## 4 Core Topology Optimisation
\begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**ID**} & **Amplitude \(A\)** & **Frequency \(f\)** & **Frequency \(\omega\)** \\  & **[mm]** & & **[mm]** \\ \hline Reference & 0 & – & – \\ A1-f2 & 1 & 2 & 50.27 \\ A1-f4 & 1 & 4 & 25.13 \\ A1-f8 & 1 & 8 & 12.57 \\ A2-f2 & 2 & 2 & 50.27 \\ A2-f4 & 2 & 4 & 25.13 \\ A2-f8 & 2 & 8 & 12.57 \\ A4-f2 & 4 & 2 & 50.27 \\ A4-f4 & 4 & 4 & 25.13 \\ A4-f8 & 4 & 8 & 12.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Geometric parameters (amplitude and frequency) for the core optimisation analysis (the reference case corresponds to the flat core). Figure 10: Sandwich panel with sinusoidal core and generic dimensions. node shows a velocity peak for all amplitudes, which is higher than the reference case. Although the trends are similar in all cases, a slight increase of the velocity with the amplitude can be seen. This is explained because the sinusoidal shape delays the contact between the core and the rear skin, allowing for the front skin to move with higher velocity. It also increases the time of arrival of the impact to the rear skin. These differences delay the stationary velocity stage in which both skins have the same velocity and, thus, the core and the front skin are allowed to deflect more when compared to the reference case. The results presented in Figure 10(b), on the other hand, show the kinematics of the plates as a function of the frequency of the core shape for a constant amplitude \(A=1\) mm. The kinematic behaviour is similar to the previous cases, and the peak velocity of the front skin reaches similar order of magnitude values. However, the increase of the geometrical frequency reduces the time of arrival of the impact to the rear skin, albeit with similar accelerations. As the frequency increases, the free distance for the core to move diminishes, allowing for the rear skin to initiate its motion sooner. The changes in kinematic and mechanical behaviour described above lead to differences in the energy absorption mechanisms within each sandwich plate. Figure 12 shows the total absorbed energy for each combination of geometrical frequency and amplitude. The results presented in Figure 11(a) show a clear increase in the energy absorption with the amplitude, visible for the whole range of analysed frequencies. Additionally, the energy absorbed by the sandwich plate is consistently higher than in the reference case. These observations are supported by the differences on velocities shown in Figure 11. Both energy absorption mechanisms--cork compression and overall bending and stretching--contribute to this. An inverse trend can be observed in Figure 11(b), where the absorbed energy decreases with the increase of the geometrical frequency, regardless of the amplitude. The rear skin starts to move sooner, as the frequency increases, and therefore Stage III initiates earlier in the process, while cork compression ends prematurely, as discussed above. Figure 11: Sandwich plate kinematics for different core shape parameters: (**a**) effect of the sinusoidal core amplitude for a constant geometrical frequency \(f=4\), and (**b**) effect of the geometrical frequency for a constant amplitude \(A=1\) mm. These observed inverse trends in amplitude and frequency, as shown in Figure 13, allow for an optimal case to be identified, where the energy absorption is maximised. This optimal case is the one with higher amplitude (\(A=4\) mm) and lower geometrical frequency (\(f=2\)). For this particular case, the energy absorbed by each individual sandwich component are listed in Table 8, together with the reference case. The overall performance of the sandwich plate improves by close to 40% in terms of energy absorption, and the skins and core are able to absorb more energy than in the reference case, as discussed above. The increase in absorbed energy is substantial in both skins. This can be explained by the higher velocity of the front skin and the cork core, which increase the bending effect, as can be seen from the corresponding maximum displacement.