# Airfoil Optimization Design Based on the Pivot Element Weighting Iterative Method
## 2 Airfoil Parameterization
### Parameter Matrix Analysis of the CST Method  
The CST method is used to analyze the parameterization matrix of RAE2822 airfoil. As the order of Bernstein polynomial increases, the condition number of the parameterization matrix rapidly increases in Figure 6. The distribution of eigenvalue moduli of the parameterization matrix is presented in Figure 7. When the numbers of parameters change from 1 to 15, maximum eigenvalue has no obvious change. But minimum eigenvalue quickly approaches 0. So the parameterization matrix is close to extraordinary. The larger the condition number of the parameterization matrix is, the closer the parameterization matrix is to morbidity. The slight fluctuations of control point coordinates or the parameterization matrix can lead to the huge change of the solution vectors and vice versa. Therefore, two completely different solution vectors may generate quite close geometry and the difference between them may be less than the size of one grid cell. Figure 4: (a) Comparison between the CST parametric airfoil and the RAE2822 airfoil (b) RAE2822 airfoil second derivatives when BPO = 6. Figure 5: (a) Comparison between the CST parametric airfoil and the NACA0012 airfoil (b) NACA0012 airfoil second derivatives when BPO = 3. CST method is selected to fit the high altitude laminar S1223 airfoil. Due to the large camber of S1223, the fitting airfoil has obvious deformation, especially the location of leading and trailing edge in Figure 8a. When BPO = 8, the S1223 airfoil second derivatives are shown in Figure 8b. The upper and lower airfoils have the obvious fluctuations. So it is necessary to increase the order of Bernstein polynomial. As mentioned above: the larger the order of the parameterization matrix is, the closer the parameterization matrix is to morbidity. And the slight fluctuations of the parameterization matrix or control point coordinates can lead to the large change of the solution vectors, which may go against the global optimum design of the airfoil. At the same time the more the parameters are, the lower the optimization efficiency is. Therefore, it is necessary to weigh the order of Bernstein polynomial and time consumed by the airfoil optimization. Figure 6: Spectral condition number of the parameterization matrix. BPO: Bernstein polynomials. Figure 7: Eigenvalue moduli of the parameterization matrix. The fitting residuals with different order Bernstein polynomial are presented in Figure 9. As the order of the polynomial parameterization increases, the residuals of S1223 airfoil decrease. So PEWI method is used to improve the coefficient matrix of the CST method. When BPO is 14, the RMSE can reach the requirement of typical wind-tunnel model tolerances. Meanwhile the variables are too many in BPO = 20. Hence, BPO = 14 is chosen. When BPO is 14, the condition number of the coefficient matrix from the parameterization of the upper airfoil is \(1.5819\times 10^{8}\) and the lower airfoil is \(2.1266\times 10^{8}\). The coefficient matrix is obviously morbid. Any two elements of \(\mathbf{Z}\) matrix change a little, such as: \(z_{3}=z_{3}+0.00001\), \(z_{10}=z_{10}+0.00001\), the element value of the parameterization matrix is shown in Figure 10. The amplitudes of the parameterization matrix have the huge changes. So it is disadvantageous for the optimization design of S1223 airfoil. Figure 8: (**a**) Comparison between the CST parametric airfoil and the S1223 airfoil (**b**) S1223 airfoil second derivatives when BPO = 8. Figure 9: CST parameterization residuals of the S1223 airfoil. Aiming at the above problem, PEWI method is used to change the condition number of the coefficient matrix. When \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are set as 0.000001 in Figure 1, \(\alpha=100\) of the upper airfoil and \(\alpha=50\) of the lower airfoil can be obtained. So the condition number of the upper airfoil change to 1.0154 and the lower is 1.0284. The Bernstein coefficients of the original CST method and the modified CST with PEWI method are showed in Figures 11 and 12.