#### Abstract

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.

#### 1. Introduction

Currently, meshless method has been applied successfully to deal with various kinds of problems in the fields of science and engineering. When solving large deformation problems and dynamic propagation of cracks, meshless method can obtain greater precision than finite element method .

As an important meshless method, the EFG method has been applied to different kinds of engineering problems. In the EFG method, the moving least-squares (MLS) approximation is employed to construct the shape function. Because the MLS approximation is based on the least-squares method [2, 3], the disadvantages of the least-squares method also exist in the MLS approximation, in which sometimes ill-conditional or singular matrices occur in the final equations.

Cheng and Chen studied the IMLS approximation  which can make up for the deficiency of the MLS approximation; then the IEFG method is applied for transient heat conduction , wave equation , fracture , elastoplasticity , and viscoelasticity  problems. The improved complex variable EFG method is presented for wave equation problem  and bending problem of thin plate on elastic foundations . The IEFG and the improved complex variable EFG method can enhance the computational speed of the EFG method.

Based on the MLS approximation with a singular weight function, the interpolating MLS method was proposed by Lancaster et al. , and the corresponding meshless method can be applied with the essential boundary condition directly. Based on the interpolating MLS method, Kaljevic et al. proposed the improved formulation of EFG method . Based on the concept of an inner production, Ren et al. improved the interpolating MLS method by using singular weight function in interpolating points and orthogonalizing some of basis functions . And the interpolating EFG method is presented for potential [15, 16], transient heat conduction , elasticity , viscoelasticity , elastoplasticity , and elastic large deformation  problems.

In order to overcome the difficulties caused by singular weight function in the interpolating MLS method, an improved interpolating least-squares (IIMLS) method based on nonsingular weight function was presented by Wang et al. Based on the IIMLS method, the improved interpolating EFG method is presented for potential , elasticity , and some complex mechanical problems .

By combining the dimension splitting method and meshless methods, the hybrid complex variable EFG method , the dimension split EFG method , and the dimension splitting reproducing kernel particle method  for 3D problems are proposed, respectively, and those new methods can improve the computational speed of traditional meshless methods for solving 3D problems greatly.

In this paper, combining the IMLS approximation and the Galerkin weak form, the IEFG method is used for solving 3D advection-diffusion problems. The 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.

In the section of numerical examples, the weight functions, the scale parameter, the penalty factor, the node distribution, and the time step are discussed, respectively. The numerical results show that the IEFG method in this paper has a higher computational speed. And the advantage that the IEFG method can avoid the singular matrix is given.

#### 2. The IMLS Approximation

The approximation of a function is defined aswhere is the basis function vector, is the basis function number, andis the corresponding coefficient vector of .

In general,

The local approximation of equation (1) is

Definewhere is a weighting function which contains compact support, and are the nodes with influence domains covering the point .

Equation (5) can be written aswhere

Fromwe havewhere

Equation (9) sometimes forms singular or ill-conditional matrix. In order to make up for this deficiency, for basis functionsusing Gram-Schmidt process, we can make the orthogonal basis functions as

Then from equation (9), can be obtained directly aswhere

Substituting equation (13) into equation (4), we havewhereis the shape function.

This is the IMLS approximation, in which can be obtained simply and directly because it is unnecessary to obtain the inverse matrix of . Therefore, the IMLS approximation can avoid ill-conditional or singular matrix; then it can improve the computational precision and efficiency of the MLS approximation .

#### 3. The IEFG Method for 3D Advection-Diffusion Problems

For 3D advection-diffusion problems, the governing equation iswith the boundary conditionsand the initial conditionwhere is the field function, is the given field function on the essential boundary , is the given value on the natural boundary , is the boundary of the problem domain , and ; is the source term; is the diffusion efficient in the direction and is the advection velocity in the direction ; is known function; is the unit outward normal to the boundary in the direction .

The equivalent functional of 3D advection-diffusion problem can be obtained as

Suppose that is the penalty factor; by introducing the penalty method, we can obtain

Fromwe can obtain that the equivalent integral weak form iswhere

In the cubic domain , we employ nodes . The function at the node is

From the IMLS approximation, at the time of , the function at an arbitrary node iswhere

From equations (25) and (27), we havewhere

Substituting equations (27), (29), and (30) into equation (23) yields

Analyzing the integral terms in equation (32), respectively, we havewhere

Substituting equations (33)–(41) into equation (32), we can obtain

is arbitrary in equation (43); then the second-order ordinary differential equations can be obtained as follows:where

Then, using the finite difference method to separate the time, the relation of and can be established aswhere is the time step; solving equation (44) for and , respectively, and substituting the results into equation (44), we havewhere is a time weighed coefficient; we select in numerical examples.

#### 4. Numerical Examples

Three numerical examples of 3D advection-diffusion problems are solved with the EFG and the IEFG method, and the computational accuracy and efficiency are compared, respectively.

Define the relative error aswhere

In this section, the linear basis function is employed and the node distribution is regular. Moreover, Gaussian points are used in an integral cell.

The first example we considered is a 3D advection-diffusion problem:with the initial conditionand the boundary conditions

The problem domain is , and is total time.

The analytical solution is

In this example, we discuss the effects of the weight functions, the scale parameter, the penalty factor, and the node distribution on the solution of the IEFG method, respectively.

##### 4.1. Weight Functions

The time step is selected as 0.01, nodes are distributed regularly, and integral cells are selected. Then the accuracy of the IEFG method is discussed.

When the cubic spline function is used, , and , then the smaller relative errors of the IEFG method are , , , , and at the times of 0.1 s, 0.3 s, 0.5 s, 0.7 s, and 0.9 s, respectively. When the quartic spline function is used, , and , the smaller relative errors of the IEFG method are , , , , and at the times of 0.1 s, 0.3 s, 0.5 s, 0.7 s, and 0.9 s, respectively.

Thus, when we use the cubic spline function as the weight function, higher accuracy can be obtained.

##### 4.2. Scale Parameter

, regular nodes and integral cells are selected, respectively, , and the cubic spline function is used as the weight function. Figure 1 shows the relative errors of the IEFG method with the increase of when is 0.1 s. It is shown that the numerical solution has greater computational precision when is 1.44.

##### 4.3. Penalty Factor

The same time step, weight function, node distribution, and background integral grid are used, respectively; . Figure 2 shows the relative errors of the IEFG method with the change of when is 0.1 s. It is shown that the numerical solution has greater computational precision when is .

##### 4.4. Node Distribution

The same weight function and time step are used, respectively; and . Figure 3 shows the relative errors of the IEFG method with the increase of nodes when is 0.1 s. Therefore, the numerical solution of the IEFG method for 3D advection-diffusion problems is convergent.

##### 4.5. Time Step

The same weight function is used; regular nodes and integral cells are selected, respectively; and . Figure 4 shows the relative errors of the IEFG method with the change of when is 0.1 s. We can see that when , not only is the greater precision obtained, but also the corresponding CPU time is saved.

The EFG method is selected to solve this example, regular nodes and integral cells are selected, respectively, , , and , and the cubic spline function is selected; then the great precision can be obtained. When the IEFG method is selected to solve it, the same parameters and the weight function are selected, respectively, then the great precision can also be obtained, and the relative errors are similar.

The comparison of the CPU time of the EFG and the IEFG methods under the same relative errors is shown in Table 1. And the numerical results are given to be compared with the analytical one when are 0.1 s, 0.5 s, and 0.9 s, respectively (see Figures 57). Comparing with the EFG method, the IEFG method can obtain the solutions with similar computational accuracy, but it has greater computational efficiency.

Moreover, the IEFG method which can avoid singular matrix is another advantage when constructing the shape functions. If , when using the EFG method to solve it, the computational result cannot be obtained, and the error “Warning: Matrix is singular to working precision” appears in MATLAB code because of the singular matrix formed. But when the IEFG method is used, , and other parameters are the same, then we can obtain the computational solution, and the relative error is when is 0.1 s. The numerical solutions are compared with the analytical one which is shown in Figure 8; it is shown that the solutions of the IEFG method are in agreement with the analytic ones.

In the second example, a 3D advection-diffusion problem with source term is given aswith the initial conditionand the boundary conditions

The problem domain is .

The analytical solution is

The EFG method is used to solve this example, regular nodes and integral cells are selected, respectively, , , and , and the cubic spline function is used; then the great precision can be obtained. When using the IEFG method to solve it, the same parameters and the weight function are selected, respectively, the great precision can also be obtained, and the relative errors of two methods are similar.

The comparison of the CPU time of the EFG and the IEFG methods under the same relative errors is shown in Table 2. And the numerical solutions are compared with the analytical one when are 0.1 s, 0.5 s, and 0.9 s, respectively (see Figures 911). We can see again that the computational speed of the IEFG method is faster.

Similarly, if , when using the EFG method to solve it, the computational result cannot be obtained, and the same error appears in MATLAB code. But when the IEFG method is used, , and other parameters are the same, then the relative error can be obtained which is when is 0.1 s. The numerical solutions are compared with the analytical one which is shown in Figure 12; the solutions of the IEFG method are in agreement with the analytic ones which are shown.

In the third example, a 3D advection-diffusion equation is given aswith the initial conditionand the boundary conditions

The problem domain is .

The analytical solution iswhere

We set , , , , and , , for simplicity. The EFG method is used to solve this example, regular nodes and integral cells are selected, respectively, , , and , and the cubic spline function is selected; then the great precision can be obtained. When using the IEFG method to solve it, the same parameters and the weight function are selected, respectively, the numerical results can also be obtained with great precision, and the relative errors of two methods are similar.

The comparison of the CPU time of the EFG and the IEFG methods under the same relative errors is shown in Table 3. And the numerical solutions are compared with the analytical one when are 0.1 s, 0.5 s, and 0.9 s, respectively (see Figures 1315).

Similarly, if , when using the EFG method to solve it, the computational result cannot be obtained, and the same error appears in MATLAB code. But when the IEFG method is used, , and other parameters are the same, then the relative error can be obtained which is when is 0.1 s. The numerical solutions are compared with the analytical one which is shown in Figure 16. Again, the solutions of the IEFG method are in agreement with the analytic ones which are shown.

For this example, we can select different parameters, , , , , and , . Using the EFG method to solve it, regular nodes and integral cells are selected, respectively, , , and , and cubic spline function is selected; then the great precision can be obtained. When the IEFG method is selected to solve it, the same parameters and the weight function are selected, respectively, then the great precision can also be obtained, and the relative errors of two methods are similar.

The comparison of the CPU time of the EFG and the IEFG methods under the same relative errors is shown in Table 4. And the numerical solutions are compared with the analytical one when are 0.1 s, 0.5 s, and 0.9 s, respectively (see Figures 1719).

Similarly, if , when using the EFG method to solve it, the computational result cannot be obtained, and the same error appears in MATLAB code. But when the IEFG method is used, , and other parameters are the same, then the relative error can be obtained which is when is 0.3 s. The numerical solutions are compared with the analytical one which is shown in Figure 20. The solutions of the IEFG method are in agreement with the analytic ones which are shown.

From this example, we can see that both diffusion dominated case and advection dominated one can be solved by using the EFG and the IEFG methods. Although two methods can obtain the solutions with similar computational accuracy, the computational speed of the IEFG is faster than the EFG method.

#### 5. Conclusions

On the basis of the IMLS approximation, the IEFG method for 3D advection-diffusion problems is proposed in this paper.

In Section 4, the influences of the weight functions, the scale parameter, the penalty factor, the node distribution, and the time step on the computational precision of the solutions of the IEFG method are discussed, respectively. We can see that the IEFG method in this paper is convergent.

From three examples, we can conclude that, compared with the EFG method for 3D advection-diffusion problems, the IEFG method has a higher computational efficiency.

The numerical solutions show that the IEFG method can avoid singular matrix when constructing the shape functions; then the deficiency of the EFG method is overcome.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11571223) and the Science and Technology Innovation Project of Shanxi Colleges and Universities (Grant no. 2020L0344).