Pattern formation in vector-valued phase fields under convex constraints

In this work, a new class of vector-valued phase field models is presented, where the values of the phase parameters are constrained by a convex set. The generated phase fields feature the partition of the domain into patches of distinct phases, separated by thin interfaces. The configuration and dynamics of the phases are directly dependent on the geometry and topology of the convex constraint set, which makes it possible to engineer models of this type that exhibit desired interactions and patterns. An efficient proximal gradient solver is introduced to study numerically their L2-gradient flow, i.e.~the associated Allen-Cahn-type equation. Applying the solver together with various choices for the convex constraint set, yields numerical results that feature a number of patterns observed in nature and engineering, such as multiphase grains in metal alloys, traveling waves in reaction-diffusion systems, and vortices in magnetic materials.