## 4 Results and Discussion
### Optimization Results for Two Degrees of Freedom
Figure 8b shows the minimization of the \((\theta_{max})_{2m}\) and the convergence to the magnitude \((\theta_{max})_{2m}=0.029969\). For all versions, it is possible to observe a large spread on the attempts to find the optimal angle \(\beta\) for the first 500 iterations in the range of \(-60^{\circ}\leq(\theta)_{20}\leq 80^{\circ}\). Figure 6: Dunn test of all DE versions for \((\theta_{max})_{2m}\) with _NP_ = 20 and _NG_ = 50. Figure 7: Dunn test of all DE versions for \((\theta_{max})_{2m}\) with _NP_ = 30 and _NG_ = 50. With the increase in the number of simulations, the algorithms converge to optimal conditions. For the DE41 version, around 1000 iterations are required for convergence of \((\beta)_{20}\), while D43 is slowly requiring near 1500 iterations for convergence. The fastest algorithms are versions DE42 and DE44, which converge for almost 700 iterations. A similar behavior is noticed in Figure 8b for convergence of the minimum magnitude of \((\theta_{max})_{2m}\). Figure 8 shows that for \(F=1\), the mutation operators _best/1/bin_ and _current-to-best/1/bin_ have the same behavior. As previously mentioned, constructal design has its main concern with the comprehension of the effect of the degrees of freedom over the optimal geometry and thermal performance of the flow system. In this sense, Figure 9 shows a comparison of the effect of the degree of freedom \(\alpha\) over the once optimized angle \(\beta\), \((\beta)_{0}\), and the once minimized dimensionless maximum excess of temperature, \((\theta_{max})_{m}\), reached by the exhaustive search (ES) method, in comparison with the results of the differential evolution algorithms. The values registered by meta-heuristics during the search space exploration were ordered according to angle \(\alpha\) and can be seen in the points with colors red and green in Figure 9. The effect of \(\alpha\) generated by the exhaustive search (ES) method is represented by the gray curve in Figure 9. The blue points in Figure 9 represent the optimal values of \((\beta)_{0}\) and minimum magnitudes of \((\theta_{max})_{m}\) reached by the meta-heuristics for each value of \(\alpha\). Results for the DE14 version in Figure 9a show that it is possible to reproduce in an approximate form the effect of \(\alpha\) over \((\theta_{max})_{m}\)for \(\alpha>-40^{\circ}\). This behavior occurs due to the slow convergence of the DE14 algorithm version, resulting in scattered data. However, the results of the DE14 in Figure 9b show that this algorithm cannot reproduce the effect of \(\alpha\) over \((\beta)_{0}\) like the ES method, mainly in the regions near the inferior and superior extremes of the search space of \(\alpha\). According to Figure 9b, it is possible to obtain an approximation of the effect of \(\alpha\) over \((\beta)_{0}\) near the optimal region of \((\alpha)_{0}=-7^{\circ}\). For values of \(\alpha>0^{\circ}\), the problem has more than one solution when different values of \(\beta\) led to similar values of \((\theta_{max})_{m}\), as can be seen in Figure 9a,c. This behavior happened due to the significant sensitivity of the angle \(\alpha\). For version DE24, the reproduction of the effect of \(\alpha\) over \((\beta)_{0}\) and \((\theta_{max})_{m}\) is more inaccurate due to the fast convergence of the algorithm and the concentration of the exploration in the optimal region of \((\alpha)_{0}\), as seen in Figure 9c,d. Figure 8: Optimization process of DEx4 versions with NP = 30 and NG = 50: (a) optimization of \((\beta)_{20}\); (b) minimization of \((\theta_{max})_{2m}\). Results indicated that different versions of DE were suitable for the prediction of the once minimized magnitude of \((\theta_{max})_{m}\) and the corresponding optimal angle \(\beta\), \((\beta)_{o}\), near the global optimal angle of \(\alpha\). This behavior is indicative that the DE version can be used for the prediction of optimal shapes.