When surface evolution meets Fokker-Planck equation: a novel tangential velocity model for uniform parametrization

TL;DR

This paper introduces a meshless artificial tangential velocity method based on Fokker-Planck equations to improve surface evolution simulations, ensuring stable point distributions and accurate results.

Contribution

It develops a novel surface density-guided tangential velocity approach using Fokker-Planck equations and a Kullback-Leibler divergence-based algorithm within a meshless framework.

Findings

Method enhances numerical stability in surface evolution simulations.

Approach achieves high accuracy across various geometric flows.

Framework is flexible for unstructured point cloud data.

Abstract

A common issue in simulating geometric evolution of surfaces is unexpected clustering of points that may cause numerical instability. We propose a novel artificial tangential velocity method for this matter. The artificial tangential velocity is generated from a surface density field governed by a Fokker-Planck equation to guide the point distribution. A target distribution matching algorithm is developed leveraging the surface Kullback-Leibler divergence of density functions. The numerical method is formulated within a fully meshless framework using the moving least squares approximation, thereby eliminating the need for mesh generation and allowing flexible treatment of unstructured point cloud data. Extensive numerical experiments are conducted to demonstrate the robustness, accuracy, and effectiveness of the proposed approach across a variety of surface evolution problems, including the mean curvature flow.