Numerical homogenization by an adaptive Fourier spectral method on non-uniform grids using optimal transport

Abstract

This article focuses on the computation of the solutions to periodic cell problems that arise in a number of homogenization schemes. Formulated in terms of a generic constitutive relation, encompassing the linear and non-linear cases, a model problem is intended to be discretized on a Fourier-basis and solved using a Fast Fourier Transform (FFT)-based iterative scheme. Such a spectral method, which makes use of a uniform grid and global basis functions, has inherent limitations for correctly capturing any localized features of the solution, such as singularities or discontinuities. An adaptive method is proposed here to overcome these shortcomings. It relies on the introduction of a non-homogeneous bijective mapping between the original physical domain and a computational domain, where the Fourier spectral method will be used to solve a transformed problem. This makes it possible both to discretize the original equations on a non-uniform grid, concentrating grid points where necessary to increase computational accuracy, and to maintain the use of standard FFT schemes. To compute the mapping, an approach based on optimal transport is proposed and a numerical scheme based on FFT tools is described. A number of adaptive strategies are then discussed, either based on a priori material-based considerations, or a posteriori using a preliminary solution. The properties of the transported problem in the computational domain are described and the applicability, up to minor modifications, of a number of standard iterative schemes is discussed. A set of numerical examples is included to assess the performances of the method, both in the case of linear conductivity and for a strongly non-linear elastoplastic problem, evaluating its results both qualitatively with full-field comparisons and quantitatively with error measurements relatively to the number of iterations, the discretization and the original material contrast. Overall, these results show that, for a given discretization and compared with a computation on a regular grid, a systematic gain in accuracy is obtained using the adaptive method on a non-uniform grid.