Modeling materials with jigsaw puzzles: Beyond periodic unit cells

The spatial arrangement of material phases plays a crucial role in the macroscopic response of materials. To increase their modeling capabilities, many contemporary numerical strategies rely on a single characteristic material sample, mostly provided in the form of a Periodic Unit Cell (PUC). While well suited for problems with clearly separated scales, the PUC notion becomes questionable whenever a material structure has to be explicitly resolved in a macroscopic domain. Aiming at these scenarios, we present an extension of the PUC concept, using the formalism of so-called Wang tiles. Our approach allows storing microstructural geometry of random heterogeneous materials in the compressed form of handful domains with predefined mutual compatibility. Following a simple sequential algorithm, stochastic samples of microstructural geometry - including its finite element discretization - can be reassembled nearly instantly from the compressed form, making the Wang tile concept appealing to problems where multiple, statistically consistent realizations of a material microstructure must be considered.
Starting with the brief introduction of the concept fundamentals and methods for compressing microstructural information into a set of tiles, I will then mainly focus on recent results in precomputing characteristic responses of the compressed material microstructure and reusing these microstructure-informed approximations for accelerating macroscopic analyses with fully resolved material details. In the end, I will briefly outline related applications of the concept (i) in a concurrent optimization of manufacturable microstructure modules and their distribution in macroscopic products, and (ii) locally tuneable meta-materials.