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 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 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.

A Meshless Method Based on the Nonsingular Weight Functions for Elastoplastic Large Deformation Problems

2019 ◽  
Vol 11 (01) ◽  
pp. 1950006 ◽  
Author(s):  
Fengbin Liu ◽  
Qiang Wu ◽  
Yumin Cheng
Keyword(s):  
Large Deformation ◽  
Numerical Solutions ◽  
Weak Form ◽  
Weight Functions ◽  
Computational Accuracy ◽  
Lagrange Formulation ◽  
Element Free Galerkin ◽  
Penalty Factor ◽  
Element Free ◽  
Large Deformation Problems

In this study, based on a nonsingular weight function, the improved element-free Galerkin (IEFG) method is presented for solving elastoplastic large deformation problems. By using the improved interpolating moving least-squares (IMLS) method to form the approximation function, and using Galerkin weak form based on total Lagrange formulation of elastoplastic large deformation problems to form the discretilized equations, which is solved with the Newton–Raphson iteration method, we obtain the formulae of the IEFG method for elastoplastic large deformation problems. In numerical examples, the influences of the penalty factor, scale parameter of influence domain and weight functions on the computational accuracy are analyzed, and the numerical solutions show that the IEFG method for elastoplastic large deformation problems has higher computational efficiency and accuracy.

scholarly journals Analyzing 3D Advection-Diffusion Problems by Using the Improved Element-Free Galerkin Method

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Heng Cheng ◽  
Guodong Zheng
Keyword(s):  
Penalty Method ◽  
Weak Form ◽  
Singular Matrix ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Advection Diffusion ◽  
Computational Speed ◽  
The Difference ◽  
Element Free ◽  
Diffusion Problems

In this paper, the improved element-free Galerkin (IEFG) method is used for solving 3D advection-diffusion problems. The improved moving least-squares (IMLS) approximation is used to form the trial function, the penalty method is applied to introduce the essential boundary conditions, the Galerkin weak form and the difference method are used to obtain the final discretized equations, and then the formulae of the IEFG method for 3D advection-diffusion problems are presented. The error and the convergence are analyzed by numerical examples, and the numerical results show that the IEFG method not only has a higher computational speed but also can avoid singular matrix of the element-free Galerkin (EFG) method.

The improved element-free Galerkin method based on the nonsingular weight functions for elastic large deformation problems

2018 ◽  
Vol 07 (04) ◽  
pp. 1850023 ◽  
Author(s):  
Fengbin Liu ◽  
Yumin Cheng
Keyword(s):  
Weight Function ◽  
Large Deformation ◽  
Weak Form ◽  
Weight Functions ◽  
Computational Accuracy ◽  
Lagrange Formulation ◽  
Element Free Galerkin ◽  
Penalty Factor ◽  
Element Free ◽  
Large Deformation Problems

In this paper, the interpolating moving least-squares (IMLS) method based on a nonsingular weight function is applied to obtain the approximation function. The penalty method is applied to impose the displacement boundary condition, and Galerkin weak form of elastic large deformation problems based on total Lagrange formulation is used to form the final equations which is solved with the Newton–Raphson iteration method, then the improved element-free Galerkin (IEFG) method based on a nonsingular weight function for elastic large deformation problems is presented. The IMLS method can overcome the disadvantage of singular weight functions in the traditional MLS method, then the IEFG method in this paper has high computational accuracy and efficiency, which are shown by numerical examples of elastic large deformation problems. And the influences of the weight functions, scale parameter of influence domain, step number and penalty factor on the numerical results are discussed.

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 Element Free Galerkin Method for Fully Developed, Open-Channel Magnetohydrodynamic Flow

2012 ◽  
Vol 232 ◽  
pp. 111-114
Author(s):  
Xing Hui Cai ◽  
Guo Xun Ji ◽  
Peng Xu ◽  
Man Lin Zhu ◽  
Jiang Ren Lu
Keyword(s):  
Liquid Metal ◽  
Galerkin Method ◽  
Open Channel ◽  
Metal Flow ◽  
Weak Form ◽  
Governing Equations ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Liquid Metal Flow ◽  
Element Free

In this paper, an element-free Galerkin method is presented to simulate the liquid metal flow in an open channel under external magnetic field. The global weak form of governing equations is obtained for the case of same size of the height of the liquid film and width of the open channel. Numerical simulations are carried out for some cases of liquid metal flow in an open channel. Results show that the element-free Galerkin method may steadily compute this kind of problem in some cases.

The Improved Interpolating Complex Variable Element Free Galerkin Method for Two-Dimensional Potential Problems

2019 ◽  
Vol 11 (10) ◽  
pp. 1950104 ◽  
Author(s):  
Yajie Deng ◽  
Xiaoqiao He ◽  
Ying Dai
Keyword(s):  
Complex Variable ◽  
Basis Function ◽  
Potential Problem ◽  
Shape Function ◽  
Weak Form ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Variable Element ◽  
Potential Problems ◽  
Element Free

In this paper, the improved interpolating complex variable moving least squares (IICVMLS) method is applied, in which the complete basis function is introduced and combined with the singular weight function to achieve the orthometric basis function. Then, the interpolating shape function is achieved to construct the interpolating trial function. Incorporating the IICVMLS method and the Galerkin integral weak form, an improved interpolating complex variable element free Galerkin (IICVEFG) method is proposed to solve the 2D potential problem. Because the essential boundary conditions can be straightaway imposed in the above method, the expressions of final dispersed matrices are more concise in contrast to the non-interpolating complex variable meshless methods. Through analyzing four specific potential problems, the IICVEFG method is validated with greater computing precision and efficiency.

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