## Methodology
### Generation of Meta-Models Using DOE
#### 3.4.3 Meta-Model Development  
The goal at this stage was to develop efficient and simple mathematical expressions that expressed the von Mises stress in Schwarz primitive lattice domains in terms of \(\{\varepsilon_{Xx},\varepsilon_{Yy},\varepsilon_{ZZ}\}\).  
We used response surface methodologies, specifically central composite face-centered design (CCF), to efficiently devise the meta-models. The shape of the devised meta-models for Schwarz primitive cells of relative densities \(\rho\in\{0.25,0.5,0.75\}\) was  
\[\hat{g}=\left(\beta_{0}+\sum_{i\leq j}\hat{\rho}_{ij}\varepsilon_{ii}\varepsilon _{jj}\right)^{2}, \tag{7}\]  
and the shape of the meta-model for \(\rho=1.0\) was  
\[\hat{g}=\sqrt{\beta_{0}\left(\sum_{i}\varepsilon_{ii}^{2}-\sum_{i<j} \varepsilon_{ii}\varepsilon_{jj}\right)}. \tag{8}\]  
We used R [33] to perform the regression analysis to estimate the coefficients \(\left(\beta_{i},\beta_{ij}\right)\) of the meta-models (see Table 2).  
To evaluate the meta-models, we ran 100 random simulations for each of the four domains and compared the experimental (result of FEA) and predicted (result of the meta-model) von Mises stress. The values of the [PERSON]'s modulus and Poisson's ratio used for the simulations with the homogeneous domains were the ones reported in Table 1. For the cell of density \(\rho=0.75\), we used \(E^{Q}=71.0\) GPa and \(\
u^{Q}=0.25\), which resulted by interpolating the corresponding moduli of the cells of densities \(\rho=0.7\) and \(\rho=0.8\). We used the displacements on the homogeneous domains to calculate the normal strains at the flat faces \(\varepsilon_{ij}\) and used them as inputs for the meta-models in Equations (7) and (8).  
The boundary conditions imposed on the four domains were prescribed displacements in the normal direction of the flat faces of the domains. The size of the cell used was \(L=1.0\) cm, so that the imposed displacements were equivalent to normal strains at the flat faces \((\varepsilon_{ij})\).  
Our analysis was limited to the elastic zone of the material. The range of the variables was \(\varepsilon_{ij}\in 10^{-5}\times[-1.0,1.0]\). The values of the variables were coded in the range \([-1,1]\) to be in concordance with the procedures found in the literature [29]. To ensure that the strains in the flat faces were in the working range \(\left(\varepsilon_{ij}\in 10^{-5}\times[-1.0,1.0]\right)\) and to explore it evenly, the values of the imposed normal strains at the flat faces were generated from a uniform distribution in the interval \((-1.0\times 10^{-5},\ 1.0\times 10^{-5})\).  
Figure 4 displays the aforementioned comparison for each cell. We can see that the meta-models for the densities \(\rho\in\{0.25,0.50,0.75\}\) tend to overestimate the von Mises stress at low stress conditions. This is due to the term \(\beta_{0}\) in Equation (7), which impedes the meta-model to predict small values of von Mises stress. Table 2 gives the average and maximum relative error of the predicted vs. the experimental von Mises stress. The maximum relative errors are associated to low stress conditions, mainly influenced by the value of \(\beta_{0}\) in Equation (7) (as previously discussed). It is clear that the meta-models are not well-suited for low stress conditions. The average and maximum relative errors in the estimations show that this methodology is not applicable in very sensitive processes where high accuracy is required.