## Numerical Simulations
### Accuracy Verification of the Finite Element Model  
The soil boundary condition of the coupling analysis method of the finite element and infinite element, which reasonably reflects the real boundary condition, gives a unified solving format, and makes the infinite element of the soil section become part of the finite element model ([PERSON] et al., 2010 [38]; [PERSON] et al., 2012 [39]). Compared with the viscoelastic boundary condition, the infinite element boundary condition can not only simulate the far field boundary to absorb seismic energy, but also correctly simulate the boundary conditions whose infinity displacement is zero. Therefore, the coupling analysis method of the finite element and infinite element is used as the soil boundary condition to simulate semi-infinite domain soil.  
For the three-dimensional finite element model, the distance value \(r\) is first calculated from the random boundary node to the structure center. Secondly, normal spring stiffness, normal damping coefficient, tangential spring stiffness, and tangential damping coefficient of the node on the boundary are calculated by Equations (7) and (8). Finally, all of the boundary nodes are calculated cyclically in order to obtain a viscoelastic artificial boundary.  
\[K_{li}=\frac{\lambda+2G}{r}\;C_{li}=\rho c_{P} \tag{7}\]  
\[K_{li}=\frac{G}{r}\;C_{li}=\rho c_{S} \tag{8}\]  
where \(c_{P}\) is the longitudinal velocity of the medium, \(c_{s}\) is the longitudinal velocity of the medium, \(\rho\) is density, \(\lambda\) and \(G\) are lame constants, and \(r\) is the distance from the artificial boundary point to the scattering source.  
The soil surface is supposed to be a free boundary, and the horizontal reaction forces at the nodes on the lateral boundary and bottom boundary are calculated according to the artificially constrained boundary assumption ([PERSON] et al., 2006 [31]; [PERSON] et al., 2010 [32]). Therefore, the fixed boundary model, viscoelastic boundary model and coupling analysis model are set up to verify different soil boundary conditions, which are simplified as follows:  
Model 1 (M1): Coupling boundary between finite elements and infinite elements  
Model 2 (M2): Viscoelastic boundary  
Model 3 (M3): Fixed boundary  
The acceleration time-history curves at the soil surface midpoint are shown in Figure 15.  
From Figure 15, it can be inferred that the curves are almost the same in three different conditions before 4 s. After 4 s, only curves of Model 1 and Model 2 are similar to each other, while the acceleration values of Model 3 gradually increase compared to those of the other two curves. The reason is that seismic energy is absorbed and the seismic wave is rarely reflected when the motions pass through the infinite element boundary or viscoelastic artificial boundary, while seismic energy is greatly reflected when the motions pass through the fixed boundary, which leads to the accumulation of acceleration  
Figure 14: Lanzhou motion in the vertical direction after inversion.  
values on the free boundary. Therefore, compared with the viscoelastic boundary condition, the infinite element boundary condition does not need to be involved in the analytical solution expression in ABAQUS software. For this reason, Model 1 is selected as the soil boundary condition in this paper.