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Numerical methods for the study of fluctuations in multi-scale materials and related problems

Abstract : This thesis is about the numerical approximation of multi-scale materials. We consider heterogeneous materials whose physical or mechanical (thermal conductivity, elasticity tensor, . . . ) vary on a small scale compared to the material length. This thesis is composed of two parts describing two different aspects of multiscale problems. In the first part, we consider the stochastic homogenization framework. The aim here is to go beyond the identification of an effective behavior, by attempting to characterize the fluctuations of the response. Generally speaking we strive to understand: (i) what parameters of the distribution of the material coefficient affect the distribution of the response and (ii) if it is possible to approximate this distribution without resorting to a costly Monte-Carlo method. On the theoretical standpoint, we consider a weakly random material (the micro-structure is periodic and presents some small random defects). We show that we are able to compute a tensor Q that governs completely the fluctuations of the response, thanks to the use of standard corrector functions from the stochastic homogenization theory. This tensor is defined by an explicit formula and allows us to estimate the fluctuation of the response without solving the fine problem for many realizations. A numerical approximation of this tensor has been proposed and numerical experiments have been performed in broader random frameworks to assess the effectiveness of the approach. In the second part, we consider a heterogeneous deterministic material where classical homogenization (periodicity, . . . ) assumptions are not satisfied. Standard methods such as Finite Elements give bad approximations. In order to solve this issue the Multiscale Finite Element Method (MsFEM) can be used. This approach proceeds in two steps: (i) design a coarse approximation space spanned by solutions to well-chosen local problems; (ii) approximate the solution by an inexpensive Galerkin approach on the space designed in (i). On this topic, we first implemented the main variants of the MsFEM methods in the Finite Element software FreeFem++ on template form. Second, many MsFEM approaches suffer from resonance error: when the size of the heterogeneities is close to the coarse mesh size the accuracy decreases. In order to circumvent this issue, we designed an enriched MsFEM method: to the classical Ms-FEM basis, we add solutions to local problems with high degree polynomial boundary conditions. The use of polynomials allows us to obtain a converging approach for a limited computational cost.
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Pierre-Loïk Rothé. Numerical methods for the study of fluctuations in multi-scale materials and related problems. Analysis of PDEs [math.AP]. Université Paris-Est Marne la Vallée, 2019. English. ⟨tel-02447725⟩

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