The topological asymptotic expansion for the Quasi-Stokes problem

Maatoug Hassine; Mohamed Masmoudi

ESAIM: Control, Optimisation and Calculus of Variations (2004)

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

Abstract

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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

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Hassine, Maatoug, and Masmoudi, Mohamed. "The topological asymptotic expansion for the Quasi-Stokes problem." ESAIM: Control, Optimisation and Calculus of Variations 10.4 (2004): 478-504. <http://eudml.org/doc/245466>.

@article{Hassine2004,
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},
language = {eng},
number = {4},
pages = {478-504},
publisher = {EDP-Sciences},
title = {The topological asymptotic expansion for the Quasi-Stokes problem},
url = {http://eudml.org/doc/245466},
volume = {10},
year = {2004},
}

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
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
UR - http://eudml.org/doc/245466
ER -

References

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  1. [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. [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. Zbl0805.73043MR1247433
  3. [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. [4] F. Brezzi and M. Fortin, Mixed and hybrid finite element method. Springer Ser. Comput. Math. 15 (1991). Zbl0788.73002MR1115205
  5. [5] G. Buttazzo and G. Dal Maso, Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991) 17-49. Zbl0762.49017MR1076053
  6. [6] J. Céa, A. Gioan and J. Michel, Quelques résultats sur l’identification de domaines. CALCOLO (1973). Zbl0303.93023
  7. [7] J. Céa, Conception optimale ou identification de forme, calcul rapide de la dérivée directionnelle de la fonction coût. ESAIM: M2AN 20 (1986) 371-402. Zbl0604.49003MR862783
  8. [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. Zbl0972.74057MR1784106
  9. [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. Zbl0769.35013MR1161483
  10. [10] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). Zbl0383.65058MR520174
  11. [11] R. Dautray et J. Lions, Analyse mathémathique et calcul numérique pour les sciences et les techniques. Masson, collection CEA (1987). Zbl0642.35001
  12. [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. Zbl0492.65051MR672564
  13. [13] M. Fortin, R. Peyret et R. Temam, Résolution numérique des équations de Navier-Stokes pour un fluide incompressible. J. Mécanique 10 (1971). Zbl0225.76016MR421338
  14. [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. [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. Zbl0990.49028MR1825864
  16. [16] P. Germain and P. Muller, Introduction à la mécanique des milieux continus. Masson (1994). Zbl0465.73001MR576236
  17. [17] V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations, Theory and Algorithms. Springer-Verlag Berlin (1986). Zbl0585.65077MR851383
  18. [18] J. Giroire, Formulations variationnelles par équations intégrales de problèmes aux limites extérieurs. Thèse, École Polytechnique, Palaiseau (1976). 
  19. [19] R. Glowinski, Numerical methods for nonlinear variational problems. J. Optim. Theory Appl. 57 (1988) 407-422. MR859924
  20. [20] R. Glowinski and O. Pironneau, Toward the computational of minimun drag profile in viscous laminar flow. Appl. Math. Model. 1 (1976) 58-66. Zbl0361.76035MR455851
  21. [21] P. Grisvard, Elliptic problems in non smooth domains. Pitman Publishing Inc., London (1985). Zbl0695.35060
  22. [22] Ph. Guillaume and M. Masmoudi, Computation of high order derivatives in optimal shape design. Numer. Math. 67 (1994) 231-250. Zbl0792.65044MR1262782
  23. [23] Ph. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet Problem. SIAM J. Control. Optim. 41 (2002) 1052-1072. Zbl1053.49031MR1972502
  24. [24] Ph. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations. Rapport MIP (2001) 01-24. 
  25. [25] M. Hassine, Contrôle des processus d’aération des lacs eutrophes. Thesis, Tunis II University, ENIT, Tunisia (2003). 
  26. [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. [27] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod (1996). Zbl0165.10801
  28. [28] M. Masmoudi, Outils pour la conception optimale de formes. Thèse d’État, Université de Nice (1987). 
  29. [29] M. Masmoudi, The topological asymptotic, in Computational Methods for Control Applications, H. Kawarada and J. Periaux Eds., International Séries GAKUTO (2002). Zbl1082.93584
  30. [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. [31] O. Pironneau, Méthode des éléments finis pour les fluides. Masson, Paris (1988). Zbl0748.76003
  32. [32] O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984). Zbl0534.49001MR725856
  33. [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). Zbl0801.76075MR1129956
  34. [34] J. Simon, Domain variation for drag Stokes flows. A. Bermudez Eds., Springer, Berlin, Lect. Notes Control Inform. Sci. 114 (1987) 277-283. Zbl0801.76075
  35. [35] A. Schumacher, Topologieoptimierung von bauteilstrukturen unter verwendung von lopchpositionierungkrieterien. Thesis, Universitat-Gesamthochschule-Siegen (1995). 
  36. [36] M. Shœnauer, L. Kallel and F. Jouve, Mechanical inclusions identification by evolutionary computation. Rev. Eur. Élém. Finis 5 (1996) 619-648. Zbl0924.73321
  37. [37] J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251-1272 (electronic). Zbl0940.49026MR1691940
  38. [38] R. Temam, Navier Stokes equations (1985). 

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