A meshfree collocation scheme for surface differential operators on point clouds

TL;DR

This paper introduces a meshfree collocation scheme called Surface DC-PSE for discretizing surface differential operators on point clouds, enabling accurate intrinsic computations on curved surfaces without meshes.

Contribution

The paper develops a novel meshfree method based on DC-PSE that reduces operator kernels along surface normals for intrinsic surface differential operator discretization.

Findings

Successfully discretized the Laplace-Beltrami operator on complex surfaces.

Achieved convergence in both explicit and implicit solvers.

Validated the method by computing curvature on various surfaces and comparing with existing methods.

Abstract

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.