## 2 Materials and Methods
### Numerical Models  
The numerical models were prepared in the commercial computing environment (COMSOL)--specifically, using heat transfer (HT) and electromagnetic waves (EMW) modules to fully analyze the phenomenon of microwave heating. The propagation of an electromagnetic wave is defined as follows [51]:  
\[\
abla^{2}\mathbf{E}+\left(\mu\epsilon\omega^{2}-i\mu\sigma\omega\right) \mathbf{E}=0, \tag{1}\]  
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. The heat flow is characterized here by the equation [16; 52]:  
\[\rho\mathrm{C_{p}}\left(\frac{\partial\mathrm{T}}{\partial\mathrm{t}}+\
abla \mathrm{T}\right)+\
abla\mathbf{q}=\mathrm{Q}, \tag{2}\]  
where \(\rho\) is the material density, \(\mathrm{C_{p}}\) 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. And the external heat source is equal to the resistive heat generated by the electromagnetic field, which is described as follows [51]:  
\[\mathrm{Q}=1/2\mathrm{Re}((\sigma\ -\ j\omega e)\mathbf{E}\cdot\mathbf{E}^{*}), \tag{3}\]  
where \(\mathrm{Re}\) indicates the real part of the value. Figure 1 presents the geometry of the modeled horn antenna, with a rectangular waveguide. 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 \(\mathrm{TE_{10}}\), the frequency was equal to 2.45 GHz, and the total power was set to 1000 W. The energy was transmitted in a continuous manner. Figure 1: The geometry of the microwave antenna used in the numerical modeling. To thoroughly examine the functioning of the tested system, the suggested model includes four distinct variations of the sample. The sample being modeled is a concrete block of 400 \(\times\) 400 \(\times\) 45 mm, which has been reinforced with a layer of CFRP measuring 220 \(\times\) 400 \(\times\) 1 mm. A 1 mm epoxy adhesive bond was simulated between the composite layer and the concrete. All material parameters important for the modeled objects have been collected in a table. Four variants of the sample were prepared (Figure 2):  
* A sample without any defects (Figure 2a) (no-defect sample--NDS);
* A sample with a defect in the form of lack of adhesive (air void in the epoxy glue layer with dimensions of 100 \(\times\) 100 mm located centrally) (Figure 2b) (central defect sample--CDS);
* A sample with defects simulating the aging of the material, located on the edges of the composite reinforcement layer (Figure 2c) (ageing defects sample--ADS);
* A sample with defects simulating aging and with a centrally located defect in the form of lack of adhesive (Figure 2d) (central and ageing defects sample--CADS). The main challenge in designing the numerical model was to model defects resulting from the aging of the composite material. Such defects first appear on the edges of the reinforcement layer and are caused by the crumbling of the glue and the fraying of the carbon fibers. A unique approach to the problem was suggested in order to replicate a similar result in a numerical model. The aging changes were represented by a stochastic structure consisting of many cylindrical air spaces that spanned both the epoxy adhesive layer and the adjacent carbon composite layer. This method replicates both the deterioration of the adhesive and the weakening of the carbon fibers at the same time. Building such a structure in the COMSOL environment is possible thanks to the automation of the geometry creation process in the application builder.