## 3 Results and Discussion
### Multi-Objective Optimization of Stress-Relief Groove Parameters
#### 3.2.2 Multi-Objective Optimization  
The three essential elements of multi-objective optimization are objective functions, constraint functions, and design variables. By applying the NSGA-II multi-objective optimization algorithm to the determined objective functions, the optimal solution set can be obtained, serving as a reference for subsequent experiments. In this process, the establishment of constraint functions is crucial and must consider practical conditions. The effectiveness of the multi-objective optimization algorithm is evaluated using the hypervolume (HV) indicator and the Spacing indicator. The spacing indicator measures the standard deviation of the minimum distance between each solution and other solutions. The smaller the spacing value, the more uniform the solution set. Hypervolume metric (HV) is volume of the objective space region enclosed by the set of non-dominated solutions obtained by the algorithm and a reference point. A higher HV value indicates the better overall performance of the algorithm. The constraints are detailed in Table 11, which aims to closely simulate real engineering conditions to ensure realistic data generation. Figure 18 illustrates the changes in HV and Spacing during the iteration process of the four-objective optimization. It can be observed that the values stabilize after a certain number of iterations, ensuring the uniformity and performance of the solution set. This stabilization confirms that the results obtained from the multi-objective optimization in this study are reliable. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Constraints** & **Range** & **Continuity** \\ \hline Groove width & [0, 0.5] & Continuous \\ Groove depth & [0, 0.5] & Continuous \\ Groove position & [0, 1] & Discrete \\ Backfilling & [0, 1] & Discrete \\ Support & [0, 1] & Discrete \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Constraint conditions. Figure 17: Predicted value—real value. Figure 18: Iterative process. Table 13 shows a subset of Pareto optimal solutions from the computations, which can be selected and validated as needed. Due to the complexity of visualizing four-objective optimization, visualizations for three-objective and two-objective optimizations are provided. Curve fitting was achieved using the PCHIP method, ensuring first derivative continuity and interval monotonicity. Figure 19 illustrates that some actual values dominated solutions in previous simulations. Multi-objective optimization helps obtain the optimal solution set and filters out dominated solutions. The Pareto front effectively filters out dominated solutions, particularly those distant from it. The screening process involves defining one objective function as the label and others as inputs, using SVR to fit a hyperplane. Monitoring scheme data are then input into the trained SVR model to assess optimization potential based on the difference between input and actual values. The figures show that some schemes are closer to the Pareto front while others need improvement. Constructing the Pareto front aids in filtering dominated solutions. After building the SVR hyperplane, stress indicator values are calculated; the closer these values are to zero, the closer they are to the hyperplane. Figure 20 demonstrates that this method builds an evaluation model, where the output difference can determine if the scheme is optimal. Random input parameters can be evaluated for feasibility based on their proximity to zero in the output. Based on the four-objective optimization solution set, a selection of scenarios was numerically simulated, focusing on floor and sidewall displacement control. The scenarios, including central and side grooves with backfilling, were chosen to compare specific differences and validate model optimization feasibility. These scenarios, detailed in Table 12, were modeled, and relevant data were extracted using numerical simulations. Groove dimensions, initially represented as percentages of roadway height, were converted into specific numerical values for simplification.