# NSGA-III-XGBoost-Based Stochastic Reliability Analysis of Deep Soft Rock Tunnel
## 3 NSGA-III Algorithm
The design parameters of the initial support, second lining and rock-bolt are as follows: [PERSON]'s modulus of shotcrete with grillage steel frame \(E_{\text{IS}}\) = 26.311 GPa; Poisson's ratio of shotcrete with grillage steel frame \(v_{\text{IS}}\) = 0.22; [PERSON]'s modulus of concrete lining \(E_{\text{CL}}\) = 28.0 GPa; [PERSON]'s ratio of concrete lining \(v_{\text{CL}}\) = 0.27; [PERSON]'s modulus of rock bolt \(E_{\text{bolt}}\) = 210.0 GPa; [PERSON]'s ratio of rock bolt \(v_{\text{bolt}}\) = 0.3. In this paper, the displacement changes of each measuring point or line under different parameters are taken as the dataset of the proposed method. The layout of measuring points or lines is shown in Figure 3.

Figure 2: Initial support and second lining of tunnel (unit: dm). In this paper, the properties of the initial support structure, second lining structure and rock bolt are assumed to be deterministic; \(E\), \(\
u\), \(\varphi\) and \(c\) are regarded as basic random variables with statistical properties, as shown in Table 1. Firstly, the LHS method is used to generate the 1000-sample dataset of geo-mechanical parameters of surrounding rock, and then the corresponding finite difference numerical calculations based on this dataset are conducted using FLAC3D. ### Performance Function and Random Variables

For simplicity, only the inward displacement of the tunnel initial concrete lining is considered to be the criterion of stability for this example. The performance function \(G_{\text{Tunnel}}(\mathbf{X}^{\prime})\) is given by Equation (8). According to the Chinese Highway Tunnel Design Code, the vault settlement limit value \(U_{\text{max}}\) in Equation (8) is set as \(u_{\text{vmax}}=0.1\) m, and the displacement convergence limit values \(U_{\text{max}}\) of Monitoring Line BC, AB, AC, AD and AE in Equation (8) are set as \(U_{\text{BCmax}}=0.072\) m, \(U_{\text{ABcmax}}=U_{\text{ACcmax}}=0.075\) m and \(U_{\text{ADcmax}}=U_{\text{AEcmax}}=0.087\) m, respectively. \[G_{\text{Tunnel}}(\mathbf{X}^{\prime})=U_{\text{max}}-U(E,c,\varphi,\
u) \tag{8}\]

where \(E\) is the Young's modulus of rock; \(\
u\) is the Poisson's ratio of rock; \(\varphi\) is the friction angle of rock; \(c\) is the cohesion of rock. The determination coefficient (\(R2\_\)score) is selected as the prediction evaluation index to measure the algorithms in this paper. \(R2\_\)score is denoted by Equation (9). \[R2\_\text{score}=1-\frac{\sum\limits_{i}{(\hat{y}_{i}-y_{i})^{2}}}{\sum\limits _{i}{(\overline{y}_{i}-y_{i})^{2}}} \tag{9}\]

where \(y_{i}\) is the actual value; \(\hat{y}_{i}\) is the prediction value; \(\overline{y}_{i}\) is the mean of the data. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Random Variables** & **Distribution** & \(\mu_{X_{i}}\) & \(\sigma_{X_{i}}\) & **COV** \\ \hline \(\
u\) & Normal & 0.33 & 0.033 & 0.1 \\ \(\varphi/^{\circ}\) & Normal & 33 & 4.95 & 0.15 \\ \(E\)/MPa & Normal & 350 & 50.5 & 0.144 \\ \(c\)/kPa & Normal & 210 & 23 & 0.11 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical properties of rock mechanical parameters. Figure 3: Schematic diagram of tunnel monitoring point layout (unit: dm). ### Results and Discussion

#### 5.3.1 Comparison of Tunnel Reliability Based on NSGA-III-XGBoost, SVR and BP

Using the same sample dataset, the nonlinear mapping surrogate model between the geo-mechanical parameters of the surrounding rock and convergence or displacement are established using NSGA-III-XGBoost, SVR (support vector regression) and BP (back propagation neural network) algorithms, respectively. Using these established surrogate models, the corresponding convergence or displacement values of Monitoring Point A, Monitoring lines BC, AB, AC, AD and AE are obtained for 30,000 samples of surrounding rock mechanical parameters generated by the LHS method.