## 2 Methodology
### Model and Simulation  
In order to simplify the computational model, a two-dimensional (2D) geometry was adopted, where the scene was assumed to be constant along the direction parallel to the floor and to the MUT sheet. The domain of the problem, as shown in Figure 6, was bounded by perfect matched layers (PMLs) [22], simulating an open environment. The antennae and center of the MUT were at \(h=0.45\) m above the floor. However, as this distance was not quite far enough, a multipath propagation should be considered. To this end, we incorporated a 10 cm layer at the bottom of the geometry, in which the complex permittivity of the building material of the floor, i.e., concrete, could be set to approximate the overall effect. A study regarding the adopted values is described in the next section.  
The 2D electromagnetic wave equation governs the propagation in the domain [29], as in Equation (1), where \(\mu_{r}\) is the relative magnetic permeability, \(\mathbf{E}\) is the electric field, \(k_{0}\) is the free-space wavenumber, \(\varepsilon_{r}\) is the relative electric permittivity, \(\omega\) is the angular frequency, \(\sigma\) is the conductivity of the material and \(\varepsilon_{0}\) is the free-space electric permittivity.  
\[\
abla\times\mu_{r}^{-1}(\
abla\times\mathbf{E})-k_{0}^{2}\bigg{(}\varepsilon _{r}-\frac{j\sigma}{\omega\varepsilon_{0}}\bigg{)}\mathbf{E}=0. \tag{1}\]  
The software COMSOL Multiphysics was used to solve the problem by the finite element method (FEM) [22], with MATLAB routines controlling the variable parameters in order to simulate different scenarios, as discussed later in this section. The solution was provided in terms of a scattered electric field in the frequency domain. The electric field was polarized along the \(z\)-axis, being the coordinates \(x\) and \(y\) in the plane of the model and the coordinate \(z\) out-of-plane. In order to simulate the transmitting antenna, an incident  
Figure 6: Computational domain of the problem: a rectangular region is bounded by PML (top, bottom, left and right), with the floor just above the bottom PML. The MUT is centered in the middle of the domain; the Tx antenna is on the left and the Rx antenna is on the right. The in\(-\)plane coordinates \(x\) (horizontal) and \(y\) (vertical) are in meters.  
Gaussian beam of unitary amplitude \(\text{E}_{bg0}=(0\hat{x},0\hat{y},1\hat{z})\) V/m, focused along the \(x\)-axis, was assumed. The beam radius \(r\) is defined by Equation (2), where \(w_{0}\) is a parameter whose value was varied to test different Gaussian radii on the model.  
\[r=\frac{w_{0}2\pi}{k_{0}}. \tag{2}\]  
The focus (origin of the beam) was slightly displaced to be just outside of the air domain, in order to avoid the singularity of a punctual source inside the domain.  
According to the experimental setup, when the configuration is AIR, i.e., when there is no material sheet placed between the antennae, the MUT region in the simulation is set to be simply air, just as the major part of the domain. On the contrary, when the configuration is MUT, its dielectric properties are used. Finally, when the configuration is MET, the left interface of the MUT region is set as PEC (perfect electric conductor) [22], corresponding to the thin metal foil used in the experiment.  
A physics-controlled mesh was used, with a 0.1 wavelength as the maximum element size. Special refinement was performed where the fields were more subject to variation due to the small parts of the geometry or interfaces between materials (see Figure 7), resulting in \(76,576\) triangular elements. The normalized quality of these elements was 0.6942 (minimum) and 0.974 (average), on a scale where the maximum value of 1 means a perfect equilateral triangle, that is, fewer errors in the FEM.