Serious progress was made within BeyondRVE on fast micromechanics solvers. We started with investigating the theoretical basics of FFT-based methods: we uncovered the superconvergence phenomenon of the effective stresses responsible for the success of regular grid based methods, we provided a sharp analysis of the convergence rate of the original Moulinec-Suquet discretization and connected the composite-voxel method to assumed/enhanced strain methods.
Aside from these findings, we extended FFT-based methods to exploit adaptivity via an octree-based coarsening, together with an elastic stabilization technique.
Traditionally, methods based on the fast Fourier transform (FFT) impose periodic boundary conditions naturally for the sought fluctuation fields under consideration. Although Dirichlet boundary conditions are readily imposed for conducting composites, the extension to small-strain mechanics seemed impossible due to problems with constructing Green's operator. We introduced a trick to side-step this inherent limitation with immediate applications, e.g. for nano-indentation.
The work on microstructure models focused on fiber composites. Fiber-orientation tensors (FOTs) serve as a critical mathematical tool for modeling injection molding. Unfortunately, filling simulations only provide the second-order FOT, whereas the fourth-order FOT is required for a mechanics. To close this apparent gap, closure approximations are used to guesstimate the fourth-order FOT based on the second-order one. Within BeyondRVE, we contributed to the understanding of FOTs and the design of closures in a number of ways. For a start, we gave a complete mathematical description of the set of realizable fourth-order FOTs in dimension three filling a decade-long gap (and requiring a result of D. Hilbert from 1890!). With this description at hand, we provided two novel closure approximations. The symmetrized implicit quadratic closure is a computationally cheap version of the maximum-entropy closure. Also, within BeyondRVE, a groundbreaking closure was introduced which couples length to orientation and captures experimental results more more accurately than length-agnostic closures. Last but not least, the project BeyondRVE made a breakthrough discovery by providing a robust LFT (long fiber thermoplastics) generator, regarding the phase space of curved fibers as a smooth manifold. With the technology at hand, LFT microstructures with industrial volume fractions may be generated.
