## 4 Numerical Method
### Interaction System and Earthquake Records  
During the GM excitation process, it is required to express the relative squeezing, reclosing, sliding, and dislocation between the surrounding rock and concrete lining. To reflect the complex dynamic contact mechanics, the Lagrange multiplier method and penalty function are adopted to establish the rock-structure interface relationship. At the same time, the interface related to the concrete lining and surrounding rock mass is also defined to simulate the normal compressive stress mechanical transferring characteristics and the shear direction contact relationship. Furthermore, a nonlinear shear-displacement-dependent wakening friction law equation can be used to decribe the ultimate shearing stress of the contact surface:  
\[\tau=\tau_{resid}+(\tau_{peak}-\tau_{resid})\bigg{(}\frac{\alpha}{\alpha+u_{ s}^{p}}\bigg{)}^{\beta}=c_{resid}+\sigma_{n}\tan\varphi_{resid}+(c_{peak}+\sigma_{n}\tan \varphi_{peak}-c_{resid}-\sigma_{n}\tan\varphi_{resid})\bigg{(}\frac{\alpha}{ \alpha+u_{s}^{p}}\bigg{)}^{\beta} \tag{12}\]  
where \(\tau_{resid}\) and \(\tau_{peak}\) are, respectively, the residual shear strength and peak shear strength; \(c_{resid}\) and \(c_{peak}\) represent the residual cohesion and peak cohesion, respectively; \(\varphi_{resid}\) and \(\varphi_{peak}\) indicate the residual and peak friction angle, respectively; \(\sigma_{n}\) is the normal stress; \(\alpha\) and \(\beta\) are the control parameters of exponential degradation; and \(u_{s}^{p}\) indicates the inelastic shear displacement. The parameters for the hanging wall and footwall are as follows: shear stiffness = 4.5 GPa/m, normal stiffness = 4.0 GPa/m, peak cohesion = 1.0 MPa, residual cohesion = 0.4, peak friction angle = 41.5\({}^{\circ}\), residual friction angle = 34.2\({}^{\circ}\), \(\alpha\) = 0.005, \(\beta\) = 1.2. For the fault fracture zone and footwall/hanging wall, the Hook and Coulomb friction law equation is used to express the normal and tangential contact relationship with the friction coefficient = 0.4. As for the fluid-structure dynamic interaction, the stress-strain relationship of the contact system can be derived as follows:  
\[ \begin{cases}\kappa\\ \kappa_{x}\\ \kappa_{y}\\ \kappa_{z}\end{cases} = \begin{bmatrix}\mathbb{C}_{11}\\ &\mathbb{C}_{22}\\ &&\mathbb{C}_{33}\\ &&&\mathbb{C}_{44}\end{bmatrix}\begin{cases}\varepsilon_{v}\\ w_{x}\\ w_{y}\\ w_{z}\end{cases}  \tag{13}\]  
where \(\kappa\) is the hydraulic pressure; \(\mathbb{C}_{11}\) and \(\varepsilon_{v}\) represent the bulk modulus and volumetric strain of fluid elements; \(\kappa_{i}\) indicates rotational stresses (\(i=x,y,z\)); \(\mathbb{C}_{22}\), \(\mathbb{C}_{33}\), \(\mathbb{C}_{44}\) are constraint parameters; and \(w_{i}\) is rotation (\(i=x,y,z\)). To express the randomness and uncertainty of seismic events, a set of 90 GM records is selected from the PEER center of the United States, including the pulse-type near-fault, near-fault without velocity pulse, and far-field, as suggested and used in the work of [PERSON]  
Figure 7: The flowchart of the static and dynamic analysis for CFHT. et al. [42]. The distribution of the magnitude and fault distance of the selected earthquakes is displayed in Figure 8. It can be seen from Figure 8 that the earthquake magnitude and fault distance of the selected GM records cover a wide range. Figure 9 shows the spectral acceleration of the equivalent damping ratio of 5% for single selected earthquake records. It is worth noting that the predominant source of aleatory uncertainty in the dynamic analysis system and seismic performance assessment is the unpredictability of the GM from the seismology features. [PERSON] et al. [40] provide a thorough list of potential earthquake IMs for CFHTs, comparing up to 20 parameters based on correlation, efficiency, proficiency, and sufficiency.