Fast calculation of interaction tensors in clustering-based homogenization

Recently proposed clustering-based methods considerably reduce numerical cost for homogenizing heterogeneous materials, while maintaining the accuracy of mechanical property predictions in an online stage. In such an algorithm, however, the calculation of interaction tensors consumes much of the total computing time. We introduce a new method that expedites the interaction tensors calculation, thereby enhancing the clustering-based methods. We first cast a cubic/rectangular coarse grid over the representative volume element. Using analytical expressions for the integral of the Green’s functions, we then calculate interaction tensors on the coarse grid. Finally, the desired interaction tensors on the clusters are approximated based on composition ratios. Moreover, in virtual clustering analysis, we derive the Lippmann–Schwinger equation for finite strain problems. Numerical tests in two and three space dimensions verify the efficiency and accuracy of the proposed method.