We present a framework for solving partial different equations on evolving surfaces. Based on the grid-based particle method (GBPM) [18], the method can naturally resample the surface even under large deformation from the motion law. We introduce a new component in the local reconstruction step of the algorithm and demonstrate numerically that the modification can improve computational accuracy when a large curvature region is developed during evolution. The method also incorporates a recently developed constrained least-squares ghost sample points (CLS-GSP) formulation, which can lead to a better-conditioned discretized matrix for computing some surface differential operators. The proposed framework can incorporate many methods and link various approaches to the same problem. Several numerical experiments are carried out to show the accuracy and effectiveness of the proposed method.

翻译：本文提出了一种在演化曲面上求解偏微分方程的框架。该方法基于网格粒子方法（GBPM）[18]，即使在运动定律导致大变形的情况下，也能自然地重采样曲面。我们在算法的局部重构步骤中引入了一个新组件，并通过数值实验证明，当演化过程中出现大曲率区域时，该改进能够提高计算精度。该方法还结合了近期发展的约束最小二乘虚拟样本点（CLS-GSP）公式，该公式能为计算某些曲面微分算子带来条件更优的离散化矩阵。所提出的框架能够整合多种方法，并将不同方法关联至同一问题。通过若干数值实验验证了所提方法的精度与有效性。