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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-

Journal of Antennas and Propagation

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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)

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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