## 3 Multi-Objective Optimization of the Body-in-White Structure
### Development and Validation of Surrogate Model  
The surrogate model offers high accuracy and low computational cost, making it widely used in multi-objective, multi-variable optimization design. Common response surface models include the Kriging model, radial basis function, and polynomial response surface model [26]. The response surface approximation model can accurately approximate functional relationships within a local range using a limited number of experiments, while expressing them in simple algebraic equations. This significantly reduces computational time.  
Polynomial response surface models are commonly used to construct surrogate models in vehicle crashworthiness optimization because they can capture nonlinear relationships for complex crash dynamics, while being computationally more efficient than finite element analysis for optimization iterations [27; 28; 29; 30]. At the same time, they strike a balance between accuracy and computational cost. Therefore, this study adopts the response surface method to establish the surrogate model and selects a third-order response surface model. Its mathematical expression is given in Equation (1).  
\[y(x)=\beta_{0}+\sum_{i=1}^{n}\alpha_{i}x_{i}+\sum_{i=1}^{n}\beta_{i}x_{i}^{2}+ \sum_{i=1}^{n}\gamma_{i}x_{i}^{3}+\sum_{i\
eq j}\theta_{ij}x_{i}x_{j} \tag{1}\]  
In this equation, \(y(x)\) represents the response objective function, while \(x_{i}\) and \(x_{j}\) denote the \(i\) and \(j\) design variables, respectively. The term \(\beta_{0}\) is a constant; \(\alpha_{i}\), \(\beta_{i}\), \(\gamma_{i}\), and \(\theta_{i}\) are the coefficients of the corresponding terms. The parameter \(n\) indicates the number of design variables.  
Based on the large number of simulation results obtained from the experimental design plan, and using Equation (1), response surface approximate models for the total mass (\(M\)) of the optimized components, total energy absorption (\(E\)) of the optimized components, B-pillar abdominal intrusion (\(L_{1}\)), and lower threshold intrusion (\(L_{2}\)) were established. The corresponding third-order response surface approximate models are shown in Equations (2)-(5).  
\[M(t)=26.467+2.133t_{1}t_{2}+0.934t_{2}t_{6}+1.694t_{3}t_{6}+1.685t_{4}t_{7} \tag{2}\]  
\[\begin{array}{l}E(t)=3823.61+3862.316t_{1}+3291.686t_{5}\\ \begin{array}{l}-663.432t_{1}^{2}-590.22t_{4}^{2}-632.87t_{1}t_{5}+313.528t_{ 2}t_{7}\\ \begin{array}{l}+215.674t_{3}t_{4}+866.421t_{4}t_{5}-407.328t_{5}^{3}\\ \end{array}\\ \begin{array}{l}L_{1}(t)=354.13-8.777t_{5}-3.827t_{1}t_{6}-4.666t_{2}t_{4}-4.459t_{2}t_{7}-1.978t_{1}^{3}\\ \begin{array}{l}L_{2}(t)=310.583-9.218t_{1}-4.032t_{5}^{2}-6.087t_{2}t_{4}-1. 675t_{3}t_{5}-1.785t_{1}^{3}\\ \end{array}\\ \end{array}\end{array} \tag{3}\]  
To verify the accuracy of the third-order response surface approximate models constructed above, the \(R^{2}\) value from variance analysis was used to assess the precision of the models. \(R^{2}\) represents the similarity between the actual values and the model's predicted values. The closer \(R^{2}\) is to 1, the higher the similarity between the actual and predicted values, indicating better model fitting. Typically, an \(R^{2}\) value greater than 0.9 is considered acceptable. The specific calculation methods are given in Equations (6)-(8).  
\[R^{2}=1-\frac{Q_{r}}{Q_{t}} \tag{4}\]  
\[Q_{r}=\sum_{i=1}^{n_{1}}(y_{i}-\hat{y}_{i})^{2} \tag{5}\]\[Q_{t}=\sum_{i=1}^{n_{1}}(y_{i}-\overline{y}_{i})^{2} \tag{8}\]  
In the equation, \(n_{i}\) is the number of samples; \(y_{i}\) is the actual simulation values; \(\overline{y}_{i}\) is the response surface prediction values; \(\overline{y}_{i}\) is the mean of actual simulation values.  
We obtained an approximate model of component energy absorption with \(R^{2}\) = 0.925; the \(R^{2}\) of the quality approximation model is 0.965. The \(R^{2}\) values of the B-pillar abdominal invasion approximation model and the lower threshold invasion approximation model are 0.954 and 0.983, respectively.