## 4 Results and Discussion
### Optimization Results for Three Degrees of Freedom  
Here, it is investigated the optimization of three degrees of freedom, more precisely the ratio \(S_{1}/H_{0}\), and the angles \(\alpha\) and \(\beta\). In this investigation, 30 runs are also performed for each version of the DE algorithm. For the height of the inferior branch of the double Y-shaped cavity, represented by rate \(S_{1}/H_{0}\), a discrete search space was investigated, \(S_{1}/H_{0}\in[0.1;0.9]\) with \(\Delta S_{1}/H_{0}=0.1\). The search space of degrees of freedom \(\alpha\) and \(\beta\) was the same as in the previous section. The versions of the DE algorithm used at this level are  
Figure 9: Comparison between the effect of \(\alpha\) over \((\theta_{max})_{m}\) and \((\beta)_{o}\) generated with exhaustive search and the exploration of the search space with differential evolution algorithms: (**a**) effect of \(\alpha\) over \((\theta_{max})\); (**b**) effect of \(\alpha\) over \((\beta)_{o}\); (**c**) effect of \(\alpha\) over \((\theta_{max})\); (**d**) effect of \(\alpha\) over \((\beta)_{o}\). the ones with better performance in the two degrees of freedom geometric optimization. The amplification factor (\(F\)) and crossover rate (\(CR\)) parameters are applied with \(F=1\) and \(CR=0.9\). The mutation operators investigated are the functions represented by Equations (12), (15) and (16). Therefore, the DE versions are named here as DE1, DE2, and DE3, respectively, according to the mutation operators _rand/1/bin_, _best/1/bin_, and _best/2/bin_. The population size (\(NP\)) and the total number of generation (\(NG\)) parameters were defined as \(NP=30\) and \(NG=100\). Table 10 shows statistical metrics obtained for different versions of DE algorithms to obtain the three times minimized \((\theta_{max})_{3m}\). More precisely, it presented versions DE1 and DE3 that conducted to the best performance and the version DE2 that conducted to the worst performance. The algorithms DE1 and DE3 also led to the same optimal magnitudes of \((S_{1}/H_{0})_{0}\), \((\alpha)_{20}\), and \((\beta)_{30}\). It is worth mentioning that, even the worst version (DE2) conducted to a similar configuration was found with the best versions of DE. Table 11 shows the statistical tests of Shapiro-Wilk (SW) and Kruskal-Wallis (KW) for the investigated DE versions. Results seen in Table 11 were calculated from the registry of \((\theta_{max})_{3m}\) in the 30 runs of each algorithm. Results of the SW test show that the data do not come from a normal distribution. KW test indicates a significant difference between the DE versions analyzed since the calculated \(p\)-value is much smaller than the significance level \(\alpha_{\mathrm{s}}\). [PERSON] test results were shown in Figure 10, being possible to classify the algorithm versions in groups according to the multiple comparisons, as well as the KW test. Figure 10 shows the classification of [PERSON]'s test for the data of exploration of the search space, Figure 10a, and the classification for the convergence to the three times minimized dimensionless maximum excess of temperature, \((\theta_{max})_{3m}\), obtained for the 30 runs of each algorithm, Figure 10b. Points in Figure 10a represent the values of \((\theta_{max})_{3m}\) reached by each algorithm during the optimization process of one run. The classification of [PERSON]'s test indicates that the DE versions have different behaviors in exploring the search space. According to the _boxplot_ and points in Figure 10a, version DE2 has a more concentrated exploration of the search space than other versions, similar to the previous analysis performed in Section 4.2. The concentrated exploration conducted DE2 version to be retained at a local point of minimum. This behavior classifies the DE2 version in group \(B\) as different from the other versions, as seen in Figure 10b.