## 2 Methodology
### RSM Model
#### 2.3.2 Numerical Analysis of RSM Model
ANOVA on Table 5 reveals that the buried depth of the small-radius curved tunnel is the primary factor influencing acceleration, as evidenced by its high F-value of 21.98. The radius of the curved tunnel is also a significant factor affecting acceleration, while the angle of the curved tunnel does not significantly impact the acceleration response. The response values were analyzed by Design-Expert software and the best fit output indicated that for the displacement and acceleration response of point B of small-radius curve tunnel, the application of quadratic and linear models was recommended and the adequacy of the developed models was tested using ANOVA methods. Tables 6 and 7 summarize the ANOVA of the responses and show the significance and adequacy measures of the models. Table 6 shows the ANOVA results derived from the simplified quadratic model of the displacement response at point B. The model exhibits an F-value of 8782.04, suggesting a significant model. The \(p\)-value of the model is less than 0.0001, indicating an extremely low likelihood of an F-value attributed to noise in this model. The prediction coefficient R\({}^{2}\) of 0.9986 corresponds well with the adjustment coefficient R\({}^{2}\) of 0.9998, with a difference between the two coefficients within 0.2. The signal-to-noise ratio of 263.2664 surpasses the 4 specified in the relevant statistical literature, indicating that the model features adequate accuracy and signal strength. There is no need to simplify the solution of the equation by removing the relatively insignificant term. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Source** & **Sum of Squares** & **Mean Square** & **F-Value** & \(p\)**-Value** & **Significance 1** \\ \hline Model & 0.0002 & 0.0000 & 8782.04 & \(<\)0.0001 & *** \\ U-Depth & 0.0002 & 0.0002 & 77,/05.23 & \(<\)0.0001 & *** \\ V-Angle & 1.326 \(\times\) 10\({}^{-7}\) & 1.326 \(\times\) 10\({}^{-7}\) & 46.18 & 0.0003 & ** \\ W-R & 9.901 \(\times\) 10\({}^{-10}\) & 9.901 \(\times\) 10\({}^{-10}\) & 0.3448 & 0.5755 & \\ UV & 2.116 \(\times\) 10\({}^{-9}\) & 2.116 \(\times\) 10\({}^{-9}\) & 0.7369 & 0.4191 & \\ UW & 2.756 \(\times\) 10\({}^{-9}\) & 2.756 \(\times\) 10\({}^{-9}\) & 0.9599 & 0.3599 & \\ VW & 3.600 \(\times\) 10\({}^{-9}\) & 3.600 \(\times\) 10\({}^{-9}\) & 1.25 & 0.2998 & \\ U\({}^{2}\) & 3.668 \(\times\) 10\({}^{-6}\) & 3.668 \(\times\) 10\({}^{-6}\) & 1277.49 & \(<\)0.0001 & *** \\ V\({}^{2}\) & 6.579 \(\times\) 10\({}^{-14}\) & 6.579 \(\times\) 10\({}^{-14}\) & 0.0000 & 0.9963 & \\ W\({}^{2}\) & 3.901 \(\times\) 10\({}^{-10}\) & 3.901 \(\times\) 10\({}^{-10}\) & 0.1358 & 0.7233 & \\ Residual & 2.010 \(\times\) 10\({}^{-8}\) & 2.871 \(\times\) 10\({}^{-8}\) & & \\ Cor. total & 0.0002 & & & \\ R\({}^{2}\) & 0.9999 & & & Coded Equation \\ Adjusted R\({}^{2}\) & 0.9998 & \(\text{Sqrt}(s)=0.1191-0.0053U+0.0001V\) \\ Predicted R\({}^{2}\) & 0.9986 & \(-0.0009U^{2}-1.25\times 10^{-7}V^{2}+9.625\times 10^{-6}W^{2}\) \\ \multicolumn{2}{c}{Adeq.} & 263 & \\ \hline \hline \end{tabular} 1 Markers of significance factors: ** means moderate significant; *** means highly significant.  
\end{table} 
Table 6: Quadratic model ANOVA for displacement at point B. A comparison between the displacement response at point B (Table 6) and that at point A (Table 4) reveals that the angular F-value at point B is 46.18 with a \(p\)-value of 0.0003, while the angular F-value at point A is 0.0063 and has a \(p\)-value of 0.9391. These findings indicate that the angular factor at point B exerts a more significant impact on the displacement response than the angular factor at point A. Furthermore, the change in the angular factor at point B is more reliable for the results of the multi-objective optimized design. The ANOVA analysis conducted on Table 6 reveals that the buried depth of the small-radius curved tunnel at point B is the primary influencing factor on displacement, as evidenced by the F-value of 77,705.23.