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 efficientThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Superconvergence of the effective Cauchy stress in computational homogenization of inelastic materials

“…where we used the Lipschitz continuity (12) and the a priori estimate (24). Thus, the bound (28) shows that the effective stresses converge at least as fast as the H 1 0 -norm of the local displacement field.…”

“…where we used the a priori estimate (24) and the definition (36) of the field 𝜉 h . This estimate has a few direct implications 1.…”

“…Such a behavior does not come unexpected for the Moulinec-Suquet discretization, 7,48 as it is prone to the well-known ringing artifacts characteristic for Fourier-type discretizations. Please keep in mind that other popular discretizations in FFT-based computational micromechanics, see Schneider 24 for a recent overview, are also characterized by high errors at interfaces, essentially due to the jagged interfaces resulting from voxel discretizations.…”

“…For a start, fully integrated finite elements on regular grids can be handled by FFT‐based methods, 21‐23 but do not give rise to the best compromise between accuracy and computational effort. Indeed, fully integrated elements require either a higher computational effort or an increased memory footprint, and computational experiments suggest their accuracy is not superior compared to more traditional discretizations (Reference 24, section 1.2). Thus, we decided to investigate the most popular discretizations employed in FFT‐based methods instead.…”

Fourier Transform approach to numerical homogenization of periodic media containing sharp insulating and superconductive cracks

“…The latter solves the discrete equations with a solver based on a suitable algorithm e.g conjugate gradient, minimal residual iterative schemes (see e.g Saad, 2003; Barrett et al, 1994, and the references therein). Depending on the discretization schemes, solvers can be constructed based on FFT techniques, e.g spectral Galerkin (Fata and Gray, 2009;Hu et al, 2022), finite difference (Feng and S., 2020;Costa, 2022;Ren et al, 2022;Willot et al, 2014), finite element (Schneider, 2022;Zeman et al, 2017), finite volume (Nunez et al, 2012), discrete element (Calvet et al, 2022),... Among the numerical methods, the class of Fourier Transform methods used in the present work has seen a fast development in the recent years, but with very few developments for problems involving cracks.…”