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%.
Highlights
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 [1,2,3,4]
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 [5,6,7],especially those problems with discontinuities or moving boundaries
Summary
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. 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. The definition of the influence domain based on non-consolidated weight function, improves the numerical efficiency of MLPG. In such case, the influence domain of each node can be determined so that the nodal overlapping decreases. Good results can be achieved with less computational efforts
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