# Gradient-Based Aerodynamic Optimization of an Airfoil with Morphing Leading and Trailing Edges
## 2 Methodologies
### Feasible Morphing Airfoil Geometry Parameterization
#### 2.2.1 Generalized CST Equations  
The CST method, a fundamental parametric airfoil geometry parameterization method [12], consists of a class function \(C(\psi)\), a shape function \(S(\psi)\), and a pair of boundary conditions at leading and trailing edges [18]. The geometric parameters describing the CST airfoil are shown in Figure 2 This method maps the non-dimensional coordinate \(\psi\in[0,\ 1]\), which is the chord-wise axis of the Cartesian coordinate system, to vertical ordinate \(\zeta=\zeta(\psi)\):  
\[\zeta(\psi)=C_{N_{2}}^{N_{1}}(\psi)S(\psi)+(1-\psi)\zeta_{\rm LE}+\psi\zeta_{ \rm TE} \tag{1}\]  
where the subscript \((\cdot)_{\rm TE}\) and \((\cdot)_{\rm LE}\) represent trailing edge and leading edge, respectively. The class function \(C(\psi)\), a single-lope curve, is a term that includes two parameters, \(N_{1}\) and \(N_{2}\). Various general shapes, such as airfoil and aircraft body cross-sectional shape, can be represented by adjusting those two parameters [12]. For example, by setting \(N_{1}=0.5\) and \(N_{2}=1\), a round leading and sharp (or blunt) trailing edge subsonic airfoil can be  
Figure 2: The geometric parameters describing the class/shape transformation (CST) airfoil and the corresponding Bernstein polynomial of the baseline shape. \(R_{\rm LE}\) is the dimensional radius of leading edge curvature, the subscripts \({}_{\rm u}\) and \({}_{\rm 1}\) represent upper and lower surfaces. \(z_{\rm LE}\) and \(z_{\rm TE}\) are the vertical positions of leading and trailing edge points. \(x_{\rm LE}\) and \(x_{\rm TE}\) are chord-wise positions. \(\beta\) is the trailing edge’s boat-tail angle. \(A\) is the coefficient that scales each corresponding Bernstein polynomial. defined. In this paper, \(N_{1}=0.5\) and \(N_{2}=1\) are default parameters. The expression of class function is:  
\[C_{N_{2}}^{N_{1}}(\psi)=\psi^{N_{1}}(1-\psi)^{N_{2}} \tag{2}\]  
The geometry is further modified by introducing the shape function \(S(\psi)\), which is composed of Bernstein polynomials \(B_{i}(\psi)\) to order \(n\), and coefficients \(A_{i}\) that adjust and scale each polynomial. The shape function can be expressed in a compact form:  
\[S(\psi)=\sum_{i=0}^{n}A_{i}B_{i}(\psi)=\sum_{i=0}^{n}A_{i}K_{i,n}\psi^{i}(1- \psi)^{n-i}=\mathbf{A}^{\intercal}\mathbf{B}(\psi) \tag{3}\]  
where \(K_{i,n}=\frac{n!}{i!(n-i)! }\) is the binomial coefficient, \(\mathbf{A}=\{A_{0}\ A_{1}\ \cdots\ A_{n}\}^{\intercal}\) is a column vector of coefficients, and \(\mathbf{B}(\psi)\) consists of Bernstein polynomials. Furthermore, a pair of boundary conditions for leading and trailing edges is added to release the vertical displacement freedom, thereby enabling the ability of representing the movement of the morphing leading and trailing edges physically:  
\[\zeta_{\text{LE}} =\frac{Z_{\text{LE}}}{c} \tag{4}\] \[\zeta_{\text{TE}} =\frac{Z_{\text{TE}}\pm 0.5\Delta Z_{\text{TE}}}{c}\]  
where \(\zeta_{\text{LE}}\) and \(\zeta_{\text{TE}}\) are the non-dimensional vertical positions of leading and trailing edge points, respectively. Both of them are non-dimensionalized by the chord from dimensional \(Z_{\text{LE}}\) and \(Z_{\text{TE}}\). \(\Delta Z_{\text{TE}}\geq 0\) is the thickness of the trailing edge tip. For those airfoils with sharp trailing edges, \(\Delta Z_{\text{TE}}=0\). Consequently, the upper and lower surfaces of an airfoil are:  
\[\zeta_{\text{u}}(\psi) =C_{N_{2}}^{N_{1}}(\psi)\mathbf{A}_{\text{u}}^{\intercal}\mathbf{B}(\psi )+(1-\psi)\zeta_{\text{LE}}+\psi\zeta_{\text{TE,u}} \tag{5}\] \[\zeta_{\text{l}}(\psi) =C_{N_{2}}^{N_{1}}(\psi)A_{1}^{\intercal}\mathbf{B}(\psi)+(1-\psi) \zeta_{\text{LE}}+\psi\zeta_{\text{TE,l}}\]  
where the subscript \((\cdot)_{\text{u}}\) and \((\cdot)_{\text{l}}\) represent the upper and lower surfaces of an airfoil, respectively. The first and the last Bernstein polynomials' coefficients have a physical interpretation.