Improved guaranteed computable bounds on homogenized properties of periodic media by the Fourier-Galerkin method with exact integration

Abstract

Moulinec and Suquet introduced FFT-based homogenization in 1994, and 20 years later, their approach is still effective for evaluating the homogenized properties arising from the periodic cell problem. This paper builds on the author's (2013) variational reformulation approximated by trigonometric polynomials establishing two numerical schemes: Galerkin approximation (Ga) and a version with numerical integration (GaNi). The latter approach, fully equivalent to the original Moulinec-Suquet algorithm, was used to evaluate guaranteed upper-lower bounds on homogenized coefficients incorporating a closed-form doublegrid quadrature. Here, these concepts, based on the primal and dual formulations, are employed for the Ga scheme. For the same computational effort, the Ga outperforms the GaNi with more accurate guaranteed bounds and more predictable numerical behaviors. The quadrature technique leading to block-sparse linear systems is extended here to materials defined via high-resolution images in a way that allows for effective treatment using the FFT. Memory demands are reduced by a reformulation of the double-grid scheme to the original grid scheme using FFT shifts. Minimization of the bounds during iterations of conjugate gradients is effective, particularly when incorporating a solution from a coarser grid. The methodology presented here for the scalar linear elliptic problem could be extended to more complex frameworks.

Double Grid Integration with Projection (DoGIP): Application to numerical homogenization by Fourier‐Galerkin method

“…defined on a trial space V = ∇H The Fourier-Galerkin method, described for homogenisation in [1][2][3][4][5], belongs to FFT-based methods introduced in [6] and investigated and developed to many different schemes such as [7][8][9]. is based on Galerkin approximation with trigonometric polynomials of uniform order N = (n, .…”

“…, which components can be evaluated using FFT algorithm for many material coefficients A, see [1,3] for details. Still, it is impossible to directly derive linear system because test functions v 2N −1 do not span the whole space…”

“…where the linear operator G : R d×N → R d×N enforces the compatibility condition (curl-free and zero-mean condition of gradient functions) and its action is again provided by FFT algorithm, for details see [1][2][3][4][5].…”

FFT‐based homogenization for microstructures discretized by linear hexahedral elements

“…Unfortunately, there are 3D structures involving pores and defects, for which all of these solvers no longer converge, see Schneider et al [27] for a simple example with strongly erratic behavior. Regarding the second item on the list, for linear problems, both the Hashin-Shtrikman scheme of Brisard-Dormieux [25,28] and the Fourier-Galerkin method of Bonnet [29], recently revitalized and extended by Vondřejc [30], produce bounds on the elastic moduli. The latter scheme requires working on a doubly fine grid, increasing the memory occupancy and computational time by a factor of eight in 3D, whereas the former even requires the evaluation of a slowly converging infinite sum.…”

An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

“…For a comparison of the different schemes, we refer to Moulinec and Silva . It were also Brisard and Dormieux and Zeman et al who used the conjugate gradient method as the primary solver; see also Vondřejc for recent applications. This led to much higher speed and numerical stability of the FFT algorithm.…”