FFT-based computational micromechanics with Dirichlet boundary conditions on the rotated staggered grid

Imposing nonperiodic boundary conditions for unit cell analyses may be necessary for a number of reasons in applications, for example, for validation purposes and specific computational setups. The work at hand discusses a strategy for utilizing the powerful technology behind fast Fourier transform (FFT)-based computational micromechanics—initially developed with periodic boundary conditions in mind—for essential boundary conditions in mechanics, as well, for the case of the discretization on a rotated staggered grid. Introduced by F. Willot into the community, the rotated staggered grid is presumably the most popular discretization, and was shown to be equivalent to underintegrated trilinear hexahedral elements. We leverage insights from previous work on the Moulinec–Suquet discretization, exploiting a finite-strain preconditioner for small-strain problems and utilize specific discrete sine and cosine transforms. We demonstrate the computational performance of the novel scheme by dedicated numerical experiments and compare displacement-based methods to implementations on the deformation gradient.