# A Dynamic Model Updating Method with Thermal Effects Based on Improved Support Vector Regression
## 5 The Numerical Examples
### Thermophysical Parameters Identification of Aluminum Alloy Plate  
This numerical example shows the general process of identifying the thermophysical parameters, which vary with the temperature of some aluminum alloy plates according to the heat transfer coefficient, with temperature dependence based on SVR. The aluminum alloy plate has a size of 2 \(\times\) 0.6 \(\times\) 0.004 m, there are 750 quadrilateral shell elements in the finite element mode. In order to verify the accuracy of the thermal parameter identification method, we define the thermal boundary conditions as a heating temperature upper bound of 300 \({}^{\circ}\)C, and an ambient temperature of 20 \({}^{\circ}\)C, which is conducted from the short side of one end of the aluminum sheet to the other. To facilitate the calculation and comparison, the following assumptions are made:  
1. Assume that the real heat transfer coefficients satisfy the distribution in Table 4.
2. The emissivity of thermal radiation has been identified in previous work [19] is 0.09 in this study. 3. The influence of the screw thermal contract on the temperature distribution is ignored. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number** & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) & **Number** & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) \\ \hline
1 & \(-\)0.7 & \(-\)0.7 & 9 & 0.3 & \(-\)0.7 \\
2 & \(-\)0.7 & \(-\)0.3 & 10 & 0.3 & \(-\)0.3 \\
3 & \(-\)0.7 & 0.3 & 11 & 0.3 & 0.3 \\
4 & \(-\)0.7 & 0.7 & 12 & 0.3 & 0.7 \\
5 & \(-\)0.3 & \(-\)0.7 & 13 & 0.7 & \(-\)0.7 \\
6 & \(-\)0.3 & \(-\)0.3 & 14 & 0.7 & \(-\)0.3 \\
7 & \(-\)0.3 & 0.3 & 15 & 0.7 & 0.3 \\
8 & \(-\)0.3 & 0.7 & 16 & 0.7 & 0.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The test sample points. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(R^{2}\) & **RMSE** \\ \hline SVR-Polynomial & 0.7374 & 0.0869 \\ SVR-Gauss & 0.8768 & 0.0184 \\ Improved SVR & 0.9995 & 0.0052 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The \(R^{2}\) and RMSE. Table 5 shows the design space of the upper and lower boundaries of the modified thermophysical parameters. The Figure 4 shows that the real temperature distribution, the initial temperature distribution and the updated temperature distribution are generally consistent according to the comparison above. However, there is a dislocation in the position on the aluminum alloy plate when comparing the initial temperature distribution in the range of 200-80 \({}^{\circ}\)C with the real temperature distribution, and the updated temperature distribution and the real temperature distribution are almost the same. To further illustrate the consistency between the real temperature distribution and the updated temperature distribution based on the thermophysical parameter identification method proposed in this paper, seven typical nodes are selected in the finite element model to analyze the deviations among the real temperature, initial temperature and updated temperature. Table 6 shows the deviation comparisons. \begin{table}
\begin{tabular}{c c} \hline \hline
**Temperature\({}^{f}\)C** & **Heat Transfer Coefficients/(W/(m\({}^{2,\circ}\)C))** \\ \hline
20 & 237 \\
125 & 246 \\
175 & 268 \\
250 & 304 \\
300 & 330 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The real heat transfer coefficients distribution. Figure 4: The comparison between real temperature distribution, initial temperature distribution and updated temperature distribution. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Low Temperature/20 \({}^{\circ}\)C** & **High Temperature/300 \({}^{\circ}\)C** \\ \hline Upper (W/(m\({}^{2,\circ}\)C)) & 220 & 320 \\ Lower (W/(m\({}^{2,\circ}\)C)) & 250 & 350 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The upper and lower boundaries of modified thermophysical parameters. It is shown in Table 6 that the deviation in the initial temperature distribution and the real temperature distribution is small in the high temperature range, and the deviation gradually increases as the temperature decreases.