## 4 Results
### Validation of the Proposed Methodology
#### 4.1.1 Material Homogenization in Schwarz Primitive Lattice Structures  
The FEA simulations of the lattice and homogeneous domains were executed in ANSYS. The lattice models were meshed in ANSYS using tetrahedral elements (SOLID285). The material properties of the lattice models correspond to the bulk material properties (\(\rho=1.0\) in Table 1). On the other hand, given the regular shape of the homogeneous domains, we used cubic elements (SOLID185) for the respective meshes. Each cubic sub-domain was isotropic and its material properties were assigned according to its relative density and the properties reported in Table 1. The homogeneous domain was then a regular 3D array of isotropic cubic sub-domains. Since the properties of each sub-domain could be different, the homogeneous domain resulted to be anisotropic. Figure 5: Studied domains and boundary conditions for the FEA simulations. Domains of eight cells of uniform density (**a**) \(\rho=0.25\); (**b**) \(\rho=0.50\); and (**c**) \(\rho=0.75\). Domains of eight cells of density (**d**) per Equation (9); and (**e**) per Equation (10). (**f**) Domain of 24 cells with density per [7]. (**g**) Domain of 112 cells with density per Equation (12). FEA simulations were executed using different hardware settings and different operative systems; therefore, it was not possible to compare execution times between different simulations in equal conditions. To overcome this difficulty, we used the number of elements in the mesh as a measurement of computational expense in each domain. The numbers of elements required for each lattice and each homogeneous domain are reported in Figure 6 and Table 3. It is noticeable that the FEA meshes of lattice domains required more elements than the homogeneous domains. Another important aspect to highlight is that the number of elements of the homogeneous domain only depend on the number of cells (particularly, we chose a mesh of \(10\times 10\times 10\) elements per unit cell). On the other hand, the number of elements for the lattice domains does not completely depend on the number of cells. Notice that five out of the six domains are conformed by eight unit cells. However, the number of elements (Figure 6 and Table 3) is different for each domain. These variations are mainly influenced by the shape of the domain, which affects the corresponding meshing algorithms of the FEA software (ANSYS). After conducting the FEA simulations, we proceeded to compare the resultant displacements of the lattice and homogeneous domains. Figures 7 and 8 (1) contrast the nodal displacement in the load direction \(X\) for the six load cases and (2) show the absolute difference between the \(X\) displacement predicted by the lattice and homogeneous approaches. The reader may observe the similarity in both the distribution and magnitude of the displacements field of the lattice and homogeneous domains. Figures 7 and 8 also show that the maximum value of the absolute error is in all cases approximately 10 times smaller than the maximum  
we used the meta-models presented in Section 3.4 to estimate the maximum von Mises stress in each cell. To apply the meta-models, we calculated the average relative density of each cell of the non-uniform (graded) density domains. Since we had only meta-models for the relative densities \(\rho\in\{0.25,0.50,0.75,1.0\}\), we used linear interpolation to do the approximations for the intermediate values of density \(\rho\). For instance, \(\left(\sigma_{VM}^{(0.4)}\right)\), the von Mises stress for a relative density \(\rho=0.4\) is approximated as  
\[\sigma_{VM}^{(0.4)}=\frac{2}{5}\sigma_{VM}^{(0.25)}+\frac{3}{5}\sigma_{VM}^{(0.5)}, \tag{11}\]  
where \(\sigma_{VM}^{(0.25)}\) and \(\sigma_{VM}^{(0.50)}\) denote the von Mises stresses for the cells of densities \(\rho=0.25\) and \(\rho=0.50\).