## 4 Engineering Application
### Verification of Discontinuity Sets' Clustering Effect
The proposed FCM-ABC method can accurately obtain the clustering center and membership degree through extensive training, which can effectively solve the problem of conventional clustering methods not being able to effectively select the appropriate initial clustering center. Accordingly, the new clustering approach can be applied to solve the problem of grouping the dominant structural planes of rock masses and reveal the geometric characteristics of the structural planes. In addition, the quality of clustering results will affect the convergence process to some extent. Figure 10 shows the process of searching \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{c}\) & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline \(H_{m}\) & 0.10 & 0.21 & 0.19 & 0.19 & 0.19 & 0.22 \\ \(F_{m}\) & 0.58 & 0.25 & 0.18 & 0.13 & 0.10 & 0.09 \\ \(F_{h}\) & 998.47 & 1288.00 & 1340.69 & 1297.49 & 1322.03 & 1459.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Obtained values of the three cluster validity test indicators under different \(c\) values. Figure 7: Distribution of joints on the surface of the original rock mass. for optimal solutions during the structural plane clustering calculation. The optimal solution is reached after around 125 cycles, and the actual time to find the optimum is approximately 0.02 s. The important feature of the ABC algorithm converging to the global optimal solution ensures that the proposed FCM-ABC method can quickly obtain the global optimal solution. To further validate the performance and advantages of the proposed FCM-ABC method, this method was compared with other clustering algorithms (the FCM algorithm, genetic algorithm, and neural network) based on the same data. The clustering results that achieve the optimal solutions obtained by different methods are listed in Table 3. Compared to the traditional clustering algorithms, the FCM-ABC method significantly reduces the convergence times, demonstrating a better algorithm performance and faster  
Figure 8: Clustering results of the field structural plane data based on different \(c\) values: (**a**) \(c=2\); (**b**) \(c=3\); (**c**) \(c=4\); (**d**) \(c=5\); (**e**) \(c=6\); and (**f**) \(c=7\). Figure 9: Relationship between the values of the cluster validity test indicators and \(c\) values. convergence speed. Consequently, the calculation time of the program can be greatly reduced in engineering applications, which is convenient for repeated calculations. Moreover, this new method recruits following bees through a roulette wheel algorithm according to the probability determined by the calculated Euclidean distance of the clustering and constantly updates the Euclidean distance, inducing the optimization process to approach the optimal solution more closely. While improving the convergence speed, it can also converge to better results, greatly improving the reliability and objectivity of the clustering results. Therefore, the approach that we developed has outstanding advantages in grouping analyses of structural planes, effectively avoids the shortcoming of conventional clustering algorithms easily falling into local optima, and has broad application prospects. This study provides an effective approach and method for the automatic identification and grouping of structural planes. In the grouping calculation of structural planes, for different geological conditions, a clustering analysis can be performed as long as there are structural plane parameters of the dip direction and dip angle. Of course, other parameters can also be selected, but they have not been verified in this paper. For low-quality or incomplete structural plane data, they can be eliminated in advance manually or by compiling a program that meets certain rules to avoid affecting the accuracy of the clustering calculation results.