The topological asymptotic expansion for the Quasi-Stokes problem

Maatoug Hassine; Mohamed Masmoudi

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 4, page 478-504
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Hassine, Maatoug, and Masmoudi, Mohamed. "The topological asymptotic expansion for the Quasi-Stokes problem." ESAIM: Control, Optimisation and Calculus of Variations 10.4 (2010): 478-504. <http://eudml.org/doc/90739>.

@article{Hassine2010,
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. },
author = {Hassine, Maatoug, Masmoudi, Mohamed},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Topological optimization; topological sensitivity; Quasi-Stokes equations; topological gradient; shape optimization.; topological optimization; quasi-Stokes equations; shape optimization},
language = {eng},
month = {3},
number = {4},
pages = {478-504},
publisher = {EDP Sciences},
title = {The topological asymptotic expansion for the Quasi-Stokes problem},
url = {http://eudml.org/doc/90739},
volume = {10},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 4
SP - 478
EP - 504
AB - 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.
LA - eng
KW - Topological optimization; topological sensitivity; Quasi-Stokes equations; topological gradient; shape optimization.; topological optimization; quasi-Stokes equations; shape optimization
UR - http://eudml.org/doc/90739
ER -

References

top
  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 R. Kohn, Optimal bounds on the effective behavior of a mixture of two well-order elastic materials. Quat. Appl. Math.51 (1993) 643-674.  
  3. M. Bendsoe, Optimal topology design of continuum structure: an introduction. Technical report, Department of mathematics, Technical University of Denmark, DK2800 Lyngby, Denmark, September (1996).  
  4. F. Brezzi and M. Fortin, Mixed and hybrid finite element method. Springer Ser. Comput. Math.15 (1991).  
  5. G. Buttazzo and G. Dal Maso, Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991) 17-49.  
  6. J. Céa, A. Gioan and J. Michel, Quelques résultats sur l'identification de domaines. CALCOLO (1973).  
  7. J. Céa, Conception optimale ou identification de forme, calcul rapide de la dérivée directionnelle de la fonction coût. ESAIM: M2AN20 (1986) 371-402.  
  8. J. Céa, S. Garreau, Ph. Guillaume and M. Masmoudi, The shape and Topological Optimizations Connection. Comput. Methods Appl. Mech. Engrg.188 (2000) 713-726.  
  9. 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.  
  10. P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978).  
  11. R. Dautray et J. Lions, Analyse mathémathique et calcul numérique pour les sciences et les techniques. Masson, collection CEA (1987).  
  12. J. Douglas and T.F. Russell, 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.  
  13. M. Fortin, R. Peyret et R. Temam, Résolution numérique des équations de Navier-Stokes pour un fluide incompressible. J. Mécanique10 (1971).  
  14. 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.  
  15. S. Garreau, Ph. Guillaume and M. Masmoudi, The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim.39 (2001) 1756-1778.  
  16. P. Germain and P. Muller, Introduction à la mécanique des milieux continus. Masson (1994).  
  17. V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations, Theory and Algorithms. Springer-Verlag Berlin (1986).  
  18. J. Giroire, Formulations variationnelles par équations intégrales de problèmes aux limites extérieurs. Thèse, École Polytechnique, Palaiseau (1976).  
  19. R. Glowinski, Numerical methods for nonlinear variational problems. J. Optim. Theory Appl. 57 (1988) 407-422.  
  20. R. Glowinski and O. Pironneau, Toward the computational of minimun drag profile in viscous laminar flow. Appl. Math. Model.1 (1976) 58-66.  
  21. P. Grisvard, Elliptic problems in non smooth domains. Pitman Publishing Inc., London (1985).  
  22. Ph. Guillaume and M. Masmoudi, Computation of high order derivatives in optimal shape design. Numer. Math.67 (1994) 231-250.  
  23. Ph. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet Problem. SIAM J. Control. Optim. 41 (2002) 1052-1072.  
  24. Ph. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations. Rapport MIP (2001) 01-24.  
  25. M. Hassine, Contrôle des processus d'aération des lacs eutrophes. Thesis, Tunis II University, ENIT, Tunisia (2003).  
  26. 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).  
  27. J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod (1996).  
  28. M. Masmoudi, Outils pour la conception optimale de formes. Thèse d'État, Université de Nice (1987).  
  29. M. Masmoudi, The topological asymptotic, in Computational Methods for Control Applications, H. Kawarada and J. Periaux Eds., International Séries GAKUTO (2002).  
  30. 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.  
  31. O. Pironneau, Méthode des éléments finis pour les fluides. Masson, Paris (1988).  
  32. O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984).  
  33. J. Simon, Domain variation for Stokes flow. X. Li and J. Yang Eds., Springer, Berlin, Lect. Notes Control Inform. Sci. 159 28-42 (1990).  
  34. J. Simon, Domain variation for drag Stokes flows. A. Bermudez Eds., Springer, Berlin, Lect. Notes Control Inform. Sci. 114 (1987) 277-283.  
  35. A. Schumacher, Topologieoptimierung von bauteilstrukturen unter verwendung von lopchpositionierungkrieterien. Thesis, Universitat-Gesamthochschule-Siegen (1995).  
  36. M. Shœnauer, L. Kallel and F. Jouve, Mechanical inclusions identification by evolutionary computation. Rev. Eur. Élém. Finis5 (1996) 619-648.  
  37. J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization. SIAM J. Control Optim.37 (1999) 1251-1272 (electronic).  
  38. R. Temam, Navier Stokes equations (1985).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.