## 2 Mathematical Model for Aerodynamic Shape Optimization
### CST Parametrization  
The class shape transformation (CST) method is a relatively powerful and effective parameterization technique that uses a Bernstein polynomial for modeling both two-dimensional and three-dimensional aerodynamic shapes. The CST method is not limited to airfoil shapes only, but it can also generate different aerodynamic shapes. The coefficients of two arrays, shown in Equations (1) and (2), were built to define the CST method and differentiate one airfoil from another. These CST array coefficients were controlled by the curvature of the lower and upper surfaces of the airfoil [12], while the curvature of the airfoil provided a set of design data that was further utilized in the main task of shape optimization [27]. This parameterization scheme covered a wide range of airfoils and made it equally applicable to any shape of the airfoil. As described later mathematically, CST consists of two main elements: a shape function and a class function. The class function of an airfoil deals with fixed parameters inside the class function itself. The smoothness of the curve plays an important role for the CST and Bernstein polynomial, which is beneficial for optimization purposes. The relation for the CST is stated in the next paragraph. The relation for the upper and lower surface of the CST airfoil is shown below in Equations (1) and (2), and where C and S are the class and shape functions, respectively. The upper and lower surfaces are defined by the following equations:  
\[\left(\frac{z}{c}\right)_{upper}=C_{N_{2}}^{N_{1}}\Big{(}\frac{x}{c}\Big{)}S_{ l}\Big{(}\frac{x}{c}\Big{)}+\frac{x}{c}\,\frac{zu_{TE}}{c} \tag{1}\]  
\[\left(\frac{z}{c}\right)_{lower}=C_{N_{2}}^{N_{1}}\Big{(}\frac{x}{c}\Big{)}S_{ l}\Big{(}\frac{x}{c}\Big{)}+\frac{x}{c}\,\frac{z_{ITE}}{c} \tag{2}\]  
where \(zu_{TE}\) and \(z_{ITE}\) are the trailing edge thicknesses of the upper and lower surfaces, respectively. \(N_{1}=0.5\) and \(N_{2}=1\), and both values remain constant. The class function describes the main profile of the aerodynamic shape while the shape function creates a fixed shape within the same geometry class. The relation of the class function is shown in Equation (3):  
\[C_{N2}^{N1}\Big{(}\frac{x}{c}\Big{)}=\Big{(}\frac{x}{c}\Big{)}^{N_{1}}\Big{(}1- \frac{x}{c}\Big{)}^{N_{2}} \tag{3}\]  
All the airfoils which were represented by the CST method of parameterization were extracted from the class function. The exponents \(N_{1}\) and \(N_{2}\), which define the type of geometry, are replaced by the values 0.5 and 1.0, respectively, to represent the upper and lower surfaces of the airfoil. To represent an airfoil, Equations (1) and (2) can be explained by  
\[\Big{(}\frac{z}{c}\Big{)}_{upper}=C_{1.0}^{0.5}\Big{(}\frac{x}{c}\Big{)}S_{U} \Big{(}\frac{x}{c}\Big{)}+\frac{x}{c}\,\frac{zu_{TE}}{c} \tag{4}\]  
\[\Big{(}\frac{z}{c}\Big{)}_{lower}=C_{1.0}^{0.5}\Big{(}\frac{x}{c}\Big{)}S_{L} \Big{(}\frac{x}{c}\Big{)}+\frac{x}{c}\,\frac{z_{TE}}{c} \tag{5}\]  
The general shape functions \(S_{U}\) (for the upper surface) and \(S_{L}\) (for the lower surface) define particular shapes within the same airfoil class. Equations (6) and (7) show the overall shape function:  
\[S_{U}\Big{(}\frac{x}{c}\Big{)}=\sum_{i=0}^{N_{U}}W_{U}(i)\ S\Big{(}\frac{x}{c},i \Big{)} \tag{6}\]  
\[S_{L}\Big{(}\frac{x}{c}\Big{)}=\sum_{i=0}^{N_{L}}W_{L}(i)\ S\Big{(}\frac{x}{c},i \Big{)} \tag{7}\]  
where \(W_{U}\) and \(W_{L}\) are the upper and lower surface weights, respectively, \(N_{U}\) and \(N_{L}\) are the upper and lower surfaces for the CST order of the Bernstein polynomial, and \(S\) denotes the component shape function, which differs for the upper and lower surfaces on the basis of the order of the Bernstein polynomial. The component shape function is written as  
\[S\Big{(}\frac{x}{c},\ i\Big{)}=K_{i}^{N}S\Big{(}\frac{x}{c}\Big{)}^{\ i}* \Big{(}1-\frac{x}{c}\Big{)}^{N-i} \tag{8}\]  
where \(K\) is the binomial coefficient, and it is related to the order of the Bernstein polynomial.