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.
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},
mrnumber = {2375755},
zbl = {1140.49029},
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 SP - 160 EP - 191 VL - 14 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007045/ DO - 10.1051/cocv:2007045 LA - en ID - COCV_2008__14_1_160_0 ER -
%0 Journal Article %A Hassine, Maatoug %A Guillaume, Philippe %T Removing holes in topological shape optimization %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 160-191 %V 14 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007045/ %R 10.1051/cocv:2007045 %G en %F COCV_2008__14_1_160_0
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] , , and , 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] and , On some recent advances in shape optimization. C. R. Acad. Sci. Paris, Ser. II B 329 (2001) 383-396. | Zbl
[3] and , Optimal bounds on the effective behavior of a mixture of two well-orded elastic materials. Quart. Appl. Math. 51 (1996) 643-674. | MR | Zbl
[4] , and , Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004) 363-393. | MR | Zbl
[5] , and , The topological asymptotic for the Helmholtz equation. SIAM J. Contr. Optim. 42 (2003) 1523-1544. | MR | Zbl
[6] , , and , 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 | Zbl
[7] , Optimal topology design of continuum structure: an introduction. Technical report, Departement of mathematics, Technical University of Denmark, DK2800 Lyngby, Denmark (1996).
[8] , and , IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials. Springer (2006). | Zbl
[9] and , Mixed and hybrid finite element method, Springer Series in Computational Mathematics 15. Springer Verlag- New York (1991). | MR | Zbl
[10] and , Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991) 17-49. | MR | Zbl
[11] , 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 | Zbl
[12] , and , Quelques résultats sur l'identification de domains. Calcolo (1973). | Zbl
[13] , , and , The shape and topological optimizations connection. Comput. Methods Appl. Mech. Engrg. 188 (2000) 713-726. | MR | Zbl
[14] and , Relaxed shape optimization: the case of nonnegative data for the Dirichlet problems. Adv. Math. Sci. Appl. 1 (1992) 47-81. | MR | Zbl
[15] , The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR | Zbl
[16] and , Analyse mathémathique et calcul numérique pour les sciences et les techniques. Masson, collection CEA (1987). | Zbl
[17] , and , The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778. | MR | Zbl
[18] and , Finite element methods for Navier-Stokes equations, Theory and Algorithms. Springer Verlag (1986). | MR | Zbl
[19] and , Toward the computational of minimun drag profile in viscous laminar flow. Appl. Math. Model. 1 (1976) 58-66. | MR | Zbl
[20] , Dérivées d'ordre supérieur en conception optimale de forme. Ph.D. thesis, Université Paul Sabatier, Toulouse, France (1994).
[21] and , Computation of high order derivatives in optimal shape design. Numer. Math. 67 (1994) 231-250. | MR | Zbl
[22] and , The topological asymptotic expansion for the Dirichlet Problem. SIAM J. Control. Optim. 41 (2002) 1052-1072. | MR | Zbl
[23] and , Topological sensitivity and shape optimization for the Stokes equations. SIAM J. Control Optim. 43 (2004) 1-31. | MR | Zbl
[24] and , 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 | Zbl
[25] and , The topological sensitivity analysis for the Quasi-Stokes problem. ESAIM: COCV 10 (2004) 478-504. | Numdam | MR | Zbl
[26] , and , 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] and , Problèmes aux limites non homogenes et applications. Dunod (1968). | Zbl
[28] , The topological asymptotic, in Computational Methods for Control Applications, H. Kawarada and J. Periaux Eds., International Séries Gakuto (2002). | Zbl
[29] , and , The topological asymptotic expansion for the Maxwell equations and some applications. Inverse Probl. 21 (2005) 547-564. | MR | Zbl
[30] , and , Asymptotic theory of elliptic boundary value problems in singularly perturbed domains2000). | Zbl
[31] , On optimum profiles in Stokes flow. J. Fluid Mech. 59 (1973) 117-128. | MR | Zbl
[32] , Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984). | MR | Zbl
[33] , Topologieoptimierung von bauteilstrukturen unter verwendung von lopchpositionierungkrieterien. Ph.D. thesis, Universitat-Gesamthochschule-Siegen (1995).
[34] , and , Mechanical inclusions identification by evolutionary computation. Revue européenne des éléments finis 5 (1996) 619-648. | MR | Zbl
[35] , Domain variation for Stokes flow, in Lect. Notes Control Inform. Sci. 159, X. Li and J. Yang Eds., Springer, Berlin (1990) 28-42. | MR | Zbl
[36] , Domain variation for drag Stokes flows, in Lect. Notes Control Inform. Sci. 114, A. Bermudez Eds., Springer, Berlin (1987) 277-283. | Zbl
[37] and , On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251-1272 (electronic) | MR | Zbl
[38] and , Modelling of topological derivatives for contact problems. Numer. Math. 102 (2005) 145-179. | MR | Zbl
[39] , Navier Stokes equations (1985).
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