# A General Integrated Method for Design Analysis and Optimization of Missile Structure
## 2 Theory  
### CST Parameterization Method  
The CST geometry representation method is a powerful parameterization that can efficiently model any airfoil in the entire design space. This method combines an analytical \"class function\" with a parametric \"shape function,\" where the class function describes an essential class of shapes, and the shape function describes the permutation around this basic shape. The CST general form of the mathematical expression representing the airfoil geometry is:  
\[\frac{z}{c}\bigg{(}\frac{x}{c}\bigg{)}=\bigg{(}\frac{x}{c}\bigg{)}^{N_{1}}\cdot \bigg{(}1-\frac{x}{c}\bigg{)}^{N_{2}}\cdot\sum\
olimits_{i=0}^{n}\bigg{[}A_{i }\bigg{(}\frac{x}{c}\bigg{)}^{i}\bigg{]}+\frac{x}{c}\frac{z_{TE}}{c}, \tag{1}\]where \(c\) is chord length, the exponents \(N_{1}\) and \(N_{2}\) define the type of geometry, \(\left(\frac{x}{c}\right)^{N_{1}}\left(1-\frac{x}{c}\right)^{N_{2}}\) is the general expression of class function, \(\sum_{i=0}^{n}\left[A_{i}\left(\frac{x}{c}\right)^{i}\right]\) is a general function that describes the unique shape of the geometry between the nose and the aft end, and the term \(\frac{z_{TE}}{c}\) provides trailing edge thickness. Correspondingly, we define the term of \(C\left(\frac{x}{c}\right)\) and \(S\left(\frac{x}{c}\right)\) to simplify the form of expression:  
\[\frac{z}{c}\left(\frac{x}{c}\right)=C\left(\frac{x}{c}\right)S\left(\frac{x}{ c}\right)+\frac{x}{c}\cdot\frac{z_{TE}}{c}, \tag{2}\]  
where  
\[C\left(\frac{x}{c}\right)=\left(\frac{x}{c}\right)^{N_{1}}\cdot\left[1-\frac {x}{c}\right]^{N_{2}} \tag{3}\]  
\[S\left(\frac{x}{c}\right)=\sum\
olimits_{i=0}^{n}\left[A_{i}\left(\frac{x}{c }\right)^{i}\right]. \tag{4}\]  
The CST parameterization method represents a two-dimensional geometry by the product of a class function \(C\left(\frac{x}{c}\right)\), and a shape function \(S\left(\frac{x}{c}\right)\), plus a term that characterizes the trailing edge thickness, as denoted in Equation (2). In addition, the first and last term of the shape function is determined by imposing boundary conditions on the airfoil shape; the result directly related to the airfoil leading edge nose radius \(R_{\text{LE}}\), trailing edge boattail angle \(\beta\), and the thickness \(z_{\text{TE}}\):  
\[S(0)=\sqrt{\frac{2R_{\text{LE}}}{c}} \tag{5}\]  
\[S(1)=\tan\beta+\frac{z_{\text{TE}}}{c}. \tag{6}\]  
As mentioned, the class function coefficients \(N_{1}\) and \(N_{2}\) determine the basic shape, whereas the shape function provides the relation that transforms the airfoil physical domain into a well behaved analytic function. The shape function is represented by a Bernstein polynomial of arbitrary order \(n\), which is composed of \(n+1\) scalable components, and the individual components in the polynomial can be scaled to define a wide variety of airfoil geometries. The definition of the order \(n\) Bernstein polynomial is:  
\[\text{BP}_{n}=\sum\
olimits_{\text{r}=0}^{n}[K_{r,n}\left(\frac{x}{c}\right)^ {r}\left(1-\frac{x}{c}\right)^{n-r}]. \tag{7}\]  
The shape function can be recast as:  
\[S\left(\frac{x}{c}\right)=\sum\
olimits_{\text{r}=0}^{n}\left[V_{\text{r}}S _{r,n}\left(\frac{x}{c}\right)\right], \tag{8}\]  
where  
\[S_{r,n}\left(\frac{x}{c}\right)=K_{r,n}\left(\frac{x}{c}\right)^{r}\left(1- \frac{x}{c}\right)^{n-r} \tag{9}\]  
\[K_{r,nwhere  
\[\mathrm{M_{r}}\left(\frac{\mathrm{x}}{\mathrm{c}}\right)=\mathrm{C}\left(\frac{ \mathrm{x}}{\mathrm{c}}\right)\mathrm{S_{r,n}}\left(\frac{\mathrm{x}}{\mathrm{ c}}\right), \tag{12}\]  
Examples of the ability of the CST method to represent typical airfoils NACA0006 and NACA4412, and the residual differences between the actual airfoil and the approximated are shown in Figures 1 and 2. In the figure of the residual differences, the range of 0-1 and -1-0 represent the upper and lower airfoil surface, respectively. Note from the figures that the differences between the actual NACA airfoil and the approximated airfoil are hardly discernible even for the Bernstein polynomial of order 5.