## 4 Verification and Comparison of Results between This Study and Other Research
### Comparison of Results between DCM and Other Studies  
The case study compared with DCM includes research such as [PERSON] (2004) [36], [PERSON] (2009) [6], and [PERSON] (2012) [23]. First of all, for analyzing the ground reaction caused by the tunnel excavation under the no support condition, the input data of the numerical calculation can be found in the aforementioned articles. According to the analysis results between DCM and the listed articles, the percentage error of the plastic zone radius is from 0.79% to 1.31%, and the percentage error of the radial displacement is from 0.24% to 1.70%. In particular, compared with the results of Rocksupport (2004), this study presents values with little error, as shown in Figure 6 and Table 3.  
Figure 4: Comparison of the interaction behavior of support-ground between FEM and DCM. (**a**) short unsupported span (\(d\) = 0.53 m), and (**b**) long unsupported span (\(d\) = 1.37 m). Figure 5: Distribution of the displacement/stress around a circular tunnel proximity under the support and no support condition by the comparison results obtained between FEM and DCM. (**a**) the tangential stress (\(\sigma_{\theta}\)), (**b**) the radial stress (\(\sigma_{r}\)), and (**c**) the radial displacement (\(u_{r}\)). \begin{table} \begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{FEM} & \multicolumn{4}{c}{DCM} \\ \hline Support condition & \begin{tabular}{c} Additional support person, \(p_{p}\) \\ (0/H) \\ \end{tabular} & \begin{tabular}{c} Radial displacement, \(\sigma_{\theta}^{p}\) \\ (mm) \\ \end{tabular} & \begin{tabular}{c} Radial displacement, \(\sigma_{\theta}^{p}\) \\ (mm) \\ \end{tabular} & \begin{tabular}{c} Radial displacement, \(\sigma_{\theta}^{p}\) \\ (mm) \\ \end{tabular} & \
\begin{tabular}{c} Radial displacement, \(\sigma_{\theta}^{p}\) \\ (mm) \\ \end{tabular} &
\begin{tabular}{c} Fluid force zero radius, \(R_{p}^{p}\) (m) \\ \end{tabular} \\ No support (Blackbody) & n/a & 21.665 & n/a & n/a & (0.017) & n/a \\ No support (Blackbody) & n/a & 136.73 & 9.62 & n/a & (1.048) & n/a \\ Support & \(\chi_{\theta}\)+0.53\({}^{a}\) & 0.481 & 11.674 & n/a & 0.487 & 11.124 & 0.533 \\ Support & \(\chi_{\theta}\) +1.37\({}^{b}\) & 0.302 & 18.504 & 5.972 & 0.279 & 18.482 & 5.832 \\ \hline \hline \end{tabular}   
* short unsupported span, \({}^{b}\) long unsupported span;
* error (%) = 100%( | FEM-DCM | / FEM). Figure 6: Percentage error of results between DCM and other studies under no support condition. (**a**) the plastic zone radius (\(R_{p}\)), and (**b**) the radial displacement (\(u_{R}\)). Regarding the comparison of DCM calculation results with published research articles, firstly, the input data of the calculation proposed by the tutorial manual of Rockssupport (2004) [36] includes two support systems that are only rock bolts and rock bolt plus shotcrete lining. The results calculated by DCM are as shown in Table 4, and one can observe that the range of percentage error of the plastic zone radius is 2.6-2.7%, and the range of the percentage error is 2.2-5.2% for the radial displacement, and 8.7-10.0% for the mobilized support pressure. Secondly, the comparison with the study of [PERSON] (2009), the results as shown in Table 5 that displays a short unsupported span, deep tunnel location, good quality of rock mass, and general support system. The percentage error of the mobilized support pressure, the radial displacement, and the plastic zone radius are 7%, 0.7%, and 1.7%, respectively. Thirdly, for a comparison with the study of [PERSON] (2012), the results are shown in Table 6. The percentage error of the mobilized support pressure, the radial displacement, and the plastic zone radius are 7.6%, 1.9%, and 4.0%, respectively. Finally, for the comparison of results shown in Figure 7, the more obvious difference is the mobilized support pressure calculated by DCM, and the range of percentage error is between 7% and 10%.