## 5 Applications to Aerodynamic Design Optimization of Wing Shapes
### Wing Shape Optimization  
In this design, eight sections were used to describe the whole wing geometry for its shape optimization, its configuration and control surface distribution, as shown in Figure 7. The parameterization method described in Section 5.1 was used to control the wing shape, with a total of 72 design variables. The optimal design of the wing for the cruise factor was considered in the cruising condition at the flow condition of Mach 0.785, a 1.92\({}^{\circ}\) angle of attack and a Reynolds number of \(2\times 10^{7}\) based on the aerodynamic mean chord. HCGA was used to optimize the shape with a population size of 150 and evolutionary iterations of 100. The objective was to maximize the cruise factor, and the constraints were that the maximum thickness of each control surface and the lift coefficient should not to be reduced. The mathematical optimization model is described as follows:  
\[max:f(x)=Ma\frac{L}{D} \tag{22}\]  
\[s.t:t_{i}\geq t_{Initial}(i=1,2,\ldots,8),C_{L}\geq C_{L0} \tag{23}\]  
where \(Ma\) is Mach number, \(L\) is lift, \(D\) is drag, \(t_{i}\) is the maximum thickness of the \(i\)th section, \(t_{initial}\) is the maximum thickness of the initial control surface, \(C_{L}\) is the lift coefficient and \(C_{L0}\) is the lift coefficient of the initial wing. The flow was modeled by the compressible full potential flow with viscous boundary layer correction, and the total number of mesh points was about 0.5 million. The pressure coefficient contours on the upper and lower surface of the initial and optimized wing are shown in Figure 8. The respective pressure distributions at each section are shown in Figure 9 and the wing section shapes before and after optimization are compared in Figure 10. It is seen that the shock waves were significantly smeared owing to the shape modification, which resulted in a considerable reduction of wave drag on the upper surface and therefore in better aerodynamic performance. The aerodynamic parameters of the wing before and after optimization are shown in Table 9. The cruise factor \(Ma\times L/D\) was significantly increased from 21.863 to 26.938 because the drag coefficient \(C_{D}\) of the optimized wing was obviously reduced from 0.01605 to 0.01302. It can be seen that the cruise factor increased by 23.21%, while the drag coefficient decreased by 18.88% and the constraints of lift coefficient and thickness were satisfied. There was also no significant change in the induced drag coefficient \(C_{DIND}\) since there was no change in the lift coefficient \(C_{L}\). The wave drag coefficient \(C_{Dwave}\) of the wing was reduced apparently from 0.00240 to 0.00030, and the profile drag coefficient \(C_{DPD}\) was also reduced from 0.00680 to 0.00602. \begin{table}
\begin{tabular}{c c} \hline Parameter & Range \\ \hline \(R_{u}/c\) & (0.004, 0.012) \\ \(\beta(rad)\) & (0.14, 0.28) \\ \(\beta(rad)\) & (0.00, 0.14) \\ \(b_{1}/c\) & (0.05, 0.20) \\ \(b_{2}/c\) & (0.10, 0.30) \\ \(b_{3}/c\) & (0.10, 0.30) \\ \(b_{1}^{\prime}/c\) & (0.05, 0.30) \\ \(b_{2}^{\prime}/c\) & (0.05, 0.30) \\ \hline \end{tabular}
\end{table} 
Table 8: Range of design parameter. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(C_{L}\) & \(C_{D}\) & \(C_{Dwave}\) & \(C_{DPD}\) & \(C_{DIND}\) & \(Ma\frac{L}{D}\) \\ \hline Baseline & 0.447 & 0.01605 & 0.0024 & 0.0068 & 0.00685 & 21.863 \\ Optimized & 0.447 & 0.01302 & 0.0003 & 0.00602 & 0.0067 & 26.938 \\ \(\Delta(\%)\) & 0 & \(-0.00303\) (\(-18.88\%\)) & \(-0.00210\) (\(-87.50\%\)) & \(-0.00078\) (\(-11.47\%\)) & \(-0.00015\) (\(-2.19\%\)) & 5.075 (\(+23.21\%\)) \\ \hline \end{tabular}
\end{table} 
Table 9: Comparison of airfoil aerodynamic parameters. Figure 8: (**a**) Pressure coefficient contours on the wing upper surface of the initial wing; (**b**) Pressure coefficient contours on the wing upper surface of the optimized wing; (**c**) Pressure coefficient contours on the wing lower surface of the initial wing; (**d**) Pressure coefficient contours on the wing lower surface of the optimized wing.