# NURBS-Enhanced Meshfree Method with an Integration Subtraction Technique for Complex Topology
## 4 Numerical Examples
### Infinite Plate with a Circular Hole  
The first example is an infinite plate with a circular hole in the center, subjected to a unidirectional tensile load of \(p=1.0\) in the x-direction, as shown in Figure 9a. Due to symmetry, only one quadrant of the plate is modeled in Figure 9b. The geometry and material properties are as follows: length \(L=5\), hole radius \(a=1\), [PERSON]'s modulus \(E=10^{3}\), [PERSON]'s ratio \(v=0.3\). The symmetry conditions \(u_{x}=0\) along CD and \(u_{y}=0\) along AB are imposed on the left and bottom edges, respectively. Traction boundary conditions given by the exact solution are imposed on the right (x = 5) and top (y = 5) edges. The exact stress solution for this problem is:  
\[\sigma_{xx}(x,y)=1-\frac{a^{2}}{r^{2}}\Big{(}\frac{3}{2}{\rm cos}2\theta+{\rm cos }4\theta\Big{)}+\frac{3a^{4}}{2r^{4}}{\rm cos}4\theta \tag{30}\]  
\[\sigma_{yy}(x,y)=-\frac{a^{2}}{r^{2}}\Big{(}\frac{1}{2}{\rm cos}2\theta-{\rm cos }4\theta\Big{)}-\frac{3a^{4}}{2r^{4}}{\rm cos}4\theta \tag{31}\]  
Figure 8: Moving subtraction when a hole varies from \(\mathbf{C}_{1}\) (red) to \(\mathbf{C}_{2}\) (blue). \[\sigma_{xy}(x,y)=-\frac{a^{2}}{r^{2}}\bigg{\{}\frac{1}{2}\text{sin}2\theta+\text{ sin}4\theta\bigg{\}}+\frac{3a^{4}}{2r^{4}}\text{sin}4\theta \tag{32}\]  
On the other hand, a plane stress case is discussed here, for which the exact displacement is:  
\[u_{x}=\frac{p}{4\eta}\bigg{\{}\frac{k+1}{2}r\text{cos}\theta+\frac{a^{2}}{r}[(1 +k)\text{cos}\theta+\text{cos}3\theta]-\frac{a^{4}}{r^{3}}\text{cos}3\theta \bigg{\}} \tag{33}\]  
\[u_{y}=\frac{p}{4\eta}\bigg{\{}\frac{k-3}{2}r\text{sin}\theta+\frac{a^{2}}{r}[(1 -k)\text{sin}\theta+\text{sin}3\theta]-\frac{a^{4}}{r^{3}}\text{sin}3\theta \bigg{\}} \tag{34}\]  
where the parameters \(\eta\) and \(k\) are given by  
\[\eta=\frac{E}{2(1+v)},\ k=\frac{3-v}{1+v} \tag{35}\]  
Convergence and efficiency are first compared in the three tested methods, which are the presented integration subtraction method, the traditional RPIM meshfree method, and the bilinear isoparametric FEM, denoted hereafter as RSUB, RPIM and FEM, respectively. In the traditional meshfree RPIM, the mesh is the same as in the FEM, in which the nodes and elements are treated as field nodes and background cells, respectively. The discretization of RSUB using integration meshes and the subtraction process is shown in Figure 10, while the discretization of RPIM and FEM is shown in Figure 11. A node interval of \(h=0.2\) is used for illustration. Figure 10: Discretization of RSUB with node interval \(h=0.2\). Figure 9: Infinite plate with a circular hole: (**a**) problem description; (**b**) quadrant numerical model. The errors in displacement and energy are evaluated by  
\[\sigma_{yy}(x,y)=-\frac{a^{2}}{r^{2}}\Big{(}\frac{1}{2}{\rm cos}2\theta-{\rm cos }4\theta\Big{)}-\frac{3a^{4}}{2r^{4}}{\rm cos}4\theta \tag{31}\]0  
\[\sigma_{yy}(x,y)=-\frac{a^{2}}{r^{2}}\Big{(}\frac{1}{2}{\rm cos}2\theta-{\rm cos }4\theta\Big{)}-\frac{3a^{4}}{2r^{4}}{\rm cos}4\theta \tag{31}\]1  
where the superscript e denotes the exact solutions, while the superscript q denotes the numerical solutions. The five different node intervals \(h=0.25\), \(0.2\), \(0.125\), \(0.1\), and \(0.0625\) are investigated. In Figure 12, the two meshfree methods, RPIM and RSUB, demonstrate higher accuracy than FEM in displacement and energy, with RSUB appearing to be the most accurate among the three tested methods. The better accuracy of RSUB with coarse mesh may be obtained by the duplicate integration for subtraction, which may be able to improve the result around the boundary of hole. When the node interval gets smaller, this advantage in precision of RSUB becomes less significant and converges to RPIM. In addition to having greater displacement accuracy, RPIM has higher convergence rates. Moreover, the presented RSUB method achieves extremely high accuracy on a rough mesh. However, with respect to energy, the convergence processes of RPIM and RSUB are not good, although their accuracy is better than that of FEM.