Research Article | Open Access

# An Improved Interpolating Element-Free Galerkin Method Based on Nonsingular Weight Functions

**Academic Editor:**Miaojuan Peng

#### 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).

#### References

- Z. Zhang, K. M. Liew, Y. Cheng, and Y. Y. Lee, “Analyzing 2D fracture problems with the improved element-free Galerkin method,”
*Engineering Analysis with Boundary Elements*, vol. 32, no. 3, pp. 241–250, 2008. View at: Publisher Site | Google Scholar - K. M. Liew, J. Ren, and J. N. Reddy, “Numerical simulation of thermomechanical behaviours of shape memory alloys via a non-linear mesh-free Galerkin formulation,”
*International Journal for Numerical Methods in Engineering*, vol. 63, no. 7, pp. 1014–1040, 2005. View at: Publisher Site | Google Scholar - D. Li, F. Bai, Y. Cheng, and K. M. Liew, “A novel complex variable element-free Galerkin method for two-dimensional large deformation problems,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 233–236, pp. 1–10, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,”
*International Journal for Numerical Methods in Engineering*, vol. 37, no. 2, pp. 229–256, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. Ju-Feng, S. Feng-Xin, and C. Rong-Jun, “Element-free Galerkin method for a kind of KdV equation,”
*Chinese Physics B*, vol. 19, no. 6, Article ID 060201, 2010. View at: Publisher Site | Google Scholar - Z. Zhang, D.-M. Li, Y.-M. Cheng, and K. M. Liew, “The improved element-free Galerkin method for three-dimensional wave equation,”
*Acta Mechanica Sinica*, vol. 28, no. 3, pp. 808–818, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Zhang, J. F. Wang, Y. M. Cheng, and K. M. Liew, “The improved element-free Galerkin method for three-dimensional transient heat conduction problems,”
*Science China Physics, Mechanics & Astronomy*, vol. 56, no. 8, pp. 1568–1580, 2013. View at: Google Scholar - Z. Zhang, S. Y. Hao, K. M. Liew, and Y. M. Cheng, “The improved element-free Galerkin method for two-dimensional elastodynamics problems,”
*Engineering Analysis with Boundary Elements*, vol. 37, no. 12, pp. 1576–1584, 2013. View at: Publisher Site | Google Scholar | MathSciNet - S. N. Atluri and T. Zhu, “A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics,”
*Computational Mechanics*, vol. 22, no. 2, pp. 117–127, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Chen and Y. M. Cheng, “Reproducing kernel particle method with complex variables for elasticity,”
*Acta Physica Sinica*, vol. 57, no. 1, pp. 1–10, 2008. View at: Google Scholar | MathSciNet - L. Chen and Y. M. Cheng, “Complex variable reproducing kernel particle method for transient heat conduction problems,”
*Acta Physica Sinica*, vol. 57, no. 10, pp. 6047–6055, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. Chen and Y. M. Cheng, “The complex variable reproducing kernel particle method for elasto-plasticity problems,”
*Science China*, vol. 53, no. 5, pp. 954–965, 2010. View at: Publisher Site | Google Scholar - L. Chen and Y.-M. Cheng, “The complex variable reproducing kernel particle method for two-dimensional elastodynamics,”
*Chinese Physics B*, vol. 19, no. 9, Article ID 090204, 2010. View at: Publisher Site | Google Scholar - Y. M. Cheng, M. J. Peng, and J. H. Li, “The complex variable moving least-square approximation and its application,”
*Chinese Journal of Theoretical and Applied Mechanics*, vol. 37, no. 6, pp. 719–723, 2005. View at: Google Scholar | MathSciNet - Y. M. Cheng and J. H. Li, “A meshless method with complex variables for elasticity,”
*Acta Physica Sinica*, vol. 54, no. 10, pp. 4463–4471, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Y. Cheng and J. Li, “Complex variable meshless method for fracture problems,”
*Science in China, Series G*, vol. 49, no. 1, pp. 46–59, 2006. View at: Publisher Site | Google Scholar - K. M. Liew, C. Feng, Y. Cheng, and S. Kitipornchai, “Complex variable moving least-squares method: a meshless approximation technique,”
*International Journal for Numerical Methods in Engineering*, vol. 70, no. 1, pp. 46–70, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Peng, P. Liu, and Y. Cheng, “The complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems,”
*International Journal of Applied Mechanics*, vol. 1, no. 2, pp. 367–385, 2009. View at: Publisher Site | Google Scholar - M. Peng, D. Li, and Y. Cheng, “The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems,”
*Engineering Structures*, vol. 33, no. 1, pp. 127–135, 2011. View at: Publisher Site | Google Scholar - F.-N. Bai, D.-M. Li, J.-F. Wang, and Y.-M. Cheng, “An improved complex variable element-free Galerkin method for two-dimensional elasticity problems,”
*Chinese Physics B*, vol. 21, no. 2, Article ID 020204, 2012. View at: Publisher Site | Google Scholar - F.-N. Bai, D.-M. Li, J.-F. Wang, and Y.-M. Cheng, “An improved complex variable element-free Galerkin method for two-dimensional elasticity problems,”
*Chinese Physics B*, vol. 21, no. 2, Article ID 020204, 2012. View at: Publisher Site | Google Scholar - Y. M. Cheng, R. X. Li, and M. J. Peng, “Complex variable element-free Galerkin (CVEFG) method for viscoelasticity problems,”
*Chinese Physics B*, vol. 21, no. 9, Article ID 090205, 2012. View at: Google Scholar - Y. M. Cheng, J. F. Wang, and R. X. Li, “The complex variable element-free Galerkin (CVEFG) method for two-dimensional elastodynamics problems,”
*International Journal of Applied Mechanics*, vol. 4, no. 4, Article ID 1250042, 2012. View at: Google Scholar - S. C. Li and Y. M. Cheng, “Meshless numerical manifold method based on unit partition,”
*Acta Mechanica Sinica*, vol. 36, no. 4, pp. 496–500, 2004. View at: Google Scholar - S. C. Li and Y. M. Cheng, “Numerical manifold method and its applications in rock mechanics,”
*Advances in Mechanics*, vol. 34, no. 4, pp. 446–454, 2004. View at: Google Scholar - S. Li, Y. Cheng, and Y.-F. Wu, “Numerical manifold method based on the method of weighted residuals,”
*Computational Mechanics*, vol. 35, no. 6, pp. 470–480, 2005. View at: Publisher Site | Google Scholar - S. C. Li, S. C. Li, and Y. M. Cheng, “Enriched meshless manifold method for two-dimensional crack modeling,”
*Theoretical and Applied Fracture Mechanics*, vol. 44, no. 3, pp. 234–248, 2005. View at: Publisher Site | Google Scholar - S.-C. Li, Y.-M. Cheng, and S.-C. Li, “Meshless manifold method for dynamic fracture mechanics,”
*Acta Physica Sinica*, vol. 55, no. 9, pp. 4760–4766, 2006. View at: Google Scholar - H. F. Gao and Y. M. Cheng, “Complex variable numerical manifold method for elasticity,”
*Chinese Journal of Theoretical and Applied Mechanics. Lixue Xuebao*, vol. 41, no. 4, pp. 480–488, 2009. View at: Google Scholar | MathSciNet - H. Gao and Y. Cheng, “A complex variable meshless manifold method for fracture problems,”
*International Journal of Computational Methods*, vol. 7, no. 1, pp. 55–81, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. J. Cheng and K. M. Liew, “Analyzing two-dimensional sine-Gordon equation with the mesh-free reproducing kernel particle Ritz method,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 245-246, pp. 132–143, 2012. View at: Publisher Site | Google Scholar | MathSciNet - E. Oñate, S. Idelsohn, O. C. Zienkiewicz, and R. L. Taylor, “A finite point method in computational mechanics. Applications to convective transport and fluid flow,”
*International Journal for Numerical Methods in Engineering*, vol. 39, no. 22, pp. 3839–3866, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Dehghan and M. Tatari, “Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions,”
*Mathematical and Computer Modelling*, vol. 44, no. 11-12, pp. 1160–1168, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. D. Dai and Y. M. Cheng, “Local boundary integral equation method based on radial basis functions for potential problems,”
*Acta Physica Sinica*, vol. 56, no. 2, pp. 597–603, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Y. M. Cheng and M. J. Chen, “A boundary element-free method for linear elasticity,”
*Acta Mechanica Sinica*, vol. 35, no. 2, pp. 181–186, 2003. View at: Google Scholar - Y. Cheng and M. Peng, “Boundary element-free method for elastodynamics,”
*Science in China G*, vol. 48, no. 6, pp. 641–657, 2005. View at: Publisher Site | Google Scholar - K. M. Liew, Y. Cheng, and S. Kitipornchai, “Boundary element-free method (BEFM) for two-dimensional elastodynamic analysis using Laplace transform,”
*International Journal for Numerical Methods in Engineering*, vol. 64, no. 12, pp. 1610–1627, 2005. View at: Publisher Site | Google Scholar - K. M. Liew and Y. Cheng, “Complex variable boundary element-free method for two-dimensional elastodynamic problems,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 198, no. 49–52, pp. 3925–3933, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Peng and Y. Cheng, “A boundary element-free method (BEFM) for two-dimensional potential problems,”
*Engineering Analysis with Boundary Elements*, vol. 33, no. 1, pp. 77–82, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Cheng, K. M. Liew, and S. Kitipornchair, “Reply to ‘Comments on Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems,”
*International Journal for Numerical Methods in Engineering*, vol. 78, no. 10, pp. 1258–1260, 2009. View at: Publisher Site | Google Scholar - J. Zhang, M. Tanaka, and T. Matsumoto, “Meshless analysis of potential problems in three dimensions with the hybrid boundary node method,”
*International Journal for Numerical Methods in Engineering*, vol. 59, no. 9, pp. 1147–1166, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. N. Atluri, J. Sladek, V. Sladek, and T. Zhu, “Local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity,”
*Computational Mechanics*, vol. 25, no. 2, pp. 180–198, 2000. View at: Google Scholar - B. Dai and Y. Cheng, “An improved local boundary integral equation method for two-dimensional potential problems,”
*International Journal of Applied Mechanics*, vol. 2, no. 2, pp. 421–436, 2010. View at: Publisher Site | Google Scholar - D. Shepard, “A two-dimensional interpolation function for irregularly spaced points,” in
*Proceeding of the 23rd ACM National Conference*, pp. 517–524, 1968. View at: Google Scholar - P. Lancaster and K. Salkauskas, “Surfaces generated by moving least squares methods,”
*Mathematics of Computation*, vol. 37, no. 155, pp. 141–158, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Zhu and S. N. Atluri, “A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method,”
*Computational Mechanics*, vol. 21, no. 3, pp. 211–222, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Mostt and C. Bucher, “A Moving Least Squares weighting function for the Element-free Galerkin Method which almost fulfills essential boundary conditions,”
*Structural Engineering and Mechanics*, vol. 21, no. 3, pp. 315–332, 2005. View at: Google Scholar - T. Most and C. Bucher, “New concepts for moving least squares: an interpolating non-singular weighting function and weighted nodal least squares,”
*Engineering Analysis with Boundary Elements*, vol. 32, no. 6, pp. 461–470, 2008. View at: Publisher Site | Google Scholar - S. L. L. Verardi, J. M. Machado, and Y. Shiyou, “The application of interpolating MLS approximations to the analysis of MHD flows,”
*Finite Elements in Analysis and Design*, vol. 39, no. 12, pp. 1173–1187, 2003. View at: Publisher Site | Google Scholar - I. Kaljević and S. Saigal, “An improved element free Galerkin formulation,”
*International Journal for Numerical Methods in Engineering*, vol. 40, no. 16, pp. 2953–2974, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H.-P. Ren, Y.-M. Cheng, and W. Zhang, “An improved boundary element-free method (IBEFM) for two-dimensional potential problems,”
*Chinese Physics B*, vol. 18, no. 10, pp. 4065–4073, 2009. View at: Publisher Site | Google Scholar - R. Hongping, C. Yumin, and Z. Wu, “An interpolating boundary element-free method (IBEFM) for elasticity problems,”
*Science China*, vol. 53, no. 4, pp. 758–766, 2010. View at: Publisher Site | Google Scholar - H. Ren and Y. Cheng, “The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems,”
*Engineering Analysis with Boundary Elements*, vol. 36, no. 5, pp. 873–880, 2012. View at: Publisher Site | Google Scholar | MathSciNet - H. Ren and Y. Cheng, “The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems,”
*Engineering Analysis with Boundary Elements*, vol. 36, no. 5, pp. 873–880, 2012. View at: Publisher Site | Google Scholar | MathSciNet - H. Netuzhylov, “Enforcement of boundary conditions in meshfree methods using interpolating moving least squares,”
*Engineering Analysis with Boundary Elements*, vol. 32, no. 6, pp. 512–516, 2008. View at: Publisher Site | Google Scholar - J. F. Wang, F. X. Sun, and Y. M. Cheng, “An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems,”
*Chinese Physics B*, vol. 21, no. 9, Article ID 090204, 2012. View at: Google Scholar - J. Wang, J. Wang, F. Sun, and Y. Cheng, “An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems,”
*International Journal of Computational Methods*, vol. 10, no. 6, Article ID 1350043, 2013. View at: Publisher Site | Google Scholar | MathSciNet - A. P. S. Selvadurai,
*Partial Differential Equations in Mechanics*, Springer, Berlin, Germany, 2000.

#### Copyright

Copyright © 2014 F. X. Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.