## 6 Engineering Application  
The Tai'an Pumped Storage Power Station in China is mainly composed of large caverns, including the main powerhouse, four omnibus bar caves, the main transformer chamber and a tailrace surge chamber. The surrounding rocks are mainly II and III granite, and there are four hydrogenerator sets arranged. The size of the main powerhouse is 180 m \(\times\) 25.9 m \(\times\) 53.675 m; the size of the main transformer chamber is 164 m \(\times\) 17.5 m \(\times\) 18.175 m; and the length of the omnibus bar cave is 35 m. Figure 11 shows the staged excavation. There are two displacement monitoring sections in the underground powerhouse. The layout of the corresponding monitoring points is shown in Figure 12. The FLAC\({}^{\text{3D}}\) calculation grid of the excavation body is shown in Figure 13, which includes 687,133 elements and 191,323 nodes, and the grid has been colored to distinguish the tunnel area and excavation sequence. The corresponding monitoring point layout is shown in Figure 14.  
The measured displacements of monitoring points M7, M19 and M24 are selected to construct the objective function, as shown in Equation (14). The selected three measurement points are based on the three points of maximum deformation that have been recorded. The geomechanical parameters to be back-analysed are as follows: elastic modulus \(E_{1}\) of the surrounding rock upstream of the main powerhouse; elastic modulus \(E_{2}\) of the surrounding rock downstream of the #1 unit of the main powerhouse; elastic modulus \(E_{3}\) of the surrounding rock downstream of the main powerhouse units #2-4; elastic modulus \(E_{4}\) of the surrounding rock of the variable chamber; the cohesion \(c_{1}\) of the surrounding rock upstream of the main powerhouse; the cohesion \(c_{2}\) of the surrounding rock downstream of the main powerhouse and the internal friction angle \(\varphi\). The parameter search intervals are set to 60 GPa \(\leq E_{1}\leq\) 80 GPa, 10 GPa \(\leq E_{2}\leq\) 30 GPa, 20 GPa \(\leq E_{3}\leq\) 40 GPa, 100 GPa \(\leq E_{4}\leq\) 300 GPa, 5 MPa \(\leq c_{1}\leq\) 7 MPa, 5 MPa \(\leq c_{2}\leq\) 7 MPa, and 40\({}^{\circ}\leq\varphi\leq\) 60\({}^{\circ}\). Figure 11: Schematic diagram of excavation by stages of the powerhouse (unit m). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & \multicolumn{3}{c}{**GOA-FLAC\({}^{\text{3D}}\)**} & \multicolumn{3}{c}{**GOA-GPR-FLAC\({}^{\text{3D}}\)**} \\ \hline Parameter & \(E/\text{GPa}\) & \(c/\text{MPa}\) & \(\varphi/^{\circ}\) & \(E/\text{GPa}\) & \(c/\text{MPa}\) & \(\varphi/^{\circ}\) \\ Real solution & 2.2 & 1.1 & 30 & 2.2 & 1.1 & 30 \\ Back-analysis result & 2.279 & 1.160 & 29.8 & 2.195 & 1.125 & 30.3 \\ Relative error/\% & 3.59 & 5.46 & 0.67 & 0.23 & 2.27 & 1.00 \\ Fitness & & 0.0426 & & & 0.0362 & \\ Fitness function evaluation times & & 9254 & & 1077 & \\ Computing time/s & & \(1.73\times 10^{5}\) & & & \(2.26\times 10^{3}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The results calculated by the two methods. Figure 12: The layout map of the displacement monitoring points section. (**a**) The monitoring points at the \(0+15\) m section to the left of the powerhouse. (**b**) The monitoring points at the \(0+54\) m section to the left of the powerhouse. Figure 13: The computing grid of excavated rock. The GOA parameter is set to \(D=7\), \(N=30\), \(c_{\rm max}=1\), and \(c_{\rm min}=1\times 10^{-5}\). The initial hyperparameters of the GPR model are set as \(\rm{In}I=[-1,-1,-1,-1,-1,-1]\), \(\rm{In}\sigma_{f}=-1\), \(\sigma_{n}=1\times 10^{-6}\). Take the number of FLAC\({}^{\rm 3D}\) calls greater than or equal to 1000 as the convergence criterion. The GOA-FLAC\({}^{\rm 3D}\) method and the method in this paper are used to invert the geomechanical parameters. The results are shown in Tables 5 and 6.