Improved Stencil Selection for Meshless Finite Difference Methods in 3D

TL;DR

This paper presents a new geometric stencil selection algorithm for 3D meshless finite difference methods that improves accuracy and efficiency in solving Laplacian problems on complex geometries.

Contribution

The paper introduces a novel octant-based geometric stencil selection algorithm for 3D meshless methods, enhancing accuracy over previous approaches.

Findings

The new method outperforms previous selection techniques in numerical experiments.

It achieves competitive results compared to finite element methods.

Effective on complex STL models with various node distributions.

Abstract

We introduce a geometric stencil selection algorithm for Laplacian in 3D that significantly improves octant-based selection considered earlier. The goal of the algorithm is to choose a small subset from a set of irregular points surrounding a given point that admits an accurate numerical differentiation formula. The subset serves as an influence set for the numerical approximation of the Laplacian in meshless finite difference methods using either polynomial or kernel-based techniques. Numerical experiments demonstrate a competitive performance of this method in comparison to the finite element method and to other selection methods for solving the Dirichlet problems for the Poisson equation on several STL models. Discretization nodes for these domains are obtained either by 3D triangulations or from Cartesian grids or Halton quasi-random sequences.