Multiscale expansion and numerical approximation for surface defects

V. Bonnaillie-Noël; D. Brancherie; M. Dambrine; F. Hérau; S. Tordeux; G. Vial

doi:10.1051/proc/201133003

Geodesic

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Multiscale expansion and numerical approximation for surface defects

V. Bonnaillie-Noël ¹ ; D. Brancherie ² ; M. Dambrine ³ ; F. Hérau ⁴ ; S. Tordeux ⁵ ; G. Vial ⁶

¹ IRMAR - UMR6625, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av Robert Schuman, 35170 Bruz, France
² Laboratoire Roberval - UMR6253, Université de Technologie de Compiègne, rue Personne de Roberval, BP 20529, 60205 Compiègne Cedex, France
³ LMAP - UMR5142, Université de Pau et des Pays de l’Adour, av de l’Université, BP 1155, 64013 Pau Cedex, France
⁴ Laboratoire de mathématiques Jean Leray - UMR6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
⁵ INRIA - IPRA, Université de Pau et des Pays de l’Adour, av de l’Université, BP 1155, 64013 Pau Cedex, France
⁶ Université de Lyon - CNRS UMR 5208, École Centrale de Lyon, Institut Camille Jordan, 36 avenue Guy de Collongue, 69134 Ecully Cedex, France

ESAIM. Proceedings, Tome 33 (2011), pp. 22-35.

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Résumé

This paper is a survey of articles [5, 6, 8, 9, 13, 17, 18]. We are interested in the influence of small geometrical perturbations on the solution of elliptic problems. The cases of a single inclusion or several well-separated inclusions have been deeply studied. We recall here techniques to construct an asymptotic expansion. Then we consider moderately close inclusions, i.e. the distance between the inclusions tends to zero more slowly than their characteristic size. We provide a complete asymptotic description of the solution of the Laplace equation. We also present numerical simulations based on the multiscale superposition method derived from the first order expansion (cf [9]). We give an application of theses techniques in linear elasticity to predict the behavior till rupture of materials with microdefects (cf [6]). We explain how some mathematical questions about the loss of coercivity arise from the computation of the profiles appearing in the expansion (cf [8]).

DOI : 10.1051/proc/201133003

Affiliations des auteurs :

V. Bonnaillie-Noël ¹ ; D. Brancherie ² ; M. Dambrine ³ ; F. Hérau ⁴ ; S. Tordeux ⁵ ; G. Vial ⁶

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     title = {Multiscale expansion and numerical approximation for surface defects},
     journal = {ESAIM. Proceedings},
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     doi = {10.1051/proc/201133003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201133003/}
}

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V. Bonnaillie-Noël; D. Brancherie; M. Dambrine; F. Hérau; S. Tordeux; G. Vial. Multiscale expansion and numerical approximation for surface defects. ESAIM. Proceedings, Tome 33 (2011), pp. 22-35. doi : 10.1051/proc/201133003. http://geodesic.mathdoc.fr/articles/10.1051/proc/201133003/

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