# Analysis of a Preliminary Design Approach for Conformal Lattice Structures
## 2 Materials and Methods
### Structural Size Optimization  
This work refers to the method firstly implemented by [PERSON] et al. [56] using a Phyton script and exploiting the Karmaba3D libraries [57], a Grasshopper plug-in for performing FEAs, and it is based on the utilization factor described in the Eurocode 3 [58]. The utilization factor considers the axial force, biaxial bending, torsion, shear force, and buckling load in compression. As an input, the method requires the wireframe model, the material properties, the boundary conditions, and the size constraints. Finally, the iterative FEA is run, varying the beam size to obtain the desired utilization for each element inside a predefined range [54].  
Two different load cases are applied, a traction load case of 7.5 kN for comparing the result with the previous work of [PERSON] et al. [54] and a compression load case of \(-\)10 kN for evaluating the structure's stability and buckling effects. The load is evenly applied to the nodes of the beams connected to the big rod's end (Table 1). All the displacements and rotations of the beams nodes connected to the small rod's end are locked; the displacements of the beams nodes connected to the big rod's end are locked along the X- and Y-direction, while they can move along the Z-direction and rotate around all the axes (Figure 4). Even if the boundary conditions do not perfectly represent the actual conditions, they may be sufficient to test and validate the method on different configurations. Due to the impossibility of producing cellular structures with traditional manufacturing technologies, the piston rod is supposed to be produced by AM; in particular, the SLM process, which belongs to the powder bed fusion category [59], is chosen due to its consolidated capability in producing almost fully dense lattices and, more in general, components characterized by mechanical properties similar to the parts obtained by conventional technologies [60; 61], and due to its diffusion in the industry [62]. The selected AM machine is the Renishaw AM 400, and the material is the aluminum AlSi10 Mg-0403 powder produced by Renishaw [63]. The mechanical proprieties declared by the producer are summarized in Table 2. The material was assumed as a linear elastic isotropic material without considering the influence of the printing direction, and the deformations are assumed to be small. The utilization factor for the optimization is set to (90 \(\pm\) 1)% of the yield strength. The beam cross-section is circular. The size constraints were defined to avoid interference between the beams, to avoid powder evacuation problems, and considering the manufacturing size constraints given by the SLM technology, especially the minimum manufacturable diameter of beams [64]: the allowed beam diameter ranges from 0.5 to 1.5 mm for the SC, BCC, and WG structures, whereas, due to the bigger cell dimension of the DA lattice, the diameter ranges from 0.95 to 4.75 mm.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Lattice Structure** & **Number of** & **Traction Load** & **Compression Load** \\
**Type** & **Loaded Nodes** & **(for Each Node) [N]** & **(for Each Node) [N]** \\ \hline SC & 55 & 136.36 & \(-\)181.82 \\ BCC & 55 & 136.36 & \(-\)181.82 \\ WG & 94 & 79.79 & \(-\)106.38 \\ DA & 20 & 375.00 & \(-\)500.00 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Load information for each lattice structure.