Adaptive finite element simulations for macroscopic and mesoscopic models

During heat treatment and other production processes, gradients of temperature and other observables may vary rapidly in narrow regions, while in other parts of the workpiece the behaviour of these quantities is quite smooth. Nevertheless, it is important to capture these fine structures during numerical simulations. Local mesh refinement in these regions is needed in order to resolve the behaviour in a sufficient way. On the other hand, these regions of special interest are changing during the process, making it necessary to move also the regions of refined meshes. Adaptive finite element methods present a tool to automatically give criteria for a local mesh refinement, based on the computed solution (and not only on a priori knowledge of an expected behaviour). We present examples from heat treatment of steel, including phase transitions with transformation induced plasticity and stress dependent phase transformations. On a mesoscopic scale of grains, similar methods can be used to efficiently and accurately compute phase field models for phase transformations.