## 3 Numerical Results
### Results from the Optimizations Methods  
This section introduces the results obtained for the proposed comparison. The convergence of the different optimization algorithms are shown in Figure 4. The results shown in the graph of Figure 4 reflect the problems encountered by the gradient based method. Most runs of the conjugate gradient initially improve faster than the Genetic Algorithm but then get stuck between \(C_{l}\approx 1.35\) and \(C_{l}\approx 1.45\) (except for two runs that get stuck at \(C_{l}\approx 0.8\) and \(C_{l}\approx 1.25\), respectively). Those lift coefficient values are achieved with almost 100 evaluations of the objective function for each optimization of the conjugate gradient. The strong dependence of the conjugate gradient on the starting point is also reflected on these results, as it presents very different solutions between runs than the other methods.  
Figure 3: Averaged streamlines and pressure field of baseline case (\(C_{l}=0.80\)).  
Figure 2: Full mesh of the domain (**a**) and Mesh nearby the Synthetic Jet actuator (**b**).  
In all cases, the conjugate gradient method was stopped because the algorithm found a local minimum and could not compute the gradient to further improve the results.  
On the other hand, the Genetic Algorithm provides optimal solutions similar to the conjugate gradient, but with a higher computational cost, approximately four times higher. Both runs of the Genetic Algorithm achieve values of the objective function in the range of the conjugate gradient results. One of the runs achieves a better objective function than all of the conjugate gradient runs, with a value of \(C_{l}=1.49\). It is important to note that the Genetic Algorithm optimizations could run additional iterations and achieve better results, but with a high computational cost.  
The Hybrid Method is the method that achieved better results, outperforming all runs of the other algorithms in both of its runs. During the first iterations, it achieved a convergence rate similar to the conjugate gradient. However, the improvement of the solution has continued, avoiding being trapped in any local minimum. It is also the most robust, as both runs are very similar in their performance. Both Hybrid Method runs outperformed all the other optimization methods with a \(C_{l}=1.52\) and \(C_{l}=1.54\). One run of the Genetic Algorithm achieved a comparable solution (\(C_{l}=1.49\)), but it took more than twice the computational cost of the Hybrid Method. Case 3, with a lift coefficient of \(C_{l}=1.49\), obtained by the Genetic Algorithm with around 375 objective function evaluations, improves the baseline lift coefficient by 86%. Cases 4 and 5, obtained by the Hybrid Method runs, achieved a lift coefficient of \(C_{l}=1.52\) and \(C_{l}=1.54\), respectively. Both runs needed around 125 objective function evaluations, which is 67% the number of evaluations of case 3, with an increase in the lift coefficient of 90% and 93%, respectively, from the baseline. The best lift coefficient achieved by each optimization is presented in Table 4. The mean (\(\overline{C_{l}}\)) and standard deviation (\(\sigma\)) of the lift coefficient achieved by each optimization method is also presented. The Hybrid Method presents the best lift coefficient mean (\(C_{l}=1.53\)) followed by the Genetic Algorithm (\(C_{l}=1.44\)) and the conjugate gradient (\(C_{l}=1.29\)). The conjugate gradient is the least robust, with a standard deviation of \(\sigma=0.2458\), but four of the six optimizations achieved similar results than the Genetic Algorithm with less computational effort.  
Looking at these results, one can conclude that the proposed Hybrid Method performs much more robustly than the conjugate gradient method and much faster than the Genetic Algorithm.  
Figure 4: Comparative of the convergence of the different optimization methods.