IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS 1

Electromagnetic Modeling of Lossy Materials witha Potential-Based Boundary Element Method

Shashwat Sharma, Graduate Student Member, IEEE, and Piero Triverio, Senior Member, IEEE

Abstract—The boundary element method (BEM) enables solv-ing three-dimensional electromagnetic problems using a two-dimensional surface mesh, making it appealing for applicationsranging from electrical interconnect analysis to the design ofmetasurfaces. The BEM typically involves the electric andmagnetic fields as unknown quantities. Formulations based onelectromagnetic potentials rather than fields have garnered in-terest recently, for two main reasons: (a) they are inherentlystable at low frequencies, unlike many field-based approaches,and (b) potentials provide a more direct interface to quantumphysical phenomena. Existing potential-based formulations forelectromagnetic scattering have been proposed primarily for per-fect conductors. We develop a potential-based BEM formulationwhich can capture both dielectric and conductive losses, andaccurately models the skin effect over broad ranges of frequency.The accuracy of the proposed formulation is validated throughcanonical and realistic numerical examples.

Index Terms—Maxwell’s equations, electromagnetic potentials,boundary element method, integral equations, lossy conductors.

I. INTRODUCTION

ELECTROMAGNETIC simulation tools based on theboundary element method (BEM) [1] have gained trac-

tion in a variety of applications, ranging from antenna model-ing [2] to high-speed interconnect analysis [3]–[7]. The BEMis based on a surface integral representation of Maxwell’sequations, which allows three-dimensional problems to besolved in terms of quantities defined on a two-dimensional sur-face mesh. Many conventional full-wave BEM formulations,which take electric and magnetic fields as unknown quantities,suffer from numerical instability at very low frequencies [8].The need to model multiscale structures at both high andlow frequencies arises in applications such as the analysisof integrated circuit components. This motivates the develop-ment of broadband BEM formulations where electromagneticpotentials, rather than fields, are taken as the unknowns [9],[10]. These formulations do not rely on the coupling betweenelectric and magnetic fields, and provide a natural interface toquantum phenomena, which is an important consideration inemerging applications such as quantum computing [11], [12].Potential-based methods may also be well-suited for coupledelectromagnetic-circuit simulations.

S. Sharma is with the Edward S. Rogers Sr. Department of Electrical& Computer Engineering, University of Toronto, Toronto, ON, M5S 3G4Canada, e-mail: shash.sharma@mail.utoronto.ca. P. Triverio is with the Ed-ward S. Rogers Sr. Department of Electrical & Computer Engineering andwith the Institute of Biomedical Engineering, University of Toronto, Toronto,ON, M5S 3G4 Canada, email: piero.triverio@utoronto.ca.

This work was supported by Advanced Micro Devices, by the Natural Sci-ences and Engineering Research Council of Canada (Collaborative Researchand Development Grants program), and by CMC Microsystems.

Manuscript received . . .; revised . . ..

Potential-based integral equation (PIE) methods for scat-tering analysis have been developed primarily for perfectconductors [9], [10], [13]–[15]. For low frequencies and sub-wavelength structures, where PIE formulations are most sorelyneeded, the skin depth in a conductor may be large comparedto its physical dimensions, and modeling the conductor asperfect may be inaccurate. Although magnetoquasistatic PIEformulations have been used for eddy current modeling inlossy conductors [16]–[20], these techniques do not apply inthe presence of dielectric inclusions, or at high frequencies.PIE-based modeling of lossless dielectric objects was consid-ered in [21], but mainly from a theoretical perspective, andin [22], but in the time domain. Also, the method in [21]involves a linear combination of integral equations writtenfor adjacent materials. As in the case of analogous field-based formulations [23]–[25], the formulation in [21] maybe inaccurate for large contrasts in material parameters ofadjacent media. To the best of our knowledge, a full-wavePIE formulation for lossy penetrable materials has not beendemonstrated in the frequency domain.

In this article, we devise a novel full-wave PIE formula-tion for electromagnetic scattering from lossy dielectrics andconductors, applicable at both low and high frequencies. Theproposed formulation couples the scalar and vector potentialintegral equations [9] in the regions internal and external toeach object. An appropriate discretization is discussed, and theaccuracy of the formulation is demonstrated numerically overwide ranges of conductivity and frequency.

II. PROPOSED FORMULATION

We consider time-harmonic scattering from an object oc-cupying volume V , with surface S and outward unit normalvector n. Symbols S− and S+ denote the internal and ex-ternal sides of S, respectively. The object has permittivity ε,permeability µ, and conductivity σ > 0. The permittivity maybe complex, ε = ε′ − jε′′, where the imaginary part representsdielectric losses [26]. The object resides in free space, V0, withpermittivity ε0 and permeability µ0.

A. Internal Region

For ~r ∈ V , the magnetic vector and electric scalar poten-tials, ~A (~r) and φ (~r), respectively, can be defined via [27]

µ ~H (~r) = ∇× ~A (~r), (1)~E (~r) = −jω ~A (~r)−∇φ (~r), (2)

where ω is the angular frequency, ~H (~r) is the magneticfield, and ~E (~r) is the electric field. Using (1) and (2) in

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS 2

Maxwell’s equations, it can be shown that ~A (~r) satisfies thehomogeneous Helmholtz equation [27]

∇2 ~A (~r) + k2 ~A (~r) = 0, (3)

when the Lorenz gauge

∇ · ~A (~r) = −µ (jωε′ + ωε′′ + σ) φ (~r) (4)

is adopted [9], [28]. In (3), k2 = −jωµ (jωε′ + ωε′′ + σ)is the wave number associated with the object’s material.Likewise, φ (~r) satisfies the Helmholtz equation [27](

γµ−1 + σ) [∇2φ (~r) + k2φ (~r)

]= 0, (5)

where γ , (jωε′ + ωε′′ + σ)µ. The use of Green’s identi-ties [29] with (3) and (5) leads to the integral equations [9]

L[n′ ×∇′ × ~A (~r ′)

]+K

[n′ × ~A (~r ′)

]+ ~A (~r) + γ L

[φ (~r ′)n′

]−∇L

[n′ · ~A (~r ′)

]= 0, (6)

γµ−1(L[n′ · ∇φ (~r ′)

]+M

[φ (~r ′)

]− φ (~r)

)= 0, (7)

where primed and unprimed coordinates denote source andobservation points, respectively, ~r ∈ V , and ~r ′ ∈ S−. Theintegral operators in (6) and (7) are defined as [1]

L[~a (~r ′)

]=

ˆSdS ′G (k, ~r, ~r ′)~a (~r ′) , (8)

K[~a (~r ′)

]=

ˆSdS ′∇G (k, ~r, ~r ′)× ~a (~r ′) , (9)

M[a (~r ′)

]=

ˆSdS ′ n′ · ∇G (k, ~r, ~r ′) a (~r ′) , (10)

where G (k, ~r, ~r ′) is the Green’s function associated with theobject’s material,

G (k, ~r, ~r ′) =e−jk|~r−~r ′|4π |~r − ~r ′| . (11)

Next, the object’s surface is discretized with a triangu-lar mesh. Quantity n×∇× ~A (~r) is expanded with Rao-Wilton-Glisson (RWG) functions [30], ~fn (~r) normalizedby edge length, while n× ~A (~r) is expanded with Buffa-Christiansen functions [31], ~gn (~r), which are defined on abarycentric refinement of the mesh. This choice of functionsstems from (1) and (2), which indicate that n × ∇× ~A (~r)and n× ~A (~r) are related to n × ~H (~r) and n × ~E (~r) on S,respectively. Since n × ~H (~r) and n × ~E (~r) can be interpretedas electric and magnetic surface current densities, respec-tively, n × ∇× ~A (~r) and n× ~A (~r) must both be expandedwith divergence-conforming basis functions [32], while re-specting their mutual orthogonality [33]. These requirementsare satisfied by the proposed expansion scheme.

Unknowns φ (~r) and n · ~A (~r) are expanded with unit-amplitude pulse functions, hn (~r), while n·∇φ (~r) is expandedwith area-normalized pulse functions, hn (~r). The choice ofnormalizing by area was based on an empirical study ofthe condition number of the final system matrix. A moresophisticated choice of basis function for φ (~r), such as the onesuggested in [22], may lead to improved accuracy. However,

as shown in Section III, the pulse functions are sufficientlyaccurate in a variety of cases.

Taking the cross product of n with (6) and letting ~r → S−gives its rotated tangential part, which is tested with n×RWGfunctions to get the matrix relation

L(ff)ac, +

(K(fg) +

1

2I(fg)×

)at,

+ γ L(fh)Φ + DTL(hh)an, = 0. (12)

Taking the dot product of n with (6) and letting ~r → S− givesits normal component, which is tested with hn (~r) to obtain

L(hf)ac, + K(hg)at, + γ L(hh)Φ

−(−(M(hh)

)T− 1

2I(hh)‖

)an, = 0, (13)

where the superscript “T ” denotes the matrix transpose. Fi-nally, taking ~r → S− in (7) and testing it with hn (~r) yields

γµ−1

[L(hh)Ψ +

(M(hh) − 1

2I(hh)‖

)Φ

]= 0. (14)

In (12), (13), and (14), L, K and M are the discretized L, Kand M operators, respectively, where a dash through a ma-trix indicates that the associated integral is computed in theprincipal value sense [1]. Term D is a sparse incidencematrix linking mesh edges and triangles [34]. Identity op-erator I

(fg)× is obtained by testing ~gn (~r) with n × ~fn (~r),

while I(hh)‖ involves testing hn (~r) with hn (~r). The super-

script labels (ij) on each discrete operator represent thetesting and basis functions involved, respectively. Column vec-tors ac,, at,, an,, Φ and Ψ contain the unknown coefficientsassociated with n×∇× ~A (~r), n× ~A (~r), n · ~A (~r), φ (~r)and n · ∇φ (~r), respectively.

B. External RegionNext, PIEs are devised to capture the physics in the re-

gion external to S, following the procedure in Section II-Afor ~r, ~r ′ ∈ S+. However, instead of using the normal compo-nent of (6) for the external region, we take its divergence [13]and test the resulting equation with hn (~r). We found that thischoice leads to better conditioning of the final system matrix,and better accuracy. The resulting discrete equations are

L(ff)0 ac,0 +

(K

(fg)0 − 1

2I(fg)×

)at,0

+ γ0 L(fh)0 Φ0 + DTL

(hh)0 an,0 = −at,inc, (15)

L(hh)0 Dac,0 + γ0

(M

(hh)0 +

1

2I(hh)‖

)Φ0

+ k20 L

(hh)0 an,0 = γ0 Φinc, (16)

L(hh)0 Ψ0 +

(M

(hh)0 +

1

2I(hh)‖

)Φ0 = −Φinc, (17)

where the subscript “0” on the matrix operators denotesthat the Green’s function associated with V0 is used,and γ0 , jωε0µ0. Subscript “0” on the column vectors ofunknowns indicates that the quantities are defined on S+.Subscript “inc” indicates incident potentials [9], [10].

IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS 3

C. Boundary Conditions

For n×∇× ~A (~r), n× ~A (~r) and φ (~r), we use boundaryconditions on S identical to those in [9], [10],

µ−10 n×∇× ~A0 (~r) = µ−1 n×∇× ~A (~r), (18)

n× ~A0 (~r) = n× ~A (~r), (19)φ0 (~r) = φ (~r), (20)

where ~A (~r) and ~A0 (~r) are the magnetic vector potentialson S− and S+, respectively. For conductive objects, a newboundary condition is required for n · ~A (~r) and n · ∇φ (~r),

− jω[γ0µ−10 n · ~A0 (~r)− γµ−1n · ~A (~r)

]−[γ0µ−10 n · ∇φ0 (~r)− γµ−1n · ∇φ (~r)

]= 0, (21)

which is used to eliminate n · ∇φ (~r) in (7). Equation (21)is derived using (2), (4) and the standard boundary condi-tions [28] for n · ~E (~r), n · ~E0 (~r) and n · ~J (~r), where ~E (~r)and ~E0 (~r) are the electric fields in V and V0, respectively,and ~J (~r) is the conduction volume current density in V .

D. Final System Matrix

Equations (15), (12), (16), (13), (14) and (17) are con-catenated in that order, and the boundary condition (21) isapplied to get the final system of equations (22) at the topof the following page, where ξ is the average mesh edgelength. In (22), the equations and unknowns have been strate-gically scaled to ensure stable direct factorization and goodaccuracy for wide ranges of frequency and conductivity. Itmay be necessary to enforce charge conservation at extremelylow frequencies. Although this point is not addressed in thepresent formulation, the results in Section III demonstrate theextremely wide ranges of frequency over which the proposedmethod still remains accurate. Also, the system matrix in (22)contains two vector quantities (ac,0, at,) and four scalarquantities (Φ, an,, an,0 Ψ0) as unknowns, while most field-based formulations for lossy conductors contain only twovector and up to one scalar unknown quantity [3], [7], [35].However, the additional unknowns in the proposed methodmay be well worth the broadband performance, particularlywhen coupled with acceleration algorithms [36]–[39]. It maybe possible to obtain a smaller system matrix, but the scope ofthis work is to establish that potential-based formulations canbe used to model lossy conductors accurately and over verywide ranges of frequency and conductivity.

III. RESULTS

The accuracy of the proposed formulation is validatedthrough comparisons with analytical solutions and an exist-ing field-based BEM formulation: the enhanced augmentedelectric field integral equation (eAEFIE) for penetrable ob-jects [35], [40]. The numerical integration routines in [3] wereused for the operators associated with the internal region,to maintain accuracy for highly conductive media. Electricand magnetic fields tangential to S were obtained as a post-processing step via (1) and (2).

A. Sphere

First, we consider a sphere with diameter 1m and relativepermittivity 2, excited by a plane wave. The sphere is meshedwith 2, 114 triangles, and the bistatic radar cross section (RCS)is compared against the analytical Mie series (Fig. 1). The RCSis reported for the plane along which the incident electric fieldis polarized (E-plane). We consider conductivities spanning 10decades from 10−3 S/m, where the sphere behaves like adielectric, to 107 S/m, corresponding to a good conductor. Ninedecades of frequencies from 1Hz to 1GHz are simulated toencompass skin depths from 1 µm to several times the sphere’sdiameter. For the 1GHz cases, a finer mesh with 3, 786triangles was used. Fig. 1 demonstrates the excellent accuracyof the proposed formulation. The results deviate slightly fromthe Mie series in the bottom panel, but identical deviationsare observed for the eAEFIE formulation. Therefore, theseerrors can be attributed to the discretization and numericalintegration, which are common to both formulations.

B. Cube

Next, we consider an FR-4 cube meshed with 3, 274triangles, with side length 1m, ε′ = 4.4 ε0, and loss tan-gent tan δ = ε′′/ε′ = 0.03 [41]. A plane wave impinges on thecube with the electric field polarized as shown in the insetin the bottom panel of Fig. 2. The E-plane bistatic RCSfor the proposed formulation is compared with the resultsobtained via the eAEFIE, for frequencies between 10 kHzand 1GHz. The top panel of Fig. 2 demonstrates the excel-lent accuracy of the proposed formulation compared to theeAEFIE. We also considered the case when ε′ = ε0, ε′′ = 0,and σ ∈

[10−3, 107

]S/m at 100MHz, corresponding to skin

depths between 1.6m and 16 µm. Excellent agreement with theeAEFIE was achieved, as shown in the bottom panel of Fig. 2.The inset shows the surface current density for σ = 10S/m.

C. Split Ring Resonator Array

Finally, we consider a 2× 2 array of split ring resonators(SRRs), which are of relevance in the design of metamaterialsand metasurfaces [42]. Each element has a relative permittivityof 11 and an electrical conductivity of 107 S/m. The elementshave side length 2 µm, width 0.2 µm, and height 0.1 µm. Thewidth of the gap is 0.2 µm. The structure is meshed with 3, 392triangles, and excited with an incident plane wave travelingalong the −z direction, with the electric field polarized inthe y direction. The geometry is shown in Fig. 3. We considerthe frequencies 10GHz, 1THz and 100THz. Fig. 3 shows theelectric surface current density for the 100THz case. Fig. 4shows the magnitude of the electric field measured along theprobe line shown in Fig. 3. The probe line is placed alongthe x axis in the xz plane bisecting the array, 0.4 µm aboveit. Excellent agreement is obtained compared to the eAEFIEfor all three frequencies, demonstrating the accuracy of theproposed PIE formulation for realistic structures.

IV. CONCLUSION

A boundary element formulation based on the electric scalarand magnetic vector potential is proposed for the accurate

IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS 4

1ξL

(ff)0 K

(fg)0 − 1

2I(fg)×

jk0ξ

L(fh)0 0 1

ξDTL

(hh)0 0

µξµ0

L(ff) K(fg) + 12I(fg)×

c0γξ

L(fh) 1ξDTL(hh) 0 0

L(hh)0 D 0 jk0

(M

(hh)0 + 1

2I(hh)

‖

)0 k2

0 L(hh)0 0

µµ0

L(hf) ξK(hg) c0γ L(hh)(M(hh)

)T − 12I(hh)

‖ 0 0

0 0 M(hh) − 12I(hh)

‖k2

γc0L(hh) − k20

γc0

µµ0

L(hh) γ0γc0

µµ0

L(hh)

0 0 M(hh)0 − 1

2I(hh)

‖ 0 0 1c0

L(hh)0

ac,0

at,/ξ

Φ/c0

an,

an,0

Ψ0

=

−at,inc/ξ

0

γ0 Φinc

0

0

−Φinc/c0

. (22)

0 25 50 75 100 125 150 175θ (degrees)

−350

−340

−330

−320

Bis

tati

cR

CS

(dB

sm)

f = 1 Hz

σ ∈ {10−3, 10−1,

101, 103} S/m

σ ∈ {10−3, 10−1,

101, 103} S/m

σ ∈ {10−3, 10−1,

101, 103} S/m

σ = 107 S/mσ = 107 S/mσ = 107 S/m

Mie

eAEFIE

Proposed

0 25 50 75 100 125 150 175θ (degrees)

−240

−220

−200

Bis

tati

cR

CS

(dB

sm)

f = 1 kHz

σ ∈ {10−3, 10−1, 101} S/mσ ∈ {10−3, 10−1, 101} S/mσ ∈ {10−3, 10−1, 101} S/m

σ = 103 S/mσ = 103 S/mσ = 103 S/m

σ = 107 S/mσ = 107 S/mσ = 107 S/m

Mie

eAEFIE

Proposed

0 25 50 75 100 125 150 175θ (degrees)

−140

−120

−100

−80

Bis

tati

cR

CS

(dB

sm)

f = 1 MHz

σ ∈ {10−3, 10−1} S/mσ ∈ {10−3, 10−1} S/mσ ∈ {10−3, 10−1} S/m

σ = 101 S/mσ = 101 S/mσ = 101 S/m

σ = 103 S/mσ = 103 S/mσ = 103 S/m

σ = 107 S/mσ = 107 S/mσ = 107 S/m

Mie

eAEFIE

Proposed

0 25 50 75 100 125 150 175θ (degrees)

−30

−20

−10

0

10

20

Bis

tati

cR

CS

(dB

sm)

f = 1 GHzσ = 10−3 S/mσ = 10−3 S/mσ = 10−3 S/m

σ = 10−1 S/mσ = 10−1 S/mσ = 10−1 S/m

σ = 101 S/mσ = 101 S/mσ = 101 S/m σ ∈ {103, 107} S/mσ ∈ {103, 107} S/mσ ∈ {103, 107} S/m

Mie

eAEFIE

Proposed

Fig. 1: Accuracy validation for the sphere in Section III-A.The same horizontal axis applies to all panels.

modeling of lossy objects over wide ranges of frequencyand conductivity. Unlike existing potential-based scatteringformulations, the proposed method accurately captures the skineffect both at low and high frequencies, and can model bothgood conductors and lossy dielectrics. The accuracy of the pro-posed formulation is validated through canonical and realisticnumerical examples, and excellent agreement with analyticalresults and an existing field-based method is observed for atleast nine decades of frequency and conductivity.

0 25 50 75 100 125 150 175θ (degrees)

−200

−150

−100

−50

0

Bis

tati

cR

CS

(dB

sm)

tan δ = 0.03

f = 10 kHzf = 10 kHz

f = 1 MHzf = 1 MHz

f = 10 MHzf = 10 MHz

f = 100 MHzf = 100 MHz

f = 1 GHzf = 1 GHz

eAEFIE

Proposed

0 25 50 75 100 125 150 175µ (degrees)

°40

°20

0

Bis

tatic

RC

S(d

Bsm

)

f = 100 MHz

æ = 10°3 S/mæ = 10°3 S/m

æ = 10°1 S/mæ = 10°1 S/m

æ = 101 S/mæ = 101 S/m

æ 2 {103, 107} S/mæ 2 {103, 107} S/m

eAEFIE

Proposed

E→

H→

Propagationdirection

Fig. 2: Accuracy validation for the cube in Section III-B.

Probe line

0.4 µm

Fig. 3: Geometry and electric surface current density magni-tude for the SRR array in Section III-C.

0 50 100 150 200 250Distance along probe line (µm)

10−1

100

101

|~ E|(

V/m

) f = 10 GHzf = 10 GHz

f = 1 THzf = 1 THz

f = 100 THzf = 100 THz

eAEFIE

Proposed

Fig. 4: Near-field | ~E (~r)| for the SRR array in Section III-C.

IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS 5

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