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.

翻译：本文提出了一类新的向量值相场模型，其中相参数的值受到凸集的约束。生成的相场将定义域划分为由薄界面分隔的不同相区。相位的构型与动力学直接依赖于凸约束集的几何与拓扑结构，从而使设计具有特定相互作用和模式的此类模型成为可能。本文引入了一种高效的近端梯度求解器，以数值研究其L2梯度流，即与之相关的Allen-Cahn型方程。将该求解器与不同凸约束集的选择相结合，所得数值结果呈现出自然界和工程中观察到的多种模式，如金属合金中的多相晶粒、反应-扩散系统中的行波以及磁性材料中的涡旋。