## 2 Method
### Parameterization Strategy  
Parameterization of the airfoil shapes for the UAS design and optimization requires a mathematical formulation, and it becomes an important part of the overall optimization process. Studies have shown that the choice of shape parameterization technique has a strong influence on the solution accuracy, robustness and computational time of the overall optimization process [69]. In order to obtain an optimal aerodynamic solution, the shape function must be directly related to the airfoil geometry, have a flexible design space, and be robust for the control of all parameters. Some of the well-known methods are the Discrete Based Approach, the Bezier curves, the B-Spline or NURBS curves, the Cubic Spline and the Free-Form Representation [69; 70]. The drawbacks of these methods are that they do not use airfoil shape parameters, they require a large number of design variables and they often provide inaccurate shapes for the leading edge and trailing edge of an airfoil. These limitations have been partly removed by using the polynomial-based functions, such as PARSEC and CST. \"PARSEC\" is characterized by eleven design coefficients that control airfoil shape parameters, and they are the: (a) edge radius, (b) upper crest abscissa, (c) upper crest ordinate, (d) upper crest curvature, (e) lower crest abscissa, (f) lower crest ordinate, (g) lower crest curvature, (h) trailing edge ordinate, (i) trailing edge thickness, (j) trailing edge direction and (k) trailing edge wedge angle. Several studies have shown PARSEC method's superior performance with respect to the Discrete Based Approach, based on their convergence rate, flexibility and epistasis, which is due to the nonlinear dependency of the objective function on the design parameters [71]. However, it has also been found that the leading edge and trailing edge design coefficients in this parameterization method do not provide accurate airfoil shape; therefore, implementing morphing design using only these PARSEC design coefficients might further increase the airfoil shape complexity. Therefore, a combined approach, the Bezier-PARSEC (BP) parameterization method, was developed [72] that included the advantages of Bezier with PARSEC combination, and it was found to be suitable for this study. The BP parameterization was implemented in this paper with the aim to increase the solution accuracy, flexibility and efficiency of morphing leading edge and trailing edge optimizations. More significantly, this technique provides the large search space needed for morphing leading edge and trailing edge design, and it reduces the computational time. The BP method has been further divided into two parameterization sub-methods: BP3333 and BP3434. Four third-order Bezier curves are used to represent the airfoil shape in the BP333 parameterization sub-method. The BP3434 parameterization has a 3 rd-degree edge thickness curve, a 4 th-degree trailing edge thickness curve, a 3 rd-degree leading edge camber curve and a 4 th-degree trailing edge camber curve. Due to the smaller number of degrees of freedom in the BP3333 method with respect to those of the BP3434 method, specifically at the trailing edge and at the leading edge, the BP3434 method was chosen in this research. A fourth-degree Bezier curve is given by:  
\[x(u)=x_{0}(1-u)^{2}+4x_{1}u(1-u)^{3}+6x_{2}u^{2}(1-u)^{2}+4x_{3}u^{3}(1-u)+x_{ 4}u^{4} \tag{5}\]  
and  
\[y(u)=y_{0}(1-u)^{2}+4y_{1}u(1-u)^{3}+6y_{2}u^{2}(1-u)^{2}+4y_{3}u^{3}(1-u)+y_{ 4}u^{4} \tag{6}\]  
The parameterization is controlled by 15 parameters: 10 aerodynamic and 5 Bezier parameters. Figure 3 shows the graphical representations of these parameters. Figure 3: BP3434 method parameter definition. The BP3434 parameterization has a 3 rd-degree edge thickness curve, a 4 th-degree trailing edge thickness curve, a 3 rd-degree leading edge camber curve and a 4 th-degree trailing edge camber curve.