## 2 Generation and Representation of Generalized Airfoil Set
### Airfoil Integration Theory and Its Airfoil Set Generation  
The Joukowski transformation can convert a circular arc into an airfoil [14]. The circular arc is defined in the z-plane, with the center coordinates at \((-\varepsilon 1,\,\varepsilon 2)\) and a radius of \(r\). Through the Joukowski transformation, it can be transformed into an airfoil in the \(\zeta\)-plane. The formula in the \(\zeta\)-plane is as follows:  
\[\zeta=f(z)=z+\frac{a^{2}}{z} \tag{1}\]  
In the equation, a represents one-fourth of the airfoil chord length. The coordinates of any point on the circular arc can be expressed as z = r-exp(i\(\theta\)), and r and z are represented by the following formulas:  
\[\mathrm{r}=a\exp(\varphi(\theta)) \tag{2}\]  
\[z=a\exp(\varphi(\theta)+i\theta) \tag{3}\]  
From the above equation, it can be seen that z is a function of \(\theta\), and different functions \(\phi(\theta)\) can yield different airfoil shapes. Based on the [PERSON] transformation theory, the generalized functional equation for \(\phi(\theta)\) can be expressed as follows:  
\[ \begin{array}{l}\varphi(\theta)=a_{1}(1-\mathrm{cos}\theta)+b_{1}\mathrm{ sin}\theta+a_{2}(1-\mathrm{cos}\theta)^{2}\\ +b_{2}\mathrm{sin}^{2}\theta+\cdots+a_{k}(1-\mathrm{cos}\theta)^{k}+b_{k} \mathrm{sin}^{k}\theta+\cdots\\ k=1,2,\cdots,n\end{array}  \tag{4}\]  
By setting different values for parameters \(a\) and \(b\) in Equation (4), various airfoil shapes can be obtained. During the airfoil fitting process, determining the appropriate values of \(a\) and \(b\) is particularly critical. In this study, the optimal airfoil fitting parameters are obtained through iterative computation using the PSO algorithm. The coefficients \(a\) and \(b\) in the theoretical airfoil integration formula Equation (4) are treated as design variables, while  
Figure 1: Airfoil Dataset Generation Process. the mean square error (MSE) of the \(y\)-coordinate values between the fitted airfoil and the original airfoil at selected chordwise positions serves as the objective function. However, when simultaneous fitting of the upper and lower surfaces is applied, certain airfoils with larger camber exhibit noticeable deviations between the fitted and original airfoil shapes, as shown in Figure 2.  
To address this problem, this study employs the airfoil integration theory combined with the PSO algorithm to fit the upper and lower surfaces of a baseline airfoil separately. The fitted upper and lower surfaces are then combined, as illustrated in Figure 3. In the PSO algorithm, two distinct populations are established: Population 1 fits the upper surface, while Population 2 fits the lower surface. These two populations iterate independently, with their respective objective functions defined as the mean square error of the y-coordinate values of the upper and lower surfaces at selected chordwise positions. During the optimization process, an inertia weight decay strategy is implemented to progressively enhance the fitting accuracy. Each time the iteration count reaches a specified step length for inertia weight reduction, the inertia weight is multiplied by a predefined decay factor. The optimization algorithm was used to fit the airfoil to obtain the optimal coefficients of fitting the upper and lower airfoil surfaces respectively, and then the optimal upper and lower airfoil surfaces were combined to obtain the final fitting airfoil. The PSO algorithm parameters are listed in Table 1, and the fitting results are presented in Figure 4. The results demonstrate that this method effectively fits the baseline airfoil and yields the corresponding fitting parameters. Figure 3: Fit the upper and lower surfaces of airfoil respectively. Figure 2: Fit highly cambered airfoils using airfoil integration. When utilizing the airfoil integration theory to generate a series of airfoils, the fitting parameters obtained in the previous step can be perturbed within a certain range, resulting in airfoils with varying camber and thickness characteristics.