AI 中文总结
本文提出一种基于定向点云变分结构的无网格方法，用于计算单相Mullins-Sekerka流的周长及其Wasserstein梯度流，通过检测隐藏边界实现组分合并，并保证周长递减与面积守恒。
AI 中文摘要
本文针对单相Mullins-Sekerka流开发了一种无网格数值方法，其中界定集合的界面作为其周长的Wasserstein梯度流移动。计算该流的一个难点在于，当组分合并时，具有相反法向的界面两段会形成隐藏边界。我们的方法通过定向点云表示界面，并将其视为定向变分结构。我们引入了一个量来度量附近法向相互抵消的程度，并利用该量检测隐藏边界。基于该量，我们构建了一个周长估计器，证明了其期望误差随点数呈代数速率衰减，即使在存在隐藏边界的情况下也是如此，并将其用于二维流的极小化移动方案中。数值实验表明，该方案能够通过隐藏边界计算组分的合并，同时方案中的周长项在每一步都减小，且面积近似守恒。由于隐藏边界也出现在其他周长驱动的界面问题中，我们预期我们的定向表示方法同样适用于这些问题。
英文摘要
This paper develops a mesh-free numerical method for the one-phase Mullins-Sekerka flow, in which the interface bounding a set moves as the Wasserstein gradient flow of its perimeter. One difficulty in computing this flow is that two pieces of the interface with opposite normals form a hidden boundary when components merge. Our approach represents the interface by an oriented point cloud, which we treat as an oriented varifold. We introduce a quantity that measures how much nearby normals cancel each other and use it to detect the hidden boundary. Based on this quantity, we construct a perimeter estimator, prove that its expected error decays at an algebraic rate in the number of points, also in the presence of a hidden boundary, and use it in a minimizing movement scheme for the flow in two dimensions. Numerical experiments show that the scheme computes mergers of components through the hidden boundary, while the perimeter term in the scheme decreases at every step and the area is nearly conserved. As hidden boundaries also appear in other perimeter-driven interface problems, we expect our oriented representation to be applicable to them as well.
Comments23 pages, 4 figures
