## 3 2.5D Finite Element Substructure Method
### A Zigzag-Paraxial Boundary Condition Suitable for 2.5D Finite Element Method  
The infinite domain includes multilayer soil and underlying half space bedrock. A zigzag-paraxial boundary condition suitable for 2.5D finite element method is presented, which can be used for efficient analysis seismic responses of long lined tunnel. The establishment process of the 2.5D zigzag-paraxial boundary condition is introduced, leading into an independent non-orthogonal local coordinate system (\(r,s,t\)) with zigzag boundaries attached to the surface of the tunnel. Due to the artificial boundary conditions on the left and right having the same building process, the detailed derivation process of the right boundary is introduced in this subsection. The conversion relationship between the global coordinate system (\(x,y,z\)) and the local coordinate system (\(r,s,t\)) is shown in Figure 3. The local coordinate system is used in each soil sublayer element to represent the zigzag shape of the boundary, in which the direction of \(r\)-axis and \(s\)-axis are the same as \(x\)-axis and \(y\)-axis of global coordinate, respectively. The \(t\)-axis is aligned with the zigzag shape of the surface boundary. The coordinate conversion relationship is as follows:  
\[ \begin{Bmatrix}x\\ y\\ z\end{Bmatrix} = \begin{bmatrix}1&0&0\\ 0&1&n_{y}\\ 0&0&n_{z}\end{bmatrix}\begin{Bmatrix}r\\ s\\ t\end{Bmatrix}  \tag{12}\]  
where \(n_{y}\) and \(n_{z}\) are the \(y\) and \(z\) components of unit directional vector along \(t\) axis, respectively. Figure 3: The coordinate system conversion relationship. According to the Equation (12), the derivatives with respective to \(x\), \(y\) and \(z\) can be obtained from the corresponding ones with respective to \(r\), \(s\) and \(t\) as follows:  
\[ \begin{cases}\frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z}\end{cases} = \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&-n_{y}/n_{z}&1/n_{z}\end{bmatrix}\begin{cases}\frac{\partial}{\partial r}\\ \frac{\partial}{\partial s}\\ \frac{\partial}{\partial t}\end{cases}  \tag{13}\]  
Substituting Equation (13) into Equation (2), the wave equation of multi-layer soil can be expressed as follows:  
\[\mathbf{L}_{k}^{\mathrm{T}}\mathbf{D}\mathbf{L}_{k}\tilde{\mathbf{u}}=-\omega^ {2}\rho\tilde{\mathbf{u}} \tag{14}\]  
where the \(\sim\) on \(u\) denotes its value in frequency domain. Since \(x\) and \(r\) have the same direction, the partial derivative along the \(r\) direction \(\partial/\partial r\) is also replaced by \(-ik_{x}\). \(\mathbf{L}_{st}\) is the differential operator in the local coordinate system. The \(\mathbf{L}_{st}\) expression is as follows:  
\[\mathbf{L}_{k}= \begin{bmatrix}-ik_{x}&0&0&\frac{\partial}{\partial s}&0&-\frac {n_{y}}{n_{z}}\frac{\partial}{\partial s}+\frac{1}{n_{z}}\frac{\partial}{ \partial t}\\ 0&\frac{\partial}{\partial s}&0&-ik_{x}&-\frac{n_{y}}{n_{z}}\frac{\partial}{ \partial s}+\frac{1}{n_{z}}\frac{\partial}{\partial t}&0\\ 0&0&-\frac{n_{y}}{n_{z}}\frac{\partial}{\partial s}+\frac{1}{n_{z}}\frac{ \partial}{\partial t}&0&\frac{\partial}{\partial s}&-ik_{x}\end{bmatrix} ^{T} \tag{15}\]  
To conveniently discretize the wave equilibrium equation along the \(s\) direction, \(\mathbf{L}_{k}\) is rewritten as  
\[\mathbf{L}_{k}=-ik_{x}\mathbf{L}_{r}+\mathbf{L}_{s}\frac{\partial}{\partial s }+\mathbf{L}_{t}\frac{\partial}{\partial t} \tag{16}\]  
with  
\[\mathbf{L}_{r}= \begin{bmatrix}1&0&0&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&0&1\end{bmatrix} ^{T} \tag{17a}\] \[\mathbf{L}_{s}=\frac{1}{n_{z}} \begin{bmatrix}0&0&0&n_{z}&0&-n_{y}\\ 0&n_{z}&0&0&-n_{y}&0\\ 0&0&-n_{y}&0&n_{z}&0\end{bmatrix} ^{T}\] (17b) \[\mathbf{L}_{t}=\frac{1}{n_{z}} \begin{bmatrix}0&0&0&0&0&1\\ 0&0&0&0&1&0\\ 0&0&1&0&0&0\end{bmatrix}