Universal Journal of Mechanical Engineering 2(1): 20-27, 2014 http://www.hrpub.org DOI: 10.13189/ujme.2014.020103 Benefits of Using Non-consolidated Domain Influence in Meshless Local Petrov-galerkin (Mlpg) Method for Solving Lefm Problems Hashim N. Al-Mahmud, Haider K. Mehbes, Ameen A. Nassar * Mechanical Engineering Department, College of Engineering, University of Basrah *Corresponding Author: ameenaledani@yahoo.com Copyright © 2014 Horizon Research Publishing All rights reserved. Abstract This paper presents an efficient meshless method in the formulation of the weak form of local Petrov-Galerkin method MLPG. The formulation is carried out by using an elliptic domain rather than conventional isotropic domain of influence. Therefore, the method involves an MLPG formulation in conjunction with an anisotropic weight function. In the elliptic weight function, each node has three characteristic indicated that were major radius, inner radius, and the direction of the local domain. Furthermore, the space that will be covered by the elliptical domain will be less than the area of the circle (isotropic) at the same main diameter. This means leaving many points of integration are not necessary. Therefore, the computational cost will be decreased. MLPG method with the elliptical domain is used in solving problems of linear elastic fracture mechanism LEFM. MATLAB and Fortran codes are used for obtaining the results of this research .The results were compared with those presented in the literature which shows a reduction in the computational cost up to 15%, and an error criteria enhancement up to 25%. Keywords Meshless Methods, Local Petrov-Galerkin Method MLPG, Elliptic Domain 1. Introduction Meshless (MFree) methods, as alternative numerical approaches to eliminate the well-known drawbacks in the finite element and boundary element methods have attracted much attention in the past decade, due to their flexibility, and due to their potential in neglecting the need for the human-labor intensive process of constructing geometric meshes in a domain. There are a number of MFree methods has been developed named according to the technique used in the formulation of the method the major differences in these meshless methods come from the interpolation techniques used [1-4]. In recent decades Mfree methods in computational mechanics have a great attention in solving practical engineering problems in heat transfer, fluid mechanics, and applied mechanics [5-7],especially those problems with discontinuities or moving boundaries. The numerical solution by the traditional finite element method (FEM) of fracture mechanics problems with arbitrary dynamic cracks is limited to simple cases. This is because solution of growing discontinuities requires time consuming remeshing at every time step. For this reason adaptive FEM has become essential. However adaptive remeshing and mapping of variables is a difficult, computationally expensive task and is a source of cumulative numerical errors. The development of meshless methods has enabled the solution of growing cracks without remeshing. Nevertheless, these methods continue to be computationally expensive because of the large nodal densities in meshless methods for an accurate solution. Therefore there is a constant effort to improve the accuracy without increasing the degrees of freedom The main objective of MFree methods is to get rid of or at least alleviate the difficulty of meshing the entire structure ,by only adding or deleting nodes in the entire structure. A truly meshless method shadow elements are inevitable as in Element-Free Galerkin Methods [8-9]called Meshless Local Petrov-Galerkin Method (MLPG) have been successfully developed in[10-16]for solving linear and non-linear boundary problems[17]. The MLPG method uses local weak forms over a local sub-domain and shape functions from the moving least-squares (MLS) approximation[18]. In the MLS approximation, each node in the global domain Ω has two sub-domains the 1 st is the domain of influence Ω x , in which a trail function of compact support is used as a weight function. The weight function determines the intensity of the effect of a node at various points in its domain of influence, the 2 nd is a sub-domain for the test function Ω s (Integral Domain) which often similar in shape but smaller than the trial function. These nodal trial and test functions are centered with maximum value at the nodes (which are the centers of the domains Ω x and Ω s ) ,