An X-FFT Solver for Two-Dimensional Thermal Homogenization Problems

Abstract

We introduce an approach to computational homogenization which unites the accuracy of interface-conforming finite elements (FEs) and the computational efficiency of methods based on the fast Fourier transform (FFT) for two-dimensional thermal conductivity problems. FFT-based computational homogenization methods have been shown to solve multiscale problems in solid mechanics effectively. However, the obtained local solution fields lack accuracy in the vicinity of material interfaces, and simple fixes typically interfere with the numerical efficiency of the solver. In the work at hand, we identify the extended finite element method (X-FEM) with modified absolute enrichment as a suitable candidate for an accurate discretization and design an associated fast Lippmann-Schwinger solver. We implement the concept for two-dimensional thermal conductivity and demonstrate the advantages of the approach with dedicated computational experiments.