## 2 Research Methods  
### Multiphysics Simulation  
The simulation analysis of electromagnetic induction heating involves the multiphysics coupling analysis of the electromagnetic heating [18], in which the analysis of the electromagnetic field is based on Maxwell field equations summarized by [PERSON] in the 19 th century, including Gaussian law (1), Gaussian electromagnetic induction law (2), the Faraday law of electromagnetic induction (3), and the Ampere circuit law (4) [19]. The differential form is as follows:  
\[\
abla\cdot\mathbf{D}=\rho \tag{1}\]  
\[\
abla\cdot\mathbf{B}=0 \tag{2}\]  
\[\
abla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t} \tag{3}\]  
\[\
abla\times\mathbf{H}=\mathbf{J}+\frac{\partial\mathbf{D}}{\partial t} \tag{4}\]  
In the above equations, \(\mathbf{D}\) represents the electric flux density, C/m\({}^{2}\); \(\mathbf{B}\) is magnetic flux density, T; \(\mathbf{E}\) is electric field strength, V/m; \(\mathbf{H}\) is magnetic field strength, A/m; \(\mathbf{J}\) is conduction current, A; \(t\) is time, s; and \(\rho\) is the charge density, C/m\({}^{3}\). Of these, \(\mathbf{D}\), \(\mathbf{H}\), and \(\mathbf{J}\) can form the constitutive equation together with the dielectric constant \(\varepsilon\) (F/m), magnetic permeability \(\mu\) (H/m), and electric conductivity \(\sigma\) (S/m) of electromagnetic material properties as follows:  
\[\mathbf{D}=\varepsilon\mathbf{E} \tag{5}\]  
\[\mathbf{B}=\mu\mathbf{H} \tag{6}\]  
\[\mathbf{J}=\sigma\mathbf{E} \tag{7}\]  
The heat transfer phenomenon occurring in induction heating is the heat conduction in the conductor, which can be expressed by the transient heat conduction Equation (8) used for all simulations as follows:  
\[\rho C_{p}\frac{\partial T}{\partial t}=k\
abla^{2}T+Q \tag{8}\]  
In this formula, \(T\) refers to temperature, K; \(\rho\) is density, kg/m\({}^{3}\); \(C_{p}\) is the specific heat capacity, J/(kg\(\cdot\)K); \(k\) is the thermal conductivity of the material, W/(m\(\cdot\)K); and \(Q\) is Joule heat. To obtain a simplified calculation model, CATIA software was used to draw the plate-type steel for the heating mold with a mold size of 300 mm \(\times\) 200 mm \(\times\) 40 mm. In addition, COMSOL Multiphysics software was introduced to simulate the mold surface temperature distribution and heating rate under three different conditions. A specific parameter range was set for a 30 KW induction heating power supply, with a maximum current of 1400 A when the equipment was running at full power, and a safety margin was set so that the current parameter was set to a maximum of 1200 A. As the induction coils needed to be fitted as close to the mold surface as possible, 5 mm to 9 mm was the ideal range to balance the heating effect with the mold structure. High frequency induction heating required an inductor frequency of 10 KHz or more, while a 30 KW induction heating power supply was set to a maximum regulation frequency of 50 KHz, so the frequency range was 20 KHz to 40 KHz. The parameters are set as follows: (1) The depths from the coil to the mold surface were set to be 5 mm, 7 mm, and 9 mm, respectively, to explore the influence of varying depths on the uniformity of the temperature distribution on the mold surface and the heating rate. The two-dimensional diagram of the mold is shown in Figure 1a. As the inductors are made of soft coils and need to be mounted inside the mold, they need to be perforated inside the mold. This is the reason for using soft coils as inductors, which can be arranged at random inside the mold and reach the desired temperature of the mold. (2) The impact of different currents on the heating efficiency of the new induction heating coil was studied at following currents: 800 A, 1000 A, and 1200 A. (3) The frequencies were set to 20 KHz, 30 KHz, and 40 KHz, respectively, to discuss the effect of varying frequencies on this efficiency.