# Airfoil Optimization Design Based on the Pivot Element Weighting Iterative Method
## 2 Airfoil Parameterization  
### Airfoil Parameterization Based on the PEWI Method  
The CST method, proposed by [PERSON] [16; 17; 18], is widely used in the airfoil optimization. The universal parameterization for the complex aircraft components is developed by using this CST method, which is not limited only to the airfoils. For the normal airfoil with a rounded leading edge and pointed trailing edge, the difficulties in representing it mathematically are due to the infinite slope and second derivative requirement at the leading edge and large variations of curvature over the shape. The CST method can overcome these limits and represent the different types of geometries in a generic way. The general mathematical expression of the two-dimensional airfoil is as follows  
\[\frac{z}{c}=\sqrt{\frac{x}{c}}\times(1-\frac{x}{c})\times\sum_{i=0}^{N}\left[ A_{i}\times(\frac{x}{c})^{i}\right]+\frac{x}{c}\times\frac{\Delta z_{te}}{c} \tag{1}\]  
This form can be rewritten as  
\[\zeta(\psi)=C(\psi)S(\psi)+\psi\zeta_{T} \tag{2}\]  
where \(\psi=x/c\), \(\xi=z/c\), \(\xi_{T}=\Delta z_{te}/c\), \(C(\psi)\) is the class function and \(S(\psi)\) is the shape function  
\[C(\psi)=(\psi)^{a}(1-\psi)^{b} \tag{3}\]  
\[S(\psi)=\frac{\xi(\psi)-\psi\zeta_{T}^{\tau}}{\sqrt{\psi}(1-\psi)}=\sum_{i=0}^ {n}A_{i}\psi_{i} \tag{4}\]The class parameters \(a\) and \(b\) for the general airfoil with a rounded leading edge and pointed trailing edge are set to 0.5 and 1.0. The Bernstein polynomial is employed as the shape function to describe the detailed shape as  
\[BP_{i,n}(\psi)=K_{i}\psi^{i}(1-\psi)^{n-i} \tag{5}\]  
where \(K_{i}\) is the binomial coefficient and \(n\) is the order of Bernstein polynomial. The shape function is  
\[S(\psi)=\sum_{i=0}^{n}A_{i}BP_{i,n}(\psi) \tag{6}\]  
The first weight coefficient of the Bernstein polynomial \(A_{0}\) is directly related to the airfoil leading edge radius by the relation:  
\[S(0)=A_{0}=\sqrt{\frac{2R_{le}}{c}} \tag{7}\]  
The last weight coefficient of the Bernstein polynomial \(A_{n}\) is directly related to the airfoil boattail angle and trailing edge thickness by the relation:  
\[S(1)=A_{n}=\tan(\theta)+\frac{\Delta z_{le}}{c} \tag{8}\]  
Therefore, \(A_{0}\) and \(A_{n}\) are two intuitive parameters of the CST method. The other coefficients of the CST method are nonintuitive. Equation (2) can be written as follows:  
\[C(\psi)\sum_{i=0}^{n}A_{i}\times\frac{n!}{i!(n-i)!}\times\psi^{i}\times(1-\psi )^{n-i}=\zeta(\psi)-\psi\zeta_{T} \tag{9}\]  
\[S_{0}(\psi)=C(\psi)\times\frac{n!}{i!(n-i)!}\times\psi^{i}\times(1-\psi)^{n-i} \tag{10}\]  
To solve for the Bernstein coefficients \(A_{i}\) to represent a given airfoil, Equation (9) can be built as follows:  
\[\left( \begin{array}{cccc}S_{0}(\psi_{0})&S_{1}(\psi_{0})&\cdots&S_{n}(\psi_ {0})\\ S_{0}(\psi_{1})&S_{1}(\psi_{1})&\cdots&S_{n}(\psi_{1})\\ \vdots&&\ddots&\vdots\\ S_{0}(\psi_{n})&S_{1}(\psi_{n})&&S_{n}(\psi_{n})\end{array} \right)\cdot\left( \begin{array}{c}A_{0}\\ A_{1}\\ \vdots\\ A_{n}\end{array} \right)=\left( \begin{array}{c}\zeta(\psi_{0})-\psi_{0}\zeta_ {T}\\ \zeta(\psi_{1})-\psi_{1}\zeta_{T}\\ \vdots\\ \zeta(\psi_{n})-\psi_{n}\zeta_{T}\end{array} \right) \tag{11}\]  
equation is given by:  
\[\textbf{{S}}\textbf{{A}}=\textbf{{Z}} \tag{12}\]  
This is an ill-conditioned linear equation. If \(\textbf{{S}}\) or \(\textbf{{Z}}\) matrix had a slight fluctuation, such as \(\delta\textbf{{S}}\) or \(\delta\textbf{{Z}}\), \(\textbf{{A}}\) might appear obvious errors. The condition number of \(\textbf{{S}}\) matrix can be used to weigh multiple of error. When the coefficient matrix \(\textbf{{S}}\) is obviously ill-conditioned, the condition number of \(\textbf{{S}}\) matrix is much larger than 1. The PEWI method [14; 15] is used to improve the condition number of \(\textbf{{S}}\). \[\textbf{{S}}+\alpha\textbf{{E}} \tag{13}\]  
where \(\textbf{{S}}\) is the \((n+1)\)th order positive matrix, \(\textbf{{E}}\) is the \((n+1)\)th order unit matrix and \(\alpha\) is the weight coefficient.