Lecture 2: Consistent discretization of the LS equation (1/2): Introduction to FFT-based numerical methods for the homogenization of random materials

In this course, we will discuss spatial discretization of the Lippmann–Schwinger using a Galerkin technique. The resulting discretization is consistent, because the linear and bilinear forms of the weak form of the Lippmann–Schwinger equation are evaluated exactly. The derivation of the discrete Green operator will clarify how “FFT-based homogenization methods” rely on... the FFT. The following topics will be discussed: weak form of the LS equation, Galerkin discretization of the LS equation, the discretized operators, applying the discrete Green operator, towards linear LS solvers.---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