## 4 Numerical Examples
### Plate Model  
The two-dimensional plate model shown in Figure 2. The size of the finite element model is \(100\) mm \(\times\)\(100\) mm, with a total of \(100\) elements, each with a side length of \(10\) mm and a plate thickness of \(1\) mm. The density of the structure is \(\rho=4000\) kg/m\({}^{3}\), the specific heat capacity is \(c=200\) J/(kg.\({}^{\circ}\)C), the thermal conductivity coefficient is \(\lambda=10\) W/(m.\({}^{\circ}\)C), the initial temperature is \(T_{0}=0\)\({}^{\circ}\)C, the calculation time is \(200\) s, and the time step is \(\Delta t=1\) s.  
Figure 1: AIFEM calculation flow chart.  
A thermal load of \(T_{\rm s}\) = 200 \({}^{\circ}\)C is applied to the left surface of the plate, a heat flow density of \(q_{\rm s}\) = 50,000 W/m\({}^{2}\) is applied to the upper surface, and the rest of the surface has adiabatic boundary conditions. The temperature gradient is not considered in the thickness direction of the plate. We set the specific heat capacity \(c\) and thermal conductivity \(\lambda\) of the material as uncertainty interval parameters, where the uncertainties were 5% and 10%, respectively. The detailed uncertainty interval parameters for the upper bounds (UB) and lower bounds (LB) are shown in Table 1. As shown in Figure 2, three red nodes are used to monitor the temperature field of the board.  
The bounds of the temperature response of plates with different uncertainties are shown in Figures 3 and 4. The green dotted line in the figure represents the median value of the temperature response, the red solid line represents the upper and lower bounds of the MCM temperature response, and the blue dashed line represents the upper and lower bounds of the AIFEM temperature response. It can be seen from Figures 3 and 4 that the temperature response boundary of the model predicted by the AIFEM is very close to the MCM, and the change rule is consistent, which demonstrates the rationality of the proposed method. The upper and lower bounds of the temperature response calculated by the AIFEM completely envelop the temperature response of the MCM, exhibiting a certain degree of conservatism. As the uncertainty level increases from 5% to 10%, the range of the model's temperature response also becomes wider. As time increases, the range of the temperature response gradually increases.  
Table 2 shows the bounds and relative errors of the temperature response of Node 2 in the plate model at different times. Comparing the AIFEM with the MCM yields relative errors. The relative errors obtained are all less than 3%. When the uncertainty reaches 10%, the AIFEM is still effective for uncertainty analysis. As the uncertainty increases, the relative error increases slightly. The accuracy of the AIFEM was verified by comparing relative errors.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Uncertainty Level (\%)** & **Bound** & \(\epsilon\) **(J/(kg\(\cdot\)\({}^{\circ}\)C))** & \(\lambda\) **(W/(m\(\cdot\)\({}^{\circ}\)C))** \\ \hline \multirow{2}{*}{5} & UB & 210 & 10.5 \\  & LB & 190 & 9.5 \\ \multirow{2}{*}{10} & UB & 220 & 11 \\  & LB & 180 & 9 \\ \hline \end{tabular}
\end{table}
Table 1: Interval parameter bounds of plate model.  
Figure 2: Plate model: ���⃝1, �⃝2, and �⃝3 respectively represent node 1, node 2, and node 3.  
Figure 4: Plate temperature response at uncertainty level 10%: (**a**) Node 1; (**b**) Node 2; (**c**) Node 3.  
Figure 3: Plate temperature response at uncertainty level 5%: (**a**) Node 1; (**b**) Node 2; (**c**) Node 3.