# Application of Tucker Decomposition in Temperature Distribution Reconstruction
## 4 Simulation Experiment
### Simulation Data Reconstruction Analysis  
The feasibility of the reconstruction algorithm is verified by using the temperature distribution data calculated from the numerical simulation model.  
The temperature of 1600 locations on the three planes of z = 30 d, 45 d and 60 d of the methane combustion simulation model were selected as the research objects of the three-dimensional temperature distribution reconstruction. The selected positions include 36 measuring points on the axial 30 d, 45 d, and 60 d published by Sandia Labs so as to facilitate the experimental data reconstruction and analysis. Based on the simulation calculation results of the prior and test conditions, a three-dimensional temperature distribution reconstruction prior dataset with dimensions of 24 \(\times\) 3 \(\times\) 40 \(\times\) 40 and a test dataset with dimensions of 5 \(\times\) 3 \(\times\) 40 \(\times\) 40 were constructed.  
The 3-D temperature distribution reconstruction algorithm is used to reconstruct the test conditions' temperature distribution. The measurement data is taken from the simulation calculation data. Based on the measurement matrix, the temperature of the corresponding position in the test dataset is extracted as the temperature measurement data, which is used for the temperature distribution reconstruction.  
Taking the test case 5 as an example, the simulation results of the temperature distribution in the three target sections are shown in Figure 13.  
\begin{table}
\begin{tabular}{c c c} \hline  & **Fuel Velocity** & **Heat Flow** \\
**Condition Name** & **(m/s)** & **Velocity** \\  & & **(m/s)** \\ \hline Test Condition 1 & 32 & 10.6 \\ Test Condition 2 & 49 & 15.3 \\ Test Condition 3 & 57 & 17.7 \\ Test Condition 4 & 68 & 14.4 \\ Test Condition 5 & 77 & 26 \\ \hline \end{tabular}
\end{table}
Table 2: Boundary conditions of combustion test dataset.  
Figure 13 shows the simulation results of the three-dimensional temperature distribution calculated by using the simulation numerical software according to the geometric parameters of the established model, combustion parameters and boundary conditions of test condition 5. The white points in the Figure 13 represent the arrangement of temperature measuring points.  
The temperatures of the 12 measuring points shown in Figure 13 are used as temperature measurement data, and the reconstruction calculation is performed using the core tensor with dimensions of \(1\times 2\times 2\times 3\).  
After reconstruction calculation, it can be seen that the reconstruction error of test case 5 is 4.13%, and the temperature deviation of the reconstruction results is less than 90 K, which realizes the effective original temperature distribution reconstruction and verifies the 3-D temperature distribution reconstruction algorithm.  
In order to better show the reconstruction results, the 3D temperature distribution reconstruction calculation and analysis process presents the reconstruction effect as a contour map of the absolute value of the deviation between the reconstructed temperature value and the calculated temperature value. The absolute value of the temperature deviation between the reconstruction calculation results and the simulation calculation results is shown in Figure 14.  
In order to analyze the adaptability of the reconstruction algorithm under different working conditions, the three-dimensional temperature distribution reconstruction algorithm was used to reconstruct the five test conditions' data, and the reconstruction errors are shown in Figure 15.  
As shown in Figures 14 and 15, the three-dimensional temperature distribution reconstruction algorithm can effectively reconstruct the five different test conditions' data, which further verifies the practicability and effectiveness.  
Figure 13: Simulation results of test case 5 (Three-dimensional).