论文标题
在点云上的表面差分运算符的网格搭配方案
A meshfree collocation scheme for surface differential operators on point clouds
论文作者
论文摘要
我们提出了一个网格搭配方案，可以在具有给定的正常矢量和非相邻的管状邻域的平滑弯曲表面上的标态弯曲表面上离散固有的表面差异算子。该方法基于离散校正后的粒子强度交换（DC-PSE），该粒子强度交换（DC-PSE）将有限差方法推广到无网状点云。所提出的表面DC-PSE方法是从嵌入定理得出的，但是我们通过分析沿表面正常的算子内核，以在表面点云上获得纯粹的内在计算方案。我们通过在圆圈和球体上离散Laplace-Beltrami操作员来基准表面DC-PSE，并为显式和隐式求解器提供收敛结果。然后，我们通过近似于正常矢量场的固有差异来展示有关计算高斯和斯坦福兔子的平均曲率问题的算法。最后，我们将表面DC-PSE与表面有限元元件（SFEM）和差异界面有限元（DI FEM）进行比较。
We present a meshfree collocation scheme to discretize intrinsic surface differential operators over scalar fields on smooth curved surfaces with given normal vectors and a non-intersecting tubular neighborhood. The method is based on Discretization-Corrected Particle Strength Exchange (DC-PSE), which generalizes finite difference methods to meshfree point clouds. The proposed Surface DC-PSE method is derived from an embedding theorem, but we analytically reduce the operator kernels along surface normals to obtain a purely intrinsic computational scheme over surface point clouds. We benchmark Surface DC-PSE by discretizing the Laplace-Beltrami operator on a circle and a sphere, and we present convergence results for both explicit and implicit solvers. We then showcase the algorithm on the problem of computing Gauss and mean curvature of an ellipsoid and of the Stanford Bunny by approximating the intrinsic divergence of the normal vector field. Finally, we compare Surface DC-PSE with Surface Finite Elements (SFEM) and Diffuse-Interface Finite Elements (DI FEM) in a validation case.
