Periodic homogenization using the Lippmann--Schwinger formalism

TL;DR

This paper reviews and analyzes numerical methods based on the Lippmann--Schwinger formalism for periodic homogenization of elliptic PDEs, providing convergence proofs and error estimates, with an application to 3D elasticity.

Contribution

It offers a unified mathematical analysis of existing schemes and introduces a rigorous convergence framework for discretized Lippmann--Schwinger equations in homogenization.

Findings

Most variants are approximations of the non-local term in the integral equation.

Proved convergence of schemes with respect to grid size.

Provided a priori error estimates for the solutions.

Abstract

When homogenizing elliptic partial differential equations, the so-called corrector problem is pivotal to compute the macroscale effective coefficients from the microscale information. To solve this corrector problem in the periodic setting, Moulinec and Suquet introduced in the mid-nineties a numerical strategy based on the reformulation of that problem as an integral equation (known as the Lippmann--Schwinger equation), which is then suitably discretized. This results in an iterative, matrix-free method, which is of particular interest for complex microstructures. Since the seminal work of Moulinec and Suquet, several variants of their scheme have been proposed. The aim of this contribution is twofold. First, we provide an overview of these methods, recast in the language of the applied mathematics community. These methods are presented as asymptotically consistent Galerkin discretizations of the Lippmann--Schwinger equation. The bilinear form arising in the weak form of this integral equation is indeed the sum of a local and a non-local term. We show that most of the variants proposed in the literature correspond to alternative approximations of this non-local term. Second, we propose a mathematical analysis of the discretized problem. In particular, we prove under mild hypotheses the convergence of these numerical schemes with respect to the grid-size. We also provide a priori error estimates on the solution. The article closes on a three-dimensional numerical application within the framework of linear elasticity.