Lecture 3: Asymptotically consistent discretizations of the LS equation (2/2) ; Lecture 3: Asymptotically consistent discretizations of the LS equation (2/2): Introduction to FFT-based numerical methods for the homogenization of random materials

In this lecture, we will discuss asymptotically consistent discretizations of the Lippmann–Schwinger equations. In other words, we won't enforce exact evaluation of the continuous variational problem over the discretization space. This additional flexibility will allow to derive more efficient numerical schemes. The following topics will be discussed: (1) discussion of the consistent discretization, (2) on asymptotically consistent discretizations, (3) asymptotically consistent discretizations of the microstructure, (4) asymptotically consistent discretizations of the Green operator, (5) comparison of some discretizations.---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 ...