ArticlePDF Available

Electromagnetic Integral Equations: Insights in Conditioning and Preconditioning

Authors:

Abstract and Figures

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.
Content may be subject to copyright.
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <1
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๏ฌ€ectiveness in several real case scenarios. Luckily,
however, it is possible to leverage e๏ฌ€ective preconditioning and regularization strategies that can cure a large majority of these
problems. Not surprisingly, integral equation preconditioning is currently a quite active ๏ฌeld 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 ๏ฌeld 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 ๏ฌeld
(EFIE), magnetic ๏ฌeld (MFIE), and combined ๏ฌeld integral equation (CFIE) and of the associated remedies.
Index Termsโ€”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]โ€“[4]. What makes these schemes so suitable for electro-
magnetic analyses is that, di๏ฌ€erently from approaches based
on di๏ฌ€erential equations such as the ๏ฌnite element method
(FEM) or the ๏ฌnite-di๏ฌ€erence time-domain method (FDTD),
they naturally incorporate radiation conditions without the
need for arti๏ฌcial 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๏ฌ€erential 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 Unionโ€™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 Rรฉgion Bretagne and the Conseil Dรฉpartemental du
Finistรจre under the project โ€œTONNERREโ€, by the ANR Labex CominLabs
under the project โ€œCYCLEโ€, and by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) โ€“ SFB 1270/1โ€“299150580.
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(๐‘log 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(๐‘iter๐‘log ๐‘)(or ๐‘‚(๐‘iter ๐‘)in the low-frequency
regime), where ๐‘iter is the number of iterations.
The number of iterations ๐‘iter 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 ๐‘and,
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‘‚(๐‘2),
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๏ฌnement (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 ๏ฌrst 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)โ€“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.
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <2
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 ๏ฌnding 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 ๏ฌeld integral equation (EFIE) and magnetic
๏ฌeld 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๏ฌ€ective 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๏ฌƒcient
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๏ฌc 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 ๏ฌeld integral equation (CFIE) and
of the associated potential remedies. In contrast to [22], our
focus will include low-frequency e๏ฌ€ects 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๏ฌ€ective 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๏ฌned 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 ๏ฌnal considerations.
II. Notation and Background
We are interested in solving the electromagnetic scattering
problem where a time-harmonic, electromagnetic wave (๐’†i,๐’‰i)
in a space with permittivity ๐œ€and permeability ๐œ‡impinges on
a connected domain ๐›บโˆ’โŠ‚R3with PEC boundary ๐›คโ‰”๐œ• ๐›บโˆ’
resulting in the scattered wave (๐’†s,๐’‰s). The total electric ๐’†B
๐’†i+๐’†sand magnetic ๐’‰B๐’‰i+๐’‰s๏ฌelds satisfy Maxwellโ€™s
equations
โˆ‡ ร— ๐’†(๐’“)=+i๐‘˜๐’‰(๐’“),for all ๐’“โˆˆ๐›บ+,(1)
โˆ‡ ร— ๐’‰(๐’“)=โˆ’i๐‘˜๐’†(๐’“),for all ๐’“โˆˆ๐›บ+,(2)
where ๐›บ+โ‰”๐›บโˆ’
c,๐‘˜โ‰”๐œ”โˆš๐œ€๐œ‡ is the wave number, ๐œ”
the angular frequency, and ๐’†,๐’‰must satisfy the boundary
conditions for PEC boundaries
ห†
๐’ร—๐’†=0,for all ๐’“โˆˆ๐›ค , (3)
ห†
๐’ร—๐’‰=๐’‹๐›ค,for all ๐’“โˆˆ๐›ค , (4)
where ๐’‹๐›คis the induced electric surface current density. In
addition, ๐’†sand ๐’‰smust satisfy the Silver-Mรผller radiation
condition [38], [39]
lim
๐‘Ÿโ†’โˆž๎€Œ๎€Œ๐’‰sร—๐’“โˆ’๐‘Ÿ๐’†s๎€Œ๎€Œ=0.(5)
We assumed (and suppressed) a time dependency of eโˆ’i๐œ”๐‘ก and
normalized ๐’‰with the wave impedance ๐œ‚โ‰”๎ฐ๐œ‡/๐œ€.
To ๏ฌnd (๐’†s,๐’‰s), we can solve the EFIE
T๐‘˜๐’‹๐›ค=โˆ’ห†
๐’ร—๐’†i(6)
for ๐’‹๐›ค, where ห†
๐’is the surface normal vector directed into ๐›บ+
and
T๐‘˜โ‰”i๐‘˜T
A,๐‘˜ +1/(i๐‘˜)T
ฮฆ,๐‘˜ (7)
is the EFIE operator composed of the vector potential operator
(T
A,๐‘˜ ๐’‹๐›ค)(๐’“)=ห†
๐’ร—๎‚น๐›ค
๐บ๐‘˜(๐’“,๐’“0)๐’‹๐›ค(๐’“0)d๐‘†(๐’“0)(8)
and the scalar potential operator
(T
ฮฆ,๐‘˜ ๐’‹๐›ค)(๐’“)=โˆ’ห†
๐’ร—grad๐›ค๎‚น๐›ค
๐บ๐‘˜(๐’“,๐’“0)div๐›ค๐’‹๐›ค(๐’“0)d๐‘†(๐’“0),
(9)
where
๐บ๐‘˜(๐’“,๐’“0)=ei๐‘˜|๐’“โˆ’๐’“0|
4ฯ€|๐’“โˆ’๐’“0|(10)
is the free-space Greenโ€™s function. A de๏ฌnition of the surface
di๏ฌ€erential operators grad๐›คand div๐›คcan be found in [40,
Appendix 3] or [41, Chapter 2]. Once ๐’‹๐›คis obtained, ๎€€๐’†s,๐’‰s๎€
can be computed using the free-space radiation operators.
Alternatively, one can solve the MFIE for the exterior
scattering problem
ห†
๐’ร—๐’‰i=M+
๐‘˜๐’‹๐›คโ‰”+๎€€I/2+K๐‘˜๎€๐’‹๐›ค,(11)
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <3
๐’—+
๐‘›
๐’—โˆ’
๐‘›
๐’“+
๐‘›
๐’“โˆ’
๐‘›
๐‘+
๐‘›
๐‘โˆ’
๐‘›
๐’†๐‘›
Fig. 1. The vector ๏ฌeld of an RWG function. The vector ๐’†๐‘›denotes the
directed edge, ๐‘+
๐‘›and ๐‘โˆ’
๐‘›denote the domains of the cells, ๐‘ฃ+
๐‘›and ๐‘ฃโˆ’
๐‘›denote
vertices on the edge ๐’†๐‘›, and ๐’“+
๐‘›and ๐’“โˆ’
๐‘›are the vertices opposite to the edge
๐’†๐‘›.
where Iis identity operator, M+
๐‘˜is the MFIE operator for the
exterior scattering problem, and
๎€€K๐‘˜๐’‹๐›ค๎€(๐’“)Bโˆ’ห†
๐’ร—๎‚น๐›คโˆ‡๐บ๐‘˜(๐’“,๐’“0) ร— ๐’‹๐›คd๐‘†(๐’“0).(12)
The MFIE operator for the interior scattering problem is
Mโˆ’
๐‘˜โ‰”โˆ’I/2+K๐‘˜and 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]
โˆ’๐›ผT๐‘˜๐’‹๐›ค+ (1โˆ’๐›ผ)ห†
๐’ร—M+
๐‘˜๐’‹๐›ค
=๐›ผห†
๐’ร—๐’†i+ (1โˆ’๐›ผ)ห†
๐’ร—ห†
๐’ร—๐’‰i(13)
which is uniquely solvable for all frequencies.
For the discretization of the EFIE, we employ Rao-Wilton-
Glisson (RWG) basis functions ๐’‡๐‘›โˆˆ๐‘‹๐’‡which are hereโ€”in
contrast to their original de๏ฌnition in [42]โ€”not normalized
with the edge length, that is,
๐’‡๐‘›=๏ฃฑ
๏ฃด
๏ฃด
๏ฃด
๏ฃด
๏ฃด๏ฃฒ
๏ฃด
๏ฃด
๏ฃด
๏ฃด
๏ฃด
๏ฃณ
๐’“โˆ’๐’“+
๐‘›
2๐ด๐‘+
๐‘›
for ๐’“โˆˆ๐‘+
๐‘›,
๐’“โˆ’
๐‘›โˆ’๐’“
2๐ด๐‘โˆ’
๐‘›
for ๐’“โˆˆ๐‘โˆ’
๐‘›
(14)
using the convention depicted in Figure 1.
Following a Petrov-Galerkin approach, we obtain the system
of equations
T๐‘˜j=๎€€i๐‘˜TA,๐‘˜ +1/(i๐‘˜)Tฮฆ, ๐‘˜ ๎€j=โˆ’ei(15)
that can be solved to obtain an approximation of the solution
in the form ๐’‹๐›คโ‰ˆ๎ƒ๐‘›[j]๐‘›๐’‡๐‘›and where
๎€‚TA,๐‘˜ ๎€ƒ๐‘› ๐‘š Bhห†
๐’ร—๐’‡๐‘›,T
A,๐‘˜ ๐’‡๐‘ši๐›ค,(16)
๎€‚Tฮฆ,๐‘˜ ๎€ƒ๐‘› ๐‘š Bhห†
๐’ร—๐’‡๐‘›,T
ฮฆ,๐‘˜ ๐’‡๐‘ši๐›ค,(17)
๎จei๎ฉ๐‘›
Bhห†
๐’ร—๐’‡๐‘›,ห†
๐’ร—๐’†ii๐›ค,(18)
with
h๐’‡,๐’ˆi๐›คB๎‚น๐›ค
๐’‡(๐’“) ยท ๐’ˆ(๐’“)d๐‘†(๐’“).(19)
Even though we are testing with ห†
๐’ร—๐’‡๐‘›, the resulting system
matrix T๐‘˜is the one from [42] (up to the fact that the
RWG functions we are using are not normalized), because
our de๏ฌnition of the EFIE operator includes an ห†
๐’ร—term (in
contrast to [42]).
For the discretization of the MFIE, functions dual to the
RWGs must be used for testing [43]. Historically, the ๏ฌrst
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๏ฌ€a and Christiansen introduced the
Bu๏ฌ€a-Christiansen (BC) functions [45], which di๏ฌ€er 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 ๎ฅ๐’‡โˆˆ๐‘‹๎ฅ
๐’‡, where the tilde indicates that
the function is de๏ฌned on the dual mesh. The analysis is,
however, applicable to CW functions as well, and thus, we
will mostly speak of โ€œdual functionsโ€ to stress the generality
of our analysis. For a de๏ฌnition of the BC functions as well
as implementation details, we refer the reader to [46]. For the
discretization of the MFIE, we obtain
M+
๐‘˜jโ‰”๎€1/2Gห†
๐’ร—๎ฅ๐’‡,๐’‡+K๐‘˜๎€‘j=hi,(20)
where
[K๐‘˜]๐‘›๐‘š Bhห†
๐’ร—๎ฅ๐’‡๐‘›,K๐‘˜๐’‡๐‘ši๐›ค,(21)
๎จhi๎ฉ๐‘›
Bhห†
๐’ร—๎ฅ๐’‡๐‘›,ห†
๐’ร—๐’‰ii๐›ค.(22)
and where the Gram matrix for any two function spaces ๐‘‹๐‘“
and ๐‘‹๐‘”is de๏ฌned as
๎€‚G๐‘“ ,๐‘” ๎€ƒ๐‘š๐‘› โ‰”h๐‘“๐‘š, ๐‘”๐‘›i๐›ค,(23)
with ๐‘“๐‘šโˆˆ๐‘‹๐‘“and ๐‘”๐‘›โˆˆ๐‘‹๐‘”.
For the discretization of the CFIE, we have
C๐‘˜jโ‰”๎€’โˆ’๐›ผT๐‘˜+ (1โˆ’๐›ผ)G๐’‡,๐’‡Gโˆ’1
ห†
๐’ร—๎ฅ
๐’‡,๐’‡M+
๐‘˜๎€“j
=๐›ผei+ (1โˆ’๐›ผ)G๐’‡,๐’‡Gโˆ’1
ห†
๐’ร—๎ฅ
๐’‡,๐’‡hi(24)
with the combination parameter 0< ๐›ผ < 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๏ฌ‚uence 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].
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <4
๐’—+
๐‘›
๐’—โˆ’
๐‘›
๐’“+
๐‘›
๐’“โˆ’
๐‘›
๐‘+
๐‘›
๐‘โˆ’
๐‘›
๐’†๐‘›
Fig. 2. The vector ๏ฌeld 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‘–)โˆ’b๎€๎€/kbk< ๐œ€,
where ๐œ€ > 0is the solver tolerance and x(๐‘–)the approximate
solution after the ๐‘–th 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 ๐‘for obtaining an
approximation of xwithin the tolerance ๐œ€, a relation between
๐‘iter and ๐‘is needed. One way to obtain such a relationship
is via the condition number of the matrix, which is de๏ฌned as
cond A=๎€๎€๎€A๎€๎€๎€2๎€๎€๎€Aโˆ’1๎€๎€๎€2
=๐œŽmax(A)
๐œŽmin(A),(26)
where kk2is the spectral norm, and ๐œŽmax/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๏ฌnite, there is an
upper bound on the error e(๐‘–)โ‰”x(๐‘–)โˆ’xgiven by [51]
๎€๎€e(๐‘–)๎€๎€Aโ‰ค2๎€ โˆšcond Aโˆ’1
โˆšcond A+1๎€ก๐‘–๎€๎€e(0)๎€๎€A,(27)
where kkAis the energy norm de๏ฌned by kxkAโ‰”๎€xโ€ 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 ๐œ€and by
considering limits for cond A๎€1, one notes [51] that
๐‘–โ‰ค๎€ฆ1
2โˆšcond Aln ๎€’2
๐œ€๎€“๎€ง(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๏ฌned, this implies that
the complexity is at most O(๐‘1.5log ๐‘).
One could argue that this is an overly simpli๏ฌed 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๏ฌnite.
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 ๏ฌnite precision can lead to a
non-converging solverโ€”despite the theory dictating that CG
should converge in at most ๐‘steps [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๏ฌcantly di๏ฌ€er: 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 (๐œ†๐‘–)). 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๏ฌnement-free Calderรณn preconditioner which
will be discussed in Section V-A2, give rise to a Hermitian,
positive-de๏ฌnite 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 ๐ถ, which is independent from the
average edge length โ„Žof the mesh. For dynamic problems,
however, we also need to study the condition number as a
function of the frequency ๐‘“โ‰”๐œ”/(2ฯ€), and one must specify
if a formulation is well-conditioned with respect to โ„Ž, to
๐‘“, to both, or only in a particular regime, for example, for
frequencies where the corresponding wavelength is larger then
the diameter of ๐›ค.
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)โ‰ค๐ถ , (31)
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <5
where ๐ถis a constant both independent of โ„Žand ๐‘“(in which
case the preconditioner is optimal). Normally, the matrix-
matrix products in (30) are not formed explicitly and, to be an
e๏ฌƒcient 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 ๐’‡de-
creases, was one of the ๏ฌrst 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 ๏ฌeld in the static limit: magnetostatic
loop currents excite the magnetic ๏ฌeld and electrostatic charges
excite the electric ๏ฌeld [20]. Both the EFIE and the MFIE
su๏ฌ€er from computational challenges at low-frequencies. As
we will see in this section, the EFIE su๏ฌ€ers from conditioning
issues when the frequencies decreases and so does, albeit
for di๏ฌ€erent 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:
๏ฌnite 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๏ฌts 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 ๏ฌeld
can be decomposed into a solenoidal, irrotational, and a
harmonic vector ๏ฌeld, which in the case of a tangential surface
vector ๏ฌeld such as ๐’‹๐›คleads to [41, p. 251]
๐’‹๐›ค=curl๐›ค๐›ท+grad๐›ค๐›น+๐‘ฏ(32)
where ๐›ทand ๐›นare su๏ฌƒciently smooth scalar functions,
curl๐›ค๐›ทโ‰”grad๐›ค๐›ทร—ห†
๐’, and div๐›ค๐‘ฏ=curl๐›ค๐‘ฏ=0;
here, curl๐›คis the adjoint operator of curl๐›ค, that is, we have
hcurl๐›ค๐‘“ , ๐’ˆi๐›ค=h๐‘“ , curl๐›ค๐’ˆi๐›ค(see [41, see (2.5.194)]). The
space of harmonic functions ๐ป๐‘ฏ(๐›ค)is ๏ฌnite dimensional with
dim ๐ป๐‘ฏ(๐›ค)=2๐‘”on a closed surface, where ๐‘”is the genus of
๐›ค. The Helmholtz subspaces are all mutually orthogonal with
respect to the ๐‘ณ2(๐›ค)-inner product.
When ๐’‹๐›คis a linear combination of div- but not curl-
conforming functions (e.g., RWG and BC functions), only
00.511.522.53
โˆ’1,400
โˆ’1,200
โˆ’1,000
Angle [rad]
Radar Cross Section [dBsm]
Mie series EFIE P-EFIE
Loop-star EFIE MFIE
Fig. 3. Radar cross sections calculated, with di๏ฌ€erent 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†
๐’™and propagation along
ห†
๐’›oscillating at ๐‘“=10โˆ’20 Hz. The โ€œEFIEโ€ and โ€œMFIEโ€ labels refer to the
standard formulations (15) and (20), while the โ€œLoop-star EFIEโ€ and โ€œP-
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 ๐’‹๐›คis
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 ๐’‹๐›ค, a quasi-Helmholtz decomposition
decomposes the discrete solution jas
๐‘
๎ƒ•
๐‘›=1[j]๐‘›๐’‡๐‘›=
๐‘V
๎ƒ•
๐‘›=1[j๐œฆ]๐‘›๐œฆ๐‘›+
๐‘C
๎ƒ•
๐‘›=1[j๐œฎ]๐‘›๐œฎ๐‘›+
2๐‘”
๎ƒ•
๐‘›=1[j๐‘ฏ]๐‘›๐‘ฏ๐‘›,
(33)
where ๐œฆ๐‘›โˆˆ๐‘‹๐œฆare solenoidal loop functions, ๐œฎ๐‘›โˆˆ๐‘‹๐œฎ
are non-solenoidal star functions, and ๐‘ฏ๐‘›โˆˆ๐‘‹๐‘ฏare quasi-
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <6
harmonic global loops [54] and where j๐œฆ,j๐œฎ, and j๐‘ฏare
the vectors containing the associated expansion coe๏ฌƒcients;
moreover, ๐‘Vis the number of vertices and ๐‘Cis 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 ๐œฆ๐‘›,๐‘ฏ๐‘›, and ๐œฎ๐‘›can be represented in terms of
RWG functions [54]. Thus the expansion coe๏ฌƒcients 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 ๐’—๐‘—=๐’—โˆ’
๐‘–
โˆ’1for ๐’—๐‘—=๐’—+
๐‘–
0otherwise,
(34)
where ๐’—๐‘—is the ๐‘—th vertex of the mesh (inner vertex if ๐›คis
open), and for the star transformation matrix
[ฮฃ]๐‘– ๐‘— =๏ฃฑ
๏ฃด
๏ฃด
๏ฃด๏ฃฒ
๏ฃด
๏ฃด
๏ฃด
๏ฃณ
1for ๐‘๐‘—=๐‘+
๐‘–
โˆ’1for ๐‘๐‘—=๐‘โˆ’
๐‘–
0otherwise,
(35)
where ๐‘๐‘—is the ๐‘—th cell of the mesh, following the conventions
depicted in Figure 1. With the de๏ฌnition of these matrices, the
quasi-Helmholtz decomposition in (33) can be equivalently
written as
j=ฮ›j๐œฆ
๎ผ๎ป๎บ๎ฝ
=๐‘—sol
+Hj๐‘ฏ
๎ผ๎ป๎บ๎ฝ
=jqhar
+ฮฃj๐œฎ
๎ผ๎ป๎บ๎ฝ
=๐‘—nsol
=jsol +jqhar +jnsol .(36)
The linear combinations of RWGs implied by the coe๏ฌƒcient
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๏ฌ€erent
coe๏ฌƒcient 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 ฮ›and ฮฃa meaning that goes beyond
merely interpreting them as basis transformation matrices.
The matrices ฮ›and ฮฃare edge-node and edge-cell incidence
matrices of the graph de๏ฌned by the mesh and they are
orthogonal, that is, ฮฃTฮ›=0. It follows that jsol and jnsol are
๐’2-orthogonal, that is, jT
nsoljsol =0. We ๏ฌnd this noteworthy
for two reasons: i) the loop ๐œฆ๐‘–and star functions ๐œฎ๐‘—are, in
general, not ๐‘ณ2-orthogonal (after all, ๐œฎ๐‘—is not irrotational);
ii) the ๐’2-orthogonality is not true for other quasi-Helmholtz
decompositions such as the loop-tree basis. In light of this
consideration, the matrices ฮ›and 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 ๐‘ฏ๐‘›, no such simple graph-based de๏ฌnition
exists. Indeed, they are, in general, not uniquely de๏ฌned and
must be constructed from a search of holes and handles. For
any global loop basis so obtained, we have ฮฃTH=0; however,
ฮ›TH=0is, in general, not true. This property can be enforced
by constructing Has the right nullspace of ๎€‚ฮ› ฮฃ๎€ƒT. Such a
construction is possible, for example, via a full singular value
decomposition (SVD), or, via more computationally e๏ฌƒcient
randomized projections [56]. However, the computational cost
is higher, in general, compared with using a global loop-
๏ฌnding algorithm, in particular, since Hwill be a dense matrix.
A similar decomposition can be obtained for dual functions
๐‘
๎ƒ•
๐‘›=1[m]๐‘›๎ฅ๐’‡๐‘›=
๐‘V
๎ƒ•
๐‘›=1๎จm๎ฅ
๐œฆ๎ฉ๐‘›๎ฅ
๐œฆ๐‘›+
๐‘C
๎ƒ•
๐‘›=1๎€‚m๎ฅ
๐œฎ๎€ƒ๐‘›๎ฅ
๐œฎ๐‘›+
2๐‘”
๎ƒ•
๐‘›=1๎€‚m๎ฅ
๐‘ฏ๎€ƒ๐‘›๎ฅ
๐‘ฏ๐‘›,
(37)
where, in contrast to the RWG case, ๎ฅ
๐œฆ๐‘›are non-solenoidal
dual star and ๎ฅ
๐œฎ๐‘›are solenoidal dual loop functions. In matrix
notation, we have
j=ฮฃm๎ฅ
๐œฎ
๎ผ๎ป๎บ๎ฝ
msol
+๎ฅ
Hm ๎ฅ
๐‘ฏ
๎ผ๎ป๎บ๎ฝ
mqhar
+ฮ›m๎ฅ
๐œฆ
๎ผ๎ป๎บ๎ฝ
mnsol
=msol +mqhar +mnsol .(38)
Note that the same matrices ฮฃand ฮ›are present both in
the decomposition of RWG functions and in the one of dual
functions. However, while for RWGs the transformation matrix
ฮ›describes 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 ฮ›describes non-solenoidal. Thus on the dual mesh, ฮ›
acts as graph gradient and ฮฃas a graph curl. This is consistent
with the de๏ฌnition of dual functions: dual basis functions can
be interpreted as a div-conforming โ€œrotationโ€ by 90ยฐof the
primal functions (note that the functions ห†
๐’ร—๐’‡๐‘–are a rotation
by 90ยฐ, which is not div-con๏ฌrming); given that curl๐›ค๐›ทโ‰”
โˆ‡๐›ค๐›ทร—ห†
๐’, it is consistent that the roles of ฮ›and ฮฃas graph
counterparts to continuous di๏ฌ€erential 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› ฮฃ๎€ƒT, a condition, which evidently
leads to the aforementioned unique de๏ฌnition of H. Even
though the construction of Has the nullspace of ๎€‚ฮ› ฮฃ๎€ƒTis
cumbersomeโ€”and by introducing quasi-Helmholtz projectors
in the following, we will sidestep itโ€”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 thatโ€”in 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
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <7
TLS
๐‘˜B๎€‚ฮ› ฮฃ๎€ƒTT๐‘˜๎€‚ฮ› ฮฃ๎€ƒcan be represented in block
matrix form as
TLS
๐‘˜=๎€”ฮ›TT๐‘˜ฮ› ฮ›TT๐‘˜ฮฃ
ฮฃTT๐‘˜ฮ› ฮฃTT๐‘˜ฮฃ๎€•,(39)
and the corresponding matrix equation now reads TLS
๐‘˜๐’‹LS =
๎€‚ฮ› ฮฃ๎€ƒT๐’†i, where ๐’‹=๎€‚ฮ› ฮฃ๎€ƒ๐’‹LS . In these de๏ฌnitions, the
ฮ›and ฮฃmatrices refer to the full-rank transformation matrices
in which linearly dependent columns have been removed:
for each connected component of ๐›คone 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๏ฌ€erent low-frequency behaviors of the
EFIE matrix on the solenoidal and non-solenoidal subspaces,
the properties ฮ›TTฮฆ, 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 Greenโ€™s function in both T
A,๐‘˜ and T
ฮฆ,๐‘˜ for ๐‘˜โ†’0. For
instance,
hห†
๐’ร—๐œฎ๐‘š,T
A,๐‘˜ ๐œฆ๐‘›i๐›ค=
๐‘˜โ†’0๎‚น๐›ค๎‚น๐›ค
๐œฎ๐‘š(๐’“) ยท ๐œฆ๐‘›(๐’“0)
4ฯ€
๎€ 1
๐‘…โˆ’๐‘˜2๐‘…
2โˆ’i๐‘˜3๐‘…2
6+ O(๐‘˜4)๎€กd๐‘†(๐’“0)d๐‘†(๐’“),(40)
where ๐‘…=|๐’“โˆ’๐’“0|and where we have used
๎‚ฏ๐›คi๐‘˜๐œฆ๐‘›(๐’“0)d๐‘†(๐’“0)=0. We can deduce that, in general,
Re hห†
๐’ร—๐œฎ๐‘š,T
A,๐‘˜ ๐œฆ๐‘›i๐›ค=
๐‘˜โ†’0O(1),(41)
Im hห†
๐’ร—๐œฎ๐‘š,T
A,๐‘˜ ๐œฆ๐‘›i๐›ค=
๐‘˜โ†’0O(๐‘˜3).(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†
๐’ร—๐œฎ๐‘š,T๐‘˜๐œฎ๐‘›i๐›ค=
๐‘˜โ†’0O(๐‘˜2),(43)
Im hห†
๐’ร—๐œฎ๐‘š,T๐‘˜๐œฎ๐‘›i๐›ค=
๐‘˜โ†’0O(๐‘˜โˆ’1),(44)
Re hห†
๐’ร—๐œฎ๐‘š,T๐‘˜๐œฆ๐‘›i๐›ค=
๐‘˜โ†’0O(๐‘˜4),(45)
Im hห†
๐’ร—๐œฎ๐‘š,T๐‘˜๐œฆ๐‘›i๐›ค=
๐‘˜โ†’0O(๐‘˜).(46)
By symmetry, both hห†
๐’ร—๐œฆ๐‘š,T๐‘˜๐œฎ๐‘›i๐›คand hห†
๐’ร—๐œฆ๐‘š,T๐‘˜๐œฆ๐‘›i๐›ค
have the same low-frequency behavior as hห†
๐’ร—๐œฎ๐‘š,T๐‘˜๐œฆ๐‘›i๐›ค.
The scaling of the behavior of the block matrix is now
straightforward to obtain
Re ๎€TLS
๐‘˜๎€‘=
๐‘˜โ†’0๎€ขO(๐‘˜4) O(๐‘˜4)
O(๐‘˜4) O(๐‘˜2)๎€ฃ,(47)
Im ๎€TLS
๐‘˜๎€‘=
๐‘˜โ†’0๎€”O(๐‘˜) O(๐‘˜)
O(๐‘˜) O(๐‘˜โˆ’1)๎€•,(48)
and the dominant behavior of TLS
๐‘˜is 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 ๎จ๐‘˜โˆ’1/2๐‘˜1/2๎ฉ
in which the block dimensions are consistent with that of the
loop star decomposition matrix. Clearly,
D๐‘˜TLS
๐‘˜D๐‘˜=
๐‘˜โ†’0๎€”O(1) O(๐‘˜)
O(๐‘˜) O(1)๎€•,(49)
is a well-conditioned matrix, in the sense that
lim๐‘˜โ†’0cond D๐‘˜TLS
๐‘˜D๐‘˜โ‰•๐›พis ๏ฌnite. It then follows
that
cond TLS
๐‘˜=cond Dโˆ’1
๐‘˜D๐‘˜TLS
๐‘˜D๐‘˜Dโˆ’1
๐‘˜
โ‰ค(cond D๐‘˜)2cond D๐‘˜TLS
๐‘˜D๐‘˜
(50)
and thus lim๐‘˜โ†’0cond TLS
๐‘˜=O(๐‘˜โˆ’2). 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
๐‘˜๎€‘โ‰ฅ๐œŽmin๐‘˜โˆ’2where ๐œŽmin 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‘˜)โˆผ๐‘˜โˆ’2when ๐‘˜โ†’0.
The second source of instability of the EFIE at low frequen-
cies is the loss of signi๏ฌcant digits in the right-hand side ei,
solution j, or radiated ๏ฌelds. To see this e๏ฌ€ect, 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†
๐’ร—๐œฆ๐‘š,ห†
๐’ร—๐’†i
PWi๐›ค=
๐‘˜โ†’0O(๐‘˜2),(51)
Im hห†
๐’ร—๐œฆ๐‘š,ห†
๐’ร—๐’†i
PWi๐›ค=
๐‘˜โ†’0O(๐‘˜),(52)
Re hห†
๐’ร—๐œฎ๐‘š,ห†
๐’ร—๐’†i
PWi๐›ค=
๐‘˜โ†’0O(1),(53)
Im hห†
๐’ร—๐œฎ๐‘š,ห†
๐’ร—๐’†i
PWi๐›ค=
๐‘˜โ†’0O(๐‘˜),(54)
where ๐’†i
PW is the electric ๏ฌeld of the incident plane-wave.
It is crucial to remember that when the standard EFIEโ€”
with no treatmentโ€”is solved numerically in ๏ฌnite precision
๏ฌ‚oating point arithmetic, the real parts (resp. imaginary parts)
of the loop and star components of the right-hand side are
stored in the same ๏ฌ‚oating point number. In particular, the
real part of the solenoidal component that behaves as O(๐‘˜2)
is summed with an asymptotically much larger non-solenoidal
component behaving as O(1). In the context of ๏ฌnite precision
arithmetic, the dynamic range of the ๏ฌ‚oating point number will
be imposed by the larger of the two components, meaning that
the ๏ฌ‚oating point number will become increasingly incapable
of storing accurately the smaller one. This loss of signi๏ฌcant
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๏ฌ€ect of this loss of accuracy
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <8
on the dominant parts of the solution. Using the well-known
relations on block matrix inverses [59], one can show that
Re ๎€TLS
๐‘˜๎€‘โˆ’1
=
๐‘˜โ†’0๎€ขO(๐‘˜2) O(๐‘˜4)
O(๐‘˜4) O(๐‘˜4)๎€ฃ,(55)
Im ๎€TLS
๐‘˜๎€‘โˆ’1
=
๐‘˜โ†’0๎€”O(๐‘˜โˆ’1) O(๐‘˜)
O(๐‘˜) O(๐‘˜)๎€•,(56)
which, in combination with the right-hand side results yields
the behavior of the solution coe๏ฌƒcients
Re (j๐œฆ)=
๐‘˜โ†’0O(1),(57)
Im (j๐œฆ)=
๐‘˜โ†’0O(๐‘˜),(58)
Re (j๐œฎ)=
๐‘˜โ†’0O(๐‘˜2),(59)
Im (j๐œฎ)=
๐‘˜โ†’0O(๐‘˜),(60)
which are indeed the behavior predicted by physics [60]. Note
that the inaccurate right-hand side component will only have a
signi๏ฌcant 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 ๏ฌeld could be quite high.
Finally, the reader should note that to numerically observe
these resultsโ€”and successfully implement the remedies that
we will see later onโ€”the vanishing of all relevant integrals
must be explicitly enforced in some way, because ๏ฌ‚oating
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๏ฌcant digits that plagues the EFIE, since the loop and star
contributions of each entity are stored in separate ๏ฌ‚oating point
numbers. In the case of the loop-star approach, the stabilized
matrix system is
D๐‘˜TLS
๐‘˜D๐‘˜jDLS =D๐‘˜๎€‚ฮ› ฮฃ๎€ƒTei,(61)
where j=๎€‚ฮ› ฮฃ๎€ƒD๐‘˜jDLS, 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 ๏ฌeld 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 =๎€‚ฮ›0๎€ƒD๐‘˜jDLS and jsol =๎€‚0ฮฃ๎€ƒD๐‘˜jDLS,
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๏ฌcance and would further compromise
the accuracy of the ๏ฌelds.
The key di๏ฌ€erence 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๏ฌne 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๏ฌned as the subset
๎€ˆ๐œฝ๐‘—๎€‰of 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› ฮฃ๎€ƒis replaced
by an RWG-to-loop-tree mapping ๎€‚ฮ› ฮ˜๎€ƒand TLS
๐‘˜becomes
TLT
๐‘˜B๎€‚ฮ› ฮ˜๎€ƒTT๐‘˜๎€‚ฮ› ฮ˜๎€ƒwhere
[ฮ˜]๐‘– ๐‘— =๎€จ1if ๐’‡๐‘–=๐œฝ๐‘—
0otherwise, (62)
is the general term of the RWG-to-tree transformation matrix.
The resulting preconditioned equation is
D๐‘˜TLT
๐‘˜D๐‘˜jDLT =D๐‘˜๎€‚ฮ› ฮ˜๎€ƒTei,(63)
where j=๎€‚ฮ› ฮ˜๎€ƒD๐‘˜jDLT.
At ๏ฌrst glance, the computational overhead of the two meth-
ods seems low, since ฮ›,ฮฃ,ฮ˜, and D๐‘˜are 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๏ฌcially 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
ฮ›and 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๏ฌ€ect of
this ill-conditioning on the preconditioned EFIE matrix is
out of the scope of this review, pseudo-di๏ฌ€erential operator
theory can be used to show that the di๏ฌ€erential strength of
the loop-star transformation operators is su๏ฌƒciently high not
to be compensated by that of the vector potential. To illustrate
this adverse e๏ฌ€ect, 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
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <9
10โˆ’44 10โˆ’33 10โˆ’22 10โˆ’11 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 โ€œLoop-tree EFIEโ€ and โ€œP-
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๏ฌcantly 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 ๏ฌ‚at spectrum that, di๏ฌ€erently 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=ฮฃj๐œฎ+ฮ›j๐œฆ+Hj๐‘ฏ,(64)
the quasi-Helmholtz projector for the non-solenoidal part is
the operator that maps jinto ฮฃj๐œฎ. Since
ฮฃTj=ฮฃTฮฃj๐œฎ,(65)
the looked for projector is
PฮฃBฮฃ(ฮฃTฮฃ)+ฮฃT,(66)
where +denotes the Mooreโ€“Penrose pseudoinverse. The pro-
jector for the solenoidal plus harmonic components can be
obtained out of complementarity as
Pฮ›Hโ‰”Iโˆ’Pฮฃ.(67)
The same reasoning for dual functions leads to the dual
de๏ฌnitions of the projector
Pฮ›Bฮ›(ฮ›Tฮ›)+ฮ›T(68)
which is the non-solenoidal projector for dual functions. The
solenoidal plus harmonic projector for dual functions is, again,
obtained by complementarity as
PฮฃHโ‰”Iโˆ’Pฮ›.(69)
It is important to note that, even though the projectors pre-
sented so far include a pseudo-inverse in their de๏ฌnition, 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๏ฌ€ects 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
PB๐›ผPฮ›H+๐›ฝPฮฃ,(70)
where, following a frequency analysis similar to the one used
for loop-star/tree decompositions, an optimal coe๏ฌƒcient choice
can be found to be ๐›ผโˆ๐‘˜โˆ’1
2and ๐›ฝโˆ๐‘˜1
2, that is,
P๐‘˜B๎ฐ๐ถ/๐‘˜Pฮ›H+i/โˆš๐ถ๐‘˜ Pฮฃ,(71)
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <10
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‘“=10โˆ’20 Hz, computedwith di๏ฌ€erent formulations.
resulting in a new system of equations
P๐‘˜T๐‘˜P๐‘˜y=P๐‘˜ei,(72)
where P๐‘˜y=j. The constant ๐ถcan 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
๐ถโ‰”๎ณkTฮฆ,๐‘˜ k
kPฮ›HTA,๐‘˜ Pฮ›Hk.(73)
The analysis of the conditioning e๏ฌ€ect 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
๐‘˜โ†’0P๐‘˜T๐‘˜P๐‘˜โ†’i๐ถPฮ›HTA,0Pฮ›H+i/๐ถTฮฆ,0,(74)
where we used that PฮฃTฮฆ,๐‘˜ Pฮฃ=Tฮฆ, ๐‘˜ and Pฮ›Tฮฆ,๐‘˜ =Tฮฆ, ๐‘˜ Pฮ›=
0and thus
lim
๐‘˜โ†’0cond (P๐‘˜T๐‘˜P๐‘˜)=๐›พ , (75)
where ๐›พis 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๏ฌ€ects 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๏ฌ€ective 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 ๐‘“=10โˆ’20 Hz, computed with di๏ฌ€erent 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 ๏ฌeld and static charge extraction, which includes
a boundary integral equation for the normal component of
magnetic ๏ฌeld [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๏ฌ€erent 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๏ฌƒces to ๏ฌnd 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
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <11
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 ๏ฌeld computation
As we have already mentioned, a well-conditioned dis-
cretization alone is not su๏ฌƒcient to accurately compute j: the
right-hand side su๏ฌ€ers typically from numerical inaccuracies
due to ๏ฌnite integration precision and from round-o๏ฌ€ errors.
The main reason for this is that the quasi-Helmholtz compo-
nents scale di๏ฌ€erently 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 ei๐‘˜ห†
๐’Œยท๐’“with ei๐‘˜ห†
๐’Œยท๐’“โˆ’1,
where ห†
๐’Œdenotes the direction of propagation. Then
๎ฐ๐ถ/๐‘˜Pฮ›Heextracted +iโˆš๐ถ๐‘˜ 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 ei๐‘˜ห†
๐’Œยท๐’“โˆ’1should be
replaced by a Taylor series, where the static part is omitted.
Similarly, the far-๏ฌeld cannot be computed by simply eval-
uating ๎‚น๐›ค
๐‘
๎ƒ•
๐‘›=1[j]๐‘›๐’‡๐‘›(๐’“0)eโˆ’i๐‘˜ห†
๐’“ยท๐‘Ÿ0d๐‘†(๐’“0),(77)
where j=P๐‘˜yfrom (72), ห†
๐’“=๐’“/|๐’“|. On the one hand,
by computing the unknown vector of the unpreconditioned
formulation j=P๐‘˜y, the di๏ฌ€erent asymptotic behavior
in ๐‘˜of 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 ๏ฌnite machine
precision. Thus for the ๏ฌeld computation, one should keep
the unpreconditioned components of jseparately, that is,
jsol-qhar =๎ฐ๐ถ/๐‘˜Pฮ›Hyand jnsol =iโˆš๐ถ๐‘˜ Pฮฃy. On the other
hand, it has been pointed out that also the far-๏ฌeld computation
su๏ฌ€ers from round-o๏ฌ€ errors [88]. To avoid these, we compute
the far-๏ฌeld in two steps: we compute the contribution of jnsol
to the far-๏ฌeld by evaluating
๐‘ฌfar
nsol (๐’“)=
๐‘
๎ƒ•
๐‘›=1[jnsol]๐‘›๎‚น๐›ค
๐’‡๐‘›(๐’“0)eโˆ’i๐‘˜ห†
๐’“ยท๐’“0d๐‘†(๐’“0)(78)
and the contribution of jsol-qhar by
๐‘ฌfar
sol-qhar (๐’“)=
๐‘
๎ƒ•
๐‘›=1๎€‚jsol-qhar๎€ƒ๐‘›๎‚น๐›ค
๐’‡๐‘›(๐’“0)๎€eโˆ’i๐‘˜ห†
๐’“ยท๐’“0โˆ’1๎€‘d๐‘†(๐’“0),
(79)
where a Taylor-series expansion should be used for small
arguments of the exponential; then
๐‘ฌfar (๐’“)=๐‘ฌfar
nsol (๐’“) + ๐‘ฌfar
sol-qhar (๐’“)(80)
Also for the near-๏ฌeld computation, the separation in jsol-qhar
and in jnsol must be maintained, the static contribution removed
from the Greenโ€™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 Greenโ€™s function kernel,
no explicit dependency on ๐‘˜and should thus be expected to
remain well-conditioned in frequency for ๐‘˜โ†’0. Indeed, for
simply-connected geometries ๐›ค, we have cond(M๐‘˜)=O(1)
when ๐‘˜โ†’0. 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‘˜)โ‰ฅ๐ถ/๐‘˜2for some constant ๐ถโˆˆR+.
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 ๐›บโˆ’has genus ๐‘”, then the space ๐ป๐‘ฏ(๐›บโˆ’)de๏ฌned by the
harmonic functions in ๐›บโˆ’and the space ๐ป๐‘ฏ(๐›บ+)de๏ฌned by
the harmonic functions in ๐›บ+have both dimension ๐‘”. The
space de๏ฌned by ๐ปห†
๐‘ฏP(๐›ค)โ‰”ห†
๐’ร—๐ป๐‘ฏ(๐›บโˆ’) |๐›คare the poloidal
loops and the space de๏ฌned by ๐ปห†
๐‘ฏT(๐›ค)โ‰”ห†
๐’ร—๐ป๐‘ฏ(๐›บ+) |๐›ค
are the toroidal loops [92], [93]. ๐ปห†
๐‘ฏP(๐›ค)has been show [89]
to be the nullspace of Mโˆ’
0and ๐ปห†
๐‘ฏT(๐›ค)the nullspace of M+
0
and that ห†
๐’ร—ห†
๐‘ฏT๐‘›โˆˆ๐‘ฏห†
๐‘ฏP(๐›ค)and ห†
๐’ร—ห†
๐‘ฏP๐‘›โˆˆ๐‘ฏห†
๐‘ฏT(๐›ค).
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 ๏ฌnd linear combinations that are in
๐ปห†
๐‘ฏP(๐›ค)or in ๐ปห†
๐‘ฏT(๐›ค), a consequence of the fact that these
functions are not curl-conforming, as mentioned in Section IV.
On the other hand, quasi-harmonic functions ๐‘ฏ๐‘›(and dual
quasi-harmonic functions ๎ฅ
๐‘ฏ๐‘›, respectively) are associated with
the holes and handles of the geometry. They are not, unlike the
locally de๏ฌned loop functions ๐œฆ๐‘›, derived from a continuous
scalar potential on ๐›ค. Together with the fact that ๐‘ฏ๐‘›and
๎ฅ
๐‘ฏ๐‘›are 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 {๐‘ฏ๐‘›}2๐‘”
๐‘›=1can be rearranged into two bases,
where one basis {๐‘ฏT๐‘›}๐‘”
๐‘›=1is orthogonal to poloidal loops and
the other basis {๐‘ฏP๐‘›}๐‘”
๐‘›=1is orthogonal to toroidal loops. In the
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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <12
following, ห†
๐‘ฏT๐‘›โˆˆ๐ปห†
๐‘ฏT(๐›ค)denote the harmonic toroidal and
ห†
๐‘ฏP๐‘›โˆˆ๐ปห†
๐‘ฏP(๐›ค)denote the harmonic poloidal basis functions
on ๐›ค, while ๐‘ฏT๐‘›,๐‘ฏP๐‘›โˆˆ๐‘‹๐’‡and ๎ฅ
๐‘ฏT๐‘›,๎ฅ
๐‘ฏP๐‘›โˆˆ๐‘‹๎ฅ
๐’‡are their
quasi-harmonic counterparts.
In [91], scalings were reported of the blocks of the system
matrix of M+
๐‘˜in terms of a quasi-Helmholtz decomposition
๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ
๐œฆ ๐œฎ ๐‘ฏT๐‘ฏP
ห†
๐’ร—๎ฅ
๐œฎO(๐‘˜2) O(1) O(๐‘˜2) O(๐‘˜2)
ห†
๐’ร—๎ฅ
๐œฆO(1) O(1) O(1) O(1)
ห†
๐’ร—๎ฅ
๐‘ฏPO(๐‘˜2) O(1) O(๐‘˜2) O(๐‘˜2)
ห†
๐’ร—๎ฅ
๐‘ฏTO(๐‘˜2) O(1) O(๐‘˜2) O(1)
๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป
.(81)
An analogous result can be obtained for Mโˆ’
๐‘˜with 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 ๐‘˜โ†’0of the block
matrices, we start by considering the Taylor series of the
Greenโ€™s function kernel of the MFIE
โˆ‡๐บ(๐’“,๐’“0)=๐‘น
4ฯ€๐‘…3(i๐‘˜ ๐‘… โˆ’1)ei๐‘˜ ๐‘… (82)
=๐‘น
4ฯ€๐‘…3๎€’โˆ’1+1
2(i๐‘˜)2๐‘…2+1
3(i๐‘˜)3๐‘…3+. . .๎€“,(83)
where ๐‘…=|๐’“โˆ’๐’“0|and ๐‘น=๐’“โˆ’๐’“0. An O(๐‘˜2)-scaling is
observed for a block matrix if the contribution due to the
static term in the Taylor series vanishes. For ๐œฎ๐‘›as expansion
function or ห†
๐’ร—๎ฅ
๐œฆas 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 ๐’‹๐›คis either solenoidal or harmonic.
In this case, we have
hห†
๐’ร—๎ฅ
๐œฎ๐‘š,Mยฑ
0๐œฆ๐‘›i๐›ค=0,(84)
hห†
๐’ร—๎ฅ
๐œฎ๐‘š,Mยฑ
0๐‘ฏT๐‘›i๐›ค=0,(85)
hห†
๐’ร—๎ฅ
๐œฎ๐‘š,Mยฑ
0๐‘ฏP๐‘›i๐›ค=0.(86)
This can be seen by considering that a solenoidal function ๐œฆ
has a corresponding scalar potential
๐œฆ=curl๐›ค๐›ท . (87)
Furthermore, we have the equality
hห†
๐’ร—๐œฆ,Mยฑ
0๐’‹๐›คi๐›ค=โˆ’h๐œฆ,ห†
๐’ร—Mยฑ
0๐’‹๐›คi๐›ค.(88)
Inserting (87) for the testing function in the right-hand side
of (88) and using the fact that curl๐›คis the adjoint operator of
curl๐›ค, we obtain
hห†
๐’ร—๐œฆ,Mยฑ
0๐’‹๐›คi๐›ค=h๐›ท, curl๐›คห†
๐’ร— (Mยฑ
0๐’‹๐›ค)i๐›ค
=โˆ’h๐›ท, curl๐›ค๐’‰โˆ“
Ti๐›ค,(89)
where ๐’‰โˆ“
Tโ‰”ห†
๐’ร—(Mยฑ
0๐’‹๐›ค)is the (rotated) tangential component
of the magnetic ๏ฌeld
๐’‰(๐’“)=curl ๎‚น๐›ค
๐บ๐‘˜(๐’“,๐’“0)๐’‹๐›ค(๐’“0)d๐‘†(๐’“0)for ๐’“โˆˆ๐›บโˆ’,(90)
that is, ๐’‰โˆ’
Tis the (rotated) tangential component of ๐’‰when ๐›คis
approached from within ๐›บโˆ’, and ๐’‰+
Twhen ๐›คis approach from
within ๐›บ+. We recall that (90) is obtained by ๏ฌnding a vector
potential ๐’‚such that ๐’‰=curl ๐’‚and noting that curl curl ๐’‚=๐’‹๐›ค
under the assumption that div ๐’‹๐›ค=0[40, see Chapter 6.1].
From
curl๐›ค๎€€๐’—ยฑ
T(๐’“)๎€=lim
๐›บยฑ3๐’“0โ†’๐’“๎€curl ๎€€๐’—(๐’“0)๎€๎€‘ยทห†
๐’(๐’“0).(91)
and from [40, (6.17)]
curl ๐’—={curl ๐’—}+ห†
๐’ร—๎€€๐’—+โˆ’๐’—โˆ’๎€ฮด๐›ค,(92)
where curly braces {} mean that this part is evaluated only in
๐›บยฑand ฮด๐›คis the surface Dirac delta function, together with
(90), we have
curl๐›ค๐’‰โˆ“
T=curl ๐’‰ยทห†
๐’=0.(93)
Next we will establish that hห†
๐’ร—๐‘ฏP๐‘š,Mยฑ
0๐œฆ๐‘›i๐›ค=0and
hห†
๐’ร—๐‘ฏT๐‘š,Mยฑ
0๐œฆ๐‘›i๐›ค=0. First, note that the exterior MFIE
operator has the mapping properties that [89]
M+
0ห†
๐‘ฏT๐‘›=0(94)
and
M+
0ห†
๐‘ฏP๐‘›=ห†
๐‘ฏP๐‘›,(95)
while for the interior MFIE operator, we have the mappings
Mโˆ’
0ห†
๐‘ฏP๐‘›=0(96)
and
Mโˆ’
0ห†
๐‘ฏT๐‘›=ห†
๐‘ฏT๐‘›.(97)
Furthermore, for any two surface functions ๐’‡,๐’ˆ, we have [89,
Section 5]
h๐’ˆ,K๐’‡i๐›ค=hโˆ’ห†
๐’ร—๎€K๎€€๐’ˆร—ห†
๐’๎€๎€‘,๐’‡i๐›ค
=hโˆ’K๎€€๐’ˆร—ห†
๐’๎€,๐’‡ร—ห†
๐’i๐›ค(98)
Then we ๏ฌnd
hห†
๐’ร—ห†
๐‘ฏT๐‘›,M+
0๐œฆi๐›ค=hMโˆ’
0ห†
๐‘ฏT๐‘›,๐œฆร—ห†
๐’i๐›ค
(97)
=0,(99)
hห†
๐’ร—ห†
๐‘ฏP๐‘›,M+
0๐œฆi๐›ค=hMโˆ’
0ห†
๐‘ฏP๐‘›,๐œฆร—ห†
๐’i๐›ค
(96)
=0,(100)
hห†
๐’ร—ห†
๐‘ฏT๐‘›,Mโˆ’
0๐œฆi๐›ค=hM+
0ห†
๐‘ฏT๐‘›,๐œฆร—ห†
๐’i๐›ค
(94)
=0,(101)
hห†
๐’ร—ห†
๐‘ฏP๐‘›,Mโˆ’
0๐œฆi๐›ค=hM+
0ห†
๐‘ฏP๐‘›,๐œฆร—ห†
๐’i๐›ค
(95)
=0,(102)
where we used (94)-(97) and the orthogonality of harmonic
and irrotational functions ห†
๐’ร—๐œฆ. Now consider that ๎ฅ
๐‘ฏP๐‘›=
ห†
๐‘ฏP๐‘›+๎ฅ
๐œฎP๐‘›and ๎ฅ
๐‘ฏT๐‘›=ห†
๐‘ฏT๐‘›+๎ฅ
๐œฎT๐‘›, where ๎ฅ
๐œฎP/T๐‘›is the respective
perturbation. Thus by taking into account (84), we have
hห†
๐’ร—๎ฅ
๐‘ฏP๐‘š,Mยฑ
0๐œฆ๐‘›i๐›ค=0,(103)
hห†
๐’ร—๎ฅ
๐‘ฏT๐‘š,Mยฑ
0๐œฆ๐‘›i๐›ค=0.(104)
From now on, we will only consider the harmonic functions
ห†
๐‘ฏT๐‘›and ห†
๐‘ฏP๐‘›instead of their quasi-harmonic counterpart
since, as we have seen, the solenoidal pertubation will always
vanish. Then for M+
0, we have
hห†
๐’ร—ห†
๐‘ฏT๐‘š,M+
0ห†
๐‘ฏT๐‘›i๐›ค=hห†
๐’ร—ห†
๐‘ฏP๐‘š,M+
0ห†
๐‘ฏT๐‘›i๐›ค
(94)
=0,(105)
hห†
๐’ร—ห†
๐‘ฏT๐‘š,Mโˆ’
0ห†
๐‘ฏP๐‘›i๐›ค=hห†
๐’ร—ห†
๐‘ฏP๐‘š,Mโˆ’
0ห†
๐‘ฏP๐‘›i๐›ค
(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
>REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER <13
due to the nullspace. Finally, we have
hห†
๐’ร—ห†
๐‘ฏP๐‘š,M+
0ห†
๐‘ฏP๐‘›i๐›ค
(95)
=hห†
๐’ร—ห†
๐‘ฏP๐‘š,ห†
๐‘ฏP๐‘›i๐›ค=0,(107)
as ห†
๐’ร—ห†
๐‘ฏP๐‘šโˆˆ๐ปห†
๐‘ฏT(๐›ค)and this space is orthogonal to ๐ปห†
๐‘ฏP(๐›ค).
Likewise, we can conclude that there is at least one ๐‘šsuch
that
hห†
๐’ร—ห†
๐‘ฏT๐‘š,M+
0ห†
๐‘ฏP๐‘›i๐›คโ‰ 0.(108)
Analogously, we obtain
hห†
๐’ร—ห†
๐‘ฏP๐‘š,Mโˆ’
0ห†
๐‘ฏT๐‘›