## 2 Extended Scaled Boundary Finite Element Method
### Scaled Boundary Finite Element Method  
X-SBFEM is used to simulate the crack tip for the high efficiency and high precision of stress singular field simulations. As shown in Figure 3, there are side-face forces at the face of the super-element at the crack tip in the case of SBFEM. Without taking the body force into account, the displacement field and the stress field given by the SBFEM are [21]  
\[\{\mathbf{u}(\xi,\ \eta)\}=[\mathbf{N}(\eta)]\{\mathbf{u}(\xi)\}=\sum\limits_{i=1}^{n}c_{i }\xi^{\lambda_{i}}[\mathbf{N}(\lambda)]\{\varphi_{i}\} \tag{8}\]  
\[\{\mathbf{\sigma}(\xi,\ \eta)\}=[\mathbf{D}][\mathbf{L}][\mathbf{N}(\eta)]\{\mathbf{u}(\xi)\}=[\mathbf{D}] \Big{[}\mathbf{B}^{1}(\eta)\Big{]}\{\mathbf{u}(\xi)\}_{\
u\xi}+\frac{1}{\xi}[\mathbf{D}] \Big{[}\mathbf{B}^{2}(\eta)\Big{]}\{\mathbf{u}(\xi)\} \tag{9}\]  
where \(\mathbf{N}(\eta)\) is the interpolation shape function for one-dimensional line elements and \(\{\varphi_{i}\}\) and \(\lambda_{i}\) are the displacement modes and the eigenvalues, respectively. Furthermore, \([\mathbf{B}^{1}(\eta)]\) and \([\mathbf{B}^{2}(\eta)]\) are determined by the geometric shape of the boundary of the element. The formulas calculated by the virtual work principle are  
\[\{\mathbf{P}\}=[\mathbf{K}]\{\mathbf{u}_{h}\}=\Big{[}\mathbf{E}^{0}\Big{]}[\mathbf{\Phi}][\lambda] [\mathbf{\Phi}]^{-1}+\Big{[}\mathbf{E}^{1}\Big{]}^{T}\{\mathbf{u}_{h}\} \tag{10}\]\[\left[\mathbf{E}^{0}\right]\xi^{2}\{\mathbf{u}(\xi)\}_{\
u\xi\xi}+\left(\left[\mathbf{E}^{0} \right]+\left[\mathbf{E}^{1}\right]^{T}-\left[\mathbf{E}^{1}\right]\right)\{\mathbf{u}(\xi) \}_{\
u\xi}-\left[\mathbf{E}^{2}\right]\{\mathbf{u}(\xi)\}=0 \tag{11}\]  
where \(\{\mathbf{P}\}\) is the equivalent boundary nodal force and \(\left[\mathbf{E}^{0}\right]\), \(\left[\mathbf{E}^{1}\right]\), and \(\left[\mathbf{E}^{2}\right]\) are the coefficient matrices of the SBFEM governing equations. Equation (11) is a second-order homogeneous ordinary differential equation and its solution is  
\[\{\mathbf{u}(\xi)\}=[\varphi]\xi^{\lambda}\{\mathbf{c}\}=\sum_{i=1}^{n}c_{i}\xi^{ \lambda_{i}}\varphi_{i} \tag{12}\]  
where \(c_{i}\) is the weight of this mode. Substitute Equation (12) into Equation (10) and get a quadratic eigenvalue problem. \[[\lambda_{i}^{2}\left[\mathbf{E}^{0}\right]+\lambda_{i}\left(\left[\mathbf{E}^{1} \right]^{T}-\left[\mathbf{E}^{1}\right]\right)-\left[\mathbf{E}^{2}\right]]\varphi_{i }=0 \tag{13}\]  
Substitute Equation (12) into Equation (10) and get the equivalent node force \(q_{i}\) on the boundary corresponding to the displacement mode. \[q_{i}=\left(\left[\mathbf{E}^{1}\right]^{T}+\lambda_{i}\left[\mathbf{E}^{0}\right] \right)\varnothing_{i} \tag{14}\]  
Combining Equations (13) and (14) and introducing auxiliary variables can transform quadratic eigenvalue problems into standard linear eigenvalue problems. \[\left[ \begin{array}{cc}-\left[\mathbf{E}^{0}\right]^{-1}\left[\mathbf{E}^{1}\right]^ {T}&\left[\mathbf{E}^{0}\right]^{-1}\\ -\left[\mathbf{E}^{1}\right]\left[\mathbf{E}^{0}\right]^{-1}\left[\mathbf{E}^{1}\right]^{T }+\left[\mathbf{E}^{2}\right]&\left[\mathbf{E}^{1}\right]\left[\mathbf{E}^{0}\right]^{-1} \end{array} \right]\left[ \begin{array}{c}\Phi\\ \mathbf{Q}\end{array} \right]=\lambda\left[ \begin{array}{c}\Phi\\ \mathbf{Q}\end{array} \right] \tag{15}\]  
where \(\left[\mathbf{\Phi}\right]\) and \(\left[\mathbf{Q}\right]\) are modal matrices of the displacement modal matrix and the force modal matrix. For the square root singular problem for homogeneous materials, the stress intensity factors can be defined as  
\[\left\{ \begin{array}{c}\mathbf{K}_{I}\\ \mathbf{K}_{II}\end{array} \right\}=\sqrt{2\pi L_{0}}=\sum_{i=1,2}\left(c_{i} \left\{ \begin{array}{c}\Psi_{yy}(\eta=\eta_{A})\\ \Psi_{xy}(\eta=\eta_{A})\end{array} \right\}_{i}\right) \tag{16}\]  
where \(L_{0}\) is the distance between the scaling center and the point of the crack surface (the segment OA in Figure 3). From Equation (8), it can be deduced that if \(\lambda_{i}\geq 1\), the distribution of the stress mode tends to 0 when \(\xi\to 0\)[27].