An embedded corrector problem for homogenization. Part II: Algorithms and discretization

JOURNAL OF COMPUTATIONAL PHYSICS

卷:407

Article

Cances, Eric^1,3  Ehrlacher, Virginie^1,3  Legoll, Frederic^2,3  Stamm, Benjamin⁴  Xiang, Shuyang⁴ 

[1] Ecole Ponts ParisTech, CERMICS, F-77455 Marne La Vallee 2, France

[2] Ecole Ponts ParisTech, Lab Navier, F-77455 Marne La Vallee 2, France

[3] Inria Paris, MATHERIALS Project Team, 2 Rue Simone Iff,CS 42112, F-75589 Paris 12, France

[4] Rhein Westfal TH Aachen, Dept Math, MATHCCES, Schinkelstr 2, D-52062 Aachen, Germany

关键词: Homogenization;    Corrector problem;    Integral equation;    Numerical discretization;    Spherical harmonics;   

DOI  :  10.1016/j.jcp.2020.109254

来源: Elsevier

【 摘 要 】

This contribution is the numerically oriented companion article of the work [9]. We focus here on the numerical resolution of the embedded corrector problem introduced in [8, 9] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [9], we have shown how to define three approximate homogenized diffusion coefficients on the basis of the embedded corrector problem. We have also proved that these approximations all converge to the exact homogenized coefficients when the size of the bounded domain increases. We show here that, under the assumption that the diffusion matrix is piecewise constant, the corrector problem to solve can be recast as an integral equation. In case of spherical inclusions with isotropic materials, we explain how to efficiently discretize this integral equation using spherical harmonics, and how to use the fast multipole method (FMM) to compute the resulting matrix-vector products at a cost which scales only linearly with respect to the number of inclusions. Numerical tests illustrate the performance of our approach in various settings. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free