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Optimal design in small amplitude homogenization

Published online by Cambridge University Press:  02 August 2007

Grégoire Allaire
Centre de Mathématiques Appliquées (UMR 7641), École Polytechnique, 91128 Palaiseau, France.
Sergio Gutiérrez
Centre de Mathématiques Appliquées (UMR 7641), École Polytechnique, 91128 Palaiseau, France. : Departamento de Ingeniería Estructural y Geotécnica, Pontificia Universidad Católica de Chile, Santiago Chile, Chile.
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This paper is concerned with optimal design problems with a special assumption on the coefficients of the state equation. Namely we assume that the variations of these coefficients have a small amplitude. Then, making an asymptotic expansion up to second order with respect to the aspect ratio of the coefficients allows us to greatly simplify the optimal design problem. By using the notion of H-measures we are able to prove general existence theorems for small amplitude optimal design and to provide simple and efficient numerical algorithms for their computation. A key feature of this type of problems is that the optimal microstructures are always simple laminates.

Research Article
© EDP Sciences, SMAI, 2007

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