## 5 Back-Analysis of Rock Parameters with GOA-GPR-FLAC\({}^{\text{3D}}\)
### Case Study
In order to carry on the proper analysis of the tunnel excavation process, it was necessary to divide the shallow tunnel excavation process into four subsequent sub-steps as follows:  
Step 1: The classical elastic solution method was adopted to ensure the stability of the in situ stress solution. In this method, the constitutive model was set as the elastic model, and the volume modulus and shear modulus were set to large values so that the model could be balanced without plastic failure, and unrealistic displacement or even failure could be prevented. Step 2: After confirming the initial in situ stress, the existing displacement and velocity were cleared to zero to avoid interfering with the subsequent analyses. In FLAC\({}^{\text{3D}}\), accurate calculations are often based on correct initial in situ stress analysis results. Therefore, in the first two steps, the initial in situ stress of the entire model was solved. Step 3: The chosen Mohr-Colomb elastoplastic constitutive model described above was assigned to the whole tunnel model and the properties have been changed from elastic to plastic. Step 4: The 'null' model that comes with FLAC\({}^{\text{3D}}\) was assigned to the part of the grid that needs to be excavated to simulate the actual excavation process. To verify the feasibility of the algorithm, we assume that the real mechanical parameters of the rock are \(E=2.2\) GPa, \(c=1.1\) MPa, \(\varphi=30^{\circ}\). Five monitoring points (1, 2, 3, 4 and 5) are selected at \(0^{\circ}\), \(22.5^{\circ}\), \(45^{\circ}\), \(67.5^{\circ}\) and \(90^{\circ}\), respectively, along the circular direction of the tunnel (see Figure 10). The displacement of the monitoring point (including horizontal displacement and vertical displacement) is calculated by FLAC\({}^{\text{3D}}\) and regarded as the \"measured displacement\". The horizontal and vertical displacements calculated by FLAC\({}^{\text{3D}}\) at each monitoring point are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Direction**} & \multicolumn{4}{c}{**Displacement Value of Each Monitoring Point (mm)**} \\ \cline{2-6}  & **1** & **2** & **3** & **4** & **5** \\ \hline Horizontal direction (\(d_{\text{T}}^{\text{z}}\)) & \(-27.0159\) & \(-23.3271\) & \(-19.6929\) & \(-14.6111\) & 0 \\ Vertical direction (\(d_{\text{z}}^{\text{z}}\)) & 0 & \(-14.5982\) & \(-19.6261\) & \(-23.2481\) & \(-26.8916\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The measured displacements of points. Figure 9: Computational grid of FLAC\({}^{\text{3D}}\). For this example, the objective function is:  
\[f(\mathbf{X})=\sqrt{\sum_{i=1}^{5}\Bigl{[}(d_{i,x}-d_{i,x}^{*})^{2}+(d_{i,z}-d_{i,z}^{ *})^{2}\Bigr{]}} \tag{14}\]  
where \(d_{i,x}\) and \(d_{i,z}\) are the calculated displacement components of the horizontal and vertical directions of the \(i\)-th monitoring point, respectively; \(d_{i,x}^{*}\) and \(d_{i,z}^{*}\) are the actual displacement components of the horizontal and vertical directions of the \(i\)-th monitoring point, respectively; \(\mathbf{X}\) is a group of decision variables; and the decision variables in this example are elastic modulus \(E\), cohesion \(c\) and internal friction angle \(\varphi\). To verify the feasibility of the proposed method, we used the GOA-FLAC\({}^{\text{3D}}\) and the GOA-GPR-FLAC\({}^{\text{3D}}\) algorithms to invert the rock mechanics parameters (decision variables) of the tunnel. The GOA parameter is set to \(D=3\), \(N=50\), \(c_{\text{max}}=1\), and \(c_{\text{min}}=1\). The initial hyperparameters of the GPR models are set as \(\text{In}I=[-1,-1,-1]\), \(\text{ln}\sigma_{f}=-1\), \(\sigma_{n}=1\times 10^{-6}\). The search interval of the above two algorithms, namely, the search range of decision variable \(\mathbf{X}\) in the objective function \(f(\mathbf{X})\), is set as \(2.0\) GPa \(\leq E\leq 2.4\) GPa, \(1.0\) MPa \(\leq c\leq 1.4\) MPa, \(28^{\circ}\leq\varphi\leq 32^{\circ}\).