## 3 Uncertainty-Based Optimization Design for the Multilayer TPS of Hypersonic Wing
### Robust Optimization Design for the Multilayer TPS  
The uncertainty-based optimization design of the multilayer TPS tiles is conducted to alleviate the effect of the uncertainties in this section. The robust optimization method proposed in Section 3 is employed here. The goal of the robust optimization is to reduce the weight of the multilayer TPS by designing the geometry parameters, while satisfying the constraints so that the upper values of the maximum temperature of each layer and the inner surface does not exceed the corresponding allowable value. In the meantime, the fluctuations of the four maximum temperature responses are not inferior to those of the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & \(\mathbf{T_{1\_\_\max}}\) & \(\mathbf{T_{2\_\_\max}}\) & \(\mathbf{T_{3\_\max}}\) & \(\mathbf{T_{4\_\max}}\) \\ \hline Quartic RSM & 0.00663 & 0.01034 & 0.01106 & 0.01003 \\ RBF & 0.00879 & 0.01021 & 0.00662 & 0.00696 \\ Kriging & 0.11527 & 0.11474 & 0.09915 & 0.09574 \\ OPM & 0.00750 & 0.01050 & 0.01086 & 0.00994 \\ \hline \hline \end{tabular}
\end{table}
Table 5: R-square error of each surrogate model.  
initial TPS structure. The mathematical expression of the robust optimization model can be written as:  
\[\left\{\begin{array}{l}find\ t_{1},t_{2},t_{3}\\ \min\ m\\ s.t.\ \overline{T_{1,\max}}\leq T_{1}^{0},\ \lambda_{T_{1,\max}}\leq\lambda_{T_{1, \max}}^{0}\\ \overline{T_{2,\max}}\leq T_{2}^{0},\ \lambda_{T_{2,\max}}\leq\lambda_{T_{2, \max}}^{0}\\ \overline{T_{3,\max}}\leq T_{3}^{0},\ \lambda_{T_{3,\max}}\leq\lambda_{T_{3, \max}}^{0}\\ \overline{T_{4,\max}}\leq T_{4}^{0},\ \lambda_{T_{4,\max}}\leq\lambda_{T_{4, \max}}^{0}\\ \text{Design Variables :}\\ t_{1}\in[0.5,1.5],\ t_{2}\in[10,30],\ t_{3}\in[20,40]\\ \text{Uncertain Parameters :}\\ \lambda_{1}\in[0.0684,0.0756],\ c_{1}\in[89.5,98.9]\\ \lambda_{2}\in[0.0523,0.0578],\ c_{2}\in[855,945]\\ \lambda_{3}\in[7.60,8.40],\ c_{3}\in[926.5,913.5],\ \varepsilon\in[0.76,0.84]\end{array}\right. \tag{18}\]  
where \(t_{1}\), \(t_{2}\), \(t_{3}\) are the thicknesses of the C/C panel, HTTI, and MTTI, respectively. \(m\) is the mass of the multilayer TPS. \(T_{1}^{0}\), \(T_{2}^{0}\), \(T_{3}^{0}\) and \(T_{4}^{0}\) are the maximum allowable temperatures of the C/C panel, HTTI, MTTI and the inner surface, respectively. The values of \(T_{1}^{0}\), \(T_{2}^{0}\), \(T_{3}^{0}\) and \(T_{4}^{0}\) are set to be 1400 K, 1400 K, 800 K and 473 K. Moreover, the allowable deviation factors for each output response, namely \(\lambda_{T_{1,\max}}^{0}\), \(\lambda_{T_{2,\max}}^{0}\), \(\lambda_{T_{3,\max}}^{0}\), \(\lambda_{T_{4,\max}}^{0}\) are calculated from the obtained uncertainty analysis results, and the values are 0.0145, 0.0159, 0.1056, 0.0585, respectively.  
mass of the multilayer TPS is reduced from 2.713 kg to 2.445 kg by 9.89%. In the meantime, for the optimized design, \(\overline{T_{3\text{ max}}}=799.9\) K, \(\overline{T_{4\text{ max}}}=472.54\) K, which satisfies the allowable temperature requirements. Moreover, the deviation factors of the uncertain responses are \(\lambda_{T_{3\text{ max}}}=0.1032\), \(\lambda_{T_{4\text{ max}}}=0.0553\), which means the robustness of maximum temperature of MTTI and the inner surface are ensured. Therefore, it can be concluded that the proposed uncertainty-based optimization method can efficiently reduce the mass of the TPS of the hypersonic wing on the premise of guaranteeing the heat-insulating performance.