## 4 Numerical Examples
### TPS Model  
We selected the three-dimensional structure of a single cell from the TPS model for research, and conducted heat transfer analysis on the main stressed structures, as shown in Figure 5. The size of the finite element model is 100 mm \(\times\) 100 mm \(\times\) 40 mm, with a total of 300 elements, each with a side length of 10 mm. The thicknesses of the top face sheet, web, and bottom face sheet are all 5 mm. The TPS model materials all use titanium alloy. The density of the titanium alloy is \(\rho\) = 4500 kg/m\({}^{3}\), the specific heat capacity is \(c\) = 611 J/(kg.\({}^{\circ}\)C), the thermal conductivity is \(\lambda\) = 6.8 W/(m.\({}^{\circ}\)C), the initial temperature is \(T_{0}\) = 0 \({}^{\circ}\)C, the calculation time is 600 s, and the time step is \(\Delta t\) = 1 s. A heat flow density of \(q_{\text{s}}\) = 1 \(\times\) 10\({}^{5}\) W/m\({}^{2}\) was applied to the top face sheet of the TPS, and adiabatic boundary conditions were applied around the periphery and underside of the model. The convective heat transfer coefficient is \(h\) = 20 W/(m\({}^{2}\)-K), the ambient temperature is \(T_{a}\) = \(-\)80 \({}^{\circ}\)C, the surface emissivity of the top face sheet is \(\varepsilon\) = 0.9, and the Stefan-Boltzmann constant is \(\sigma\) = 5.67 \(\times\) 10\({}^{-8}\) W/(m\({}^{2}\)-K\({}^{4}\)). We set the specific heat capacity \(c\), heat conduction coefficient \(\lambda\), heat flow density \(q_{\text{s}}\), and ambient temperature \(T_{a}\) as uncertainty interval parameters, where the uncertainties were 5% and 10%, respectively. The detailed uncertainty interval parameter bounds are shown in Table 3. As shown in Figure 5, three red nodes are used to monitor the temperature field of the TPS. Node 1 is at the connection between the web and the top face sheet, and Nodes 2 and 3 are both on the web. The bounds of the TPS temperature responses with different uncertainties are shown in Figures 6 and 7. It can be seen from Figures 6 and 7 that the obtained interval temperature response boundary has a significant deviation from the median value. As the uncertainty increases, the deviation becomes more obvious. Therefore, it is necessary to analyze the uncertainty of the TPS under transient heat transfer. The TPS temperature response boundary calculated by the AIFEM is consistent with and very close to the boundary change rule of the MCM, which proves the rationality of the AIFEM. The temperature response boundary of the AIFEM calculations is wider, completely enveloping the temperature response of the MCM calculations. The AIFEM is conservative and can provide a safer theoretical solution for the design of TPSs under transient heat transfer conditions. As the uncertainty increases from 5% to 10%, the temperature response range of the TPS becomes wider. As time increases, the range of the TPS temperature response gradually increases. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Uncertain Level (\%)** & **Time (s)** & **Bound** & **AIFEM (\({}^{\circ}\)C)** & **MCM (\({}^{\circ}\)C)** & **Error (\%)** \\ \hline \multirow{4}{*}{5} & \multirow{4}{*}{100} & UB & 101.71 & 99.90 & 1.81 \\  & & LB & 83.52 & 85.14 & 1.90 \\  & & UB & 179.61 & 176.42 & 1.81 \\  & & LB & 156.16 & 159.11 & 1.85 \\  & & UB & UB & 110.26 & 107.61 & 2.46 \\  & & LB & 76.08 & 78.02 & 2.49 \\  & & UB & UB & 190.43 & 185.26 & 2.79 \\  & & LB & LB & 146.33 & 150.59 & 2.83 \\ \hline \end{tabular}
\end{table} 
Table 2: Temperature response bounds and relative error at Node 2 of plate model. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Uncertainty Level (\%)** & **Bound** & \(c\) **(J/(kg.\({}^{\circ}\)C))** & \(\lambda\) **(W/(m.\({}^{\circ}\)C))** & \(q_{\text{s}}\) **(W/m\({}^{2}\))** & \(T_{a}\) (\({}^{\circ}\)C)** \\ \hline \multirow{4}{*}{5} & UB & 641.55 & 7.14 & \(1.05\times 10^{5}\) & \(-\)76 \\  & LB & 580.45 & 6.46 & \(0.95\times 10^{5}\) & \(-\)84 \\  & UB & 672.10 & 7.48 & \(1.1\times 10^{5}\) & \(-\)72 \\  & LB & 549.90 & 6.12 & \(0.9\times 10^{5}\) & \(-\)88 \\ \hline \end{tabular}
\end{table}