## 3 Optimization Results
At the LE region, the droop nose part corresponds to the 25% of the chord length, both on the upper and lower sides, whilst at the TE region the deformable flap corresponds to the 20% of the chord length. In some of the previous investigations [4; 5; 6], more conservative approaches have been adopted to avoid possible discontinuities, especially on the upper side of the droop nose part. The present algorithm can be easily adapted to such approaches by keeping the control points 1 and 7 fixed at the LE part (Figure 2a), as well as the points 8 and 14 at the TE part (Figure 2b). In this way, the deformable parts of the LE and TE regions are restricted to 20% and 15%, respectively. Moreover, an even smaller deformation is permitted for the upper side of the droop nose part. The results of this more conservative approach for the three optimization algorithms are summarized in Table 3. It can be observed that, keeping the same number of generations, the PSO algorithm again provides the airfoil with the maximum \(\mathrm{C_{L,max}}\). Compared to the more \"aggressive\" approach, there is a 2.25% reduction in the \(\mathrm{C_{L,max}}\) plus the fact that the double number of generations are needed to reach the optimum design. On the other hand,  
Figure 4: Final solutions Particle Swarm Optimizer (PSO) (four generations, first type) and ES (eight generations, second type) along with the original airfoil. Figure 5: Comparison of the morphing achieved using the PSO (four generations) and ES (eight generations) algorithms (**a**) LE region and (**b**) TE region. smaller deformations are required to achieve that \(\text{C}_{\text{L,max}}\), as shown in Table 4. Figure 6 compares the two different designs, whilst Figure 7a,b focuses on the deformed parts at the LE and TE regions, respectively. These plots confirm the smallest deformations when the more conservative setup is used. It should be noted though, that in both setups, proper constraints ensuring the continuity of the geometry have been imposed, as described in Section 2. The more conservative design could prove to be more efficient in cases where limitations of the permitted deformations are imposed by structural analysis. In the present work, such an analysis is not included, therefore the more aggressive design (design 1) with the higher \(\text{C}_{\text{L,max}}\) is adopted in the sequel. A closer look at the connection points between the morphing and the undeformed parts of the airfoil reveals small discontinuities of the geometry, more evident at the lower side of the LE region and the upper side of the TE region. These discontinuities can be smoothed out by imposing additional fixed control points between those already fixed (green points in Figure 2) and the points with numbers 1, 7, 8, and 14 (Figure 2). The effect of such a treatment is shown in Figure 8. It is observed that a smoother geometry is achieved locally, through the addition of the intermediate control points, which keep the continuity of the first derivative at the connections. As expected, the effect is more pronounced at the lower side of the LE and the upper side of the TE, where the larger discontinuities were present. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Airfoil**} & \multicolumn{3}{c}{**Deformations**} \\ \cline{2-4}  & **Leading Edge** & **Trailing Edge** & **Overall** \\ \hline PSO, design 1 (4 generations) & 5.68\% & 3.54\% & 4.74\% \\ PSO, design 2 (8 generations) & 4.70\% & 2.73\% & 3.84\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Deformations of the LE and TE regions and of the total geometry (length increase with respect to the original geometry) for the optimum design of the aggressive (design 1) and the conservative (design 2) approaching the PSO algorithm. Figure 6: Final solutions of PSO algorithm using the aggressive (design 1, four generations) and the conservative (design 2, eight generations) setup.