## 2 Generation and Representation of Generalized Airfoil Set
### CST Theory and Its Airfoil Set Generation  
The Class function/Shape function Transformation (CST) method is a technique proposed by [PERSON] and colleagues from Boeing [15]. The upper and lower surfaces of an airfoil are expressed using the following two formulas:  
\[(\zeta)_{Upper}=C_{N_{2}}^{N_{1}}(\psi)\cdot Su(\psi)+\psi\cdot\triangle\zeta_{ Upper} \tag{6}\]  
\[(\zeta)_{Lower}=C_{N_{2}}^{N_{1}}(\psi)\cdot Sl(\psi)+\psi\cdot\triangle\zeta_{ Lower} \tag{7}\]  
In the equations: \(\psi\)--the dimensionless x-coordinate of the airfoil, \(\psi=\) x/c; \(\zeta\)--the dimensionless y-coordinate of the airfoil, \(\zeta=\) z/c; \(\Delta\)EUpper--the leading-edge thickness ratio of the upper surface, \(\Delta\)EUpper = zuTE/C; \(\Delta\)ELower--the trailing-edge thickness ratio of the lower surface, \(\Delta\)ELower = zLTE/C.  
The class function is defined as follows:  
\[C_{N_{1}}^{N_{2}}(\psi)=\psi^{N_{1}}\cdot(1-\psi)^{N_{2}} \tag{8}\]  
The shape function \(S(\psi)\) is defined as follows:  
\[S_{u}(\psi)=\sum_{i=1}^{n}Vu_{i}\cdot S_{i}(\psi) \tag{9}\]  
\[S_{l}(\psi)=\sum_{i=1}^{n}Vl_{i}\cdot S_{i}(\psi) \tag{10}\]  
\[S_{i}(\psi)=\frac{n!}{\vec{n}(n-\vec{i})!}\;\psi^{i}(1-\psi)^{n-i} \tag{11}\]  
In the equations: Vui, Vli--Coefficients of the shape function components for the upper and lower surfaces of the airfoil; Si(\(\psi\))--Bernstein polynomials; n--Order of the Bernstein polynomials for the upper and lower surfaces of the airfoil.  
When fitting the baseline airfoil using the CST theory, the PSO algorithm is employed to determine the fitting parameters. The design variables include the shape function coefficients \(V_{u}\), \(V_{l}\), as well as the parameters \(N_{1}\) and \(N_{2}\) from CST theory. The mean squared error of the \(y\)-coordinate values between the fitted airfoil and the original airfoil at selected chordwise positions is used as the objective function. The algorithm parameters are provided in Table 1, and the fitting performance is shown in Figure 6. The results indicate that this method effectively fits the baseline airfoil and yields the corresponding fitting parameters.  
Figure 6: Fit airfoils using CST.  
To enrich the airfoil set with more diverse geometric features, interpolation methods are utilized to generate interpolated airfoils based on two baseline airfoils created using the CST method. By applying the aforementioned optimization algorithm to fit the two airfoils, two sets of shape function coefficients and \(N_{1}\), \(N_{2}\) values can be obtained. The parameterization formula for the interpolated airfoil is expressed as follows:  
\[Q_{new}=Q_{A}+random(0\sim 1)\cdot(Q_{B}-Q_{A}) \tag{12}\]  
In the formula: QA, QB--represent the fitting parameters of the two baseline airfoils, where Q = [Vu, Vl, N1, N2]. Here, Vu and Vl are the shape function coefficients described in Equations (9) and (10), while N1 and N2 are the parameters defined in Equations (6) and (7).  
Using this method, a set of airfoils with intermediate characteristics between any two airfoils can be obtained, as illustrated in Figure 7. This approach allows interpolation both within the same airfoil family and across different airfoil families, resulting in an enriched set of airfoils with more diverse geometric features.