Fast Fourier transform based homogenization with mixed uniform boundary conditions

The fast Fourier transform (FFT) based homogenization method of Moulinec and Suquet has been established as a fast, accurate, and robust tool for periodic homogenization in solid mechanics. In a finite element context, Pahr and Zysset have introduced nonperiodic boundary conditions (PMUBC) for homogenization problems. We show how to implement PMUBC efficiently in an FFT-based code using discrete sine and cosine transforms. Compared with the domain mirroring approach, we reduce the runtime by a factor of 2 to 3, and the memory requirements by a factor of 8. We show that the use of periodic boundary conditions for nonperiodic geometries yields vastly different results than with PMUBC. Furthermore, we examine the influence of the discretization method by comparing the staggered grid discretization with a finite element discretization.