## 3 Results and Discussion
### Mechanical Characterization of Cellular Structures and Insoles
#### 3.1.2 Deformation and Stress Analysis via FEM of Insoles  
The comparison of the displacement and Von Mises stress results of the solid insole, with those of the BCC and TPMS cellular structures with different relative densities, are presented in Table 4.  
Figure 10 shows the local deformation of the cellular structure within the insole when applying the boundary conditions described in Section 2.3. Note in Figure 10 that the displacement of the BCC structures within the insole was higher than the TPMS structures; this result was related to the information presented in Figure 8, where it was shown that the BCC structure is more flexible than the TPMS structure. Figure 10a-c show the displacement in the BCC structures with a relative density of 10.33%, 14.67%, and 20.19%, respectively. Bending is observed in the unit-cell struts and a slight displacement of the unit cells to the right. Figure 10d-f show the displacement in the TPMS structures with a relative density of 10.33%, 14.67%, and 20.19%, respectively. It can be seen that the unit cell tends to shorten as the circular feature of the TPMS unit cell is deformed (losing its circular form). The displacement results are reported in Table 4. Note that the highest displacement in the insole resulted in the upper surface in all insole models. The upper surface of the insole deformed mostly over the voids formed by the cellular structure; this deformation was related to the geometry of the unit cell and not to the stiffness of the cellular structure. For example, the BCC structure has its struts at 45\({}^{\circ}\), forming a smaller gap between the unit cells compared to the TPMS structures; due to this, the maximum insole displacement is lower in the BCC structure than in the TPMS. Despite that, the BCC structure is still more flexible. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Topology** & **Relative Density (\%)** & **Displacement (mm)** & **Von Mises Stress (MPa)** \\ \hline BCC & 10.33 & 2.06 & 22.17 \\ BCC & 14.67 & 1.61 & 13.08 \\ BCC & 20.19 & 1.29 & 11.99 \\ TPMS & 10.33 & 2.7 & 23.84 \\ TPMS & 14.67 & 1.77 & 7.99 \\ TPMS & 20.19 & 1.44 & 14.22 \\ SOLID & 100 & 0.04 & 0.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Displacement and Von Mises stress resulting from simulation for the insole with BCC and TPMS topologies and different relative densities. Figure 10: Static simulation displacement results in cellular structures in the insole: (**a**) insole with BCC cellular structure, 10.33%; (**b**) insole with BCC cellular structure, 14.67%; (**c**) insole with BCC cellular structure, 20.19%; (**d**) insole with TMPS cellular structure, 10.33%; (**e**) insole with TPMS cellular structure, 14.67%; and (**f**) insole with TPMS cellular structure, 20.19%. Figure 11a-c show maximum displacement in the hallux and medial forefoot areas. The upper surface of the insole was deformed in the gaps generated by the distribution of cellular structure in the internal part of the insole. The upper surface of the insole was in contact with the struts of the BCC structure when the insole was deformed. When these parts came into contact, the struts prevented further surface displacement. Figure 11d-f show maximum displacement in the medial midfoot and medial rearfoot areas. When the upper surface of the insole was deformed, contact of the deformed area with the walls of the TPMS unit cells occurred; therefore, even where the TPMS cellular had the highest compression stiffness, the insole with TPMS cellular structure exhibited larger displacements due to the geometry of the unit cell. Figure 11g shows that the maximum displacement occurred on the solid insole in the medial forefoot and medial rearfoot areas. The maximum Von Mises stress in the insole with BCC cellular structure occurred in the medial forefoot (see Figure 12a-c), the stress was concentrated in the node of the unit cell, and the union of the top surface of the insole with the unit cell.