## 3 Results
### FEA Optimization Results  
SEA values resulting from the optimization runs are summarized in Table 5; their thickness distributions are displayed in Figure 15. For the Simple Cubic lattice, the thickness value curve of the linear law approach shows an inflexion point between the T3 and T4 regions, situating the thickest part of the lattice in its upper half. The thickness values do not deviate strongly from the initial distribution, varying from 1.4 mm at its thickest part to 1.2 mm at its thinnest. The quadratic law approach shows a much larger spread, with a thickness of 1.05 mm in the first region, 1.6 mm in the middle, and 0.82 mm in the last region. Given its definition, the power law approach shows a symmetric thickness distribution, with the thinnest regions situated in the middle section of the lattice. Here, the outer regions' diameters increase to 1.32 mm and reduce to 1.26 mm in the middle section. They remain 3.4% below the mass limit, while the other approaches make full use of it. All three approaches attain significant SEA value improvement predictions, with the linear law approach reaching a 29% increase over the uniform lattice. On the other hand, the same thickness optimization approaches yield only marginal changes in the Isotruss lattice structure. The quadratic law approach allows for a 5.1% increase in SEA, while the power law approach yields a slightly negative result. The reaction force evolutions of each lattice type and optimisation strategy plotted versus the displacements of the compression plate are plotted in Figure 16. It appears that the maximal response force levels for the Simple Cubic lattice structure reach values that are comparable to those of the uniform base model. Only the quadratic thickness distribution approach model peaks at a much lower level. The increase in energy absorption is caused by a considerably greater mean reaction force level in the plateau phase for all thickness optimisation techniques. It becomes visible when comparing the plateau regions of the linear and quadratic law approaches that the linear law approach gains the majority of the absorbed energy in the first half, up to a displacement of 12.5 mm, while the quadratic law approach gains it in the second. All of the Isotruss models' reaction force curves peak in the linear elastic phase at about 3000 N to 3250 N of force. The force levels of the thickness optimisation methods \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Simple Cubic Lattice**} & \multicolumn{3}{c}{**Isotruss Lattice**} \\ \cline{2-6}  & **Absorbed Energy (J)** & **Weight (g)** & **SEA (J/kg)** & **Absorbed Energy (J)** & **Weight (g)** & **SEA (J/kg)** \\ \hline Sample 1 & 4.6 & 5.9 & 777 & 56.2 & 20.0 & 2809 \\ \hline Sample 2 & 5.2 & 6.0 & 866 & 58.2 & 18.7 & 3114 \\ \hline Sample 3 & 4.6 & 5.7 & 810 & 42.3 * & 20.3 & 2084 \\ \hline Avg. samples & 4.8 & 5.9 & 818 & 52.2 & 19.7 & 2669 \\ \hline Simulated & 6.3 & 6.74 & 940 & 61.8 & 22.22 & 2780 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Total absorbed energy and specific energy absorption (SEA) values of the Simple Cubic and Isotruss lattice structures with uniform beam diameters. Sample 3 of the Isotruss model is considered an outlier and marked by a star (*). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Grading Approach**} & \multicolumn{3}{c}{**Simple Cubic Lattice**} & \multicolumn{3}{c}{**Isotruss Lattice**} \\ \cline{2-7}  & **Absorbed Energy (J)** & **Weight (g)** & **SEA (J/kg)** & **Absorbed Energy (J)** & **Weight (g)** & **SEA (J/kg)** \\ \hline Baseline value & 6.3 & 6.74 g & 940 & 61.8 & 22.22 g & 2780 \\ \hline Linear law & 8.1 & 6.71 g & 1213 (+29.0\%) & 62.6 & 22.20 g & 2819 (+1.4\%) \\ \hline Quadratic law & 8.2 & 6.74 g & 1210 (+28.7\%) & 65.1 & 22.27 g & 2921 (+5.1\%) \\ \hline Power law & 7.2 & 6.51 g & 1105 (+17.5\%) & 61.7 & 22.23 g & 2775 (\(-0.2\%\)) \\ \hline \hline \end{tabular}
\end{table}