## 4 Numerical Simulation Analysis
### Determining the Step Height Parameters Based on the NSGA-II Multi-Objective Optimization Algorithm
#### 4.2.3 Determining the Optimal Step Parameters Based on the NSGA-II Optimization Algorithm  
The variable to be determined in the numerical simulation was the lower step height x. The objective function values primarily involved three factors: the maximum deformation y\({}_{1}\) after 20 m of lower step excavation, the maximum deformation y\({}_{2}\) after 40 m of lower step excavation, and the maximum deformation y\({}_{3}\) after 60 m of lower step excavation. To integrate the NSGA-II optimization algorithm with FLAC3D numerical simulation, secondary development was carried out using the built-in Python programming language in FLAC3D. Using Python, the entire process was implemented, including step height parameter setting, command file generation, retrieval of numerical simulation results, integration of numerical simulation results, and updating of the new step parameters based on the current step parameters. The numerical simulation process was automated without manual intervention, and when the termination condition was met, the simulation ended, yielding the optimal step height parameters [44; 45; 46; 47]. Step 1: Initialize the hyperparameters of the NSGA-II multi-objective optimization algorithm, with a population size of N = 20 and a maximum iteration count of Max_iter = 10,000, and initialize the current iteration count Now_iter to 0. Step 2: Initialize the population. Since the population size N is 20, generate 20 lower step height parameters x, with the parameter values ranging from 0.1 to 4.85. Step 3: Based on the 20 sets of step parameters, generate 20 FLAC3D command flow files using a Python script. Perform numerical simulations for each file to obtain the maximum deformation values y\({}_{1}\), y\({}_{2}\), and y\({}_{3}\) of the tunnel after 20 m, 40 m, and 60 m of lower step excavation, respectively. To accelerate the computation speed, conduct 20 FLAC3D numerical simulations simultaneously using parallel processing. Step 4: Based on the 20 lower step height parameters and the corresponding maximum deformation values y\({}_{1}\), y\({}_{2}\), and y\({}_{3}\) of the tunnel after 20 m, 40 m, and 60 m of lower step excavation, update the 20 new lower step height parameter values using the NSGA-II optimization algorithm. Step 5: Increment the current iteration count Now_iter by 1. If Now_iter exceeds the maximum iteration count Max_iter, end the calculation (Step 6); otherwise, proceed to Step 2. Step 6: End the calculation. The optimization variable in this study was the lower step height (x), with a search range of 0.1 m to 4.85 m, corresponding to the tunnel height minus a minimal construction clearance. The population size was set to 20, and the maximum generation number was 10,000, resulting in up to 200,000 solution candidates evaluated through NSGA-II. These parameter settings were chosen to balance optimization accuracy and computational efficiency. While the NSGA-II algorithm yields a Pareto front of optimal solutions in terms of objective functions (deformations at 20 m, 40 m, and 60 m), engineering validation is also required. Therefore, the final chosen value (3 m) was not only optimal in the simulation but was also verified through field monitoring results, confirming that the predicted  
about 5.25 \(\times\) 10\({}^{6}\) Pa, and the compressive stress at the top is about 1.01 \(\times\) 10\({}^{6}\) Pa. After 60 m of tunnel excavation, the maximum compressive stress on both sides is approximately 6.56 \(\times\) 10\({}^{6}\) Pa, the minimum compressive stress at the bottom is about 5.46 \(\times\) 10\({}^{6}\) Pa, and the compressive stress at the top is about 1.56 \(\times\) 10\({}^{6}\) Pa.  
From the simulation results of both the full-face excavation and step excavation techniques, it can be seen that compared to the step excavation technique (with a lower step height of 3 m), after tunnel excavation using the full-face method, the larger excavation cross-section leads to a larger loosening range of the tunnel free face and surrounding rock.