## 2 The Numerical Tool Development
### Mathematical Models: Development of the Numerical Model's Logic
#### 2.2.2 The Cure Shrinkage of the Resin
Our calculations show that it is possible to define a linear dependence as A = 0.02, a concave dependence as A = 0.001, or a convex dependence as A = 0.04. Based on [31], we considered that the dependence between the cure shrinkage and the degree of cure is concave, as presented in Figure 12.  
The Boghetti-Gillespie model recommends a more conservative value of 2% less than the experiments in Figure 13, which presents the effect of variation in parameter A. In our simulation, we considered A = 0.04, corresponding to a concave dependence shape. The Boghetti-Gillespie model also offers the possibility of using the \(\alpha c1\) and \(\alpha c2\) parameters to insert a delay phase at the start and the end of the shrinking process. This is especially important for the calibration phase of the model, as two important observations indicate:  
Figure 12: The shape of variations in resin volumetric shrinkage with cure [31]. Figure 13: Cure shrinkage as a function of the degree of cure for a parallel plate geometry [32]. 1. The shrinkage does not start at a degree of cure equal to 0, but later in the polymerization process, at a higher parameter value;
2. The shrinkage does not end at a degree of cure equal to 1, but earlier in the process for a smaller parameter value. The data presented in Figure 13 and based on the experiments from [32] confirm this assumption. In the White-Hahn model implemented in Marc, the \"B\" parameter was defined to tune the resin shrinkage, like the \"A\" parameter in the Bogetti-Gillespie implementation. Applying values presented in Table 5, the variations in resin volumetric shrinkage vs. cure degree based on the White-Hahn equation are shown in Figure 14.  
In the Bogetti-Gillespie model implemented in MARC, the \"A\" parameter was defined to tune the resin shrinkage versus the degree of cure dependence over the experimental data. Our calculations show that it is possible to define a linear dependence as A = 0.02, a concave dependence as A = 0.001, or a convex dependence as A = 0.04. Based on reference [31], we considered that the dependence between the cure shrinkage and the degree of cure is concave, as shown in the blue plot in Figure 11:0  
Figure 11:1 Variations in resin volumetric shrinkage with cure based on the White–Hahn equation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Units** & **Value** \\ \hline \(\alpha_{C}\) & - & 1 \\ \hline \(V_{r}^{S\infty}\) & - & 0.02 (the same value as above) \\ \hline \(B\) & - & 1 \\ \cline{2-3}  & & 2 \\ \cline{2-3}  & & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Starting values for the White–Hahn cure shrinkage model. In the White-Hahn model implemented in MARC, the \"B\" parameter tunes the resin shrinkage, like the \"A\" parameter in the Bogetti-Gillespie implementation. Figure 15 plots the variations in resin volumetric shrinkage vs. cure degree based on the White-Hahn equation. Based on these data, we decided to use the Bogetti-Gillespie mathematical model for the following reasons:  
* The model mentioned above can simulate concave and convex shapes of the variation between shrinkage and degree of cure. Based on experimental data, changing the theoretical \"A\" parameter to tune the model is easy. We chose a value that corresponds to a convex variation;
* The Bogetti-Gillespie model is more complex than the White-Hahn model because, by using the \(\alpha_{\text{c1}}\) and \(\alpha_{\text{c2}}\) parameters, it is possible to obtain a mathematical model closer to an actual experiment;
* Finally, the Bogetti-Gillespie model can be tuned closer to experimental trials for calibration.