# Analysis of a Preliminary Design Approach for Conformal Lattice Structures
## 3 Results and Discussions
### Structural Size Optimization  
Figure 6 shows the size-optimized lattice structures. In both the traction and compression load cases, the optimization algorithm converged after the first iteration for the SC, BCC, and WG lattice structures, whereas it required 20 iterations for the DA structure. Figure 5: Conformal wireframe obtained by the NURBS-FFD modeling method. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Lattice** & \multirow{2}{*}{\begin{tabular}{c} **Number of** \\ **Nodes** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Number of** \\ **Beams** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Total Beams** \\ **Length [m]** \\ \end{tabular} } & \multirow{2}{*}{**Maxwell’s Rule**} \\
**Type** & & & & \\ \hline SC & 880 & 2141 & 7.86 & Bending-dominated \\ BCC & 1480 & 6937 & 23.27 & Stretching-dominated \\ WG & 3841 & 18,124 & 48.22 & Stretching-dominated \\ DA & 623 & 1078 & 3.96 & Bending-dominated \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Geometric information of the wireframes and Maxwell law results. The increase in the number of iterations given by the DA structure is due to the diameter variation mechanism of the algorithm, which calculates the optimized diameter based on Hooke's equation derived for the tensional stress of a uniform bar. The stretching-dominated structures size optimization process converges immediately, as the characteristic feature of these structures is the predisposition for axial loads. Consequently, the large number of iterations required for the DA is due to the algorithm that tends to increase the size of certain beams without increasing the diameters of the surrounding ones, leading to divergence, especially in the case of bending-dominated structures, in line with previous studies [56; 76]. The bar chart in Figure 7 shows that the median value of the diameters of all the configurations equals the lower diameter constraint, i.e., 0.5 mm for the SC, BCC, and WG lattices and 0.95 mm for the DA lattice; this is explained by the presence of several oversized beams that, according to the size optimization, should have a diameter smaller than the minimum allowable size, but in the final configuration present the minimum cross-section to guarantee the printability of the part. The mean value of the diameter of the SC structure is higher than the BCC and WG ones due to the lower number of beams not aligned to the load direction. The DA lattice structure is the only one that requires beams with a cross-section as high as the upper manufacturing constraint, i.e., 4.75 mm. Figure 6: Optimized lattice structures for the compression load case inside Rhinoceros 7 design environment. Figure 7: Overall results from the size optimization of the four lattice structures: maximum, average, and median diameter for traction and the compression load cases. Figure 8 shows the values of the slenderness ratio and the relative density of the structures. The relative density is calculated through the Karamba3D plug-in, which does not consider the nodal connection between the beams, overestimating the value. The presence of a high number of oversized beams (beams with the minimum manufacturable diameter, i.e., 0.5 mm, even though the optimized size would be lower) affects the relative density of the structures with a stretching-dominated behavior, namely the SC, the BCC, and WG, which increases according to the total number of elements. Moreover, due to [PERSON]'s criterion, the BCC and WG structures are over-constrained, whereas the SC and DA structures are under-constrained; this reflects on the different ways the lattices react when changing the load case. Moving from the traction to the compression load case, the relative density of the BCC and WG increases by about 13%, whereas the relative density of the SC and DA increases by about 24%.