## 3 Uncertainty-Based Optimization Design for the Multilayer TPS of Hypersonic Wing
### Parametric Modeling of the Multilayer TPS  
As shown in Figure 7a, the typical hypersonic wing can be generally divided into three regions according to the aerodynamic heating effects. Correspondingly, three types of materials are involved. Region I is the leeside area, where the aerodynamic heating is relatively weak; the Flexible Reusable Surface Insulation (FRSI) is adopted here. Region II is the windward area, where the aerodynamic heating is relatively drastic; the Ceramic Matrix Composite (CMC) tile is adopted here. Region III is the leading-edge area, which suffers from the most severe aerodynamic heating; the multilayer TPS is adopted here [1]. Figure 7b describes the internal structure of the multilayer TPS, which is composed of a heat-resisting C/C panel, high temperature thermal insulation (HTTI) and middle temperature thermal insulation (MTTI).  
Note that a huge mesh amount is needed when the TPS of the whole hypersonic wing is modeled. Under this circumstance, the cost of the computation is very high and even unaffordable for the uncertainty-based optimization design process. Therefore, some reasonable simplifications are made as follows:  
1. Only the multilayer TPS is chosen for the optimization design. One reason is that Region III takes up most of the mass compared with region I and Region II; thus, Region III possesses a greater potential for the lightweight design. Another reason is that the design space of the multilayer TPS is bigger than that of either CMC or FRSI, which is a simple single-layered structure.
2. To avoid the stress concentration problem caused by the thermal deformation, the multilayer TPS is always paved as individual tiles. Flexible insulation material is filled in the gaps between the different multilayer TPS tiles, to prevent the transverse heat conduction and maintain the aerodynamics. In this case, it is rational to take one typical multilayer TPS tile for the analysis and optimization design.  
Then, the finite element analysis (FEA) parametric modeling of the typical multilayer tile is further conducted. Here, the geometry driven FEA parametric modeling method is adopted. Firstly, as shown in Figure 7c, the geometry of the multilayer tile is parametrically constructed based on the optimized aerodynamic shape in Section 3, and the thickness of each layer is chosen as the independent geometry variable. The initial thicknesses of the C/C panel, HTTI and MTTI are 2 mm, 20 mm and 30 mm, respectively. Then, as  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Parameters** & \begin{tabular}{c} **Initial** \\ **Design** \\ \end{tabular} & \
\begin{tabular}{c} **Deterministic** \\ **Optimization** \\ \end{tabular} &
\begin{tabular}{c} **Robust** \\ **Optimization** \\ \end{tabular} \\ \hline \multirow{4}{*}{Design Variables} & \(AR\) & 1.9483 & 1.9519 & 1.9403 \\  & \(TR\) & 0.1517 & 0.1524 & 0.1531 \\  & \(LR\) (m) & 0.0127 & 0.0128 & 0.0127 \\  & \(D\) (deg) & 3.8703 & 3.3023 & 3.3631 \\ \hline \multirow{4}{*}{Output Responses} & \(Q^{\prime}\) (Kw/m\({}^{2}\)) & 265.5743 & 245.8037 & 246.1267 \\  & \(Q^{\prime}\) (Kw/m\({}^{2}\)) & 11.8280 & 11.0870 & 11.0999 \\ \cline{1-1}  & \(\lambda_{Q}\) & 0.0445 & 0.0451 & 0.0451 \\ \cline{1-1}  & \(K^{\prime}\) & 1.0229 & 1.0243 & 1.0242 \\ \cline{1-1}  & \(K^{\prime}\) & 0.0245 & 0.0253 & 0.0238 \\ \cline{1-1}  & \(\lambda_{K}\) & 0.0239 & 0.0247 & 0.0233 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparisons of the three aerodynamic shapes for the hypersonic wing.  
shown in Figure 7d, by utilizing the APDL language in ANSYS product, the FEA model is automatically constructed based on the parametrical geometry. This includes mesh generation, boundary condition definition, material assignment, and so on. Specially, in the parametric FEA model, the material properties (density, conductivity, specific heat) of each layer are selected as the independent variables. The nominal values of the material properties for the multilayer TPS are listed in Table 4.