The topological asymptotic for the Navier-Stokes equations

Samuel Amstutz

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

  • Volume: 11, Issue: 3, page 401-425
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Amstutz, Samuel. "The topological asymptotic for the Navier-Stokes equations." ESAIM: Control, Optimisation and Calculus of Variations 11.3 (2010): 401-425. <http://eudml.org/doc/90770>.

@article{Amstutz2010,
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. },
author = {Amstutz, Samuel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Shape optimization; topological asymptotic; Navier-Stokes equations.; asymptotic expansion of a shape functional; shape optimization; Navier-Stokes equations; no-slip condition},
language = {eng},
month = {3},
number = {3},
pages = {401-425},
publisher = {EDP Sciences},
title = {The topological asymptotic for the Navier-Stokes equations},
url = {http://eudml.org/doc/90770},
volume = {11},
year = {2010},
}

TY - JOUR
AU - Amstutz, Samuel
TI - The topological asymptotic for the Navier-Stokes equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 3
SP - 401
EP - 425
AB - 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.
LA - eng
KW - Shape optimization; topological asymptotic; Navier-Stokes equations.; asymptotic expansion of a shape functional; shape optimization; Navier-Stokes equations; no-slip condition
UR - http://eudml.org/doc/90770
ER -

References

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  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.  
  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.  
  3. G. Allaire, Shape optimization by the homogenization method. Springer, Appl. Math. Sci.146 (2002).  
  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).  
  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.  
  8. G. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vols. I and II, Springer-Verlag 39 (1994).  
  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.  
  10. Ph. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control. Optim.41 (2002) 1052–1072.  
  11. Ph. Guillaume and K. Sid Idris, Topological sensitivity and shape optimization for the Stokes equations. Rapport MIP No. 01–24 (2001).  
  12. M. Hassine and M. Masmoudi, The topological asymptotic expansion for the quasi-Stokes problem. ESAIM: COCV10 (2004) 478–504.  
  13. A.M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems. Translations Math. Monographs102 (1992).  
  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.  
  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).  
  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.  
  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.  
  19. S. Nazarov and M. Specovius-Neugebauer, Approximation of exterior boundary value problems for the Stokes system. Asymptotic Anal.14 (1997) 223–255.  
  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.  
  21. B. Samet, S. Amstutz and M. Masmoudi, The topological asymptotic for the Helmholtz equation. SIAM J. Control Optim.42 (2003) 1523–1544.  
  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.  
  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.  
  26. R. Temam, Navier-Stokes equations. Elsevier (1984).  

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