## 2 Robust Optimization Design for the Aerodynamic Shape of Hypersonic Vehicle
### Aerodynamic Force and Heating Analysis of the Hypersonic Wing  
To guarantee the efficiency in the optimization design, the engineering methods are applied in this work to calculate the aerodynamic force and heating of the hypersonic wing. Firstly, for the aerodynamic force, the tangent wedge method and the modified Dahlem-Buck method are utilized to compute the windward and leeside surface pressure of the hypersonic wing, in accordance with the Moore principle [27]. The pressure coefficient \(C_{p\_w}\) and \(C_{p\_J}\) of the windward and leeside surface of a wing can be determined as follows [28; 29]:  
\[C_{p\_w}=\frac{4}{\gamma+1}\times\frac{Ma_{\theta_{i}}^{2}-1}{Ma_{\infty}^{2}} \tag{6}\]  
\[C_{p\_l}=-K^{\prime}\sin^{2}\theta_{i}(1+a_{1}\theta_{i}^{n_{1}}) \tag{7}\]  
where \(Ma_{\infty}\) is the Mach number of the free-stream, \(\gamma\) is the specific heat ratio, \(a_{1}\) and \(n_{1}\) are Mach number functions, while \(Ma_{\theta_{i}}^{2}\), \(K^{\prime}\), \(\theta_{i}\) are defined in ref. [30].  
For the aerodynamic heat, the leading edge of the wing is regarded as a cylinder with an inclined angle. The two-dimension approximation method based on the three-dimension Fay-Riddell method can then be used to calculate the heat flux at the stagnation point of the hypersonic wing.  
The Fay-Riddell method [31] is utilized to estimate the aerodynamic heatflux at the three-dimension stagnation point, with the radius of the curvature of the cylinder edges as:  
\[(q_{w}^{s})_{\rm 3D}=0.763\mathrm{Pr}^{-0.6}\left(\frac{\rho_{w}^{s}\mu_{w}^{s}}{ \rho_{e}^{s}\mu_{e}^{s}}\right)^{0.1}\sqrt{\rho_{e}^{s}\mu_{e}^{s}}\left(\frac {\mathrm{d}u_{e}}{\mathrm{d}x}\right)\left[1+\left(\mathrm{L}e^{0.52}-1\right) \frac{h_{d}}{h_{e}^{s}}\right](h_{e}^{s}-h_{w}^{s}) \tag{8}\]  
where \(\mu_{w}^{s}\), \(\rho_{w}^{s}\) and \(h_{w}^{s}\) represent the viscosity, density and enthalpy on the wall, \(\mu_{e}^{s}\), \(\rho_{e}^{s}\) and \(h_{e}^{s}\) are the corresponding coefficients at the edge of boundary layer, \(h_{d}\) denotes the formation enthalpy, \(\frac{\mathrm{d}u_{e}}{\mathrm{d}x}\) is the velocity gradient.  
The aerodynamic heat flux at the two-dimension stagnation point of the hypersonic wing can then be approximated as [30; 32]:  
\[(q_{w}^{s})_{\rm 2D}=\sqrt{\frac{2}{1+k}}(q_{w}^{s})_{\rm 3D} \tag{9}\]  
where \(k=R_{1}/R_{2}\) denotes the three-dimension effect factor, \(R_{1}\) and \(R_{2}\) are the maximum and minimum radius of the curvature around the stagnation point.  
Furthermore, the aerodynamic heating on the downstream region of the stagnation point can be calculated by the plate heat flow method, while that of the transition region is obtained by the empirical hemispheric heat flow method.  
In this work, the flight attack angle of the hypersonic wing is 40\({}^{\circ}\). The other flight conditions, including the flight altitude and velocity, are shown in Figure 5. The results show that the harshest aerodynamic heating appears around 600 s, and at this moment the heat flux on the stagnation point rises to 265.6 kW/m\({}^{2}\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**No.** & **Parameters** & **Values** & **No.** & **Parameters** & **Values** \\ \hline
1 & \(S\) & 12.705 m\({}^{2}\) & 5 & \(D\) & 3.8703 deg \\
2 & \(AR\) & 1.94830 & 6 & \(x_{0}\) & 1300 mm \\
3 & \(TR\) & 0.15165 & 7 & \(y_{0}\) & 0 mm \\
4 & \(LR\) & 0.012656 m & 8 & \(z_{0}\) & 0 mm \\ \hline \end{tabular}
\end{table}
Table 1: Initial values of the independent parameters.