## 2 Materials and Methods
### Numerical Simulation Method
#### 2.3.3 Comparison of DEEPSOIL and ABAQUS Software Boundary Conditions  
DEEPSOIL software is a 1-D infinite half-space program. Because of the limitations of the DEEPSOIL software, modeling is done only in the flat ground model, modeling and boundary settings are shown in Figure 4a. ABAQUS software is a multi-dimensional modeling program. Artificial boundary conditions have a direct impact on the accuracy of numerical simulation of near-field fluctuations, so the study of artificial boundary conditions is of great significance. In order to eliminate the effect of boundary on model analysis results, the two sides of boundary are redesigned infinite boundaries, as shown in Figure 4b,c.  
Constitutive Models  
The DEEPSOIL software uses the Darendeli nonlinear model, while ABAQUS software uses the Mohr-Coulomb linear model. In this study, the main difference between the two software is the constitutive model.  
[PERSON] [12] proposed a four-parameter model (\(\gamma_{\text{r}}\), a, b and D\({}_{\text{min}}\)) that can characterize the normalized modulus reduction and material damping curves.  
The Mohr-Coulomb model includes five parameters, elasticity modulus E and Poisson's ratio, internal friction angle, cohesion, and dilatancy angle. The Mohr-coulomb model is a first-order approximation result to describe the behavior of rock and soil. It is often used in the preliminary analysis or simplified analysis of the problem. The average stiffness calculated by the Mohr-Coulomb model is constant, and the estimated value of the complex deformation problem can be obtained. Because the Mohr-Coulomb yield  
Figure 4: Boundary settings: (**a**) 1-D site model in DEEPSOIL; (**b**) Infinite boundary flat model in ABAQUS; (**c**) Infinite boundary slope model in ABAQUS.  
surface has singular points such as spires and corners, the numerical calculation becomes complicated and the convergence is slow in ABAQUS.  
Figure 5a,b show the shear stress-shear strain curves of Darendeli model and Mohr-Coulomb model. Figure 5c,d are the G/G\({}_{\text{max}}\)-shear strain curve of [PERSON] model and Mohr-Coulomb model. In the Darendeli model, G/G\({}_{\text{max}}\) decreases with an increase of shear strain, from linear elasticity to non-linear elasticity to unlimited plasticity. The G/G\({}_{\text{max}}\)-shear strain of the Mohr-Coulomb model does not change with an increase of strain. Figure 5e,f show the Darendeli model and the Mohr-Coulomb model using hysteresis curves to calculate the damping ratio. In the Darendeli model, the nonlinearity of shear stress and shear strain will increase energy consumption. Therefore, the damping ratio will increase as shear strain increases. The Mohr-Coulomb model stress-strain curve is linear, so the damping ratio is basically unchanged.  
Figure 5: Comparison under different models: (**a**) Shear modulus in Darendeli model; (**b**) Shear modulus in Mohr-Coulomb model; (**c**) Normalized shear modulus in Darendeli model; (**d**) Normalized shear modulus in Mohr-Coulomb model; (**e**) Material damping ratio with shearing strain in Darendeli model; (**f**) Material damping ratio with shearing strain in Mohr-Coulomb model.