Lectures 8 & 9 : FFT solvers for transport problems in heterogeneous media (2/4) ; Lectures 8 & 9 : FFT solvers for transport problems in heterogeneous media (2/4): Introduction to FFT-based numerical methods for the homogenization of random materials

In this course, we address various FFT methods developed for treating other physics and multi-physical couplings, including conductivity, diffusion, optics in the electrostatic regime, viscoelasticity, phase-field for damag mechanics and Stokes flow. We also give a brief review of comparison between FFT and finite element methdos in terms of accuracy and overall performances.---Analysis at the macroscopic scale of a structure that exhibits heterogeneities at the microscopic scale requires a first homogenization step that allows the heterogeneous constitutive material to be replaced with an equivalent, homogeneous material.Approximate homogenization schemes (based on mean field/effective field approaches) as well as rigorous bounds have been around for several decades; they are extremely versatile and can address all kinds of material non-linearities. However, they rely on a rather crude description of the microstructure. For applications where a better account of the finest details of the microstructure is desirable, the solution to the so-called corrector problem (that delivers the homogenized properties) must be computed by means of full-field simulations. Such simulations are complex, and classical discretization strategies (e.g., interface-fitting finite elements) are ill-suited to the task.During the 1990s, Hervé Moulinec and Pierre Suquet introduced a new numerical method for solving the corrector problem. This method is based on the discretization of an integral equation that is equivalent to the original boundary-value problem. Observing that the resulting linear system has a very simple structure (block-diagonal plus block-circulant), Moulinec and Suquet used the fast Fourier transform (FFT) to compute the matrix-vector products that are required to find the solution efficiently.During the last decade, the resulting method has gained in popularity (the initial Moulinec Suquet paper is cited 134 times over the 1998–2009 period and 619 times over the 2010–2020 period — source: Scopus). Significant ...