scholarly journals A Modification of the Moving Least-Squares Approximation in the Element-Free Galerkin Method

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Yang Cao ◽  
Jun-Liang Dong ◽  
Lin-Quan Yao
Keyword(s):  
Least Squares ◽  
Singular System ◽  
Basis Functions ◽  
Moving Least Squares ◽  
Shape Functions ◽  
Monomial Basis ◽  
Element Free Galerkin ◽  
Small Matrix ◽  
Mls Approximation ◽  
Element Free

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.

THE INTERPOLATING ELEMENT-FREE GALERKIN (IEFG) METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS

2011 ◽  
Vol 03 (04) ◽  
pp. 735-758 ◽  
Author(s):  
HONGPING REN ◽  
YUMIN CHENG
Keyword(s):  
Least Squares ◽  
Shape Function ◽  
Weak Form ◽  
New Method ◽  
Moving Least Squares ◽  
Two Dimensional ◽  
Element Free Galerkin ◽  
Elasticity Problems ◽  
Mls Approximation ◽  
Element Free

In this paper, a new method for deriving the moving least-squares (MLS) approximation is presented, and the interpolating moving least-squares (IMLS) method proposed by Lancaster is improved. Compared with the IMLS method proposed by Lancaster, a simpler formula of the shape function is given in the improved IMLS method in this paper so that the new method has higher computing efficiency. Combining the shape function constructed by the improved IMLS method with Galerkin weak form of the elasticity problems, the interpolating element-free Galerkin (IEFG) method for the two-dimensional elasticity problems is presented, and the corresponding formulae are obtained. In the IEFG method, the boundary conditions can be applied directly which makes the computing efficiency higher than the conventional EFG method. Some numerical examples are presented to demonstrate the validity of the method.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
F. X. Sun ◽  
C. Liu ◽  
Y. M. Cheng
Keyword(s):  
Least Squares ◽  
Shape Function ◽  
Weak Form ◽  
Moving Least Squares ◽  
Weight Functions ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Node Distribution ◽  
Mls Approximation ◽  
Element Free

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.

The influence of selectable parameters in the element-free Galerkin method: A one-dimensional beam-in-bending problem

2009 ◽  
Vol 223 (7) ◽  
pp. 1579-1590 ◽  
Author(s):  
O F Valencia ◽  
F J Gómez-Escalonilla ◽  
J López-Díez
Keyword(s):  
Galerkin Method ◽  
Least Squares ◽  
Least Squares Method ◽  
Moving Least Squares ◽  
Shape Functions ◽  
Element Free Galerkin Method ◽  
One Dimensional ◽  
Element Free Galerkin ◽  
Axially Loaded ◽  
Element Free

Continuing with the analysis performed for the one-dimensional axially loaded bar problem, a beam in bending is analysed to understand the influence of the characteristic parameters that have any influence in the solution of this problem using the element-free Galerkin method (EFGM), one of the most popular meshless methods. Both accuracy and time cost are considered as the evaluation functions to perform such an analysis. Both functions provide a reasonable idea to consider EFGM as an adequate method to solve the problem considered in this article. As in a one-dimensional axially loaded bar problem, the parameters to be considered will be those that affect the solution: number of nodes in which the domain is modelled, the nodes scatter, the order of the polynomial base to generate shape functions, the order of the quadrature to solve integrals, and the support radius. Besides, as in a one-dimensional axially loaded problem, some cases with different loading and stiffness conditions are considered. However, in this analysis a generalized moving least squares method is used to create shape functions instead of the moving least squares.

An Improved Interpolating Element-Free Galerkin Method for Elastoplasticity via Nonsingular Weight Functions

2016 ◽  
Vol 08 (08) ◽  
pp. 1650096 ◽  
Author(s):  
Fengxin Sun ◽  
Jufeng Wang ◽  
Yumin Cheng
Keyword(s):  
Least Squares ◽  
Numerical Solutions ◽  
Weak Form ◽  
Moving Least Squares ◽  
Weight Functions ◽  
Shape Functions ◽  
Computational Accuracy ◽  
Displacement Boundary ◽  
Element Free Galerkin ◽  
Element Free

An improved interpolating element-free Galerkin (IIEFG) method for elastoplasticity is proposed in this paper. In the IIEFG method, the shape functions are constructed by the improved interpolating moving least-squares (IIMLS) method, and the final system equations are obtained by using the Galerkin weak form of elastoplasticity. Compared with the interpolating moving least-squares (IMLS) method, the weight functions are not singular in the IIMLS method, in which the shape functions have the interpolating property. The IIMLS method has fewer unknown coefficients to be solved in the trial functions than the moving least-squares (MLS) approximation. Hence, the IIEFG method is able to directly enforce the displacement boundary condition and obtain numerical solutions with high computational accuracy and efficiency. To show advantages of the IIEFG method, some selected elastoplastic examples are given.

The Interpolating Complex Variable Element-Free Galerkin Method for Temperature Field Problems

2015 ◽  
Vol 07 (02) ◽  
pp. 1550017 ◽  
Author(s):  
Yajie Deng ◽  
Chao Liu ◽  
Miaojuan Peng ◽  
Yumin Cheng
Keyword(s):  
Temperature Field ◽  
Least Squares ◽  
Complex Variable ◽  
Weak Form ◽  
Equation System ◽  
Moving Least Squares ◽  
Element Free Galerkin ◽  
Variable Element ◽  
Mls Approximation ◽  
Element Free

In this paper, an interpolating complex variable moving least-squares (ICVMLS) method is presented. In the ICVMLS method, the trial function of a two-dimensional problem is formed with a one-dimensional basis function, and the shape function of the ICVMLS method satisfies the property of Kronecker δ function. The ICVMLS method has greater computational efficiency than the moving least-squares (MLS) approximation. Then combining the ICVMLS method with the Galerkin weak form of temperature field problems, an interpolating complex variable element-free Galerkin (ICVEFG) method is proposed. In the ICVEFG method, we can obtain the equation system by applying the essential boundary conditions directly. Compared with the element-free Galerkin (EFG) method and the complex variable element-free Galerkin (CVEFG) method, the ICVEFG method in this paper has higher accuracy and efficiency.

A NEW ELEMENT-FREE GALERKIN METHOD BASED ON IMPROVED COMPLEX VARIABLE MOVING LEAST-SQUARES APPROXIMATION FOR ELASTICITY

2012 ◽  
Vol 01 (01) ◽  
pp. 1250011 ◽  
Author(s):  
HONGPING REN ◽  
YUMIN CHENG
Keyword(s):  
Least Squares ◽  
Complex Variable ◽  
Weak Form ◽  
Moving Least Squares ◽  
Two Dimensional ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Variable Element ◽  
Elasticity Problems ◽  
Element Free

In this paper, by constructing a new functional, an improved complex variable moving least-squares (ICVMLS) approximation is presented. Based on element-free Galerkin (EFG) method and the ICVMLS approximation, a new complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems is presented. Galerkin weak form is used to obtain the discretized equations and the essential boundary conditions are applied with Lagrange multiplier. Then the formulae of the new CVEFG method for two-dimensional elasticity problems are obtained. Compared with the conventional EFG method, the new CVEFG method has greater computational precision and efficiency. For the purposes of demonstration, some selected numerical examples are solved using the ICVEFG method.

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