Abstract
The element-free Galerkin (EFG) method is one of the widely used meshfree methods for solving partial differential equations. In the EFG method, shape functions are derived from a moving least-squares (MLS) approximation, which involves the inversion of a small matrix for every point of interest. To avoid the calculation of matrix inversion in the formulation of the shape functions, an improved MLS approximation is presented, where an orthogonal function system with a weight function is used. However, it can also lead to ill-conditioned or even singular system of equations. In this paper, aspects of the IMLS approximation are analyzed in detail. The reason why singularity problem occurs is studied. A novel technique based on matrix triangular process is proposed to solve this problem. It is shown that the EFG method with present technique is very effective in constructing shape functions. Numerical examples are illustrated to show the efficiency and accuracy of the proposed method. Although our study relies on monomial basis functions, it is more general than existing methods and can be extended to any basis functions.
Highlights
In recent years, the meshfree method has been developed rapidly as a computational technique for solving partial differential equations
A group of meshfree methods have been proposed and developed, such as the diffuse element method (DEM) [1], the element-free Galerkin method (EFG) [2, 3], the meshfree point interpolation method [4], the meshless method based on radial basis functions [5,6,7], the meshless local PetrovGalerkin method [8], and the meshfree weak-strong (MWS) method [9]
In the sequel and unless mentioned otherwise, EFG method indicates the element-free Galerkin method with the original moving least-squares (MLS) approximation (10) and MEFG method stands for the element-free Galerkin method with the modified MLS approximation (21)
Summary
The meshfree (meshless) method has been developed rapidly as a computational technique for solving partial differential equations. With the use of weighted orthogonal basis functions, the moment matrix becomes a diagonal matrix; the burden of inverting A at each computational point is totally eliminated This idea is further studied and applied in elasticity problems [12, 16], dynamics analysis [12], fracture problems [13], and so on. In [18], the reason for singularity problem was discussed and a method to solve the singularity problem by finding optimal radius of the support domain was proposed This method is useful for regular nodes distribution and monomial basis functions.
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