# NSGA-III-XGBoost-Based Stochastic Reliability Analysis of Deep Soft Rock Tunnel
## 3 NSGA-III Algorithm
### Nsga-iii-XgBoost-Based MCS Reliability Analysis

The whole calculation process of the NSGA-III-XGBoost-based MCS reliability analysis method is depicted in the flowchart in Figure 1. The detailed calculation steps of the proposed method are as follows. _Step 1_ The dataset of geo-mechanical parameters is generated using the LHS method. _Step 2_ Use the finite difference method (FDM) such as FLAC3D to calculate the displacement of each monitoring point. _Step 3_ Establish the surrogate model by XGBoost training on the dataset:

First, the data of displacements and geo-mechanical parameters are standardized to form the sample dataset. Second, establish the model to map the nonlinear relationship between displacements and their corresponding geo-mechanical parameters. Finally, the nine hyper-parameters (max_depth, learning_rate, n_estimators, gamma, min_child_weight, subsample, colsample_bytesree, reg_alpha, and reg_lambda) of the XGBoost-based surrogate model are tuned by the NSGA-III algorithm, and then the optimized surrogate model is obtained. NSGA-III is a fast nondominated objective optimization algorithm based on the Pareto optimal solution with an elite retention strategy [29; 30; 31]. Therefore, it is selected as the method used for tuning the hyperparameter of the XGBoost-based surrogate model. The specific hyperparameter tuning process for of XGBoost-based surrogate model was depicted in reference [28]. _Step 4_ The geo-mechanical parameters sample data are obtained using the Monte Carlo method (MCM). Then, the displacements of the tunnel are calculated by the optimized NSGA-III-XGBoost-based surrogate model established in Step 3. Figure 1: Calculation procedure of NSGA–III–XGBoost–MCS reliability analysis. _Step 5_ According to the tunnel displacement allowable design value, the failure probability of the tunnel primary support structure is calculated by Equation (7). \[P_{\text{f}}\approx\frac{N_{\text{f}}}{N} \tag{7}\]

where \(P_{\text{f}}\) is the failure probability of tunnel primary support structure; \(N_{\text{f}}\) is the number of \(G_{\text{tunnel}}^{j}(\mathbf{X}^{\prime})=U_{\text{max}}-U_{j}(\mathbf{X}^{\prime})<0\) in all \(G_{\text{tunnel}}^{j}(\mathbf{X}^{\prime})(j=1,2,\cdots,N)\); \(N\) is the total number of random samples. ## 5 Case Study

### Problem Description

In this paper, the reliability of a horseshoe-shaped soft rock tunnel is analyzed using the proposed procedure of NSGA-III-XGBoost-MCS reliability. As shown in Figure 2, the tunnel, with a height of 10.23 m and a span of 12.46 m, is located in Gansu Province, People's Republic of China. Surrounding rock of the tunnel is fully strongly weathered granite gneiss. The design parameters of the initial support of the tunnel: length of rock bolt 25-5 = 3.5 m; spacing of grille steel frame 25 = 0.6 m; thickness of sprayed concrete = 25 cm; mesh spacing of \(\varphi\)8 steel = 20 \(\times\) 20 cm. Design parameters of second lining of the tunnel: thickness of reinforcement = 33-45 cm. Three-dimensional numerical simulation is performed using FLAC3D. The top of the model is considered as a free surface. The bottom boundary, the horizontal direction (X direction) boundaries and the Y direction (longitudinal direction of the tunnel) boundaries are constrained using displacement. The FDM model is 100.0 m in the horizontal direction, 20.0 m in the longitudinal direction of the tunnel and 149.2 m in the vertical direction. The lateral pressure coefficient \(\lambda=1.2\); the bulk density of surrounding rock = 24.0 kN/m\({}^{3}\). The self-weight stress field is applied in the vertical direction of the model. The elastic-plastic constitutive model and the Mohr-Coulomb failure criterion are used to simulate surrounding rock. The concrete lining combined with rock bolts is installed as support.