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A multiscale correction method for local singular perturbations of the boundary

Published online by Cambridge University Press:  26 April 2007

Marc Dambrine  and
Grégory Vial
Marc Dambrine
Affiliation:
LMAC, Université de Technologie de Compiègne, France.
Grégory Vial
Affiliation:
IRMAR, Antenne de Bretagne de l'ENS Cachan, France. [email protected]
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Abstract

In this work, we consider singular perturbations of the boundary of a smooth domain. We describe the asymptotic behavior of the solution uE of a second order elliptic equation posed in the perturbed domain with respect to the size parameter ε of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of uE based on a multiscale superposition of the unperturbed solution u 0 and a profile defined in a model domain. We conclude with numerical results.

Keywords

Multiscale asymptotic analysisshape optimizationpatch of elements.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 1 , January 2007 , pp. 111 - 127
Copyright
© EDP Sciences, SMAI, 2007

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References

G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences 146. Springer-Verlag, New York (2002).
D. Brancherie and A. Ibrahimbegović, Modélisation `macro' de phénomènes localisés à l'échelle `micro' : formulation et implantation numérique. Revue européenne des éléments finis, numéro spécial Giens 2003 13 (2004) 461–473. CrossRef
D. Brancherie, M. Dambrine, G. Vial and P. Villon, Ultimate load computation, effect of surfacic defect and adaptative techniques, in 7th World Congress in Computational Mechanics, Los Angeles (2006).
Caloz, G., Costabel, M., Dauge, M. and Vial, G., Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. Asymptotic Anal. 50 (2006) 121–173.
Dambrine, M. and Vial, G., On the influence of a boundary perforation on the dirichlet energy. Control Cybern. 34 (2005) 117–136.
Engquist, B. and Majda, A., Absorbing boundary conditions for the numerical simulation of waves. Math. Comp. 31 (1977) 629–651. CrossRef
Givoli, D., Nonreflecting boundary conditions. J. Comput. Phys. 94 (1991) 1–29. CrossRef
A.M. Il'lin, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Translations of Mathematical Monographs 102, Amer. Math. Soc., Providence, R.I. (1992).
Kondrat'ev, V.A., Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16 (1967) 227–313.
D. Leguillon and E. Sanchez-Palencia, Computation of singular solutions in elliptic problems and elasticity. Masson, Paris (1987).
Lenoir, M. and Tounsi, A., The localized finite element method and its application to the two-dimensional sea-keeping problem. SIAM J. Numer. Anal. 25 (1988) 729–752. CrossRef
T. Lewiński and J. Sokołowski, Topological derivative for nucleation of non-circular voids. The Neumann problem, in Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999), Contemp. Math. 268, Amer. Math. Soc., Providence, RI (2000) 341–361.
M. Masmoudi, The Topological Asymptotic, in Computational Methods for Control Applications, International Séries GAKUTO (2002).
Maz'ya, V.G. and Nazarov, S.A., Asymptotic behavior of energy integrals under small perturbations of the boundary near corner and conic points. Trudy Moskov. Mat. Obshch. 50 (1987) 79–129, 261.
V.G. Maz'ya, S.A. Nazarov and B.A. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Birkhäuser, Berlin (2000).
Nazarov, S.A. and Olyushin, M.V., Perturbation of the eigenvalues of the Neumann problem due to the variation of the domain boundary. Algebra i Analiz 5 (1993) 169–188.
Nazarov, S.A. and Sokołowski, J., Asymptotic analysis of shape functionals. J. Math. Pures Appl. 82 (2003) 125–196. CrossRef
S. Tordeux and G. Vial, Matching of asymptotic expansions and multiscale expansion for the rounded corner problem. SAM Research Report, ETH, Zürich (2006).