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Electromagnetic Integral Equations:
Insights in Conditioning and Preconditioning
Simon B. Adrian, Member, IEEE, Alexandre Dรฉly, 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 e๏ฌectiveness in several real case scenarios. Luckily,
however, it is possible to leverage e๏ฌective preconditioning and regularization strategies that can cure a large majority of these
problems. Not surprisingly, integral equation preconditioning is currently a quite active ๏ฌeld 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 ๏ฌeld 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 ๏ฌeld
(EFIE), magnetic ๏ฌeld (MFIE), and combined ๏ฌeld integral equation (CFIE) and of the associated remedies.
Index TermsโIntegral 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]โ[4]. What makes these schemes so suitable for electro-
magnetic analyses is that, di๏ฌerently from approaches based
on di๏ฌerential equations such as the ๏ฌnite element method
(FEM) or the ๏ฌnite-di๏ฌerence time-domain method (FDTD),
they naturally incorporate radiation conditions without the
need for arti๏ฌcial 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 di๏ฌerential 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]โ[12] are used at high frequency or
the adaptive cross approximation (ACA)/H-matrix methods
and related schemes [13]โ[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 Unionโs 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 Rรฉgion Bretagne and the Conseil Dรฉpartemental du
Finistรจre under the project โTONNERREโ, by the ANR Labex CominLabs
under the project โCYCLEโ, and by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) โ SFB 1270/1โ299150580.
S. B. Adrian is with Universitรคt Rostock, Rostock, Germany (e-mail:
simon.adrian@uni-ro).
A. Dรฉly 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(๐log ๐)or even ๐(๐)complexity, where ๐
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(๐iter๐log ๐)(or ๐(๐iter ๐)in the low-frequency
regime), where ๐iter is the number of iterations.
The number of iterations ๐iter 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 ๐and,
when the BEM formulation is set in the frequency domain,
of the wavenumber ๐. This can potentially result in a solution
complexity greater, and sometimes much greater, than ๐(๐2),
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-re๏ฌnement (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 ๏ฌrst 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)โiv) 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, Ylรค-Oฤณala et al [23]
discussed issues in ๏ฌnding 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 ๏ฌeld integral equation (EFIE) and magnetic
๏ฌeld 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 e๏ฌective preconditioned methods have been presented
for modelling penetrable bodies both homogeneous and in-
homogeneous [26]โ[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 insu๏ฌcient
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]โ[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 speci๏ฌc 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 ๏ฌeld integral equation (CFIE) and
of the associated potential remedies. In contrast to [22], our
focus will include low-frequency e๏ฌects and wideband stable
formulations as well Calderรณn 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 e๏ฌective 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
re๏ฌned 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 ๏ฌnal considerations.
II. Notation and Background
We are interested in solving the electromagnetic scattering
problem where a time-harmonic, electromagnetic wave (๐i,๐i)
in a space with permittivity ๐and permeability ๐impinges on
a connected domain ๐บโโR3with PEC boundary ๐คโ๐ ๐บโ
resulting in the scattered wave (๐s,๐s). The total electric ๐B
๐i+๐sand magnetic ๐B๐i+๐s๏ฌelds satisfy Maxwellโs
equations
โ ร ๐(๐)=+i๐๐(๐),for all ๐โ๐บ+,(1)
โ ร ๐(๐)=โi๐๐(๐),for all ๐โ๐บ+,(2)
where ๐บ+โ๐บโ
c,๐โ๐โ๐๐ is the wave number, ๐
the angular frequency, and ๐,๐must satisfy the boundary
conditions for PEC boundaries
ห
๐ร๐=0,for all ๐โ๐ค , (3)
ห
๐ร๐=๐๐ค,for all ๐โ๐ค , (4)
where ๐๐คis the induced electric surface current density. In
addition, ๐sand ๐smust satisfy the Silver-Mรผller radiation
condition [38], [39]
lim
๐โโ๎๎๐sร๐โ๐๐s๎๎=0.(5)
We assumed (and suppressed) a time dependency of eโi๐๐ก and
normalized ๐with the wave impedance ๐โ๎ฐ๐/๐.
To ๏ฌnd (๐s,๐s), we can solve the EFIE
T๐๐๐ค=โห
๐ร๐i(6)
for ๐๐ค, where ห
๐is the surface normal vector directed into ๐บ+
and
T๐โi๐T
A,๐ +1/(i๐)T
ฮฆ,๐ (7)
is the EFIE operator composed of the vector potential operator
(T
A,๐ ๐๐ค)(๐)=ห
๐ร๎น๐ค
๐บ๐(๐,๐0)๐๐ค(๐0)d๐(๐0)(8)
and the scalar potential operator
(T
ฮฆ,๐ ๐๐ค)(๐)=โห
๐รgrad๐ค๎น๐ค
๐บ๐(๐,๐0)div๐ค๐๐ค(๐0)d๐(๐0),
(9)
where
๐บ๐(๐,๐0)=ei๐|๐โ๐0|
4ฯ|๐โ๐0|(10)
is the free-space Greenโs function. A de๏ฌnition of the surface
di๏ฌerential operators grad๐คand div๐คcan be found in [40,
Appendix 3] or [41, Chapter 2]. Once ๐๐คis obtained, ๎๐s,๐s๎
can be computed using the free-space radiation operators.
Alternatively, one can solve the MFIE for the exterior
scattering problem
ห
๐ร๐i=M+
๐๐๐คโ+๎I/2+K๐๎๐๐ค,(11)
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๐+
๐
๐โ
๐
๐+
๐
๐โ
๐
๐+
๐
๐โ
๐
๐๐
Fig. 1. The vector ๏ฌeld of an RWG function. The vector ๐๐denotes the
directed edge, ๐+
๐and ๐โ
๐denote the domains of the cells, ๐ฃ+
๐and ๐ฃโ
๐denote
vertices on the edge ๐๐, and ๐+
๐and ๐โ
๐are the vertices opposite to the edge
๐๐.
where Iis identity operator, M+
๐is the MFIE operator for the
exterior scattering problem, and
๎K๐๐๐ค๎(๐)Bโห
๐ร๎น๐คโ๐บ๐(๐,๐0) ร ๐๐คd๐(๐0).(12)
The MFIE operator for the interior scattering problem is
Mโ
๐โโI/2+K๐and 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]
โ๐ผT๐๐๐ค+ (1โ๐ผ)ห
๐รM+
๐๐๐ค
=๐ผห
๐ร๐i+ (1โ๐ผ)ห
๐รห
๐ร๐i(13)
which is uniquely solvable for all frequencies.
For the discretization of the EFIE, we employ Rao-Wilton-
Glisson (RWG) basis functions ๐๐โ๐๐which are hereโin
contrast to their original de๏ฌnition in [42]โnot normalized
with the edge length, that is,
๐๐=๏ฃฑ
๏ฃด
๏ฃด
๏ฃด
๏ฃด
๏ฃด๏ฃฒ
๏ฃด
๏ฃด
๏ฃด
๏ฃด
๏ฃด
๏ฃณ
๐โ๐+
๐
2๐ด๐+
๐
for ๐โ๐+
๐,
๐โ
๐โ๐
2๐ด๐โ
๐
for ๐โ๐โ
๐
(14)
using the convention depicted in Figure 1.
Following a Petrov-Galerkin approach, we obtain the system
of equations
T๐j=๎i๐TA,๐ +1/(i๐)Tฮฆ, ๐ ๎j=โei(15)
that can be solved to obtain an approximation of the solution
in the form ๐๐คโ๎๐[j]๐๐๐and where
๎TA,๐ ๎๐ ๐ Bhห
๐ร๐๐,T
A,๐ ๐๐i๐ค,(16)
๎Tฮฆ,๐ ๎๐ ๐ Bhห
๐ร๐๐,T
ฮฆ,๐ ๐๐i๐ค,(17)
๎จei๎ฉ๐
Bhห
๐ร๐๐,ห
๐ร๐ii๐ค,(18)
with
h๐,๐i๐คB๎น๐ค
๐(๐) ยท ๐(๐)d๐(๐).(19)
Even though we are testing with ห
๐ร๐๐, the resulting system
matrix T๐is the one from [42] (up to the fact that the
RWG functions we are using are not normalized), because
our de๏ฌnition of the EFIE operator includes an ห
๐รterm (in
contrast to [42]).
For the discretization of the MFIE, functions dual to the
RWGs must be used for testing [43]. Historically, the ๏ฌrst
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, Bu๏ฌa and Christiansen introduced the
Bu๏ฌa-Christiansen (BC) functions [45], which di๏ฌer 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 ๎ฅ๐โ๐๎ฅ
๐, where the tilde indicates that
the function is de๏ฌned on the dual mesh. The analysis is,
however, applicable to CW functions as well, and thus, we
will mostly speak of โdual functionsโ to stress the generality
of our analysis. For a de๏ฌnition of the BC functions as well
as implementation details, we refer the reader to [46]. For the
discretization of the MFIE, we obtain
M+
๐jโ๎1/2Gห
๐ร๎ฅ๐,๐+K๐๎j=hi,(20)
where
[K๐]๐๐ Bhห
๐ร๎ฅ๐๐,K๐๐๐i๐ค,(21)
๎จhi๎ฉ๐
Bhห
๐ร๎ฅ๐๐,ห
๐ร๐ii๐ค.(22)
and where the Gram matrix for any two function spaces ๐๐
and ๐๐is de๏ฌned as
๎G๐ ,๐ ๎๐๐ โh๐๐, ๐๐i๐ค,(23)
with ๐๐โ๐๐and ๐๐โ๐๐.
For the discretization of the CFIE, we have
C๐jโ๎โ๐ผT๐+ (1โ๐ผ)G๐,๐Gโ1
ห
๐ร๎ฅ
๐,๐M+
๐๎j
=๐ผei+ (1โ๐ผ)G๐,๐Gโ1
ห
๐ร๎ฅ
๐,๐hi(24)
with the combination parameter 0< ๐ผ < 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 in๏ฌuence 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]โ[50].
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๐+
๐
๐โ
๐
๐+
๐
๐โ
๐
๐+
๐
๐โ
๐
๐๐
Fig. 2. The vector ๏ฌeld 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(๐)โb๎๎/kbk< ๐,
where ๐ > 0is the solver tolerance and x(๐)the approximate
solution after the ๐th 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 ๐for obtaining an
approximation of xwithin the tolerance ๐, a relation between
๐iter and ๐is needed. One way to obtain such a relationship
is via the condition number of the matrix, which is de๏ฌned as
cond A=๎๎๎A๎๎๎2๎๎๎Aโ1๎๎๎2
=๐max(A)
๐min(A),(26)
where kk2is the spectral norm, and ๐max/min denotes the
maximal and minimal singular value.
In the case of the conjugate gradient (CG) method, which
requires Ato be Hermitian and positive de๏ฌnite, there is an
upper bound on the error e(๐)โx(๐)โxgiven by [51]
๎๎e(๐)๎๎Aโค2๎ โcond Aโ1
โcond A+1๎ก๐๎๎e(0)๎๎A,(27)
where kkAis the energy norm de๏ฌned by kxkAโ๎xโ Ax ๎1/2
and xโ denotes the conjugate transpose of x. If the objective
is to reduce the relative error ๎๎e(๐)๎๎/๎๎e(0)๎๎below ๐and by
considering limits for cond A๎1, one notes [51] that
๐โค๎ฆ1
2โcond Aln ๎2
๐๎๎ง(28)
iterations are at most needed (assuming an exact arithmetic).
If the condition number grows linearly in ๐, as observed for
the EFIE when the mesh is uniformly re๏ฌned, this implies that
the complexity is at most O(๐1.5log ๐).
One could argue that this is an overly simpli๏ฌed 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 de๏ฌnite.
One strategy to still obtain a bound on the number of iterations
is to use the CG method on the normal equation
Aโ Ax =Aโ b.(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-o๏ฌ errors due to ๏ฌnite precision can lead to a
non-converging solverโdespite the theory dictating that CG
should converge in at most ๐steps [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
signi๏ฌcantly di๏ฌer: 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 (๐๐)). 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 re๏ฌnement-free Calderรณn preconditioner which
will be discussed in Section V-A2, give rise to a Hermitian,
positive-de๏ฌnite 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 ๐ถ, which is independent from the
average edge length โof the mesh. For dynamic problems,
however, we also need to study the condition number as a
function of the frequency ๐โ๐/(2ฯ), and one must specify
if a formulation is well-conditioned with respect to โ, to
๐, to both, or only in a particular regime, for example, for
frequencies where the corresponding wavelength is larger then
the diameter of ๐ค.
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)โค๐ถ , (31)
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where ๐ถis a constant both independent of โand ๐(in which
case the preconditioner is optimal). Normally, the matrix-
matrix products in (30) are not formed explicitly and, to be an
e๏ฌcient 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 ๐de-
creases, was one of the ๏ฌrst 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 ๏ฌeld in the static limit: magnetostatic
loop currents excite the magnetic ๏ฌeld and electrostatic charges
excite the electric ๏ฌeld [20]. Both the EFIE and the MFIE
su๏ฌer from computational challenges at low-frequencies. As
we will see in this section, the EFIE su๏ฌers from conditioning
issues when the frequencies decreases and so does, albeit
for di๏ฌerent 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:
๏ฌnite machine precision and inaccuracies due to numerical
integration that result in catastrophic round-o๏ฌ 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 bene๏ฌts 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 ๏ฌeld
can be decomposed into a solenoidal, irrotational, and a
harmonic vector ๏ฌeld, which in the case of a tangential surface
vector ๏ฌeld such as ๐๐คleads to [41, p. 251]
๐๐ค=curl๐ค๐ท+grad๐ค๐น+๐ฏ(32)
where ๐ทand ๐นare su๏ฌciently smooth scalar functions,
curl๐ค๐ทโgrad๐ค๐ทรห
๐, and div๐ค๐ฏ=curl๐ค๐ฏ=0;
here, curl๐คis the adjoint operator of curl๐ค, that is, we have
hcurl๐ค๐ , ๐i๐ค=h๐ , curl๐ค๐i๐ค(see [41, see (2.5.194)]). The
space of harmonic functions ๐ป๐ฏ(๐ค)is ๏ฌnite dimensional with
dim ๐ป๐ฏ(๐ค)=2๐on a closed surface, where ๐is the genus of
๐ค. The Helmholtz subspaces are all mutually orthogonal with
respect to the ๐ณ2(๐ค)-inner product.
When ๐๐คis a linear combination of div- but not curl-
conforming functions (e.g., RWG and BC functions), only
00.511.522.53
โ1,400
โ1,200
โ1,000
Angle [rad]
Radar Cross Section [dBsm]
Mie series EFIE P-EFIE
Loop-star EFIE MFIE
Fig. 3. Radar cross sections calculated, with di๏ฌerent 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 ห
๐and propagation along
ห
๐oscillating at ๐=10โ20 Hz. The โEFIEโ and โMFIEโ labels refer to the
standard formulations (15) and (20), while the โLoop-star EFIEโ and โP-
EFIEโ 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 ๐๐คis
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 ๐๐ค, a quasi-Helmholtz decomposition
decomposes the discrete solution jas
๐
๎
๐=1[j]๐๐๐=
๐V
๎
๐=1[j๐ฆ]๐๐ฆ๐+
๐C
๎
๐=1[j๐ฎ]๐๐ฎ๐+
2๐
๎
๐=1[j๐ฏ]๐๐ฏ๐,
(33)
where ๐ฆ๐โ๐๐ฆare solenoidal loop functions, ๐ฎ๐โ๐๐ฎ
are non-solenoidal star functions, and ๐ฏ๐โ๐๐ฏare quasi-
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harmonic global loops [54] and where j๐ฆ,j๐ฎ, and j๐ฏare
the vectors containing the associated expansion coe๏ฌcients;
moreover, ๐Vis the number of vertices and ๐Cis 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 ๐ฆ๐,๐ฏ๐, and ๐ฎ๐can be represented in terms of
RWG functions [54]. Thus the expansion coe๏ฌcients are linked
by linear transformation matrices ฮ,H, and ฮฃ. For the loop
transformation matrix, we have
[ฮ]๐ ๐ =๏ฃฑ
๏ฃด
๏ฃด
๏ฃด๏ฃฒ
๏ฃด
๏ฃด
๏ฃด
๏ฃณ
1for ๐๐=๐โ
๐
โ1for ๐๐=๐+
๐
0otherwise,
(34)
where ๐๐is the ๐th vertex of the mesh (inner vertex if ๐คis
open), and for the star transformation matrix
[ฮฃ]๐ ๐ =๏ฃฑ
๏ฃด
๏ฃด
๏ฃด๏ฃฒ
๏ฃด
๏ฃด
๏ฃด
๏ฃณ
1for ๐๐=๐+
๐
โ1for ๐๐=๐โ
๐
0otherwise,
(35)
where ๐๐is the ๐th cell of the mesh, following the conventions
depicted in Figure 1. With the de๏ฌnition of these matrices, the
quasi-Helmholtz decomposition in (33) can be equivalently
written as
j=ฮj๐ฆ
๎ผ๎ป๎บ๎ฝ
=๐sol
+Hj๐ฏ
๎ผ๎ป๎บ๎ฝ
=jqhar
+ฮฃj๐ฎ
๎ผ๎ป๎บ๎ฝ
=๐nsol
=jsol +jqhar +jnsol .(36)
The linear combinations of RWGs implied by the coe๏ฌcient
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 di๏ฌerent
coe๏ฌcient 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 ฮand ฮฃa meaning that goes beyond
merely interpreting them as basis transformation matrices.
The matrices ฮand ฮฃare edge-node and edge-cell incidence
matrices of the graph de๏ฌned by the mesh and they are
orthogonal, that is, ฮฃTฮ=0. It follows that jsol and jnsol are
๐2-orthogonal, that is, jT
nsoljsol =0. We ๏ฌnd this noteworthy
for two reasons: i) the loop ๐ฆ๐and star functions ๐ฎ๐are, in
general, not ๐ณ2-orthogonal (after all, ๐ฎ๐is not irrotational);
ii) the ๐2-orthogonality is not true for other quasi-Helmholtz
decompositions such as the loop-tree basis. In light of this
consideration, the matrices ฮand ฮฃcould be interpreted as the
graph curl (ฮ) and graph gradient (ฮฃ) of the standard mesh,
an interpretation that further increases the correspondence with
the continuous decomposition (32).
For global loops ๐ฏ๐, no such simple graph-based de๏ฌnition
exists. Indeed, they are, in general, not uniquely de๏ฌned and
must be constructed from a search of holes and handles. For
any global loop basis so obtained, we have ฮฃTH=0; however,
ฮTH=0is, in general, not true. This property can be enforced
by constructing Has the right nullspace of ๎ฮ ฮฃ๎T. Such a
construction is possible, for example, via a full singular value
decomposition (SVD), or, via more computationally e๏ฌcient
randomized projections [56]. However, the computational cost
is higher, in general, compared with using a global loop-
๏ฌnding algorithm, in particular, since Hwill be a dense matrix.
A similar decomposition can be obtained for dual functions
๐
๎
๐=1[m]๐๎ฅ๐๐=
๐V
๎
๐=1๎จm๎ฅ
๐ฆ๎ฉ๐๎ฅ
๐ฆ๐+
๐C
๎
๐=1๎m๎ฅ
๐ฎ๎๐๎ฅ
๐ฎ๐+
2๐
๎
๐=1๎m๎ฅ
๐ฏ๎๐๎ฅ
๐ฏ๐,
(37)
where, in contrast to the RWG case, ๎ฅ
๐ฆ๐are non-solenoidal
dual star and ๎ฅ
๐ฎ๐are solenoidal dual loop functions. In matrix
notation, we have
j=ฮฃm๎ฅ
๐ฎ
๎ผ๎ป๎บ๎ฝ
msol
+๎ฅ
Hm ๎ฅ
๐ฏ
๎ผ๎ป๎บ๎ฝ
mqhar
+ฮm๎ฅ
๐ฆ
๎ผ๎ป๎บ๎ฝ
mnsol
=msol +mqhar +mnsol .(38)
Note that the same matrices ฮฃand ฮare present both in
the decomposition of RWG functions and in the one of dual
functions. However, while for RWGs the transformation matrix
ฮdescribes solenoidal functions and the transformation matrix
ฮฃdescribes non-solenoidal functions, the opposite is true for
the dual functions: it is ฮฃthat describes solenoidal functions,
while ฮdescribes non-solenoidal. Thus on the dual mesh, ฮ
acts as graph gradient and ฮฃas a graph curl. This is consistent
with the de๏ฌnition of dual functions: dual basis functions can
be interpreted as a div-conforming โrotationโ by 90ยฐof the
primal functions (note that the functions ห
๐ร๐๐are a rotation
by 90ยฐ, which is not div-con๏ฌrming); given that curl๐ค๐ทโ
โ๐ค๐ทรห
๐, it is consistent that the roles of ฮand ฮฃas graph
counterparts to continuous di๏ฌerential 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 ๎ฅ
H=H. This equality is only true
if His the nullspace of ๎ฮ ฮฃ๎T, a condition, which evidently
leads to the aforementioned unique de๏ฌnition of H. Even
though the construction of Has the nullspace of ๎ฮ ฮฃ๎Tis
cumbersomeโand by introducing quasi-Helmholtz projectors
in the following, we will sidestep itโit 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 thatโin the case of the EFIEโ
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
๐B๎ฮ ฮฃ๎TT๐๎ฮ ฮฃ๎can be represented in block
matrix form as
TLS
๐=๎ฮTT๐ฮ ฮTT๐ฮฃ
ฮฃTT๐ฮ ฮฃTT๐ฮฃ๎,(39)
and the corresponding matrix equation now reads TLS
๐๐LS =
๎ฮ ฮฃ๎T๐i, where ๐=๎ฮ ฮฃ๎๐LS . In these de๏ฌnitions, the
ฮand ฮฃmatrices refer to the full-rank transformation matrices
in which linearly dependent columns have been removed:
for each connected component of ๐คone star basis function
(column of ฮฃ) must always be removed and one loop basis
function must be removed (column of ฮ) if the component is
closed [57].
To evidence the di๏ฌerent low-frequency behaviors of the
EFIE matrix on the solenoidal and non-solenoidal subspaces,
the properties ฮTTฮฆ, ๐ =0and Tฮฆ,๐ ฮ=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 Greenโs function in both T
A,๐ and T
ฮฆ,๐ for ๐โ0. For
instance,
hห
๐ร๐ฎ๐,T
A,๐ ๐ฆ๐i๐ค=
๐โ0๎น๐ค๎น๐ค
๐ฎ๐(๐) ยท ๐ฆ๐(๐0)
4ฯ
๎ 1
๐
โ๐2๐
2โi๐3๐
2
6+ O(๐4)๎กd๐(๐0)d๐(๐),(40)
where ๐
=|๐โ๐0|and where we have used
๎ฏ๐คi๐๐ฆ๐(๐0)d๐(๐0)=0. We can deduce that, in general,
Re hห
๐ร๐ฎ๐,T
A,๐ ๐ฆ๐i๐ค=
๐โ0O(1),(41)
Im hห
๐ร๐ฎ๐,T
A,๐ ๐ฆ๐i๐ค=
๐โ0O(๐3).(42)
This process can be repeated for both T
A,๐ and T
ฮฆ,๐ when
both expansion and testing functions are non-solenoidal and
when at least one of the two is solenoidal. In summary,
Re hห
๐ร๐ฎ๐,T๐๐ฎ๐i๐ค=
๐โ0O(๐2),(43)
Im hห
๐ร๐ฎ๐,T๐๐ฎ๐i๐ค=
๐โ0O(๐โ1),(44)
Re hห
๐ร๐ฎ๐,T๐๐ฆ๐i๐ค=
๐โ0O(๐4),(45)
Im hห
๐ร๐ฎ๐,T๐๐ฆ๐i๐ค=
๐โ0O(๐).(46)
By symmetry, both hห
๐ร๐ฆ๐,T๐๐ฎ๐i๐คand hห
๐ร๐ฆ๐,T๐๐ฆ๐i๐ค
have the same low-frequency behavior as hห
๐ร๐ฎ๐,T๐๐ฆ๐i๐ค.
The scaling of the behavior of the block matrix is now
straightforward to obtain
Re ๎TLS
๐๎=
๐โ0๎ขO(๐4) O(๐4)
O(๐4) O(๐2)๎ฃ,(47)
Im ๎TLS
๐๎=
๐โ0๎O(๐) O(๐)
O(๐) O(๐โ1)๎,(48)
and the dominant behavior of TLS
๐is 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 D๐=diag ๎จ๐โ1/2๐1/2๎ฉ
in which the block dimensions are consistent with that of the
loop star decomposition matrix. Clearly,
D๐TLS
๐D๐=
๐โ0๎O(1) O(๐)
O(๐) O(1)๎,(49)
is a well-conditioned matrix, in the sense that
lim๐โ0cond D๐TLS
๐D๐โ๐พis ๏ฌnite. It then follows
that
cond TLS
๐=cond Dโ1
๐D๐TLS
๐D๐Dโ1
๐
โค(cond D๐)2cond D๐TLS
๐D๐
(50)
and thus lim๐โ0cond TLS
๐=O(๐โ2). 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
๐, which proves that
cond ๎TLS
๐๎โฅ๐min๐โ2where ๐min is the smallest singular
value of (ฮฃTTฮฆ,0ฮฃ). Considering these results and that the
loop-star transformation matrix is invertible and frequency
independent, we conclude that cond (T๐)โผ๐โ2when ๐โ0.
The second source of instability of the EFIE at low frequen-
cies is the loss of signi๏ฌcant digits in the right-hand side ei,
solution j, or radiated ๏ฌelds. To see this e๏ฌect, 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 hห
๐ร๐ฆ๐,ห
๐ร๐i
PWi๐ค=
๐โ0O(๐2),(51)
Im hห
๐ร๐ฆ๐,ห
๐ร๐i
PWi๐ค=
๐โ0O(๐),(52)
Re hห
๐ร๐ฎ๐,ห
๐ร๐i
PWi๐ค=
๐โ0O(1),(53)
Im hห
๐ร๐ฎ๐,ห
๐ร๐i
PWi๐ค=
๐โ0O(๐),(54)
where ๐i
PW is the electric ๏ฌeld of the incident plane-wave.
It is crucial to remember that when the standard EFIEโ
with no treatmentโis solved numerically in ๏ฌnite precision
๏ฌoating point arithmetic, the real parts (resp. imaginary parts)
of the loop and star components of the right-hand side are
stored in the same ๏ฌoating point number. In particular, the
real part of the solenoidal component that behaves as O(๐2)
is summed with an asymptotically much larger non-solenoidal
component behaving as O(1). In the context of ๏ฌnite precision
arithmetic, the dynamic range of the ๏ฌoating point number will
be imposed by the larger of the two components, meaning that
the ๏ฌoating point number will become increasingly incapable
of storing accurately the smaller one. This loss of signi๏ฌcant
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 e๏ฌect 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
๐๎โ1
=
๐โ0๎ขO(๐2) O(๐4)
O(๐4) O(๐4)๎ฃ,(55)
Im ๎TLS
๐๎โ1
=
๐โ0๎O(๐โ1) O(๐)
O(๐) O(๐)๎,(56)
which, in combination with the right-hand side results yields
the behavior of the solution coe๏ฌcients
Re (j๐ฆ)=
๐โ0O(1),(57)
Im (j๐ฆ)=
๐โ0O(๐),(58)
Re (j๐ฎ)=
๐โ0O(๐2),(59)
Im (j๐ฎ)=
๐โ0O(๐),(60)
which are indeed the behavior predicted by physics [60]. Note
that the inaccurate right-hand side component will only have a
signi๏ฌcant 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 ๏ฌeld could be quite high.
Finally, the reader should note that to numerically observe
these resultsโand successfully implement the remedies that
we will see later onโthe vanishing of all relevant integrals
must be explicitly enforced in some way, because ๏ฌoating
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
signi๏ฌcant digits that plagues the EFIE, since the loop and star
contributions of each entity are stored in separate ๏ฌoating point
numbers. In the case of the loop-star approach, the stabilized
matrix system is
D๐TLS
๐D๐jDLS =D๐๎ฮ ฮฃ๎Tei,(61)
where j=๎ฮ ฮฃ๎D๐jDLS, 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 ๏ฌeld 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 =๎ฮ0๎D๐jDLS and jsol =๎0ฮฃ๎D๐jDLS,
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 signi๏ฌcance and would further compromise
the accuracy of the ๏ฌelds.
The key di๏ฌerence 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 de๏ฌne 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 de๏ฌned as the subset
๎๐ฝ๐๎of 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 ๎ฮ ฮฃ๎is replaced
by an RWG-to-loop-tree mapping ๎ฮ ฮ๎and TLS
๐becomes
TLT
๐B๎ฮ ฮ๎TT๐๎ฮ ฮ๎where
[ฮ]๐ ๐ =๎จ1if ๐๐=๐ฝ๐
0otherwise, (62)
is the general term of the RWG-to-tree transformation matrix.
The resulting preconditioned equation is
D๐TLT
๐D๐jDLT =D๐๎ฮ ฮ๎Tei,(63)
where j=๎ฮ ฮ๎D๐jDLT.
At ๏ฌrst glance, the computational overhead of the two meth-
ods seems low, since ฮ,ฮฃ,ฮ, and D๐are 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 arti๏ฌcially 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
ฮand ฮฃcan 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 e๏ฌect of
this ill-conditioning on the preconditioned EFIE matrix is
out of the scope of this review, pseudo-di๏ฌerential operator
theory can be used to show that the di๏ฌerential strength of
the loop-star transformation operators is su๏ฌciently high not
to be compensated by that of the vector potential. To illustrate
this adverse e๏ฌect, 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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10โ44 10โ33 10โ22 10โ11 1001011
100
103
106
Frequency ๐[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 ๐[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 โLoop-tree EFIEโ and โP-
CMP-EFIEโ refer to the EFIE stabilized with the loop-tree approach (63) and
the Calderรณn EFIE stabilized with quasi-Helmholtz projectors (128).
frequencies, the conditioning of the of the loop-star and loop-
tree preconditioned matrices is signi๏ฌcantly 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 ๏ฌat spectrum that, di๏ฌerently 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=ฮฃj๐ฎ+ฮj๐ฆ+Hj๐ฏ,(64)
the quasi-Helmholtz projector for the non-solenoidal part is
the operator that maps jinto ฮฃj๐ฎ. Since
ฮฃTj=ฮฃTฮฃj๐ฎ,(65)
the looked for projector is
PฮฃBฮฃ(ฮฃTฮฃ)+ฮฃT,(66)
where +denotes the MooreโPenrose pseudoinverse. The pro-
jector for the solenoidal plus harmonic components can be
obtained out of complementarity as
PฮHโIโPฮฃ.(67)
The same reasoning for dual functions leads to the dual
de๏ฌnitions of the projector
PฮBฮ(ฮTฮ)+ฮT(68)
which is the non-solenoidal projector for dual functions. The
solenoidal plus harmonic projector for dual functions is, again,
obtained by complementarity as
PฮฃHโIโPฮ.(69)
It is important to note that, even though the projectors pre-
sented so far include a pseudo-inverse in their de๏ฌnition, 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 e๏ฌects 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
PB๐ผPฮH+๐ฝPฮฃ,(70)
where, following a frequency analysis similar to the one used
for loop-star/tree decompositions, an optimal coe๏ฌcient choice
can be found to be ๐ผโ๐โ1
2and ๐ฝโ๐1
2, that is,
P๐B๎ฐ๐ถ/๐PฮH+i/โ๐ถ๐ Pฮฃ,(71)
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0 100 200 300 400 500
10โ35
10โ24
10โ13
10โ2
Coe๏ฌcient index
Current density ๎A mโ1๎
EFIE Loop-star EFIE P-EFIE MFIE
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 ๐=10โ20 Hz, computedwith di๏ฌerent formulations.
resulting in a new system of equations
P๐T๐P๐y=P๐ei,(72)
where P๐y=j. The constant ๐ถcan 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
๐ถโ๎ณkTฮฆ,๐ k
kPฮHTA,๐ PฮHk.(73)
The analysis of the conditioning e๏ฌect 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
๐โ0P๐T๐P๐โi๐ถPฮHTA,0PฮH+i/๐ถTฮฆ,0,(74)
where we used that PฮฃTฮฆ,๐ Pฮฃ=Tฮฆ, ๐ and PฮTฮฆ,๐ =Tฮฆ, ๐ Pฮ=
0and thus
lim
๐โ0cond (P๐T๐P๐)=๐พ , (75)
where ๐พis a frequency independent constant. This approach
can be proved to simultaneously solve the problem of catas-
trophic round-o๏ฌ 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 e๏ฌects 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 e๏ฌective cures for one
0 100 200 300 400 500
10โ64
10โ43
10โ22
Coe๏ฌcient index
Current density ๎A mโ1๎
EFIE Loop-star EFIE P-EFIE MFIE
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 ๐=10โ20 Hz, computed with di๏ฌerent 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-o๏ฌ 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 ๏ฌeld and static charge extraction, which includes
a boundary integral equation for the normal component of
magnetic ๏ฌeld [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 di๏ฌerent 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 Calderรณn-type preconditioners. With a
few exceptions [78], hierarchical basis preconditioners are
based on explicit quasi-Helmholtz decomposition [79]โ[84],
since it then su๏ฌces to ๏ฌnd 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]. Calderรณn-type preconditioners will
be discussed in the next section in greater detail. At this point,
we are content to say that standard Calderรณn 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 Calderรณn preconditioned EFIE. Initially remedies relied
on combining Calderรณn 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 ๏ฌeld computation
As we have already mentioned, a well-conditioned dis-
cretization alone is not su๏ฌcient to accurately compute j: the
right-hand side su๏ฌers typically from numerical inaccuracies
due to ๏ฌnite integration precision and from round-o๏ฌ errors.
The main reason for this is that the quasi-Helmholtz compo-
nents scale di๏ฌerently in frequency. As an example, for the
case of the plane-wave excitation, the asymptotic behavior is
noted in (51)โ(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 ei๐ห
๐ยท๐with ei๐ห
๐ยท๐โ1,
where ห
๐denotes the direction of propagation. Then
๎ฐ๐ถ/๐PฮHeextracted +iโ๐ถ๐ Pฮฃe(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 ei๐ห
๐ยท๐โ1should be
replaced by a Taylor series, where the static part is omitted.
Similarly, the far-๏ฌeld cannot be computed by simply eval-
uating ๎น๐ค
๐
๎
๐=1[j]๐๐๐(๐0)eโi๐ห
๐ยท๐0d๐(๐0),(77)
where j=P๐yfrom (72), ห
๐=๐/|๐|. On the one hand,
by computing the unknown vector of the unpreconditioned
formulation j=P๐y, the di๏ฌerent asymptotic behavior
in ๐of the quasi-Helmholtz components of jas denoted
in (57)โ(60) would lead to a loss of the solenoidal/quasi-
harmonic components in the static limit due to ๏ฌnite machine
precision. Thus for the ๏ฌeld computation, one should keep
the unpreconditioned components of jseparately, that is,
jsol-qhar =๎ฐ๐ถ/๐PฮHyand jnsol =iโ๐ถ๐ Pฮฃy. On the other
hand, it has been pointed out that also the far-๏ฌeld computation
su๏ฌers from round-o๏ฌ errors [88]. To avoid these, we compute
the far-๏ฌeld in two steps: we compute the contribution of jnsol
to the far-๏ฌeld by evaluating
๐ฌfar
nsol (๐)=
๐
๎
๐=1[jnsol]๐๎น๐ค
๐๐(๐0)eโi๐ห
๐ยท๐0d๐(๐0)(78)
and the contribution of jsol-qhar by
๐ฌfar
sol-qhar (๐)=
๐
๎
๐=1๎jsol-qhar๎๐๎น๐ค
๐๐(๐0)๎eโi๐ห
๐ยท๐0โ1๎d๐(๐0),
(79)
where a Taylor-series expansion should be used for small
arguments of the exponential; then
๐ฌfar (๐)=๐ฌfar
nsol (๐) + ๐ฌfar
sol-qhar (๐)(80)
Also for the near-๏ฌeld computation, the separation in jsol-qhar
and in jnsol must be maintained, the static contribution removed
from the Greenโs 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 Greenโs function kernel,
no explicit dependency on ๐and should thus be expected to
remain well-conditioned in frequency for ๐โ0. Indeed, for
simply-connected geometries ๐ค, we have cond(M๐)=O(1)
when ๐โ0. 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]โ[91]. This leads to an ill-
conditioned system matrix [91]; in the following, we are going
to show that cond (M๐)โฅ๐ถ/๐2for some constant ๐ถโR+.
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 ๐บโhas genus ๐, then the space ๐ป๐ฏ(๐บโ)de๏ฌned by the
harmonic functions in ๐บโand the space ๐ป๐ฏ(๐บ+)de๏ฌned by
the harmonic functions in ๐บ+have both dimension ๐. The
space de๏ฌned by ๐ปห
๐ฏP(๐ค)โห
๐ร๐ป๐ฏ(๐บโ) |๐คare the poloidal
loops and the space de๏ฌned by ๐ปห
๐ฏT(๐ค)โห
๐ร๐ป๐ฏ(๐บ+) |๐ค
are the toroidal loops [92], [93]. ๐ปห
๐ฏP(๐ค)has been show [89]
to be the nullspace of Mโ
0and ๐ปห
๐ฏT(๐ค)the nullspace of M+
0
and that ห
๐รห
๐ฏT๐โ๐ฏห
๐ฏP(๐ค)and ห
๐รห
๐ฏP๐โ๐ฏห
๐ฏT(๐ค).
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 ๏ฌnd linear combinations that are in
๐ปห
๐ฏP(๐ค)or in ๐ปห
๐ฏT(๐ค), a consequence of the fact that these
functions are not curl-conforming, as mentioned in Section IV.
On the other hand, quasi-harmonic functions ๐ฏ๐(and dual
quasi-harmonic functions ๎ฅ
๐ฏ๐, respectively) are associated with
the holes and handles of the geometry. They are not, unlike the
locally de๏ฌned loop functions ๐ฆ๐, derived from a continuous
scalar potential on ๐ค. Together with the fact that ๐ฏ๐and
๎ฅ
๐ฏ๐are 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 {๐ฏ๐}2๐
๐=1can be rearranged into two bases,
where one basis {๐ฏT๐}๐
๐=1is orthogonal to poloidal loops and
the other basis {๐ฏP๐}๐
๐=1is orthogonal to toroidal loops. In the
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following, ห
๐ฏT๐โ๐ปห
๐ฏT(๐ค)denote the harmonic toroidal and
ห
๐ฏP๐โ๐ปห
๐ฏP(๐ค)denote the harmonic poloidal basis functions
on ๐ค, while ๐ฏT๐,๐ฏP๐โ๐๐and ๎ฅ
๐ฏT๐,๎ฅ
๐ฏP๐โ๐๎ฅ
๐are their
quasi-harmonic counterparts.
In [91], scalings were reported of the blocks of the system
matrix of M+
๐in terms of a quasi-Helmholtz decomposition
๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ
๐ฆ ๐ฎ ๐ฏT๐ฏP
ห
๐ร๎ฅ
๐ฎO(๐2) O(1) O(๐2) O(๐2)
ห
๐ร๎ฅ
๐ฆO(1) O(1) O(1) O(1)
ห
๐ร๎ฅ
๐ฏPO(๐2) O(1) O(๐2) O(๐2)
ห
๐ร๎ฅ
๐ฏTO(๐2) O(1) O(๐2) O(1)
๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป
.(81)
An analogous result can be obtained for Mโ
๐with 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 ๐โ0of the block
matrices, we start by considering the Taylor series of the
Greenโs function kernel of the MFIE
โ๐บ(๐,๐0)=๐น
4ฯ๐
3(i๐ ๐
โ1)ei๐ ๐
(82)
=๐น
4ฯ๐
3๎โ1+1
2(i๐)2๐
2+1
3(i๐)3๐
3+. . .๎,(83)
where ๐
=|๐โ๐0|and ๐น=๐โ๐0. An O(๐2)-scaling is
observed for a block matrix if the contribution due to the
static term in the Taylor series vanishes. For ๐ฎ๐as expansion
function or ห
๐ร๎ฅ
๐ฆas 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 ๐.
We now consider the static MFIE, which models the mag-
netostatic problem, where ๐๐คis either solenoidal or harmonic.
In this case, we have
hห
๐ร๎ฅ
๐ฎ๐,Mยฑ
0๐ฆ๐i๐ค=0,(84)
hห
๐ร๎ฅ
๐ฎ๐,Mยฑ
0๐ฏT๐i๐ค=0,(85)
hห
๐ร๎ฅ
๐ฎ๐,Mยฑ
0๐ฏP๐i๐ค=0.(86)
This can be seen by considering that a solenoidal function ๐ฆ
has a corresponding scalar potential
๐ฆ=curl๐ค๐ท . (87)
Furthermore, we have the equality
hห
๐ร๐ฆ,Mยฑ
0๐๐คi๐ค=โh๐ฆ,ห
๐รMยฑ
0๐๐คi๐ค.(88)
Inserting (87) for the testing function in the right-hand side
of (88) and using the fact that curl๐คis the adjoint operator of
curl๐ค, we obtain
hห
๐ร๐ฆ,Mยฑ
0๐๐คi๐ค=h๐ท, curl๐คห
๐ร (Mยฑ
0๐๐ค)i๐ค
=โh๐ท, curl๐ค๐โ
Ti๐ค,(89)
where ๐โ
Tโห
๐ร(Mยฑ
0๐๐ค)is the (rotated) tangential component
of the magnetic ๏ฌeld
๐(๐)=curl ๎น๐ค
๐บ๐(๐,๐0)๐๐ค(๐0)d๐(๐0)for ๐โ๐บโ,(90)
that is, ๐โ
Tis the (rotated) tangential component of ๐when ๐คis
approached from within ๐บโ, and ๐+
Twhen ๐คis approach from
within ๐บ+. We recall that (90) is obtained by ๏ฌnding a vector
potential ๐such that ๐=curl ๐and noting that curl curl ๐=๐๐ค
under the assumption that div ๐๐ค=0[40, see Chapter 6.1].
From
curl๐ค๎๐ยฑ
T(๐)๎=lim
๐บยฑ3๐0โ๐๎curl ๎๐(๐0)๎๎ยทห
๐(๐0).(91)
and from [40, (6.17)]
curl ๐={curl ๐}+ห
๐ร๎๐+โ๐โ๎ฮด๐ค,(92)
where curly braces {} mean that this part is evaluated only in
๐บยฑand ฮด๐คis the surface Dirac delta function, together with
(90), we have
curl๐ค๐โ
T=curl ๐ยทห
๐=0.(93)
Next we will establish that hห
๐ร๐ฏP๐,Mยฑ
0๐ฆ๐i๐ค=0and
hห
๐ร๐ฏT๐,Mยฑ
0๐ฆ๐i๐ค=0. First, note that the exterior MFIE
operator has the mapping properties that [89]
M+
0ห
๐ฏT๐=0(94)
and
M+
0ห
๐ฏP๐=ห
๐ฏP๐,(95)
while for the interior MFIE operator, we have the mappings
Mโ
0ห
๐ฏP๐=0(96)
and
Mโ
0ห
๐ฏT๐=ห
๐ฏT๐.(97)
Furthermore, for any two surface functions ๐,๐, we have [89,
Section 5]
h๐,K๐i๐ค=hโห
๐ร๎K๎๐รห
๐๎๎,๐i๐ค
=hโK๎๐รห
๐๎,๐รห
๐i๐ค(98)
Then we ๏ฌnd
hห
๐รห
๐ฏT๐,M+
0๐ฆi๐ค=hMโ
0ห
๐ฏT๐,๐ฆรห
๐i๐ค
(97)
=0,(99)
hห
๐รห
๐ฏP๐,M+
0๐ฆi๐ค=hMโ
0ห
๐ฏP๐,๐ฆรห
๐i๐ค
(96)
=0,(100)
hห
๐รห
๐ฏT๐,Mโ
0๐ฆi๐ค=hM+
0ห
๐ฏT๐,๐ฆรห
๐i๐ค
(94)
=0,(101)
hห
๐รห
๐ฏP๐,Mโ
0๐ฆi๐ค=hM+
0ห
๐ฏP๐,๐ฆรห
๐i๐ค
(95)
=0,(102)
where we used (94)-(97) and the orthogonality of harmonic
and irrotational functions ห
๐ร๐ฆ. Now consider that ๎ฅ
๐ฏP๐=
ห
๐ฏP๐+๎ฅ
๐ฎP๐and ๎ฅ
๐ฏT๐=ห
๐ฏT๐+๎ฅ
๐ฎT๐, where ๎ฅ
๐ฎP/T๐is the respective
perturbation. Thus by taking into account (84), we have
hห
๐ร๎ฅ
๐ฏP๐,Mยฑ
0๐ฆ๐i๐ค=0,(103)
hห
๐ร๎ฅ
๐ฏT๐,Mยฑ
0๐ฆ๐i๐ค=0.(104)
From now on, we will only consider the harmonic functions
ห
๐ฏT๐and ห
๐ฏP๐instead of their quasi-harmonic counterpart
since, as we have seen, the solenoidal pertubation will always
vanish. Then for M+
0, we have
hห
๐รห
๐ฏT๐,M+
0ห
๐ฏT๐i๐ค=hห
๐รห
๐ฏP๐,M+
0ห
๐ฏT๐i๐ค
(94)
=0,(105)
hห
๐รห
๐ฏT๐,Mโ
0ห
๐ฏP๐i๐ค=hห
๐รห
๐ฏP๐,Mโ
0ห
๐ฏP๐i๐ค
(96)
=0(106)
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
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
hห
๐รห
๐ฏP๐,M+
0ห
๐ฏP๐i๐ค
(95)
=hห
๐รห
๐ฏP๐,ห
๐ฏP๐i๐ค=0,(107)
as ห
๐รห
๐ฏP๐โ๐ปห
๐ฏT(๐ค)and this space is orthogonal to ๐ปห
๐ฏP(๐ค).
Likewise, we can conclude that there is at least one ๐such
that
hห
๐รห
๐ฏT๐,M+
0ห
๐ฏP๐i๐คโ 0.(108)
Analogously, we obtain
hห
๐รห
๐ฏP๐,Mโ
0ห
๐ฏT๐