# Numerical Simulation of Seismic-Wave Propagation in Specific Layered Geological Structures
## 3 Numerical Simulation  
### Numerical Modeling of Deep Earthquake  
The LS-DYNA 970 module of ANSYS-19.0 is employed for numerical simulation [47; 48]. For computational efficiency, the geological structure is modeled as an axisymmetric spatial representation. Figure 4 provides a detailed illustration of the model structure and parameters. This model spans a horizontal length of 400 km and descends to a depth of 730 km. These dimensions were specifically selected to effectively observe seismic-wave propagation while optimizing computational resources. Vertically, it is segmented into six layers. The crust has two layers: a granite layer (20 km) followed by a basalt layer (16 km). The upper mantle is divided into four layers, from top to bottom: the lithospheric mantle (24 km), the asthenosphere (160 km), the B-layer transition zone (190 km), and the C-layer transition zone (320 km). The epicenter of the earthquake is located in the C-layer transition zone at a depth of 578 km.  
The left boundary of the model represents the axis of symmetry. Axisymmetric boundary conditions are defined by Axisymmetric Solid Element Formulation in the *SECTION_SHELL keyword. Free boundary conditions are set on the top edge, while non-reflecting boundary conditions are applied to the right and bottom edges using the *BOUNDARY_NON_REFLECTING keyword. The energy release source in the model corresponds to a TNT explosion characterized by a pressure density of 1.63 g/cm\({}^{3}\) and a total mass of 336,000 tons [49]. In the TNT simulations, the Arbitrary Lagrangian-Eulerian (ALE) algorithm elements are employed to accurately simulate the dynamic response of materials [50; 51]. The fluid-structure coupling method is utilized to govern interactions within the C-layer transition zone. Except for the TNT region, Lagrangian elements are employed in other areas. The interaction between layers is managed using common nodes, thereby ensuring continuity in metrics of displacement, velocity, and acceleration. It takes about 30 h on a 20-core computer to run the model in this study.  
To characterize the seismic wave generated by the detonation of TNT explosives, the high explosive intrinsic model (keyword *MAT_HIGH_EXPLOSIVE_BURN) and the  
Figure 4: Schematic diagram of the numerical model.  
Jones-Wilkins-Lee equation of state (keyword *EOS_JWL) are employed. The pressure can be expressed as  
\[P=A_{j\mid t}\Bigg{(}1-\frac{\omega_{j\mid t\mid}}{R_{1j\mid t\mid}V}\Bigg{)}e^{- R_{1j\mid t\mid}V_{j\mid t}}+B_{j\mid t}\Bigg{(}1-\frac{\omega_{j\mid t\mid}}{R_{2 j\mid t\mid}V_{j\mid t}}\Bigg{)}e^{-R_{2j\mid t}V_{j\mid t}}+\frac{\omega_{j\mid t \mid}E_{j\mid t\mid}}{E_{j\mid t}} \tag{16}\]  
where \(P\) is the pressure, \(E_{j\mid t\mid}\) is the initial internal energy, \(V_{j\mid t\mid}\) is the relative volume, and \(A_{j\mid t\mid}\), \(B_{j\mid t\mid}\), \(R_{1j\mid t\mid}\), \(R_{2j\mid t\mid}\), and \(\omega_{j\mid t\mid}\) are the material constants. The relevant parameters of the TNT model are shown in Table 1[52].  
The velocities \(V_{p}\) and \(V_{s}\) are determined using Equation (17),  
\[\begin{array}{l}V_{p}=\sqrt{\frac{K}{\rho}},\\ V_{s}=\sqrt{\frac{G}{\rho}},\end{array} \tag{17}\]  
where \(V_{p}\) is the longitudinal wave velocity; \(V_{s}\) is the shear wave velocity; \(K\) is the elastic modulus; \(G\) is the shear modulus; and \(\rho\) is the density [53].