## 3 Optimization Results
As an example, for the maximum number of generations (17), the number of fitness evaluations is 17 \(\times\) 112 + 112 = 2016. For the current setup, PSO algorithm outperforms the other two algorithms as it predicts the solution with the maximum \(\text{C}_{\text{L,max}}\), while it requires the minimum number of generations (4). ES algorithm requires a significantly larger number of generations (17) to reach about the same \(\text{C}_{\text{L,max}}\) level (2% lower). On the other hand, the optimum design predicted by the GA algorithm after 17 generations, has a maximum \(\text{C}_{\text{L,max}}\) value 7% lower than that of the PSO algorithm. Comparing ES against GA, it is observed that the ES algorithm achieves a higher \(\text{C}_{\text{L,max}}\) value at the expense of computational time, as it requires 17 generations in order to reach that solution. GA reaches a solution with a slightly higher \(\text{C}_{\text{L,max}}\) (2.5% difference) at a lower number of generations (after only eight generations). The resulting optimum morphing shapes produced by the three algorithms are then compared in terms of the required airfoil shape deformation in order to reach their maximum \(\text{C}_{\text{L,max}}\). Shape deformation is measured in terms of the percentage of increase of the airfoil skin length at the morphing regions with respect to the corresponding lengths of the baseline undeformed airfoil. Table 2 summarizes these percentages separately at the LE and TE regions as well as for the overall skin length. In general, two shape types of geometries are produced by the three algorithms, the first type with the highest \(\text{C}_{\text{L,max}}\) (>2.2) presenting larger deformations at the LE region (\(\approx\)5.7%), and the second type with \(\text{C}_{\text{L,max}}\approx\) 2.1 presenting smaller deformations at the LE region (\(\approx\)4.5%). Similar TE deformations are predicted for both types of morphed designs.The solution of GA after eight generations and the solution of ES after eight generations belong to the first type. The solution of ES after 17 generations and the solution of PSO after 4 generations belong to the 2 nd type. Figure 4 presents one morphed geometry per airfoil type; the PSO first type after four generations and the ES second type after eight generations. In the subsequent analysis, the ES solution is preferred instead of the corresponding GA solution, since it is considered as more representative of the second type (smaller deformation at the LE region). Differences between the two shapes are highlighted in the focus plot of Figure 5. The second type solution could qualify as preferred design, e.g., in the case of an upper limit on allowable \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Optimization Algorithm**} & \multicolumn{4}{c}{**C\({}_{\text{L,max}}\)**} \\ \cline{2-5}  & **2 Generations** & **4 Generations** & **8 Generations** & **17 Generations** \\ \hline GA & 2.02 & 2.07 & 2.12 & 2.12 \\ PSO & 2.2 & 2.27 & 2.27 & - \\ ES & 2.0 & 2.01 & 2.07 & 2.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Maximum lift coefficients of the modified airfoils created during the parametric study of the number of generations. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Airfoil**} & \multicolumn{4}{c}{**Deformations**} \\ \cline{2-5}  & **Leading Edge** & **Trailing Edge** & **Overall** \\ \hline PSO (4 generations, 1 st type) & 5.68\% & 3.54\% & 4.74\% \\ GA (8 generations, 2 nd type) & 4.49\% & 3.31\% & 3.98\% \\ ES (8 generations, 2 nd type) & 4.28\% & 3.48\% & 3.93\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Deformations of the LE and TE regions and of the total geometry (length increase with respect to the original geometry) for the derived airfoils of first and second type. LE deformation, imposed by constraints on material properties (maximum stress level) or actuator limitations. It is noted that in the current setup, a rather \"aggressive\" design has been chosen in terms of the permitted deformations.