The overarching goal of the MATT project (Multiscale Analysis of Phase Transformations in Thermoelasticity) is to rigorously derive, analyze, and simulate efficient and accurate mathematical models for complex multiscale processes. These processes, which require careful consideration of evolving small-scale internal surfaces over time and space, are called multiscale moving boundary problems or MultMBPs. A prime example is the modeling of mechanical microstructural changes in steel under rapid temperature changes (Bainite formation from Austenite). Other real-world applications leading to the same class of mathematical problems include swelling of porous media, growth of tumors, and thawing of glaciers/permafrost - issues increasingly relevant in engineering and environmental contexts. In a world, where the need for specialized, dynamic, and complex materials (e.g. composites, meta-materials, bio-engineered materials) is ever increasing, increasing the mathematical understanding of their effective material behavior is particularly relevant.
The scale heterogeneity, introduced by numerous small, evolving internal boundaries, renders numerical simulations infeasible. For that reason, 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 mathematical derivation of such models is made complicated by the inherent non-linear structure of MultMBPs and requires new results regarding uniform estimates and compactness arguments. But even after a homogenization limit procedure, the limit models still suffer from complex scale-interactions making numerical simulations computationally expensive. As a consequence, smart and efficient numerical schemes are needed to tackle these limit model. Here, the project proposes a novel precomputing approach where certain calculations are shifted and parallelized into an offline phase. With this approach, simulations can be done much faster without sacrificing much accuracy.
