Mathematical analysis of a coupling method for the practical computation of homogenized coefficients

TL;DR

This paper provides a rigorous mathematical analysis of a coupling method for computing homogenized coefficients, proving its well-posedness and asymptotic correctness for media with fine-scale structures.

Contribution

It establishes the mathematical validity and convergence properties of Cottereau's coupling approach for homogenized coefficient computation.

Findings

The method is mathematically well-posed.

It converges to the true homogenized coefficient as the structure becomes infinitely fine.

The approach complements existing numerical methods.

Abstract

We present the mathematical study of a computational approach originally introduced by R. Cottereau in [R. Cottereau, IJNME 2013]. The approach aims at evaluating the effective (a.k.a. homogenized) coefficient of a medium with some fine-scale structure. It combines, using the Arlequin coupling method, the original fine-scale description of the medium with an effective description and optimizes upon the coefficient of the effective medium to best fit the response of an equivalent purely homogeneous medium. We prove here that the approach is mathematically well-posed and that it provides, under suitable assumptions, the actual value of the homogenized coefficient of the original medium in the limit of asymptotically infinitely fine structures. The theory presented here therefore usefully complements our numerical developments of [O. Gorynina, C. Le Bris and F. Legoll, SIAM J. Sci. Computing 2021].