IEEE TRANSACTIONS ON MAGNETICS, VOL. 56, NO. 11, NOVEMBER 2020 6300312

Boundary Element Solution of Electromagnetic Fields forNon-Perfect Conductors at Low Frequencies

and Thin Skin DepthsNail A. Gumerov 1, Ross N. Adelman 2, and Ramani Duraiswami3

1Institute for Advanced Computer Studies (UMIACS), University of Maryland, College Park, MD 20742 USA2Army Research Laboratory, Adelphi, MD 20783 USA

3Department of Computer Science, Institute for Advanced Computer Studies (UMIACS), University of Maryland,College Park, MD 20742 USA

A novel boundary element formulation for solving problems involving eddy currents in the thin skin depth approximationis developed. It is assumed that the time-harmonic magnetic field outside the scatterers can be described using the quasistaticapproximation. A two-term asymptotic expansion with respect to a small parameter characterizing the skin depth is derived for themagnetic and electric fields outside and inside the scatterer, which can be extended to higher order terms if needed. The introductionof a special surface operator (the inverse surface gradient) allows the reduction of the computational complexity of the solution.A method to compute this operator is developed. The obtained formulation operates only with scalar quantities and requires thecomputation of surface integral operators that are customary in boundary element (method of moments) solutions to the Laplaceequation. The formulation can be accelerated using the fast multipole method. The resulting method is much faster than solving thevector Maxwell equations. The obtained solutions are compared with the Mie solution for scattering from a sphere, and the errorof the solution is studied. Computations for much more complex shapes of different topologies, including for magnetic and electricfield cages used in testing, are also performed and discussed.

Index Terms— Asymptotic methods, boundary element method, boundary integral equations, computational electromagnetics, eddycurrents, method of moments.

I. INTRODUCTION

MANY systems of practical interest consist of conductorsand dielectric materials. The modeling of time-varying

electromagnetic fields in such systems is an important engi-neering problem where scattering from antennas, buildings,and various other objects of arbitrary shape must be computed.In many cases, the conductors can be modeled as perfectelectric conductors, and this approximation is widely used.However, there are a number of situations where this approx-imation is invalid. When eddy currents appear due to finiteconductivity (“non-perfectness” of the conductor), which iscaused by the diffusion of the magnetic field into the conductorand must be accounted for in the modeling the fields.

Scattering from a non-perfect conductor can be modeledusing Ohm’s law and Maxwell’s equations for the electro-magnetic fields inside and outside of the conductor, withthe fields coupled together by transmission boundary condi-tions on the surface of the conductor [1]. For time-harmonicelectromagnetic fields, Ohm’s law leads to the concept ofcomplex electric permittivity, and the well-known boundaryintegral equations for Maxwell’s equations [2] can be used.The complex electric permittivity of a conductor results ina complex-valued wavenumber, and the reciprocal of theimaginary part of this number defines the skin depth, δ,or the depth of penetration of the magnetic field inside the

Manuscript received May 22, 2020; revised July 12, 2020; acceptedAugust 18, 2020. Date of publication August 26, 2020; date of currentversion October 19, 2020. Corresponding author: N. A. Gumerov (e-mail:ngumerov@umd.edu).

Color versions of one or more of the figures in this article are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2020.3019634

conductor. Several numerical challenges with the boundaryintegral solvers appear in this case where the skin depth ismuch smaller than the characteristic length of the scatterer, a(i.e., δ � a). Indeed, the size of the boundary elements, �,should be much smaller than the skin depth (i.e., � � δ)to enable a valid discrete representation of the continuouselectromagnetic fields. Such a requirement may lead to hugenumbers of boundary elements and drastically increase thecomputational complexity of the problems and even makethem practically unsolvable.

In this article, we consider an approximation that is notapplicable either at very high or at very low frequencies.We assume that the wavelength in the air or some othercarrier dielectric medium is much larger than the wavelength,so the displacement current in that medium can be neglected.Combined with the thin skin depth approximation this createsopportunities for a simplified but computationally efficientapproach, which is based on perturbation theory.

The perturbation approach with respect to small parameterδ/a is not new. An asymptotic method was first introducedby Rytov [3] and also can be found in Jackson’s book [1].Mitzner [4] derived a boundary integral equation for this,which enables computing the scattered fields from bodies withthin skin depths with no additional assumptions imposed onthe carrier media. In the last two decades, a number of authorsproposed the use of boundary element methods for solvingproblems with eddy currents [5]–[18]. It can be noted that inmany studies (see [5], [8]), the boundary integral equationsfor eddy current simulations are derived from the generalboundary integral equations for Maxwell’s equations in termsof vector quantities that require special surface basis functions(such as the RWG-basis [19]) and substantially complicates

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6300312 IEEE TRANSACTIONS ON MAGNETICS, VOL. 56, NO. 11, NOVEMBER 2020

the method. There exist studies treating the problem in 2-Dapproximation and using isogeometric discretization requiringa relatively low number of boundary elements [15]. In thesestudies, the surface impedance boundary conditions (SIBC)were applied for the external solvers. In the latter cited andsome other studies, a quasistatic approximation for the mag-netic field is accepted. In the 3-D case considered in [10], onlythe magnetic field is computed using the zero-order approxi-mation for the internal problem. In the magnetostatic approx-imation for the external field, the solution to the scatteringproblem can be reduced to computing only one scalar quantity,the surface magnetic potential [10]. Combinations of theboundary element method and finite element method, as wellas fast multipole accelerations, can also be found [11]. Mathe-matical aspects of the problem are considered in several studies(see [14], [17]).

We develop and demonstrate a new approach to solvelow-frequency electromagnetic problems in the case of thinskin depth inside conductors. This approach is free of thelimitation, � � δ, allowing for much smaller numericalproblem sizes. The approach reduces to the solution of a fewelectrostatic and magnetostatic problems for scalar potentials,and it is much more efficient compared with conventionalboundary element (or method of moments) Maxwell solvers.This is achieved as the problem can be expressed via scalarpotentials for the external problem and standard methods forthe solution of the scalar Laplace equations can be applied.This presents a new boundary integral solution for the Maxwellequations in the absence of the displacement current and,generally, the divergence constraint vector Laplace equation(DCVLE) [21] (the electric field satisfies this equation), whichcan be used in a number of different problems. Moreover,for large problems, the method can be accelerated using thefast multipole method (FMM) for the Laplace equation [22],[23], which makes it scalable and suitable for parallelizationon large computational clusters [24]. In this article, a two-term asymptotic solution is obtained, which can be extendedto higher orders if needed. The unique feature of the presentmethod is the use of special surface operators, such as theinverse surface gradient, which enables the reduction of vectorproblems to scalar problems. The basic formulation is pre-sented in Section II, while the asymptotic approach, integralequations, and validation for the case of a sphere are describedin Section III. Section IV provides the results for a numberof more complex cases, including practical cases. Section Vconcludes this article.

II. PROBLEM STATEMENT

Consider, as shown in Fig. 1, two low-frequency,time-harmonic (∼ e−iωt ) electromagnetic fields where the dis-placement current in Maxwell’s equations has been neglected.The first field is contained in the domain, V1, possibly extend-ing to infinity, of negligible conductivity, and the second fieldis contained in the domain, V2, of finite conductivity, σ2,so Maxwell’s equations become

∇ · E1 = 0, ∇ · H1 = 0 (1)

∇ × E1 = iωμ1H1, ∇ × H1 = 0 (2)

Fig. 1. Problem and notation. A known incident electromagnetic fieldEin

1 , Hin1 from domain V1 with magnetic permeability μ1 interacts with a

scatterer occupying a volume V2 with electrical conductivity σ2 and magneticpermeability μ2 and separated from V1 by the boundary S. The interactioncauses a field in domain V1 that is the sum of the incident fields and scatteredfields Ein

1 + Esc1 and Hin

1 + Hsc1 , and a field inside the scatterer E2, H2. The

characteristic size of domain V2 is a, while the parameter δ is the skin depth,defined in (11). We seek to develop fast approximate integral equation basedalgorithms to compute Esc

1 , Hsc1 and E2, H2 for small values of δ/a.

∇ · E2 = 0, ∇ · H2 = 0 (3)

∇ × E2 = iωμ2H2, ∇ × H2 = σ2E2. (4)

In these equations, E and H are the electric- and magnetic-fieldvectors, μ is the magnetic permeability, and ω is the circularfrequency. The field in the domain V1 is composed of the givenincident field and the scattered field

E1 = E(in)1 + E(sc)

1 , H1 = H(in)1 + H(sc)

1 (5)

each of which satisfies (1) and (2). The incident field may alsocontain some given sources. The problem is to determine thescattered and internal fields that satisfy the following boundaryconditions on the surface, S, with the normal, n, separatingthe domains V1 and V2

n · E2 = 0 (6)

n ×(

E2 − E(sc)1

)= n × E(in)

1 (7)

n ·(μ2H2 − μ1H(sc)

1

)= μ1n · H(in)

1 (8)

n ×(

H2 − H(sc)1

)= n × H(in)

1 . (9)

The first condition above, that is n · E2 = 0, has an intuitiveexplanation: no eddy currents that are induced in the conductorcan leave through the surface. We assume further that thesurface S is closed and smooth and that the domain V2 is finite.For the infinite domain V1 the scattered field decays at infinity

limr→∞ E(sc)

1 = 0, limr→∞ H(sc)

1 = 0. (10)

The obtained solution can easily be modified for the case whenV1 is finite and V2 is infinite, and we will later show how to dothis. Furthermore, we characterize the domain V2 using sometypical length scale a, which can be related to the domain sizeor to the rate of change of n (e.g., reciprocal curvature). Thenotion of the length scale is actually only needed to ensurethat the skin depth in the conductor is small compared withthe conductor size

δ =(

2

σ2μ2ω

)1/2

� a. (11)

Note also that such a constraint can hold in the case whenσ2 and μ2 are spatially varying. Such a situation can occur

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GUMEROV et al.: BOUNDARY ELEMENT SOLUTION OF ELECTROMAGNETIC FIELDS 6300312

when the conductor is composed of several materials withdifferent properties or there are several different scatterers inthe domain, as will arise in some of our application problems.

III. METHOD

A. Boundary Integral Formulation for External Problem

First, consider the external problem in the domain V1. It isclear here that we are in the magnetostatic regime

H(sc)1 = −∇�

(sc)1 , ∇2�

(sc)1 = 0. (12)

The electric field, however, cannot be written this way becausethe curl of E(sc)

1 is not zero. Instead, by the general Helmholtzdecomposition, this field can be represented as

E(sc)1 = −∇�(sc)

1 + iωμ1∇ × C(sc)1 , ∇2�(sc)

1 = 0 (13)

where C(sc)1 is some vector field. Substituting (13) into (2),

we see that C(sc)1 satisfies the equation

∇ × ∇ × C(sc)1 = H(sc)

1 = −∇�(sc)1 . (14)

Denoting

φ(sc)1 = ∇ · C(sc)

1 + �(sc)1 (15)

and using the identity, ∇2 = ∇(∇·) − ∇ × ∇×, we obtain

∇2C(sc)1 = ∇φ(sc)

1 . (16)

As any particular solution of this equation can be used, we use

C(sc)1 (r) = −

∫V1

G(r, r�)∇r�φ

(sc)1 dV . (17)

Here and below, integration over volume and surface is takenover variable r� (dV = dV

(r�), d S = d S

(r�)), G

(r, r�) is the

free-space Green’s function for the Laplace equation

G(r, r�) = 1

4π |r − r�| , ∇2r� G

(r, r�) = −δ

(r, r�) (18)

and δ(r, r�) is the delta function in 3-D space. Taking the

divergence of both sides of ( 17) and using (15), we obtain anintegral equation for the unknown scalar function φ

(sc)1

φ(sc)1 − �(sc)

1 = −∇r ·∫

V1

G(r, r�)∇r�φ(sc)

1 dV . (19)

The right-hand side of this equation can be transformed usingGreen’s identity as follows:

−∇r ·∫

V1

G(r, r�)∇r�φ

(sc)1 dV (20)

= −∫

V1

∇rG(r, r�) · ∇r�φ

(sc)1 dV (21)

=∫

V1

∇r� G(r, r�) · ∇r�φ

(sc)1 dV (22)

=∫

V1

{∇r� G(r, r�)·∇r�φ

(sc)1 (23)

+ [∇2r� G(r, r�)+ δ(r, r�)

]φ

(sc)1

(r�)}dV

=−∫

Sφ

(sc)1

(r�)∂G(r, r�)

∂n(r�)d S + α(r)φ(sc)

1 (r) (24)

where the normal is directed from V2 to V1 (which causesa negative sign to appear in front of the surface integral inthe last expression), α(r) = 1 for points internal to V1 (i.e.,r ∈ V1), and α(r) = 1/2 for points on S (i.e., r ∈ S). Inthe latter case, the surface integral is singular and should betreated in terms of its principal value ( p.v.). We now introducethe following notation for single- and double-layer potentials:

L[s](r) =∫

Ss(r�)G

(r, r�)d S (25)

M[s](r) = p.v.

∫S

s(r�)∂G

(r, r�)

∂n(r�)d S. (26)

This shows that the unknown variable φ(sc)1 on the surface can

be found by solving the boundary integral equation

M[φ

(sc)1

](r) + 1

2φ

(sc)1 (r) = �

(sc)1 (r), r ∈ S. (27)

Note that ∇ ×C(sc)1 can then be expressed via the scalar, φ(sc)

1 .Indeed, using (17), we have

C(sc)1 (r) = −

∫V1

G(r, r�)∇r�φ

(sc)1 dV (28)

= −∫

V1

∇r�[G

(r, r�)φ(sc)

1

]dV (29)

+∫

V1

φ(sc)1

(r�)∇r� G

(r, r�)dV

=∫

SG

(r, r�)φ(sc)

1 n(r�)d S (30)

−∫

V1

φ(sc)1

(r�)∇rG

(r, r�)dV

= L[nφ

(sc)1

]− ∇

∫V1

φ(sc)1

(r�)G

(r, r�)dV . (31)

Taking the curl and plugging the result into (13), we obtain

E(sc)1 = −∇�

(sc)1 + iωμ1∇ × L

[nφ

(sc)1

]. (32)

This equation shows that only the boundary value of the scalarfunction φ(sc)

1 is needed to compute the rotational part of E(sc)1 ,

and (27) provides a method to determine this function usingthe magnetic potential. Note that, for the Laplace equation,the operator M + (1/2)I in (27) is degenerate. However, thisproblem can be solved by applying the additional conditionthat the average of φ

(sc)1 over the surface is zero (or in the

case of several objects, the average over each object surfaceis zero). In fact, any constant added to φ

(sc)1 does not affect

the value of ∇ × L[nφ(sc)1 ].

B. Asymptotic Expansions for Thin Skin Depth

Now, consider the internal problem in the domain V2. Themagnetic field here satisfies the vector Helmholtz equation

∇2H2 + 2i

δ2H2 = 0. (33)

At small δ, the field will be nonzero only in some vicinity ofthe surface, so we separate the ∇ operator into parts relatedto differentiation along and normal to the surface

∇ = ∇s + n∂

∂n. (34)

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6300312 IEEE TRANSACTIONS ON MAGNETICS, VOL. 56, NO. 11, NOVEMBER 2020

The Laplacian can then be written as

∇2 = ∇ ·(

∇s + n∂

∂n

)(35)

= ∇2s + ∇ ·

(n

∂

∂n

)(36)

= ∇2s + (∇ · n)

∂

∂n+ (n · ∇)

∂

∂n(37)

= ∇2s + 2κ

∂

∂n+ ∂2

∂n2(38)

where ∇2s is the surface Laplace–Beltrami operator and

κ = 1

2∇ · n (39)

is the mean surface curvature. Let us introduce curvilinearcoordinates, (ξ, η, ζ ), fitted to the surface so that the surfacecorresponds to ζ = 0 and ζ is directed opposite to theexternal normal to the center, i.e., ∂/∂ζ = −∂/∂n. Since theelectromagnetic field changes only within the skin depth fromthe value on the surface to zero, we have

E2(r) = E2(ξ �, η�, ζ �), H2(r) = H2

(ξ �, η�, ζ �) (40)

ξ � = ξ

a, η� = η

a, ζ � = ζ

δ(41)

so we obtain

∂2

∂ζ �2 H2 + 2iH2 = 2κ � δa

∂H2

∂ζ � − δ2

a2∇�2

s H2 (42)

∇�s = a∇s, κ � = κa (43)

where ∇�s is the dimensionless surface del operator in the

coordinates marked by primes, and κ � is the dimensionlesssurface curvature, which is assumed to be of the order of unity.

Furthermore, we consider expansions over the small para-meter δ/a of the form

H2 = H(0)2 + δ

aH(1)

2 +(

δ

a

)2

H(2)2 +

(δ

a

)3

H(3)2 + · · · (44)

and, similarly, for all other internal and external fields. Notethat the expansions in this form were considered earlier byseveral authors (see [12], [20]) who also provided relatedexpressions of the differential operators in curvilinear coor-dinates.

Substituting (44) into (42) and collecting terms of the samepower of δ/a, we obtain the following recurrence relations:

∂2

∂ζ �2 H(0)2 + 2iH(0)

2 = 0 (45)

H(0)2

∣∣∣ζ �=0

= H(0)2S , H(0)

2

∣∣∣ζ �=∞

= 0 (46)

∂2

∂ζ �2 H(1)2 + 2iH(1)

2 = 2κ � ∂H(0)2

∂ζ � (47)

H(1)2

∣∣∣ζ �=0

= H(1)2S , H(1)

2

∣∣∣ζ �=∞

= 0 (48)

∂2

∂ζ �2 H( j)2 + 2iH( j)

2 = 2κ � ∂H( j−1)2

∂ζ � − ∇�2s H( j−2)

2

j = 2, 3 . . . (49)

In this article, we focus on the first-order approximation, whichrequires the first two terms. The boundary conditions relate the

fields to the conditions on the surface, marked by the subscriptS and to the conditions far from the surface, where the fieldshould decay to zero. Solutions to (45) and (47) are

H(0)2 = H(0)

2S e−(1−i)ζ �(50)

H(1)2 =

(H(1)

2S + κ �H(0)2S ζ �

)e−(1−i)ζ �

. (51)

1) Zero-Order Approximation: The zero-order approxima-tion corresponds to the case of vanishing skin depth, i.e.,δ = 0. This means that the scattered field in the zero-orderapproximation is equal to the field scattered by a perfectconductor. Thus, the magnetic field can be found from thesolution to the Laplace equation with the Neumann boundaryconditions

H(sc)(0)1 = −∇�

(sc)(0)1 , ∇2�

(sc)(0)1 = 0 (52)

∂

∂n�

(sc)(0)1

∣∣∣∣S

= − ∂

∂n�

(in)1

∣∣∣∣S

. (53)

This shows that the normal component of the total externalfield is zero, so, from the boundary conditions in (8), thismeans that the normal component of H(0)

2 on the surface isalso zero. The tangential component can be found from thesolution to (52)

H(0)2S = −n × n × H(0)

2S = −n × n×(

H(in)1 + H(sc)(0)

1

)∣∣∣S

(54)

= −∇s

(�(in)

1 + �(sc)(0)1

). (55)

Using this value, we obtain H(0)2 from (50).

As soon as �(sc)(0)1 is available, the auxiliary function φ

(sc)(0)1

responsible for the rotational part of the scattered electric fieldcan be found from the equation

M[φ

(sc)(0)1

](r) + 1

2φ

(sc)(0)1 (r) = �

(sc)(0)1 (r), r ∈ S. (56)

Since the electric field inside a perfect conductor is zero,

E(0)2 = 0 (57)

the tangential components of the incident and scattered fieldson the surface are simply related, and according to (32),we have

−n × n × E(in)(0)1

= n × n × E(sc)(0)1 (58)

= n × n ×{−∇�

(sc)(0)1 + iωμ1∇ × L

[nφ

(sc)(0)1

]}(59)

= −n × n × ∇�(sc)(0)1 + iωμ1n × n × ∇ × L

[nφ

(sc)(0)1

](60)

= ∇s�(sc)(0)1 + iωμ1n × n × ∇ × L

[nφ

(sc)(0)1

], r ∈ S.

(61)

The problem here is to obtain �(sc)(0)1 from its surface gradient.

Such a problem is solvable, and we introduce the inverse sur-face del operator ∇−1

s and propose an algorithm for computingit, later on in this article. Using this, we get

�(sc)(0)1 (r)

= ∇−1s

{−n×n×

{E(in)(0)

1 +iωμ1∇×L[nφ

(sc)(0)1

]}}, r ∈ S.

(62)

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GUMEROV et al.: BOUNDARY ELEMENT SOLUTION OF ELECTROMAGNETIC FIELDS 6300312

This determines the boundary conditions for the Dirichletproblem. After the Laplace equation is solved with theseboundary conditions, we have �(sc)(0)

1 (r) for any point in V1

and can compute the scattered electric field using (32)

E(sc)(0)1 (r) = −∇�

(sc)(0)1 (r)

+iωμ1∇ × L[nφ

(sc)(0)1

](r), r ∈ V1. (63)

2) First-Order Approximation: The first-order approxima-tion can be constructed by considering Faraday’s law, which,using the primed coordinates in (41) and (43), takes the form

1

a∇�

s × E2 − 1

δn × ∂E2

∂ζ � = iωμ2H2. (64)

Plugging in the first-order approximation, we get

1

a∇�

s ×(

E(0)2 + δ

aE(1)

2

)− 1

δn ×

(∂E(0)

2

∂ζ � + δ

a

∂E(1)2

∂ζ �

)

= iωμ2

(H(0)

2 + δ

aH(1)

2

). (65)

Using (57) and collecting remaining terms near the samepowers of δ, we obtain

n × ∂E(1)2

∂ζ � = −iωμ2aH(0)2 (66)

H(1)2 = 1

iωμ2a∇�

s × E(1)2 . (67)

Since E2 is tangential on the surface [see (6)], we have

∂E(1)2

∂ζ � = −n × n × ∂E(1)2

∂ζ � (68)

= iωμ2a(

n × H(0)2

)(69)

= −ωμ2a

2

∂2

∂ζ �2(

n × H(0)2

). (70)

Using (50), we get

E(1)2 = E(1)

2S e−(1−i)ζ �(71)

E(1)2S = 1 − i

2ωμ2a

(n × H(0)

2S

). (72)

Now having this result, we can use (67) to determine thenormal component of the internal magnetic field on the surface

n · H(1)2S = 1

iωμ2an · ∇�

s × E(1)2S . (73)

This determines the boundary conditions for the Neumannproblem to solve for the scattered magnetic field via (7)

∂� (sc)(1)1

∂n

∣∣∣∣∣S

= −n · H(sc)(1)1

∣∣∣S

= −μ2

μ1n · H(1)

2S (74)

= − 1

iωμ1an · ∇�

s × E(1)2S . (75)

This then gives us the tangential component of the internalmagnetic field via (8)

−n × n × H(1)2S = −n × n × H(sc)(1)

1

∣∣∣S

(76)

= −∇s�(sc)(1)1

∣∣∣S. (77)

All together, the internal magnetic field on the surface is

H(1)2S = −n × n × H(1)

2S + n(

n · H(1)2S

). (78)

According to (51), we now have the first-order approximationfor the internal problem.

The last thing to determine in the first-order approximationis the scattered electric field. The steps to get this fromthe known magnetic potential and the boundary conditionsare similar to those for the zero-order approximation. First,we determine the auxiliary potential

M[φ

(sc)(1)1

](r) + 1

2φ

(sc)(1)1 (r) = �

(sc)(1)1 (r). (79)

Then, we calculate the surface electric potential

�(sc)(1)1 = ∇−1

s

[−E(1)

2S (80)

−n × n × iωμ1∇ × L[nφ

(sc)(1)1

]], r ∈ S. (81)

Finally, we solve the Dirichlet problem to determine thescattered field

E(sc)(1)1 = −∇�

(sc)(1)1 + iωμ1∇ × L

[nφ

(sc)(1)1

]. (82)

C. Analytical Solution for the Sphere

It is not difficult to construct an analytical solution, whichcan be used for tests. Perhaps, the simplest solution is thesolution for a sphere of radius a illuminated by a plane wave.Assume that the plane wave propagates in the x-direction andis polarized in the y-direction for the electric field and thez-direction for the magnetic field

E(in)1 = E0iyeik1 x , H(in)

1 = H0izeik1 x, k1 = ω

c1(83)

where k1 and c1 are the wavenumber and the speed of light inmedium 1. In the low-frequency approximation, k1 → 0, thiscorresponds to the following exact solution of (1) and (2):

E(in)1 = iy(E0 + iωμ1 H0x) (84)

H(in)1 = H0iz, �(in)

1 = −H0z. (85)

Note that, in (1) and (2), while H(in)1 can be a constant, E(in)

1cannot be a constant in the presence of the magnetic field andnon-zero frequency (according to Faraday’s law ∇×E(in)

1 = 0).Let us introduce spherical coordinates referenced to the centerof the sphere

(x, y, z) = r(sin θ cos ϕ, sin θ sin ϕ, cos θ). (86)

The first-order (two-term) approximation of the problem isprovided in the following, and its validity can be checkedby verifying that it satisfies the equations and the boundary

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6300312 IEEE TRANSACTIONS ON MAGNETICS, VOL. 56, NO. 11, NOVEMBER 2020

Fig. 2. Mie solution (left) and the first-order (two-term) approximationfrom (91) to (95) (right) for the imaginary part of the magnetic field alongthe slice, y = 0. The computations were carried out for a copper ball in airat 100 kHz (a = 1 mm, δ/a = 0.2061).

conditions

E(sc)(0)1 = −E0a3

(iy

1

r3− ir

3y

r4

)+ iωμ1 H0

×[

a3

2r2ir × iz − a5

2

(yix + x iy

r5− 5

xy

r6ir

)](87)

E(0)2 = 0 (88)

E(sc)(1)1 = −iωμ1c1

H0a3

2r2ir × iz (89)

E(1)2 =3(1 − i)

4ωμ2a H0(ir × iz)e

−(1−i)(a−r)/δ (90)

�(sc)(0)1 = −H0

a3

2r3z (91)

H(sc)(0)1 = H0

a3

2r3

(iz − ir

3z

r

)(92)

H(0)2 = 3

2H0

(iz − ir

z

r

)e−(1−i)(a−r)/δ (93)

�(sc)(1)1 = −c1�

(sc)(0)1 H(sc)(1)

1 = −c1H(sc)(0)1 (94)

H(1)2 =

(a − r

δ− 1

3c1

)H(0)

2 (95)

+3

2H0(1 + i)ir

z

re−(1−i)(a−r)/δ

c1 = 3(1 + i)μ2

2μ1. (96)

Apart from this solution, which reflects the approach of thisarticle, there exists an exact solution of the full problem, whichis the Mie solution [25]. To compare that solution to thepresent one, the external and internal wavenumbers in thatproblem should be set to

k1 = ωμ1 H0

E0, k2 = (1 + i)

√ωμ2σ2

2= 1 + i

δ. (97)

Figs. 2 and 3 compare the Mie solution and the analytical,first-order (two-term) approximation from (87) to (96) for theimaginary part of the magnetic and electric fields, respectively.The computations were carried out for a copper ball in air at100 kHz (a = 1 mm, δ/a = 0.2061). The lines show themagnetic and electric field lines. It is seen that the two-termsolution is, qualitatively, similar to the Mie solution (the maxrelative error in the domain shown is roughly 9%). The colorsshow the magnitude of the field (the imaginary and real parts

Fig. 3. Mie solution (left) and the first-order (two-term) approximationfrom (87) to (90) (right) for the imaginary part of the electric field alongthe slice, z = 0. The computations were carried out for a copper ball in airat 100 kHz (a = 1 mm, δ/a = 0.2061).

Fig. 4. Relative error in Im{H(sc)1 } of the asymptotic solution computed at

the surface point z = a at different values of δ/a for copper (μ2/μ1 = 1)and stainless steel (μ2/μ1 = 4) balls.

are both taken into account), which achieves its maximum ina relatively narrow zone near the boundary. It is also seenthat the magnitude of the electric field is substantially smallerinside the sphere than outside, which is due to E(0)

2 = 0.It is also seen that the internal magnetic field has a non-zeronormal component, which is a manifestation of the first-orderterm, H(1)

2 (recall that H(0)2 has only a tangential component).

On the other hand, the internal electric field is tangential tothe surface, which is also clearly seen in the Mie solution.

Fig. 4 illustrates the error of the two-term solution for theimaginary part of the magnetic field computed at the surfacepoint, z = a. According to (92), (94), and (96), we have

Im{

H(sc)1

}∣∣∣z=a

= δ

aIm

{H(sc)(1)

1

}∣∣∣z=a

(98)

= 3

2

δ

a

μ2

μ1H0iz. (99)

Values were compared for copper (μ2/μ1 = 1, i.e., non-magnetic) and stainless steel (μ2/μ1 = 4, i.e., slightlymagnetic) balls in the range, 0.01 � δ/a � 1. Accordingto the expansions, the relative error should be O(δ/a). Thegraph shows that this holds. Moreover, for μ2/μ1 = 1,the asymptotic constant in O(δ/a) is close to one, but, for

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GUMEROV et al.: BOUNDARY ELEMENT SOLUTION OF ELECTROMAGNETIC FIELDS 6300312

μ2/μ1 = 4, it is about 4. This means that the residual shouldbe written rather as O((δ/a)(μ2/μ1)) than O(δ/a).

The reason for factor μ2/μ1 can be understood by consider-ation of subsequent approximations. Denoting the normal andtangential components of the fields with subscripts n and tan,we have the following orders of magnitude:

H( j+1)2n ∼ H( j)

2 tan, H(sc)( j+1)1n = (μ2/μ1)H

( j+1)2n (100)

H(sc)( j+1)1 tan ∼ H(sc)( j+1)

1n , H( j+1)2 tan = H(sc)( j+1)

1 tan j = 0, 1, . . .

(101)

Here, the second relations (100)–(101) are due to the boundaryconditions (8)–(9) (H(in)( j+1)

1 = 0, j = 0, 1, . . .). The first rela-tion in (100) follows from Faraday’s law. Indeed, the normaland tangential projections of (64) show that

H( j+1)2n = 1

iωμ2an · ∇�

s × E( j+1)2 (102)

H( j)2 tan = − 1

iωμ2an × ∂E( j+1)

2

∂ζ � . (103)

The first relation in (101) reflects the fact that in magnetosta-tics the normal component of the magnetic field completelydetermines the full field and so its tangential component (theNeumann problem for the Laplace equation). Hence, rela-tions (100) and (101) show that H( j+2)

2n ∼ (μ2/μ1)H( j+1)2n and

H( j+1)2 ∼ (μ2/μ1)

j H(0)2 tan ∼ (μ2/μ1)

j H0, j = 0, 1, . . . Thisshows that, indeed, if we expand up to the nth approximation,the residual can be estimated as O

((δ/a)n+1(μ2/μ1)

n+1).

This estimate shows that for ferromagnetic materials withlarge μ2/μ1 ratios, the obtained solution is applicable only atvery small values of δ/a. For example, the error for carbonsteel (μ2/μ1 = 100, i.e., very magnetic) is on the order of oneat δ/a ≈ 10−2, which is consistent with the abovementionedobservation.

IV. NUMERICAL SIMULATIONS

To implement the first-order (two-term), low-frequencyapproximation described in this article, we need severalnumerical tools. First, solvers for the Laplace equation withthe Dirichlet and Neumann boundary conditions are needed.A solver for the auxiliary equation in (27) is needed as well.There should also be routines available for the computation ofthe curl of the L operator and the computation of the surfacecurl, which can be computed using the Stokes theorem. Allthese are available in our previously developed FMM/GPU-accelerated boundary element method software [23]. However,the inverse surface gradient is new, and we describe a methodfor computing it in the following.

A. Dirichlet and Neumann Solvers

The solution of the Laplace equation can be represented inthe form of a single-layer potential

�(r) = L[s](r). (104)

Thus, the problem is to determine the single-layer density sfrom the boundary conditions. For the Dirichlet problem, thisreduces to solving of boundary integral equation

L[s](r) = �(r), r ∈ S. (105)

For the Neumann problem, the boundary integral equationsturns to

L �[s](r) − 1

2s(r) = ∂

∂n�(r), r ∈ S (106)

L �[s](r) = p.v.

∫S

s(r�)∂G

(r, r�)

∂n(r)d S. (107)

B. Computation of the Inverse Surface Gradient

The BEM that we used for the examples in this article isbased on the center panel approximation, that is, the solutionon the boundary is piecewise constant on each panel, and theboundary conditions are enforced at the panel centers. Onestep in the low-frequency approximation is to compute theinverse surface gradient, φ = ∇−1

s v. In other words, given asurface gradient, v, at the panel centers of a triangular mesh,we want to determine the potential, φ, on the panels such that∇sφ = v.

If φ1 and φ2 are the values of the potential at neighboringvertices, x1 and x2, of the mesh, then the directional gradientbetween them along the edge, l, can be approximated as

l|l| · v = l

|l| · ∇φ = dφ

dl≈ φ2 − φ1

|l| , l = x2 − x1. (108)

Ideally, in this formula, v should be evaluated at the point,x = (x1 + x2)/2, but, since v are available only at the panelcenters, we compute v simply as v = (v1 + v2)/2, where v1

and v2 are the values of v on the faces sharing l. Hence, eachedge produces a linear equation

φ2 − φ1 = 1

2(x2 − x1) · (v1 + v2). (109)

Since the number of edges in the mesh is larger than the num-ber of vertices, this forms an overdetermined system, whichcan be solved using least squares. However, this problem doesnot have a unique solution. Any constant can be added to φ,and its gradient will still equal v. Thus, the obtained systemcan be poorly conditioned. For a simple connective surface,an additional constraint setting the average of φ over thesurface to zero can be added to the system. This constraint hasan intuitive explanation: if the scattered field is generated byan external field, it should not have any monopole component,meaning that φ should average to zero over the surface. Theequation enforcing this additional constraint is

Nv∑j=1

w jφ j = 0, w j = 1

3

∑i

A(i)j (110)

where w j is the weight (surface area) associated with the j thvertex, A(i)

j are the areas of the triangles sharing the j th vertex,and Nv is the total number of vertices. For multi-connectivesurfaces (several objects), each object should be supplied by asimilar condition, as the rank of the system is Nv − M , whereM is the number of single connective surfaces constituting S.Note also that, technically, it is simpler to assign some valueto some vertex, say φ1 = 0, find a solution, and then correctall of the values to a given average. In any event, as soonas the values at the vertices are determined, the values at

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6300312 IEEE TRANSACTIONS ON MAGNETICS, VOL. 56, NO. 11, NOVEMBER 2020

Fig. 5. Relative L2-norm errors for the pairs of solutions for Im{H(sc)1 } on the

surface obtained by different methods in the text to the solution listed secondin the legend. Computations performed for the copper (μ2/μ1 = 1) andstainless steel (μ2/μ1 = 4) balls of radius a = 1 mm. The dashed anddashed–dotted straight lines show linear dependence.

Fig. 6. Imaginary parts of the scattered and internal fields (copper,f = 0.1 MHz, δ = 0.2061 mm). The pear shape shown in the figure wasgenerated as a convex hull of two spheres of radii 1 and 0.3 mm whose centersare separated by the distance 2 mm.

the panel centers can be found using a simple vertex-to-faceinterpolation

φ = 1

3(φ1 + φ2 + φ3). (111)

We also mention that this method can naturally be extended tothe case when the BEM uses vertex collocation. In this case,it becomes even simpler.

C. Examples

We computed and analyzed several cases using the method,some of which are briefly described in the following. In allexamples, the incident field was generated in air, which is adielectric with the permittivity �1 = 8.85 ×10−12 F/m and thepermeability μ1 = 1.257 × 10−6 H/m. In the cases illustratedin Figs. 5–12, the incident electric field is a plane wave of unitintensity (e.g., E0 = 1 V/m), which results in the value H0 =((�1/μ1))

1/2 E0 = 2.65 × 10−3 A/m. Also, in these figures,the pictures show the internal fields inside the scatterers andthe scattered fields outside.

Fig. 7. Imaginary parts of the scattered and internal fields (copper,f = 0.1 MHz, δ = 0.2061 mm). The torus’ larger and small cross sectionradii are 2.5 and 1 mm, respectively.

Fig. 8. Real and imaginary parts of the electric field on the torus surfacemade of copper, f = 0.1 MHz, δ = 0.2061 mm. The torus’ larger and smallcross section radii are 2.5 and 1 mm, respectively.

Fig. 9. Imaginary parts of the scattered and internal fields for two spheresof radii 1 mm (copper, f = 0.1 MHz, δ = 0.2061 mm).

1) Sphere: First, we computed the benchmark case for thesphere and compared it with the analytical solution describedearlier and also with the Mie solution. The obtained resultsshow errors on the order of a few percent for low discretiza-tion, and the accuracy of the solution increases for higherdiscretization, consistent with the BEM accuracy described inour recent article [23]. In the tests, we used δ/a in the range0.01–1 and meshes with 103–106 faces. A typical example isshown in Fig. 5, which provides cross-estimation of the errorin Im{H(sc)

1 } (which is the same as Im{H1}). In contrast toFig. 4, we evaluated the relative error not at the single surface

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GUMEROV et al.: BOUNDARY ELEMENT SOLUTION OF ELECTROMAGNETIC FIELDS 6300312

Fig. 10. Same as in Fig. 9, but for spheres made of different materials shownon the pictures. The field frequency is 100 MHz.

Fig. 11. Computational mesh for the ARL electric cage (20 loops,120 000 faces, and the circular cross section radius for a rod is 2.54 cm).

Fig. 12. Imaginary parts of the scattered fields for the cage mesh shownin Fig. 11 (aluminum, f = 1 kHz, δ = 2.6 mm).

point but over the entire surface. The legend here lists pairsof solutions obtained by different methods, where the secondsolution is considered as a reference solution. The relative

error is computed as a ratio of the L2-norm of the differenceof solutions and the L2-norm of the reference solution. Theerror check was performed for copper (μ2/μ1 = 1) and thestainless steel (μ2/μ1 = 4) balls.

The results show that the error between the analyticalsolution and the BEM does not depend neither on δ/a, norμ2/μ1 due to the fact that the compared terms are simplyproportional to (μ2/μ1)(δ/a) [see (92), (94), and (96)] and,in all cases, is about 5%, which is smaller than the errorbetween the analytical and the Mie solutions. Note that this5% can be reduced using, say, linear panel approximationinstead of constant panel BEM used in this article, but thereis no substantial need for this as this error is smaller than theintrinsic errors of the model.

Comparison of the BEM or analytical solution with theMie solution shows three regions for the behavior of theresidual. At relatively low and high δ/a, the errors are relatedto the model errors, while, in some medium range, the relativeerror behaves approximately as (μ2/μ1)(δ/a), which is alsoconsistent with Fig. 4. While the deviation from the lineardependence at larger δ/a can be simply explained by the factthat the model is developed for the thin skin depth, the increaseof the error at low δ/a is also due to another model assumptionthat the external field can be computed using magentostatics.Indeed, this LF assumption is valid when k1a � 1 with theerror of approximation O(k1a). In terms of the consideredrelative error for the imaginary part of the scattered field, thisproduces the error

�(LF)2 ∼ k1a

(μ2/μ1)(δ/a). (112)

In our example, we used a = 10−3 m, and δ/a = 0.01corresponds to f ≈ 42 MHz for copper and 440 MHzfor stainless steel, in which k1 = 2π f/c1 ≈ 0.9 m−1 and0.09 m−1, respectively. Thus, (112) predicts 9% of the errorfor copper and 23% for stainless steel, which is in an excellentagreement with that in Fig. 4. Note that we did not see thisLF effect on Fig. 4 since the error was estimated for a singlepoint, which is not representative for the entire surface. Also,using (11), we can rewrite (112) in the form

�(LF)2 ∼ 2

c1σ2μ2(μ2/μ1)(δ/a)3

1

a(113)

which shows that, for fixed materials, δ/a can be made assmall as desired by increasing the object size a. Thus, Fig. 4can be also considered as the limiting case of very large a.

2) Non-Spherical Shapes of Different Topologies: Next,we used simple, non-spherical shapes of different topologies,including a pear-shaped body and a torus, which are bodiesof rotation. Figs. 6 and 7 illustrate the imaginary part ofthe scattered and internal fields in response to the incidentfield given by (84). The pear shape was generated as theconvex hull of two spheres of radii 1 and 0.3 mm with centersseparated by 2 mm. The major and minor radii of the torusare 2.5 and 1 mm, respectively. The material for both shapeswas copper, and the frequency of the field was 0.1 MHz,so δ = 0.2061 mm. Note that on this and the following

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6300312 IEEE TRANSACTIONS ON MAGNETICS, VOL. 56, NO. 11, NOVEMBER 2020

figures showing the field lines, there can be some asymme-try introduced by the plotting software (which automaticallydetermines the position of the field lines to fill the picturemore or less evenly). In these pictures, the shades of grayshow the magnitude of the field on a logarithmic scale (this isalso applicable to the figures in the next subsection). The meshused contains 6514 faces for the pear shape and 6480 facesfor the torus.

One peculiarity of the pear shape is that the curvaturechanges significantly, which, according to (51), has an effecton the internal magnetic field. The internal field is computedwell within the skin depth. Closer to the center of the body,the field lines of the asymptotic solution are substantiallydistorted. However, since the magnitude of the field decaysexponentially toward the center, such deviations from the trueshape are not so important. It is seen that while the magneticfield penetrates the body surface smoothly (at μ2 = μ1),the magnitudes of the electric field inside and outside theconductor are substantially different. Also, the internal electricfield, and so the electric current, is tangential to the surface.Fig. 8 shows the real and imaginary parts of the electricfield on the surface. It is remarkable that, in the presentexample, the magnitudes of the real and imaginary parts differby a factor of 105. Nevertheless, the present method handlesthis situation and produces robust results: the error in eachcomponent is on the order of 1%.

3) Multi-Connective Domains: To illustrate that the methodworks for domains consisting of several disconnected objects(multiply connective domains), we conducted computationsfor several spheres with different material properties. As itwas mentioned earlier, the modifications of the code here arerelated to the fact that the rank deficiency of the matrices,M + (1/2)I , and the inverse surface gradient is exactly thenumber of the disconnected objects in the domain, so addi-tional equations, such as specifying the zero average over eachobject, should be added. The results obtained show that thisapproach works well, and there were no difficulties to computesuch cases.

Note that, in the case of multiply-connected domains, onecan treat σ2 and μ2 as piecewise constants on S, which maytake different values in different connected regions. Indeed,separated objects do not interact directly, but only via thecarrier medium, for which the boundary conditions depend onthe properties of the scatterer and can be different for differentscatterers.

In the cases illustrated in Figs. 9 and 10, the spheres areof the same radius (a = 1 mm) and placed at the samedistance apart. In Fig. 9, both spheres are made of copper( f = 0.1 MHz, δ/a = 0.2061). Fig. 10 is computed atf = 100 MHz for spheres of different conductivities (on theleft) and also of different permeabilities (on the right). Here,we have δ/a = 0.0326, 0.0228, and 0.0209 for titanium, lead,and stainless steel, respectively. Since μ2/μ1 = 1 for bothtitanium and lead, the scattered magnetic field is stronger forlarger δ/a and “pushed” from the titanium to the lead sphere,which is seen by the turn points of the magnetic field lines,which happens at z > 1.5 mm, not at z = 1.5 mm as it shouldbe for the balls of the same properties (see Fig. 9). However,

Fig. 13. Computational mesh for the ARL magnetic cage (six coils,60 512 faces).

in the case of the stainless steel (μ2/μ1 = 4), the field is“pushed” in the opposite direction, i.e., from the stainless steelto the titanium, and turn points are located here at z < 1.5 mm.The explanation here is that the scattered field is proportionalto (μ2/μ1) (δ/a), not (δ/a) alone.

4) Real-World Examples: Apart from these canonical andsimple shapes, we can also use the methods in this articleto solve real-world problems. The only purpose of the casesshown in the following is to demonstrate that the resultsobtained are consistent with the physical intuition and seewhat kind of potential problems with the developed methodone can face. The Army Research Laboratory (ARL) has twofacilities for generating low-frequency electric and magneticfields for sensor calibration and characterization, as well asfor hardware-in-the-loop experiments. The electric-field cage,constructed in 2006, generates a uniform, single-axis electricfield to a high degree of accuracy [26]. It is composed of twolarge parallel plates separated by 4.2 m with 20 equally space“guard tubes” between them to control the fringing fields. Theguard tubes are made of aluminum and are 2 in thick (cross-sectional radius of 2.54 cm). The mesh of the electric-fieldcage, as shown in Fig. 11, contains 120 000 faces (20 guardtubes with 6000 faces per guard tube). For testing purposes,we illuminated the electric-field cage with the incident fieldin (84) and (85). The magnetic and electric fields for the cageare shown in Fig. 12. Computations were performed at 1 kHz(δ = 2.6 mm).

The magnetic-field cage, constructed in 2017, generates auniform, three-axis magnetic field, and is used for similarpurposes. It is a coil system, similar to a Helmholtz orMerritt coil, with six coils in the x-direction and two coilsin the y- and z-directions (carbon steel, μ2/μ1 = 100, andσ2 = 6.99 · 106 S/m). The size, spacing, and drive currentswere optimized to account for the steel walls in the lab andproduce a highly accurate field. The mesh of the magnetic-fieldcage is shown in Fig. 13. For the incident field, we used theBiot–Savart law to compute the fields produced by the sixcoils in the x-direction. This provides, for straight segmentsC j connecting end points r(1)�

j and r(2)�j and forming closed

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GUMEROV et al.: BOUNDARY ELEMENT SOLUTION OF ELECTROMAGNETIC FIELDS 6300312

Fig. 14. Incident magnetic field, the total magnetic field, and the real andimaginary parts of the scattered magnetic field for the cage mesh shownin Fig. 13 (carbon steel, f = 60 Hz, currents in the coils are 2.0520,0.8250, 0.6006, 0.6006, 0.8250, and 2.0520 A, as x increases), and clockwiseorientation for a viewer located at x = −4 m.

loops C , the following expressions for the electric and mag-netic fields:

H(in)1 (r) =

N∑j=1

H(in)1 j (r) (114)

E(in)1 (r) =

N∑j=1

E(in)1 j (r), C = ∪N

j=1C j (115)

H(in)1 j (r) = I j

4π

∫Ci

dr� × (r − r�)

|r − r�|3 (116)

= I j

4π

r(1)j × r(2)

j

r (1)j r (2)

j + r(1)j · r(2)

j

(1

r (1)j

+ 1

r (2)j

)

E(in)1 j (r) = iωμ1 I j

4π

∫C j

dr�

|r − r�| (117)

= iωμ1 I j

4πe j ln

∣∣∣∣∣r (2)j − r(2)

j · e j

r (1)j − r(1)

j · e j

∣∣∣∣∣ (118)

r(m)j =r − r(m)�

j , r (m)j =

∣∣∣r(m)j

∣∣∣, m = 1, 2 (119)

e j = r(2)�j − r(1)�

j∣∣∣r(2)�j − r(1)�

j

∣∣∣ . (120)

Here, I j is the current in the j th line element. Fig. 14illustrates the incident and scattered magnetic fields in thecage at frequency 60 Hz. As discussed earlier, because thesteel walls have a high permeability (μ2/μ1 = 100), this willcause the error to be approximately 100 times higher than ifthey were non-magnetic. It should be noted that, though, forsuch a frequency, we have δ = 2.5 mm, which is much smallerthan the wall thickness (d = 10 cm), and while the skin depthcriterion is satisfied, the accuracy of computations for thiscase is questionable due to large ratio μ2/μ1, so parameter(μ2/μ1)(δ/d) is not small, while (μ2/μ1)(δ/ l), where l is thecharacteristic length of the walls (meters), is small. This is anexample of a problem that would benefit from expanding tothe second- or third-order approximation, which would allowus to investigate much lower frequencies in the presence ofhighly magnetic materials. Such a study is beyond the scope

Fig. 15. Magnitude (colors) and the field lines of the surface currents |J2| =|σ2E2| on the ground (the upper row) and on the lower layer of the ionosphere(the lower row) for different orientations of a dipole antenna located at height1 km in the Earth’s equatorial region. The dipole orientation vector p is shownfor each plot.

of this article and, hopefully, will be conducted in future.Computations with this cage are shown in Fig. 14.

5) Domain Inversion: In all of the cases considered earlier,the conductor occupied a finite domain and was surroundedby an infinite dielectric. In practice, however, there are manycases where the opposite situation holds: the incident fieldis generated in a finite dielectric surrounded by an infiniteconductor. One such case is an antenna emitting in Earth’swaveguide between the ground and the ionosphere. To reducethe scale of the problem, we considered a rectangular box500 km wide, 500 km long, and 100 km high. We assumed thatboth the ground and the ionosphere were uniform conductors(σ2 = 1 mS/m for the ground and σ2 = 0.1 mS/m forthe ionosphere) and non-magnetic (μ2/μ1 = 1). At 100 Hz,the wavelength in air is 3000 km, while the skin depthsin the ground and ionosphere are 1.6 and 5 km, respec-tively. Thus, the specified domain size, material properties,and frequency satisfy the conditions for which the presentapproximation is derived. Note that modeling the ionospherewith a constant isotropic conductivity is questionable sincethe Hall conductivity introduces anisotropy, especially in thepolar regions, where it is strongest [27]. Near the equator,though, the Hall conductivity plays a much lesser role, so weassumed that the Pedersen and parallel conductivities are ofthe same order. In addition, we assumed that the thicknessof the most conductive layer of the ionosphere is largerthan 5 km. In any event, the case considered here is notdesigned for accurate predictions, but rather as an illustra-tion that the present method can be used to consider suchproblems.

As the normal to the surface is now directed outside of thedomain occupied by air, the sign near the 1/2 in (27) and (106)should be flipped, and that’s it! Particularly, M + (1/2)Ibecomes M − (1/2)I , which is now non-singular. We alsonote that the boundary conditions on the open sides of therectangular domain can be naturally handled by the acceptedsingle-layer representation of the potentials. In this case,we simply specify zero charge on those surfaces.

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6300312 IEEE TRANSACTIONS ON MAGNETICS, VOL. 56, NO. 11, NOVEMBER 2020

In the computations, the antenna was modeled as a Hertziandipole

E(in)1 = pG1(r − rs) + 1

k21

∇[p · ∇G1(r − rs)

](121)

H(in)1 = 1

iωμ1∇G1(r − rs) × p (122)

G1(r) = eik1r

4πr, r = |r|, k1 = ω

c1(123)

where rs are the coordinates of the dipole, p is the dipolemoment, and G1 is the free-space Green function for theHelmholtz equation with wavenumber k1 corresponding to thespeed of light c1. The surface of the computational domain wasdiscretized using 7488 triangles. Fig. 15 shows the magnitudeof the surface current, |σ2E2|, and the field lines on the groundand ionosphere surfaces for different values of p. The antennais located at the center of the ground surface at a height of1 km. The plotted cases correspond to the dipole, monopole,and mixed type of antennas.

V. CONCLUSION

The effects of eddy currents are well known, but theircomputation can be challenging due to the need for veryhigh-resolution meshes in the full-wave Maxwell solvers.The method developed in this article is substantially simplerthan the full-wave solver and produces physically meaningfulresults. Comparisons with the exact Mie solution for the sphereshow that the computational errors stay within the error boundsof the method and are mostly determined by the errors of theapproximation rather than the errors of the boundary elementmethod or the FMM. While this article provides a frameworkof how to construct a general asymptotic expansion, onlythe zero- and first-order approximations have been explic-itly derived. This is sufficient for a number of practicallyimportant problems when solutions for perfect conductorsneed to be corrected to account for the imperfectness of realconductors. However, if the accuracy of the first-order (two-term) approximation of this article is insufficient, higher-orderapproximations could be performed. Such work would beespecially useful for highly magnetic materials, when theaccuracy of the approximation is determined by the parameter,(μ2/μ1)(δ/a), rather than δ/a alone.

ACKNOWLEDGMENT

This work was supported by the Cooperative ResearchAgreement between the University of Maryland and the ArmyResearch Laboratory, with David Hull and Steven Vinci asTechnical monitors under Grant W911NF1420118.

REFERENCES

[1] J. D. Jackson, Classical Electrodynamics. Hoboken, NJ, USA: Wiley,1998, pp. 352–356.

[2] W. C. Chew, Waves and Fields in Inhomogeneous Media. Piscataway,NJ, USA: IEEE Press, 1995.

[3] S. M. Rytov, “Computation of the skin effect by the perturbationmethod,” Zhur. Eksp. i Teoret. Fiz., vol. 10, no. 2, pp. 180–189, 1940.

[4] K. M. Mitzner, “An integral equation approach to scattering from abody of finite conductivity,” Radio Sci., vol. 2, no. 12, pp. 1459–1470,Dec. 1967.

[5] T. Fawzi, M. Ahmed, and P. Burke, “On the use of the impedanceboundary conditions in eddy current problems,” IEEE Trans. Magn.,vol. 21, no. 5, pp. 1835–1840, Sep. 1985.

[6] A. Nicolas, “3D eddy current solution by BIE techniques,” IEEE Trans.Magn., vol. 24, no. 1, pp. 130–133, Jan. 1988.

[7] W. M. Rucker and K. R. Richter, “A BEM code for 3-D eddy current cal-culations,” IEEE Trans. Magn., vol. 26, no. 2, pp. 462–465, Mar. 1990.

[8] A. Bendali, M. B. Fares, and J. Gay, “A boundary-element solutionof the Leontovitch problem,” IEEE Trans. Antennas Propag., vol. 47,no. 10, pp. 1597–1605, Oct. 1999.

[9] L. Kettunen and S. Yuferev, “Implementation of high order surfaceimpedance boundary conditions using vector approximating functions,”IEEE Trans. Magn., vol. 36, no. 4, pp. 1606–1609, Jul. 2000.

[10] K. Sun, K. O’Neill, F. Shubitidze, S. A. Haider, and K. D. Paulsen,“Simulation of electromagnetic induction scattering from targets withnegligible to moderate penetration by primary fields,” IEEE Trans.Geosci. Remote Sens., vol. 40, no. 4, pp. 910–927, Apr. 2002.

[11] Y. Takahashi and S. Wakao, “Large-scale analysis of eddy-current prob-lems by the hybrid finite element-boundary element method combinedwith the fast multipole method,” IEEE Trans. Magn., vol. 42, no. 4,pp. 671–674, Apr. 2006.

[12] S. Yuferev and L. Di Rienzo, “Surface impedance boundary conditionsin terms of various formalisms,” IEEE Trans. Magn., vol. 46, no. 9,pp. 3617–3628, Sep. 2010.

[13] S. Engleder and O. Steinbach, “Boundary integral formulations for theforward problem in magnetic induction tomography,” Math. MethodsAppl. Sci., vol. 34, no. 9, pp. 1144–1156, Jun. 2011.

[14] L. Weggler, “Stabilized boundary element methods for low-frequencyelectromagnetic scattering,” Math. Methods Appl. Sci., vol. 35, no. 5,pp. 574–597, Mar. 2012.

[15] R. Vazquez, A. Buffa, and L. Di Rienzo, “NURBS-based BEM imple-mentation of high-order surface impedance boundary conditions,” IEEETrans. Magn., vol. 48, no. 12, pp. 4757–4766, Dec. 2012.

[16] M. Issa, “Asymptotic modeling and discretisation of magnetic compo-nents in eddy current problems. Electromagnetism,” Ph.D. dissertation,Univ. Paul Sabatier Toulouse III, Toulouse, France, 2019.

[17] M. Bonnet and E. Demaldent, “The eddy current model as a low-frequency, high-conductivity asymptotic form of the Maxwell transmis-sion problem,” Comput. Math. Appl., vol. 77, no. 8, pp. 2145–2161,Apr. 2019.

[18] Q.-A. Phan, O. Chadebec, G. Meunier, J.-M. Guichon, andB. Bannwarth, “3D BEM formulations for eddy current problems withmultiply connected domains and circuit coupling,” IEEE Trans. Magn.,early access, Jan. 13, 2020, doi: 10.1109/TMAG.2019.2954323.

[19] S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering bysurfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30,no. 3, pp. 409–418, May 1982.

[20] H. Haddar, P. Joly, and H.-M. Nguyen, “Generalized impedance bound-ary conditions for scattering by strongly absorbing obstacles: The scalarcase,” Math. Models Methods Appl. Sci., vol. 15, no. 8, pp. 1273–1300,Aug. 2005.

[21] N. A. Gumerov and R. Duraiswami, “Efficient FMM accelerated vortexmethods in three dimensions via the Lamb–Helmholtz decomposition,”J. Comput. Phys., vol. 240, pp. 310–328, May 2013.

[22] L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,”J. Comput. Phys., vol. 73, no. 2, pp. 325–348, Dec. 1987.

[23] R. Adelman, N. A. Gumerov, and R. Duraiswami, “FMM/GPU-accelerated boundary element method for computational magnetics andelectrostatics,” IEEE Trans. Magn., vol. 53, no. 12, pp. 1–11, Dec. 2017.

[24] Q. Hu, N. A. Gumerov, and R. Duraiswami, “Scalable fast multipolemethods on distributed heterogeneous architectures,” in Proc. Int. Conf.High Perform. Comput., Netw., Storage Anal. (SC), New York, NY, USA,2011, pp. 36:1–36:12.

[25] N. A. Gumerov and R. Duraiswami, “A scalar potential formulation andtranslation theory for the time-harmonic Maxwell equations,” J. Comput.Phys., vol. 225, no. 1, pp. 206–236, Jul. 2007.

[26] D. M. Hull, S. J. Vinci, and Y. Zhang, “ARL electric-field cage modeling,design and calibration,” in Proc. 12th Biennial IEEE Conf. Electromagn.Field Comput., Apr. 2006, p. 89.

[27] K. Papadopoulos, N. A. Gumerov, X. Shao, I. Doxas, and C. L. Chang,“HF-driven currents in the polar ionosphere,” Geophys. Res. Lett.,vol. 38, no. 12, Jun. 2011, Art. no. L12103.

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