Automatic phase field regularisation of interfacial problems

Phase field models are a useful approximation method for sharp interfacial problems. Sharp interfacial problems aim to model physical phenomena by describing the problem as different phases separated by infinitely thin hypersurfaces. Phase field models introduce a (phase field) variable which varies smoothly between phases and has the effect of smoothing or diffusing the interfaces. A small parameter is present in the phase field equations that can be shown to be proportional to the diffuse interfacial width; when this is sent to 0 the “sharp” interfacial problem is recovered. Often one formally derives a phase field problem from thermodynamic principles. Formal asymptotics are performed to recover the sharp interfacial problem. Carrying out these asymptotics can be laborious, we show that the limit problem is often obvious using a formal gradient flow structure. Utilising this structure we implement a software interface which allows users to go in the converse direction. That is, the user postulates a sharp interfacial problem, the software then automatically regularises the sharp equations to produce a phase field approximation. This is done symbolically using the Unified Form Language. Unified Form Language is a domain specific language used by a wide range of software packages, consequently the phase field equations can be exported and solved using other finite element packages. We demonstrate the effectiveness of this package using a number of models found in the literature, ranging from two and three phase mean curvature flow problems, to more complex problems involving partial differential equations defined away from the interface e.g. dendritic growth, tumour growth and crack propagation models.