## 2 Generation and Representation of Generalized Airfoil Set
### B-Spline Theory and Airfoil Set Generation  
B-spline theory is widely applied in airfoil parameterization and optimization tasks [16]. The B-spline function is expressed as follows:  
\[P_{k,n}(t)=\sum\limits_{k=0}^{n}P_{i+k}G_{i,n}(t),t\in[0,1] \tag{13}\]  
where \(P_{k,n}(t)\) represents the \(k\)-th segment of the \(n\)-degree B-spline curve, and \(P_{i}\) (\(i\) = 0, 1, \(\ldots\), \(n+m\)) are the vertices of the B-spline characteristic polygon. The expression for the basis function \(G(t)\) is as follows:  
\[ \begin{split}& G_{i,n}(t)=\frac{1}{n!}\sum\limits_{j=0}^{n-j}(-1)^{ j}C_{n+1}^{j}(t+n-i-j)^{n}\\ & t\in[0,1],i=0,1,\cdots,n\end{split}  \tag{14}\]  
For a cubic B-spline curve, the basis function expression becomes:  
\[\left\{ \begin{array}{c}G_{0,3}(t)=\frac{1}{6}(-t^{3}+3t^{2}-3t+1)\\ G_{1,3}(t)=\frac{1}{6}(3t^{3}-3t^{2}+4)\\ G_{2,3}(t)=\frac{1}{6}(-3t^{3}+3t^{2}+3t+1)^{\prime}\\ G_{0,3}(t)=\frac{1}{6}t^{3}\end{array} \right.,t\in[0,1] \tag{15}\]  
Figure 7: Generate Airfoil set through Interpolation Using the CST Theory. (**a**) Interpolation with RISOA30 and DU91W2250 (**b**) Interpolation with FFAW3211 and S814. The matrix form of the cubic B-spline function is given as:  
\[\frac{1}{6}[1\quad t\quad t^{2}\quad t^{3}] \begin{bmatrix}1&4&1&0\\ -3&0&3&0\\ 3&-6&3&0\\ -1&3&-3&1\end{bmatrix}\begin{bmatrix}P_{0}\\ P_{1}\\ P_{2}\\ P_{3}\end{bmatrix} ,t\in[0,1] \tag{16}\]  
where \(P_{0}\), \(P_{1}\), \(P_{2}\), \(P_{3}\) are the control points, and \(t\) represents the horizontal coordinate of the B-spline curve. In this study, the airfoil is expressed by positioning control points on the airfoil curve. By setting different control points \(P\), various airfoil geometries can be obtained. The process of fitting the baseline airfoils using B-spline theory is illustrated in Figure 1, with the relevant parameter design provided in Table 1. The fitting results, as shown in Figure 8, demonstrate that the method accurately fits the baseline airfoils and extracts the corresponding fitting parameters. Similar to the airfoil interpolation method using CST theory, the fitting of two airfoils using the above optimization algorithm yields two sets of control point coordinates. After obtaining the new control point coordinates using Equation (17), the new airfoil is generated from these new control points. \[K_{new}=K_{A}+random(0\sim 1)\cdot(K_{B}-K_{A}) \tag{17}\]  
where \(K_{A}\) and \(K_{B}\) represent the control point coordinates of the two baseline airfoils. The process of fitting the baseline airfoils using B-spline theory is illustrated in Figure 1, with the relevant parameter design provided in Table 1. The fitting results, as shown in Figure 8, demonstrate that the method accurately fits the baseline airfoils and extracts the corresponding fitting parameters. Using this method, a set of airfoils with intermediate characteristics between any two airfoils can be obtained, as illustrated in Figure 9.  
In order to enrich the airfoil dataset as much as possible, the above three expression are used to generate new airfoils. The sharp trailing edge airfoils with different bend and thickness characteristics are generated by the airfoil integrated theory. Also, some new  
Figure 8: Fit airfoils using CST B-spline. airfoils are generated by the interpolation method based on CST and B-spline theory. A program is made by using MATLAB(R2023a) software to generate airfoil datasets. When calculating airfoil noise, the RFOIL software is used to compute the boundary layer parameters of the airfoil [17]. To ensure that the airfoil set successfully converges during RFOIL calculations, each newly generated airfoil is imported into RFOIL for calculation. If the calculation fails to converge, the airfoil is regenerated until successful convergence is achieved, ensuring that each new airfoil serves as valid input data for noise calculations. Using the above method, a generalized airfoil set containing 18,300 airfoils was generated.