# Airfoil Optimization Design Based on the Pivot Element Weighting Iterative Method
## 1 Introduction  
Shape parameterization, which can represent the geometries by the coefficients of the parameterization, is a crucial step in the airfoil aerodynamic optimization [1]. The range of the design variables and the order of the parameterization have a significant effect on the design space which can determine the optimum geometries obtainable. Therefore, it is very important for the designer to choose a proper shape parameterization method.  
The Bezier polynomial method [2; 3], the B-spline method [4], and the class function/shape function transformation method [1; 5; 6; 7; 8] are widely used in the airfoil parameterization design. In order to fit the curves of airfoils, the least square method is generally adopted to obtain control parameters in these three methods. The CST method is based on analytical expressions to represent and modify the various shapes [9]. The CST method is widely used in the airfoil aerodynamic optimization.  
For the symmetric NACA0012 airfoil and small camber RAE2822 airfoil, the low order Bernstein polynomials (BPO) can reach the requirement of the airfoil fitting accuracy. But for some large camber airfoils, it is necessary to increase the order of Bernstein polynomials. Because the fitting errors of low order Bernstein polynomials are large. [PERSON] [10] found that the coefficient matrix was obviously ill-conditioned for the 36 th Bernstein polynomial of CST. [PERSON] et al. [11; 12] analyzed the coefficient matrix of the Bernstein polynomial and found that the higher the order of control parameters of the polynomial was, the worse the mathematic behavior of control parameters was. [PERSON] et al. [13]discovered that the higher the order of the Bernstein polynomial was, the closer the coefficient matrix was to morbidity.  
Essentially the parameterization equation of the CST method is a linear equation. The coefficient matrix of the parameterization equation is ill-conditioned in most cases. And the higher the order of the coefficient matrix is, the larger the condition number is. The larger the condition number of the coefficient matrix is, the closer the coefficient matrix is to morbidity. The solution of the linear equation is very sensitive to the preturbations of the coefficient matrix. These preturbations can lead to the inaccurate solution. Therefore, an inaccurate Bernstein coefficient solution of the CST method can be obtained for the morbidity.  
Aiming at the ill-conditioned linear equation, [PERSON] et al. [14] proposed a new iterative improvement which was more effective for the ill-conditioned equation. The ill-conditioned problem could be converted into a series of relatively well-conditioned problems. [PERSON] et al. [15] put forward a new iterative refinement to solve the ill-conditioned problem and the two-step iterative method could refine the ill-conditioned equation.  
The CST method is adopted to reproduce the NACA0012, RAE2822, and S1223 airfoils in this paper. When the number of control parameters for the S1223 airfoil increase, the Bernstein coefficient matrix is close to ill-conditioned and the condition number of this matrix increases. The slight fluctuations of the control point coordinates or the Bernstein coefficient matrix can lead to the huge change of Bernstein coefficients. So two similar shape airfoils have entirely different control parameters and it is unfavorable for the global optimization of airfoils to select the control parameters.  
The CST with the pivot element weighting iterative method is proposed to improve the ill-conditioned Bernstein coefficient matrix. And the ill-conditioned Bernstein coefficient matrix trends to be well-conditioned. So accurate control parameters can be calculated and the robustness of the CST method is also improved. The range of control parameters has a great influence on the airfoil optimization. Therefore normalization method is put forward to control the range of parameters. Genetic algorithm is adopted in the S1223 airfoil optimization.