## 4 Results and Discussion
### Validation Using a FEA  
The models that were obtained with the three methods: (a) using a fixed Bezier curve degree (starting with a degree of 4 with increments of 2 up to a degree of 12) for all of the graphite nodules contours; (b) using the proposed methodology; and, (c) using a commercial software (img2 CAD), were imported into WORKBENCH from ANSYS(r) and an extrusion of 10 um was applied with the aim to perform a FEA in this platform. The geometric models were composed of two phases: graphite nodules and a ferritic matrix. The behavior of both these phases, the ferritic matrix and the graphite, was considered as isotropic. The values of the Young's modulus of the graphite and the ferritic matrix used were 15 GPa and 235 GPa, respectively, and the Poisson ratio of 0.28 for both of the phases [16]. The geometric models were discretized by means of solid tetrahedral elements, which are the basic three-dimensional elements that are used by the abovementioned software, producing a total number of 88658 elements. The models were subjected to an applied loading of 50 mN, which is distributed to the top surface in the positive y-direction with a clamped bottom surface. Based on the obtained results, the concentration of stresses, that is, the maximum stresses of Von-Mises are obtained in the vicinity of the graphite nodules. The model that is obtained with the proposed methodology is shown in Figure 8a. Table 2 shows the maximum stresses and relative errors that are obtained by the FEA for the models developed with the three abovementioned methods. The relative errors (\(E_{\text{s}}\)) of the stresses were calculated by the following equation:  
\[E_{\text{s}}=\frac{\sigma_{\text{ref}}-\sigma_{\text{e}}}{\sigma_{\text{ref} }}\times 100\% \tag{4}\]  
where, \(\sigma_{\text{ref}}\), is the maximum stress obtained with the proposed methodology, whose value is 64 MPa (Figure 8b), and was taking as reference since it is desired to estimate the error (variation)in the stresses obtained by the other methods in relation to the proposed methodology, and \(\sigma_{\text{e}}\) is the stress computed from other methods.  
Based on Table 2, the relative error that is obtained with the commercial software model (CSM) was \(-35.9\%\). The negative value means that the stress that is produced for the CSM is greater than the reference, as the curves of the graphite nodules contours that are obtained by the CSM have characteristics that reproduce the geometric errors generated in the discretization process. On the other hand, for the models that used a fixed Bezier curve degree, errors of \(25\%\) and up to \(-34.4\%\) were obtained for the Bezier curve degrees of \(4\) and \(12\), respectively. In particular, for the model that used a Bezier curve degree of \(4\), the positive relative error is due to the maximum stress was smaller than the reference since the curves of the graphite nodules contours have values of \(CSF_{\text{est}}\) near to the unit, which represents curves with an almost spherical geometry. Therefore, based on the obtained results it is confirmed that the discretization process generates significant errors in the estimation of the stresses produced when a FEA is applied for the micro-scale model of a DCI microstructure, therefore the errors generated in the discretization process should not be ignored when developing geometric models.