## 6 Case Study Validation
### Applicable Boundaries and Limitations of the Method  
While the proposed method has demonstrated robust performance in simulating the seismic responses of underground structures, its current applicability is bounded by the following conditions:  
1. Geological Complexity The method is theoretically applicable to deep underground structures and heterogeneous or layered geological media. However, the current validation cases focus on homogeneous soils. Additional verification for stratified or spatially variable soil profiles will strengthen its generalizability.
2. Earthquake Type: The present verification is based solely on horizontal ground motions. The influences of vertical earthquake components and their coupling effects have not yet been considered. To fully capture three-dimensional seismic behavior, the method must be extended to three-dimensional analyses.
3. Soil Constitutive Model: The current framework assumes elastic soil behavior. Thus, it does not account for soil nonlinearity or liquefaction phenomena, which may occur under strong seismic excitation.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(E_{\rm s}/E^{*}\) & **Method** & \begin{tabular}{c} **A Section Axial** \\ **Force N (\&N)** \\ \end{tabular} & \begin{tabular}{c} **E Section Shear** \\ **Force Q (\&N)** \\ \end{tabular} & \
\begin{tabular}{c} **F Section Bending** \\ **Moment M (\&N\(\cdot\)m)** \\ \end{tabular} &
\begin{tabular}{c} **Deformation of Central** \\ **Columns (mm)** \\ \end{tabular} \\ \hline
0.5 &
\begin{tabular}{c} Dynamic time history method \\ Proposed method \\ \end{tabular} & \(-32.07\) & \(-6.02\) & \(-19.75\) & \(-0.8016\) \\ \cline{2-6}
1 &
\begin{tabular}{c} Dynamic time history method \\ Proposed method \\ \end{tabular} & \(-32.87\) (\(2.49\%\)) & \(-5.83\) (\(-3.16\%\)) & \(-19.11\) (\(-3.24\%\)) & \(-0.7798\) (\(-2.72\%\)) \\ \hline
1 &
\begin{tabular}{c} Dynamic time history method \\ Proposed method \\ \end{tabular} & \(-32.54\) & \(-10.85\) & \(-31.87\) & \(-0.6587\) \\ \cline{2-6}
1.5 &
\begin{tabular}{c} Dynamic time history method \\ Proposed method \\ \end{tabular} & \(-33.09\) (\(1.71\%\)) & \(-10.62\) (\(-2.12\%\)) & \(-31.38\) (\(-1.54\%\)) & \(-0.6542\) (\(-0.69\%\)) \\ \hline
1.5 &
\begin{tabular}{c} Dynamic time history method \\ Proposed method \\ \end{tabular} & \(-34.11\) & \(-14.27\) & \(-4.04\) & \(-0.5718\) \\ \cline{2-6} Proposed method & \(-32.75\) (\(-3.99\%\)) & \(-13.24\) (\(-7.22\%\)) & \(-39.04\) (\(-3.58\%\)) & \(-0.5539\) (\(-3.13\%\)) \\ \hline
2 &
\begin{tabular}{c} Dynamic time history method \\ Proposed method \\ \end{tabular} & \(-34.63\) & \(-16.83\) & \(-46.92\) & \(-0.5114\) \\ \cline{2-6} Proposed method & \(32.98\)(\(-4.76\%\)) & \(-16.00\) (\(-4.93\%\)) & \(-44.91\) (\(-4.28\%\)) & \(-0.5011\) (\(-2.01\%\)) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Internal forces at selected sections and deformation of the central columns under different structural stiffnesses.  
The incorporation of nonlinear constitutive models is necessary for applications involving severe ground motions or soft soils.  
Structural Type:  
The validation is currently limited to single-story underground structures. Structures with complex cross-sectional configurations, such as multicell tunnels or irregular geometries, have not yet been validated and require further verification to confirm the generalizability of the method.  
Boundary Representation Fidelity  
Current boundary conditions are based on static artificial boundaries without the incorporation of specialized infinite elements. In future work, the proposed method will be integrated with infinite elements to synergistically enhance both computational efficiency and accuracy.  
In addition, a comparative summary table is summarized to enhance clarity and allow better evaluation of the advantages and limitations of each method. The results are presented in Table 5.