## 2 Numerical Model
### Boundary Conditions and Material Properties and Meshing  
As previously stated, the modeling of microwave heating necessitates a comprehensive examination of the issue through a combination of two distinct fields of physics: electromagnetic wave propagation and heat transfer. The model presented in this study described the propagation of an electromagnetic wave through the utilization of the electromagnetic wave equation, specifically in the frequency domain [45]:  
\[\
abla^{2}\mathbf{E}+\Big{(}\mu\epsilon\omega^{2}-i\mu\sigma\omega\Big{)} \mathbf{E}=0 \tag{1}\]  
where \(\mu\) denotes the magnetic permeability, \(\epsilon\) is the electric permittivity, \(\sigma\) represents the electric conductivity, \(\omega\) is the angular frequency, and \(\mathbf{E}\) is the electric field vector. This equation is solved in all domains within the model. Conversely, the heat transfer was characterized by the heat balance equation defined as follows [12]:  
\[\rho C_{p}\bigg{(}\frac{\partial\mathrm{T}}{\partial t}+\
abla\mathrm{T} \bigg{)}+\
abla\mathbf{q}=\mathrm{Q} \tag{2}\]  
where \(\rho\) is the material density, \(C_{p}\) denotes the material heat capacity, \(\mathbf{q}\) is the heat flux associated with the convection phenomenon, \(\mathrm{T}\) is the temperature, and \(\mathrm{Q}\) denotes the external heat source. This phenomenon was exclusively taken into account within domains that were associated with the sample. The external heat source was equal to the resistive heat generated by the electromagnetic field, which is described as follows (derived from [PERSON]'s theorem [45]):  
\[\mathrm{Q}=1/2\mathrm{Re}((\sigma-j\omega\epsilon)\mathbf{E}\cdot\mathbf{E}^{ *}) \tag{3}\]  
Figure 3: Geometry of the model simulated in COMSOL: (**a**) the view on the full model and (**b**) half of the model used in the simulations. Figure 2: Models of the samples: (**a**) case without the defect, (**b**) case with the defect located in the center of the sample, and (**c**) case with the defect located away from the center of the sample (dimensions in mm). where Re indicates the real part of the value. Accurately simulating the phenomenon under investigation necessitates the identification of suitable boundary conditions for both electromagnetic waves and heat transfer. The simulated antenna's walls were subjected to an impedance boundary condition, which is defined as follows [46]:  
\[\sqrt{\frac{\mu}{\epsilon-j\sigma/\omega}}\mathbf{n}\times\mathbf{H}=\mathbf{n} \times\mathbf{E} \tag{4}\]  
where \(\mathbf{H}\) is the magnetic field strength vector and \(\mathbf{n}\). is the normal vector. The walls of the air-filled domain surrounding the sample were subjected to a scattering boundary condition [47]:  
\[\mathbf{n}\times(\
abla\times\mathbf{E})-j\mathbf{k}\mathbf{n}\times(\mathbf{ E}\times\mathbf{n})=0 \tag{5}\]  
where \(k\) is the wavenumber. An electromagnetic wave was produced through the application of the port boundary condition on a specific wall within the waveguide. Here, the assumed mode was TE\({}_{10}\) and the propagation constant was equal to [48]  
\[\beta=\frac{\omega}{c}\sqrt{1-\frac{\omega_{c}^{2}}{\omega^{2}}}, \tag{6}\]  
where \(\omega_{c}=c\sqrt{\left(\frac{n\omega}{a}\right)^{2}-\left(\frac{n\omega}{b} \right)^{2}}\) is the cut-off frequency of a waveguide, where \(n=1\) and \(m=0\). Finally:  
\[\beta=\frac{2\pi}{c}\sqrt{f^{2}-\frac{c^{2}}{4a^{2}}}, \tag{7}\]  
where \(a=62\) mm was the width of the rectangular waveguide, \(c\) denotes the speed of light, and \(f\) denotes the frequency.