## 3 Optimization Results  
For the optimization process, NACA 64A010 is used, as it is a common airfoil for aircraft applications. The ambient flow conditions employed in the numerical analyses are M = 0.22 and Re = 6 \(\times\) 10\({}^{6}\), corresponding to typical take-off and landing conditions of a short to medium range aircraft with flight speed of 75 m/s and a chord length of 1.2 m (close to the tip of the aircraft wing). Separate Bezier curves are produced for the TE and the LE regions considering a separate group of control points for each one of them. Both the suction and pressure side geometry, either of the LE or the TE region, are represented by the same curve. The x-coordinates of the control points used in the present analysis are pre-selected as illustrated in Figure 2 and they are not modified throughout the optimization procedure. A preliminary study is made to determine the optimum number of control points for the LE and TE regions. A total of nine control points is found sufficient for reproducing  
Figure 1: Flowchart of the optimization process. CP stands for Control Point. The superscript 0 on C\({}_{\text{L,max}}\) denotes the previous ensemble of CPs and geometry. the sharp TE shape of the airfoil section with reasonable detail. Less points result in significant geometrical irregularities, demanding additional constraints that is not obvious to determine. More points increase the possible combinations and the computational cost of the optimization procedure. Two of the control points are placed at the connections between the undeformed and the morphing parts of the airfoil (one on the upper and another on the lower side respectively) and remain fixed throughout the optimization process. The remaining points are divided between the upper and lower side (three on the upper side, three on the lower side, and one at the TE point) and they are only allowed to move vertically. As shown in Figure 2b, expected deflections in the TE region will be relatively small and therefore it is sufficient to simply distribute control points at equal distances along the \(x\)-axis (Figure 2b). The y-coordinates of the seven movable control points constitute the design variables of the optimization loop for the TE morphing shape. A similar approach is followed for the LE region. Again, two control points remain fixed, connecting the undeformed with the morphing part, and seven control points are distributed on the two sides, three on the upper and four on the lower side, as shown in (Figure 2a). Again, the y-coordinates of the seven control points constitute the design variables of the optimization loop. However, there are some differences compared to the approach followed at the TE region: First, more control points are needed close to the LE in order to maintain the rounded shape of the LE region. Therefore, the control point 4 of the lower side (CP4) must be located very close to the LE control point (CP5). Furthermore, on the upper side two extra control points ex1, ex2 are added (Figure 2a). The x,y-coordinates of the ex1 additional control point are derived through linear extrapolation to the x,y-coordinates of CP4 and CP5. The x,y-coordinates of the ex2 additional control point are defined so that the curvature of the lower side (points CP3,CP4,CP5) is similar to that of the upper side (points CP5,ex1,ex2), maintaining thus the roundness of the LE. Since the y-coordinates of the ex1, ex2 points depend on the y-coordinates of the control points 3, 4, 5, they are not considered as independent design variables. Second, an additional control point, CP1, is needed at the lower side, close to the quarter of chord, in order to avoid abrupt gradient changes between the morphing and the undeformed geometries that may cause flow disturbances. An overall number of 11 control points are used for determining the Bezier curve of the LE region (Figure 2a). Figure 2: Definition of the x-coordinates of the control points in (**a**) the leading edge (LE)region and (**b**) the trailing edge (TE) region.