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.
Classification : 35J60, 49Q10, 49Q12, 76D05, 76D55
Mots clés : shape optimization, topological asymptotic, Navier-Stokes equations
@article{COCV_2005__11_3_401_0,
author = {Amstutz, Samuel},
title = {The topological asymptotic for the {Navier-Stokes} equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {401--425},
publisher = {EDP-Sciences},
volume = {11},
number = {3},
year = {2005},
doi = {10.1051/cocv:2005012},
zbl = {1123.35040},
mrnumber = {2148851},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2005012/}
}
TY - JOUR AU - Amstutz, Samuel TI - The topological asymptotic for the Navier-Stokes equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2005 DA - 2005/// SP - 401 EP - 425 VL - 11 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2005012/ UR - https://zbmath.org/?q=an%3A1123.35040 UR - https://www.ams.org/mathscinet-getitem?mr=2148851 UR - https://doi.org/10.1051/cocv:2005012 DO - 10.1051/cocv:2005012 LA - en ID - COCV_2005__11_3_401_0 ER -
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] , 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] , 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] , Shape optimization by the homogenization method. Springer, Appl. Math. Sci. 146 (2002). | MR 1859696 | Zbl 0990.35001
[4] , The topological asymptotic for the Helmholtz equation: insertion of a hole, a crack and a dielectric object. Rapport MIP No. 03-05 (2003).
[5] , Optimal topology design of continuum structure: an introduction. Technical report, Departement of mathematics, Technical University of Denmark, DK2800 Lyngby, Denmark (1996).
[6] and, Analyse mathématique et calcul numérique pour les sciences et les techniques. Masson, collection CEA 6 (1987). | MR 918560 | Zbl 0642.35001
[7] and, 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] , 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] , and, The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778. | Zbl 0990.49028
[10] and, The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control. Optim. 41 (2002) 1052-1072. | Zbl 1053.49031
[11] and, Topological sensitivity and shape optimization for the Stokes equations. Rapport MIP No. 01-24 (2001). | Zbl 1093.49029
[12] and, The topological asymptotic expansion for the quasi-Stokes problem. ESAIM: COCV 10 (2004) 478-504. | Numdam | Zbl 1072.49027
[13] , Matching of asymptotic expansions of solutions of boundary value problems. Translations Math. Monographs 102 (1992). | Zbl 0754.34002
[14] , and, 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] , 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] , and, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Birkhäuser Verlag, Oper. Theory Adv. Appl. 101 (2000). | Zbl 1127.35301
[17] , and, 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] , and, 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] and, Approximation of exterior boundary value problems for the Stokes system. Asymptotic Anal. 14 (1997) 223-255. | Zbl 0884.35130
[20] , and, Nonlinear artificial boundary conditions for the Navier-Stokes equations in an aperture domain. Math. Nachr. 265 (2004) 24-67. | Zbl 1042.35050
[21] , and, The topological asymptotic for the Helmholtz equation. SIAM J. Control Optim. 42 (2003) 1523-1544. | Zbl 1051.49029
[22] and, 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] , Topologieoptimisierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien. Thesis, Universität-Gesamthochschule-Siegen (1995).
[24] , Sensibilité topologique en optimisation de forme. Thèse de l'INSA Toulouse (2001).
[25] and, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1241-1272. | Zbl 0940.49026
[26] , Navier-Stokes equations. Elsevier (1984). | MR 769654
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