## 2 Calculation and Experiment of Induction Heating
### Comparison between Experiment and Finite Element Method  
Several sections and points in Figure 5 are selected to compare the temperature and deformation from the finite element calculation with the test results. The sections include the \(S_{L1}\)-\(S_{L1}\) section, which coincides with the heating line, the \(S_{L2}\)-\(S_{L2}\) section at the plate end, which is parallel to the heating line, the \(S_{B2}\)-\(S_{B2}\) section at the plate end, which is vertically oriented to the heating line and the \(S_{B1}\)-\(S_{B1}\) section at \(L/2\). The point selected is point \(T_{1}\) at the plate's center. Figure 6 compares the temperature variation from the finite element calculations with the test results at the plate's central point, \(T_{1}\), during heating. The diagram shows that based on the coil's dimensions, time record and speed, the relative position of the coil versus \(T_{1}\) at any moment can be derived. The finite element calculation results show that before 50 s, the temperature at \(T_{1}\) does not rise. Next, the temperature fluctuates in a small range and rises continuously until approximately 97 s. Thereafter, it declines steadily. The coil center passes \(T_{1}\) approximately 80 s after the beginning of the test. When the coil center and \(T_{1}\) coincide, the temperature at \(T_{1}\) is not at the highest level. Figure 4: Induction coil heating deformation analysis of the experiment: (**a**) local heating contraction force and (**b**) effect of the resultant force of contractions along the \(L\) direction. Figure 5: Point and sections for comparing the finite element calculation results with the test results. The finite element calculation shows that the temperature at \(T_{1}\) fluctuates slightly before reaching its highest level. There are two reasons for this: the coil panel is not continuous. Before passing \(T_{1}\), there is a large amount of heat dissipation from the coil toward the unheated \"cold plate\" in the front. After the coil passes \(T_{1}\), the \"cold plate\" before \(T_{1}\) becomes a \"hot plate\", and the temperature at \(T_{1}\) declines steadily after a short period of fluctuation and increase. Additionally, because the temperature monitoring point at \(T_{1}\) is blocked by the moving coil near 80 s, the measurement data during this period are invalid. Because the temperature is measured at regular intervals (a discontinuous measurement method), small temperature fluctuations are not reflected. However, the finite element calculation results and the test data are closely related throughout the process. To further validate the finite element calculation method, data are collected from the temperature sensor \(C_{2}\) for comparison. The temperature variation versus time at \(C_{2}\) is the variation of the highest temperature at the plate heating line after 4 s (\(C_{2}\) is 20 mm from the coil edge, and the coil travel speed is 5 mm/s). The test results are compared with the finite element calculation and shown in Figure 7, which confirms that the two match closely. Figure 8 shows the heat distribution from the finite element model of heat transmission after heating for 115 s. The plate deformation shapes from the test and the finite element calculation are shown in Figure 9. Both results are saddle-shaped. Figures 10 and 11 show the bending deformations of the heated plate in the transversal and longitudinal directions. The diagrams show that the finite element calculation and the test result have an excellent correlation. The maximum relative errors of  
Figure 6: Comparison of temperature history at \(T_{1}\). Finite Element Method (FEM), Exponential (EXP). Figure 7: Comparison of temperature at the induction coil heating line \(S_{L1}\)-\(S_{L1}\). the transversal and longitudinal deflections are approximately 10% and 9%, respectively, and both are located at the center of bending. Figure 8: Heat distribution in the finite element model of heat transmission after 115 s.  
Figure 10: Comparison of plate transversal bending deformation.