# Fast Temperature Field Extrapolation Under Non-Periodic Boundary Conditions
## 4 Results and Discussions
### Accuracy Analysis of POD-FVM Reduced-Order Extrapolation  
The key thermal indicators of a structure with explosives are temperature and explosion time. Figure 8 shows the temperatures of the cut surface of different mode sets after 1500 s. Except for the Z positions of 5 mm and 355 mm for mode set 1, the other mode sets have similar temperatures to the FVM result and show good agreement across all the cuts. This is due to the inability to capture slight energy thermal field trends in the case of low mode sets. As a result, the high-temperature areas are additionally highlighted. The deviation is gradually eliminated as the number of mode sets increases. From the temperature of the cut surfaces, the comprehensiveness of the thermal field in the extrapolation results of the algorithm can be demonstrated. By harnessing the POD-FVM reduced-order extrapolation method, the precise moment of detonation is swiftly forecast. Mode sets 2, 3, and 4 accurately predict the explosion time, whereas mode set 1 does not. Figure 9 compares the explosion time predicted by the reduced-order method with the experimental data and delineates the relative errors for the various mode sets. Among the four mode sets, mode set 3 achieves the lowest  
Figure 8: Temperature of the cut surfaces with different mode sets after 1500 s. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode Set** & \(\gamma\)**Value** & **Number of POD Modes** & **Energy Ratio** \\ \hline
1 & 99\% & 6 & 99.2004\% \\
2 & 99.9\% & 13 & 99.9029\% \\
3 & 99.99\% & 26 & 99.9906\% \\
4 & 99.999\% & 55 & 99.9990\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Selection of the POD mode set. relative error at 3.46%, with the maximum error remaining below 10%. This evidence supports the conclusion that the reduced-order model accurately predicts the explosion time. Moreover, the analysis indicates that the limited number of snapshots hinders the method's ability to accurately capture the strong nonlinear heat transfer processes inherent in the explosion. This insufficiency results in the excessive smoothing of localized hot spots within the explosive material, ultimately causing mode sets that incorporate more modes to predict a delayed explosion time. The analysis in Section 3.2 and the simulated explosion time results show that the temperature change on the facing-fire surface of the structure is much more drastic. Specifically, the temperature change at the junction of the facing-fire surface and the side of the structure is critical and will reflect the detonation point of the explosive. Therefore, in order to further analyze the temperature field in each layer of the structure, three representative points are selected. Figure 10 shows the location of these three points. Figure 11 shows the temperatures of three representative points predicted by the POD under each mode set, calculated by FVM. The temperatures are calculated for the first 1700 s.  
Figure 10: The representative points made in the structure with explosives as shown in points 1-3. Figure 9: Comparison of the explosion times of different mode sets. The graph in the upper left corner shows the temperature cloud at the moment of the explosion; the colors are only indicative of the temperature. It is evident that the temperature predictions at points 1 and 2 are successfully captured by four POD mode sets, and the temperature pulsation characteristics at point 3 are also successfully captured. Each layer of the structure with explosives achieves thermal protection and the temperature decreases from the outside to the inside. In order to quantitatively examine the influence of different mode sets on the prediction results, the relative error formula is used to calculate the error of each point versus time:  
\[\delta_{e}=\frac{|T_{\mathrm{POD}}-T_{\mathrm{FVM}}|}{T_{\mathrm{FVM}}}\times 1 00\% \tag{19}\]  
where \(T_{\mathrm{POD}}\) represents the temperature obtained by the POD of each POD mode set, and \(T_{\mathrm{FVM}}\) represents the temperature obtained by the FVM standard solution.