A Meshfree Collocation Scheme for Surface Differential Operators on Point Clouds

Research output: Contribution to journalResearch articleContributedpeer-review

Contributors

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.

Details

Original languageEnglish
Article number89
Number of pages19
JournalJournal of scientific computing
Volume96 (2023)
Issue number3
Publication statusPublished - 7 Aug 2023
Peer-reviewedYes

External IDs

ORCID /0000-0003-4414-4340/work/159608266

Keywords

Keywords

  • Collocation, Intrinsic calculus, Meshfree methods, Partial differential equations on surfaces, Surface differential operators

Library keywords