The topological asymptotic for the Navier-Stokes equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 3, pp. 401-425.

The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional and a numerical application is presented.

DOI : https://doi.org/10.1051/cocv:2005012
Classification : 35J60,  49Q10,  49Q12,  76D05,  76D55
Mots clés : shape optimization, topological asymptotic, Navier-Stokes equations
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Amstutz, Samuel. The topological asymptotic for the Navier-Stokes equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 3, pp. 401-425. doi : 10.1051/cocv:2005012. http://www.numdam.org/articles/10.1051/cocv:2005012/

[1] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. Arch. Rational Mech. Anal. 113 (1990) 209-259. | Zbl 0724.76020

[2] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes. Arch. Rational Mech. Anal. 113 (1990) 261-298. | Zbl 0724.76021

[3] G. Allaire, Shape optimization by the homogenization method. Springer, Appl. Math. Sci. 146 (2002). | MR 1859696 | Zbl 0990.35001

[4] S. Amstutz, The topological asymptotic for the Helmholtz equation: insertion of a hole, a crack and a dielectric object. Rapport MIP No. 03-05 (2003).

[5] M. Bendsoe, Optimal topology design of continuum structure: an introduction. Technical report, Departement of mathematics, Technical University of Denmark, DK2800 Lyngby, Denmark (1996).

[6] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Masson, collection CEA 6 (1987). | MR 918560 | Zbl 0642.35001

[7] A. Friedman and M.S. Vogelius, Identification of small inhomogeneities of extreme conductivity byboundary measurements: a theorem of continuous dependence. Arch. Rational Mech. Anal. 105 (1989) 299-326. | Zbl 0684.35087

[8] G. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vols. I and II, Springer-Verlag 39 (1994). | MR 1284205 | Zbl 0949.35005

[9] S. Garreau, Ph. Guillaume and M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778. | Zbl 0990.49028

[10] Ph. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control. Optim. 41 (2002) 1052-1072. | Zbl 1053.49031

[11] Ph. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations. Rapport MIP No. 01-24 (2001). | Zbl 1093.49029

[12] M. Hassine and M. Masmoudi, The topological asymptotic expansion for the quasi-Stokes problem. ESAIM: COCV 10 (2004) 478-504. | Numdam | Zbl 1072.49027

[13] A.M. Il'In, Matching of asymptotic expansions of solutions of boundary value problems. Translations Math. Monographs 102 (1992). | Zbl 0754.34002

[14] J. Jacobsen, N. Olhoff and E. Ronholt, 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).

[15] M. Masmoudi, The Toplogical Asymptotic, Computational Methods for Control Applications, R. Glowinski, H. Kawarada and J. Periaux Eds. GAKUTO Internat. Ser. Math. Sci. Appl. 16 (2001) 53-72. | Zbl 1082.93584

[16] V. Mazya, S. Nazarov and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Birkhäuser Verlag, Oper. Theory Adv. Appl. 101 (2000). | Zbl 1127.35301

[17] S. Nazarov, A. Sequeira and J. Videman, Steady flows of Jeffrey-Hamel type from the half plane into an infinite channel. Linearization on an antisymmetric solution. J. Math. Pures Appl. 80 (2001) 1069-1098. | Zbl 1042.76013

[18] S. Nazarov, A. Sequeira and J. Videman, Steady flows of Jeffrey-Hamel type from the half plane into an infinite channel. Linearization on a symmetric solution. J. Math. Pures Appl. 81 (2001) 781-810. | Zbl 1043.76015

[19] S. Nazarov and M. Specovius-Neugebauer, Approximation of exterior boundary value problems for the Stokes system. Asymptotic Anal. 14 (1997) 223-255. | Zbl 0884.35130

[20] S. Nazarov, M. Specovius-Neugebauer and J. Videman, Nonlinear artificial boundary conditions for the Navier-Stokes equations in an aperture domain. Math. Nachr. 265 (2004) 24-67. | Zbl 1042.35050

[21] B. Samet, S. Amstutz and M. Masmoudi, The topological asymptotic for the Helmholtz equation. SIAM J. Control Optim. 42 (2003) 1523-1544. | Zbl 1051.49029

[22] B. Samet and J. Pommier, The topological asymptotic for the Helmholtz equation with Dirichlet condition on the boundary of an arbitrary shaped hole. SIAM J. Control Optim. 43 (2004) 899-921. | Zbl 1080.49028

[23] A. Schumacher, Topologieoptimisierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien. Thesis, Universität-Gesamthochschule-Siegen (1995).

[24] K. Sid Idris, Sensibilité topologique en optimisation de forme. Thèse de l'INSA Toulouse (2001).

[25] J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1241-1272. | Zbl 0940.49026

[26] R. Temam, Navier-Stokes equations. Elsevier (1984). | MR 769654

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