## 5 Conclusions  
In this work, a methodology that is capable of generating microscale-models of a DCI microstructure is proposed. The models that are generated are characterized by the smoothness in their curves with the objective of avoiding or mitigating the discretization errors produced when the graphite nodules contours are obtained using image processing, since the discretization process by which the graphite nodules contours are obtained, generates significant errors up to \(35.9\%\) in the estimation of the stresses that are produced in a simulation by means of a FEA.  
The microscale-models were generated using Bezier curves, where the selection of the degree of the approximated curves for each graphite nodule contour was optimized by means of a genetic  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Geometric Models** & **Equivalent Stress (MPa)** & **Relative Error (\%)** \\ \hline Proposed methodology (GA) & \(64\) & Ref. \\ Commercial software (SC) & \(87\) & \(-35.9\) \\ Fixed & \(4\) & \(48\) & \(25.0\) \\ Bézier & \(6\) & \(55\) & \(14.1\) \\ curve & \(8\) & \(56\) & \(12.5\) \\ degree & \(10\) & \(68\) & \(-6.3\) \\
12 & \(86\) & \(-34.4\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Stresses and relative errors obtained by the Finite Element Analysis (FEA) for the different methods used.  
Figure 8: (a) Micro-scale model obtained by the proposed methodology from a ductile Cast Irons (DCI) micrography (Figure 6a); (b) Results from the Finite Element Analysis of the model in Figure 8a.  
algorithm-based strategy. The selection of the optimal degree was based on the parameters \(CSF_{\text{dig}}\) digitally obtained, the estimated \(CSF_{\text{est}}\) for each curve obtained with the genetic algorithm, and the coordinates of the graphite nodules contours. This advantage is possible to achieve since the GA is capable of deciding which Bezier curve degree is the appropriate according to the characteristics of each nodule.  
The commercial software models reproduce the geometric errors that are obtained in the discretizacion process; this effect can be seen in the obtained values of \(E_{\text{r}}\), as they have the lowest values compared with the ones obtained using the proposed methodologies. Therefore, the obtained models using the proposed methodology have the advantage, in comparison to the commercial software, that realistic geometries are estimated, thus allows for a reduction of the geometric errors that are generated in the discretization process.  
When the graphite nodules are modeled with a very small Bezier curve degree, the generated curves have high \(CSF_{\text{est}}\) values; which, in consequence, produce stresses of low magnitudes. In contrast, as the Bezier curve degree increases, curves are approximated to the contours that are obtained at the digital level, and therefore their \(CSF_{\text{est}}\) decrease noticeably, causing stress concentrators with increased magnitudes of the maximum stresses.  
The first author would like to thank CONACYT for the scholarship (CVU: 419770) given. This publication was funded with CONACYT-PI 243152.  
All the authors contributed equally to the present work. [PERSON] performed the analysis of the simulation results, designed genetic algorithm method, analyzed data and wrote the paper; [PERSON] analyzed data and edited the paper; [PERSON] designed genetic algorithm method and analyzed data; [PERSON] performed simulation experiments and collected data; [PERSON] designed image processing algorithms, gave technical support and conceptual advice. All authors discussed the results and implications and commented on the manuscript at all stages.  
The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.