Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T03:45:01.492Z Has data issue: false hasContentIssue false
  • English
  • Français

The topological asymptotic for the Navier-Stokes equations

Published online by Cambridge University Press:  15 July 2005

Samuel Amstutz
Samuel Amstutz*
Affiliation:
Mathématiques pour l'Industrie et la Physique, UMR 5640, CNRS-Université Paul Sabatier-INSA, 118 route de Narbonne, 31062 Toulouse Cedex 4, France; amstutz@mip.ups-tlse.fr
Get access

Abstract

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.

Keywords

Shape optimizationtopological asymptoticNavier-Stokes equations.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 3 , July 2005 , pp. 401 - 425
Copyright
© EDP Sciences, SMAI, 2005

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

Allaire, G., 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. CrossRef
Allaire, G., 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. CrossRef
G. Allaire, Shape optimization by the homogenization method. Springer, Appl. Math. Sci. 146 (2002).
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).
M. Bendsoe, Optimal topology design of continuum structure: an introduction. Technical report, Departement of mathematics, Technical University of Denmark, DK2800 Lyngby, Denmark (1996).
R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Masson, collection CEA 6 (1987).
Friedman, A. and Vogelius, M.S., Identification of small inhomogeneities of extreme conductivity byboundary measurements: a theorem of continuous dependence. Arch. Rational Mech. Anal. 105 (1989) 299–326. CrossRef
G. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vols. I and II, Springer-Verlag 39 (1994).
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
Ph. Guillaume, K. Sid Idris, The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control. Optim. 41 (2002) 1052–1072. CrossRef
Ph. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations. Rapport MIP No. 01–24 (2001).
Hassine, M. and Masmoudi, M., The topological asymptotic expansion for the quasi-Stokes problem. ESAIM: COCV 10 (2004) 478–504. CrossRef
A.M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems. Translations Math. Monographs 102 (1992).
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).
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.
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).
Nazarov, S., Sequeira, A. and Videman, J., 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. CrossRef
Nazarov, S., Sequeira, A. and Videman, J., 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. CrossRef
Nazarov, S. and Specovius-Neugebauer, M., Approximation of exterior boundary value problems for the Stokes system. Asymptotic Anal. 14 (1997) 223–255.
Nazarov, S., Specovius-Neugebauer, M. and Videman, J., Nonlinear artificial boundary conditions for the Navier-Stokes equations in an aperture domain. Math. Nachr. 265 (2004) 24–67. CrossRef
Samet, B., Amstutz, S. and Masmoudi, M., The topological asymptotic for the Helmholtz equation. SIAM J. Control Optim. 42 (2003) 1523–1544. CrossRef
Samet, B. and Pommier, J., 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.
A. Schumacher, Topologieoptimisierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien. Thesis, Universität-Gesamthochschule-Siegen (1995).
K. Sid Idris, Sensibilité topologique en optimisation de forme. Thèse de l'INSA Toulouse (2001).
Sokolowski, J. and Zochowski, A., On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1241–1272. CrossRef
R. Temam, Navier-Stokes equations. Elsevier (1984).