FFT‐based homogenization using a reduced set of frequencies and a clustered microstructure

Abstract

To capture the material behavior of composite microstructures, Moulinec and Suquet [5] proposed a homogenization scheme making use of fast Fourier transforms (FFT) and fixed-point iterations. To reduce the computational effort of this spectral method, Kochmann et al.[3] introduced a model order reduction technique, which is based on using a fixed reduced set of frequencies for the computations in Fourier space. Within the current work, we improved the accuracy of the approach by use of a geometrically adapted set of frequencies, see [1]. Since the constitutive relations are still evaluated in real space, the technique is most beneficial for a linear material behavior. Considering nonlinear material behavior, most of the computing time is related to solving the constitutive relations. Therefore, the total speed-up is lower. To achieve a further reduction of the computational effort for a nonlinear material behavior, the earlier proposed model order reduction technique is coupled with a clustering analysis [4]. The whole microstructure is thus divided into clusters, which show a similar material behavior. Within these clusters, the micromechanical fields are assumed to be constant which leads to a significant reduction of computational costs compared to the highly resolved solution.