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Electromagnetic Integral Equations: Insights in Conditioning and Preconditioning

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Integral equation formulations are a competitive strategy in computational electromagnetics but, lamentably, are often plagued by ill-conditioning and by related numerical instabilities that can jeopardize their effectiveness in several real case scenarios. Luckily, however, it is possible to leverage effective preconditioning and regularization strategies that can cure a large majority of these problems. Not surprisingly, integral equation preconditioning is currently a quite active field of research. To give the reader a propositive overview of the state of the art, this paper will review and discuss the main advancements in the field of integral equation preconditioning in electromagnetics summarizing strengths and weaknesses of each technique. The contribution will guide the reader through the choices of the right preconditioner for a given application scenario. This will be complemented by new analyses and discussions which will provide a further and more intuitive understanding of the ill-conditioning of the electric field (EFIE), magnetic field (MFIE), and combined field integral equation (CFIE) and of the associated remedies.
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Electromagnetic Integral Equations:
Insights in Conditioning and Preconditioning
Simon B. Adrian, Member, IEEE, Alexandre Dély, Davide Consoli, Student Member, IEEE,
Adrien Merlini, Member, IEEE, and Francesco P. Andriulli, Senior Member, IEEE
Integral equation formulations are a competitive strategy in computational electromagnetics but, lamentably, are often plagued by
ill-conditioning and by related numerical instabilities that can jeopardize their effectiveness in several real case scenarios. Luckily,
however, it is possible to leverage effective preconditioning and regularization strategies that can cure a large majority of these
problems. Not surprisingly, integral equation preconditioning is currently a quite active field of research. To give the reader a
propositive overview of the state of the art, this paper will review and discuss the main advancements in the field of integral
equation preconditioning in electromagnetics summarizing strengths and weaknesses of each technique. The contribution will guide
the reader through the choices of the right preconditioner for a given application scenario. This will be complemented by new
analyses and discussions which will provide a further and more intuitive understanding of the ill-conditioning of the electric field
(EFIE), magnetic field (MFIE), and combined field integral equation (CFIE) and of the associated remedies.
Index Terms—Integral Equations, Boundary Element Method, Computational Electromagnetic, Preconditioning, EFIE, MFIE.
I. Introduction
Integral equation formulations, solved by the boundary
element method (BEM), have become a well established tool
to solve scattering and radiation problems in electromagnet-
ics [1]–[4]. What makes these schemes so suitable for electro-
magnetic analyses is that, differently from approaches based
on differential equations such as the finite element method
(FEM) or the finite-difference time-domain method (FDTD),
they naturally incorporate radiation conditions without the
need for artificial absorbing boundary conditions, they only set
unknowns on boundary surfaces (two-dimensional manifolds)
instead of discretizing the entire volume, and they are mostly
free from numerical dispersion. On the other hand, linear sys-
tem matrices arising from differential equations schemes are
sparse [5], while those arising in BEM are, in general, dense.
This drawback, however, can be overcome if a fast method
such as the multilevel fast multipole method (MLFMM) [6],
the multilevel matrix decomposition algorithm (MLMDA) [7]
and later equivalents [8]–[12] are used at high frequency or
the adaptive cross approximation (ACA)/H-matrix methods
and related schemes [13]–[18] are used at lower frequencies.
These schemes are often capable of performing matrix-vector
This work was supported in part by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 724846, project 321), by the Italian Ministry
of University and Research within the Program PRIN2017, EMVISION-
ING, Grantno. 2017HZJXSZ, CUP:E64I190025300, by the Italian Ministry
of University and Research within the Program FARE, CELER, Grantno.
R187PMFXA4, by the Région Bretagne and the Conseil Départemental du
Finistère under the project “TONNERRE”, by the ANR Labex CominLabs
under the project “CYCLE”, and by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) – SFB 1270/1–299150580.
S. B. Adrian is with Universität Rostock, Rostock, Germany (e-mail:
simon.adrian@uni-ro).
A. Dély is with the Politecnico di Torino, Turin, Italy (e-mail: alexan-
dre.dely@polito.it).
D. Consoli is with the Polytechnic University of Turin, Turin, Italy (e-
mail: davide.consoli@polito.it).
A. Merlini is with IMT Atlantique, Brest, France (e-mail: adrien.mer-
lini@imt-atlantique.fr).
F. P. Andriulli is with the Politecnico di Torino of Turin, Turin, Italy
(e-mail: francesco.andriulli@polito.it).
products in O(𝑁log 𝑁)or even 𝑂(𝑁)complexity, where 𝑁
denotes the number of unknowns (the linear system matrix
dimension). Thus the complexity to obtain the BEM solution
of the electromagnetic problem is, when an iterative solver
is used, O(𝑁iter𝑁log 𝑁)(or 𝑂(𝑁iter 𝑁)in the low-frequency
regime), where 𝑁iter is the number of iterations.
The number of iterations 𝑁iter is generally correlated with
the condition number of the linear system matrix, that is,
the ratio between the largest and smallest singular values of
the matrix [19]. This number is often a function of 𝑁and,
when the BEM formulation is set in the frequency domain,
of the wavenumber 𝑘. This can potentially result in a solution
complexity greater, and sometimes much greater, than 𝑂(𝑁2),
something that would severely jeopardize the other advantages
of using BEM approaches.
For this reason it is of paramount importance to address
and solve all sources of ill-conditioning for integral equations
and, not surprisingly, this has been the target of substantial
research in the last decade that this work will analyze, review,
and summarize.
For surface integral equations (SIEs) that model scatter-
ing or radiation problems for perfect electrical conductors
(PEC) geometries, we can typically distinguish the following
sources of ill-conditioning: i) the low-frequency breakdown, ii)
the h-refinement (dense-discretization) breakdown, iii) high-
frequency issues (including internal resonances and the high-
frequency breakdown), and iv) the lack of linear independence
in the basis elements (including lack of orthogonality and mesh
irregularities).
Some of the first methods explicitly addressing electromag-
netic integral equation ill-conditioning date back to the 1980s,
when the focus was on the low-frequency breakdown [20]
and on the problem of interior resonances [21]. Since then,
a plethora of schemes and strategies addressing one or more
of the issues i)–iv) have been presented and some of these
strategies are still the topic of intense research. In the past, a
few review articles have appeared that dealt with aspects of
stabilizing ill-conditioned electromagnetic integral equations.
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Most recently, Antoine and Darbas [22] presented an extensive
review on operator preconditioning with a focus on high-
frequency issues. A few years ago, Ylä-Oijala et al [23]
discussed issues in finding a stable and accurate integral
equation formulation and they addressed certain open issues
in preconditioning, and Carpentieri discussed preconditioning
strategies with a focus on large-scale problems [24], [25].
Finally, although for space limitation this paper will focus
on the electric field integral equation (EFIE) and magnetic
field integral equation (MFIE) operators (which are the fun-
damental building blocks for several other formulations), the
reader should not that a substantial amount of literature and
quite effective preconditioned methods have been presented
for modelling penetrable bodies both homogeneous and in-
homogeneous [26]–[29]. The reader should also be aware
that domain decomposition schemes can play a fundamen-
tal role in managing and solving electromagnetic problems
containing even severely ill-conditioned operators [30], [31].
These approaches, however, are per se a discipline within
Computational Electromagnetics and any brief treatment out-
side of a dedicated review would inevitably be insufficient
and partial. Moreover, domain decomposition algorithms are
not competing with the strategies discussed here but, often
times, complementary [32]. For these reasons, we will not
treat domain decomposition strategies in this review, but
rather refer the interested reader to the excellent contributions
in literature [33]–[35]. Similarly, discontinuous Galerkin and
related methods for handling non-conformal meshes will not
be treated here, as extensive additional treatments would be
required; the reader can refer to [36], [37] and references
therein for specific discussions on this family of methods.
The purpose of this article is two-fold: on the one hand,
we review and discuss the strategies that have been devised
in the past to overcome the sources of ill-conditioning i)-
iv) summarizing strengths and weaknesses, guiding the reader
through the choices of the right preconditioner for a given
application scenario. On the other hand, we complement the
overview with new results that contribute to better character-
izing the ill-conditioning of the EFIE and MFIE. Finally, we
will complement our discussions with a spectral analysis of
the formulations on the sphere, which will provide a further
and more intuitive understanding of the ill-conditioning of the
EFIE, MFIE, and combined field integral equation (CFIE) and
of the associated potential remedies. In contrast to [22], our
focus will include low-frequency effects and wideband stable
formulations as well Calderón and quasi-Helmholtz projection
strategies. Moreover, whenever appropriate, we will provide
implementational considerations and details that will enable
the reader to dodge all practical challenges that are usually
faced when engineering the most effective preconditioning
schemes.
This paper is organized as follows: Section II introduces
the background material and sets up the notation, Section III
reviews the connection of the spectrum of matrices and the
role of condition number in the solutions of the associated
linear systems. Section IV focuses on low-frequency scenarios
analyzing their main challenges and solution strategies. Sec-
tion V presents problems and solutions associated with highly
refined meshes, while Section VI focuses on scenarios in the
high-frequency regime. Section VII considers the low of mesh
and basis functions quality on the overall conditioning and
Section VIII presents the conclusions and final considerations.
II. Notation and Background
We are interested in solving the electromagnetic scattering
problem where a time-harmonic, electromagnetic wave (𝒆i,𝒉i)
in a space with permittivity 𝜀and permeability 𝜇impinges on
a connected domain 𝛺R3with PEC boundary 𝛤𝜕 𝛺
resulting in the scattered wave (𝒆s,𝒉s). The total electric 𝒆B
𝒆i+𝒆sand magnetic 𝒉B𝒉i+𝒉sfields satisfy Maxwell’s
equations
∇ × 𝒆(𝒓)=+i𝑘𝒉(𝒓),for all 𝒓𝛺+,(1)
∇ × 𝒉(𝒓)=i𝑘𝒆(𝒓),for all 𝒓𝛺+,(2)
where 𝛺+𝛺
c,𝑘𝜔𝜀𝜇 is the wave number, 𝜔
the angular frequency, and 𝒆,𝒉must satisfy the boundary
conditions for PEC boundaries
ˆ
𝒏×𝒆=0,for all 𝒓𝛤 , (3)
ˆ
𝒏×𝒉=𝒋𝛤,for all 𝒓𝛤 , (4)
where 𝒋𝛤is the induced electric surface current density. In
addition, 𝒆sand 𝒉smust satisfy the Silver-Müller radiation
condition [38], [39]
lim
𝑟→∞𝒉s×𝒓𝑟𝒆s=0.(5)
We assumed (and suppressed) a time dependency of ei𝜔𝑡 and
normalized 𝒉with the wave impedance 𝜂𝜇/𝜀.
To find (𝒆s,𝒉s), we can solve the EFIE
T𝑘𝒋𝛤=ˆ
𝒏×𝒆i(6)
for 𝒋𝛤, where ˆ
𝒏is the surface normal vector directed into 𝛺+
and
T𝑘i𝑘T
A,𝑘 +1/(i𝑘)T
Φ,𝑘 (7)
is the EFIE operator composed of the vector potential operator
(T
A,𝑘 𝒋𝛤)(𝒓)=ˆ
𝒏×𝛤
𝐺𝑘(𝒓,𝒓0)𝒋𝛤(𝒓0)d𝑆(𝒓0)(8)
and the scalar potential operator
(T
Φ,𝑘 𝒋𝛤)(𝒓)=ˆ
𝒏×grad𝛤𝛤
𝐺𝑘(𝒓,𝒓0)div𝛤𝒋𝛤(𝒓0)d𝑆(𝒓0),
(9)
where
𝐺𝑘(𝒓,𝒓0)=ei𝑘|𝒓𝒓0|
4π|𝒓𝒓0|(10)
is the free-space Green’s function. A definition of the surface
differential operators grad𝛤and div𝛤can be found in [40,
Appendix 3] or [41, Chapter 2]. Once 𝒋𝛤is obtained, 𝒆s,𝒉s
can be computed using the free-space radiation operators.
Alternatively, one can solve the MFIE for the exterior
scattering problem
ˆ
𝒏×𝒉i=M+
𝑘𝒋𝛤+I/2+K𝑘𝒋𝛤,(11)
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𝒗+
𝑛
𝒗
𝑛
𝒓+
𝑛
𝒓
𝑛
𝑐+
𝑛
𝑐
𝑛
𝒆𝑛
Fig. 1. The vector field of an RWG function. The vector 𝒆𝑛denotes the
directed edge, 𝑐+
𝑛and 𝑐
𝑛denote the domains of the cells, 𝑣+
𝑛and 𝑣
𝑛denote
vertices on the edge 𝒆𝑛, and 𝒓+
𝑛and 𝒓
𝑛are the vertices opposite to the edge
𝒆𝑛.
where Iis identity operator, M+
𝑘is the MFIE operator for the
exterior scattering problem, and
K𝑘𝒋𝛤(𝒓)Bˆ
𝒏×𝛤𝐺𝑘(𝒓,𝒓0) × 𝒋𝛤d𝑆(𝒓0).(12)
The MFIE operator for the interior scattering problem is
M
𝑘I/2+K𝑘and will be used later in the construction
of preconditioners.
The EFIE and the MFIE have non-unique solutions for
resonance frequencies. A classical remedy is the use of the
CFIE [21]
𝛼T𝑘𝒋𝛤+ (1𝛼)ˆ
𝒏×M+
𝑘𝒋𝛤
=𝛼ˆ
𝒏×𝒆i+ (1𝛼)ˆ
𝒏׈
𝒏×𝒉i(13)
which is uniquely solvable for all frequencies.
For the discretization of the EFIE, we employ Rao-Wilton-
Glisson (RWG) basis functions 𝒇𝑛𝑋𝒇which are here—in
contrast to their original definition in [42]—not normalized
with the edge length, that is,
𝒇𝑛=
𝒓𝒓+
𝑛
2𝐴𝑐+
𝑛
for 𝒓𝑐+
𝑛,
𝒓
𝑛𝒓
2𝐴𝑐
𝑛
for 𝒓𝑐
𝑛
(14)
using the convention depicted in Figure 1.
Following a Petrov-Galerkin approach, we obtain the system
of equations
T𝑘j=i𝑘TA,𝑘 +1/(i𝑘)TΦ, 𝑘 j=ei(15)
that can be solved to obtain an approximation of the solution
in the form 𝒋𝛤𝑛[j]𝑛𝒇𝑛and where
TA,𝑘 𝑛 𝑚 Bhˆ
𝒏×𝒇𝑛,T
A,𝑘 𝒇𝑚i𝛤,(16)
TΦ,𝑘 𝑛 𝑚 Bhˆ
𝒏×𝒇𝑛,T
Φ,𝑘 𝒇𝑚i𝛤,(17)
ei𝑛
Bhˆ
𝒏×𝒇𝑛,ˆ
𝒏×𝒆ii𝛤,(18)
with
h𝒇,𝒈i𝛤B𝛤
𝒇(𝒓) · 𝒈(𝒓)d𝑆(𝒓).(19)
Even though we are testing with ˆ
𝒏×𝒇𝑛, the resulting system
matrix T𝑘is the one from [42] (up to the fact that the
RWG functions we are using are not normalized), because
our definition of the EFIE operator includes an ˆ
𝒏×term (in
contrast to [42]).
For the discretization of the MFIE, functions dual to the
RWGs must be used for testing [43]. Historically, the first
dual basis functions for surface currents where introduced by
Chen and Wilton for a discretization of the Poggio-Miller-
Chang-Harrington-Wu-Tsai (PMCHWT) equation [44]. Later
and independently, Buffa and Christiansen introduced the
Buffa-Christiansen (BC) functions [45], which differ from the
Chen-Wilton (CW) functions in that the charge on the dual
cells is not constant. Figure 2 shows a visualization of a BC
function. In our implementation, we are using BC functions
and denote them as 𝒇𝑋
𝒇, where the tilde indicates that
the function is defined on the dual mesh. The analysis is,
however, applicable to CW functions as well, and thus, we
will mostly speak of “dual functions” to stress the generality
of our analysis. For a definition of the BC functions as well
as implementation details, we refer the reader to [46]. For the
discretization of the MFIE, we obtain
M+
𝑘j1/2Gˆ
𝒏×𝒇,𝒇+K𝑘j=hi,(20)
where
[K𝑘]𝑛𝑚 Bhˆ
𝒏×𝒇𝑛,K𝑘𝒇𝑚i𝛤,(21)
hi𝑛
Bhˆ
𝒏×𝒇𝑛,ˆ
𝒏×𝒉ii𝛤.(22)
and where the Gram matrix for any two function spaces 𝑋𝑓
and 𝑋𝑔is defined as
G𝑓 ,𝑔 𝑚𝑛 h𝑓𝑚, 𝑔𝑛i𝛤,(23)
with 𝑓𝑚𝑋𝑓and 𝑔𝑛𝑋𝑔.
For the discretization of the CFIE, we have
C𝑘j𝛼T𝑘+ (1𝛼)G𝒇,𝒇G1
ˆ
𝒏×
𝒇,𝒇M+
𝑘j
=𝛼ei+ (1𝛼)G𝒇,𝒇G1
ˆ
𝒏×
𝒇,𝒇hi(24)
with the combination parameter 0< 𝛼 < 1.
III. Condition Numbers, Iterative Solvers, and
Computational Complexity
To solve the linear system of equations arising from bound-
ary element discretizations, such as (15), one can resort either
to (fast) direct or to iterative solvers. For direct solvers, the time
to obtain a solution is independent from the right-hand side,
whereas for iterative solvers, the right-hand side as well as the
spectral properties of the system matrix influence the solution
time. Standard direct solvers such as Gaussian elimination
have a cubic complexity, which renders them unattractive for
large linear systems. Recent progress in the development of
fast direct solvers has improved the overall computational
cost [47]–[50].
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𝒗+
𝑛
𝒗
𝑛
𝒓+
𝑛
𝒓
𝑛
𝑐+
𝑛
𝑐
𝑛
𝒆𝑛
Fig. 2. The vector field of a BC function.
Iterative solvers, on the other hand, start from an initial
guess of the solution, x(0), and compute a sequence of ap-
proximate solutions, where the following element of such a
sequence is based on the previously computed one, until a
desired accuracy is achieved. Formally, given a linear system
of equations
Ax =b,(25)
an iterative solver should stop when Ax(𝑖)b/kbk< 𝜀,
where 𝜀 > 0is the solver tolerance and x(𝑖)the approximate
solution after the 𝑖th iteration. Whether an iterative solver
will converge or not, depends on the chosen solver and the
properties of A, as we will discuss in the following.
To assess the overall complexity in 𝑁for obtaining an
approximation of xwithin the tolerance 𝜀, a relation between
𝑁iter and 𝑁is needed. One way to obtain such a relationship
is via the condition number of the matrix, which is defined as
cond A=A2A12
=𝜎max(A)
𝜎min(A),(26)
where kk2is the spectral norm, and 𝜎max/min denotes the
maximal and minimal singular value.
In the case of the conjugate gradient (CG) method, which
requires Ato be Hermitian and positive definite, there is an
upper bound on the error e(𝑖)x(𝑖)xgiven by [51]
e(𝑖)A2cond A1
cond A+1𝑖e(0)A,(27)
where kkAis the energy norm defined by kxkAxAx 1/2
and xdenotes the conjugate transpose of x. If the objective
is to reduce the relative error e(𝑖)/e(0)below 𝜀and by
considering limits for cond A1, one notes [51] that
𝑖1
2cond Aln 2
𝜀(28)
iterations are at most needed (assuming an exact arithmetic).
If the condition number grows linearly in 𝑁, as observed for
the EFIE when the mesh is uniformly refined, this implies that
the complexity is at most O(𝑁1.5log 𝑁).
One could argue that this is an overly simplified picture
of the situation; indeed, the CG method is not applicable to
standard frequency domain integral equations as the resulting
system matrices are neither Hermitian nor positive definite.
One strategy to still obtain a bound on the number of iterations
is to use the CG method on the normal equation
AAx =Ab.(29)
The price for this, however, is that the condition number of the
resulting system matrix is (cond A)2and thus this approach
is, for the standard formulations, of little practical value. In
addition, round-off errors due to finite precision can lead to a
non-converging solver—despite the theory dictating that CG
should converge in at most 𝑁steps [52], [53]. Thus, the
condition number bound is relevant in practice often only in
the case that cond Ais small.
The problem with other popular Krylov methods such as
the generalized minimal residual (GMRES) or the conjugate
gradient squared (CGS) method is that, for general matrices,
no bound on the number of iterations in terms of the condition
number alone is available. In fact, even if two matrices have
the same condition number, their convergence behavior can
significantly differ: the distribution of the eigenvalues in the
complex plane impacts the convergence behavior as well [22].
Typically, a better convergence can be observed if all the
eigenvalues are located on either the real and or imaginary
axis and are either strictly positive or negative (if they are
on the imaginary axis, then positive or negative with respect
to Im (𝜆𝑖)). We will see in the following that, under certain
conditions, for low-frequency electromagnetic problems it is
possible to cluster the eigenvalues on the real axis and that
the condition number becomes a good indicator of the con-
vergence behavior. Moreover, some preconditioning strategies,
such as the refinement-free Calderón preconditioner which
will be discussed in Section V-A2, give rise to a Hermitian,
positive-definite system, and thus the CG and the associated
convergence theory is applicable.
For frequency-independent problems, it is customary to call
a formulation well-conditioned if cond Ais asymptotically
bounded by a constant 𝐶, which is independent from the
average edge length of the mesh. For dynamic problems,
however, we also need to study the condition number as a
function of the frequency 𝑓𝜔/(2π), and one must specify
if a formulation is well-conditioned with respect to , to
𝑓, to both, or only in a particular regime, for example, for
frequencies where the corresponding wavelength is larger then
the diameter of 𝛤.
The classical remedy to overcome ill-conditioning and thus
improve the convergence behavior of iterative solvers is to
use a preconditioning strategy. Such a strategy results, in the
general case, in a linear system
PLAPRy=PLb,(30)
where x=PRyand the matrices should be chosen such that,
if possible,
cond (PLAPR)𝐶 , (31)
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where 𝐶is a constant both independent of and 𝑓(in which
case the preconditioner is optimal). Normally, the matrix-
matrix products in (30) are not formed explicitly and, to be an
efficient preconditioner, the cost of a matrix-vector product
should not jeopardize the lead complexity set by the fast
method. In practice, to obtain an optimal preconditioner, the
nature of the underlying operators must be taken into account.
Thus, in the following sections, we will analyze the spectral
properties of the (discretized) EFIE, MFIE, and CFIE operator,
discuss the causes of their ill-conditioning as well as potential
remedies.
IV. Low-Frequency Scenarios
The low-frequency breakdown of the EFIE, that is, the
growth of the condition number when the frequency 𝒇de-
creases, was one of the first sources of ill-conditioning of the
EFIE to be studied. From a physical point of view, several
problems at low-frequency are rooted in the decoupling of the
electric and the magnetic field in the static limit: magnetostatic
loop currents excite the magnetic field and electrostatic charges
excite the electric field [20]. Both the EFIE and the MFIE
suffer from computational challenges at low-frequencies. As
we will see in this section, the EFIE suffers from conditioning
issues when the frequencies decreases and so does, albeit
for different reasons, the MFIE when applied to non-simply
connected geometries (i.e., geometries containing handles like
the torus illustrated in Figure 4, for example). The condition
number growth is, however, only one of the possible problems:
finite machine precision and inaccuracies due to numerical
integration that result in catastrophic round-off errors are
also plaguing the otherwise low-frequency well-conditioned
integral equations such as the MFIE on simply-connected
geometries. Together, these issues make the two formulations
increasingly inaccurate as the frequency decreases, which is
attested by the low-frequency radar cross sections illustrated
in Figure 3 that show wildly inaccurate results for the standard
formulations.
The low-frequency analysis of electromagnetic integral
equations benefits from the use of Helmholtz and quasi-
Helmholtz decompositions that we will summarize here for
the sake of completeness and understanding. The well-known
Helmholtz decomposition theorem states that any vector field
can be decomposed into a solenoidal, irrotational, and a
harmonic vector field, which in the case of a tangential surface
vector field such as 𝒋𝛤leads to [41, p. 251]
𝒋𝛤=curl𝛤𝛷+grad𝛤𝛹+𝑯(32)
where 𝛷and 𝛹are sufficiently smooth scalar functions,
curl𝛤𝛷grad𝛤𝛷׈
𝒏, and div𝛤𝑯=curl𝛤𝑯=0;
here, curl𝛤is the adjoint operator of curl𝛤, that is, we have
hcurl𝛤𝑓 , 𝒈i𝛤=h𝑓 , curl𝛤𝒈i𝛤(see [41, see (2.5.194)]). The
space of harmonic functions 𝐻𝑯(𝛤)is finite dimensional with
dim 𝐻𝑯(𝛤)=2𝑔on a closed surface, where 𝑔is the genus of
𝛤. The Helmholtz subspaces are all mutually orthogonal with
respect to the 𝑳2(𝛤)-inner product.
When 𝒋𝛤is a linear combination of div- but not curl-
conforming functions (e.g., RWG and BC functions), only
00.511.522.53
1,400
1,200
1,000
Angle [rad]
Radar Cross Section [dBsm]
Mie series EFIE P-EFIE
Loop-star EFIE MFIE
Fig. 3. Radar cross sections calculated, with different formulations, for the
sphere of unit radius discretized with an average edge length of 0.15 m, and
excited by a plane wave of unit polarization along ˆ
𝒙and propagation along
ˆ
𝒛oscillating at 𝑓=1020 Hz. The “EFIE” and “MFIE” labels refer to the
standard formulations (15) and (20), while the “Loop-star EFIE” and “P-
EFIE” refer to the EFIE stabilized with the loop-star (61) and quasi-Helmholtz
projectors (72), respectively.
Fig. 4. Illustration of a torus and the corresponding toroidal (in blue) and
poloidal (in orange) loops.
a quasi-Helmholtz decomposition is possible, where 𝒋𝛤is
decomposed into a solenoidal, a non-solenoidal, and a quasi-
harmonic current density. It is not possible to obtain irrota-
tional or harmonic current densities, since the curl of div-
conforming (but not curl-conforming) functions such as the
RWGs (or their dual counterparts) is, in general, not existing as
a classical derivative; therefore, it is termed quasi-Helmholtz
decomposition. Next we introduce the quasi-Helmholtz de-
compositions for primal (i.e., RWGs) and dual (i.e., BCs)
functions that we will use for our analysis in the next section.
Just as the Helmholtz decomposition (32) decomposes the
continuous solution 𝒋𝛤, a quasi-Helmholtz decomposition
decomposes the discrete solution jas
𝑁
𝑛=1[j]𝑛𝒇𝑛=
𝑁V
𝑛=1[j𝜦]𝑛𝜦𝑛+
𝑁C
𝑛=1[j𝜮]𝑛𝜮𝑛+
2𝑔
𝑛=1[j𝑯]𝑛𝑯𝑛,
(33)
where 𝜦𝑛𝑋𝜦are solenoidal loop functions, 𝜮𝑛𝑋𝜮
are non-solenoidal star functions, and 𝑯𝑛𝑋𝑯are quasi-
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harmonic global loops [54] and where j𝜦,j𝜮, and j𝑯are
the vectors containing the associated expansion coefficients;
moreover, 𝑁Vis the number of vertices and 𝑁Cis the number
of cells of the mesh.
We highlight some of the properties which we are going
to use throughout this article. First, and most importantly, the
functions 𝜦𝑛,𝑯𝑛, and 𝜮𝑛can be represented in terms of
RWG functions [54]. Thus the expansion coefficients are linked
by linear transformation matrices Λ,H, and Σ. For the loop
transformation matrix, we have
[Λ]𝑖 𝑗 =
1for 𝒗𝑗=𝒗
𝑖
1for 𝒗𝑗=𝒗+
𝑖
0otherwise,
(34)
where 𝒗𝑗is the 𝑗th vertex of the mesh (inner vertex if 𝛤is
open), and for the star transformation matrix
[Σ]𝑖 𝑗 =
1for 𝑐𝑗=𝑐+
𝑖
1for 𝑐𝑗=𝑐
𝑖
0otherwise,
(35)
where 𝑐𝑗is the 𝑗th cell of the mesh, following the conventions
depicted in Figure 1. With the definition of these matrices, the
quasi-Helmholtz decomposition in (33) can be equivalently
written as
j=Λj𝜦

=𝑗sol
+Hj𝑯

=jqhar
+Σj𝜮

=𝑗nsol
=jsol +jqhar +jnsol .(36)
The linear combinations of RWGs implied by the coefficient
vectors jsol,jnsol, and jqhar are solenoidal, non-solenoidal, and
quasi-harmonic current densities. These decompositions are
not unique: if we were to use, for example, the loop-tree
quasi-Helmholtz decomposition, we would obtain different
coefficient vectors jsol,jnsol, and jqhar . The decomposition is,
however, unique with respect to the loop-star space, that is,
when the linear dependency of loop and of star functions
(see [55] and references therein) is not resolved by arbitrarily
eliminating a loop and a star function; what sets the loop-star
basis apart from other quasi-Helmholtz decompositions is the
symmetry with respect to dual basis functions. A symmetry
that we are now going to further highlight.
First, we give to Λand Σa meaning that goes beyond
merely interpreting them as basis transformation matrices.
The matrices Λand Σare edge-node and edge-cell incidence
matrices of the graph defined by the mesh and they are
orthogonal, that is, ΣTΛ=0. It follows that jsol and jnsol are
𝒍2-orthogonal, that is, jT
nsoljsol =0. We find this noteworthy
for two reasons: i) the loop 𝜦𝑖and star functions 𝜮𝑗are, in
general, not 𝑳2-orthogonal (after all, 𝜮𝑗is not irrotational);
ii) the 𝒍2-orthogonality is not true for other quasi-Helmholtz
decompositions such as the loop-tree basis. In light of this
consideration, the matrices Λand Σcould be interpreted as the
graph curl (Λ) and graph gradient (Σ) of the standard mesh,
an interpretation that further increases the correspondence with
the continuous decomposition (32).
For global loops 𝑯𝑛, no such simple graph-based definition
exists. Indeed, they are, in general, not uniquely defined and
must be constructed from a search of holes and handles. For
any global loop basis so obtained, we have ΣTH=0; however,
ΛTH=0is, in general, not true. This property can be enforced
by constructing Has the right nullspace of Λ ΣT. Such a
construction is possible, for example, via a full singular value
decomposition (SVD), or, via more computationally efficient
randomized projections [56]. However, the computational cost
is higher, in general, compared with using a global loop-
finding algorithm, in particular, since Hwill be a dense matrix.
A similar decomposition can be obtained for dual functions
𝑁
𝑛=1[m]𝑛𝒇𝑛=
𝑁V
𝑛=1m
𝜦𝑛
𝜦𝑛+
𝑁C
𝑛=1m
𝜮𝑛
𝜮𝑛+
2𝑔
𝑛=1m
𝑯𝑛
𝑯𝑛,
(37)
where, in contrast to the RWG case,
𝜦𝑛are non-solenoidal
dual star and
𝜮𝑛are solenoidal dual loop functions. In matrix
notation, we have
j=Σm
𝜮

msol
+
Hm
𝑯

mqhar
+Λm
𝜦

mnsol
=msol +mqhar +mnsol .(38)
Note that the same matrices Σand Λare present both in
the decomposition of RWG functions and in the one of dual
functions. However, while for RWGs the transformation matrix
Λdescribes solenoidal functions and the transformation matrix
Σdescribes non-solenoidal functions, the opposite is true for
the dual functions: it is Σthat describes solenoidal functions,
while Λdescribes non-solenoidal. Thus on the dual mesh, Λ
acts as graph gradient and Σas a graph curl. This is consistent
with the definition of dual functions: dual basis functions can
be interpreted as a div-conforming “rotation” by 90°of the
primal functions (note that the functions ˆ
𝒏×𝒇𝑖are a rotation
by 90°, which is not div-confirming); given that curl𝛤𝛷
𝛤𝛷׈
𝒏, it is consistent that the roles of Λand Σas graph
counterparts to continuous differential surface operators are
swapped on the dual mesh with respect to the primal mesh.
Regarding the quasi-harmonic functions, it must be empha-
sized that we cannot identify
H=H. This equality is only true
if His the nullspace of Λ ΣT, a condition, which evidently
leads to the aforementioned unique definition of H. Even
though the construction of Has the nullspace of Λ ΣTis
cumbersome—and by introducing quasi-Helmholtz projectors
in the following, we will sidestep it—it suggests that these
global loops are capturing the analytic harmonic Helmholtz
subspace better than arbitrarily chosen global loops.
A. Electric Field Integral Equation
To put into light the low-frequency challenges that plague
the EFIE, its behavior on both the solenoidal and the non-
solenoidal subspaces must be analyzed. The following de-
velopments focus on geometries that do not contain global
loops, however the results can be immediately extended to the
general case by considering that—in the case of the EFIE—
global and local loops have similar properties. While they have
practical limitations, loop-star bases are a convenient tool to
perform this analysis. The loop-star transformed EFIE matrix
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TLS
𝑘BΛ ΣTT𝑘Λ Σcan be represented in block
matrix form as
TLS
𝑘=ΛTT𝑘Λ ΛTT𝑘Σ
ΣTT𝑘Λ ΣTT𝑘Σ,(39)
and the corresponding matrix equation now reads TLS
𝑘𝒋LS =
Λ ΣT𝒆i, where 𝒋=Λ Σ𝒋LS . In these definitions, the
Λand Σmatrices refer to the full-rank transformation matrices
in which linearly dependent columns have been removed:
for each connected component of 𝛤one star basis function
(column of Σ) must always be removed and one loop basis
function must be removed (column of Λ) if the component is
closed [57].
To evidence the different low-frequency behaviors of the
EFIE matrix on the solenoidal and non-solenoidal subspaces,
the properties ΛTTΦ, 𝑘 =0and TΦ,𝑘 Λ=0, which follow di-
rectly from the divergence-free nature of solenoidal functions,
must be enforced. In addition, the behavior of the matrix terms
must be derived by performing a Taylor series expansion of
the Green’s function in both T
A,𝑘 and T
Φ,𝑘 for 𝑘0. For
instance,
hˆ
𝒏×𝜮𝑚,T
A,𝑘 𝜦𝑛i𝛤=
𝑘0𝛤𝛤
𝜮𝑚(𝒓) · 𝜦𝑛(𝒓0)
4π
1
𝑅𝑘2𝑅
2i𝑘3𝑅2
6+ O(𝑘4)d𝑆(𝒓0)d𝑆(𝒓),(40)
where 𝑅=|𝒓𝒓0|and where we have used
𝛤i𝑘𝜦𝑛(𝒓0)d𝑆(𝒓0)=0. We can deduce that, in general,
Re hˆ
𝒏×𝜮𝑚,T
A,𝑘 𝜦𝑛i𝛤=
𝑘0O(1),(41)
Im hˆ
𝒏×𝜮𝑚,T
A,𝑘 𝜦𝑛i𝛤=
𝑘0O(𝑘3).(42)
This process can be repeated for both T
A,𝑘 and T
Φ,𝑘 when
both expansion and testing functions are non-solenoidal and
when at least one of the two is solenoidal. In summary,
Re hˆ
𝒏×𝜮𝑚,T𝑘𝜮𝑛i𝛤=
𝑘0O(𝑘2),(43)
Im hˆ
𝒏×𝜮𝑚,T𝑘𝜮𝑛i𝛤=
𝑘0O(𝑘1),(44)
Re hˆ
𝒏×𝜮𝑚,T𝑘𝜦𝑛i𝛤=
𝑘0O(𝑘4),(45)
Im hˆ
𝒏×𝜮𝑚,T𝑘𝜦𝑛i𝛤=
𝑘0O(𝑘).(46)
By symmetry, both hˆ
𝒏×𝜦𝑚,T𝑘𝜮𝑛i𝛤and hˆ
𝒏×𝜦𝑚,T𝑘𝜦𝑛i𝛤
have the same low-frequency behavior as hˆ
𝒏×𝜮𝑚,T𝑘𝜦𝑛i𝛤.
The scaling of the behavior of the block matrix is now
straightforward to obtain
Re TLS
𝑘=
𝑘0O(𝑘4) O(𝑘4)
O(𝑘4) O(𝑘2),(47)
Im TLS
𝑘=
𝑘0O(𝑘) O(𝑘)
O(𝑘) O(𝑘1),(48)
and the dominant behavior of TLS
𝑘is that of its imaginary part.
These results can be used to demonstrate the issues plaguing
the EFIE at low frequencies, starting with its ill-conditioning.
Consider the block diagonal matrix D𝑘=diag 𝑘1/2𝑘1/2
in which the block dimensions are consistent with that of the
loop star decomposition matrix. Clearly,
D𝑘TLS
𝑘D𝑘=
𝑘0O(1) O(𝑘)
O(𝑘) O(1),(49)
is a well-conditioned matrix, in the sense that
lim𝑘0cond D𝑘TLS
𝑘D𝑘𝛾is finite. It then follows
that
cond TLS
𝑘=cond D1
𝑘D𝑘TLS
𝑘D𝑘D1
𝑘
(cond D𝑘)2cond D𝑘TLS
𝑘D𝑘
(50)
and thus lim𝑘0cond TLS
𝑘=O(𝑘2). A lower bound for the
condition number of interest can be obtained through the
application of the Gershgorin disk theorem after diagonal-
ization of the bottom right block of TLS
𝑘, which proves that
cond TLS
𝑘𝜎min𝑘2where 𝜎min is the smallest singular
value of (ΣTTΦ,0Σ). Considering these results and that the
loop-star transformation matrix is invertible and frequency
independent, we conclude that cond (T𝑘)𝑘2when 𝑘0.
The second source of instability of the EFIE at low frequen-
cies is the loss of significant digits in the right-hand side ei,
solution j, or radiated fields. To see this effect, the behavior of
the right-hand side of the EFIE must be considered. Here we
will restrict our developments to the plane-wave excitation,
but similar results can be obtained for other problems [58].
Following the same procedure as for the matrix elements, we
can determine the behavior of the loop and star right-hand side
elements
Re hˆ
𝒏×𝜦𝑚,ˆ
𝒏×𝒆i
PWi𝛤=
𝑘0O(𝑘2),(51)
Im hˆ
𝒏×𝜦𝑚,ˆ
𝒏×𝒆i
PWi𝛤=
𝑘0O(𝑘),(52)
Re hˆ
𝒏×𝜮𝑚,ˆ
𝒏×𝒆i
PWi𝛤=
𝑘0O(1),(53)
Im hˆ
𝒏×𝜮𝑚,ˆ
𝒏×𝒆i
PWi𝛤=
𝑘0O(𝑘),(54)
where 𝒆i
PW is the electric field of the incident plane-wave.
It is crucial to remember that when the standard EFIE—
with no treatment—is solved numerically in finite precision
floating point arithmetic, the real parts (resp. imaginary parts)
of the loop and star components of the right-hand side are
stored in the same floating point number. In particular, the
real part of the solenoidal component that behaves as O(𝑘2)
is summed with an asymptotically much larger non-solenoidal
component behaving as O(1). In the context of finite precision
arithmetic, the dynamic range of the floating point number will
be imposed by the larger of the two components, meaning that
the floating point number will become increasingly incapable
of storing accurately the smaller one. This loss of significant
digits will worsen until the solenoidal component has com-
pletely vanished from the numerical value. This phenomenon
is not necessarily damageable per se, but can lead to drastic
losses in solution accuracy. In the particular case of the plane-
wave excitation, we will study the effect of this loss of accuracy
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on the dominant parts of the solution. Using the well-known
relations on block matrix inverses [59], one can show that
Re TLS
𝑘1
=
𝑘0O(𝑘2) O(𝑘4)
O(𝑘4) O(𝑘4),(55)
Im TLS
𝑘1
=
𝑘0O(𝑘1) O(𝑘)
O(𝑘) O(𝑘),(56)
which, in combination with the right-hand side results yields
the behavior of the solution coefficients
Re (j𝜦)=
𝑘0O(1),(57)
Im (j𝜦)=
𝑘0O(𝑘),(58)
Re (j𝜮)=
𝑘0O(𝑘2),(59)
Im (j𝜮)=
𝑘0O(𝑘),(60)
which are indeed the behavior predicted by physics [60]. Note
that the inaccurate right-hand side component will only have a
significant contribution to the imaginary part of the solenoidal
component of the solution, which is non-dominant. As such,
although the error of the current could be low, the error of the
charge or field could be quite high.
Finally, the reader should note that to numerically observe
these results—and successfully implement the remedies that
we will see later on—the vanishing of all relevant integrals
must be explicitly enforced in some way, because floating
point arithmetic and numerical integration are not capable of
obtaining an exact zero in their computation and will saturate
at machine precision, in the best case scenarios. Indeed, had
they not been enforced, the solenoidal and non-solenoidal parts
of the solution would have had the same behavior and, as such,
would not yield a solution behaving as predicted by physics.
1) Loop-Star/Tree Approaches
Historically, the loop-star and loop-tree decompositions
have been used to cure the low-frequency breakdown of the
EFIE [20], [54] and as such are well-known and studied [55].
The fundamental curing mechanism of these approaches is
to decompose the EFIE system using a RWG-to-loop-star or
RWG-to-loop-tree mapping and isolate the solenoidal and non-
solenoidal parts of the system. This separation allows for a
diagonal preconditioning of the decomposed matrix to cure
its ill-conditioning (as was done in Section IV-A). In addition,
this separation makes it possible to enforce that the required
integrals and matrix products vanish and cures the loss of
significant digits that plagues the EFIE, since the loop and star
contributions of each entity are stored in separate floating point
numbers. In the case of the loop-star approach, the stabilized
matrix system is
D𝑘TLS
𝑘D𝑘jDLS =D𝑘Λ ΣTei,(61)
where j=Λ ΣD𝑘jDLS, following the notations of Sec-
tion IV-A. Once the intermediate solution jDLS has been ob-
tained, it must be handled with particular care. If, for instance,
the quantity of interest is the field radiated by the solution,
the radiation operators must be applied separately on the
solenoidal and non-solenoidal parts of the solution that can be
retrieved as jsol =Λ0D𝑘jDLS and jsol =0ΣD𝑘jDLS,
because additional vanishing integrals must be enforced in the
scattering operators when applied to solenoidal functions. In
addition, any explicit computation of jwould be subject to a
numerical loss of significance and would further compromise
the accuracy of the fields.
The key difference between loop-tree and loop-star tech-
niques is that, in the former, the quasi-Helmholtz decompo-
sition leverages a tree basis in place of the star basis, as
indicated by their names. To define this tree basis consider
the connectivity graph joining the centroids of all adjacent
triangle cells of the mesh. To each edge of this graph cor-
responds a unique RWG function. Then, given a spanning
tree of this graph, a tree basis can be defined as the subset
𝜽𝑗of the RWG functions whose corresponding edge in the
connectivity graph is included the spanning tree [54], [61].
The rationale behind the technique is that, by construction,
such a basis will not be capable of representing any loop
function. Clearly, the construction of this basis in not unique,
since it depends on the choice of spanning tree. In practice, the
loop-tree approach results in a matrix system similar to (61),
in which the RWG-to-loop-star mapping Λ Σis replaced
by an RWG-to-loop-tree mapping Λ Θand TLS
𝑘becomes
TLT
𝑘BΛ ΘTT𝑘Λ Θwhere
[Θ]𝑖 𝑗 =1if 𝒇𝑖=𝜽𝑗
0otherwise, (62)
is the general term of the RWG-to-tree transformation matrix.
The resulting preconditioned equation is
D𝑘TLT
𝑘D𝑘jDLT =D𝑘Λ ΘTei,(63)
where j=Λ ΘD𝑘jDLT.
At first glance, the computational overhead of the two meth-
ods seems low, since Λ,Σ,Θ, and D𝑘are sparse matrices.
However, while both methods adequately address the low-
frequency breakdown of the EFIE, in the sense that they yield
the correct solution (Figure 3) and prevent the conditioning
of the system to grow unbounded as the frequency decreases
(Figure 5), they cause the conditioning of the system matrix
to artificially worsen because the loop-star and loop-tree bases
are ill-conditioned [62]. This has led to the development of a
permutated loop-star and loop-tree bases to reduce the number
of iterations required to solve the preconditioned system using
iterative solvers [61]. In general, the loop-tree preconditioned
EFIE was observed to converge faster than the loop-star
preconditioned [63], which can be explained by the fact that
Λand Σcan be interpreted as the discretizations of the graph
curl and graph gradient [55], [62], that are ill-conditioned
derivative operators. While a rigorous proof of the effect of
this ill-conditioning on the preconditioned EFIE matrix is
out of the scope of this review, pseudo-differential operator
theory can be used to show that the differential strength of
the loop-star transformation operators is sufficiently high not
to be compensated by that of the vector potential. To illustrate
this adverse effect, the conditioning of the system matrices
has been obtained numerically and is presented in Figure 6.
Clearly, the standard EFIE matrix shows a condition number
growing as the frequency decreases. However, at moderate
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1044 1033 1022 1011 1001011
100
103
106
Frequency 𝑓[Hz]
Condition number
EFIE MFIE P-EFIE Loop-star EFIE
Fig. 5. Comparison of the conditioning of the system matrices for several
formulations on a sphere of radius 1 m discretized with an average edge length
of 0.15 m, for varying frequency.
105106107
100
102
104
106
108
Frequency 𝑓[Hz]
Condition number
EFIE Loop-star EFIE Loop-tree EFIE
P-EFIE P-CMP-EFIE
Fig. 6. Comparison of the conditioning of the loop-star, loop-tree, and
projector-based preconditioned EFIE matrices on a spheres of radius 1 m
discretized with an average edge length of 0.3 m (solid lines) and 0.2 m (dotted
lines) as a function of the frequency. The labels “Loop-tree EFIE” and “P-
CMP-EFIE” refer to the EFIE stabilized with the loop-tree approach (63) and
the Calderón EFIE stabilized with quasi-Helmholtz projectors (128).
frequencies, the conditioning of the of the loop-star and loop-
tree preconditioned matrices is significantly higher than that
of the original matrix.
2) Quasi-Helmholtz Projectors
From the previous sections it is clear that although the loop-
star/tree decompositions are helpful in analyzing the reasons
behind of the low-frequency breakdown and that historically
provided a cure for it, they still give rise to high condition
numbers since they introduce an ill-conditioning related to
the mesh discretization. Moreover, for non-simply connected
geometries, loop-star decompositions require a search for the
mesh global cycles, an operation that can be computationally
cumbersome.
A family of strategies to overcome the drawbacks of loop-
star/tree decompositions while still curing the low-frequency
breakdown is the one based on quasi-Helmholtz projec-
tors [62], [64]. Quasi-Helmholtz projectors can decompose the
current and the operators into solenoidal and non-solenoidal
components (just like a loop-star/tree decomposition does) but,
being projectors, have a flat spectrum that, differently from
loop-star/tree decompositions, do not alter the spectral slopes
of the original operators and thus do not introduce further
ill-conditioning.
Starting from the quasi-Helmholtz decomposition (36)
j=Σj𝜮+Λj𝜦+Hj𝑯,(64)
the quasi-Helmholtz projector for the non-solenoidal part is
the operator that maps jinto Σj𝜮. Since
ΣTj=ΣTΣj𝜮,(65)
the looked for projector is
PΣBΣ(ΣTΣ)+ΣT,(66)
where +denotes the Moore–Penrose pseudoinverse. The pro-
jector for the solenoidal plus harmonic components can be
obtained out of complementarity as
PΛHIPΣ.(67)
The same reasoning for dual functions leads to the dual
definitions of the projector
PΛBΛ(ΛTΛ)+ΛT(68)
which is the non-solenoidal projector for dual functions. The
solenoidal plus harmonic projector for dual functions is, again,
obtained by complementarity as
PΣHIPΛ.(69)
It is important to note that, even though the projectors pre-
sented so far include a pseudo-inverse in their definition, they
can be applied to arbitrary vectors in quasi-linear complexity
by leveraging algebraic multigrid preconditioning [62], [65],
[66] and, as such, are fully compatible with standard fast
solvers.
Quasi-Helmholtz projectors can be used to cure the dif-
ferent deleterious effects of the low-frequency breakdown by
isolating the solenoidal and non-solenoidal parts of the system
matrix, unknowns, and right-hand side and rescaling them
appropriately. Thus they are an alternative to loop-star/tree de-
compositions that presents several advantages when compared
to these schemes. Quasi-Helmholtz projectors have been used
to cure the low-frequency breakdowns of several formulations,
however for the sake of readability and conciseness, we will
only detail their application to the standard EFIE where it
is more straightforward, but will point to relevant papers
describing their applications to other well-known formulations.
Preconditioning the original system (15) with matrices of the
form
PB𝛼PΛH+𝛽PΣ,(70)
where, following a frequency analysis similar to the one used
for loop-star/tree decompositions, an optimal coefficient choice
can be found to be 𝛼𝑘1
2and 𝛽𝑘1
2, that is,
P𝑘B𝐶/𝑘PΛH+i/𝐶𝑘 PΣ,(71)
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Fig. 7. Comparison of the solenoidal part of the surface current density
induced on a sphere of radius 1 m discretized with an average edge length of
0.3 m at 𝑓=1020 Hz, computedwith different formulations.
resulting in a new system of equations
P𝑘T𝑘P𝑘y=P𝑘ei,(72)
where P𝑘y=j. The constant 𝐶can be obtained by maximizing
the components of the solution current that are recovered [60]
and by enforcing an equal contribution of the vector and scalar
potential components; this results in
𝐶kTΦ,𝑘 k
kPΛHTA,𝑘 PΛHk.(73)
The analysis of the conditioning effect of the projector can
mimic the strategy used for the loop-star decomposition. In
particular, the EFIE preconditioned with the projectors has a
frequency-independent limit
lim
𝑘0P𝑘T𝑘P𝑘i𝐶PΛHTA,0PΛH+i/𝐶TΦ,0,(74)
where we used that PΣTΦ,𝑘 PΣ=TΦ, 𝑘 and PΛTΦ,𝑘 =TΦ, 𝑘 PΛ=
0and thus
lim
𝑘0cond (P𝑘T𝑘P𝑘)=𝛾 , (75)
where 𝛾is a frequency independent constant. This approach
can be proved to simultaneously solve the problem of catas-
trophic round-off errors in both the current and the right-
hand side of the EFIE [60]. Finally, the use of the projectors
has clear advantages in terms of conditioning with respect
to the use of loop-star or related decompositions that can be
seen in Figure 6. The impact on current and right-hand side
cancellation effects can be observed in Figures 7 and 8.
3) Other Strategies for the EFIE Low-Frequency Regular-
ization
From previous sections it is clear that the main drawbacks
of loop-star/tree decompositions reside in their constant-in-
frequency, but still high, condition number and also in the
need to be enriched with global loop functions [67], [68].
Both of these drawbacks can be overcome by the use of
quasi-Helmholtz projectors, as explained above, but other
schemes can alternatively be used as effective cures for one
Fig. 8. Comparison of the non-solenoidal part of the surface current density
induced on a sphere of radius 1 m discretized with an average edge length of
0.3 m at 𝑓=1020 Hz, computed with different formulations.
or both of the drawbacks above. By using a rearranged non-
solenoidal basis, for example, the conditioning of a loop-
star or a loop-tree preconditioned EFIE could be further
improved [61]. Moreover, to avoid the construction of global
loops on multiply-connected geometries, formulations have
been presented that consider the saddlepoint formulation of the
EFIE [69], [70], where the charge is introduced as unknown,
in addition to the current in the RWG basis. The most notable
are the current-charge formulation [71] and the augmented
EFIE [72]. However, these formulations are, in general, not
free from round-off errors in the current or the right-hand side
so that, for example, perturbation methods need to be used [58]
for further stabilization. An alternative to the perturbation
method is the augmented EFIE with normally constrained
magnetic field and static charge extraction, which includes
a boundary integral equation for the normal component of
magnetic field [73]. A disadvantage of current-charge formula-
tions is the introduction of an additional unknown, the charge;
hence, methods have been presented to save memory by
leveraging nodal functions [74]. An entirely different approach
is used in [75], where a closed-form expression of the inverse
of the EFIE system matrix is derived based on eigenvectors
and eigenvalues of the generalized eigenvalue problem.
Another class of strategies forfeits the EFIE approaches;
instead, they are based on potential formulations [76], [77].
These formulations are low-frequency stable on simply- and
multiply-connected without the need for searching global
loops. The potential-based approaches [76], [77] are also
dense-discretization stable. This property is shared with hi-
erarchical basis and Calderón-type preconditioners. With a
few exceptions [78], hierarchical basis preconditioners are
based on explicit quasi-Helmholtz decomposition [79]–[84],
since it then suffices to find a hierarchical basis for scalar-
valued functions. While they yield an overall improved con-
dition number with respect to classical loop-star and loop-
tree approaches, they require the search for global loops on
multiply-connected geometries; a suitable combination with
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quasi-Helmholtz projectors has been shown to alleviate the
need for this search [85]. Calderón-type preconditioners will
be discussed in the next section in greater detail. At this point,
we are content to say that standard Calderón preconditioned
EFIEs have a spectral behavior similar to that of the MFIE
operator: thus certain low-frequency issues that plague the
MFIE (and which we discuss in the next subsection) persist
in the Calderón preconditioned EFIE. Initially remedies relied
on combining Calderón preconditioners with loop-star precon-
ditioners. However, the Gram matrix becomes ill-conditioned
and global loops must be explicitly recovered [86], [87]. As
we will show in Section V, this can be avoided by using quasi-
Helmholtz projectors [64].
B. Handling of the right-hand side and field computation
As we have already mentioned, a well-conditioned dis-
cretization alone is not sufficient to accurately compute j: the
right-hand side suffers typically from numerical inaccuracies
due to finite integration precision and from round-off errors.
The main reason for this is that the quasi-Helmholtz compo-
nents scale differently in frequency. As an example, for the
case of the plane-wave excitation, the asymptotic behavior is
noted in (51)–(54).
Strategies have been presented in the past to yield stable
discretizations of the right-hand side [20], [61], which work
with arbitrary right-hand side excitations. For the plane-wave
excitation, a simple solution is to not only compute eas in
(18), but also an eextracted, where the static contribution is
extracted. We obtain this by replacing ei𝑘ˆ
𝒌·𝒓with ei𝑘ˆ
𝒌·𝒓1,
where ˆ
𝒌denotes the direction of propagation. Then
𝐶/𝑘PΛHeextracted +i𝐶𝑘 PΣe(76)
is a stable discretization of the preconditioned right-hand side.
To obtain a stable discretization for small arguments of the
exponential function, the subtraction in ei𝑘ˆ
𝒌·𝒓1should be
replaced by a Taylor series, where the static part is omitted.
Similarly, the far-field cannot be computed by simply eval-
uating 𝛤
𝑁
𝑛=1[j]𝑛𝒇𝑛(𝒓0)ei𝑘ˆ
𝒓·𝑟0d𝑆(𝒓0),(77)
where j=P𝑘yfrom (72), ˆ
𝒓=𝒓/|𝒓|. On the one hand,
by computing the unknown vector of the unpreconditioned
formulation j=P𝑘y, the different asymptotic behavior
in 𝑘of the quasi-Helmholtz components of jas denoted
in (57)–(60) would lead to a loss of the solenoidal/quasi-
harmonic components in the static limit due to finite machine
precision. Thus for the field computation, one should keep
the unpreconditioned components of jseparately, that is,
jsol-qhar =𝐶/𝑘PΛHyand jnsol =i𝐶𝑘 PΣy. On the other
hand, it has been pointed out that also the far-field computation
suffers from round-off errors [88]. To avoid these, we compute
the far-field in two steps: we compute the contribution of jnsol
to the far-field by evaluating
𝑬far
nsol (𝒓)=
𝑁
𝑛=1[jnsol]𝑛𝛤
𝒇𝑛(𝒓0)ei𝑘ˆ
𝒓·𝒓0d𝑆(𝒓0)(78)
and the contribution of jsol-qhar by
𝑬far
sol-qhar (𝒓)=
𝑁
𝑛=1jsol-qhar𝑛𝛤
𝒇𝑛(𝒓0)ei𝑘ˆ
𝒓·𝒓01d𝑆(𝒓0),
(79)
where a Taylor-series expansion should be used for small
arguments of the exponential; then
𝑬far (𝒓)=𝑬far
nsol (𝒓) + 𝑬far
sol-qhar (𝒓)(80)
Also for the near-field computation, the separation in jsol-qhar
and in jnsol must be maintained, the static contribution removed
from the Green’s function, and, in addition, the divergence of
the scalar potential explicitly enforced by omitting it.
C. Magnetic Field Integral Equation
The MFIE has, other than in the Green’s function kernel,
no explicit dependency on 𝑘and should thus be expected to
remain well-conditioned in frequency for 𝑘0. Indeed, for
simply-connected geometries 𝛤, we have cond(M𝑘)=O(1)
when 𝑘0. In the case of multiply-connected geometries,
the MFIE operator exhibits a nullspace associated with the
toroidal (for the exterior MFIE) or poloidal loops (for the
interior MFIE) in the static limit [89]–[91]. This leads to an ill-
conditioned system matrix [91]; in the following, we are going
to show that cond (M𝑘)𝐶/𝑘2for some constant 𝐶R+.
To prove this result on the condition number, we will
consider the low-frequency behavior of block matrices that
result from a discretization of the MFIE with a loop-star
basis. For the analysis, we must however distinguish two types
of harmonic functions, the poloidal and the toroidal loops.
If 𝛺has genus 𝑔, then the space 𝐻𝑯(𝛺)defined by the
harmonic functions in 𝛺and the space 𝐻𝑯(𝛺+)defined by
the harmonic functions in 𝛺+have both dimension 𝑔. The
space defined by 𝐻ˆ
𝑯P(𝛤)ˆ
𝒏×𝐻𝑯(𝛺) |𝛤are the poloidal
loops and the space defined by 𝐻ˆ
𝑯T(𝛤)ˆ
𝒏×𝐻𝑯(𝛺+) |𝛤
are the toroidal loops [92], [93]. 𝐻ˆ
𝑯P(𝛤)has been show [89]
to be the nullspace of M
0and 𝐻ˆ
𝑯T(𝛤)the nullspace of M+
0
and that ˆ
𝒏׈
𝑯T𝑛𝑯ˆ
𝑯P(𝛤)and ˆ
𝒏׈
𝑯P𝑛𝑯ˆ
𝑯T(𝛤).
We need to address how quasi-harmonic functions formed
from primal (RWG) or dual (CW/BC) functions are related to
harmonic functions. On the one hand, neither with RWG nor
with BC functions we can find linear combinations that are in
𝐻ˆ
𝑯P(𝛤)or in 𝐻ˆ
𝑯T(𝛤), a consequence of the fact that these
functions are not curl-conforming, as mentioned in Section IV.
On the other hand, quasi-harmonic functions 𝑯𝑛(and dual
quasi-harmonic functions
𝑯𝑛, respectively) are associated with
the holes and handles of the geometry. They are not, unlike the
locally defined loop functions 𝜦𝑛, derived from a continuous
scalar potential on 𝛤. Together with the fact that 𝑯𝑛and
𝑯𝑛are solenoidal but not irrotational (since the RWG/BC
functions are not curl-conforming), (32) implies that quasi-
harmonic loops are linear combinations of solenoidal and har-
monic functions (i.e., quasi-harmonic functions are harmonic
functions with solenoidal perturbation). Clearly, any quasi-
harmonic basis {𝑯𝑛}2𝑔
𝑛=1can be rearranged into two bases,
where one basis {𝑯T𝑛}𝑔
𝑛=1is orthogonal to poloidal loops and
the other basis {𝑯P𝑛}𝑔
𝑛=1is orthogonal to toroidal loops. In the
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following, ˆ
𝑯T𝑛𝐻ˆ
𝑯T(𝛤)denote the harmonic toroidal and
ˆ
𝑯P𝑛𝐻ˆ
𝑯P(𝛤)denote the harmonic poloidal basis functions
on 𝛤, while 𝑯T𝑛,𝑯P𝑛𝑋𝒇and
𝑯T𝑛,
𝑯P𝑛𝑋
𝒇are their
quasi-harmonic counterparts.
In [91], scalings were reported of the blocks of the system
matrix of M+
𝑘in terms of a quasi-Helmholtz decomposition
𝜦 𝜮 𝑯T𝑯P
ˆ
𝒏×
𝜮O(𝑘2) O(1) O(𝑘2) O(𝑘2)
ˆ
𝒏×
𝜦O(1) O(1) O(1) O(1)
ˆ
𝒏×
𝑯PO(𝑘2) O(1) O(𝑘2) O(𝑘2)
ˆ
𝒏×
𝑯TO(𝑘2) O(1) O(𝑘2) O(1)
.(81)
An analogous result can be obtained for M
𝑘with exchanged
roles for poloidal and toroidal loops. To observe such a
frequency behavior, it is necessary that the testing functions
are curl-conforming. Indeed, for the historical MFIE tested
with RWG functions, the scalings are not observed [94].
To derive the asymptotic behavior for 𝑘0of the block
matrices, we start by considering the Taylor series of the
Green’s function kernel of the MFIE
𝐺(𝒓,𝒓0)=𝑹
4π𝑅3(i𝑘 𝑅 1)ei𝑘 𝑅 (82)
=𝑹
4π𝑅31+1
2(i𝑘)2𝑅2+1
3(i𝑘)3𝑅3+. . .,(83)
where 𝑅=|𝒓𝒓0|and 𝑹=𝒓𝒓0. An O(𝑘2)-scaling is
observed for a block matrix if the contribution due to the
static term in the Taylor series vanishes. For 𝜮𝑛as expansion
function or ˆ
𝒏×
𝜦as testing function, the static contribution
does not vanish and thus we conclude that the scalings of the
blocks in the second column and the second row of (81) are
constant in 𝑘.
We now consider the static MFIE, which models the mag-
netostatic problem, where 𝒋𝛤is either solenoidal or harmonic.
In this case, we have
hˆ
𝒏×
𝜮𝑚,M±
0𝜦𝑛i𝛤=0,(84)
hˆ
𝒏×
𝜮𝑚,M±
0𝑯T𝑛i𝛤=0,(85)
hˆ
𝒏×
𝜮𝑚,M±
0𝑯P𝑛i𝛤=0.(86)
This can be seen by considering that a solenoidal function 𝜦
has a corresponding scalar potential
𝜦=curl𝛤𝛷 . (87)
Furthermore, we have the equality
hˆ
𝒏×𝜦,M±
0𝒋𝛤i𝛤=−h𝜦,ˆ
𝒏×M±
0𝒋𝛤i𝛤.(88)
Inserting (87) for the testing function in the right-hand side
of (88) and using the fact that curl𝛤is the adjoint operator of
curl𝛤, we obtain
hˆ
𝒏×𝜦,M±
0𝒋𝛤i𝛤=h𝛷, curl𝛤ˆ
𝒏× (M±
0𝒋𝛤)i𝛤
=−h𝛷, curl𝛤𝒉
Ti𝛤,(89)
where 𝒉
Tˆ
𝒏×(M±
0𝒋𝛤)is the (rotated) tangential component
of the magnetic field
𝒉(𝒓)=curl 𝛤
𝐺𝑘(𝒓,𝒓0)𝒋𝛤(𝒓0)d𝑆(𝒓0)for 𝒓𝛺,(90)
that is, 𝒉
Tis the (rotated) tangential component of 𝒉when 𝛤is
approached from within 𝛺, and 𝒉+
Twhen 𝛤is approach from
within 𝛺+. We recall that (90) is obtained by finding a vector
potential 𝒂such that 𝒉=curl 𝒂and noting that curl curl 𝒂=𝒋𝛤
under the assumption that div 𝒋𝛤=0[40, see Chapter 6.1].
From
curl𝛤𝒗±
T(𝒓)=lim
𝛺±3𝒓0𝒓curl 𝒗(𝒓0)·ˆ
𝒏(𝒓0).(91)
and from [40, (6.17)]
curl 𝒗={curl 𝒗}+ˆ
𝒏×𝒗+𝒗δ𝛤,(92)
where curly braces {} mean that this part is evaluated only in
𝛺±and δ𝛤is the surface Dirac delta function, together with
(90), we have
curl𝛤𝒉
T=curl 𝒉·ˆ
𝒏=0.(93)
Next we will establish that hˆ
𝒏×𝑯P𝑚,M±
0𝜦𝑛i𝛤=0and
hˆ
𝒏×𝑯T𝑚,M±
0𝜦𝑛i𝛤=0. First, note that the exterior MFIE
operator has the mapping properties that [89]
M+
0ˆ
𝑯T𝑛=0(94)
and
M+
0ˆ
𝑯P𝑛=ˆ
𝑯P𝑛,(95)
while for the interior MFIE operator, we have the mappings
M
0ˆ
𝑯P𝑛=0(96)
and
M
0ˆ
𝑯T𝑛=ˆ
𝑯T𝑛.(97)
Furthermore, for any two surface functions 𝒇,𝒈, we have [89,
Section 5]
h𝒈,K𝒇i𝛤=h−ˆ
𝒏×K𝒈׈
𝒏,𝒇i𝛤
=h−K𝒈׈
𝒏,𝒇׈
𝒏i𝛤(98)
Then we find
hˆ
𝒏׈
𝑯T𝑛,M+
0𝜦i𝛤=hM
0ˆ
𝑯T𝑛,𝜦׈
𝒏i𝛤
(97)
=0,(99)
hˆ
𝒏׈
𝑯P𝑛,M+
0𝜦i𝛤=hM
0ˆ
𝑯P𝑛,𝜦׈
𝒏i𝛤
(96)
=0,(100)
hˆ
𝒏׈
𝑯T𝑛,M
0𝜦i𝛤=hM+
0ˆ
𝑯T𝑛,𝜦׈
𝒏i𝛤
(94)
=0,(101)
hˆ
𝒏׈
𝑯P𝑛,M
0𝜦i𝛤=hM+
0ˆ
𝑯P𝑛,𝜦׈
𝒏i𝛤
(95)
=0,(102)
where we used (94)-(97) and the orthogonality of harmonic
and irrotational functions ˆ
𝒏×𝜦. Now consider that
𝑯P𝑛=
ˆ
𝑯P𝑛+
𝜮P𝑛and
𝑯T𝑛=ˆ
𝑯T𝑛+
𝜮T𝑛, where
𝜮P/T𝑛is the respective
perturbation. Thus by taking into account (84), we have
hˆ
𝒏×
𝑯P𝑚,M±
0𝜦𝑛i𝛤=0,(103)
hˆ
𝒏×
𝑯T𝑚,M±
0𝜦𝑛i𝛤=0.(104)
From now on, we will only consider the harmonic functions
ˆ
𝑯T𝑛and ˆ
𝑯P𝑛instead of their quasi-harmonic counterpart
since, as we have seen, the solenoidal pertubation will always
vanish. Then for M+
0, we have
hˆ
𝒏׈
𝑯T𝑚,M+
0ˆ
𝑯T𝑛i𝛤=hˆ
𝒏׈
𝑯P𝑚,M+
0ˆ
𝑯T𝑛i𝛤
(94)
=0,(105)
hˆ
𝒏׈
𝑯T𝑚,M
0ˆ
𝑯P𝑛i𝛤=hˆ
𝒏׈
𝑯P𝑚,M
0ˆ
𝑯P𝑛i𝛤
(96)
=0(106)
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/OJAP.2021.3121097, IEEE Open
Journal of Antennas and Propagation
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due to the nullspace. Finally, we have
hˆ
𝒏׈
𝑯P𝑚,M+
0ˆ
𝑯P𝑛i𝛤
(95)
=hˆ
𝒏׈
𝑯P𝑚,ˆ
𝑯P𝑛i𝛤=0,(107)
as ˆ
𝒏׈
𝑯P𝑚𝐻ˆ
𝑯T(𝛤)and this space is orthogonal to 𝐻ˆ
𝑯P(𝛤).
Likewise, we can conclude that there is at least one 𝑚such
that
hˆ
𝒏׈
𝑯T𝑚,M+
0ˆ
𝑯P𝑛i𝛤0.(108)
Analogously, we obtain
hˆ
𝒏׈
𝑯P𝑚,M
0ˆ
𝑯T𝑛