## 4 Main Causes and Response Surface Analysis
### Response Surface Modelling and Agent Modelling  
In order to obtain the three-dimensional cloud map of each output response, the two variables with less influence were set to the midpoint of their variation ranges. The other two design variables were both used as independent variables to construct each response surface, as shown in Figure 7, in accordance with the calculation results of the sample points in the experimental design. Figure 7 shows the response surface model of the peak force \(F_{\text{PC}}\), mean force \(F_{\text{MC}}\) and specific energy absorption \(E_{\text{SA}}\) of the energy-absorbing structure, from which we can see the impact of each variable on the response index. Both the peak force and the mean force of the shrink tube increased with the increase in the cone angle \(\alpha\) and thickness \(T\). The degree of influence of the two factors was relatively close, with the influence of the two factors having a greater effect on the peak force than on the average force. The specific energy absorption increased positively with the influence of the two factors. It grew faster with the increase in the cone angle \(\alpha\) and relatively slower with the increase in the thickness \(T\). Within the variable design interval, the mean force increased by 476.42% with thickness and by 515.72% with the angle, and the specific energy absorption increased by 58.59% with thickness and by 531.19% with the angle. \begin{table}
\begin{tabular}{c c c c c} \hline
**Variable Names** & **Cone Angle \(\alpha^{l}\)** & **Thickness \(T\)/mm** & **Friction Coefficient \(u\)** & **Axial Length \(S\)/mm** \\ \hline Variable values & 5, 15, 25 & 3, 6, 9, 12 & 0.02, 0.04, 0.06, 0.08, 0.1 & 21, 23, 25, 27, 29 \\ \hline \end{tabular}
\end{table} 
Table 2: Variable values. Practical engineering problems are often complex, usually do not have an explicit functional relationship between design variables and target performance, and manifest themselves as multi-parameter, high-dimensional, strongly nonlinear problems, which are difficult to compute with a simplified physical model and often need to be fitted to discrete data through a proxy model. In order to obtain optimization results efficiently and accurately, proxy methods are used to fit discrete data and build efficient models to replace the actual model without reducing the accuracy of the model. These models are also referred to as proxy models. Among them, the radial basis function is an efficient surrogate method, especially effective in predicting \(E_{\text{SA}}\) with good accuracy. The mechanical behavior of the shrink tube anti-climb device during the impact energy absorption process is highly nonlinear; therefore, we used the radial basis function (RBF) to construct a proxy model for response surface analysis. In order to evaluate the accuracy of the response model, corresponding error analysis and evaluation of the output indicators need to be performed using the coefficient of determination \(R^{2}\), the mean absolute error \(E_{\text{raa}}\) and the root mean square error \(E_{\text{rms}}\) as evaluation parameters to verify the effectiveness of the proxy model [44]. These are calculated as follows:  
\[R^{2}=1-\frac{\sum{(\hat{y}i{-}yi)^{2}}}{\sum{(\hat{y}i{-}\hat{y}i)^{2}}} \tag{5}\]  
\[E_{\text{raa}}=\frac{\frac{1}{N}\sum_{i=1}^{N}{\left|y^{i}-\hat{y}i\right|}}{ \sqrt{\frac{1}{N}\sum_{i=1}^{N}{\left|y^{i}-\overline{y}i\right|^{2}}}} \tag{6}\]  
Figure 6: Main cause analysis: (**a**) effect on peak force; (**b**) effect on mean force; (**c**) effect on specific energy absorption. \[E_{\text{rms}}=\sqrt{\frac{\sum(\hat{y}i-yi)^{2}}{N}} \tag{7}\]  
where \(\hat{y}i\) is the predicted value of the proxy model at the point \(i\); \(yi\) is the actual finite element calculation value of the point; \(\overline{y}i\) is the average of the actual response values of all sample points; and \(N\) is the number of sample points.