## 3 Establishing the Numerical Model  
### General Model Information  
Figures 3 and 4 demonstrate the dimensions of the one-sided mountain-plain site tunnel model and the corresponding mesh, respectively. The one-sided mountain-plain site tunnel model was 210 m, 100 m, and 50 m in length, width, and depth, respectively, while the mountain was 90 m, 40 m, and 50 m in length, width, and height. A circular underground tunnel was set in the model, and the length, outer radius, lining thickness, and burial depth (the distance between the site surface and the tunnel center) of the tunnel were 210 m, 2.5 m, 0.3 m, and 20 m, respectively. In terms of mesh, the C3 DSR element was used to discretize the model, and the maximum size of the mesh for the whole model should be set to 1/6-1/12 of the wavelength in soft soil [32]. Herein, eight elements for the wavelength were assumed and employed (i.e., \(l_{\text{max}}\leq\text{Vs}/8f_{\text{max}}\), where \(l_{\text{max}}\) is the maximum element length; Vs denotes the shear wave velocity; and \(f_{\text{max}}\) is the maximum accurate frequency, set to 25 Hz in this study). The element size was set to 1.5 m for the whole site model, which can assure the adequate transmission of a wave in soil and rock. Although this size is much too small for rock material and lowers the calculation efficiency, it can make sure that seismic waves can propagate in a rock medium more adequately and brings about no influence on the results. The element number of the one-sided mountain-plain site tunnel model was 371340. Moreover, the maximum time increment was set to 0.00005 s. As for the tunnel lining, the C3D8R element was also used, and the element size was set to 1.5 m as well so that the nodes of the soil overlapped those of the lining. The coupled Lagrangian-Eulerian method was employed to describe the water-tunnel lining interaction and the sloshing effect of water under seismic waves [33]. The Eulerian element was utilized to discrete the water domain. In this model, the normal direction contact passes via the contact compressive stress transfer interaction, and the element nodes on the surfaces satisfy Hook's Law and the Harmonized Condition of Displacement. Tangential contact obeys [PERSON]'s law of friction. To obtain the ultimate stress to satisfy [PERSON]'s law, the following formula was used:  
\[\tau_{\text{crit}}=\mu\cdot P, \tag{1}\]  
where \(\mu\) and \(P\) denote the coefficient of friction and the normal contact pressure, respectively. Herein, the soil-tunnel lining interface friction coefficient was set to 0.4 [33; 34].  
In this study, the SV-wave was considered, and two sets of numerical simulations were designed, each one consisting of four cases with different incidence angles, as shown in Table 1. The incidence angles that were taken into consideration were 11.31\({}^{\circ}\), 18.43\({}^{\circ}\), 26.57\({}^{\circ}\), and 33.69\({}^{\circ}\), which corresponded to following vector directions: (x, y) = (1, 5), (1, 3), (1, 2), and (1, 1.5), respectively. Two different incidence directions were set, i.e., left-side incidence and right-side incidence. In the cases with left-side incidence, the wave passed through the rock mountain first and then the tunnel (see Figure 2a), while the wave passed through the tunnel first and then the rock mountain in the cases with right-side incidence (see Figure 2b).  
Figure 4: Mesh of the one-sided mountain tunnel model.  
Figure 3: Dimensions of the one-sided mountain tunnel model.