## 2 Forward Model
### Finite Element Model  
In order to simulate the temperature distribution in the graphite disk for specific current and frequency values, a Finite Element (FE) model, with cylindrical symmetry, is set up. A weakly coupled magneto-thermal problem is solved: the magnetic problem is solved in time-harmonic conditions with the A-V formulation. In particular, the magnetic problem, solved in terms of the phasor of the magnetic vector potential, \(\hat{\mathbf{A}}\), involves the equation [25; 26]:  
\[\
abla^{2}\hat{\mathbf{A}}-j\omega\mu\sigma\hat{\mathbf{A}}=-\mu\hat{\mathbf{J}} \tag{1}\]  
where \(\hat{\mathbf{J}}\) is the phasor of the current density, \(\mu\) and \(\sigma\) are the material permeability and conductivity, respectively, and \(\omega\) the angular frequency relevant to the frequency \(f\) of the current. The model is solved for each applied frequency, in this case, 2 frequencies were considered. Finally, the power in the disk is evaluated using the superposition approach. In turn, the thermal problem is solved in the steady-state condition, considering the power density in the disk computed after the magnetic field analysis, to calculate the heat source. Specifically, the thermal problem is solved only in the graphite disk domain and  
Figure 1: Geometry of the induction heating device. takes into account both convective and radiative contributions to heat exchange with the surrounding environment. The power density \(\sigma\omega^{2}A^{2}\) calculated by solving the magnetic problem is the thermal source, i.e., the Joule term in the Fourier equation. The latter is used to evaluate the distribution of the temperature \(T\) in the material [27]:  
\[-\
abla(k\
abla\mathrm{T})=\omega^{2}\sigma\
orm{\dot{A}}^{2} \tag{2}\]  
where \(k\) is the thermal conductivity of the material. Along the disk boundary, the thermal model is subject to the following boundary conditions:  
\[\frac{\partial T}{\partial n}=0\quad at\quad r\ =\ 0 \tag{3}\]  
where \(n\) is the normal direction and  
\[-k\frac{\partial T}{\partial n}=h(T-T_{0})+\epsilon k_{B}\Big{(}T^{4}-T_{0}^{ 4}\Big{)} \tag{4}\]  
where \(k_{B}\) is the Stefan-Boltzmann constant, \(T_{0}\) is the external temperature, and \(h\) and \(\epsilon\) are the convective exchange coefficient and emissivity coefficient, respectively. As far as the convective exchange coefficient \(h\) and emissivity coefficient \(\epsilon\) are concerned, their identification is not trivial due to several factors affecting them, for instance, the surface temperature. For the purpose of modeling, values of \(h\) = 10 Wm\({}^{-2}\)K\({}^{-1}\) and \(\epsilon\) = 0.8 have been considered, identified by means of some measurement experiences on a prototype. The external temperature, \(T_{0}\), is set to 850 \({}^{\circ}\)C. The coupled problem is solved with the commercial software COMSOL 6.0 [28]. The values of the electrical, magnetic, and thermal characteristics of the model materials are shown in Table 1.  
The mesh used for solving the FE problem is composed of both mapped domains and domains with tetrahedral mesh; it exhibits around 19,000 elements with a quadratic variation in potential. Figure 2 shows details of the mesh. For the graphite and ferrite parts, a mapped mesh has been used, which means that a structured mesh composed of rectangular elements discretizes relevant sub-domains. Typical magnetic field and temperature maps are shown in Figure 3.  
The solution of a Finite Element model with two coils fed by currents at different frequencies is not straightforward [29], hence, the superposition approach has been chosen. Two magnetic simulations have been implemented, each with just one coil on, fed by a single frequency. This way, the mutual inductance between the two coils is neglected. The power densities calculated from each simulation are superposed and they are the heat source for the thermal simulation: only one thermal simulation is subsequently solved.