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 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.

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.

The Improved Element-Free Galerkin Method Based on the Nonsingular Weight Functions for Inhomogeneous Swelling of Polymer Gels

2018 ◽  
Vol 10 (04) ◽  
pp. 1850047 ◽  
Author(s):  
Fengbin Liu ◽  
Yumin Cheng
Keyword(s):  
Numerical Solutions ◽  
Weak Form ◽  
Polymer Gels ◽  
Weight Functions ◽  
Element Free Galerkin Method ◽  
Displacement Boundary ◽  
Element Free Galerkin ◽  
Node Distribution ◽  
Penalty Factor ◽  
Element Free

In this paper, the interpolating moving least-squares (IMLS) method based on a nonsingular weight function is used to construct the approximation function, the weak form of the problem of inhomogeneous swelling of polymer gels is used to obtain the final discretized equations, and penalty method is applied to impose the displacement boundary condition, then an improved element-free Galerkin (IEFG) method for the problem of the inhomogeneous swelling of polymer gels is presented. Three selected examples of inhomogeneous swelling of polymer gels solved with the IEFG method are given in this paper. The accuracy of the numerical solutions of the IEFG method are discussed by using different weight functions, penalty factor, scale parameter of influence domain, node distribution and step number. Numerical results of the IEFG method for inhomogeneous swelling of polymer gels show that this method has great precision, and it can solve large deformation problems of polymer gels effectively.

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.

scholarly journals The Interpolating Element-Free Galerkin Method for 2D Transient Heat Conduction Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Na Zhao ◽  
Hongping Ren
Keyword(s):  
Heat Conduction ◽  
Least Squares ◽  
Shape Function ◽  
Weak Form ◽  
Transient Heat Conduction ◽  
Delta Function ◽  
Moving Least Squares ◽  
Computational Accuracy ◽  
Element Free Galerkin ◽  
Element Free

An interpolating element-free Galerkin (IEFG) method is presented for transient heat conduction problems. The shape function in the moving least-squares (MLS) approximation does not satisfy the property of Kronecker delta function, so an interpolating moving least-squares (IMLS) method is discussed; then combining the shape function constructed by the IMLS method and Galerkin weak form of the 2D transient heat conduction problems, the interpolating element-free Galerkin (IEFG) method for transient heat conduction problems is presented, and the corresponding formulae are obtained. The main advantage of this approach over the conventional meshless method is that essential boundary conditions can be applied directly. Numerical results show that the IEFG method has high computational accuracy.

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.

scholarly journals The Improved Element-Free Galerkin Method for 3D Helmholtz Equations

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 14
Author(s):  
Heng Cheng ◽  
Miaojuan Peng
Keyword(s):  
Weak Form ◽  
Weight Functions ◽  
Computational Accuracy ◽  
Element Free Galerkin Method ◽  
Helmholtz Equations ◽  
Scale Parameters ◽  
Element Free Galerkin ◽  
Node Distribution ◽  
Computational Speed ◽  
Element Free

The improved element-free Galerkin (IEFG) method is proposed in this paper for solving 3D Helmholtz equations. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty technique is used to enforce the essential boundary conditions. Thus, the final discretized equations of the IEFG method for 3D Helmholtz equations can be derived by using the corresponding Galerkin weak form. The influences of the node distribution, the weight functions, the scale parameters of the influence domain, and the penalty factors on the computational accuracy of the solutions are analyzed, and the numerical results of three examples show that the proposed method in this paper can not only enhance the computational speed of the element-free Galerkin (EFG) method but also eliminate the phenomenon of the singular matrix.

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.

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