Issue |
ESAIM: ProcS
Volume 45, September 2014
Congrès SMAI 2013
|
|
---|---|---|
Page(s) | 18 - 31 | |
DOI | https://doi.org/10.1051/proc/201445003 | |
Published online | 13 November 2014 |
Homogenization theory and multiscale numerical approaches for disordered media: some recent contributions*,**
Ecole des Ponts and
INRIA, 6 & 8 avenue Blaise
Pascal, 77455
Marne-la-Vallée Cedex 2,
France.
lebris@cermics.enpc.fr
We overview a series of recent works related to some multiscale problems motivated by practical problems in Mechanics. The common denominator of all these works is that they address multiscale problems where the geometry of the microstructures is not periodic. Random modelling, as well as other types of nonperiodic modelling, can then be used to account for the imperfections of the medium. The theory at play is that of homogenization, in its many variants (stochastic, general deterministic, periodic). The numerical methods developed and adapted are finite element type methods. A special emphasis is laid on situations where the amount of randomness is small, or, put differently, when the disorder is limited. Then, specific, computationally efficient techniques can be designed and employed.
Résumé
Nous présentons un panorama d’une série de travaux récents sur des problèmes multi-échelles motivés par la science des matériaux. Le dénominateur commun de ces travaux est qu’ils traitent tous de problèmes où la géométrie des microstructures n’est pas périodique. La modélisation aléatoire, ainsi que d’autres types de modélisations non periodiques, sont alors utilisées pour rendre compte des imperfections du matériau. Le cadre théorique est celui de l’homogénéisation dans ses nombreuses variantes (stochastique, déterministique générale, périodique). Les méthodes numériques développées et adaptées sont des méthodes de type éléments finis. Une attention toute particulière est portée sur les problèmes où la “quantité de hasard” est petite ou, autrement dit, le désordre est limité. Des techniques spécifiques, efficaces pour le calcul, peuvent alors être définies et utilisées.
The work of the author is partially supported by EOARD under Grant (No. FA8655-13-1-3061) and by ONR under Grant (No. N00014-12-1-0383).
The author would like to thank his many friends and collaborators on the issues presented here, in particular X. Blanc (Paris 7), P.-L. Lions (Collège de France), F. Legoll, W. Minvielle (Ecole des Ponts and Inria), A. Lozinski (Besançon). The material covered in this survey article has been presented as a plenary lecture at SMAI 2013. The author is grateful to the organizing and scientific committees for providing this opportunity to present his work.
© EDP Sciences, SMAI 2014
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