Quasi-Stokes problem. ESAIM: COCV 10 (2004) 478-504. | Numdam | MR 2111076 | Zbl 1072.49027

[26] J. Jacobsen, N. Olhoff and E. Ronholt, Generalized shape optimization of three-dimensionnal structures using materials with optimum microstructures. Technical report, Institute of Mechanical Engineering, Aalborg University, DK-9920 Aalborg, Denmark (1996).

[27] J.L. Lions and E. Magenes, Problèmes aux limites non homogenes et applications. Dunod (1968). | Zbl 0165.10801

[28] M. Masmoudi, The topological asymptotic, in Computational Methods for Control Applications, H. Kawarada and J. Periaux Eds., International Séries Gakuto (2002). | Zbl 1082.93584

[29] M. Masmoudi, J. Pommier and B. Samet, The topological asymptotic expansion for the Maxwell equations and some applications. Inverse Probl. 21 (2005) 547-564. | MR 2146276 | Zbl 1070.35129

[30] V. Mazja, S.A. Nazarov and B.A. Plamenevski, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains2000). | Zbl 1127.35301

[31] O. Pironneau, On optimum profiles in Stokes flow. J. Fluid Mech. 59 (1973) 117-128. | MR 331973 | Zbl 0274.76022

[32] O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984). | MR 725856 | Zbl 0534.49001

[33] A. Schumacher, Topologieoptimierung von bauteilstrukturen unter verwendung von lopchpositionierungkrieterien. Ph.D. thesis, Universitat-Gesamthochschule-Siegen (1995).

[34] M. Shœnauer, L. Kallel and F. Jouve, Mechanical inclusions identification by evolutionary computation. Revue européenne des éléments finis 5 (1996) 619-648. | MR 1436837 | Zbl 0924.73321

[35] J. Simon, Domain variation for Stokes flow, in Lect. Notes Control Inform. Sci. 159, X. Li and J. Yang Eds., Springer, Berlin (1990) 28-42. | MR 1129956 | Zbl 0801.76075

[36] J. Simon, Domain variation for drag Stokes flows, in Lect. Notes Control Inform. Sci. 114, A. Bermudez Eds., Springer, Berlin (1987) 277-283. | Zbl 0801.76075

[37] J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251-1272 (electronic) | MR 1691940 | Zbl 0940.49026

[38] J. Sokolowski and A. Zochowski, Modelling of topological derivatives for contact problems. Numer. Math. 102 (2005) 145-179. | MR 2206676 | Zbl 1077.74039

[39] R. Temam, Navier Stokes equations (1985).

Cité par Sources :