## 2 Mathematical Model  
### CST Parametrization of Airfoil  
CST is an effective parameterization technique and relatively powerful as well for modeling two-dimensional and three-dimensional aerodynamic shapes using Bernstein polynomials. The CST method is not limited to the shape of the airfoil and can also produce other aerodynamic shapes. Equations (1) and (2) are the coefficients of two arrays which are designed to show the CST method which differentiates the airfoils from one another. The curvature coefficients of the lower and upper surfaces of the airfoil are controlled by the coefficient of CST arrays [26]. These curvature coefficients further provide a set of design variables that are utilized in the optimization of the airfoil shape. This parametric scheme covers a wide range of airfoils, making it equally suitable for any shape of airfoils. Later in the paper, it is described mathematically that CST consists of two main elements, namely the shape function and the class function. The class function of the airfoil handles the fixed parameters inside the class function itself. For CST and Bernstein Polynomial the smoothness of the curve plays a crucial role for optimization purposes. CST is explained mathematically in the coming paragraph. Equations (1) and (2) shows the relation between the upper and lower surface of CST, where '\(C\)' = class function and '\(S\)' = shape function. The upper and lower surfaces are expressed as follows:  
\[\Big{(}\frac{y}{c}\Big{)}_{upper}=C_{N_{2}}^{N_{1}}\Big{(}\frac{x}{c}\Big{)}S_{ l}\Big{(}\frac{x}{c}\Big{)}+\frac{x}{c}\,\frac{z_{u,TE}}{c} \tag{1}\]  
\[\Big{(}\frac{y}{c}\Big{)}_{lower}=C_{N_{2}}^{N_{1}}\Big{(}\frac{x}{c}\Big{)}S_ {L}\Big{(}\frac{x}{c}\Big{)}+\frac{x}{c}\,\frac{z_{l,TE}}{c} \tag{2}\]  
where, \(z_{u,TE}\) and \(z_{l,TE}\) are trailing edge thickness of upper and lower surface, respectively. \(N_{1}=0.5\) and \(N_{2}=1\) is constant. The class function draws the main contours of the aerodynamic shape, while the shape function creates a fixed shape within the same geometric category. The class function can be expressed by the following equation  
\[C_{N_{2}}^{N_{1}}\Big{(}\frac{x}{c}\Big{)}=\Big{(}\frac{x}{c}\Big{)}^{N_{1}} \Big{(}1-\frac{x}{c}\Big{)}^{N_{2}} \tag{3}\]The CST parameterization method airfoils are derived from class function. The exponent \(N_{1}\) and \(N_{2}\) are replaced by the values 0.5 and 1.0, respectively to represent the upper and lower surface of an airfoil [26]. To show an airfoil, Equations (1) and (2) can be written as:  
\[\left(\frac{y}{c}\right)_{upper}=C_{1.0}^{0.5}\left(\frac{x}{c}\right)S_{U} \left(\frac{x}{c}\right)+\frac{x}{c}\;\frac{zu_{LTE}}{c} \tag{4}\]  
\[\left(\frac{y}{c}\right)_{lower}=C_{1.0}^{0.5}\left(\frac{x}{c}\right)S_{L} \left(\frac{x}{c}\right)+\frac{x}{c}\;\frac{z_{l,TE}}{c} \tag{5}\]  
The general shape function for \(S_{U}\) (for upper surface) and \(S_{L}\) (for lower surface) defines specific shapes within the same class of airfoil as shown in Equations (6) and (7):  
\[S_{U}\left(\frac{x}{c}\right)=\sum_{i=0}^{N_{U}}W_{U}(i)\;S\left(\frac{x}{c},i\right) \tag{6}\]  
\[S_{L}\left(\frac{x}{c}\right)=\sum_{i=0}^{N_{L}}W_{L}(i)\;S\left(\frac{x}{c},i\right) \tag{7}\]  
where \(W_{U}=\) upper surface weights and \(W_{L}=\) lower surface weights, \(N_{U}\) and \(N_{L}\) denote the number of shape functions for the upper and lower surface, respectively. \(S\) is a component shape function that is different for the upper and lower surface.