Lecture 1: the Green operator and the Lippmann–Schwinger equation (2/2): Introduction to FFT-based numerical methods for the homogenization of random materials

This lecture provides an introduction to all the basic tools that are required by “FFT-based homogenization methods”, namely: the “corrector” problem for homogenization, the Green operator and the Lippmann–Schwinger equation.---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 advances have been made on various topics: theoretical analysis of the convergence, discretization strategies, innovative linear and non-linear solvers, etc.Nowadays, FFT-based homogenization methods have become state-of-the-art techniques in materials science and are used for industry with increasing frequency. A 5-day introductory course to FFT-based homogenization methods was held on 14-18 march 2022 at Univ Gustave Eiffel, Champs sur Marne, France. The intent of this workshop was to provide an accessible introduction to FFT-based computational homogenization methods and also have a glimpse at the current research frontier.The workshop was open to research students (M2 onwards) as well as researchers from both academia and industrial R&D. Each of the nine sessions of this workshop was composed of a theoretical lecture followed by hands-on applications (mostly on computers). Some of these lectures were recorded. We are happy to share these videos.April 2022,S. Brisard, M. Schneider and F. Willo