Homogenization and numerical methods for models appearing in grinding processes and phase transitions

In this thesis, we use homogenization to derive and analyze models in the context of grinding processes and phase transitions.
In the first part, we provide an introduction to the fundamentals of partial differential equations (PDEs), numerical algorithms for their solution, and different homogenization strategies, with a particular focus on two-scale convergence. Additionally, we give an overview of the state-of-the-art in modeling grinding processes and place the presented results within the existing literature.

The second part comprises five research papers addressing the analysis, homogenization, and simulation of various subproblems motivated by these applications. Two papers derive effective heat equations for porous grinding wheels and granular structures along an interface, accounting for convection by a prescribed flow field. A third paper rigorously derives an effective model for a nonlinearly coupled temperature-fluid system in a thin rough layer using two-scale convergence. Multiple numerical simulations accompany the results of all three papers. The fourth paper integrates the previously derived models with a microscopic Dexel removal approach to develop a multiscale workflow for simulating grinding processes. The presented model is validated by comparison with different experiments.
The final paper investigates a two-scale model for phase transitions featuring moving domains dependent on the solution. By employing the Hanzawa transformation, we map the problem to a fixed domain, and the existence of solutions is established via a fixed-point theorem. Furthermore, we introduce and show the stability of a precomputing scheme to speed up the numerical simulations.