A Cylindrical Basis Function for Solving Partial Differential Equations on Manifolds

TL;DR

This paper introduces a novel cylindrical basis function approach for efficiently solving PDEs on 3D surface point clouds by transforming surface operators into Cartesian equivalents, demonstrated on the Laplace Beltrami operator.

Contribution

It presents a new cylindrical basis function method that simplifies PDE solutions on manifolds represented as point clouds, enhancing computational efficiency.

Findings

Effective discretization of the Laplace Beltrami operator

Simplifies PDE solving on surface point clouds

Demonstrates numerical efficiency of the method

Abstract

Numerical solutions of partial differential equations (PDEs) on manifolds continues to generate a lot of interest among scientists in the natural and applied sciences. On the other hand, recent developments of 3D scanning and computer vision technologies have produced a large number of 3D surface models represented as point clouds. Herein, we develop a simple and efficient method for solving PDEs on closed surfaces represented as point clouds. By projecting the radial vector of standard radial basis function(RBF) kernels onto the local tangent plane, we are able to produce a representation of functions that permits the replacement of surface differential operators with their Cartesian equivalent. We demonstrate, numerically, the efficiency of the method in discretizing the Laplace Beltrami operator.