## 3 Multi-Objective Optimization of the Body-in-White Structure
### Multi-Objective Optimization Model and Results  
The genetic algorithm follows the principle of natural selection. Recent years have witnessed the emergence of various novel optimization algorithms such as whale optimization algorithm (WOA), gradient-based Newton direction optimization (GNDO), and arithmetic optimization algorithm (AOA), which have been successfully applied in different fields including wireless communication systems [31] and fluid mechanics [32]. Among these optimization methods, the second-generation non-dominated sorting genetic algorithm (NSGA-II) is widely used due to its efficiency in solving multi-objective optimization problems. The NSGA-II algorithm offers strong robustness and excellent global search capabilities [33; 34]. It enhances optimization accuracy through an elite strategy, ensuring a uniform distribution of Pareto optimal solutions.  
In this study, the NSGA-II algorithm parameters are set as follows: population size of 40, 40 iterations, crossover probability of 0.9, crossover distribution index of 10, and mutation distribution index of 20. Multi-objective optimization is performed based on Equation (9), considering practical manufacturing precision. The final optimized thicknesses for the seven components are determined as 2.0 mm, 1.0 mm, 0.8 mm, 0.9 mm, 2.0 mm, 0.5 mm, and 0.6 mm.  
\[\left\{\begin{array}{l}find\ \ \ t_{\text{imin}}\leq t_{i}\leq t_{\text{ imax}}(i=1,2,\cdots,7)\\ \ \ \ \ \text{min}\ M(t),\text{min}\ L_{1}(t),\text{min}\ L_{2}(t)\\ \ \ \ \ \ \text{max}\ E(t)\\ \ \ \ \ s.t.\ \ M(t)\leq 33.6\ \text{kg}\\ \ \ \ \ \ \ L_{1}(t)\leq 310\ \text{mm}\\ \ \ \ \ \ L_{2}(t)\leq 270\ \text{mm}\end{array}\right. \tag{9}\]  
The optimized solution was re-submitted to LS-DYNA for validation. A comparison between the multi-objective optimization results and simulation data, as shown in Table 4, reveals that the relative error does not exceed 2%, confirming the accuracy of both the surrogate model and the optimization method.  
Table 5 presents a comparison between the original and optimized designs. The maximum intrusion of the belly of the B-pillar is reduced from 397.5 mm to 313.0 mm, a decrease of 21.3%. The maximum intrusion of the lower sill decreases from 343.5 mm to 257.0 mm, representing a 25.2% reduction. The absorbed energy of the optimized  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & \(L_{1}\)**/mm** & \(L_{2}\)**/mm** & \(E\)**/J** & \(M\)**/kg** \\ \hline Predicted value & 309.3 & 252.3 & 11,032.0 & 32.9 \\ Simulated value & 313.0 & 257.0 & 11,126.6 & 33.2 \\ Error/\% & 1.2 & 1.9 & 0.9 & 0.9 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of predicted results and simulation results.  
components increases from 8045.0 J to 11,126.6 J, an improvement of 38.3%, while the total mass is slightly reduced from 33.6 kg to 33.2 kg, a 1.2% decrease.  
After applying multi-objective optimization, the deformation of the vehicle's bottom structure remains significant. Observation points were established on both sides of the battery pack, as shown in Figure 11. The maximum intrusion of the battery pack reached 81.9 mm, approximately 12.4% of its total size. This level of deformation could potentially threaten the safety and performance of the vehicle's bottom-mounted battery pack [25]. Therefore, from the perspective of battery pack collision safety, the optimized body structure still has limitations. Further exploration of additional protective solutions is needed.