#### Abstract :

In the field of numerical homogenization, the Fourier-based
homogenization solvers were introduced by Moulinec and Suquet in their
seminal work [1]. Since then, they have established themselves as a
compe titive alternative to finite elements in terms of accuracy,
efficiency, versatility, and simplicity of implementation. In its basic
version, the method works as a fixed-point iterative solution to a periodic
Lippman-Schwinger integral equation, whose kernel can be efficiently
handled by the Fast Fourier Transform (FFT).
In our recent work [2], we interpreted FFT-based methods in a Galerkin
framework that involves the four standard steps, namely (i) introducing a
weak form of the governing equations, (ii) projecting the weak form to an
approximation space, (iii) applying a numerical quadrature, and (iv)
solving the ensuing system of linear equations by a suitable iterative
solver.
Specifically, the basic Moulinec-Suquet scheme is obtained when (i) the
weak form involves the gradients of the potential, (ii) the approximation
space is spanned by trigonometric polynomials, (iii) the trapezoidal rule is
employed for numerical integration, and step (iv) involves the Richardson
iteration.
The purpose of this talk is twofold: to summarize these developments and
to explain how they can be used to develop more efficient FFT-based
solvers, considering scalar elliptic problems for simplicity. Specifically,
(i) I will focus on a formulation of a new exact integration based
discretization technique along with its efficiency to estimate homogenized
properties (ii) influence of Krylov subspace methods used to solve nonsymmetric
rank-deficient linear systems and convergence of an aposteriori
bound on the solution during iterations.
[1] Moulinec, H. and Suquet, P., 1994. A fast numerical method for
computing the linear and nonlinear mechanical properties of composites.
C. R. Acad. Sci. II, 318(11), pp.1417-1423.
[2] Nachiketa Mishra, Jan Zeman and Jaroslav Vondejc, 2016. A
comparative study on low-memory iterative solvers for FFT-based
homogenization of periodic media, Journal of Computational Physics ,
321, 151-168.