Application of a High Order Accurate Meshless Method to Solution of Heat Conduction in Complex Geometries

TL;DR

This paper presents a high-order meshless method using polyharmonic splines for solving heat conduction problems in complex geometries, demonstrating exponential convergence and potential for broader thermal applications.

Contribution

It introduces a novel meshless collocation method with PHS-RBFs and polynomial augmentation for high-accuracy heat conduction simulations in complex domains.

Findings

Exponential convergence observed with increased polynomial degree

Method effectively applied to various complex geometries

Potential to solve multiple thermal PDEs

Abstract

In recent years, a variety of meshless methods have been developed to solve partial differential equations in complex domains. Meshless methods discretize the partial differential equations over scattered points instead of grids. Radial basis functions (RBFs) have been popularly used as high accuracy interpolants of function values at scattered locations. In this paper, we apply the polyharmonic splines (PHS) as the RBF together with appended polynomial and solve the heat conduction equation in several geometries using a collocation procedure. We demonstrate the expected exponential convergence of the numerical solution as the degree of the appended polynomial is increased. The method holds promise to solve several different governing equations in thermal sciences.