## 2 Calculation and Experiment of Induction Heating
### Induction Heating Experiment  
To verify the accuracy of the finite element method, an induction heating test is designed. In this test, an experimental platform made of triaxial motion of the coil is developed that can easily control the coil's positioning, motion direction and travel speed over the test plate. The positioning precision of the experimental platform is 1 mm; and the controllable speed precision is 1 mm/s. To ensure a stable distance between the coil and the plate after plate deformation, the coil is embedded in the frame of a small vehicle, and the trolley wheel of the vehicle maintains rolling contact with the plate, which ensures no contact between the coil and plate, as shown in Figure 2b. For the initial condition, the lowest position of the coil center is 5 mm from the bottom of the trolley wheel. The experimental platform and test scenario are shown in Figure 3.  
Figure 2: Shape and dimension of the induction coil: (**a**) coil model; (**b**) coil image; (**c**) coil dimension and (**d**) tubular dimension of the coil. The test plate is carbon steel, of which major thermal, physical and mechanical properties are listed in Tables 1 and 2 which is similar to Reference [13]. The plate dimensions are as follows: the length \(L=1000\) mm, the width \(B=800\) mm, and the thickness \(t=16\) mm. \(L\) is the coil movement direction over the plate. Here, \(L_{\textrm{C}}/B=0.275\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Temperature (\({}^{\circ}\)C)** & **Density (kg/m\({}^{3}\))** & **Specific Heat (J/(kg.\({}^{\circ}\)C))** &
\begin{tabular}{c} **Heat Conductivity** \\ **Coefficient (W/(m.\({}^{\circ}\)C))** \\ \end{tabular} \\ \hline
0 & 7842 & 450.36 & 66.97 \\
50 & - & 464.6 & 65.21 \\
200 & 7822 & 498.1 & 57.38 \\
250 & - & 502.26 & 54.91 \\
300 & - & 514.82 & 53 \\
400 & 7802 & 537.42 & 47.92 \\
450 & - & 623.64 & 45.83 \\
500 & - & 707.35 & 43.53 \\
600 & 7782 & 812 & 39.3 \\
650 & - & 904.07 & 36.37 \\
700 & - & 967.69 & 34.74 \\
800 & 7761 & 1026.32 & 31.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermal parameters of low carbon steel. Figure 3: Plate heating test using an induction coil. In the test, the heating line is the upper surface central line along the plate width direction (i.e., \(B/2\)). The coil length \(L_{B}\) is aligned with the plate length \(L\) and is moved along the heating line. In the test, the electromagnetic induction frequency is 15 kHz, the heat source output power is 60 kW, and the coil travel speed is 5 mm/s. As the coil moves over the plate surface in a small rolling vehicle, the distance between the coil and the plate during heating is essentially constant. In the test, the infrared temperature sensor (with a measurement range of 250-1450 \({}^{\circ}\)C) detects the plate temperature change during heating, as shown in Figure 3. Temperature sensors are deployed on one side and at the rear of the coil, and they move together with the coil, i.e., temperature sensors \(C_{1}\) and \(C_{2}\) in Figure 3. The point measured by \(C_{2}\) is 20 mm from the coil rear edge. A fixed temperature measurement point is deployed on one side of the plate to measure the temperature variation at the plate center \(T_{1}\). The final deformation of the plate is measured via a three-dimensional laser scanner measurement system after the entire plate cools to room temperature. At the initial stage of the test, the entire coil is placed over the plate, and the trailing edge of the movement direction coincides with the plate's rear end. As the coil moves continuously, the first notable phenomenon is the upward bending deformation of the plate along the \(B\) direction. In contrast, the plate deformation along the \(L\) direction is smaller and demonstrates a downward-bending trend. This phenomenon continues until the coil has completed its movement over the entire heating line. After the entire coil moves away from the plate and the plate temperature drops, the upward-bending deformation of the plate along the \(B\) direction changes slightly.