# Reconstruction Optimization Algorithm of 3D Temperature Distribution Based on Tucker Decomposition
## 3 Verification and Analysis Based on Numerical Simulation
### Data Analysis and Discussion  
According to the calculation process of three-dimensional temperature profile reconstruction, the reconstruction coefficient matrix is obtained by the tensor decomposition of the prior data. The temperature values of 30 measuring points are preliminarily determined as the temperature measurement data. The core tensor with dimensions of \(9\times 4\times 5\times 3\) is used to reconstruct 1000 times repeatedly. The error of 1000 reconstruction results of test condition 5 is as follows (Test condition 5's boundary conditions are outside the data in the dataset):  
It can be seen from Figure 3 that the reconstruction errors of different measuring points are very different. The temperature field results calculated by test condition 5 and the temperature distribution results with 1.75% obtained by reconstruction are shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Working Condition** & \begin{tabular}{c} **Boundary** \\ **Condition (a)** \\ \end{tabular} & \begin{tabular}{c} **Boundary** \\ **Condition (b)** \\ \end{tabular} & \
\begin{tabular}{c} **Boundary** \\ **Condition (c)** \\ \end{tabular} &
\begin{tabular}{c} **Boundary** \\ **Condition (d)** \\ \end{tabular} \\ \hline Test Condition 1 & 821 & 1159 & 0.6 & 1.5 \\ Test Condition 2 & 1394 & 1569 & 1.1 & 2.2 \\ Test Condition 3 & 1521 & 17.7 & 2.5 & 3.6 \\ Test Condition 4 & 1777 & 1353 & 3.5 & 2.8 \\ Test Condition 5 & 1827 & 1973 & 1.2 & 3.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Boundary conditions of test dataset.  
Figure 3: 1000 reconstruction errors of condition 5.  
Figure 2: Bimodal temperature field model in the computational domain at z = 1.5 m: (a) a = 1200, b = 1200, c = 1.0, d = 3.0; (b) a = 1700, b = 900, c = 2.0, d = 3.0.  
As shown in Figure 4, the reconstruction result after preliminary selection can reconstruct the temperature field well, but this result is one of the smaller relative errors selected from the 1000 reconstructions with the different arrangements of measuring points. In order to ensure the high-precision reconstruction of the temperature field, it is necessary to further improve the accuracy on the basis of the first reconstruction calculation and to determine the number and location of the measuring points.  
In order to analyze the adaptability of the reconstruction algorithm to different operating conditions, five different operating conditions were calculated with other conditions being equal. Reconstruction error is shown in Figure 5.  
Figure 4: Bimodal temperature distribution model in the computational domain: (**a**) calculation results; (**b**) reconstruction results.  
Figure 5: Reconstruction errors of bimodal temperature distribution model.  
It can be seen from Figure 5 that, under the same calculation conditions, the error of test condition 1 is relatively large and the error of test condition 2 is small because of the different boundary conditions. In the calculation domain, the flame of test condition 1 is more inclined to the boundary, while the flame of test condition 2 is relatively located in the center of the calculation domain. Therefore, in practical engineering applications, the reconstruction error of large eccentric flames is relatively large.  
The analysis of Figures 3-5 shows that the proposed algorithm can effectively reconstruct the temperature distribution under different conditions of the bimodal three-dimensional temperature distribution model. However, it is necessary to optimize the number and arrangement of the measuring points.