## 5 Conclusions  
The present work analyzed the association between the canonical form of differential evolution (DE) algorithm and the constructal design (CD) for geometrical optimization of a double Y-Shaped cavity inserted in a solid domain with constant heat generation per unit volume. Several parameters of the DE algorithm were investigated to rationalize the search for the global point of optimum design, and mainly for the reproduction of the effect of degrees of freedom over the thermal performance and corresponding optimal configurations, which is essential for the physical analysis of the design. The parameters of the DE algorithm analyzed were the differential amplification factor (F), crossover rate (_CR_), and four mutation operators (_rand/1/bin_, _best/1/bin_, _best/2/bin_, and _current-to-best/1/bin_). The combinations of the investigated parameter values generated 16 different versions of the DE algorithm. For statistical analysis of each algorithm at each level of optimization, 30 runs for each version were performed. Results were analyzed with the statistical methods of Shapiro-Wilk, Kruskal-Wallis, and the Dunn test for multiple comparisons for non-parametric samples. With these statistical tools, the work identifies the best and worst sets of DE algorithm parameters when applied to the problem of interest. In most cases observed, the DE algorithm versions with mutation operator _rand/1/bin_ achieved the best performance, at all different optimization levels. On the other hand, the worst mutation operator was the _best/2/bin_ for the conditions investigated in this study. For constant values of _NP_ and _NG_, the parameter F significantly influenced the search for the best geometrical configuration. For the present conditions, the _CR_ parameter did not show an important influence. For the first level of optimization, the best parameters to be applied, according to the results of this study, were a combination of parameters \(F\) = 1 with _CR_ = 0.9 and the mutations operators _rand/1/bin_ and _best/1/bin_. Results of the second level of optimization show that for correct reproduction of the effect of a determined degree of freedom over the other geometric parameters and thermal performance, is recommendable an association of the meta-heuristic with the exhaustive search method. In other words, the exhaustive search should be used at the level where the effect of degree of freedom over thermal performance and corresponding optimal shapes is investigated. It was also noticed that versions DE1 and DE3 have similar performance but different behavior to explore the search space. The DE2 version presented more errors and was classified as a different group from other methods. Despite the DE2 version error, it can be seen through the results that this algorithm has a fast convergence to the optimal region. Consequently, in most cases, this version reaches local optimal values. The last experiment carried out here was the optimization process to reproduce the effect of the ratio \(H_{0}/L_{0}\) over the three times minimized maximum excess of temperature, \((\theta_{max})_{3 mt}\), and the mean square errors (MSE) for \((\theta_{max})_{3m}\) and the corresponding optimal configurations, \((S_{1}/H_{0})_{0}\), \((\alpha)_{20}\), and \((\beta)_{30}\), for different versions of DE algorithm and parameters \(NP\) and \(NG\). Results showed that the increase in parameters \(NP\) and \(NG\) generally improve the algorithm results, except for some cases of the DE2 version for degrees of freedom \(\alpha\) and \(S_{1}/H_{0}\). The DE3 version presents a significant reduction of standard deviation with the increase in \(NP\) and \(NG\) parameters reaching an intermediate performance. The performance of the DE1 version was the best in comparison with the other investigated methods. Even for \(NP=20\) and \(NG=100\), the method has obtained very close values to the optimal solutions and almost zero standard deviation.