Phase field modelling of free boundary problems

Diffuse interface models based on the phase field methodology have been developed and investigated in various applications such as solidification processes, tumour growth, or multi-phase flow. The interfaces are represented by thin layers, across which quantities rapidly but smoothly change their values. These interfacial layers are described in terms of order parameters, the equations for which can be solved using relatively straightforward methods, such as finite elements with adaptive mesh refinement, as no tracking of any interface is required. The interface motion is usually coupled to other fields and equations adjacent or on the interface, for instance, diffusion equations in alloys or the momentum equation in fluid flow. We discuss how such systems can be incorporated into phase field models in a generic way. Furthermore, we present a computational framework where specific models can be implemented and later on conveniently amended, if desired, in a high-level language, and which then bind to efficient software backends. A couple of code listings and numerical simulations serve to illustrate the approach