标题： 向量值相场在凸约束下的模式形成 显示英文标题

标题： Pattern formation in vector-valued phase fields under convex constraints

摘要： 在本工作中，提出了一类新的向量值相场模型，其中相参数的取值受到凸集的约束。生成的相场特征是将区域划分为由薄界面分隔的不同相的块。相的配置和动力学直接依赖于凸约束集的几何和拓扑结构，这使得可以设计出具有所需相互作用和图案的此类模型。引入了一种高效的近端梯度求解器来数值研究其L2梯度流，即相关的Allen-Cahn型方程。将求解器与各种凸约束集的选择相结合，得到的数值结果呈现出自然界和工程中观察到的多种图案，例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。 显示英文摘要

摘要： 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.