Enforce the Dirichlet boundary condition by volume constraint in Point Integral method

TL;DR

This paper introduces a novel approach to enforce Dirichlet boundary conditions in the Point Integral Method by using volume constraints, improving the discretization of Laplace-Beltrami operators on point clouds.

Contribution

The paper proposes a new volume constraint technique for Dirichlet boundary conditions in PIM, enhancing boundary treatment accuracy over previous Robin boundary approximations.

Findings

Effective enforcement of Dirichlet boundary conditions in PIM

Improved accuracy in Laplace-Beltrami operator discretization

Demonstrated advantages over previous boundary treatment methods

Abstract

Recently, Shi and Sun proposed Point Integral method (PIM) to discretize Laplace-Beltrami operator on point cloud. In PIM, Neumann boundary is nature, but Dirichlet boundary needs some special treatment. In our previous work, we use Robin boundary to approximate Dirichlet boundary. In this paper, we introduce another approach to deal with the Dirichlet boundary condition in point integral method using the volume constraint proposed by Du et.al.