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Numerical methods for homogenization : applications to random media

Abstract : In this thesis we investigate numerical methods for the homogenization of materials the structures of which, at fine scales, are characterized by random heterogenities. Under appropriate hypotheses, the effective properties of such materials are given by closed formulas. However, in practice the computation of these properties is a difficult task because it involves solving partial differential equations with stochastic coefficients that are additionally posed on the whole space. In this work, we address this difficulty in two different ways. The standard discretization techniques lead to random approximate effective properties. In Part I, we aim at reducing their variance, using a well-known variance reduction technique that has already been used successfully in other domains. The works of Part II focus on the case when the material can be seen as a small random perturbation of a periodic material. We then show both numerically and theoretically that, in this case, computing the effective properties is much less costly than in the general case
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https://pastel.archives-ouvertes.fr/pastel-00674957
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Ronan Costaouec. Numerical methods for homogenization : applications to random media. General Mathematics [math.GM]. Université Paris-Est, 2011. English. ⟨NNT : 2011PEST1012⟩. ⟨pastel-00674957⟩

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