MATT · Multiscale Analysis of Phase Transformations in Thermoelasticity

MATT wants to investigate new multiscale mathematical problems where the evolution of fluctuating internal surfaces with respect to time and space has to be considered. Such evolutions happen at small, unobservable spatial scales, as the evolving surfaces are typically contact interfaces between microscopic material phases. A prime example is the modeling of mechanical microstructural changes in steel (e.g. Bainite formation from Austenite) under fast temperature changes (pointwise sensor measurements are here unavailable). Besides thermoelasticity, other real-world examples leading to the same class of mathematical problems include swelling of porous media, growth of tumors, and thawing of glaciers/permafrost (now, a global problem). As the scale heterogeneity renders numerical simulations impossible, one must identify simplified models that are able to accurately describe and predict the material behavior while still being simple enough to allow for fast numerical simulations within the expected physical range. The objectives of MATT are: (a) develop a general mathematical framework for rigorously connecting different scales crossed by free boundaries, (b) design multiscale numerical schemes to simulate and validate the produced models, (c) facilitate the Researcher a quick development towards scientific independence, (d) boost the Researcher's awareness of the role the applied mathematician must play in science, technology, and society. Mathematical homogenization (two-scale convergence/periodic unfolding) is the main working tool. Due to the inherent non-linearity of moving boundary problems, several new results regarding uniform estimates and compactness arguments will be established and used to ensure convergence. Multiscale numerical schemes will be designed and implemented in Python/FEniCS. Experimental data for the Bainite transformation from Austenite will be used to validate our findings.