Determination of the E ﬀ ective Electromagnetic Parameters of Complex Building Materials for Numerical Analysis of Wireless Transmission Networks

: In this paper, we present the method for determination of e ﬀ ective electromagnetic parameters of complex building materials. By application of the proposed algorithm, it is possible to analyze electromagnetic ﬁeld distribution for large-scale problems with heterogeneous materials. The two-dimensional numerical model of building components (hollow brick) with periodic boundary conditions was solved using the ﬁnite-di ﬀ erence time-domain method (FDTD) and discussed. On this basis, the resultant transmission coe ﬃ cient was found and then the equivalent relative permeability and electric conductivity of heterogeneous dielectric structures, in the developed homogenization algorithm, were identiﬁed. The homogenization of material properties was achieved by performing a multi-variant optimization scheme and ﬁnally, selecting optimal electric parameters. Despite the analysis of heterogeneous building materials, the presented algorithm is shown as a tool for the homogenization of complex structures when scattering of a high-frequency electromagnetic ﬁeld is considered.


Introduction
Wireless networks are common, modern solutions used in telecommunications and data transfer systems and operate in the high-frequency range, e.g., in the microwave range [1][2][3][4][5][6]. Nowadays, these technologies have become even more important than wired networks and are used almost everywhere, particularly when technical or environmental restrictions make it difficult to use cables or optical fibers. Wireless data transmission systems allow for relatively easy and quick connection of many devices, preserving flexible, dynamic and automatic configuration of the group of communication elements [1,7,8]. The appropriate quality of communication in the selected area, including a connection stability and transmission speed, is an essential issue when it comes to installing wireless communication systems. The structure of a wireless network requires taking into account the number of access points and their locations [5]. The discussed problems become particularly important when wireless networks are installed inside buildings, where complex geometry [3,4,[9][10][11][12][13][14] and electrical properties of materials [15,16] influence the propagation of electromagnetic (EM) waves. The impact of these viabilities was confirmed by the measurements of mobile phone signal attenuation by different building structures [12] in Global System for Mobile Communications (GDM) networks. Therefore, the age of In order to determine the effective electrical parameters of the complex material, an example of a model with four types of walls composed of clinker bricks was used. The model presented in this article was considered for single-layer walls ( Figure 1) at frequency 2.4 GHz [16]. The analysis of the presented model was extended to the variant with two-layer walls [5]. In order to compare the results at different frequencies, numerical analysis was used at typical frequencies for Wi-Fi with the grid size [24] equal to 0.0016667 m [6]. In this article, the single-layer walls and the double-layer walls at two frequencies (2.4 and 5 GHz) were calculated with the assumption that the Yee cell size is 0.001 m for more accurate results. Also, in this article, the search area for the maximum value of the electric field intensity has been doubled. The receipt of accurate data guaranteed the correctness of the obtained results for the effective parameters of the complex material. It is worth noting that the developed optimization algorithm is universal because it determines the effective parameters for other heterogeneous structures, e.g., concrete with reinforcement. The discussed computational problem is limited to the analysis of porous elements, where the electromagnetic wave propagates through a multi-layered structure composed of an air and a non-ideal dielectric. The dimensions of the brick and holes (at frequency f = 2.4 or 5.0 GHz) are comparable to the EM wavelength in air (λa = 0.125 m at f = 2.4 GHz), as shown in Figures 1 and 2, hence it is necessary to take into account the distribution and size of the holes.
The dimensions given in Figure 1 are average values calculated based on the performed measurements, wherein the method of determining the test parameters is specified in the standard [25][26][27][28]. The standard specifies the number of samples required to test each property of a masonry element. In this study, the measurements were performed on 50 randomly selected samples of each type of brick. The main reason was the lack of regulations related to the dimensions of the hollows inside the bricks. The standard defines the permissible deviations of the external dimensions of the bricks, resulting from forming, drying or firing ceramic material [26].
In order to determine the influence of the drillings on the EM field distribution, calculations were performed for the width of drillings, which was changed along the axis of the longest dimension, s, of the brick (Figure 1), s ∈ {0, 0.005, 0.007, 0.011, 0.013, 0.015, 0.017, 0.019 m}. The size of the hole affects the percentage of the lossy dielectric (clay mass) in the brick (Table 1). This percentage was estimated as: where Vd represents volume of all holes in a given brick, and Vc = h × w × l is the volume of a brick.   The crucial part in modeling large-scale systems is a precise reproduction of complex materials, e.g., hollow bricks [2,5] or concrete with reinforcement [11]. For this reason, it is necessary to replace a composite material with a homogeneous structure possessing equivalent electrical parameters [19]. The determination of properties and formulation of the homogeneous models of complex materials is an important and constantly developed topic [2,15,18]. Effective electromagnetic parameters of complex materials (e.g., bricks) can be taken into account when calculating large-scale models, in which, due to the size of the numerical models, the discrete arrangement of air and ceramic mass cannot be fully reproduced.
In this article, a method of homogenization of complex building structures is presented. In Section 2, the developed, two-dimensional model of considered systems is introduced. Using the FDTD method, the numerical models of several exemplary walls made of hollow bricks are solved. The phenomena are analyzed until reaching a steady state and taking into account multiple reflections of EM waves. Then, the calculated transmission coefficient is used for the identification of equivalent permittivity and conductivity of the homogeneous wall, by comparing the obtained value with the analytical solution (transmission coefficient) of a homogeneous slab.
In Section 3, in this article, the whole diagram of the developed optimization algorithm, the adopted limitations and assumptions, the resolution of the solution search and the results for exemplary variants of the model (Section 1) composed of different bricks for single-layer and double-layer walls at 2.4 and 5 GHz are presented. The tables also include some of the best variants of solutions for bricks with 18 and 30 vertical hollows, taking into account the conductivity values most commonly used in the literature. The solution of the homogenization problem and identification of the properties, presented in Section 3, are achieved by finding a pair of optimal parameters: relative permeability and electric conductivity. The results obtained for f = 2.4 GHz and f = 5 GHz are presented and discussed.

Complex Building Materials-Hollow Clay Brick
One of the commonly used building materials is a hollow brick made of clay or other materials with the addition of sand, then dried and fired at high temperatures [25][26][27][28]. This material will be used to present the developed algorithm for determining the effective electrical parameters. The dimensions of the bricks vary greatly, but they are based on the ratio 1:2:4 of height (h), width (w) and length (l), respectively [5,25,28] (Figure 1). In the macroscopic approach, building materials (hollow bricks) are classified as complex, heterogeneous structures. The presented analysis is related with heterogeneous building walls made of two types of commonly used building materials presented in References [6,16]: (1) Hollow clay brick with 18 vertical holes (marked as B18) (Figure 1a), (2) Hollow clay brick with 30 vertical holes (B30) (Figure 1b).
The discussed computational problem is limited to the analysis of porous elements, where the electromagnetic wave propagates through a multi-layered structure composed of an air and a non-ideal dielectric. The dimensions of the brick and holes (at frequency f = 2.4 or 5.0 GHz) are comparable to the EM wavelength in air (λ a = 0.125 m at f = 2.4 GHz), as shown in Figures 1 and 2, hence it is necessary to take into account the distribution and size of the holes. Mapping of the relative dimensions of analyzed bricks, assuming that relative permittivity of the material is r' = 4.44 and conductivity  = 0 S/m, is shown in Figure 2. The dimensions of the ceramic elements are related to the wavelength inside the brick (λb = 0.0593 m at f = 2.4 GHz). In real cases, the dimensions of the building materials are not so strict, and their variability may also affect the propagation of the EM wave. Still, they can be taken into account using the discussed homogenization procedure. In this article, we have assumed averaged values based on 50 test samples. While considering the influence of building materials on the field intensity in wireless communication, a model with exact dimensions, even up to millimeters, is widely used [2,28,[35][36][37][38][39]. This is due to the fact that only external dimensions are subject to standardization of bricks and changes in the structure inside the bricks are not standardized. In the literature, the polynomial chaos expansion has been described to estimate the influence of uncertainties in geometrical and physical properties of a building facade on the scattered electric field [13].
Four types of wall in an area, S, in the base model were analyzed (Figure 3). Discussed variants were taking into account the number of brick layers and the type of brick. For example, 1w_B18 refers to a model of a single-layer wall made of B18 bricks. Based on the literature, for the analysis of The dimensions given in Figure 1 are average values calculated based on the performed measurements, wherein the method of determining the test parameters is specified in the standard [25][26][27][28]. The standard specifies the number of samples required to test each property of a masonry element. In this study, the measurements were performed on 50 randomly selected samples of each type of brick. The main reason was the lack of regulations related to the dimensions of the hollows inside the bricks. The standard defines the permissible deviations of the external dimensions of the bricks, resulting from forming, drying or firing ceramic material [26].
In order to determine the influence of the drillings on the EM field distribution, calculations were performed for the width of drillings, which was changed along the axis of the longest dimension, s, of the brick (Figure 1), s ∈ {0, 0.005, 0.007, 0.011, 0.013, 0.015, 0.017, 0.019 m}. The size of the hole affects the percentage of the lossy dielectric (clay mass) in the brick (Table 1). This percentage was estimated as: where V d represents volume of all holes in a given brick, and V c = h × w × l is the volume of a brick. The value of V %mc for the considered brick is shown in Table 1. The symbol ( # ) signifies values of the typical size of hole inside the brick, while symbol (*) was used to signify the two closest V %mc values in various sizes of holes. The electrical parameters of bricks, based on available literature, are presented in Table 2. Mapping of the relative dimensions of analyzed bricks, assuming that relative permittivity of the material is ε r ' = 4.44 and conductivity σ = 0 S/m, is shown in Figure 2. The dimensions of the ceramic elements are related to the wavelength inside the brick (λ b = 0.0593 m at f = 2.4 GHz).
In real cases, the dimensions of the building materials are not so strict, and their variability may also affect the propagation of the EM wave. Still, they can be taken into account using the discussed homogenization procedure. In this article, we have assumed averaged values based on 50 test samples. While considering the influence of building materials on the field intensity in wireless communication, a model with exact dimensions, even up to millimeters, is widely used [2,28,[35][36][37][38][39]. This is due to the fact that only external dimensions are subject to standardization of bricks and changes in the structure inside the bricks are not standardized. In the literature, the polynomial chaos expansion has been described to estimate the influence of uncertainties in geometrical and physical properties of a building facade on the scattered electric field [13].
Four types of wall in an area, Ω S , in the base model were analyzed (Figure 3). Discussed variants were taking into account the number of brick layers and the type of brick. For example, 1w_B18 refers to a model of a single-layer wall made of B18 bricks. Based on the literature, for the analysis of considered variants, the constant value of permittivity of the ceramic mass was assumed (ε r ' = 4.44) and the conductivity, σ, was changing from 0 to 0.2 S/m. Electronics 2020, 9, x FOR PEER REVIEW 6 of 19 considered variants, the constant value of permittivity of the ceramic mass was assumed (r' = 4.44) and the conductivity, σ, was changing from 0 to 0.2 S/m.

Analytical Solution
In the case of walls with a homogeneous material structure and perpendicular incidence of the wave on its boundary, the description of phenomena corresponds to the problem of plane wave propagation in an open space and its interaction with a dielectric slab [40]. Due to the time-harmonic field, vectors of electric field density of the incident wave E1+, transmitted E+1 and reflected E1-are written as complex numbers. The model shown in Figure 4 can be simplified to a one-dimensional layer model, where the area of a homogeneous, isotropic lossy dielectric S with a width, w, is surrounded by air. The properties of the surrounding medium are characterized by wave impedance. Determining the electrical properties is a fundamental condition for the correct and reliable mapping of field phenomena in analyzed building structures. Ceramic and concrete materials are generally non-ideal dielectrics. For this reason, the values of their parameters are discussed in detail, including permittivity ( = 0·r), permeability ( = 0·r) and conductivity () [40,41]. In a high-frequency regime, the relative permeability of dielectrics may be assumed as r ≈ 1. Taking into account the time-harmonic field, it is necessary to use complex permittivity of the material.

Analytical Solution
In the case of walls with a homogeneous material structure and perpendicular incidence of the wave on its boundary, the description of phenomena corresponds to the problem of plane wave propagation in an open space and its interaction with a dielectric slab [40]. Due to the time-harmonic field, vectors of electric field density of the incident wave E 1+ , transmitted E +1 and reflected E 1− are written as complex numbers. The model shown in Figure 4 can be simplified to a one-dimensional layer model, where the area of a homogeneous, isotropic lossy dielectric Ω S with a width, w, is surrounded by air. The properties of the surrounding medium are characterized by wave impedance.
Electronics 2020, 9, x FOR PEER REVIEW 6 of 19 considered variants, the constant value of permittivity of the ceramic mass was assumed (r' = 4.44) and the conductivity, σ, was changing from 0 to 0.2 S/m.

Analytical Solution
In the case of walls with a homogeneous material structure and perpendicular incidence of the wave on its boundary, the description of phenomena corresponds to the problem of plane wave propagation in an open space and its interaction with a dielectric slab [40]. Due to the time-harmonic field, vectors of electric field density of the incident wave E1+, transmitted E+1 and reflected E1-are written as complex numbers. The model shown in Figure 4 can be simplified to a one-dimensional layer model, where the area of a homogeneous, isotropic lossy dielectric S with a width, w, is surrounded by air. The properties of the surrounding medium are characterized by wave impedance. Determining the electrical properties is a fundamental condition for the correct and reliable mapping of field phenomena in analyzed building structures. Ceramic and concrete materials are generally non-ideal dielectrics. For this reason, the values of their parameters are discussed in detail, including permittivity ( = 0·r), permeability ( = 0·r) and conductivity () [40,41]. In a high-frequency regime, the relative permeability of dielectrics may be assumed as r ≈ 1. Taking into account the time-harmonic field, it is necessary to use complex permittivity of the material. Determining the electrical properties is a fundamental condition for the correct and reliable mapping of field phenomena in analyzed building structures. Ceramic and concrete materials are generally non-ideal dielectrics. For this reason, the values of their parameters are discussed in detail, including permittivity (ε = ε 0 ·ε r ), permeability (µ = µ 0 ·µ r ) and conductivity (σ) [40,41]. In a high-frequency regime, the relative permeability of dielectrics may be assumed as µ r ≈ 1. Taking into account the time-harmonic field, it is necessary to use complex permittivity of the material.
The real component ε' determines the ability of the dielectric to accumulate energy in the electric field, while the imaginary component ε" is responsible for energy loss related to displacement currents. Due to changes of polarization depending on the field frequency, the values of the permeability Electronics 2020, 9, 1569 7 of 18 components in the medium are not constant. When describing the properties of the dielectric medium, the effective parameters are defined as: Substituting the effective permittivity of lossy dielectric (Equation (4)), its wave impedance will be expressed by the formula: With the perpendicular incidence of the wave on boundary, the transmission coefficient in the Ω S area is given by the following equation [40]: where the respective field coefficients are and the wavenumber is defined as: Equation (6) makes it possible to verify the results obtained in numerical calculations using the FDTD, FDFD and FEM methods. Its applicability is limited to cases with isotropic, homogeneous materials such as concrete or brick. Maximum values of the electric field in the area behind the wall are used for determining the transmission coefficient based on Equation (6). When time domain methods are considered, the analysis in Ω 2 area must be performed in a steady state, where the multiple reflections of the wave in the wall area are present.

Mathematical Model
The dimensions of modeled structures are comparable or greater than wavelength in the microwave range ( Figure 2). The considered task comes to a solution of the boundary value problem described by partial differential equations [40,41]. The constituent materials present in the systems are continuous, isotropic media characterized by parameters ε r , µ r and σ. The finite-difference time-domain method (FDTD) was used to determine the distribution of the electromagnetic field. The FDTD is based on Maxwell's equations in the time-domain [20,24,42]: where: E-electric field in (V/m), and H-magnetic field in (A/m). The propagation of the EM wave in building structures can be simplified to 2D space. Equations (10) and (11) are discretized on the Electronics 2020, 9, 1569 8 of 18 rectangular differential grid [24]. In the time-domain, the calculations of instantaneous values of magnetic field components H x and H y are interlaced by calculations of electric field component E z .

Numerical Model
The system consisting of the wall made of building materials (hollow bricks) is analyzed. An open space with the air properties is attached to both sides of the wall, as shown in Figure 5. It was also assumed that the dimensions of the wall perpendicular to the direction of wave propagation (wall width and height) are much greater than the wavelength (λ 0 = 0.125 m at f = 2.4 GHz and λ 0 = 0.06 m for f = 5 GHz). Therefore, apart from the phenomena occurring at the ends of the wall, near edges or at the contact with another wall, it was possible to reduce the size of the model by application of periodic boundary conditions. These assumptions allowed for determining the influence of the considered materials on the distribution of the electromagnetic field in an isolated system. Electronics 2020, 9, x FOR PEER REVIEW 8 of 19 rectangular differential grid [24]. In the time-domain, the calculations of instantaneous values of magnetic field components Hx and Hy are interlaced by calculations of electric field component Ez.

Numerical Model
The system consisting of the wall made of building materials (hollow bricks) is analyzed. An open space with the air properties is attached to both sides of the wall, as shown in Figure 5. It was also assumed that the dimensions of the wall perpendicular to the direction of wave propagation (wall width and height) are much greater than the wavelength (0 = 0.125 m at f = 2.4 GHz and 0 = 0.06 m for f = 5 GHz). Therefore, apart from the phenomena occurring at the ends of the wall, near edges or at the contact with another wall, it was possible to reduce the size of the model by application of periodic boundary conditions. These assumptions allowed for determining the influence of the considered materials on the distribution of the electromagnetic field in an isolated system. The analysis of the electric field distribution in individual variants was carried out on the basis of observation of the maximum value of the Ez component in a specific area behind the wall, marked in Figure 5 by the green rectangle. The source field in the model was a harmonic, linearly polarized wave propagating in the direction of the Oy axis (k = 1y) [20,24,42] The phenomena of wave propagation in open space were imitated by adopting PML (perfectly matched layer) conditions at the edges perpendicular to the direction of plane wave propagation [42,43]. Periodic conditions were applied on the edges parallel to the wave vector. The area of the considered models was discretized using uniform Yee cells. To determine the effective electromagnetic properties of complex materials, it is important to obtain correct results from the numerical model, since the transmission coefficient calculated numerically (Te,FDTD) will be compared with the value found analytically (Te). Numerical analysis of the influence of the variability of The analysis of the electric field distribution in individual variants was carried out on the basis of observation of the maximum value of the E z component in a specific area behind the wall, marked in Figure 5 by the green rectangle. The source field in the model was a harmonic, linearly polarized wave propagating in the direction of the Oy axis (k = 1 y ) [20,24,42] E(x, y, t) = E z 1 z = sin(ωt) · 1(t) · 1 z .
The phenomena of wave propagation in open space were imitated by adopting PML (perfectly matched layer) conditions at the edges perpendicular to the direction of plane wave propagation [42,43]. Periodic conditions were applied on the edges parallel to the wave vector. The area of the considered models was discretized using uniform Yee cells. To determine the effective electromagnetic properties of complex materials, it is important to obtain correct results from the numerical model, since the transmission coefficient calculated numerically (T e,FDTD ) will be compared with the value found analytically (T e ). Numerical analysis of the influence of the variability of electrical conductivity and the Electronics 2020, 9, 1569 9 of 18 size of holes in the brick on the electric field was performed earlier in Reference [6], where the size of mesh in the models was ∆ = 0.0016667 m. In this article, calculations of T e,FDTD were performed for the grid with smaller size, ∆ = 0.001 m. The area of the considered model was discretized using uniform Yee cells in order to fulfill the required conditions [24,42] and as a result, due to the frequency of the EM field (and the wavelength in the air and brick), the maximum linear size of the Yee cell was ∆ x = ∆ y = 1 mm.

Initial Results of Numerical Analysis
The main results of the FDTD method are electric field distributions for all the time stamps. For this reason, it was necessary to develop an additional algorithm, where the aim was to determine the map describing the maximum values of the field by processing the sequence of instantaneous field distributions calculated at equal intervals of time. Figure 6 shows maximum values of the E z component (relative to source E z value) depending on the conductivity of the brick material and the size of the holes (s).
Electronics 2020, 9, x FOR PEER REVIEW 9 of 19 electrical conductivity and the size of holes in the brick on the electric field was performed earlier in Reference [6], where the size of mesh in the models was Δ = 0.0016667 m. In this article, calculations of Te,FDTD were performed for the grid with smaller size, Δ = 0.001 m. The area of the considered model was discretized using uniform Yee cells in order to fulfill the required conditions [24,42] and as a result, due to the frequency of the EM field (and the wavelength in the air and brick), the maximum linear size of the Yee cell was Δx = Δy = 1 mm.

Initial Results of Numerical Analysis
The main results of the FDTD method are electric field distributions for all the time stamps. For this reason, it was necessary to develop an additional algorithm, where the aim was to determine the map describing the maximum values of the field by processing the sequence of instantaneous field distributions calculated at equal intervals of time. Figure 6 shows maximum values of the Ez component (relative to source Ez value) depending on the conductivity of the brick material and the size of the holes (s). While passing through different areas of the brick, a local change of the speed of the electromagnetic wave leads to the temporary images of the field and proves the occurrence of interference. This effect is especially visible behind the wall made of B30 bricks (the field behind the wall has higher minimum and maximum values). Additionally, small changes in the size of holes resulted in a change of the electric field. Both high percentage of clay and sizes of holes may cause a change in transmission speed. Some waves passing through the area with ε > 1 (clay) may be delayed with relation to waves passing through mixed areas (air holes-clay). Due to these delays, especially, double-layer walls will decrease the original transmission speed. In order to evaluate possible data transfer problems, a precise visualization of the EM field distribution will be required. While passing through different areas of the brick, a local change of the speed of the electromagnetic wave leads to the temporary images of the field and proves the occurrence of interference. This effect is especially visible behind the wall made of B30 bricks (the field behind the wall has higher minimum and maximum values). Additionally, small changes in the size of holes resulted in a change of the electric field. Both high percentage of clay and sizes of holes may cause a change in transmission speed. Some waves passing through the area with ε > 1 (clay) may be delayed with relation to waves passing through mixed areas (air holes-clay). Due to these delays, especially, double-layer walls will decrease the original transmission speed. In order to evaluate possible data transfer problems, a precise visualization of the EM field distribution will be required. For this reason, it is necessary to accurately reproduce brick's structure. On the other hand, when large-scale models are analyzed, it is the biggest problem to reproduce all details of the structure, and for this reason, the homogenization is necessary.
Since the discussed systems operate at typical Wi-Fi frequencies, it is worth to consider an impact of the wall structure on the stability of communication. Firstly, the electrical conductivity has an influence on the wave absorption. If a receiver is located behind the wall, for less conductive material (σ ∈ <0, 0.08> S/m), the maximum value of E z is greater than 0.1 E source , while for higher σ, it tends to be 0.01 E source . Conductivity of the brick may vary throughout the year, thus walls with higher content of the absorbed water may cause significant loss of the signal and decrease communication stability. A similar conclusion may come from the comparison of one-and two-layer walls. For all single wall (1w) cases (Figure 6a,b) E z behind the wall is higher than 0.05 E source , however for all cases of double-layer (2w) wall, electric field intensity is lower (for the corresponding conductivity). Secondly, a higher percentage of clay material in the wall causes smaller distortion of the wave front in the area behind the wall. When discussing the wall in which the damping is insignificant (σ < 0.01 S/m) or equal to zero, interference effects play a much greater role. This may explain the non-monotonous course of characteristics and visible minimal values for some sizes of holes. A local destructive interference of electric field also increases the probability that the receiver will be present in an area where E z is very low.

Identification of Effective Electromagnetic Parameters
In Section 2, the model and results related to heterogeneous walls made of B18 and B30 hollow bricks were presented [5,6]. The results of this type of analysis will be used to develop an algorithm for determination of effective electromagnetic properties of heterogeneous material. Here, the transmission coefficient calculated numerically is the crucial parameter used for identification of effective permittivity and conductivity of the homogenized wall. The solution of the homogenization problem can be achieved by choosing optimal values of ε r ' ,opt and σ opt based on optimization task.

General Algorithm for Determining the Effective Parameters
The identification of the effective electric parameters by optimization algorithm (Figure 7) is based on minimization of the error between transmission coefficient, calculated numerically (T e,FDTD ) and defined analytically (T e ). As an example, the wall made of clinker bricks was used. For this reason, it is necessary to accurately reproduce brick's structure. On the other hand, when large-scale models are analyzed, it is the biggest problem to reproduce all details of the structure, and for this reason, the homogenization is necessary. Since the discussed systems operate at typical Wi-Fi frequencies, it is worth to consider an impact of the wall structure on the stability of communication. Firstly, the electrical conductivity has an influence on the wave absorption. If a receiver is located behind the wall, for less conductive material (σ ∈ ⟨0, 0.08⟩ S/m), the maximum value of Ez is greater than 0.1 Esource, while for higher σ, it tends to be 0.01 Esource. Conductivity of the brick may vary throughout the year, thus walls with higher content of the absorbed water may cause significant loss of the signal and decrease communication stability. A similar conclusion may come from the comparison of one-and two-layer walls. For all single wall (1w) cases (Figure 6a,b) Ez behind the wall is higher than 0.05 Esource, however for all cases of double-layer (2w) wall, electric field intensity is lower (for the corresponding conductivity). Secondly, a higher percentage of clay material in the wall causes smaller distortion of the wave front in the area behind the wall. When discussing the wall in which the damping is insignificant (σ < 0.01 S/m) or equal to zero, interference effects play a much greater role. This may explain the non-monotonous course of characteristics and visible minimal values for some sizes of holes. A local destructive interference of electric field also increases the probability that the receiver will be present in an area where Ez is very low.

Identification of Effective Electromagnetic Parameters
In Section 2, the model and results related to heterogeneous walls made of B18 and B30 hollow bricks were presented [5,6]. The results of this type of analysis will be used to develop an algorithm for determination of effective electromagnetic properties of heterogeneous material. Here, the transmission coefficient calculated numerically is the crucial parameter used for identification of effective permittivity and conductivity of the homogenized wall. The solution of the homogenization problem can be achieved by choosing optimal values of εr',opt and σopt based on optimization task.

General Algorithm for Determining the Effective Parameters
The identification of the effective electric parameters by optimization algorithm (Figure 7) is based on minimization of the error between transmission coefficient, calculated numerically (Te,FDTD) and defined analytically (Te). As an example, the wall made of clinker bricks was used.  The effective parameters ε r ' ,opt and σ opt are determined assuming identical wall thickness in the model under consideration and the equivalent medium (w). The optimization algorithm (A opt ) aims to determine the values of isotropic effective parameters of homogenized material (ε r ' ,opt and σ opt ) based on a non-heterogeneous material (e.g., hollow bricks): where: ε r ' ,min and ε r ' ,max -lower and upper limits on the domain of optimization for relative electric permittivity, σ min and σ max -lower and upper limits on the domain of optimization for conductivity, ∆ ε and ∆ σ -resolution of the domain optimization, δ A -objective function and f g -function of generating sequential solution variants. The objective function δ A which classifies calculated variants, depends on assumed values of parameters {ε r ', σ}: where: T e -transmission coefficient module determined by the analytical method (Equation (6)) for a given, iterative variable value of material parameters {ε r ', σ} and wall thickness (w), and T e,FDTD -transmission coefficient calculated based on the numerical solution (FDTD) for given properties of the ceramic material ε r ' ,FDTD and σ FDTD , wall thickness and brick's structure. The search for the optimal solution was carried out taking into account ( Figure 8): 1.

Algorithm Implementation
To identify effective electric parameters, an optimization algorithm was developed (Figure 9) based on the analytical [40,41] and numerical methods [20,24]. The overall structure was divided into two main stages, where stage I is related to the elements defined before the start of calculations, while stage II presents the process of optimization calculations. Stage II relies on iteratively performing calculations for assumed, subsequent values of εr' and :  Calculating temporary value of transmission coefficient for the successively adopted values of effective parameters (εr' and ), assuming that the wall is made of a solid brick (Equation (6)).

Algorithm Implementation
To identify effective electric parameters, an optimization algorithm was developed (Figure 9) based on the analytical [40,41] and numerical methods [20,24]. The overall structure was divided into two main stages, where stage I is related to the elements defined before the start of calculations, while stage II presents the process of optimization calculations.
Stage I:

•
Reading and processing input data related to the considered wall variant.

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Reading the value of the maximum electric field determined in Section 3 and calculating T e,FDTD .

•
Reading the data defining the range of the search domain of effective parameters and determining the accuracy of their selection.

•
Creating vectors in order to find the best solutions' effective parameters (ε r ' ,opt and σ opt ), for which the value of relative error (δ A ) is lowest.
Stage II relies on iteratively performing calculations for assumed, subsequent values of ε r ' and σ: • Calculating temporary value of transmission coefficient for the successively adopted values of effective parameters (ε r ' and σ), assuming that the wall is made of a solid brick (Equation (6)).

•
Finding the value of relative error (δ A ) using Equation (14).

•
Sorting solutions using the sort algorithm to create a list of calculated variants with the smallest approximation error (δ A ).
Electronics 2020, 9, x FOR PEER REVIEW 12 of 19  Finding the value of relative error (A) using Equation (14).


Sorting solutions using the sort algorithm to create a list of calculated variants with the smallest approximation error (A).

Effective Parameters of Hollow Bricks at f = 2.4 GHz
At specific frequency f = 2.4 GHz and conductivities FDTD ∈ ⟨0.01, 0.04⟩ S/m, usually used for the description of ceramic materials (Table 2), the effective electric parameters were found.

Effective Parameters of Hollow Bricks at f = 2.4 GHz
At specific frequency f = 2.4 GHz and conductivities σ FDTD ∈ <0.01, 0.04> S/m, usually used for the description of ceramic materials (Table 2), the effective electric parameters were found. Figures 10-13 show the distribution of the relative error, δ A , depending on the values of ε r ' and σ. The position of the optimal variant, in the global sense described by the minimum value of the objective function, is presented as a point (ε r ' ,opt , σ opt ) marked with a white dot. The figures present a whole set of local minima that give solutions with greater error. Four models of walls were considered: 1w_B18, 1w_B30, 2w_B18 and 2w_B30 (as described in Section 2).
Regardless of the number of brick layers in the wall and the conductivity of the material, variants with the smallest error, δ A , were found within the area σ opt ∈ <0.01, 0.03> S/m. When the increase of conductivity, σ FDTD , in the numerical model occurred, the point of optimal solution was also moved towards higher values of σ opt . Still, the selected equivalent conductivity in exemplary models was lower by 0.01 S/m than the input data. Parameters describing single-layer walls were characterized by similar dependencies within the same conductivity (Figures 10 and 11) and a similar tendency was also noticed for double-layer walls (Figures 12 and 13). It was found that in the analyzed range ε r '∈ <2, 6>, optimal effective conductivity (σ opt ) was higher than for single-layer walls. Tables 3 and 4 show the best solutions (min δ A ) obtained using the optimization algorithm. The calculated values of ε r ', opt and σ opt are given as well as the relative error. The obtained results indicate that for a single-layer walls, the equivalent value of ε r ', opt has an average value of 3.3 and the conductivity is σ opt ∈ <0.004, 0.02> S/m. On the other hand, the double-layer structure of the walls reduced the permittivity value and increased conductivity for numerically modeled structures, assuming that significant values are σ ∈ <0.03, 0.04> S/m (Table 4).

Effective Parameters of Hollow Bricks at f = 2.4 GHz
At specific frequency f = 2.4 GHz and conductivities FDTD ∈ ⟨0.01, 0.04⟩ S/m, usually used for the description of ceramic materials (Table 2), the effective electric parameters were found. Figures 10-13 show the distribution of the relative error, A, depending on the values of r' and . The position of the optimal variant, in the global sense described by the minimum value of the objective function, is presented as a point (εr',opt, opt) marked with a white dot. The figures present a whole set of local minima that give solutions with greater error. Four models of walls were considered: 1w_B18, 1w_B30, 2w_B18 and 2w_B30 (as described in Section 2).   Regardless of the number of brick layers in the wall and the conductivity of the material, variants with the smallest error, A, were found within the area opt ∈ ⟨0.01, 0.03⟩ S/m. When the increase of conductivity,  FDTD, in the numerical model occurred, the point of optimal solution was also moved towards higher values of opt. Still, the selected equivalent conductivity in exemplary models was lower by 0.01 S/m than the input data. Parameters describing single-layer walls were characterized by similar dependencies within the same conductivity (Figures 10 and 11) and a similar tendency was also noticed for double-layer walls (Figures 12 and 13). It was found that in the analyzed range r'∈ ⟨2, 6⟩, optimal effective conductivity (opt) was higher than for single-layer walls. Tables 3 and 4 show the best solutions (min A) obtained using the optimization algorithm. The calculated values of εr',opt and opt are given as well as the relative error. The obtained results indicate that for a single-layer walls, the equivalent value of εr',opt has an average value of 3.3 and the conductivity is opt ∈ ⟨0.004, 0.02⟩ S/m. On the other hand, the double-layer structure of the walls reduced the permittivity value and increased conductivity for numerically modeled structures, assuming that significant values are  ∈ ⟨0.03, 0.04⟩ S/m (Table 4). Table 3. Calculated effective electric parameters, one-layer wall (B18 and B30).

Relative Error
A (%) Regardless of the number of brick layers in the wall and the conductivity of the material, variants with the smallest error, A, were found within the area opt ∈ ⟨0.01, 0.03⟩ S/m. When the increase of conductivity,  FDTD, in the numerical model occurred, the point of optimal solution was also moved towards higher values of opt. Still, the selected equivalent conductivity in exemplary models was lower by 0.01 S/m than the input data. Parameters describing single-layer walls were characterized by similar dependencies within the same conductivity (Figures 10 and 11) and a similar tendency was also noticed for double-layer walls (Figures 12 and 13). It was found that in the analyzed range r'∈ ⟨2, 6⟩, optimal effective conductivity (opt) was higher than for single-layer walls. Tables 3 and 4 show the best solutions (min A) obtained using the optimization algorithm. The calculated values of εr',opt and opt are given as well as the relative error. The obtained results indicate that for a single-layer walls, the equivalent value of εr',opt has an average value of 3.3 and the conductivity is opt ∈ ⟨0.004, 0.02⟩ S/m. On the other hand, the double-layer structure of the walls reduced the permittivity value and increased conductivity for numerically modeled structures, assuming that significant values are  ∈ ⟨0.03, 0.04⟩ S/m (Table 4). Table 3. Calculated effective electric parameters, one-layer wall (B18 and B30).

Model of the Wall
Electric Parameters of Brick Adopted in Numerical Analysis (FDTD)

Relative Error
A (%)   Table 3, it was decided to distinguish the cases with σ FDTD ∈ <0.04, 0.1> S/m, including the values adopted for modeling ceramic materials. More than a two-fold increase in the conductivity taken in the analysis by the FDTD method results in a more than four-fold increase in the selected range for σ opt (Figure 14b).  Figures 14-16 show the distribution of the relative error (A) depending on the assumed equivalent values of the relative permittivity and conductivity. Based on Table 3, it was decided to distinguish the cases with FDTD ∈ ⟨0.04, 0.1⟩ S/m, including the values adopted for modeling ceramic materials. More than a two-fold increase in the conductivity taken in the analysis by the FDTD method results in a more than four-fold increase in the selected range for opt (Figure 14b).    Table 5 shows the values of effective parameters (εr',opt and opt) determined with the use of the optimization algorithm (Aopt) and characterized by the smallest relative error, A. Assuming the initial conductivity value FDTD = 0.04 S/m, it was found that for the description of B30 bricks used in a single-layer wall, a value lower by approximately 30% can be taken as effective conductivity, with   Table 5 shows the values of effective parameters (εr',opt and opt) determined with the use of the optimization algorithm (Aopt) and characterized by the smallest relative error, A. Assuming the initial conductivity value FDTD = 0.04 S/m, it was found that for the description of B30 bricks used in a single-layer wall, a value lower by approximately 30% can be taken as effective conductivity, with a simultaneous reduction of the relative permittivity to εr',opt = 3.66 S/m.  Table 5 shows the values of effective parameters (ε r ' ,opt and σ opt ) determined with the use of the optimization algorithm (A opt ) and characterized by the smallest relative error, δ A . Assuming the initial conductivity value σ FDTD = 0.04 S/m, it was found that for the description of B30 bricks used in a single-layer wall, a value lower by approximately 30% can be taken as effective conductivity, with a simultaneous reduction of the relative permittivity to ε r ' ,opt = 3.66 S/m. Comparing the effective electrical parameters for a single-layer wall made of bricks with a smaller number of hollows (B18), we noticed that values of ε r ' ,opt were close to 3.0. However, for B30 bricks, regardless of the type of wall, ε r ' ,opt ≈ 4. As the initial conductivity, σ FDTD , of the ceramic increases, the value of equivalent conductivity, σ opt , was also higher. Additionally, the analysis of the double-layer wall showed that regardless of the initial σ FDTD value in the model, the effective conductivity of hollow bricks (σ opt ) did not exceed 0.1 S/m.

Conclusions
The effective electromagnetic parameters of heterogeneous materials, identified by the proposed algorithm, can be applied in the modeling of large-scale systems, when reproduction of the complex structure of walls made of hollow bricks is not possible. The analysis of the obtained results has shown that at frequency f = 2.4 GHz and σ FDTD = 0.01 S/m, the effective conductivity was less than 0.008 S/m. At higher frequency and for σ FDTD = 0.04 S/m, determined values were smaller (σ opt ∈ <0.001, 0.033> S/m). Furthermore, at f = 2.4 GHz and for single-layer walls (and all types of bricks), the average effective relative permittivity was ε r ' ,opt = 3.3, while the effective conductivity was in the range σ opt ∈ <0.004, 0.02> S/m. For the double-layer walls, a decrease of ε r ' ,opt and an increase in the effective conductivity, σ opt , were observed.
The developed algorithm can be used to determine the effective parameters of complex structures (e.g., hollow brick) and multi-layer structures. The variability and possible range of the permittivity and conductivity as well as size of holes or inclusions, frequency and wall thickness were simultaneously taken into account. By the application of the presented methodology, it is possible to numerically analyze models in two-or three-dimensional space, with much larger sizes as well as number of elements. The algorithm is a general scheme, that can be applied to similar homogenization problems of various complex structures (e.g., clinker bricks, concretes with admixtures or multi-layer materials, etc.), with different electrical parameters, size of drillings or admixtures or wall thickness.