Journal of Applied Mathematics

Volume 2014 (2014), Article ID 219538, 14 pages

http://dx.doi.org/10.1155/2014/219538

## Radial Basis Point Interpolation Method with Reordering Gauss Domains for 2D Plane Problems

^{1}School of Mathematical Sciences, Soochow University, Suzhou 215006, China^{2}School of Urban Rail Transportation, Soochow University, Suzhou 215137, China

Received 13 June 2014; Revised 28 August 2014; Accepted 28 August 2014; Published 23 December 2014

Academic Editor: Guan H. Yeoh

Copyright © 2014 Shi-Chao Yi 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.

#### Abstract

We present novel Gauss integration schemes with radial basis point interpolation method (RPIM). These techniques define new Gauss integration scheme, researching Gauss points (RGD), and reconstructing Gauss domain (RGD), respectively. The developments lead to a curtailment of the elapsed CPU time without loss of the accuracy. Numerical results show that the schemes reduce the computational time to 25% or less in general.

#### 1. Introduction

In the past two decades, the development and application of meshfree methods have attracted much attention. One of the reasons is the versatility of meshfree methods for complex geometry of solids and flexibility for different engineering problems [1]. Element-free Galerkin (EFG) method, which is originated by Belytschko et al. [2–5], is one of the most widely used meshfree methods. The key advantage of EFG method is that only nodal data is required and no element connectivity is needed, when moving least-squares (MLS) interpolation is used to construct trial and test functions. It is currently widely used in computational mechanics and other areas, such as by Sarboland and Aminataei [6] for nonlinear nonhomogeneous Burgers Equation and Pirali et al. [7] for crack discontinuities problem. In the meantime, some new techniques are used to improve the performance of the MLS, complex variable moving least-squares [8–10] and improved complex variable moving least-squares [11], the moving least-squares with singular weight function [12], and so forth. But shape functions constructed by MLS interpolation do not possess Kronecker delta function property; the treatment of essential boundary conditions is one of the typical drawbacks. Thus, many special techniques have been proposed to impose essential boundary conditions [13–15], such as point collocation [13], Lagrange multipliers [2], singular weighting functions [14], and penalty method [15]. None of these methods is fully satisfactory, as they still need additional efforts to enforce essential boundary conditions.

In order to totally eliminate the drawback associated with EFG method for imposing essential boundary conditions, Liu and Gu have developed the point interpolation methods (PIM) by using polynomial basis or/and radial basis function (RBF) [16–19]. However, singularity may occur if arrangement of the nodes is not consistent with the order of polynomial basis, while the inverse of moment matrix always exists for arbitrarily scattered nodes with radial basis [20]. In this paper, we mainly care about the radial point interpolation method (RPIM) studied by Wang and Liu in [21], which is based on the global weak form.

On the other hand, high computational cost is still one of the main drawbacks in meshfree method with Galerkin weak form. In the RPIM method for a given computational point, an linear system (the coefficient matrix is called the moment matrix) should be solved to construct shape functions if radial basis functions are selected, where is the number of nodes in the support domain. Furthermore, if monomial basis functions are added, the linear system will be extended to . This is very time-consuming especially for the meshless methods based on weak forms, where integration should be employed and the number of computational points is much bigger than the number of nodes.

In fact, common meshless method is based on the numerical integration for Gauss domains with RPIM. It is a tedious process when the integration points are extremely more than the nodes. In practical work, some Gauss point has the same nodes as some neighboring computational points. Thus, they only need one shape function with them. A process will be needed to find these nodes. This scheme is called researching Gauss points method (RGP). But the storage of searching program will increase quickly with the increasing number of nodes and Gauss points. Thus, more computational time is spent. To avoid this, reconstructing Gauss domains (RGD) is presented. All computational points share one same influence domain (the same nodes) in the domain. This scheme offsets the RGP’s disadvantage without the searching program. The reason why we call it researching Gauss points method is that the RGP method needs a searching nodes program same as in the RPIM. The two techniques are collectively named reordering Gauss domain (RG) method.

The remainder of the paper is arranged as follows. In Section 2, the radial basis point interpolation method (RPIM) is presented to get the shape functions. In Section 3, a Galerkin weak form and its numerical algorithm are studied for 2D solid mechanics problems. The reordering Gauss domains methods are then constructed in Section 4. In Section 5, several examples are presented to show the effectiveness of the proposed method and some parameters’ performances of the proposed method are also investigated. Finally, we end this paper with some conclusions in Section 6.

#### 2. Radial Basis Point Interpolation Method (RPIM)

The RPIM interpolation of , for all , can be defined by with the constraint condition where is the radial basis function (RBF), is the number of nodes in the neighborhood of , is the monomial in the space coordinates , is the number of polynomial basis functions, and coefficients and are interpolation constants. The variable of the radial basis function is the distance between the interpolation point and a node . It is necessary to construct the interpolation function here to solve the equations.

There are a number of types of radial basis functions. Characteristics of radial basis functions have been widely investigated in [20, 22]. In this paper, the following multiquadrics (MQ) radial basis function is used: where and and are two parameters. is defined as where is a dimensionless coefficient and is a parameter of the nodal distance. For regularly distributed nodal case, is the shortest distance between the node and its neighbor nodes. Effects of and have been studied in detail in [22]. In static analysis of 2D solid problems, it has been found that and lead to good results. Hence, these numbers are used in this paper.

The second term in (1) consists of polynomials. To ensure invertible interpolation matrix of RBF, the polynomial added into the RBF cannot be arbitrary. A low-degree polynomial is often needed to augment RBF to guarantee the nonsingularity of the moment matrix. In addition, the linear polynomial added into the RBF can also ensure linear consistency and improve the interpolation accuracy [22]. Thus, the linear polynomial is added into the MQ RBF in the following discussion.

Coefficients and in (1) can be determined by enforcing that (1) and (2) be satisfied at the nodes surrounding point . Equations (1) and (2) can be rewritten in the matrix form: where The matrix is symmetric, and thus the matrix is symmetric. Then, the interpolation equation (1) is finally expressed as where the shape function is defined by And , . It can be found from the above discussion that RPIM passes through the nodal values. Therefore, RPIM shape functions given in (8) satisfy the Kronecker delta condition. Thus,

#### 3. Variational Form of 2D Plane Problem

Consider the 2D problem of the deformation of a linear elastic medium from an undeformed domain , enclosed by : where is the stress tensor corresponding to the displacement field and is a body force vector, and boundary conditions are as follows: in which and are prescribed tractions and displacements, respectively, on the traction boundary and on the displacement boundary and is the unit outward normal matrix to the boundary .

Using the standard principle of minimum potential energy for (10)-(11), that is, to find such that is stationary, where denotes the Sobolev space of order , and are strain-stress vectors, and is the strain-stress matrix. RPIM equation (1) is used to approximate the displacements in the Galerkin procedure. Then we can get Substituting (13) into (12) leads to the following total potential energy in the matrix form: and invoking results in the following linear system of : in which the stiffness matrix is built from matrices and the right-hand side vector is built from the vectors . These matrices and vectors are defined by where

Whether the RPIM method or other meshless methods based on global weak form, background cells are necessary to obtain the numerical integration of (16). Different from the finite element method (FEM), which uses the same nodes for both interpolation and numerical integration, background cells in meshless methods are independent of the interpolations. In this paper, quadrilateral cells and Gauss quadrature are used for the numerical integration.

#### 4. Reconstructing Gauss Domains Methods

A weak form, in contrast to a strong form (collocation method in general), requires weaker consistency on the assumed field functions. The consistency requirement on the assumed functions for field variables is very different from the strong form. For a th-order differential governing system equation, the strong formulation requires a consistency of the th order, while the weak formulation requires a consistency of only the th order. But the meshfree method usually uses the integral representation of field variable functions for solving strong form system equations and the numerical integration is extremely time-consuming. Numerical accuracy mainly depends on the number of Gauss points in the corresponding domain; the more the Gauss points, the better the results in general. Thus, one of the most time-consuming steps in the meshless method is the construction of shape functions, since for every point of interest a linear system should be computed. As discussed in Section 2, using radial basis functions to construct shape functions, an linear system should be computed for every computational point. If monomials are added, an linear system should be solved. This is very time-consuming especially for meshless methods based on weak form, where a large number of integration points are used. In this section, we propose the researching Gauss points (RGP) and reconstructing Gauss domains (RGD) methods, which are together named reordering Gauss domains (RG) methods and can partly reduce the computation cost for the meshless methods compared with the RPIM.

##### 4.1. Researching Gauss Points (RGP) with the Same Nodes

In the RPIM approximation, shape function consists of two parts: and (8). The time consumption of is less than that of , because the computational complexities are and , respectively.

Every Gauss point has its own and is different from the rest, but it may have the same as the other. Thus, we need a storage place, containing the public nodes and the Gauss points’ information. The data is obtained by searching the Gauss points. Thus, the method is called researching Gauss points (RGP) method.

The red area represents the nodes set (involved red nodes ) and the blue area represents the Gauss points set (involved blue gauss points ) in Figure 1(a). It should be pointed out that one-to-one mapping exists between the Gauss points sets and the nodes points sets (can be verified by logical deduction). Thus, the key work is how to get the relationship. First of all, we construct matrix , where denote the number of the nodes and the number of the Gauss points, respectively. The element is function: