A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.

Numerical analysis of coupled finite element with element-free Galerkin in sheet flexible-die forming

A numerical method for coupled deformation between sheet metal and flexible-die was proposed. Based on the updated Lagrangian （UL） formulation, the elastoplastic deformation of sheet metal was analyzed with finite element method （FEM） and the bulk deformation of flexible-die was analyzed with element-free Galerkin method （EFGM）. The frictional contact between sheet metal and flexible-die was treated by the penalty function method. The sheet elastic flexible-die bulging process was analyzed with the FEM-EFGM program for coupled deformation between sheet metal and bulk flexible-die, called CDSB-FEM-EFGM for short. Compared with finite element code DEFORM-2D and experiment results, the CDSB-FEM-EFGM program is feasible. This method provides a suitable numerical method to analyze sheet flexible-die forming.

The dimension split element-free Galerkin method for three-dimensional potential problems

This paper presents the dimension split element-free Galerkin （DSEFG） method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares （IMLS） approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than 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.

An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems

In this paper, an improved interpolating moving least-square （IIMLS） method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker 5 function. The weight function used in the IIMLS method is nonsingular. Then the IIMLS method can overcome the difficulties caused by the singularity of the weight function in the IMLS method. The number of unknown coefficients in the trial function of the IIMLS method is less than that of the moving least-square （MLS） approximation. Then by combining the IIMLS method with the Galerkin weak form of the potential problem, the improved interpolating element-free Galerkin （IIEFG） method for two-dimensional potential problems is presented. Compared with the conventional element-free Galerkin （EFG） method, the IIEFG method can directly use the essential boundary conditions. Then the IIEFG method has higher accuracy. For demonstration, three numerical examples are solved using the IIEFG method.

An improved boundary element-free method (IBEFM) for two-dimensional potential problems

The interpolating moving least-squares （IMLS） method is discussed first in this paper. And the formulae of the IMLS method obtained by Lancaster are revised. Then on the basis of the boundary element-free method （BEFM）, combining the boundary integral equation （BIE） method with the IMLS method, the improved boundary element-free method （IBEFM） for two-dimensional potential problems is presented, and the corresponding formulae of the IBEFM are obtained. In the BEFM, boundary conditions are applied directly, but the shape function in the MLS does not satisfy the property of the Kronecker ~ function. This is a problem of the BEFM, and must be solved theoretically. In the IMLS method, when the shape function satisfies the property of the Kronecker 5 function, then the boundary conditions, in the meshless method based on the IMLS method, can be applied directly. Then the IBEFM, based on the IMLS method, is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily, thus it gives a greater computational precision. Some numerical examples are presented to demonstrate the method.

A new complex variable element-free Galerkin method for two-dimensional potential problems

In this paper, based on the element-free Galerkin （EFG） method and the improved complex variable moving least- square （ICVMLS） approximation, a new meshless method, which is the improved complex variable element-free Galerkin （ICVEFG） method for two-dimensional potential problems, is presented. In the method, the integral weak form of control equations is employed, and the Lagrange multiplier is used to apply the essential boundary conditions. Then the corresponding formulas of the ICVEFG method for two-dimensional potential problems are obtained. Compared with the complex variable moving least-square （CVMLS） approximation proposed by Cheng, the functional in the ICVMLS approximation has an explicit physical meaning. Furthermore, the ICVEFG method has greater computational precision and efficiency. Three numerical examples are given to show the validity of the proposed method.

Topology optimization using the improved element-free Galerkin method for elasticity

The improved element-free Galerkin （IEFG） method of elasticity is used to solve the topology optimization problems. In this method, the improved moving least-squares approximation is used to form the shape function. In a topology opti- mization process, the entire structure volume is considered as the constraint. From the solid isotropic microstructures with penalization, we select relative node density as a design variable. Then we choose the minimization of compliance to be an objective function, and compute its sensitivity with the adjoint method. The IEFG method in this paper can overcome the disadvantages of the singular matrices that sometimes appear in conventional element-free Galerkin （EFG） method. The central processing unit （CPU） time of each example is given to show that the IEFG method is more efficient than the EFG method under the same precision, and the advantage that the IEFG method does not form singular matrices is also shown.

Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method

In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin（ICVEFG） method is presented for two-dimensional（2D） elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.

An improved complex variable element-free Galerkin method for two-dimensional elasticity problems

In this paper, the improved complex variable moving least-squares （ICVMLS） approximation is presented. The ICVMLS approximation has an explicit physics meaning. Compared with the complex variable moving least-squares （CVMLS） approximations presented by Cheng and Ren, the ICVMLS approximation has a great computational precision and efficiency. Based on the element-free Galerkin （EFG） method and the ICVMLS approximation, the improved complex variable element-free Galerkin （ICVEFG） method is presented for two-dimensional elasticity problems, and the corresponding formulae are obtained. Compared with the conventional EFC method, the ICVEFG method has a great computational accuracy and efficiency. For the purpose of demonstration, three selected numerical examples are solved using the ICVEFG method.

The improved element-free Galerkin method forthree-dimensional wave equation

The paper presents the improved element-free Galerkin （IEFG） method for three-dimensional wave propa- gation. The improved moving least-squares （IMLS） approx- imation is employed to construct the shape function, which uses an orthogonal function system with a weight function as the basis function. Compared with the conventional moving least-squares （MLS） approximation, the algebraic equation system in the IMLS approximation is not ill-conditioned, and can be solved directly without deriving the inverse matrix. Because there are fewer coefficients in the IMLS than in the MLS approximation, fewer nodes are selected in the IEFG method than in the element-free Galerkin method. Thus, the IEFG method has a higher computing speed. In the IEFG method, the Galerkin weak form is employed to obtain a dis- cretized system equation, and the penalty method is applied to impose the essential boundary condition. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the wave equations and the boundary-initial conditions depend on time, the scal- ing parameter, number of nodes and the time step length are considered for the convergence study.

Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems

We first give a stabilized improved moving least squares （IMLS） approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.

Element-free Galerkin (EFG) method for a kind of two-dimensional linear hyperbolic equation

The present paper deals with the numerical solution of a two-dimensional linear hyperbolic equation by using the element-free Galerkin （EFG） method which is based on the moving least-square approximation for the test and trial functions. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for hyperbolic problems needs only the scattered nodes instead of meshing the domain of the problem. It neither requires any element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for two-dimensional hyperbolic problems is investigated by two numerical examples in this paper.

An improved interpolating element-free Galerkin method for elasticity

Based on the improved interpolating moving least-squares （ⅡMLS） method and the Galerkin weak form, an improved interpolating element-free Galerkin （ⅡEFG） method is presented for two-dimensional elasticity problems in this paper. Compared with the interpolating moving least-squares （IMLS） method presented by Lancaster, the ⅡMLS method uses the nonsingular weight function. The number of unknown coefficients in the trial function of the ⅡMLS method is less than that of the MLS approximation and the shape function of the ⅡMLS method satisfies the property of Kronecker δ function. Thus in the ⅡEFG method, the essential boundary conditions can be applied directly and easily, then the numerical solutions can be obtained with higher precision than those obtained by the interpolating element-free Galerkin （IEFG） method. For the purposes of demonstration, four numerical examples are solved using the ⅡEFG method.

An element-free Galerkin method for ground penetrating radar numerical simulation

An element-free Galerkin method(EFGM) is used to solve the two-dimensional(2D) ground penetrating radar(GPR)modelling problems, due to its simple pre-processing, the absence of elements and high accuracy. Different from element-based numerical methods, this approach makes nodes free from the elemental restraint and avoids the explicit mesh discretization. First, we derived the boundary value problem for the 2D GPR simulation problems. Second, a penalty function approach and a boundary condition truncated method were used to enforce the essential and the absorbing boundary conditions, respectively. A three-layered GPR model was used to verify our element-free approach. The numerical solutions show that our solutions have an excellent agreement with solutions of a finite element method(FEM). Then, we used the EFGM to simulate one more complex model to show its capability and limitations. Simulation results show that one obvious advantage of EFGM is the absence of element mesh, which makes the method very flexible. Due to the use of MLS fitting, a key feature of EFM, is that both the dependent variable and its gradient are continuous and have high precision.

Application in metal rheological forming of element-free Galerkin method

The element-free method is a new numerical technique presented in recent years.It uses the moving least square(MLS) approximation as its shape function,and it is determined by the basic function and weight function.The weight function is the mainly determining factor,so it greatly affects the accuracy of the computational results.The element-free Galerkin method(EFGM) was applied for the solution to plastic large deformation.The simulation of metal rheological forming was successfully done by programming and its results were visualized by using the plotting and data analyses software Tecplot.Then plastic strain under different stages during rheological forming and the three principal stresses at the last deformation were obtained.The example shows the feasibility of EFGM used for metal rheological forming and provides a new method for numerical simulation of rheological forming of complex parts.

NEW ALGORITHM OF COUPLING ELEMENT-FREE GALERKIN WITH FINITE ELEMENT METHOD

Through the construction of a new ramp function, the element-flee Galerkin method and finite element coupling method were applied to the whole field, and was made fit for the structure of element nodes within the interface regions, both satisfying the essential boundary conditions and deploying meshless nodes and finite elements in a convenient and flexible way, which can meet the requirements of computation for complicated field. The comparison between the results of the present study and the corresponding analytical solutions shows this method is feasible and effective.

An improved element-free Galerkin method for solving the generalized fifth-order Korteweg-de Vries equation

In this paper, an improved element-free Galerkin （IEFG） method is proposed to solve the generalized fifth-order Korteweg-de Vries （gfKdV） equation. When the traditional element-free Galerkin （EFG） method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares （MLS） shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares （IMLS） approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.

The element-free Galerkin method of numerically solving a regularized long-wave equation

The element-free Galerkin （EFG） method is used in this paper to find the numerical solution to a regularized long-wave （RLW） equation. The Galerkin weak form is adopted to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. The effectiveness of the EFG method of solving the RLW equation is investigated by two numerical examples in this paper.

Element-free Galerkin method for a kind of KdV equation

The present paper deals with the numerical solution of the third-order nonlinear KdV equation using the element-free Calerkin （EFG） method which is based on the moving least-squares approximation. A variational method is used to obtain discrete equations, and the essential boundary conditions are enforced by the penalty method. Compared with numerical methods based on mesh, the EFG method for KdV equations needs only scattered nodes instead of meshing the domain of the problem. It does not require any element connectivity and does not suffer much degradation in accuracy when nodal arrangements are very irregular. The effectiveness of the EFG method for the KdV equation is investigated by two numerical examples in this paper.