Improved Element-free Galerkin Method Research Articles

Analyzing Three-Dimensional Laplace Equations Using the Dimension Coupling Method

Due to the low computational efficiency of the Improved Element-Free Galerkin (IEFG) method, efficiently solving three-dimensional (3D) Laplace problems using meshless methods has been a longstanding research direction. In this study, we propose the Dimension Coupling Method (DCM) as a promising alternative approach to address this challenge. Based on the Dimensional Splitting Method (DSM), the DCM divides the 3D problem domain into a coupling of multiple two-dimensional (2D) problems which are handled via the IEFG method. We use the Finite Element Method (FEM) in the third direction to combine the 2D discretized equations, which has advantages over the Finite Difference Method (FDM) used in traditional methods. Our numerical verification demonstrates the DCM’s convergence and enhancement of computational speed without losing computational accuracy compared to the IEFG method. Therefore, this proposed method significantly reduces computational time and costs when solving 3D Laplace equations with natural or mixed boundary conditions in a dimensional splitting direction, and expands the applicability of the dimension splitting EFG method.

The dimension coupling method for 3D steady convection-diffusion-reaction problems with variable coefficients

In order to solve 3D convection-diffusion-reaction problems accurately and efficiently, we study the dimension coupling method (DCM) by combining the finite element method (FEM) with the improved element-free Galerkin (IEFG) method. The idea of the dimension splitting method is introduced into the governing equation of such problems, thus a series of 2D forms can be obtained by splitting the original 3D problem. The IEFG method is used to discretize these 2D problems, thus the discretized equations of 2D forms can be obtained by using the corresponding weak form. In the dimension splitting direction, the FEM is used to couple these discretized equations, thus final linear equations of original 3D problem can be obtained. The formula of the relative error is given, and the convergence of the DCM is proved numerically. The results of four numerical examples show that the advantage of the DCM is its highly computational efficiency.

An improved meshless method based on the dimension splitting moving least-squares method for elasticity problems

An improved element-free Galerkin method (IEFGM) is proposed in this paper to solve the two-dimensional elasticity problems. In the IEFGM, the dimension-splitting moving least squares (DS-MLS) method is used to construct the trial functions, and the Galerkin variational weak form coupled with the integral coordinate transformation is applied to derive the final discrete equations of the elastic problems. The DS-MLS method is developed from the dimensional splitting method and the moving least squares (MLS) approximation. Since the shape function of the DS-MLS method is derived independently from the direction of dimensional splitting and the splitting subdivision surface, the dimension and complexity of matrix operations are greatly reduced in solving the shape function of the MLS approximation, thereby improving the calculation efficiency. Some typical examples are discussed to show the effectiveness of the improved meshless method in this paper. From the numerical results, because the DS-MLS method reduces the dimensionality when solving the shape functions, the improved meshless method in this paper can consume less CPU time and acquire a higher accuracy solution than the EFG method.

The Improved Element-Free Galerkin Method for 3D Steady Convection-Diffusion-Reaction Problems with Variable Coefficients

In order to obtain the numerical results of 3D convection-diffusion-reaction problems with variable coefficients efficiently, we select the improved element-free Galerkin (IEFG) method instead of the traditional element-free Galerkin (EFG) method by using the improved moving least-squares (MLS) approximation to obtain the shape function. For the governing equation of 3D convection-diffusion-reaction problems, we can derive the corresponding equivalent functional; then, the essential boundary conditions are imposed by applying the penalty method; thus, the equivalent integral weak form is obtained. By introducing the IMLS approximation, we can derive the final solved linear equations of the convection-diffusion-reaction problem. In numerical examples, the scale parameter and the penalty factor of the IEFG method for such problems are discussed, the convergence is proved numerically, and the calculation efficiency of the IEFG method are verified by four numerical examples.

An Improved Element-Free Galerkin Method Based on the Dimension Splitting Moving Least-Squares Method for 2D Potential Problems in Irregular Domains

By introducing the dimension splitting (DS) method into the moving least-squares (MLS) approximation, a dimension splitting moving least-squares (DS-MLS) method is proposed in this paper. In the DS-MLS method, the operator splitting and independent variable splitting of the DS method are used to reduce the dimension, thereby reducing the computational complexity of the matrix. The shape function of the DS-MLS method has the advantages of simple derivation and high computational efficiency. Then, by coupling DS-MLS method and Galerkin weak form, and performing the coordinate transformation, an improved element-free Galerkin method (IEFGM) based on the DS-MLS method is proposed for two-dimensional (2D) potential problems on irregular domains. The effectiveness of the method in this paper is verified by some numerical examples. The numerical results show that, compared with the element-free Galerkin (EFG) method, the IEFGM based on the DS-MLS method in this paper consumes less CPU time and has higher computational accuracy for some 2D potential problems on irregular domains.

The Hybrid Element-Free Galerkin Method for 3D Helmholtz Equations

In this study, the hybrid element-free Galerkin (HEFG) method is studied to solve the 3D Helmholtz equations. The idea of the dimension splitting method (DSM) is introduced into the improved element-free Galerkin (IEFG) method, thus a sequence of 2D forms can be obtained by splitting the problem domain of 3D Helmholtz equation, each 2D form can be discretized by using the IEFG method, thus the corresponding 2D discretized equations can be obtained, and these equations can be coupled by employing the finite difference method (FDM) in the dimension splitting direction, thus final formulae of numerical solution for 3D Helmholtz equation can be obtained. In Sec. 4, the relative errors and the convergence are analyzed, respectively, and the numerical results show that the calculation resources can be saved greatly without losing the computational accuracy when using the HEFG method.

The Improved Element-Free Galerkin Method for Anisotropic Steady-State Heat Conduction Problems

In this paper, we considered the improved element-free Galerkin (IEFG) method for solving 2D anisotropic steady-state heat conduction problems. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty method is applied to enforce the boundary conditions, thus the final discretized equations of the IEFG method for anisotropic steady-state heat conduction problems can be obtained by combining with the corresponding Galerkin weak form. The influences of node distribution, weight functions, scale parameters and penalty factors on the computational accuracy of the IEFG method are analyzed respectively, and these numerical solutions show that less computational resources are spent when using the IEFG method.

A Fast Element-Free Galerkin Method for 3D Elasticity Problems

In this paper, a fast element-free Galerkin (FEFG) method for three-dimensional (3D) elasticity problems is established. The FEFG method is a combination of the improved element-free Galerkin (IEFG) method and the dimension splitting method (DSM). By using the DSM, a 3D problem is converted to a series of 2D ones, and the IEFG method with a weighted orthogonal function as the basis function and the cubic spline function as the weight function is applied to simulate these 2D problems. The essential boundary conditions are treated by the penalty method. The splitting direction uses the finite difference method (FDM), which can combine these 2D problems into a discrete system. Finally, the system equation of the 3D elasticity problem is obtained. Some specific numerical problems are provided to illustrate the effectiveness and advantages of the FEFG method for 3D elasticity by comparing the results of the FEFG method with those of the IEFG method. The convergence and relative error norm of the FEFG method for elasticity are also studied.

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

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.

The Improved Interpolating Dimension Splitting Element-Free Galerkin Method for 3D Potential Problems

This paper proposes the improved interpolating dimension splitting element-free Galerkin (IIDSEFG) method based on the nonsingular weight function for three-dimensional (3D) potential problems. The core of the IIDSEFG method is to transform the 3D problem domain into a series of two-dimensional (2D) problem subdomains along the splitting direction. For the 2D problems on these 2D subdomains, the shape function is constructed by the improved interpolating moving least-squares (IIMLS) method based on the nonsingular weight function, and the finite difference method (FDM) is used to couple the discretized equations in the direction of splitting. Finally, the calculation formula of the IIDSEFG method for a 3D potential problem is derived. Compared with the improved element-free Galerkin (IEFG) method, the advantages of the IIDSEFG method are that the shape function has few undetermined coefficients and the essential boundary conditions can be executed directly. The results of the selected numerical examples are compared by the IIDSEFG method, IEFG method and analytical solution. These numerical examples illustrate that the IIDSEFG method is effective to solve 3D potential problems. The computational accuracy and efficiency of the IIDSEFG method are better than the IEFG method.

The dimension splitting interpolating element-free Galerkin method for solving three-dimensional transient heat conduction problems

In this paper, based on dimension splitting method and the improved interpolating moving least-squares (IMLS) method, a dimension splitting interpolating element-free Galerkin (DSIEFG) method for three-dimensional (3D) transient heat conduction problems is proposed. The main idea of the DSIEFG method is to split a 3D problem domain into a series of related two-dimensional (2D) subdomains. The improved IMLS method is used to construct shape function on 2D subdomains. Finite difference method is used to couple these discretized equations on 2D subdomains and employed in the time domain. Compared with the improved element-free Galerkin (IEFG) method, the advantage of the DSIEFG method is that the essential boundary conditions can be applied directly, which can improve computational accuracy and efficiency. Six examples are chosen to verify the effectiveness and efficiency of the DSIEFG method. The results of DSIEFG are compared with the numerical solutions of the IEFG method, and it is shown that the efficiency and precision of the DSIEFG method are greater than ones of the IEFG method for 3D transient heat conduction problems.

The Improved Element-Free Galerkin Method for Diffusional Drug Release Problems

By using the improved moving least-square (IMLS) approximation to present the shape function, the improved element-free Galerkin (IEFG) method is investigated to solve diffusional drug release problems in this paper. In order to get the discretized equation system, Galerkin weak form of a diffusional drug release problem is used with applying essential boundary conditions using the penalty method. The difference method is applied for discretization of time domain. Then the formulae of IEFG method for solving diffusional drug release problems are presented. Three numerical example problems are given to study the convergence of solutions of IEFG method in this paper. The influences of scale parameters of influence domain, penalty factor and node distribution on the accuracy of the solutions of IEFG method are discussed. Compared with finite element method, the correctness of IEFG method in this paper is shown.

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

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.

A Coupled Improved Element Free Galerkin-Finite Strip (IEFG-FS) Method for Free Vibration Analysis of Plate

In this paper, a coupling of improved element-free Galerkin (IEFG) method and semi-analytical finite strip (FS) method is presented for free vibration analysis of thin plates. This method is very easy to implement and has advantages of both IEFG and FS methods, so that IEFG method is used in sub-domain with complex geometry, and FS method is used for the remaining domains. The use of the FS method considerably reduces the analysis time, and the essential boundary conditions are easily enforced in FS sub-domain. In the IEFG method, the shape function does not have the Kronecker delta function property. Therefore, Lagrange multipliers method is used to satisfy the boundary conditions. Finally, five examples are presented to show the effectiveness of this work.

Analyzing 3D advection-diffusion problems by using the dimension splitting element-free Galerkin method

By combining the dimension splitting method with the improved element-free Galerkin (IEFG) method, the dimension splitting element-free Galerkin (DSEFG) method is proposed to solve 3D advection-diffusion problems. By using the dimension splitting method, a 3D advection-diffusion problem is transformed a series of 2D ones. The IEFG method is used for solving these related 2D advection-diffusion problems, and the finite difference method (FDM) is applied in the time space and dimension splitting direction. The discretized system equation is obtained depended on the Galerkin weak form of 2D problems, and the essential boundary conditions are imposed with penalty method, then the formulae of the DSEFG method for 3D advection-diffusion problem are obtained. Numerical examples are given to verify the efficiency of the DSEFG method. And the numerical results show that the DSEFG method can improve the computational efficiency and precision of the IEFG method greatly under the similar computational accuracy.

The interpolating element-free Galerkin (IEFG) method for three-dimensional potential problems

In this paper, an interpolating element-free Galerkin (IEFG) method for three-dimensional potential problems is presented based on the improved interpolating moving least-squares (IIMLS) method. The shape function of the IIMLS method satisfies the property of Kronecker δ function, which results in that the boundary conditions are applied directly. Compared with the improved element-free Galerkin (EFG) method, the IEFG method in this paper gives a higher computational accuracy. Moreover, this method also has greater computational efficiency than the improved EFG method under the similar accuracy. For the purposes of demonstration, four numerical examples are solved using the IEFG method to show the advantages of the IEFG method with higher computational accuracy and computational efficiency.

The improved element-free Galerkin method for three-dimensional elastoplasticity problems

In this study, the improved element-free Galerkin (IEFG) method is presented to the three-dimensional elastoplasticity problems. The improved least-squares (IMLS) approximation is used to obtain the shape function, the Galerkin weak form of three-dimensional elastoplasticity problems considering the nonlinear stress–strain relationship is used to form the discrete equation system, and penalty method is applied to the displacement boundary conditions, then the formula of the IEFG method for three-dimensional elastoplasticity problems are obtained. Some numerical examples are given to discuss the convergence of the IEFG method in this paper and the influences of the weight function, the scaling parameter, the penalty factor, the node distribution and the step number on the computational accuracy of the numerical solutions of the IEFG method. Comparing with the element-free Galerkin (EFG) method, the method in this study has a greater computational efficiency.

A Meshless Method Based on the Nonsingular Weight Functions for Elastoplastic 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.

The improved element-free Galerkin method based on the nonsingular weight functions for elastic 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.

The dimension splitting element-free Galerkin method for 3D transient heat conduction problems

By transforming a 3D problem into some related 2D problems, the dimension splitting element-free Galerkin (DSEFG) method is proposed to solve 3D transient heat conduction problems. The improved element-free Galerkin (IEFG) method is used for 2D transient heat conduction problems, and the finite difference method is applied in the splitting direction. The discretized system equation is obtained based on the Galerkin weak form of 2D problem; the essential boundary conditions are imposed with the penalty method; and the finite difference method is employed in the time domain. Four exemplary problems are chosen to verify the efficiency of the DSEFG method. The numerical solutions show that the efficiency and precision of the DSEFG method are greater than ones of the IEFG method for 3D problems.