A Computationally Efficient Meshless Local Petrov-Galerkin Method for Axisymmetric Problems

The Meshless Local Petrov-Galerkin (MLPG) method is one of the recently developed element-free methods. The method is convenient and can produce accurate results with continuous secondary variables, but is more computationally expensive than the finite element method. To overcome this disadvantage, a simple Heaviside test function is chosen. The computational effort is significantly reduced by eliminating the domain integral for the axisymmetric potential problems and by simplifying the domain integral for the axisymmetric elasticity problems. The method is evaluated through several patch tests for axisymmetric problems and example problems for which the exact solutions are available. The present method yielded very accurate solutions. The sensitivity of several parameters of the method is also studied. INTRODUCTION The Meshless Local Petrov-Galerkin (MLPG) method is a promising numerical method to analyze potential and elasticity problems and is shown to yield accurate results [1-6]. In this method, a set of arbitrarily distributed nodes is used to interpolate the field variables. The method does not use either ‘elements’ or a back-ground mesh for integration and hence is truly meshless. The method, however, appears to be more computationally expensive than the finite element method. One of the reasons for the high computational cost is that accurate numerical integration is required to integrate the weak form of the governing equations. Thus, if the domain integrations are eliminated or simplified, the method can be made efficient. A choice of a “Heaviside” test function [5, 6] leads to the elimination of the domain integral in the weak form of the axisymmetric potential problems and substantially simplifies the integral for axisymmetric elasticity problems. The purpose of this paper is to present such a method that utilizes the Heaviside test function and evaluate its effectiveness for axisymmetric problems. The outline of the paper is as follows. First, a brief overview of MLPG method is presented for axisymmetric potential and elasticity problems. The MLPG formulation with Heaviside function is presented, and various issues related to the use of a Heaviside test function are studied. Finally the effectiveness of the method is demonstrated using several numerical problems. OVERVIEW OF MLPG METHOD In this section, several basic concepts that are used in the MLPG method are briefly reviewed. The development of the weak forms of the solution for potential and elasticity problems is first presented. Next, the choice of the trial function for the primary variables using the moving least squares method is reviewed. In the classical element-free Galerkin methods, the test functions are chosen from the same space as the trial functions. In the current MLPG method, the test functions are chosen to be different from the trial functions, and the choice of the test functions is discussed next. Potential Problem Consider Poisson’s equation for an axisymmetric problem bounded by a toroidal domain with its cross section defined by Ω as shown in Figure 1, u g ∇ = in Ω (1) with boundary conditions u u = on Γu and q q = on Γq (2) where Γ =Γu + Γq and / q du dn = . The Laplacian in the cylindrical coordinate system is AIAA-2003-1673 * Senior Technologist, Structures and Materials Competency, Fellow AIAA. † Army Research Laboratory, MS-240, Analytical and Computational Methods Branch, NASA Langley Research Center. This material is a work of the U.S. Government and is not subject to copyright protection in the United States.