Solving Heat Conduction Problems by the Direct Meshless Local Petrov-Galerkin (DMLPG) method

TL;DR

The paper introduces the DMLPG method, an efficient meshless approach for solving transient heat conduction problems that reduces computational costs by directly recovering test functionals without MLS shape functions.

Contribution

It presents a novel DMLPG technique that simplifies numerical integration and improves efficiency over traditional MLS-based meshless methods.

Findings

Reduces computational cost of heat conduction simulations

Eliminates need for MLS shape functions in numerical integration

Achieves comparable accuracy with lower computational effort

Abstract

As an improvement of the Meshless Local Petrov-Galerkin (MLPG), the Direct Meshless Local Petrov-Galerkin (DMLPG) method is applied here to the numerical solution of transient heat conduction problem. The new technique is based on direct recoveries of test functionals (local weak forms) from values at nodes without any detour via classical moving least squares (MLS) shape functions. This leads to an absolutely cheaper scheme where the numerical integrations will be done over low-degree polynomials rather than complicated MLS shape functions. This eliminates the main disadvantage of MLS based methods in comparison with finite element methods (FEM), namely the costs of numerical integration.