## 5 Energy Absorption Effect Optimization
### Surrogate Model Construction and Evaluation  
Response surface models utilize polynomial functions to fit the design space, allowing for a relatively accurate approximation of the actual functional relationships with a limited number of input  
(1) Surrogate Model Establishment  
To establish the mathematical relationship between different parameters and specific energy absorption, as well as to obtain predicted values through input parameters, a second-order response surface method is used to fit 30 experimental samples. The most commonly used polynomial surrogate model is a second-order algebraic polynomial function as follows:  
\[\hat{f}(x)=b_{0}+\sum\limits_{i=1}^{L}b_{i}x_{i}+\sum\limits_{i=1}^{L}b_{i}x_{ i}^{2}+\sum\limits_{i=1}^{L-1}\sum\limits_{j=i+1}^{L}b_{ij}x_{i}x_{j} \tag{4}\]  
In the formula, \(\hat{f}\) represents the approximate response surface function of the true response function \(f\), \(L\) is the number of variables in the input vector \(x\), and \(b_{0}\), \(b_{i}\), \(b_{ii}\), and \(b_{ij}\) are the unknown coefficients that need to be determined using the least squares method.  
By approximating the model using a second-order polynomial response surface method, a polynomial surrogate model that characterizes the energy absorption characteristics and crashworthiness of the anti-climb energy-absorbing device, in terms of the specific energy absorption \(SEA\) and the maximum peak force \(F\), is obtained as follows:  
\[\begin{array}{l}SEA=-0.0007\text{c}^{2}-0.1707{t_{1}}^{2}-0.2579{t_{3}}^{2}+0.0362C*t_{2}+0.1181~{}t_{1}*t_{2}\\ -2.783t_{2}+2.4174t_{3}+14.6641\end{array} \tag{5}\]  
\[\begin{array}{l}F=-25.4432{t_{1}}^{2}-37.49661{t_{3}}^{2}-10.8294t_{1}*t_{2} +16.2734t_{2}*t_{3}\\ +250.9143t_{1}+149.2\end{array} \tag{6}\]  
(2) Surrogate Model Evaluation  
The accuracy of the second-order polynomial surrogate model established based on the design of experiments (DOE) methodology can be assessed using the maximum relative error (_MRE_), the average relative error (_ARE_), and the goodness of fit \(R^{2}\). The goodness of fit is defined as follows:  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}}{\sum \limits_{i=1}^{n}\left(y_{i}-\overline{y}_{i}\right)^{2}} \tag{7}\]  
The average relative error (_AER_) is defined as follows:  
\[ARE=\frac{\sum\limits_{i=1}^{n}\frac{|y_{i}-\hat{y}_{i}|}{y_{i}}\times 100\%}{n} \tag{8}\]  
The maximum relative error (_MRE_) is defined as follows:  
\[MRE=Max\frac{|y_{i}-\hat{y}_{i}|}{y_{i}}\times 100\% \tag{9}\]  
In the formula, \(\hat{y}_{i}\) represents the approximate value for the \(i^{th}\) sample point, \(\overline{y}_{i}\) is the average value of \(i\) sample points, and \(n\) is the number of test points. Generally speaking, a surrogate model with a smaller average relative error, a smaller maximum relative error, and a larger goodness of fit (\(R^{2}\) value closer to 1) indicates the model has a higher accuracy.  
A cross-validation of the surrogate models for energy absorption and peak force, established with five random data sample points, was conducted. The accuracy of the obtained surrogate models is presented in Table 3.  
From Table 3, it can be observed that the accuracy of the surrogate model established based on the response surface method meets the precision requirements of the model in terms of the goodness of fit, average error, and maximum relative error. This indicates that the constructed surrogate model can provide predictions for the energy absorption and the maximum initial peak force of the anti-climb energy-absorbing device.