## 7 Conclusions  
This study systematically investigates the influence of boundary truncation distances on the computational accuracy of the conventional IRDM. By introducing static artificial boundary conditions simulating semi-infinite medium constraints, an improved IRDM considering the far-field effect is developed. The key findings are as follows:  
1. The boundary distance in the IRDM significantly affects the computational accuracy. When boundaries are positioned close to the structure, excessive constraining effects degrade precision. To ensure reliable results, the lateral boundary distance must be at least three times the structural span, and the vertical bottom boundary distance must be at least three times the structural height. 2. By degenerating viscoelastic dynamic artificial boundaries to static conditions, static artificial boundary conditions are derived to simulate both static foundation reactions and far-field constraint effects. These boundaries are integrated into the IRDM, resulting in an improved methodology that accounts for far-field medium interactions. 3. The computational accuracy of the IRDM considering far-field effects is influenced by the static artificial boundary parameters \(\alpha\)T and \(\alpha\)N and the dimensionless boundary distance ratio R. Through comprehensive numerical analyses, the following optimal parameters are recommended: \(\alpha\)T = 0.5, \(\alpha\)N =1.0, and R = 0.5 or 1.0. Compared with conventional approaches, the proposed method achieves significantly greater accuracy in terms of these parameters. This advancement provides an efficient and precise solution for seismic response analysis of large-scale underground structures. 4. The proposed method facilitates rapid evaluation of multiple structural design schemes--such as variations in cross-sectional geometry or reinforcement layout \begin{table}
\begin{tabular}{c c c c} \hline
**Criteria** & **Dynamic Time History Method** & **Conventional IRDM** & **Improved IRDM** \\ \hline Accuracy & High (benchmark) & Moderate (errors up to 30\% for small R) & High (errors \(<\) 5\% for R = 0.5) \\ Computational Demands & Very high (full dynamic model) & High (large domain, R \(\geq\) 3.0) & Low (80-90\% reduction vs. conventional) \\ \hline Boundary Treatment & Requires artificial boundaries (e.g., viscoelastic boundary) & Fixed boundaries at R \(\geq\) 3.0 & Static artificial boundaries (R \(\geq\) 0.5) \\ Applicability & Complex 3D/nonlinear analyses & Homogeneous/layered soils & Homogeneous/layered soils \\ \hline \end{tabular}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{Base of Use}} & \multirow{2}{*}{Complex (time-step integration)} & Moderate (empirical boundary placement) & \multirow{2}{*}{High (small model)} \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of different methods. during early-stage design. For example, designers can now assess and compare the seismic performance of a three-cell metro station with that of a five-cell metro station within 2-3 h, a significant reduction from the 2-3 days typically required for full dynamic finite element simulations. This acceleration in decision-making efficiency is achieved without compromising compliance with the safety margins stipulated in codes. Conceptualization, X.B. ; methodology, X.B. and D.W.; software, S.W. and B.Z. ; validation, S.W. and D.W.; formal analysis, S.W. ; investigation, S.W. ; resources, X.B. and B.Z. ; data curation, S.W. ; writing--original draft preparation, S.W. and B.Z. ; writing--review and editing, S.W. and D.W.; visualization, B.Z. ; supervision, X.B. and B.Z. ; project administration, X.B. ; funding acquisition, X.B. All authors have read and agreed to the published version of the manuscript. This study is supported by the National Key Research and Development Program of China (no. 2022 YFC3003603), the National Natural Science Foundation of China (no. 52478497), and the Fundamental Research Funds for the Central Universities (DUT23 RC(3)069).