# Crasworthiness Analysis and Multi-Objective Optimization for Concave I-Shaped Honeycomb Structure
## 4 Multi-Objective Optimization of Concave I-Shaped Honeycomb Structure
### Radial Basis Function Model (RBF) and Error Metrics  
To streamline the multi-objective optimization process, it is required to determine how to fit an approximative model from the sampled points after the sampling issue has been resolved. The kriging model [34], the orthogonal polynomial model [35], the Gaussian radial basis function model [36], and the response surface model [37] are common fitting techniques used in data processing software. When using the kriging model, it is assumed that all data are unbiased and follow a normal distribution. Depending on the weight function, orthogonal polynomial models may be divided into Lejeune polynomials, Chebyshev polynomials, etc  
solely on the distance from the origin. This procedure may be compared to a basic neural network, which has the benefits of a simple structure, straightforward training, rapid learning convergence, the ability to approximate arbitrary nonlinear functions, and the ability to overcome local minimum values. Response surface models are fitted using polynomials, which are less efficient than neural network approaches in approximating extremely complicated functional connections but may be utilized to produce a more accurate model on a local scale with fewer sample points. The errors of the different approximation models are compared to choose an appropriate fitted model.  
The R-squared (\(R^{2}\)), root mean squared error (RMSE), maximum absolute relative error (MARE), and average absolute relative error (AARE) are used to assess the approximation model's accuracy [38]. As demonstrated in Tables 7 and 8, the suitable proxy model approach is chosen based on the accuracy of the crashworthiness measures under each method.  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{n}{(y_{i}-\hat{y}_{i})}^{2}}{\sum\limits_{i=1} ^{n}{(y_{i}-\overline{y})}^{2}}, \tag{29}\]  
\[RMSE=\sqrt{\frac{\sum\limits_{i=1}^{n}{(y_{i}-\hat{y}_{i})}^{2}}{n}}, \tag{30}\]  
\[MARE=\max_{i=1,2,\ldots,n}\biggl{(}\frac{|y_{i}-\hat{y}_{i}|}{|y_{i}|}\biggr{)}, \tag{31}\]  
\[AARE=\frac{1}{n}\sum\limits_{i=1}^{n}{\frac{|y_{i}-\hat{y}_{i}|}{|y_{i}|}}, \tag{32}\]  
where \(n\) is the number of test points, \(y_{i}\) and \(\hat{y}_{i}\) are the function value and the prediction value of the RBF model at the \(i\)th test point, and \(\overline{y}\) represents the mean value of \(y_{i}\).  
Generally, a larger value of \(R^{2}\) means a more accurate metamodel. Smaller values of RMSE, MARE, and AARE correspond to a more accurate metamodel.  
A parameter \(T\) relating to the four error values is defined. When \(T\) is smaller, it means that the error of the fitting method used is smaller.  
\[T=(1-R^{2})+\text{RMSE}+\text{MARE}+\text{RAAE}, \tag{33}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Response**} & \multirow{2}{*}{**Method**} & \multicolumn{4}{c}{**Accuracy Evaluation Coefficient**} \\ \cline{3-6}  & & \(R^{2}\) & RMSE & MARE & AARE \\ \hline \multirow{3}{*}{M} & Kriging & 0.93033 & 0.07312 & 0.31327 & 0.04141 \\  & OP & 0.14552 & 0.25607 & 1.11814 & 0.07808 \\  & RBF & 0.99878 & 0.00982 & 0.03629 & 0.00663 \\ \hline \multirow{3}{*}{SEA} & Kriging & 0.86948 & 0.10178 & 0.33218 & 0.06905 \\  & OP & 0 & 5.88261 & 25.6863 & 1.79374 \\  & RBF & 0.94909 & 0.06341 & 0.13234 & 0.05146 \\ \hline \multirow{3}{*}{PCF} & Kriging & 0.81532 & 0.11409 & 0.26187 & 0.09522 \\  & OP & 0 & 27.83448 & 121.5384 & 8.48736 \\ \cline{1-1}  & RBF & 0.92577 & 0.07289 & 0.1382 & 0.06317 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Accuracy assessment of the surrogate model.  
\[\begin{array}{l}T(Kriging)=1.78686\\ T(OP)=195.5296\\ T(RBF)=0.70057\\ T(Order.1)=1.37095,\\ T(Order.2)=1.06566\\ T(Order.3)=1.34652\\ T(Order.4)=4.86487\\ \end{array} \tag{34}\]  
Analysis of the comparative data shows that the RBF model has relatively small errors and is suitable for selection as an approximate model, so it is chosen as the surrogate model for subsequent calculations.