Voxel‐based finite elements with hourglass control in fast Fourier transform‐based computational homogenization

Abstract The power of fast Fourier transform (FFT)‐based methods in computational micromechanics critically depends on a seamless integration of discretization scheme and solution method. In contrast to solution methods, where options are available that are fast, robust and memory‐efficient at the same time, choosing the underlying discretization scheme still requires the user to make compromises. Discretizations with trigonometric polynomials suffer from spurious oscillations in the solution fields and lead to ill‐conditioned systems for complex porous materials, but come with rather accurate effective properties for finitely contrasted materials. The staggered grid discretization, a finite‐volume scheme, is devoid of bulk artifacts in the solution fields and works robustly for porous materials, but does not handle anisotropic materials in a natural way. Fully integrated finite‐element discretizations share the advantages of the staggered grid, but involve a higher memory footprint, require a higher computational effort due to the increased number of integration points and typically overestimate the effective properties. Most widely used is the rotated staggered grid discretization, which may also be viewed as an underintegrated trilinear finite element discretization, which does not impose restrictions on the constitutive law, has fewer artifacts than Fourier‐type discretizations and leads to rather accurate effective properties. However, this discretization comes with two downsides. For a start, checkerboard artifacts are still present. Second, convergence problems occur for complex porous microstructures. The work at hand introduces FFT‐based solution techniques for underintegrated trilinear finite elements with hourglass control. The latter approach permits to suppress local hourglass modes, which stabilizes the convergence behavior of the solvers for porous materials and removes the checkerboards from the local solution field. Moreover, the hourglass‐control parameter can be adjusted to “soften” the material response compared to fully integrated elements, using only a single integration point for nonlinear analyses at the same time. To be effective, the introduced technology requires a displacement‐based implementation. The article exposes an efficient way for doing so, providing minimal interfaces to the most commonly used solution techniques and the appropriate convergence criterion.