## 2 Materials and Methods
### Validation of the Numerical Model  
To further validate the robustness of the simulation and assess the accuracy of the homogenized cell material properties derived from the reference paper [13], a physical compression experiment was conducted on a battery pack. This pack consisted of nine lithium-ion cells, identical in size and configuration to those used in the referenced study [13], where 18650 Li-ion batteries from a Tesla Model S were subjected to compression tests to validate the simplified finite element method (FEM) models. The study focused on analyzing the force-deformation curve to ensure accurate predictions of mechanical behavior. To isolate the core structural response of the jelly roll together with the case, the battery's upper and lower caps were precisely removed. The results emphasized the interplay between the steel casing and the jelly roll components, providing critical insights for optimizing battery safety, performance, and design. The cells were arranged in a 3 \(\times\) 3 matrix without and with printed PLA spacers between them and printed on the CAD shown in Figure 4c. The experimentally obtained force-displacement curves were compared against the simulated results to verify the model's predictive accuracy. As illustrated in Figure 5, the experiment was designed to replicate the boundary conditions applied in the simulation to ensure consistency in comparison. The setup included a rigid base and side walls, with a movable upper plate acting as a compression press. The experiment was carried out using an Instron 8801 (Instron, Norwood, MA, USA) servo hydraulic testing machine, which provides precise displacement control and force measurement capabilities. The machine was operated at a 2 mm/min to ensure quasi-static conditions, consistent with the simulation parameters. Due to the limited number of experiments and the fact that only force and displacement were measured, the machine's measurement errors were approximately \(\pm\)1 kN for force and \(\pm\)0.001 mm for displacement. The battery pack was positioned on the rigid base within the testing apparatus, ensuring proper alignment with the movable upper plate. During the test, the compression force was applied vertically, and the force-displacement data were recorded to capture the mechanical response of the pack under loading. As illustrated in Figure 6, in the case of the pack without the spacers, the close alignment between the experimental and simulation force-displacement curves demonstrates the robustness of the homogenized cell material model when applied to a pack of cells. Key critical stiffness points, including the end of the elastic region at 0.5 mm and the onset of densification at 8 mm, show near-perfect agreement with a zero error margin. Furthermore, the force-displacement response observed between these points is consistently maintained \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Spacer** \\ **Configuration** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Aluminum Spacer** \\ **Mass** \\ **[g]** \\ \end{tabular} } & \begin{tabular}{c} **Aluminum Spacer** \\ **Proportion to Pack of** \\ **50 g Cells** \\ **[\%]** \\ \end{tabular} & \multirow{2}{*}{\
\begin{tabular}{c} **PLA Spacer Total** \\ **Mass [g]** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **PLA Spacer** \\ **Proportion to Pack** \\ **of 50 g Cells [\%]** \\ \end{tabular} } \\ \hline V1 & 130 & 26 & 42 & 8.4 \\ V2 & 134 & 26.8 & 3 & 8.6 \\ V3 & 84 & 18.7 & 27 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mass added by the spacers. in both the experimental and simulated results, underscoring the reliability of the model for representing the mechanical behavior of the battery pack under compression. When comparing the pack with the printed PLA spacers to the pack without spacers in terms of load-displacement behavior, it is evident that the inclusion of spacers results in increased stiffness, as predicted by the simulations.