## 3 FEM Simulation and Experimental Validation
### Results of the FEM Simulation and Validation  
This section demonstrates the validation of the model with the experimental data, which are obtained from [21]. The experimental results include two heating conditions: single-surface and double-surface heating conditions, as shown in Figure 6a,b, respectively. The heated specimen had dimensions of 180 mm \(\times\) 20 mm \(\times\) 1.2 mm. In this experiment, the DP 980 sheets were heated until the center of the heated area reached 800 \({}^{\circ}\)C. Figure 7 presents the FE modeling of the focused IR heating conditions. The temperatures of the air and initial material were assumed as 27 \({}^{\circ}\)C (300 K). Note that the heating simulations were conducted based on the absolute temperature unit (K). The degree Celsius unit (\({}^{\circ}\)C) was used only to compare the simulation results to the experimental results. The heated sheet was modeled using the linear tetrahedron element based on the dimensions of the test specimen, as shown in Figure 7. The convective heat transfer coefficient of air \(h=35\)\(\big{(}\text{W}/\text{m}^{2}\cdot\text{K}\big{)}\) was applied, and the emissivity of the material was \(\varepsilon=0.8\). In the FE simulation, the electric-thermal analysis of the lamp was first conducted to obtain \(I_{0}\) with the model parameters listed in Table 2. \(I_{0}\) was calculated as 1777 W, and the transient simulation of the IR heating was conducted with the material properties of DP 980 in Table 3. The values for material properties are consistent with the values in [21; 39]. In the simulation of IR heating, the values in Table 1 were used for the parameters of the new model according to the focal length of the reflector. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline Elliptical reflectors & & & & & & \\ \hline \multirow{2}{*}{Type} & Focal Length & Length of Major & Length of Minor Axis & Entrance & & \\  & \((f)\)(mm) & Axis \((a)\) (mm) & \((b)\) (mm) & \((l)\) (mm) & \(\gamma\) & Lamp \(q_{0}\) (W) \\ \hline Reflector 1 & 20 & 21.6 & 9.1 & 33.94 & 0.85 & 2000 \\ Reflector 2 & 40 & 34 & 27.5 & 44.47 & 0.85 & 2000 \\ \hline \multirow{3}{*}{IR lamp} & Electrical & Diameter & Diameter of the quartz tube & Length of the quartz tube & & \\  & Resistivity & \((\Omega\cdot\text{m})\) & \((\text{mm})\) & \((\text{mm})\) & \((\text{mm})\) & \\ \cline{1-1}  & \(5.6\times 10^{-8}\) & 1.3 & 11 & 90 & 220 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the IR lamp. heating method (Figure 9b) provides much faster heating than the single-surface heating condition (Figure 8b). Figure 8c,d present the temperature distribution and heating rate, respectively, of the 40 mm focal length method in the single-surface heating condition. By comparing Figure 8b,d, it can be seen that the reflector with the 20 mm focal length provides a faster heating rate to reach the same temperature compared to the 40 mm focal length because the 20 mm focal length reflector focuses more energy on the center, as shown in Figures 4 and 5. Figure 9c,d shows the temperature distribution and heating rate, respectively, in the double-surface heating with the 40 mm focal length. The simulation predicts the experimental results well. The multiphysics simulation results with the new model agree well with the measured data in Figures 8 and 9. Figure 10 presents the simulated temperature surfaces with respect to the position in the heated area and heating time. Figure 10a-d present the temperature surfaces of the single-surface heating for the 20 mm focal length, double-surface heating for the 20 mm focal length, single-surface heating for the 40 mm focal length, and double-surface heating for the 40 mm focal length, respectively. If the position or time is fixed in the temperature surface, the heating rate or temperature distribution curve can be obtained, respectively, as shown in Figures 8 and 9; the simulation results show good agreement with the experimental results, as shown in Figures 8 and 9.