Numerical assessment regarding the time-dependent heat conduction equation using Meshless Local Petrov-Galerkin and Radial Point Interpolation Method

This study presents and evaluates four distinct case studies concerning the transient (unsteady) heat conduction equation. The Meshless Local Petrov-Galerkin (MLPG) method is employed, and a direct comparison is performed between the distinct approximation functions: the Moving Least-Squares (MLS), the Radial Point Interpolation Method (RPIM) and the Interpolating Moving Least-Squares(IMLS). The first numerical example investigates a more complex two-dimensional geometry, specifically a quarter of a circular plate, to which a Dirichlet boundary condition is applied. The second example involves a square domain, utilizing both a Neumann boundary condition and a Dirichlet boundary condition that varies over time, thus constituting a time-dependent problem. The third example analyzes a plate with a circular arc at its bottom edge. The four example assess a three dimensional case and the response of the MLPG Method. Higher order MLS and IMLS bases are evaluated, such as quadratic and cubic bases. A regular and irregular node distribution for each study case investigates how this parameter affect the solution and convergence.The results derived from the MLPG implementation are compared against a reference solution obtained from the commercial software ABAQUS, which utilizes the consolidated Finite Element Method (FEM). The primary objective is to assess the performance and efficacy of the MLPG method when applied to the transient heat conduction equation.