## 3 Results and Discussion
### Discussions  
#### 3.3.1 Comparative Analysis  
A comparison between the Pareto front and selected design points was conducted to evaluate improvements achieved through the NSGA-II optimization. First, the baseline design, marked with a red triangle, is located at the lower right of the Pareto front, as shown in Figure 15. The optimization results in an improvement of 12.55-35.60% in \(\delta_{p}\) reduction for the same SEA and 26.21-83.68% in SEA enhancement for the same \(\delta_{p}\). Therefore, the Pareto front offers superior performance in at least one objective. Next, the balanced design, marked by the purple pentagon in Figure 15, is compared with the baseline design. A balanced design is defined as a point on the Pareto front that represents the balanced trade-off among conflicting objectives. This point is identified as the one with the minimum distance to the utopia point. The utopia point, an idealized design point represented by dark blue squares in Figure 15, corresponds to the optimal objective values (in this study, the optimal objective values are the lowest \(\delta_{p}\) and highest SEA). Therefore, the utopia point is linked to the two ideal design points, marked by green \"+\" symbols. As given in Table 8, although the balanced designs for REH and DAH have larger \(\delta_{p}\) than the baseline design by 22.97% and 2.36%, respectively, they offer superior blastworthiness in SEA, with improvements of 98.06% and 47.84%, respectively. Meanwhile, the SH and CH models outperform the baseline design in both \(\delta_{p}\) and SEA by 4.63-11.86% and 13.14-49.37%, respectively. These results indicate that the balanced models yield optimal design points that balance multiple objectives and still surpass the baseline design in at least one objective. The optimized design of ASP from Table 8 was selected by comparing it with other optimized designs. This is carried out to assess the effectiveness of the optimization results. In this study, the optimized design used for comparison is the optimized aluminum foam sandwich panel (AFSP) researched by [PERSON] et al. [52]. The optimized AFSP shares the same dimensions as the ASP with a relative density of 0.6. The material of aluminum foam was modeled as CRUSHABLE_FOAM. The design point from the Pareto front with the same mass as the optimized AFSP was selected as the optimized design for ASP. Figure 16 shows the comparison between the optimized ASP and AFSP. The selected optimized design of each auxetic geometry is marked with a black \"X,\" while the optimized AFSP is marked with a yellow diamond in Figure 15.  
As shown in Figure 15 and Table 8, the optimized ASP for all auxetic geometries outperforms the optimized AFSP in SEA by 133.33-156.86%. However, some auxetic models exhibit worse \(\delta_{p}\), specifically REH and CH, by 16.21% and 0.13%, respectively. In contrast, DAH and SH models produce a better \(\delta_{p}\) by 36.71% and 6.49%, respectively. Overall, the optimized ASP results from the Pareto front, with the same mass as the optimized AFSP, demonstrate superior blastworthiness. Furthermore, the DAH configuration  
Figure 16: Comparison of optimized ASP models: (**a**) REH, (**b**) DAH, (**c**) SH, (**d**) CH; and (**e**) optimized aluminum foam sandwich panel (AFSP) model [52]. in Table 8 shows promising performance compared to the optimized AFSP and other auxetic geometries. Figure 17 shows the comparison of the Pareto front for all auxetic models. For small values of \(\delta_{p}\) or SEA, the DAH exhibits the best blastworthiness performance, whereas the REH shows the poorest performance. The SH slightly outperforms the CH in terms of smaller \(\delta_{p}\) or SEA. For large vales of \(\delta_{p}\) or SEA, the REH and DAH share a similar Pareto front, while the CH performs better than the SH for very large values of \(\delta_{p}\) or SEA. Overall, the DAH structure demonstrates the best blastworthiness performance among the structures, followed by the REH, SH, and CH.