## 2 Topology Optimization Method
### Thermo-Mechanical Analysis Model  
Thermal-mechanical analysis models were used to determine the structural responses of ITPSs under aerothermal heat and aerodynamic pressure load. In this paper, a corrugated sandwich-structure ITPS was selected as reference design. The initial ITPS unit cell consists of: an Inconel 718 top face sheet (TFS) panel; two Ti-6 Al-4V webs; an Al 2024 bottom face sheet (BFS) panel; and Saffil insulation that fills the space between the TFS and the BFS. The material properties are plotted in Appendix A (Figures A1-A3). Half of the unit-cell model and corresponding geometry parameters are shown in Figure 1. L is the half-width of the unit cell, h is the height of the panel, \(\theta\) is the inclination of the web, and \(t_{T}\), \(t_{B}\) and \(t_{W}\) are the thickness of the TFS, BFS, and web, respectively. The initial parameters were obtained from [1], as shown in Table 1.  
Heat transfer analysis was conducted to derive the thermal response of the ITPS. A typical incident heat flux, as presented in Figure 2, was imposed on the external surface of the TFS. The governing equation is as follows, where \(x\) and \(z\) are coordinates, \(t\) is time, \(k\) is conductivity, \(\rho\) is density, \(c\) is specific heat and \(T\) is temperature. \[\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right)+\frac{ \partial}{\partial z}\left(k\frac{\partial T}{\partial z}\right)=\rho c\frac{ \partial T}{\partial t} \tag{1}\]  
A large portion of aerothermal heat is radiated out to the ambient space by the TFS; still, part of that heat is conducted to the BFS through webs and insulation material. It is assumed that there is no lateral heat flow out of the unit cell and the BFS is perfectly insulated, which indicates the BFS temperature would rise to a maximum after vehicle touchdown. During this period, in addition to radiation, convective heat transfer is imposed on the TFS as boundary conditions. Ambient temperatures during the re-entry period are listed in Table 2. Boundary conditions on S\({}_{1}\) are:  
\[-k\frac{\partial T}{\partial z}=q-\varepsilon_{s}\sigma_{s}\left(T_{T}^{4}-T_ {A}^{4}\right)0s\leq t\leq 2175\text{s} \tag{2}\]  
\[-k\frac{\partial T}{\partial z}=q-\varepsilon_{s}\sigma_{s}\left(T_{T}^{4}-T_ {A}^{4}\right)-h\left(T-T_{A}\right)2175\text{s}\leq t\leq 5175\text{s} \tag{3}\]  
In the above equations, \(T_{T}\) and \(T_{A}\) are TFS temperature and environment temperature, respectively, h is convective heat transfer coefficient and \(q\) is the heat flux depicted in Figure 2. \(\sigma_{s}\) is the Stefan-Boltzmann constant, and \(\varepsilon_{s}\) is emissivity; a typical value of 0.85 is selected in this study according to [3]. Thermo-mechanical coupling analysis was carried out to derive mechanical performance of the ITPS. Temperature fields derived from heat transfer analysis were applied to the coupling analysis model as body loads. The stress-strain relationships are:  
\[\varepsilon_{xx}=\frac{\partial U_{x}}{\partial x}=\frac{1}{E}(\sigma_{xx}- \mu\sigma_{zz})+\alpha\Delta T \tag{4}\]  
\[\varepsilon_{zz}=\frac{\partial U_{z}}{\partial z}=\frac{1}{E}(\sigma_{zz}- \mu\sigma_{xx})+\alpha\Delta T \tag{5}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Time Period/s** & **Ambient Temperature/K** \\ \hline
0–450 & 213 \\
450–1575 & 243 \\
1575–2175 & 273 \\
2175–5175 & 295 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ambient temperatures. Figure 2: Aerodynamic load during re-entry time [44]. \[\gamma_{xz}=\frac{\partial U_{z}}{\partial x}+\frac{\partial U_{x}}{\partial z}= \frac{\tau_{xz}}{G} \tag{6}\]  
\(\varepsilon_{xx}\), \(\varepsilon_{zz}\) and \(\gamma_{xz}\) are strains, \(U_{x}\) and \(U_{z}\) are displacements, \(\alpha\) is the coefficient of thermal expansion, \(\Delta T\) is temperature difference, and E, G and \(\mu\) are tension modulus, shear modulus and Poisson's ratio, respectively. Besides the thermal loads, aerodynamic pressure load shown in Figure 2 was imposed on the outer surface of the TFS.