## 4 Discussion
### Uniform Thickness (UT) Design  
Table 6 and Figure 10 summarize the results on the performance metrics for the cellular structure with the UT design configuration. To understand the meaning of these findings, Figures 11-13 will be used.  
\begin{table}
\begin{tabular}{l l c c c c c c} \hline \hline \multirow{2}{*}{**ID**} & \multirow{2}{*}{**Type**} & \(\mathbf{T_{1}}\) & \(\mathbf{T_{2}}\) & \(\mathbf{L}\) & \(\mathbf{F_{P}}\) & \(\mathbf{F_{\mu}}\) & \(\mathbf{E_{A}}\) & \(\mathbf{E_{SEA}}\) \\ \cline{3-8}  & & **[mm]** & **[mm]** & **[mm]** & **[kN]** & **[kN]** & **[kJ]** & **[kJ]/kg]** \\ \hline S1 & & 0.2 & 0.2 & 50 & 13 & 7.7 & 0.23 & 8.7 \\ S2 & UT (\(50\times 50\times 50\)) & 0.4 & 0.4 & 50 & 69 & 38 & 1.1 & 21.4 \\ S3 & & 0.6 & 0.6 & 50 & 122 & 92 & 2.7 & 34 \\ S4 & & 0.2 & 0.2 & 100 & 20 & 8 & 0.5 & 9.2 \\ S5 & UT (\(50\times 50\times 100\)) & 0.4 & 0.4 & 100 & 56 & 39 & 2.3 & 22.2 \\ S6 & & 0.6 & 0.6 & 100 & 121 & 95 & 5.6 & 35.8 \\ S7 & & 0.2 & 0.2 & 150 & 16 & 8 & 0.7 & 8.9 \\ S8 & UT (\(50\times 50\times 150\)) & 0.4 & 0.4 & 150 & 64 & 39 & 3.4 & 22.3 \\ S9 & & 0.6 & 0.6 & 150 & 121 & 93 & 8.2 & 35.3 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Uniform thickness (UT) design factors and performance metrics.  
Figure 10: Performance metrics for the UT design. (**a**) Peak and mean forces, (**b**) energy absorption metrics.  
\begin{table}
\begin{tabular}{c c c} \hline \hline _I/t_ & **Reported SEA [46], (MJ/m\({}^{3}\))** & **Simulated Results, (MJ/m\({}^{3}\))** & **[Error] \%** \\ \hline _250_ & 0.30982 & 0.31753 & 2.49 \\ _125_ & 0.77395 & 0.75346 & 2.65 \\ _83.3_ & 1.38624 & 1.33263 & 3.87 \\ _62.5_ & 2.11718 & 2.03104 & 4.07 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of \(E_{SEA}\) between our finite element (FE) model study vs. the reported ones in [46].  
Figure 11: Interaction plots between design parameters and performance metrics for the UT design.  
Figure 12: Mean effect plots for each performance metric with respect to the design factors.  
To better understand the effect of design factors (thicknesses and length) on the responses (performance metrics) interaction plots are used. From Figure 11, it is evident that there are no interaction effects between these design factors. Meaning, these design factors are independent of each other. This is because on interaction plots, parallel lines indicate that there is no interaction effect while different slopes (positive versus negative slopes) suggest that one might be present. This finding allows us to safely interpret the _mean plots_ for each response based on the design factors as shown in Figure 12. From Figure 12, it is evident that (i) the peak and mean forces increase as the thickness \(T_{1}\) and \(T_{2}\) increases. However, these forces are not affected by the length of the structure. The same applies to the specific energy absorption (\(E_{SEA}\)) and (ii) the energy absorption (\(E_{A}\)) is affected by all three factors, including length. This is obvious since there is more deformation distance before densification is reached. From these performance metrics, the mean force and \(E_{SEA}\) are most crucial to be looked at. Hence, it can be concluded that the length of the structure is not a crucial factor to be considered. This will have an implication on the design space and manufacturing requirements. Figure 13 depicts a typical force-displacement graph along with the deformation mode for these structures. These diagrams are for \(T_{1}=T_{2}=0.6\) mm. It can be seen that the deformation mode is stable indicating good crash absorber characteristics. This figure also depicts that the average force and peak force is almost similar for all lengths, indicating once again that the length is not a crucial factor.