In this paper, we propose a topological sensitivity analysis for the Quasi-Stokes equations. It consists in an asymptotic expansion of a cost function with respect to the creation of a small hole in the domain. The leading term of this expansion is related to the principal part of the operator. The theoretical part of this work is discussed in both two and three dimensional cases. In the numerical part, we use this approach to optimize the locations of a fixed number of air injectors in an eutrophized lake.
Classification : 49Q10, 49Q12, 74P05, 74P10, 74P15
Mots clés : topological optimization, topological sensitivity, quasi-Stokes equations, topological gradient, shape optimization
@article{COCV_2004__10_4_478_0,
author = {Hassine, Maatoug and Masmoudi, Mohamed},
title = {The topological asymptotic expansion for the {Quasi-Stokes} problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {478--504},
publisher = {EDP-Sciences},
volume = {10},
number = {4},
year = {2004},
doi = {10.1051/cocv:2004016},
zbl = {1072.49027},
mrnumber = {2111076},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2004016/}
}
TY - JOUR AU - Hassine, Maatoug AU - Masmoudi, Mohamed TI - The topological asymptotic expansion for the Quasi-Stokes problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2004 DA - 2004/// SP - 478 EP - 504 VL - 10 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2004016/ UR - https://zbmath.org/?q=an%3A1072.49027 UR - https://www.ams.org/mathscinet-getitem?mr=2111076 UR - https://doi.org/10.1051/cocv:2004016 DO - 10.1051/cocv:2004016 LA - en ID - COCV_2004__10_4_478_0 ER -
Hassine, Maatoug; Masmoudi, Mohamed. The topological asymptotic expansion for the Quasi-Stokes problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 4, pp. 478-504. doi : 10.1051/cocv:2004016. http://www.numdam.org/articles/10.1051/cocv:2004016/
[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, Optimal bounds on the effective behavior of a mixture of two well-order elastic materials. Quat. Appl. Math. 51 (1993) 643-674. | MR 1247433 | Zbl 0805.73043
[3] , Optimal topology design of continuum structure: an introduction. Technical report, Department of mathematics, Technical University of Denmark, DK2800 Lyngby, Denmark, September (1996).
[4] and, Mixed and hybrid finite element method. Springer Ser. Comput. Math. 15 (1991). | MR 1115205 | Zbl 0788.73002
[5] and, Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991) 17-49. | MR 1076053 | Zbl 0762.49017
[6] , and, Quelques résultats sur l'identification de domaines. CALCOLO (1973). | Zbl 0303.93023
[7] , Conception optimale ou identification de forme, calcul rapide de la dérivée directionnelle de la fonction coût. ESAIM: M2AN 20 (1986) 371-402. | Numdam | MR 862783 | Zbl 0604.49003
[8] ,, and, The shape and Topological Optimizations Connection. Comput. Methods Appl. Mech. Engrg. 188 (2000) 713-726. | MR 1784106 | Zbl 0972.74057
[9] and, 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
[10] , The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR 520174 | Zbl 0383.65058
[11] et, Analyse mathémathique et calcul numérique pour les sciences et les techniques. Masson, collection CEA (1987). | Zbl 0642.35001
[12] and, Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element methods or finite difference method. SIAM J. Numer. Anal. 19 (1982) 871-885. | MR 672564 | Zbl 0492.65051
[13] , et, Résolution numérique des équations de Navier-Stokes pour un fluide incompressible. J. Mécanique 10 (1971). | MR 421338 | Zbl 0225.76016
[14] , and, The topological sensitivity for linear isotropic elasticity. European Conferance on Computationnal Mechanics (1999) (ECCM99), report MIP 99.45.
[15] , and, The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778. | MR 1825864 | Zbl 0990.49028
[16] and, Introduction à la mécanique des milieux continus. Masson (1994). | MR 576236 | Zbl 0465.73001
[17] and, Finite element methods for Navier-Stokes equations, Theory and Algorithms. Springer-Verlag Berlin (1986). | MR 851383 | Zbl 0585.65077
[18] , Formulations variationnelles par équations intégrales de problèmes aux limites extérieurs. Thèse, École Polytechnique, Palaiseau (1976).
[19] , Numerical methods for nonlinear variational problems. J. Optim. Theory Appl. 57 (1988) 407-422. | MR 859924
[20] and, Toward the computational of minimun drag profile in viscous laminar flow. Appl. Math. Model. 1 (1976) 58-66. | MR 455851 | Zbl 0361.76035
[21] , Elliptic problems in non smooth domains. Pitman Publishing Inc., London (1985). | Zbl 0695.35060
[22] and, Computation of high order derivatives in optimal shape design. Numer. Math. 67 (1994) 231-250. | MR 1262782 | Zbl 0792.65044
[23] and, The topological asymptotic expansion for the Dirichlet Problem. SIAM J. Control. Optim. 41 (2002) 1052-1072. | MR 1972502 | Zbl 1053.49031
[24] and, Topological sensitivity and shape optimization for the Stokes equations. Rapport MIP (2001) 01-24.
[25] , Contrôle des processus d'aération des lacs eutrophes. Thesis, Tunis II University, ENIT, Tunisia (2003).
[26] , and, Generalized shape optimization of three-dimensional 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 homogènes et applications. Dunod (1996). | Zbl 0165.10801
[28] , Outils pour la conception optimale de formes. Thèse d'État, Université de Nice (1987).
[29] , The topological asymptotic, in Computational Methods for Control Applications, H. Kawarada and J. Periaux Eds., International Séries GAKUTO (2002). | Zbl 1082.93584
[30] and, Calcul des variations et homogénéisation, in Les méthodes de l'homogénéisation : Théorie et applications en physique. Eyrolles (1985) 319-369.
[31] , Méthode des éléments finis pour les fluides. Masson, Paris (1988). | Zbl 0748.76003
[32] , Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984). | MR 725856 | Zbl 0534.49001
[33] , Domain variation for Stokes flow. X. Li and J. Yang Eds., Springer, Berlin, Lect. Notes Control Inform. Sci. 159 28-42 (1990). | MR 1129956 | Zbl 0801.76075
[34] , Domain variation for drag Stokes flows. A. Bermudez Eds., Springer, Berlin, Lect. Notes Control Inform. Sci. 114 (1987) 277-283. | Zbl 0801.76075
[35] , Topologieoptimierung von bauteilstrukturen unter verwendung von lopchpositionierungkrieterien. Thesis, Universitat-Gesamthochschule-Siegen (1995).
[36] , and, Mechanical inclusions identification by evolutionary computation. Rev. Eur. Élém. Finis 5 (1996) 619-648. | Zbl 0924.73321
[37] and, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251-1272 (electronic). | MR 1691940 | Zbl 0940.49026
[38] , Navier Stokes equations (1985).
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