Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T17:16:04.280Z Has data issue: false hasContentIssue false
  • English
  • Français

The topological asymptotic expansion for the Quasi-Stokes problem

Published online by Cambridge University Press:  15 October 2004

Maatoug Hassine  and
Mohamed Masmoudi
Maatoug Hassine
Affiliation:
ENIT-LAMSIN & FSM, Campus Universitaire, Le Belvédaire BP 37, 1002 Tunis, Tunisia; maatoug.hassine@enit.rnu.tn.
Mohamed Masmoudi
Affiliation:
Mathématiques pour l'Industrie et la Physique, UMR 5640, Université de Paul Sabatier, 118 route de Narbonne, 31032 Toulouse Cedex 4, France.
Get access

Abstract

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.

Keywords

Topological optimizationtopological sensitivityQuasi-Stokes equationstopological gradientshape optimization.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 4 , October 2004 , pp. 478 - 504
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
Allaire, G. and Kohn, R., Optimal bounds on the effective behavior of a mixture of two well-order elastic materials. Quat. Appl. Math. 51 (1993) 643-674. CrossRef
M. Bendsoe, Optimal topology design of continuum structure: an introduction. Technical report, Department of mathematics, Technical University of Denmark, DK2800 Lyngby, Denmark, September (1996).
F. Brezzi and M. Fortin, Mixed and hybrid finite element method. Springer Ser. Comput. Math. 15 (1991).
G. Buttazzo and G. Dal Maso, Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991) 17-49.
J. Céa, A. Gioan and J. Michel, Quelques résultats sur l'identification de domaines. CALCOLO (1973).
Céa, J., Conception optimale ou identification de forme, calcul rapide de la dérivée directionnelle de la fonction coût. ESAIM: M2AN 20 (1986) 371-402. CrossRef
Céa, J., Garreau, S., Guillaume, Ph. and Masmoudi, M., The shape and Topological Optimizations Connection. Comput. Methods Appl. Mech. Engrg. 188 (2000) 713-726. CrossRef
Chipot, M. and Dal Maso, G., Relaxed shape optimization: the case of nonnegative data for the Dirichlet problems. Adv. Math. Sci. Appl. 1 (1992) 47-81.
P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978).
R. Dautray et J. Lions, Analyse mathémathique et calcul numérique pour les sciences et les techniques. Masson, collection CEA (1987).
Douglas, J. and Russell, T.F., 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. CrossRef
M. Fortin, R. Peyret et R. Temam, Résolution numérique des équations de Navier-Stokes pour un fluide incompressible. J. Mécanique 10 (1971).
S. Garreau, Ph. Guillaume and M. Masmoudi, The topological sensitivity for linear isotropic elasticity. European Conferance on Computationnal Mechanics (1999) (ECCM99), report MIP 99.45.
Garreau, S., Guillaume, Ph. and Masmoudi, M., The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778. CrossRef
P. Germain and P. Muller, Introduction à la mécanique des milieux continus. Masson (1994).
V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations, Theory and Algorithms. Springer-Verlag Berlin (1986).
J. Giroire, Formulations variationnelles par équations intégrales de problèmes aux limites extérieurs. Thèse, École Polytechnique, Palaiseau (1976).
R. Glowinski, Numerical methods for nonlinear variational problems. J. Optim. Theory Appl. 57 (1988) 407-422.
Glowinski, R. and Pironneau, O., Toward the computational of minimun drag profile in viscous laminar flow. Appl. Math. Model. 1 (1976) 58-66. CrossRef
P. Grisvard, Elliptic problems in non smooth domains. Pitman Publishing Inc., London (1985).
Ph. Guillaume, M. Masmoudi, Computation of high order derivatives in optimal shape design. Numer. Math. 67 (1994) 231-250. CrossRef
Ph. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet Problem. SIAM J. Control. Optim. 41 (2002) 1052-1072.
Ph. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations. Rapport MIP (2001) 01-24.
M. Hassine, Contrôle des processus d'aération des lacs eutrophes. Thesis, Tunis II University, ENIT, Tunisia (2003).
J. Jacobsen, N. Olhoff and E. Ronholt, 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).
J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod (1996).
M. Masmoudi, Outils pour la conception optimale de formes. Thèse d'État, Université de Nice (1987).
M. Masmoudi, The topological asymptotic, in Computational Methods for Control Applications, H. Kawarada and J. Periaux Eds., International Séries GAKUTO (2002).
F. Murat and L. Tartar, Calcul des variations et homogénéisation, in Les méthodes de l'homogénéisation : Théorie et applications en physique. Eyrolles (1985) 319-369.
O. Pironneau, Méthode des éléments finis pour les fluides. Masson, Paris (1988).
O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984).
J. Simon, Domain variation for Stokes flow. X. Li and J. Yang Eds., Springer, Berlin, Lect. Notes Control Inform. Sci. 159 28-42 (1990).
J. Simon, Domain variation for drag Stokes flows. A. Bermudez Eds., Springer, Berlin, Lect. Notes Control Inform. Sci. 114 (1987) 277-283.
A. Schumacher, Topologieoptimierung von bauteilstrukturen unter verwendung von lopchpositionierungkrieterien. Thesis, Universitat-Gesamthochschule-Siegen (1995).
Shœnauer, M., Kallel, L. and Jouve, F., Mechanical inclusions identification by evolutionary computation. Rev. Eur. Élém. Finis 5 (1996) 619-648. CrossRef
Sokolowski, J. and Zochowski, A., On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251-1272 (electronic). CrossRef
R. Temam, Navier Stokes equations (1985).