# Numerical Simulation of Seismic-Wave Propagation in Specific Layered Geological Structures
## 2 Seismic Event Conditions
### Boundary Condition Analysis  
Assuming that the depths of the various geological stratifications within the Earth are the same, the geometry of the Earth's interior, the constraints, and the loads acting on it are all symmetrical on a fixed axis pointing towards the center of the Earth from the overshooting source. Axisymmetric cylindrical coordinates \((r,\theta,z)\) are typically employed in this context, as illustrated in Figure 3. With the axis of symmetry designated as the \(z\)-axis, all stresses, strains, and displacements are independent of the \(\theta\)-direction and are solely a function of \(r\) and \(z\). The displacement at any given point can be expressed as a sum of two directional components: a radial displacement along the \(r\)-direction and an axial displacement along the \(z\)-direction.  
If the positive stress on the inner cylindrical face of a hexahedron is \(\sigma_{r}\), the positive stress on the outer cylindrical face is \(\sigma_{r}+(\partial\sigma_{r}/\partial r)dr\). Given the axisymmetry, it can be demonstrated that \(\sigma_{\theta}\) has no increment in the annular direction. If the positive stress below the hexahedron is designated as \(\sigma_{z}\), then the positive stress above should be expressed as \(\sigma_{z}+(\partial\sigma_{z}/\partial z)dz\). Similarly, the shear stresses on the inner and outer faces of the hexahedron are \(\tau_{zr}\) and \(\tau_{zr}+(\partial\tau_{zr}/\partial r)dr\), respectively. The shear stresses on the underside and the top are \(\tau_{rz}\) and \(\tau_{rz}+(\partial\tau_{rz}/\partial z)dz\), respectively. Furthermore, the radial body force is represented by \(F_{r}\), and the axial body force is represented by \(F_{z}\). If all the forces on the hexahedron are projected onto the radial axis at the center of the hexahedron and are taken to be s \(\sin(d\theta/2)\approx d\theta/2\) and \(\cos cos(d\theta/2)\approx 1\). The equilibrium equation in the \(r\) direction can be obtained:  
\[\begin{array}{c}\Big{(}\sigma_{r}+\frac{\partial\tau_{r}}{\partial r}dr \Big{)}(r+dr)d\theta dz-\sigma_{r}rd\theta dz-\sigma_{\theta}drdz\frac{d\theta }{2}-\sigma_{\theta}drdz\frac{d\theta}{2}\\ +\Big{(}\tau_{rz}+\frac{\partial\tau_{rz}}{\partial z}dz\Big{)}rd\theta dr- \tau_{rz}rd\theta dr+F_{r}rd\theta drdz=0\end{array} \tag{13}\]  
Similarly, the equilibrium equation in the \(z\)-direction is obtained:  
\[\begin{array}{c}\Big{(}\tau_{rz}+\frac{\partial\tau_{rz}}{\partial r}dr \Big{)}(r+dr)d\theta dz-\tau_{rz}rd\theta dz\\ +\Big{(}\sigma_{z}+\frac{\partial\tau_{z}}{\partial z}dz\Big{)}rd\theta dr- \sigma_{z}rd\theta dr+F_{z}rd\theta drdz=0\end{array}. \tag{14}\]  
Ignoring the second-order small quantities, the equilibrium differential equation can be expressed as  
\[\begin{cases}\frac{\partial\sigma_{r}}{\partial r}+\frac{\partial\tau_{zr}}{ \partial z}+\frac{\sigma_{r}-\sigma_{\theta}}{r}+F_{r}=0\\ \frac{\partial\sigma_{z}}{\partial z}+\frac{\partial\tau_{zr}}{\partial r}+ \frac{\tau_{zr}}{r}+F_{z}=0\end{cases} \tag{15}\]  
According to Equations (13)-(15), numerical simulations can be calculated using the axisymmetric model.  
Figure 3: Analyzing microelements for spatial axisymmetric problems.