A computational method based on Fast Fourier Transforms for composites with complex microstructure Pierre M. Suquet Mechanics and Acoustics Laboratory Centre National de la Recherche Scientifique, Marseille, France A common practice in structural mechanics when dealing with composite structures is to replace their highly nonhomogeneous constituents with "homogenized" materials. Some of their overall properties can be determined experimentally, but in practice many of them are difficult to measure and modelling becomes essential. A whole area of Solid Mechanics has been developed in the past thirty years to predict theoretically the "effective properties'' of composite materials, directly from the properties of their individual constituents (or "phases") and their distribution (or "microstructure''). An impressive body of literature exists when the constitutive behavior of the phases that make up the composite is linear elastic. By comparison, much less is known about composites exhibiting a nonlinear behavior such as plasticity or creep. Besides effective properties of nonhomogeneous bodies, there is an increasing need for more precise information about local fields (stress and strain) to better understand small-scale nonlinear mechanisms including failure. This has motivated in recent years the development of direct numerical simulations for composites mostly for "simple" microstructures (one or two inclusions in a block of matrix). More recently efforts have been directed towards modeling composites with "complex" microstructures and/or containing many inclusions. The difficulties arising then are the description of complex geometries (meshing) and the size of the resulting problems. This lecture will present a method based on Fast Fourier Transforms which makes direct use of digital images of the "real" microstructure in the numerical simulation. The proposed method avoids the difficulty due to meshing. It makes use of Fast Fourier Transforms (FFT) to solve the unit cell problem even when the constituents are nonlinear. The second difficulty (size of the problem) is partially overcome by a fixed-point method not requiring the formation of a stiffness matrix. The method is based on the exact expression of the Green function for a linear elastic and homogeneous comparison material. The problem for elastic nonhomogeneous phases is reduced to an integral equation (Lippman-Schwinger equation) which is solved iteratively. A nice feature of the method is that it involves a multiplication in Fourier space, a multiplication in real space, a FFT and an inverse FFT. The two first operations can be easily parallelized. The method is extended to nonlinear constituents by a step-by-step integration in time. The comparison with the Finite Element Method shows that, in many instances, the method based on FFT is often faster and more flexible.