## 2 Robust Optimization Design for the Aerodynamic Shape of Hypersonic Vehicle
### Uncertainty Propagation Alysis of Aerodynamic Force and Heating  
In this section, the uncertainty propagation analysis of the aerodynamic force and heating for the hypersonic wing are conducted. It is well acknowledged that various types of uncertainties are involved in the aerodynamic properties of the hypersonic vehicle. Therefore, the independent uncertainties are supposed to be determined first. In the work, the flight attitude \(h\), flight speed \(v\), and flight angle of attack \(\alpha\) are served as uncertain-but-bounded parameters, while the other parameters are assumed to be constant. Meanwhile, the heat flux on the stagnation point of the wing's leading edge \(Q\), as well as the lift-drag ratio \(K\), are selected to be the output responses. Over the past years, several methods, including the first order interval Taylor expansion method [33], the interval vertex method [34], the iterative dimension-wise method [35], the direct optimization method [36], etc., have been developed to deal with the interval uncertainty propagation analysis. Although the interval vertex method, the iterative dimension-wise method, and the direct optimization method can obtain more accurate interval responses than the Taylor expansion method for situations where the strong nonlinear systems or wide range interval parameters are involved, the calculation amount would exponentially increase with the number of uncertain parameters. The calculation amount is not bearable when those three uncertainty propagation analysis methods are incorporated into the optimization design. In contrast, the interval Taylor series expansion method is an approximation method that possesses high efficiency. [PERSON] and [PERSON] [32] have proved that this method is a good choice to accomplish the uncertainty propagation analysis of aerodynamic force and heating with high efficiency and accuracy. Therefore, in this paper, the first order Taylor expansion algorithm is adopted to acquire the upper and lower bounds of the output responses, as follows:  
\[\overline{Q(h,v,\alpha)} =Q(h^{c},v^{c},\alpha^{c})+\left|\frac{\partial Q(h^{c},v^{c}, \alpha^{c})}{\partial h}\right|h^{r}+\left|\frac{\partial Q(h^{c},v^{c},\alpha ^{c})}{\partial v}\right|v^{r}+\left|\frac{\partial Q(h^{c},v^{c},\alpha^{c}) }{\partial\alpha}\right|a^{r} \tag{10}\] \[\underline{Q(h,v,\alpha)} =Q(h^{c},v^{c},\alpha^{c})-\left|\frac{\partial Q(h^{c},v^{c}, \alpha^{c})}{\partial h}\right|h^{r}-\left|\frac{\partial Q(h^{c},v^{c},\alpha ^{c})}{\partial v}\right|v^{r}-\left|\frac{\partial Q(h^{c},v^{c},\alpha^{c}) }{\partial\alpha}\right|a^{r}\]  
Figure 5: The heat flux on the stagnation point over the whole trajectory. \[\overline{K(h,v,\alpha)} =K(h^{c},v^{c},\alpha^{c})+\left|\frac{\partial K(h^{c},v^{c},\alpha^{ c})}{\partial h}\right|h^{r}+\left|\frac{\partial K(h^{c},v^{c},\alpha^{c})}{ \partial v}\right|v^{r}+\left|\frac{\partial K(h^{c},v^{c},\alpha^{c})}{ \partial\alpha}\right|\alpha^{r}\] (11) \[\underline{K(h,v,\alpha)} =K(h^{c},v^{c},\alpha^{c})-\left|\frac{\partial K(h^{c},v^{c}, \alpha^{c})}{\partial h}\right|h^{r}-\left|\frac{\partial K(h^{c},v^{c},\alpha ^{c})}{\partial v}\right|v^{r}-\left|\frac{\partial K(h^{c},v^{c},\alpha^{c})} {\partial\alpha}\right|\alpha^{r}\] where overline and underline denote the upper and lower responses, superscript \(c\) and \(r\) denote the median and interval radius of the uncertain-but-bound parameters.