## 2 Calculation and Experiment of Induction Heating  
### Finite Element Method Calculation  
The entire calculation process is divided into three steps: electromagnetic calculation, heat transfer calculation and deformation calculation. The electromagnetic calculation is to obtain the heat flux distribution of the induction coil under specific conditions. COMSOL Multiphysics (Version 5.2, COMSOL Co., Ltd., Stockholm, Sweden) is used in this step to simulate the heat flux density along the two directions of the induction coil. The heat flux distribution calculated is then used as an input during heat transfer calculation in order to obtain the temperature field of the plate. The deformation calculation is to calculate the mechanical response of the plate under a specific temperature field. ABAQUS (Version 6.13-4, Dassault Systemes, Velizy-Villacoublay, France) is used during the last two steps. The overall process is illustrated in Figure 1.  
The induction heating frequency used for ship hull forming is usually 8-50 kHz. When current flows through an inductor that is near a steel workpiece, an eddy current is induced at the surface of the plate. Induction heat is generated in the plate. The eddy current during induction heating can be given by [PERSON]'s equations in the frequency domain as [12]:  
\[\frac{1}{\mu}\
abla^{2}A+j\omega\sigma A=J_{S} \tag{1}\]  
where \(\mu\) stands for permeability, \(A\) is the magnetic vector potential, \(\omega\) is the angular velocity, \(\sigma\) is the electric conductivity and \(J_{S}\) is the external current. The magnetic vector potential \(A\) can be obtained from Equation (1) when the external current \(J_{S}\) is given. The resistance heat generated by the eddy current in the plate can be obtained as follows:  
\[q=\frac{\left|J_{e}\right|^{2}}{\sigma} \tag{2}\]  
where \(J_{e}\) refers to eddy current density and can be described as \(J_{e}=\sigma(-\frac{\partial A}{\partial t})\). For induction heating, the eddy current generated inside the steel plate is mainly concentrated on the plate surface. For a plate during induction heating, the heat flux can be considered as a surface heat source and obtained as follows:  
\[\overline{q}=\int_{0}^{t}qdz \tag{3}\]  
where \(\overline{q}\) refers to the heat flux on the surface of the steel plate and \(t\) denotes the plate thickness. The heat flux calculated before is used as the initial input for thermal analysis. The heat flow in a plate can be expressed as follows:  
\[\rho c\frac{\partial T}{\partial t}=\frac{\partial}{\partial t}\left(\lambda \frac{\partial T}{\partial x}\right)+\frac{\partial}{\partial t}\left(\lambda \frac{\partial T}{\partial y}\right)+\frac{\partial}{\partial t}\left(\lambda \frac{\partial T}{\partial z}\right)+\overline{q} \tag{4}\]  
where \(\rho\) refers to density, \(c\) is the specific heat capacity, \(T\) is the temperature of the plate and \(\lambda\) is the thermal conductivity. Thermal deformation calculation is followed after the temperature field, which is treated as the thermal load obtained from Equation (4). The basic equations related to mechanical analysis are considered as follows:  
\[\sigma_{ij,j}+F_{bi}=0 \tag{5}\]  
where \(\sigma_{ij}\) refers to stress tensor and \(F_{bi}\) represents the body force. The stress-strain relationship can be expressed as:  
\[\{d\sigma\}=[D]\{d\epsilon\}-\{C\}dT \tag{6}\]  
Figure 1: Overall calculation process of the induction heating. where \(\{d\sigma\}\) refers to the stress increment, \([D]\) is the elastoplastic matrix, \(\{d\varepsilon\}\) is the strain increment, \(\{C\}\) is the thermal stiffness matrix and \(dT\) is the temperature increment. Deformation of the plate can be obtained by solving the equations under actual boundary conditions. The induction coil used in this study is shown in Figure 2. The entire coil is divided equally into three internal frames, as shown in Figure 2a. Considering plate bending deformation, the coil is designed to be slightly curved, and the horizontal curvature radius is 1000 mm.