## 3 Case Study Analysis  
### Verification of the Proposed Method  
Case I is a typical example that has been used by several scholars to verify the rationality of their respective methods [33; 34; 35]. This paper analyzes this case and compares the calculation results of two methods to verify that the PSO algorithm can significantly improve computational efficiency while ensuring calculation accuracy. The geometric model of the slope in Case I is shown in Figure 2. The calculation parameters are shown in Table 1.  
The failure probability of the 3D slope calculated by the proposed method in Case I, along with the calculation results of relevant references [33; 34; 35], is shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & **Density** & **Friction Angle** & \multicolumn{2}{c}{**Cohesion**} & \multicolumn{2}{c}{**Horizontal**} & \multicolumn{2}{c}{**Vertical**} \\  & \(\boldsymbol{\gamma}\) **(kN/m3)** & \(\boldsymbol{\varphi}\) (°)** & \multicolumn{2}{c}{**Mean Value**} & \(\boldsymbol{Cov}\) & \(\boldsymbol{l_{h}}\) **(m)** & \(\boldsymbol{l_{v}}\) **(m)** \\ \hline Clay 1 & 18 & 0 & 18 & 0.3 & 20 & 2 \\ Clay 2 & 18 & 0 & 20 & 0.2 & 20 & 2 \\ Clay 3 & 18 & 0 & 25 & 0.3 & 20 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Calculation parameters of Case I.  
Figure 2: The geometric model of Case I.  
The search process for the critical 3D sliding surface is depicted in Figure 3. As shown in Figure 3a, based on the generated 2D circular arc sliding surface, a search range (\(S_{\text{min}}-S_{\text{max}}\)) is set in the \(y\)-direction. Then, the search range is divided into \(n_{4}\) points to determine the failure surface width \(B\). A 3D ellipsoidal sliding surface is generated through four parameters, i.e., entry point, exit point, tangent point, and failure surface width. Finally, \(n_{1}\times n_{2}\times n_{3}\times n_{4}\) ellipsoidal sliding surfaces are tried to find out the 3D critical sliding surface.  
The pursuit of the minimum \(F_{s}\) through the entry and exit method presents an opposition between computational efficiency and precision. Achieving higher precision necessitates evaluating an extensive number of potential sliding surfaces [36]. The number of sliding surface trials used in this study with the entry and exit method is 48,510 (\(n_{1}\times n_{2}\times n_{3}\times n_{4}=21\times 21\times 11\times 10\)). In contrast, the PSO algorithm treats each particle as a potential sliding surface. The number of particles multiplied by the maximum number of iterations gives the total number of sliding surfaces that need to be trial-calculated. For this case, the PSO algorithm demanded 1200 sliding surface trials (40 \(\times\) 30), representing only 2.5% of the computational load of the entry and exit method.  
By comparing the calculation results and the number of sliding surface trials required for the PSO algorithm and the entry and exit method, one can find that the PSO algorithm significantly improves computational efficiency while ensuring the accuracy of the \(P_{f}\) calculation in the reliability analysis of 3D slopes.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Sample Numbers** &
\begin{tabular}{c} **Failure Probability** \\ \(P_{f}\) \\ \end{tabular} & **Dimension** & **Source** \\ \hline MCS & 10,000 & 18.7\% & 2D & [PERSON] et al. (2013) [33] \\ MCS & 100,000 & 18.6\% & 2D & [PERSON] et al. (2015) [34] \\ (RSM based on GPR) & 10,000 & 2.17\% & 3D & [PERSON] et al. (2021) [35] \\ MCS & 50,000 & 2.25\% & 3D & [PERSON] et al. (2021) [35] \\ RBFN + MCS & 10,000 & 2.35\% & 3D & [PERSON] et al. (2021) [35] \\ RBFN + MCS & 50,000 & 2.25\% & 3D & [PERSON] et al. (2021) [35] \\ MCS & 10,000 & 2.07\% & 3D & [PERSON] et al. (2023) [36] \\ MCS & 10,000 & 2.45\% & 3D & This paper (PSO) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Failure probabilities of Case I.  
Figure 3: Search for the 3D critical slip surface: (**a**) 3D geometric model; (**b**) 2D cross section P-P.