Meshless Local Petrov-Galerkin Method for 3D Steady-State Heat Conduction Problems

M. J. Mahmoodabadi,A. Bagheri,R. Abedzadeh Maafi,G. H. Baradaran

doi:10.1155/2011/251546

Abstract

The Meshless Local Petrov-Galerkin (MLPG) method is applied for solving the three-dimensional steady state heat conduction problems. This method is a truly meshless approach; also neither the nodal connectivity nor the background mesh is required for solving the initial boundary-value problems. The penalty method is adopted to enforce the essential boundary conditions. The moving least squares (MLS) approximation is used for interpolation schemes and the Heviside step function is chosen for representing the test function. The numerical results are compared with the exact solutions of the problem and Finite Difference Method (FDM). This comparison illustrates the accuracy as well as the capability of this method.

Highlights

In the recent years, the meshless methods have evoked considerable interest among researches and engineers for finding solutions of initial- and boundary-value problems
The Meshless Local Petrov-Galerkin (MLPG) method has been demonstrated to be quite successful in solving various partial differential equations
Wu et al [3] coupled the MLPG method with either the finite-element or the boundary-element method to enhance the efficiency of the MLPG method

Summary

Introduction

The meshless methods have evoked considerable interest among researches and engineers for finding solutions of initial- and boundary-value problems. The MLPG method has been demonstrated to be quite successful in solving various partial differential equations The concept of this approach was first introduced by Atluri and Zhu [1]. They solved elastostatic problems in 2D domains. The bending of a thin plate has been studied by Gu and Liu [7] and Long and Atluri [8] These works and several other published works in the literature were dealing with 2D boundary-value problems, and fewer researches were conducted on 3D problems. Han and Atluri [9] used the MLPG approach for 3D problems arising in elastostatics They applied the MLPG method for solving 3D elastic fracture [10] and 3D elastodynamic problems [11]. Comparisons are made through some illustrative examples to show the accuracy of the present approach

MLPG Formulation

The MLS Approximation Scheme

Results of Numerical Examples

Method MLPG

Results

Conclusion