## 4 Numerical Method  
### Numerical Model  
The numerical model of the typical geometric features of the HT crossing fault fracture zone is built based on the FE code \"ABAQUS\", which is frequently utilized to perform a series of FE dynamic analyses of underground structures, as displayed in Figure 5. As can be seen, the scale of the numerical model is 290 m \(\times\) 150 m \(\times\) 127 m (length \(\times\) width \(\times\) height), and the angle and width of the fault fracture zone are, respectively, 60\({}^{\circ}\) and 10 m. The external width of the HT is 15.30 m, and the thickness of the concrete lining and the exterior height is 1.00 m and 17.15 m, separately. Eight-node reduced integration elements C3D8R in the element library are adopted to model the footwall, hanging wall, fault fracture zone, and structures. The corresponding total number of elements and nodes is 61,440 and 69,986, respectively. Moreover, the eight-node fluid element in the dynamic analysis is used to model the fluid-structure interaction dynamics, which is regarded as a linearly elastic, inviscid, irrotational, and compressible medium. Simultaneously, the maximum C3D8R size is set to 7.0 m, according to the work of [PERSON] and [PERSON] [38], which complies with the transmitted wavelengths and may consequently reduce the element mesh size's influence in the dynamic analysis. Considering the linear/nonlinear dynamic behavior and plastic deformation of rock, the Drucker-Prager model is introduced to express the mechanism of failure at which the state of the surrounding rock mass reaches an inelastic state [39]. The material properties for the hanging wall and footwall are as follows: mass density = 2700 kg/m\({}^{3}\), elastic modulus = 5.0 GPa, Poisson's ratio = 0.29, friction angle = 35\({}^{\circ}\), cohesion = 0.6 MPa. The material properties for the fault fracture zone are as follows: mass density = 2000 kg/m\({}^{3}\), elastic modulus = 0.3 GPa, Poisson's ratio = 0.33, friction angle = 24.2\({}^{\circ}\), cohesion = 0.1 MPa. Furthermore, the concrete plastic damage model supplemented in the ABAQUS material library is used to describe the damage and cracking of the concrete lining induced by the stiffness reduction laws, as shown in Figure 6. The material properties for the concrete lining are as follows: mass density = 2450 kg/m\({}^{3}\), elastic modulus = 28 GPa, Poisson's ratio = 0.167, compressive yield stress = 16.7 MPa, tensile yield stress = 1.78 MPa. Figure 4: Framework of seismic vulnerability analysis for CFHTs. The applied loads during the static analysis of the FE model are as follows: (1) self-weight load; (2) internal hydrostatic pressure; (3) external hydrostatic pressure. Note that the initial stress state of the rock without the tunnel is balanced. Subsequently, the CFHT considering the stress release of the rock is constructed with 40% of the equivalent nodal force, as shown in Figure 7. During Step 1 and Step 2 of the static analysis process, it can be seen that the bottom truncated boundary condition is fixed in the three directions. The lateral truncated boundary conditions are constant on the normal vector. After the entire tunnel is built, the initial stress state of the system and the truncated boundary node reaction forces are extracted for dynamic analysis. In addition to the loads imposed by the static analysis, hydrodynamic pressure and seismic loads are considered in the dynamic analysis. The dynamic analysis is \"restarted\" (only the displacement is reset to zero, without affecting the extracted starting stress) after the first static one to simplify the viewing of the deformed shape. Additionally, the viscous-spring artificial boundary (VSAB)  
Figure 5: Schematic of the FE model of CFHT. Figure 6: Stress–strain curve of the concrete lining. (**a**) Damage factor and compressive stress–strain; (**b**) Damage factor and tensile stress–strain [40]. is activated to absorb the outward-traveling wave and is used to reflect the recovery ability of the medium outside the FE model (see Figure 7) [41].