# Effect of Plate Curvature on Heat Source Distribution in Induction Line Heating for Plate Forming
## 4 Results and Analysis
### Effect on Different Coil Widths  
We then applied coils with different widths to the calculation and found that the coils with different widths are affected by the curvature of the plate. In the process of plate bending deformation, the change of curvature of the plate will cause a change to the air gap. When the curvature changes are the same, the wider the coil, the higher the change of the air gap. However, at the same time, a wider coil can achieve a higher forming efficiency. Thus, comparing the effect of the initial curvature on coils with different widths can help with choosing the coil's dimensions. We chose a square coil with widths ranging from 60 mm to 180 mm for the simulation. The shape of the coil is shown in Figure 12. Calculation schemes are listed in Table 4. We compared the heat source distribution on the plate with an initial curvature equal to 0 and 1. The moving velocity was 5 mm/s. A square plate of 1 m \(\times\) 1 m was modeled in this calculation, with thickness settings of 20 mm and 16 mm. In order to make the highest temperature produced by different coils around 600 \({}^{\circ}\)C, different input powers were used on different coils. Figure 12: The shape of the inductor: (**a**) main dimensions of the coil; (**b**) cross section of the coil. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Thickness (mm)** & \(K\) = 0.25 & \(K\) = 0.5 & \(K\) = 0.75 & \(K\) = 1 \\ \hline \multirow{2}{*}{out-of-plane strain in the X direction} & 20 & 0.2505 & 0.3144 & 0.3463 & 0.3808 \\  & 16 & 0.1767 & 0.282 & 0.3537 & 0.4346 \\ \multirow{2}{*}{out-of-plane strain in the Y direction} & 20 & 0.0845 & 0.1722 & 0.2384 & 0.2939 \\  & 16 & 0.0723 & 0.1442 & 0.2233 & 0.2971 \\ \multirow{2}{*}{in-plane strain in the X direction} & 20 & 0.0925 & 0.1699 & 0.2681 & 0.3476 \\  & 16 & 0.1192 & 0.165 & 0.2377 & 0.2426 \\ \multirow{2}{*}{in-plane strain in the Y direction} & 20 & 0.0592 & 0.1164 & 0.1571 & 0.2032 \\  & 16 & 0.062 & 0.1266 & 0.1845 & 0.2433 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Maximum relative variation. Heat source distributions with different initial curvatures and different coil widths were calculated. The normalized results are shown below in Figure 13. We chose a cross section vertical to the symmetry axis (y = 0) S\({}_{2}\)-S\({}_{2}\) to compare the distribution of the heat generation (see Figure 14). The heat generation concentrated in the area under the coil, and the shape of the area was similar to the coil. When the coil width was smaller than 100 mm, the variation of Joule heat generation was not obvious. When the coil width was over 100 mm, the Joule heat generation was obviously reduced at the center of the workpiece. At a coil width of 180 mm and initial curvature \(K\) = 1, the Joule heat generation at the center of the workpiece was reduced by about 20%. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Initial Curvature** & **Coil Width (m)** & **Air Gap** & **Power (kW)** & **Velocity** \\  & \(K\) **(m\({}^{-1}\))** & **Coil Width (m)** & **Increment (mm)** & **Power (kW)** & **(mm/s)** \\ \hline case 6 & 0 & 0.06 & 0 & 18 & 5 \\ case 7 & 0 & 0.10 & 0 & 30 & 5 \\ case 8 & 0 & 0.14 & 0 & 42 & 5 \\ case 9 & 0 & 0.18 & 0 & 54 & 5 \\ case 10 & 1 & 0.06 & 0.45 & 18 & 5 \\ case 11 & 1 & 0.10 & 1.25 & 30 & 5 \\ case 12 & 1 & 0.14 & 2.45 & 42 & 5 \\ case 13 & 1 & 0.18 & 4.06 & 54 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculation schemes. Figure 13: Heat source distribution comparison. We performed the eight heat source models on the heat transfer and mechanical analysis, and the plastic strain distribution and displacement were calculated. Figure 15 shows the plastic strain on the bottom surface (without the inductor side) of the plate along the section \(S_{B}\)-\(S_{B}\), which is vertical to the heating line, and the thickness was 20 mm. The strain was divided into the in-plane strain and out-of-plane strain, according to Equations (12) and (13). Strain components in both the X and Y directions are shown.