Effective permeability estimation of a composite magnetic shielding mortar by using swarm intelligence

Abstract

A Finite Element based strategy is proposed for the evaluation of magnetic permeability for the equivalent homogenized material of a magnetic shielding mortar containing ferromagnetic particles. The representation of the composite mixture is done by using elements of the mesh as 3-D inclusions. A statistical analysis is performed on thousands of meshes representing the same sample geometry, with different inclusions distribution. The homogenization process is posed in form of a non-linear inverse problem. To solve it, a powerful parallel algorithm based on swarm intelligence is used. The obtained results are validated through an experimental setup.

Keywords

Homogenization technique finite element method magnetic shielding effective permeability

1. Introduction

The shielding of electromagnetic field (EMF) has been largely addressed in literature, especially considering the exposition of human beings, and electrical/electronic devices, to high frequency EMF. On the other hand, epidemiologic study has proven a possible interaction for magnetic fields acting at extremely low frequencies (ELFs) as well, especially in the industrial range (0–300 Hz) [1, 2, 3], widely diffused both in domestic and work environment. Consequently, the development of shielding screens for indoor environments has been object of investigation: in last years, the research has moved through the development and adoption of new building materials [4, 5, 6, 7, 8, 9] with absorption capabilities. In particular, the use of composite building materials, obtained by adding to common mortars iron grain or synthetic polymeric structures have been tested, and a magnetic shielding material, based on the addition of ferromagnetic particles to the mortars usually used to refine the wall of our houses, has been patented [9]. A complete mathematical theory to determine the performance of these mixtures is still lacking, thus making the shielding assessment an experimental characterization. On the other hand, the simulation of the shielding properties for these materials is a difficult problem. Indeed, the fine structure characterizing the material requires a small scale simulation. Furthermore, simulating the shielding properties of a building requires a large simulation domain. These two scales are not compatible with the use of an electromagnetic CAD for the design of magnetic shield screen.

A possible approach to solve the problem is to establish a relationship between the mixture composition parameters (i.e. the volume fraction) and its electromagnetic properties, by representing the material under study with an equivalent homogenous one. In this paradigm, a homogenization technique would allow the representation of the shielding medium without the necessity of a fine scale. The relationship could be used as a design tool to assess the shielding properties of a structure. Several techniques exist in literature for homogenization, mostly borrowed from optics (e.g. Maxwell-Garnett and Bruggerman formulas). These give an analytical formula for the evaluation of electrical parameters of the inhomogeneous medium. However, the strictness of the conditions imposed in terms of geometries, and the necessity to represent the ferromagnetic behavior of the material either by models [10] or black-box approaches [11, 12, 13] (omitted in these approaches) makes the direct application of such methods impossible for the estimation of effective permeability for this particular problem. Other approaches [14, 15, 16, 17, 18, 19, 20, 21] employ numerical techniques, such as Finite Element Method (FEM), to calculate the effective permeability, but present limitations in terms of the geometry adopted, which in some cases introduce errors in the evaluation of the homogeneous equivalent parameters as well. Nevertheless, those models revolve around the concept of energy balance employed numerically, which, although valid as principle, can lead to remarkable errors in the assessment. For these reasons, in this paper we propose an evaluation of effective permeability by exploiting an inverse problem involving a set of points lying on an assigned domain. The method is still based on an analysis through FEM of an equivalent elementary cell, which allows the characterization of the material under study by using a statistical approach. The novelty introduced by this approach lies in the quantity that is used as comparison between the homogenous and the non-homogenous materials. Whereas for the energy balance methods the criterion was to consider two materials that holds the same magnetic energy given the same external field, the herein proposed method formulate an inverse problem imposing the same field on the two material given the same boundary conditions. The advantages of this approach will be presented in the following, however, a simple drawback is plainly apparent: it is necessary to solve an inverse problem. To solve the inverse problem in this work an efficient optimization algorithm, based on swarm intelligence, is proposed [22]. The algorithm implements a hybrid strategy, and can exploit efficiently the computational advantages of running on a parallel cluster. The results of the proposed method will be in the form of an analytic (polynomial) relationship that allows to represent a non-homogenous material with a homogenous, equivalent one. To validate such relationship we use measured data available from an experimental setup, already described in our previous works [20, 21]. The paper will be structured as follows. First, the homogenization technique will be presented, starting from the classic energy-balance approach, and then deriving the proposed equivalent field method. Second, the optimization algorithm will be briefly described. Third, the tests performed and the found analytic relationship will be presented, and compared against experimental data. Conclusions and final remarks will follow.

2. Effective permeability estimation by using FEM

All the numerical methods proposed in literature for the evaluation of the effective permeability are based on simple geometries (square or cubic), which contains the ferromagnetic inclusions. The same geometries are also used for the homogeneous equivalent material, and the effective permeability is computed by imposing that the magnetic energy should be the same for both cases, assuming the same applied external field [14, 20, 21]:

$U=\frac{1}{2}\int_{V}{{\bm{B}}\cdot{\bm{H}}dV=U_{hom}=\frac{1}{2}\mu_{0}\mu_{% \textit{eff}}\int_{V}{\left|{\bm{H}}_{{hom}}\right|^{2}dV}}$ (1)

A first limit of these approaches regards the shape of the inclusions, which seldom is arbitrary, whereas another important drawback is related to the energetic approach used. In order to overcome the first issue, the authors have formulated a statistical analysis. Indeed, our technique studies the characteristic of an equivalent homogeneous material by simulating a set of elementary cubic cells (side L $=$ 1 cm) that features a random distribution of ferromagnetic inclusions. The inclusions can also be represented by single elements in the elementary cell (i.e. no particular shape is chosen, considering a tetrahedral finite element). In addition to having chosen particularly small tetrahedra, groups of them can form arbitrarily shaped inclusions. In order to avoid problem arising from the use of energy approach, in the performed FEM analysis, we use an external equivalent criterion: given the same field boundary conditions, the value of the H field outside the cubic cells containing inclusions should be equal to those given if a homogenous cube/cell of the same size was present. To account for different inclusions distribution in the cell, as stated before, a statistical approach was followed. The analysis was repeated, for the same volume fraction $f$ , several times with different distribution of the inclusions. The FEM formulation is based on the scalar magnetic potential $\psi_{m}$ . The magnetic field source is external to the domain, and the magnetic field intensity is irrotational ( $\nabla\times{\bm{H}}=$ 0) inside it. The field intensity can then be computed from the magnetic scalar potential:

${\bm{H}}=\mathrm{\nabla}\psi_{m}$ (2)

given suitable boundary conditions (Dirichelet and homogeneous Neumann). Differently from the approaches previously proposed [20, 21] by the same authors, in this case we use a larger domain, which includes the geometry under study. This is done for two reasons: first, we are considering an equivalence approach where the equivalence must be expressed in terms of external quantities (that is, the two geometries, with inhomogeneous or homogeneous cube, must be included in a larger domain to be analyzed); second, the solution of numerical problems formulated by imposition of the boundary conditions directly on the “examined cube”, determines a numerical error in the energy evaluation, due to the presence of components of the H field incompatible with boundary conditions. Indeed, even by imposing directly the Dirichlet/Neumann conditions on the small cube (for example by imposing the external field with only x component, as normally done in literature) we find, on the geometry, y and z components for H field as well. This also occurs for elements in the boundary, which means the numerical problem is not well posed and the solution could be consequently not consistent (unphysical). Clearly, these components are not present in the homogeneous case, and consequently, the “energy” equivalence can lead to error due to these components. As will be shown, such errors are not negligible in many cases. In spite of these considerations, the geometry taken into account consists of a larger cubic domain (side 5 cm) containing the cubic sample of (side 1 cm) as shown in Fig. 1. Being the constitutive law

${\bm{B}}=\mu{\bm{H}}$ (3)

Figure 1.

Schematization of the analyzed geometry.

Figure 2.

Isosurfaces in the geometry for a case with 30% of inclusions in volume (different colors in the cube at the center of the geometry).

This is a generalized Poisson problem

$\nabla\cdot{\bm{B}}=\nabla\cdot\left(\mu\nabla\psi_{M}\right)=0$ (4)

with Dirichelet conditions ( $\psi_{m}=$ $-$ M/2 for $x=$ 2.5 cm, and $\psi_{m}=$ $+$ M/2 for $x=$ 2.5 cm, with $M=$ 100) and Neumann conditions (for the remaining faces of the cubes). If one wish to take into account the nonlinear effects due to hysteresis in ferromagnetic material the problem [10, 11, 12, 13], being not linear, must be solved by an iterative scheme exploiting the Newton-Raphson or the Fixed Point techniques, whereas the non-linear model should also take into account the hysteresis effects. Herein we suppose to neglect this nonlinearity, in order to avoid the introduction of further unknowns in our statistical approach. In addition, it is important to note that, for flux density field levels present in domestic environments, the nonlinear effect due to material saturation could be neglected in first approximation. Moreover, the small volume fraction of ferromagnetic inclusions in the mixture with respect to the linear material in the mortars tends to linearize the response. This assumption is supported by results where the effective relative permeability does not exceeds the value of 200, even with remarkable volume fractions (about 30%) and a relative permeability ( $\mu_{rel}$ ) of inclusions supposed value of 1000. In addition, considering the statistical approach followed (that is thousands of FEM simulations must be executed), using a precise description for the magnetic material of the inclusion is inconvenient due to the computational resources required. For the same reason, the frequency effects are omitted in our analysis. This is not a considerable limit since our aim is to investigate the behavior of the building mixture at industrial frequency (50 Hz and its harmonics). In Figs 2 and 3, the deformation of the contour surfaces for the scalar magnetic potential is shown due to the presence of the ferromagnetic inclusions. It is apparent that if Neumann boundary conditions are directly imposed on the small examined cube, they force an unphysical solution. From Fig. 3 it is also possible to observe the size and shape of the inclusions (colored) for one of the 100 cases analyzed with a volume fraction equal to 20%. As stated above, instead of using an energetic approach, we formulate the problem as an optimization/inverse problem with the aim to identify the effective permeability of the mixture for varying volume fraction and relative permeability of the ferromagnetic inclusions. In particular we define the following fitness function, which is the square error (SE), which we minimize:

$\displaystyle SE=\sum\limits_{i}^{Np}\left(\psi_{i}^{h}-\langle\psi_{i}^{nh}% \rangle\right)^{2}$ (5)

Figure 3.

Close up of the isosurfaces around the cubic sample, showing the presence of additional field components not parallel to imposed one.

In Eq. (5), Np is a set of test points distributed outside the inhomogeneous cube, $\psi_{i}^{h}$ is the value of the magnetic scalar potential for homogeneous material at point $i$ , and $\langle\psi^{nh}_{i}\rangle$ is the mean value of the magnetic scalar potential at the same point considering the different distributions of the inclusions for a given volume fraction and a given relative permeability. The spatial distribution of the points can be seen in Fig. 4. Two sets of points exist, both forming surfaces lying midway between the external and the internal cube. One is parallel to one of the Dirichelet faces of the boundary, the other to one of the Neumann faces. The choice of these points is justified by the inherent symmetry found in this problem. The inverse problem herein proposed is a non-linear problem that may feature several local minima: this is since it is in the form of a non-linear least square problem, with noisy measurements (that is, the FEM solutions). Therefore, solution through deterministic algorithms, although possible, could be sensitive to initial guess values. In addition, deterministic algorithms require the computation of the derivative of the objective function. Since underneath the objective function lies a numerical FEM simulation, the numerical computation of the derivative is inconvenient as well, since it would require the computation of neighbor points for each solution. For this reason, an approach through a stochastic algorithm, either swarm intelligence based or evolutionary, is the best course of action for the solution of this inverse problem. The typical drawback of these algorithms, which features elevated exploration capabilities, is the lack of exploitation capabilities. The best solution requires the use of a hybrid strategy, where the solution is found by the combination (usually in cascade) of an explorative and a local algorithm [23]. Hybrid strategies can be naturally implemented on a parallel architectures, since it is possible to perform exploration by the master whereas the exploitation task is distributed among the slaves. For the minimization of this functional, a hybrid supervised technique, named CFSO ${}^{3}$ [22] was employed, that will be briefly described in the next section, in order to exploit the possibility to refine the solutions.

Figure 4.

Distribution of the sampling points in the domain. The circles denotes points sampling the Dirichelet side of the problem, squares denotes points sampling the Neumann one.

3. The CFSO

{}^{3}

algorithm

The Continuous Flock of Starlings Optimization Cube (CFSO ${}^{3}$ ) is a supervised optimization strategy that is based on a single algorithm that can be configured to be either explorative or local. This algorithm belongs to the swarm-intelligence class of optimizers, and at its core, it has a variation of the well-known Particle Swarm Optimization algorithm featuring a “topological” component, called the Flock of Starling Optimization (FSO) . The topological component is responsible for an inter-particle interaction beyond the one coming from the global best. In the FSO algorithm, every particle takes into account the velocity of the other particles in the swarm, as shown in Eq. (6)

$\displaystyle v_{k}^{j}\left(t+1\right)=\omega v_{k}^{j}(t)+\lambda\left(p_{% \textit{best}_{k}}^{j}\left(t\right)-x_{k}^{j}\left(t\right)\right)+\gamma% \left(g_{\textit{best}}^{j}\left(t\right)-x_{k}^{j}\left(t\right)\right)+\sum% \limits_{m=1}^{N}{h_{km}v_{m}^{j}\left(t\right)}x_{k}^{j}\left(t+1\right)=x_{k% }^{j}\left(t\right)+v_{k}^{j}\left(t+1\right)$ (6)

By the index $j=1\ldots\Delta$ the dimension of the velocity/position is taken into account ( $\Delta$ is the dimension of the solution), $\omega$ , $\lambda$ and $\gamma$ express the inertial (tendency to maintain the direction), cognitive (tendency to follow the personal best $p_{\textit{best}}$ ) and social (tendency to follow the global best $g_{\textit{best}}$ ) coefficients. This expression would be the basic PSO update law, with the added sum term $\sum_{m=1}^{N}{h_{km}v_{m}^{j}}$ that takes into account particle-to-particle interaction. Every particle in the swarm has a component, in its velocity, that is the average of the velocities for N other particles. The effect of this is a dramatic increase in collective behavior, and a considerable robustness towards local minima entrapment. It is possible to interpret Eq. (6), by means of an infinitesimal time-step, as a continuous state equation for a dynamic system. This was introduced in the CFSO [22]. The differential equation are:

$\displaystyle\dot{\nu}_{k}^{j}(t)=\omega v_{k}^{j}(t)+\lambda(p_{\textit{best}% _{k}}^{j}(t)-x_{k}^{j}(t))+\gamma(g_{\textit{best}}^{j}(t)-x_{k}^{j}(t))+\sum% \limits_{m=1}^{N}\,h_{km}v_{m}^{j}(t)\dot{x}^{j}_{k}(t)=v^{j}_{k}(t)$ (7)

Figure 5.

Possible hybrid strategies to be implemented with CFSO. On the left, the parallel implementation (CFSO ${}^{3}$ ), the right, the serial implementation.

Since the terms linked to personal and global best are time dependent (since they depend on the position ${x}(t)$ assumed by the particles), the integration of such system requires a time windowing approach: terms are evaluated at the beginning of each time window, with duration $\tau$ , and are assumed as constant for the duration of it. This way, by exploiting a tailored mathematical theorem involving circulant matrices [24], it is possible to integrate the trajectories of the particles and express them in closed analytical forms in the time domain. These close forms are the rules for the direct update of the positions and of the velocities of the particles in the CFSO algorithm. Still, the most important advantage of the CFSO lies in the availability of analytical expression for the poles of the dynamical system, which allows, with a suitable choice of parameters, to force the behavior of the particles. Indeed, the parameters can be tuned to create poles that are either real part negative, real part positive, or complex conjugates. This corresponds to a converging, diverging or oscillating behavior of the dynamic system. These behavior can be used to create a strategy that combines the different behaviors of this algorithm to obtain the sequence of exploration/exploitation that is implemented in hybrid techniques. The simpler way consists in running the algorithm with sequentially alternating configurations, as shown in Fig. 5 (right). The algorithm cycles through a diverging configuration (CFSOus), an explorative one (CFSOpi) and an exploitative one (CFSOas). The explorative configuration is used to coarsely move the particles in the neighbor of a good solution, that is later refined with the exploitative one. After convergence, all the particles lies in a minimum. The diverging configuration is then used for few iterations to escape from it. A similar strategy was implemented on a parallel architecture as well (CFSO ${}^{3}$ ), which is shown in Fig. 5 (left). The master node performs exclusively the coarse exploration through CFSOpi and, eventually, runs the CFSOus if an entrapment is detected. Every time a candidate area for local search is found, the coordinates are sent to one of the slave nodes to perform optimization through CFSOas. The three parametric configurations (i.e. the values for $\omega$ , $\lambda$ , $\gamma$ , h and $\tau$ ) for the converging, oscillating and diverging behavior are reported in Table 1.

Table 1

Parametric configuration of the CFSO

Parameters	CFSOpi	CFSOas	CFSOus
$\omega$	$-$ 0.8147	$-$ 2.1260	0.1260
$\lambda$	0.4210	0.4820	0.4820
$\gamma$	0.5790	0.9130	0.4130
h	0.0950	$-$ 0.6680	$-$ 0.6680
$\tau$	0.2000	0.0500	0.0200

Figure 6.

Comparison of the results achieved by using energetic approach and inverse problem approach (left), and a close-up around volume fraction of 0.2 (right).

Table 2

Polynomial coefficients for the interpolant function

Coefficient	Value	Boundaries
$p_{00}$	1.3420E $+$ 00	9.6870E-01	1.7160E $+$ 00
$p_{01}$	2.1450E-03	7.9710E-04	3.4930E-03
$p_{02}$	$-$ 1.2850E-07	$-$ 1.2980E-06	1.0410E-06
$p_{10}$	2.6290E-03	$-$ 1.5470E-01	1.6000E-01
$p_{11}$	$-$ 2.5060E-03	$-$ 2.9130E-03	$-$ 2.0990E-03
$p_{12}$	$-$ 5.4040E-08	$-$ 3.8600E-07	2.7790E-07
$p_{20}$	4.6870E-02	2.2760E-02	7.0980E-02
$p_{21}$	9.0310E-04	8.6630E-04	9.3980E-04
$p_{22}$	$-$ 9.4080E-08	$-$ 1.1960E-07	$-$ 6.8590E-08
$p_{30}$	$-$ 5.7760E-03	$-$ 7.5200E-03	$-$ 4.0320E-03
$p_{31}$	$-$ 2.7860E-05	$-$ 2.9190E-05	$-$ 2.6530E-05
$p_{32}$	2.0440E-09	1.4850E-09	2.6030E-09
$p_{40}$	2.4260E-04	1.8270E-04	3.0240E-04
$p_{41}$	3.2240E-07	3.0290E-07	3.4200E-07
$p_{50}$	$-$ 3.4140E-06	$-$ 4.1960E-06	$-$ 2.6320E-06

Figure 7.

Effective permeability $\mu_{\textit{eff}}$ of a mixture of concrete and inclusions with $\mu_{\textit{rel}}$ permeability, for a volume fraction between 1% and 30%.

4. Analysis results

The FEM analysis has been performed considering an irregular mesh consisting of more than 450000 elements and about 75000 nodes. Among the elements constituting the small cube inside the geometry a variable percentage is assigned to be ferromagnetic with a fixed relative permeability ${\mu}_{rel}$ . This randomness allows us to represent the ferromagnetic inclusions with a good accuracy. The distribution of the inclusions/elements is uniform since during the realization of these mixtures no magnetic force acts on the constituting components; on the other hand, the mixture before the application of the wall is enough dense to avoid any influence by external forces such as gravity. For any given volume fraction value in the range 1–30% (at step of 1%), a statistical analysis is performed. To do this, 100 different inclusions distributions are generated, to take into account the effects of a completely random distribution. The simulations were also done for different values of maximum relative differential permeability $\mu_{rel}$ , ranging from 100 to 1000. Overall more than 30000 simulations were performed to prepare the data for the successive statistical analysis. Indeed for each value of relative permeability and each value of volume fraction the mean value of the scalar potential in $Np=$ 200 points was computed. The Np distribution of points is shown in Fig. 4. As regards the results we refer to Figs 6–8. In Fig. 6 we compare the effective permeability achieved by using directly the energy balance approach proposed in literature and our inverse problem approach. As it is possible to observe in this figure, if we use the total energy we have a small overrate of the effective permeability with respect to the inverse problem solution, whereas if we use only the x component (as FEM solution) we remarkably underrate the value. It is worth noticing that in the energetic approach we compute the fitting curve as post processing (in fact in the close-up of Fig. 6 it is possible to view the single computations/points coming from energetic approach, from which a mean value can be extracted). If we try to interpolate (in least squares sense) the solutions points of our inverse problem at varying $\mu_{rel}$ and volume fraction $f$ (expressed in %), we find that a 5th degree polynomial is able to fit the data with enough accuracy, resulting in a maximum error lower than 0.5 for effective permeability value up to 250 (relative error lower than 1%). Using a polynomial of higher degree does not give a sensible accuracy improvement.

$\displaystyle{P}\left(f,\mu_{\textit{rel}}\right)=p_{00}+p_{10}\cdot f+p_{20}% \cdot f^{2}+p_{30}\cdot f^{3}+p_{40}\cdot f^{4}+p_{50}\cdot f^{5}+\mu_{\textit% {rel}}\cdot\left(p_{01}+p_{11}\cdot f+p_{21}\cdot f^{2}+p_{31}\cdot f^{3}+p_{4% 1}\cdot f^{4}\right)+\mu_{\textit{rel}}^{2}\cdot(p_{02}+p_{12}\cdot f+p_{22}% \cdot f^{2}+p_{32}\cdot f^{3})$ (8)

Figure 8.

Experimental data (red dots) compared to the polynomial relationship proposed. The dotted red line is the closest (LSQ) polynomial representing the experimental data, corresponding to a $\mu_{\textit{rel}}$ of the inclusions of 130.

The plot of the interpolating function resulting by the solution of the least square problem is drawn in Fig. 7: it clearly increases proportionally to the volume fraction as expected. The coefficients found with their values for 95% confidence bounds are reported in Table 2. As it is possible to view from Fig. 7, reasonable values for the effective permeability of these mixtures are lower than 50–100. The polynomial relation found was compared to experimental data obtained from a measurement setup already used in [23, 24]. The functional relationship that we are trying to validate express the effective permeability of a mixture $\mu_{\textit{eff}}$ in function of the volume fraction $f$ of the inclusions and their relative permeability $\mu_{rel}$ . Measuring the permeability for a mixture (and its inclusions) is very difficult. For this reason, validation was performed through the indirect measurement of the shielding effectiveness. A two-step approach was followed: a relationship between shielding effectiveness and magnetic permeability of a homogenous shield was computed; a relationship between shielding effectiveness and volume fraction was measured. Details about the measurement procedure can be found in [20, 21]. In Fig. 8 it is possible to observe a comparison between the experimental measurements and the family of curves given by Eq. (8) for different values of $\mu_{rel}$ . Indeed, it is difficult to quantitatively validate this approach, because the experimental measurements of shielding effectiveness, although performed according to IEEE Std. 299, carry a considerable uncertainty. Moreover, they are indirect measurements, and one can only estimate the magnetic permeability of the inclusions used to build the prototype shields. In red it is possible to see the polynomial that best fit the three experimental points, which corresponds to set of inclusions with ${\mu}_{rel}=$ 130, that is sensible considering the materials used for their construction.

5. Conclusions

An approach based on FEM for the evaluation of the effective magnetic permeability for mixtures of concrete and iron ore was presented. The advantages of the proposed method over the classic one based on energy balance involve a more realistic representation of the magnetic field (i.e. accounting for the transversal components) and fewer computational steps (i.e. not requiring numerical differentiation and integration to compute the flux density and the volumetric energy). Indeed, this comes with the cost of formulating the homogenization technique through the solution of an inverse problem. To account for this, an efficient parallel algorithm based on swarm intelligence was proposed. Further development can be considered starting from frequency dependence, hysteretic behavior under specific excitation and vector hysteresis modeling. Also a different typology of measurement apparatus should be considered in order to evaluate all the possible effects [25, 26]. In any case, only homogenization techniques like the one proposed in this work makes possible to lighten the computational costs of a simulation, thus allowing larger scale simulations that are critical for ELF magnetic field pollution modeling.

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