## 3 Results and Discussion
### Metamodel and Multi-Objective Optimization
#### 3.2.3 Global Sensitivity Analysis  
In this section, the sensitivities of each input variable are analyzed using the SHapley Additive exPlanations (SHAP) method on 50,000 generated samples. The average SHAP values for \(\delta_{p}\) are presented in Figure 12a. It can be observed that the variables \(N_{y}\) and \(t\) exhibit very high average SHAP values, indicating a significant influence on \(\delta_{p}\). On average, varying \(N_{y}\) and \(t\) results in changes in \(\delta_{p}\) of 5.06 mm and 5.68 mm, respectively. In contrast,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Geometry**} & \multicolumn{3}{c}{\(\delta_{p}\)} & \multicolumn{3}{c}{**SEA**} \\ \cline{2-7}  & **MAX** & **MSE** & \(R^{2}\) & **MAX** & **MSE** & \(R^{2}\) \\  & **(mm)** & **(mm2)** & **(-)** & **(J/g)** & **(J\({}^{2}\)/g\({}^{2}\))** & **(-)** \\ \hline REH & 3.059 & 0.713 & 0.991 & 0.418 & 0.009 & 0.996 \\ DAH & 3.307 & 0.684 & 0.994 & 0.302 & 0.006 & 0.997 \\ SH & 4.524 & 1.188 & 0.992 & 0.250 & 0.006 & 0.997 \\ CH & 4.309 & 1.779 & 0.982 & 0.498 & 0.012 & 0.996 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Error parameters of all auxetic metamodels.  
Figure 11: Scatter plots of predicted values from metamodel versus simulation results of \(\delta_{p}\) **(left)** and SEA **(right)** for **(a)** REH, **(b)** DAH, **(c)** SH, and **(d)** CH models. The black diagonal line represents the ideal case where predicted values match simulation values, indicating perfect accuracy.  
the variables \(N_{z}\) and \(\theta/r\) have relatively low average SHAP values in \(\delta_{p}\) of 2.07 mm and 1.42 mm, respectively. The averaged SHAP of SEA is given in Figure 12b. Similar to \(\delta_{p}\), the variable \(t\) has the highest average SHAP value, contributing an average change of 1.21 J/g, followed by \(N_{z}\) with 0.89 J/g, \(N_{y}\) with 0.77 J/g, and \(\theta/r\) with 0.43 J/g. Overall, the variable \(t\) has the greatest influence on both blastworthiness performance metrics, while \(\theta/r\) has the least influence.  
To provide a more intuitive understanding of how each input variable affects the output, summary plots of SHAP for each auxetic geometry are presented in Figure 13. These summary plots display the SHAP values for each variable alongside the input values for each sample. The sign of SHAP values indicates the change in output value with respect to the reference value. In this case, the reference value is the baseline design. The baseline design configuration is determined by selecting the midpoint value of the design variable range, with the thickness set to 1 mm. In general, higher input values result in lower outputs for both \(\delta_{p}\) and SEA, as indicated by the gradual change in color. However, the \(\theta\) variable shows a nonlinear influence on \(\delta_{p}\), as indicated by disordered coloring. Figure 14 shows the scatter plots of the influence of each feature for all auxetic geometries. As illustrated in Figure 14, high input values, especially for variables \(N_{y}\) and \(t\), correspond to the lower-left region of the plot, indicating small \(\delta_{p}\) and small SEA, and vice versa. The SHAP values from Figure 13 are consistent with the trends observed in Figure 14, as both depict the same influence of inputs on outputs. This consistency between SHAP values and scatter plots highlights the robustness of the input-output relationship analysis.  
Figure 12: Averaged SHapley Additive exPlanations (SHAP) values of (**a**) \(\delta_{p}\) and (**b**) SEA for each auxetic model.  
Figure 13: SHAP summary plot of \(\delta_{p}\) (**left**) and SEA (**right**) for (**a**) REH, (**b**) DAH, (**c**) SH, and (**d**) CH models.