## 4 Optimization Design of Vehicle-Side Crashworthiness
### Results and Error Analysis  
High-precision mathematical agent modelling is the basic premise of optimization calculation. For validation of agent model in this paper, this article selects R\({}^{2}\) (determination coefficient) and RMSE (root-mean-square error) precision for the model of the evaluation index. The specific calculation formula is as follows:  
\[\text{R}^{2}=\frac{\sum_{i=1}^{n}(\hat{y}_{i}-\overline{y})^{2}}{\sum_{i=1}^{n} (y_{i}-\overline{y})^{2}} \tag{6}\]  
where \(y\) is the value to be fitted; \(\overline{y}\) is the mean; \(\hat{y}\) is the fit value; \(\sum_{i=1}^{n}(\hat{y}_{i}-\overline{y})^{2}\) is regression sum of squares (SSR); \(\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}\) is total sum of squares (SST).  
\[\text{RSME}=\sqrt{\frac{\sum_{i=1}^{n}(\hat{y}-\overline{y})^{2}}{n}} \tag{7}\]  
In general, the closer R\({}^{2}\) is to 1, the smaller the value of RMSE, and the higher the accuracy of the mathematical model.  
We carried out mathematical analysis and verification on the accuracy of the model. The following table shows the accuracy verification results of the optimal approximate front when the model converges:  
As can be seen from the Table 4, after six iterations of point selection, the approximate model established through ARBF has reached a very high accuracy and can completely replace the finite element model for optimization design.  
Figure 16: The iteration process of Pareto optimal front in finding solution of vehicle-side multi-objective crash worthiness using ARBF-MLPA method.  
Figure 17: The iteration process of HV in finding solution of crash worthiness using ARBF-MLPA method.  
After 6 iterations, the model has converged, and it is now completely possible to select the optimal solution on the basis of this model. Figure 18 shows the optimal Pareto front edge of iteration 6. Researchers should then select the most satisfactory Pareto scheme, namely the Knee point [29], as the final design value of the optimization. The target response values obtained in the optimization design are maximum chest intrusion (\(d_{max}\)) is 153.92 mm and peak collision force (\(F_{max}\)) is 159.12 KN.  
Table 5 shows the comparison between the corresponding response value of optimal design point and the response value before optimization, and the computation results of finite element for the optimization scheme. The optimal design results obtained after optimization are: t\({}_{1}\) = 1.76 mm, t\({}_{2}\) = 0.93 mm, t\({}_{3}\) = 1.12 mm, t\({}_{4}\) = 0.86 mm, t\({}_{5}\) = 1.20 mm. As shown in Table 5, both the errors of finite element and the errors of ARBF-MLPA are within the acceptable range. After the optimization, in whole-vehicle-side crash, the peak collision force decreased by 2.11%, chest maximum intrusion decreased by 4.32%, while total mass decreased by 14.05%.  
According to Figures 17 and 18, the ARBF-MLPA method converges in the sixth generation, and the convergence speed and accuracy are improved compared with the traditional point method. According to Table 5 and the optimization results, it can be seen that the ARBF-MLPA method can be well used in vehicle-side crashworthiness assessments. Compared with the traditional finite element calculation, it can not only greatly reduce the calculation time, but also ensure the accuracy. Therefore, the results prove that the hypothesis proposed in this paper is valid, and the method in this paper can also be generalized and used in other disciplines.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Response Value** & **RMSE** & **R2** \\ \hline The amount of chest intrusion & 0.1875 & 0.9865 \\ Peak value of collision force & 0.1924 & 0.9737 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The RMSE and R2 of ARBF model at convergence.  
Figure 18: Pareto optimal approximate front and Knee point of multi-objective optimization design for vehicle-side crash.