#### Abstract

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 Kronecker function, 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.

#### 1. Introduction

In recent years, meshless (or meshfree) method has become very attractive to solve science and engineering problems without meshes. The most important common feature of meshless methods is that its approximation function is constructed entirely based on a set of scattered nodes without employing a mesh. Then some complicated problems, such as the large deformation and crack growth problems in mechanics, can be simulated with the method without the remeshing techniques [1–3].

Many meshless methods have been developed, such as element-free Galerkin (EFG) method [4–8], meshless local Petrov-Galerkin (MLPG) method [9], reproducing kernel particle method (RKPM) [10–13], complex variable meshless method [14–23], meshless manifold method [24–30], the mesh-free reproducing kernel particle Ritz method [31], finite point method (FPM) [32], radial basis functions (RBF) method [33, 34], boundary element-free method (BEFM) [35–40], boundary node method [41], and local boundary integral equation (LBIE) method [42, 43].

Moving least-square (MLS) approximation is an important method to form the shape functions in meshless methods, such as the EFG, LBIE, and MLPG methods. The MLS approximation was firstly introduced by Shepard [44] and then extended by Lancaster and Salkauskas for surface generation problems [45]. The shape function that is formed with MLS approximation can obtain a solution with high precision.

There exists a disadvantage in the MLS approximation in which its shape function does not satisfy the property of Kronecker function. Then the meshless methods based on the MLS approximation cannot apply the essential boundary conditions directly and easily. The essential boundary conditions need to be introduced by additional approaches, such as Lagrange multipliers [4] and penalty methods [46]. However, for Lagrange multipliers, the corresponding discrete system will introduce additional unknowns which are not directly associated with the solution themselves. And for penalty methods, the optimal value of penalty factor always affects the accuracy of the final solution.

To overcome this disadvantage, Most and Bucher designed a regularized weight function with a regularization parameter , with which the MLS approximation can almost fulfill the interpolation and boundary conditions with high accuracy [47]. Thomas enhanced the regularized weight function to obtain a true interpolation of the MLS approximation [48]. Sergio obtained a special weight function using a normalization based on the Shepard interpolation to fulfill the interpolation [49].

Another possible approach for this disadvantage is the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas [45]. The IMLS method is established based on the MLS approximation by using singular weight functions. The shape function of the IMLS method satisfies the property of Kronecker function. Thus, the meshless methods based on the IMLS method can apply the essential boundary condition directly without any additional numerical effort. Based on the IMLS method, Kaljević and Saigal [50] presented an improved formulation of the element-free Galerkin (EFG) method, in which the boundary condition is applied directly. Ren simplified the expression of the shape function of the IMLS method and then presented the interpolating element-free Galerkin (IEFG) method and interpolating boundary element-free (IBEF) method for two-dimensional potential and elasticity problems [51–54].

Certainly a disadvantage of the IMLS method is that its weight function is singular at nodes. It complicates the computation of the inverse of the singular matrix, and it causes many difficulties to obtain the derivatives of the approximation function in the IMLS method. To overcome the singularity, Netuzhylov presented the perturbation technique in the IMLS method by using a small positive number within the weight function matrix [55]. However, the correct value of number is also hard to be given, and it always affects the accuracy of the final solution. Based on the IMLS method, Cheng and so forth improved the interpolating moving least-squares method with nonsingular weight function [56, 57]. However, there exists a complicated function in the improved method, and the computation of this function needs much CPU time. And then the computing speed of the improved method is lower than that of the IMLS method and MLS approximation.

In this paper, based on the MLS approximation and IMLS method, an improved interpolating moving least-squares (IIMLS) method with nonsingular weight functions is presented. Compared with the IMLS method presented by Lancaster and Salkauskas, the weight function used in the IIMLS method is nonsingular at any points, and any weight function used in the MLS approximation can be chosen as the weight function of the IIMLS method. Then the IIMLS method can overcome the difficulties caused by singularity of the weight function as in the IMLS method. Compared with the shape function of MLS approximation, the shape function of the IIMLS method satisfies the property of Kronecker function. Then, the meshless method based on the IIMLS method can apply the essential boundary condition directly and easily without any additional approaches. And the number of unknown coefficients in the trial function of the IIMLS method is less than that in the trial function of the MLS approximation. Then fewer nodes are needed in the local influence domain in the IIMLS method than in the MLS approximation. Therefore, under the same node distribution, the IIMLS method has higher computational precision than the MLS approximation.

Based on the IIMLS method and the Galerkin weak form, an improved interpolating element-free Galerkin (IIEFG) method is presented for two-dimensional potential problems. Compared with the conventional EFG method, the essential boundary conditions in the IIEFG method are applied naturally and directly. As there are fewer coefficients in the trial function of the IIMLS method than that in the MLS approximation, fewer nodes are selected in the entire domain in the IIEFG method than in the conventional EFG method; 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.

#### 2. The Improved Interpolating Moving Least-Squares Method

The improved interpolating moving least-squares (IIMLS) method is presented in this section based on nonsingular weight functions.

Suppose , be given basis functions. In order to let the shape function of the IIMLS method, which can apply any nonsingular weight function used in the MLS approximation, satisfy the property of Kronecker function, we firstly generate a set of new local basis functions from these given basis functions.

Let denote the radius of the domain of influence of node and let , where satisfying for and for . For a given point , let where is the point in the local approximation of are the nodes with domains of influence that cover the point , and where the parameter is an even positive integer and denotes the distance from to the closest support node whose domain of influence covers .

The function in this paper is easier than the corresponding function in [56, 57]. Then the improved method in this paper has high computing speed. The function and satisfies(a);(b);(c), and if and only if and ;(d).

And then a local function is proposed by performing the same transformation to ; that is,

Next the approximation function of the new local function is obtained with the MLS approximation with the new local basis functions.

A local approximation function of is defined as where are the unknown coefficients of the new basis.

From (2), we have Then there exists Then

The unknown coefficients , , can be obtained by using the weighted least-square method. By the similar derivation as in [56], we have where and is an identity matrix.

Then the approximation function of can be obtained as where is a matrix of shape function

Equation (11) is the shape function of the IIMLS method, and then the IIMLS method is presented.

From properties of the function , the shape function of the IIMLS method certainly satisfies the property of Kronecker function; that is,

The linear combination of the basis functions can be exactly reproduced with the IIMLS method; that is, if we let where are arbitrary constants, there exists

Compared with the MLS approximation, the shape function of the IIMLS method can satisfy the property of Kronecker function, and then the meshless method based on the IIMLS method can apply the essential boundary conditions directly. From (7), it can also be seen that the number of the unknown coefficients in the trial function of the IIMLS method is less than that in the trial function of the MLS approximation. Therefore, we can select fewer nodes in the meshless method based on the IIMLS method than that based on the MLS approximation. Hence, under the same node distribution, the IIMLS method has higher computational precision than the MLS approximation.

Compared with the IMLS method presented by Lancaster and Salkauskas, the nonsingular weight function is used in the IIMLS method. Then any weight function used in the MLS approximation can also be used in the IIMLS method. Then the IIMLS method can overcome the difficulties caused by the singularity of the weight function as in the IMLS method.

#### 3. The Improved Interpolating Element-Free Galerkin Method

In this section, combining the IIMLS method and Galerkin weak form of the potential problems, the improved interpolating element-free Galerkin (IIEFG) method for the two-dimensional potential problems is presented. Since the shape function of the IIMLS method satisfies the property of Kronecker function, the IIEFG method can apply the essential boundary conditions directly and easily.

Consider the following two-dimensional Poisson’s equation: with boundary conditions of the Dirichlet type, that is, or the Neumann type, that is, where is an unknown function, is a known function, is the unit outward normal to the boundary , and and are, respectively, the prescribed values of the function and its normal derivative over the boundary . Notice that .

The Galerkin weak form of (18)–(20) is where

We employ nodes in the domain , and the union of their compact support domains , , must cover the whole domain .

From the IIMLS method, the unknown potential at arbitrary field point in the domain can be expressed as where is the number of nodes whose compact support domains cover the point .

Then we have where

Substituting (23) and (33) into (21) yields Then it follows from (26) that that is,

Because the nodal test function is arbitrary, the final discretized equation is obtained as that is, where

The shape function of the IIMLS method satisfies the property of Kronecker function, and then the essential boundary conditions can be applied directly. Substituting the boundary conditions of (19) into (30) directly, we can obtain the unknowns at nodes by solving (30).

Compared with the conventional EFG method based on the MLS approximation, the IIEFG method based on the IIMLS method can apply the essential boundary conditions directly and easily. And the number of the unknown coefficients in the trial function of the IIMLS method is less than that in the trial function of the MLS approximation. Hence, under the same node distribution, the IIEFG method has higher computational precision.

Compared with the IEFG method based on the IMLS method presented by Lancaster and Salkauskas, the IIEFG method applies the nonsingular weight function. Then the IIEFG method can overcome the difficulties caused by the singularity of the weight function as in the IEFG method. Then the IIEFG method also has higher computational precision.

#### 4. Numerical Examples

The weight function plays an important role in the IIMLS method. Any weight function used in the MLS approximation can also be used in the IIMLS method. Then the cubic spline weight function, that is, is used in the present analysis. Here And the is also chosen to be the cubic spline weight function.

##### 4.1. Examples of the IIMLS Method

In this section, two numerical examples are presented to show the advantages of the IIMLS method of this paper. In our examples, the interpolating approximation function is constructed from a given function with the IIMLS method. Let , and the linear basis function is used in these examples of this section. Define the error norm where is the number of nodes investigated, is a scalar constant and in the one-dimensional space, and is a multi-index and in the two-dimensional space.

The first example is considered in the one-dimensional space, and the given function is chosen to be .

The numerical values of obtained with the MLS approximation and IIMLS method are shown in Figure 1 under regular and irregular node distributions, where the irregular node distribution is generated by adding a random perturbation on the regular node distribution. And the corresponding first derivatives are shown in Figure 2. It can be seen that the numerical results of the IIMLS method are in good agreement with the exact ones.

The error norms of the MLS, IMLS, and IIMLS methods under the regular and irregular node distributions are shown in Figures 3 and 4, respectively. It is shown that the shape functions of the IMLS and IIMLS methods all satisfy the property of Kronecker function.

The error norms of the MLS, IMLS, and IIMLS methods under the regular and irregular node distributions are shown in Figures 5 and 6, respectively. It is shown that the rates of convergence of the MLS, IMLS, and IIMLS methods are almost the same, and the error of the IIMLS method is less than that of the IMLS method. Hence, the IIMLS method has higher precision.

The second example is considered in the two-dimensional space, and the given function is , .

Under regular and irregular node distributions, the numerical results of and at with the MLS approximation and IIMLS method are shown in Figures 7 and 8, respectively. The irregular node distribution is generated by adding a random perturbation on the regular node distribution. It can also be seen that the numerical results of the IIMLS method are all in good agreement with the exact ones.

Under , , , and irregular and regular node distributions, the error norms of the MLS, IMLS, and IIMLS methods are shown in Figures 9 and 10, respectively. It is again evident that the shape functions of the IMLS and IIMLS methods all satisfy the property of Kronecker function.

Under , , , and regular node distributions, the error norms of the MLS, IMLS, and IIMLS methods are shown in Figure 11. Then by adding a random perturbation on the regular node distribution, the corresponding error norms under irregular node distribution are shown in Figure 12. It is shown that the rates of convergence of the MLS, IMLS, and IIMLS methods are almost the same. However, the error of the IIMLS method is less than that of the IMLS method and MLS approximation. Again, the IIMLS method has high precision.

##### 4.2. Examples of the IIEFG Method

Two examples are selected to demonstrate the advantages of the IIEFG method for two-dimensional potential problems. The results obtained with the IIEFG method of this paper for these examples are compared with that obtained with the EFG and IEFG methods and analytical solutions. Define the error norm where is the number of nodes and and are, respectively, the numerical and analytical solutions at nodes.

The third example considered is a temperature field of a rectangular plate governed by Laplace’s equation

The boundary conditions are

The analytical solution for the temperature of this problem is

The linear basis function is used for analysis. When regular node distribution is employed as shown in Figure 13(a), the temperatures at obtained with the IIEFG, EFG, and IEFG methods are shown in Figure 14. Then employing the irregular node distribution as shown in Figure 13(b), which is generated by adding a random perturbation on the regular node distribution, the temperatures at are shown in Figure 15, and the absolute errors at the inner nodes are shown in Figure 16. It is evident that the IIEFG method in this paper has higher computational precision than the EFG and IEFG methods.

**(a) Regular**

**(b) Irregular**

Employing , , , and regular node distributions, the error norms of temperature obtained with the IIEFG, EFG, and IEFG methods are shown in Figure 17. The average CPU times needed to furnish these results by using the IIEFG, EFG, and IEFG methods are respectively 10.95 s, 11.39 s, and 11.04 s. Then by adding a random perturbation on the regular node distribution, the error norms under the irregular node distribution are shown in Figure 18, and the average CPU times spent with the IIEFG, EFG, and IEFG methods are, respectively, 10.87 s, 11.47 s, and 11.09 s. It can be seen that the IIEFG method has higher computational precision.

The fourth example considered is a temperature field of an annulus plate with inner radius and outer radius governed by Laplace’s equation [58]:

Due to the symmetry of the model, only a quarter of the problem domain is modeled as shown in Figure 19. In the polar coordinate system , the boundary conditions are where and are given parameters.

**(a) Regular**

**(b) Irregular**

The analytical solution is

The parameters are taken as , , , and in the computation. The regular and irregular node distributions that are used for the solution of this example are shown in Figure 19. Then the numerical solutions under the regular node distribution at are shown in Figure 20, and the average CPU times for obtaining the solutions with the IIEFG, EFG, and IEFG methods are, respectively, 0.81 s, 1.16 s, and 0.80 s. Under the irregular node distribution, the temperatures along the arc and are shown in Figures 21 and 22, respectively. The average CPU time with the irregular node distribution is almost equal to that with the regular node, and their average CPU times of the IIEFG, EFG, and IEFG methods are, respectively, 0.81 s, 1.17 s, and 0.80 s. It can be seen that the IIEFG method has higher efficiency than the EFG method. And from these figures, it can also be observed that the IIEFG method in this paper has higher computational precision than the EFG and IEFG methods.

#### 5. Conclusions

In this paper, based on the MLS approximation, the IIMLS method is presented. The shape function of the IIMLS method satisfies the property of Kronecker function. Then the meshless method based on the IIMLS method can apply the essential boundary condition directly and easily. Moreover, the number of unknown coefficients in the trial function of the IIMLS method is less than that in the trial function of the MLS approximation. Then fewer nodes are needed in the local influence domain in the IIMLS method. Therefore, the IIMLS method has high computational precision. Compared with the IMLS method presented by Lancaster and Salkauskas, the weight function in the IIMLS method is nonsingular, and any weight function in the MLS approximation can be chosen as the weight function of the NMLS method. Then IIMLS method can overcome the difficulty caused by singularity of the weight function as in the IMLS method.

Based on the IIMLS and EFG methods, an IIEFG method for two-dimensional potential problems is presented. In the IIEFG method, the weight function is not singular, and the essential boundary conditions are applied naturally and directly. And there are fewer coefficients in the IIMLS method than in the MLS approximation; fewer nodes are needed in the entire domain in the IIEFG method formed from the IIMLS method than in the conventional EFG method. Then under the same node distribution, the IIEFG method in this paper has higher computational precision than the EFG and IEFG methods.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11171208), Shanghai Leading Academic Discipline Project (no. S30106), and the Natural Science Foundation of Ningbo (no. 2013A610103).