## 2 Radial Basis Function with Compact Support (RBFCS)  
RBFCS [15] is an independent variable function based on the distance between the calculation point and the node, and is one method for constructing shapes using the meshless method. Below summary the general thought of RBFCS. Assume that \(f(x)\) is a real value of the physical field. The approximate value \(\widetilde{f}(x)\) of \(f(x)\) is obtained by some interpolation nodes in the defined field of the calculation point \(x\):  
\[f(x)\approx\widetilde{f}(x)=\sum_{i=1}^{m}R(x,x_{i})a_{i} \tag{1}\]  
where \(m\) is the number of a set of scattered interpolation nodes and \(a_{i}\) is the unknown coefficient. \(R(x,x_{i})\) is the basis function of \(r_{i}\), that is, the gap between interpolating point \(x\) and the node \(x_{i}\), and is defined as follows in this paper. \[R(x,x_{i})= \begin{cases}\dfrac{2}{3}-4r^{2}+4r^{3}&r\leq 0.5\\ \dfrac{4}{3}-4r+4r^{2}-\dfrac{4}{3}r^{3}&0.5<r\leq 1\\ 0&\text{\emph{other}}\end{cases}  \tag{2}\]where \(r=\|x-x_{i}\|/d\), \(d\) is the radius of the local compact support domain of the point \(x\). Using the matrix form, Equation (1) is given as  
\[\widetilde{f}(x)=Ba \tag{3}\]  
where the vector of undetermined coefficients is  
\[a= \begin{bmatrix}a_{1}&a_{2}&\cdots&a_{m}\end{bmatrix} ^{\mathrm{T}} \tag{4}\]  
The vector about the known basis functions is  
\[B= \begin{bmatrix}R(x,x_{1})&R(x,x_{2})&\cdots&R(x,x_{m})\end{bmatrix}  \tag{5}\]  
\(a_{i}\) in Equation (1) are determined by forcing \(\widetilde{f}(x)\) through \(m\) scattered interpolation nodes within the influence boundary. The following formula can be obtained. \[B_{S}a=f_{S} \tag{6}\]  
where the moment matrix \(B_{S}\) is  
\[B_{S}= \begin{bmatrix}R(x_{1},x_{2})&R(x_{1},x_{2})&\cdots&R(x_{1},x_{m})\\ R(x_{2},x_{1})&R(x_{2},x_{2})&\cdots&R(x_{2},x_{m})\\ \vdots&\vdots&\ddots&\vdots\\ R(x_{m},x_{1})&R(x_{m},x_{2})&\cdots&R(x_{m},x_{m})\end{bmatrix} _{m\times m} \tag{7}\]  
and the vector \(f_{S}\) is  
\[f_{s}= \begin{bmatrix}f_{1}&f_{2}&\cdots&f_{m}\end{bmatrix} ^{\mathrm{T}} \tag{8}\]  
The vector \(a\) is achieved  
\[a=B_{S}{}^{-1}f_{S} \tag{9}\]  
Using Equation (9) and Equation (3), we can obtain  
\[\widetilde{f}(x)=BB_{S}{}^{-1}f_{S}=N^{\mathrm{T}}(x)f_{S} \tag{10}\]  
where  
\[N^{\mathrm{T}}(x)=BB_{S}{}^{-1}= \begin{bmatrix}N(x,x_{1})&N(x,x_{2})&\cdots&N (x,x_{m})\end{bmatrix}  \tag{11}\]  
RBFCS is employed to interpolate the virtual load in the paper. The Virtual Boundary Meshless Galerkin Method with the Partial Differential Equation on the Weak Term  
The following is an overview of VBEM. Suppose that \(\Omega\) indicates the computational domain, whose boundaries are recorded as \(\Gamma\) in Figure 1. The external normal of the point \(x\) on \(\Gamma\) is \(n(x)\). Expanding \(\Omega\) along \(n(x)\), a virtual domain \(\Omega^{\prime}\) can be obtained, whose boundaries are defined as \(S\). Assume the continuous virtual load \(\varphi_{j}(\xi)(j=1,2)\) at any point \(\xi\) on \(S\). The \(i\)th direction displacement and surface force of the point \(x\) are \(u_{i}(x)\) and \(p_{i}(x)\), respectively. The following equations are used. \[u_{i}(x)=\int_{S}u_{ij}^{*}(x,\xi)\varphi_{j}(\xi)dS \tag{12}\]  
\[p_{i}(x)=\int_{S}p_{ij}^{*}(x,\xi)\varphi_{j}(\xi)dS \tag{13}\]  
where \(u_{ij}^{*}\) and \(p_{ij}^{*}\) are the Kevin fundamental solutions of displacement and surface force of 2D elastic problem. The boundary conditions of Equations (12) and (13) are  
\[u_{i}(x) =\overline{u}_{i}(x)\ \ (x\in\Gamma_{u}) \tag{14}\] \[p_{i}(x) =\overline{p}_{i}(x)\ \ (x\in\Gamma_{p}) \tag{15}\]  
where, \(\overline{u}_{i}(x)\) is the known displacement on the real boundary \(\Gamma_{u}\) with known displacement. \(\overline{p}_{i}(x)\) is the known surface force on the real boundary \(\Gamma_{p}\) with known surface force. \(\Gamma_{u}\cup\Gamma_{p}=\Gamma\) and \(\Gamma_{u}\cap\Gamma_{p}=\ (\text{null set})\).