An Improved Interpolating Element-Free Galerkin Method Based on Nonsingular Weight Functions

F. X. Sun,C. Liu,Y. M. Cheng

doi:10.1155/2014/323945

Abstract

Based on the moving least-squares (MLS) approximation, an improved interpolating moving least-squares (IIMLS) method based on nonsingular weight functions is presented in this paper. Then combining the IIMLS method and the Galerkin weak form, an improved interpolating element-free Galerkin (IIEFG) method is presented for two-dimensional potential problems. In the IIMLS method, the shape function of the IIMLS method satisfies the property of Kroneckerδfunction, and there is no difficulty caused by singularity of the weight function. Then in the IIEFG method presented in this paper, the essential boundary conditions are applied naturally and directly. Moreover, the number of unknown coefficients in the trial function of the IIMLS method is less than that of the MLS approximation; then under the same node distribution, the IIEFG method has higher computational precision than element-free Galerkin (EFG) method and interpolating element-free Galerkin (IEFG) method. Four selected numerical examples are presented to show the advantages of the IIMLS and IIEFG methods.

Highlights

In recent years, meshless method has become very attractive to solve science and engineering problems without meshes
Compared with the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas, the weight function used in the interpolating moving least-squares (IIMLS) method is nonsingular at any points, and any weight function used in the Moving least-square (MLS) approximation can be chosen as the weight function of the IIMLS method
Compared with the IMLS method presented by Lancaster and Salkauskas, the nonsingular weight function is used in the IIMLS method

Summary

Introduction

Meshless (or meshfree) method has become very attractive to solve science and engineering problems without meshes. For penalty methods, the optimal value of penalty factor always affects the accuracy of the final solution To overcome this disadvantage, Most and Bucher designed a regularized weight function with a regularization parameter ε, with which the MLS approximation can almost fulfill the interpolation and boundary conditions with high accuracy [47]. Sergio obtained a special weight function using a normalization based on the Shepard interpolation to fulfill the interpolation [49] Another possible approach for this disadvantage is the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas [45]. In this paper, based on the MLS approximation and IMLS method, an improved interpolating moving leastsquares (IIMLS) method with nonsingular weight functions is presented. Four selected numerical examples are presented to show the advantages of the IIMLS and IIEFG methods

The Improved Interpolating Moving Least-Squares Method

The Improved Interpolating Element-Free Galerkin Method

Numerical Examples

Conclusions