## 3 Model Description for Water Conveyance Tunnel
### Boundary Conditions and Input Seismic Wave  
In dynamic analyses, the existence of artificial boundaries will create reflections of waves and contaminate the results. To solve this problem, the Perfectly Matched Layer (PML) is used in this study, which has been proved to be an efficient way for the bounded-domain modeling of wave propagation in unbounded domains. The PML is appropriate for explicit time integration in large-scale dynamic finite element analyses with accurate results and low computational costs [36]. As shown in Figure 9, the external manifestation of PML is a wave absorbing layer placed at the boundary of a truncated model which absorbs and attenuates waves propagating outward of the truncated model [37]. Assuming that the coordinate system \(\left\{\psi_{i}\right\}\) is adopted to describe the 3D space and the coordinates system \(\left\{\psi_{i^{\prime}}\right\}\) transformed from \(\left\{\psi_{i}\right\}\) is supposed as the PML's wave absorbing space, the stretching function can be expressed as (no summation) [31; 33]:  
\[\lambda_{i}\left(\psi_{i}^{\prime}\right):=\left[1+f_{i}^{e}\left(\psi_{i}^{ \prime}\right)\right]-\mathrm{i}\frac{f_{i}^{p}\left(\psi_{i}^{\prime}\right) }{\alpha_{0}},\left(i=1,2,3\right) \tag{17}\]  
where \(\alpha_{0}\) signifies the dimensionless frequency. The functions \(f_{i}^{e}\) and \(f_{i}^{p}\) work for attenuating the evanescent waves and the propagating waves, respectively. In practice, they can be chosen as \(f_{i}^{e}=f_{i}^{p}=f_{i}\), with \(f_{i}\left(\psi_{i}\right):=f_{0i}\left(\psi_{i}/L_{pi}\right)^{p}\), where \(f_{0i}\) is an optimal coefficient estimated from a pre-analysis; \(\psi_{i}\) represents the distance into the PML and the \(L_{pi}\) is the \(i\)th direction of the PML depth. In LS-DYNA, the PML can be used with the material defined by the keyword *MAT_PML_HYSTERETIC (Type 237) which is ad hoc for the material Type 232 used for soil in this finite element model.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Layers**} & **Density** & **Young’s** & **Poisson** & **Damping** & **Dominant Excitation** & **Shear Wave** \\  & **(kg/m\({}^{3}\))** & **Modulus (MPa)** & **Ratio** & **Ratio** & **Frequency (Hz)** & **Velocity (m/s)** \\ \hline
1 & 1897.96 & 37.98 & 0.26 & 0.055 & 3.5 & 89.11 \\
2 & 1908.16 & 68.94 & 0.28 & 0.063 & 3.5 & 118.80 \\
3 & 1795.92 & 67.40 & 0.33 & 0.078 & 3.5 & 118.78 \\
4 & 1714.29 & 116.13 & 0.35 & 0.078 & 3.5 & 158.40 \\
5 & 1836.73 & 230.00 & 0.32 & 0.070 & 3.5 & 217.79 \\
6 & 1887.76 & 371.30 & 0.28 & 0.054 & 3.5 & 277.18 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of the soil layers.  
Pore water pressure is considered in this study. In LS-DYNA, it can be treated as boundary conditions. In this study, the saturated and undrained conditions with normal mean high tide of 3.3 m are adopted, and they are applied in the model by using the corresponding keywords in LS-DYNA.  
The input excitation in this study is the Shanghai artificial seismic wave record with the duration time of 20 s. The curve of acceleration _versus_ time for the seismic motion imposed at the bedrock is shown in Figure 10. Since the seismic precautionary intensity of the tunnel is 7 degrees, the peak acceleration of 0.1 g, which can be obtained through the amplitude modulation, is required. Seismic impacts due to wave propagation--so called non-uniform seismic excitation--are considered here. As for the propagation direction, only the transverse one is taken into consideration. The seismic wave will propagate along the model from left to right. The horizontal acceleration excitation is imposed at the bottom of the model. The excitation has been imposed on the nodes at the bottom of the model at 5 m intervals with corresponding phase shifts, which reflect an apparent wave propagation velocity of 0.5 km/s from the island part to the land part.