## 4 Results and Discussion
### Optimization Results for Two Degrees of Freedom  
For the second level optimization of the double Y-shaped cavity geometry, the algorithms searched for the best angles of \((\alpha)_{0}\), and \((\beta)_{20}\), independently. Two experiments were performed, one with \(NP\) = 20 and \(NG\) = 50, for all sixteen versions of DE, and a second experiment with \(NP\) = 30 and \(NG\) = 50. Table 6 shows the statistical metrics for the first experiment, with \(NP\) = 20. The first line of the Table 6 presents the algorithm versions that reached the optimal geometry of \((\alpha)_{0}=-7^{\circ}\), and \((\beta)_{20}=36^{\circ}\) for all 30 runs. Therefore, the values of variance, standard deviation, and RMSE are null. The DE versions seem in second line of Table 6 also have a suitable performance (not good as DE12, DE32, and DE34) with low magnitudes of variance, standard deviation and RMSE. In general, the best versions consider the parameter \(F\) = 1, except for DE13, that uses \(F\) = 1.5. The other algorithms led to a poorer performance in comparison with those of first two lines of Table 6, especially with the increase in the magnitudes of variance, standard deviation, and RMSE. The impact of the change in population size on the DE algorithm version can be seen in the statistical metrics of Table 7. In comparison with the experiment where \(NP\) = 20, the population with \(NP\) = 30 led to more versions of DE able to reach the optimal geometric configuration on all runs (100%). The worst results continue to emerge from the same versions (DE21, DE23, and DE33). Even with a significant reduction in the variance, standard deviation, and RMSE, the last versions in Table 6 do not reach a similar performance as that achieved for the best codes. The statistical analysis performed by the SW and KW methods validate the significant difference between the investigated algorithm versions. The results of statistical test for both experiments, \(NP\) = 20, and \(NP\) = 30, are shown in Tables 8 and 9, respectively. Tables 8 and 9 show the SW and KW tests for all algorithms and the results \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**DE Versions** & **Average** & **Variance** & **Standard Deviation** & **RMSE** \\  & \((\theta_{max})_{2m}\) & **(\(\times\) 10\({}^{-7}\))** & **(\(\times\)10\({}^{-5}\))** & **(\(\times\)10\({}^{-3}\))** \\ \hline DE12; DE32; DE34 & 0.029969 & - & - & - \\ DE13; DE14 & 0.029974 & 0.00344 & 1.86 & 1.89 \\ DE24; DE44 & 0.030169 & 1.94 & 44.1 & 47.7 \\ DE33 & 0.030410 & 1.66 & 40.7 & 59.6 \\ DE21 & 0.030322 & 6.53 & 80.8 & 86.9 \\ DE23 & 0.030472 & 11.2 & 106.0 & 115.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistical metrics of the results achieved for achievement of \((\theta_{max})_{2m}\) obtained with \(NP\) = 20 and \(NG\) = 50.  
presented a non-normal distribution, except those that reached the same magnitude of \((\theta_{max})_{2m}\) for all runs. Figures 6 and 7 show, respectively, the non-parametric multiple comparisons of Dunn tests for the achievement of \((\theta_{max})_{2m}\) considering \(NP\) = 20 and 30. In Figure 6, three groups were identified by the test of Dunn: A, B and C. Again, group A consists of the algorithms with the best performance, i.e., those where the minimum magnitude of \((\theta_{max})_{2m}\) is achieved in almost 100% of the runs, including the algorithms DE12, DE32, and DE34 which obtained the global optimum in all runs. The DE versions in group A have heterogeneous parameters with varied combinations of \(F\), \(CR\), and mutation operators.