AI 中文总结
本文提出带三叉点阻力网络曲率流的相场方法，从变分视角推导耦合偏微分方程组，验证其收敛性并保留自动处理拓扑变化的特性。
AI 中文摘要
我们开发了一种新的多相曲率运动的相场（弥散界面）近似方法，该方法包含三叉点阻力，这是多晶材料在热处理过程中微观结构演化的重要尖锐界面模型。该尖锐界面模型源于界面网络总长度相对于某一度量的梯度流。因此，我们从变分视角，以最小化运动的形式推导了该弥散界面近似，即一个耦合的偏微分方程组，它是Allen-Cahn方程组的变体。在推导过程中，我们提出了一个简单的积分表达式，用于用序参数计算节点数量，即使对于标准多相Allen-Cahn方程组，这一表达式似乎也是新的。随后，通过匹配渐近展开法验证了所得流收敛到期望的尖锐界面极限。针对已知精确解和通过前沿追踪得到的高精度基准解的数值收敛研究，为该收敛性提供了明确的进一步证据。此外，该方法保留了弥散界面方法最理想的特性：自动处理界面网络中的拓扑变化。
英文摘要
We develop a new phase-field (diffuse interface) approximation of multiphase curvature motion with triple junction drag - an important sharp interface model for the evolution of microstructure in polycrystalline materials during heat treatment. This sharp interface model arises as gradient flow for the total length of the interfacial network with respect to a certain metric. Accordingly, we derive our diffuse interface approximation - a coupled system of partial differential equations that is a variant of the Allen-Cahn system - from a variational perspective, in the style of minimizing movements: starting from a discrete in time approximation that entails a convex optimization problem to advance from one time step to the next. In the process, we propose a simple integral expression that counts the number of junctions using the order parameter that appears to be new even for the standard multiphase Allen-Cahn system. The convergence of the resulting flow to the desired sharp interface limit is then verified via the method of matched asymptotic expansions. Numerical convergence studies against both known exact solutions as well as highly accurate benchmark solutions obtained via front tracking provide clear further evidence for this convergence. Moreover, the method retains the most desirable feature of diffuse interface methods: Automatic handling of topological changes in the network of interfaces.
