## 4 Results and Discussion
### Optimization Results for Four Degrees of Freedom
The increase in \(NP\) and \(NG\) led to a decrease in MSE \(S_{1}/H_{0}\) in a similar form that happened for MSE \(\alpha\), Figure 12. In other words, the MSE \(S_{1}/H_{0}\) had a strong decrease with \(NP\) and \(NG\) for the versions DE1 and DE3, Figure 14b-d, with more difficulties to reduce the MSE \(S_{1}/H_{0}\) for \(H_{0}/L_{0}\) < 14.6. The performance of DE2 was not strongly influenced by variation of \(NP\) and \(NG\), leading to poor performance, regardless of the magnitude of \(NP\) and \(NG\) used. It is worth mentioning that the DE1 version shows zero error for all ratios of \(H_{0}/L_{0}\) investigated, i.e., the algorithm reaches the optimal values of \(S_{1}/H_{0}\) in all runs. The error to achieve the three times minimized dimensionless maximum excess of temperature, \((\theta_{max})_{3m}\), generated by each DE algorithm version for different ratios of \(H_{0}/L_{0}\) and magnitudes of \(NP\) and \(NG\) is shown in Figure 15. In Figure 15a, the MSE of \((\theta_{max})_{3m}\) for \(NP\) = 10 and \(NG\) = 50 can be seen, showing the highest magnitude errors for the performance indicator, regardless of the DE version used. For the next population size, \(NP\) = 20, and total generation of \(NG\) = 100, Figure 15b, DE1 version conducted to the minimum error for all ratios of \(H_{0}/L_{0}\), which is a strong indicator that this algorithm is able to reproduce the effect of \(H_{0}/L_{0}\) over \((\theta_{max})_{3m}\) and corresponding optimal configurations, \((S_{1}/H_{0})_{0}\), \((\alpha)_{20}\), and \((\beta)_{30}\), since the search is performed for fixed magnitudes of \(H_{0}/L_{0}\) (the curve it is desired to reproduce). For the DE3 version, a similar prediction can be found, but more computational effort is required, since the lowest magnitudes of MSE \((\theta_{max})_{3m}\) were obtained with higher values of \(NP\) and \(NG\) in comparison with DE1. For the DE2 version, the increase in the \(NP\times NG\) parameters implies a significant reduction in MSE \((\theta_{max})_{3m}\), but still presents the highest errors among the analyzed algorithm, which is an indicative of the prediction of local points of minimum and local optimal configurations. Figure 14: Effect of \(H_{0}/L_{0}\) over mean square error (MSE) of \(S_{1}/H_{0}\) for different DE versions and combinations of \(NP\times NG\): (**a**) \(NP\) = 10, \(NG\) = 50; (**b**) \(NP\) = 20, \(NG\) = 100; (**c**) \(NP\) = 30, \(NG\) = 100; (**d**) \(NP\) = 40, \(NG\) = 112. Figure 16 shows the effect of the ratio \(H_{0}/L_{0}\) over \((\theta_{max})_{3m}\) considering the overall magnitude, represented by the point, and the inferior and superior limits established by the standard deviation, represented by the bars. The effects of \(H_{0}/L_{0}\) on the thermal performance reproduced by different versions of DE and combinations of \(NP\) and \(NG\) parameters are analyzed. The first experiment with \(NP=10\) and \(NG=50\), shown in Figure 16a, presents the highest average and standard deviation values of \((\theta_{max})_{3m}\) for all DE algorithm versions and ratios of \(H_{0}/L_{0}\). These \(NP\) and \(NG\) parameters are insufficient to reproduce the effect of \(H_{0}/L_{0}\) over \((\theta_{max})_{3m}\) for any version of the DE algorithm analyzed. For the next experiment, Figure 16b, with \(NP=20\) and \(NP=100\), it is possible to note an improvement in the reproduction of the effect of \(H_{0}/L_{0}\) over \((\theta_{max})_{3m}\), mainly for DE1 where low magnitudes of standard deviation are noticed. The DE3 version also has adequate performance, but some magnitudes for overall \((\theta_{max})_{3m}\) and standard deviations are higher than those reached with DE1. The performance of the DE versions was improved with the increase in \(NP\) and \(NG\) up to the configuration with \(NP=40\) and \(NG=112\), where the standard deviation for DE1 was null and the global optimal configuration is found for each magnitude of \(H_{0}/L_{0}\), reproducing in a most adequate form the effect of \(H_{0}/L_{0}\) over  
\((\theta_{max})_{3m}\).