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On the Effectiveness of Quasi-Helmholtz Projectors

in Preconditioning Problems with Junctions

Johann Bourhis(1), Andrien Merlini(2) , and Francesco P. Andriulli(1)

(1) Politecnico di Torino, Turin, Italy

(2) IMT Atlantique, Brest, France

Abstract—The electric ﬁeld integral equation (EFIE) is known

to suffer from ill-conditioning and numerical instabilities at low

frequencies (low-frequency breakdown). A common approach to

solve this problem is to rely on the loop and star decomposition of

the unknowns. Unfortunatelly, building the loops is challenging

in many applications, especially in the presence of junctions.

In this work, we investigate the effectiveness of quasi-Helmholtz

projector approaches in problems containing junctions for curing

the low-frequency breakdown without detecting the global loops.

Our study suggests that the performance of the algorithms

required to obtain the projectors in the presence of junctions

is maintained while keeping constant the number of sheets per

junction. Finally, with a sequence of numerical tests, this work

shows the practical impact of the technique and its applicability

to real case scenarios.

I. INTRODUCTION

The electric ﬁeld integral equation (EFIE) provides the

solution of electromagnetic scattering and radiation problems

for perfectly electrically conducting (PEC) objects. At low-

frequency, the unbalanced coefﬁcients scaling of the EFIE lead

to ill-conditioned linear systems and to numerical instabili-

ties [1]. This is the so-called low-frequency breakdown and it

is standardly solved using quasi-Helmholtz approaches such

as loop and star decomposition [1] where the unknowns are

separated into their solenoidal and non-solenoidal parts. These

decompositions require to search for global loops, a challenge

even when junctions are not present. This search however,

is substantially more complicated when junctions are present,

since the number of global cycles grows fast [2].

In this work we analyse and extend the use of quasi-

Helmholtz projectors to low frequency EFIE problems con-

taining junctions. We ﬁrst show that the orthogonality between

the solenoidal and non-solenoidal projectors is not compro-

mised by the presence of junctions. Hence, the non-solenoidal

projector obtained in the case of junctions gives rise by com-

plementarity to the correct solenoidal space. A computation

of the projectors using algebraic multigrid (AGMG) [3] is

analyzed numerically. Our results show that the efﬁciency of

this algorithm is not jeopardized when the number of junctions

grows, provided that the number of insertions per junction

remains constant. This is in agreement with the spectral

properties of graph Laplacians [4]. Finally, we give numerical

results to illustrate the effectiveness of the proposed approach

and its applicability to relevant scenarios.

II. BACKGROU ND A ND NOTATI ON S

We want to determine the electric ﬁeld Esct scattered by

a perfect electric conducting (PEC) boundary Γon which

impinges a ﬁeld Einc in a medium of characteristic impedance

η. We assume the ﬁelds to be time-harmonic with wavenum-

ber κ. The EFIE is derived from Maxwell’s equations con-

sidering a surfacic electric current density Jinduced over

Γfrom which Esct can be computed outside Γ. Follow-

ing the boundary element method, we discretize the current

J(r)≈PN

i=1 jiφi(r), where φiare Rao-Wilton-Glisson

(RWG) basis functions deﬁned on a triangular mesh. After

testing, we obtain a N×Nlinear system

iκZs+1

iκZhj=e,(1)

where Zsij =⟨n×φi,TA,κφj⟩,Zhij =⟨n×

φi,Tϕ,κφj⟩and e]i=−1

η⟨n×φi,n×Einc⟩, with ⟨f , g⟩=

RΓf(r)g(r) dS(r). The operators TA,κ and Tϕ,κ are the rotated

vector and scalar potentials on the surface [1] and nis the

oriented unit normal over Γ. On edges that form a junction

between ntriangles, we deﬁne n−1independent RWGs on

n−1couples of triangles as is often done [2].

The loop and star basis functions can be mapped from the

RWGs by constructing transformation matrices [1]. Even if

the matrix mapping the local loops can always be deﬁned by

association to the vertices of the mesh, its construction requires

particular attention with junctions and open boundaries. By

analogy to the RWGs, we have to eliminate the external nodes

and to deﬁne additional loops for each plan formed around

a junction. Furthermore, it is known that the seeking of the

global loops often represents a substantial computational cost.

Unfortunately, the presence of multiple junctions can increase

the number of “non-harmonic” global cycles (see the structures

in Fig. 1 and Fig. 2) that being associated to an open boundary

instead of a point still need to be detected.

III. QUAS I-HE LM HO LTZ PRO JE CTORS WITH JUNCTIONS

The transformation matrix which maps the stars into the

RWG space can be deﬁned in the case of junctions like in the

standard case as

[Σ]ij =

1if ∇ · φi(r)>0on the triangle j,

−1if ∇ · φi(r)<0on the triangle j,

0otherwise.

(2)

With these functions, the orthogonality between the solenoidal

and non-solenoidal spaces is preserved by construction and

their completeness can equally be shown. This can be seen

considering the graph associated to Σ. This graph represents

the connexions between the triangles (vertices) formed by the

DOI: 10.1109/AP-S/USNC-URSI47032.2022.9886184 — © 2022 IEEE. Personal use of this material is permitted.

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TABLE I

NUMBER OF ITERATIONS ACHIEVED BY AGM G FOR INCREASING

DISCRETIZATION WITH A RESIDUAL ERROR BELOW 1·10−8.

Structure Unknowns Iter. Structure Unknowns Iter.

7.2·10321 2.9·10318

6.9·10422 2.8·10524

7.0·10522 2.7·10622

6.9·10623 2.8·10723

1.2·10318 3.2·10319

1.2·10522 3.0·10522

1.2·10623 3.0·10623

1.2·10723 3.0·10725

RWGs (edges). The vertices (mesh triangles) and the faces

(internal mesh nodes and cycles along open boundaries) are

respectively associated to the stars and to the loops (this is

a dual graph). We denote by N,S, and Lthe number of

edges, vertices and faces of this graph. To work with the same

general formula for open structures, the exterior domain is

counted as one face and does not correspond to any loop. For

closed structures, we also have to account for the Hhandles

that correspond to the discrete harmonic subspace. The Euler-

Poincar´

e formula establishes N=S+L+2(H−1). Moreover,

the number of independent stars is always S−1which impacts

the rank of Σand the dimension of its orthogonal space L−

1+2Hwhich is the right number of independent loops.

The orthogonality between the quasi-Helmholtz spaces

eliminates the need for seeking loops and cycles. In fact,

the quasi-Helmholtz projectors of the non-solenoidal and

solenoidal spaces are respectively calculated as PΣ=

ΣΣTΣ+ΣTand PΛH =I−PΣ, where “+” denotes

the Moore-Penrose pseudoinverse. Subsequently, we cure the

low-frequency breakdown by deﬁning a preconditioner P=

1

√κPΛH + i√κPΣwhich yields to a system with a bounded

condition number at low-frequencies

PiκZs+1

iκZhPy=Pe,with j=Py.(3)

IV. GRA PH LAPLACIANS WITH JUNCTIONS

Preconditioning the EFIE with the quasi-Helmholtz projec-

tors requires the iterative inversion of the graph Laplacian

ΣTΣ. This can be efﬁciently done via AGMG. In this work,

we have numerically veriﬁed that AGMG remains accurate

with a complexity in ONlog(N)operations even for really

dense discretized problems containing several junctions. This

is illustrated in Table I which shows that the number of

iterations of the AGMG process increases logarithmically

with the number of unknowns when the maximal number

of sheets per junction remains constant. It should be noted,

however, that the largest eigenvalue of ΣTΣincreases with

the maximal degree of the graph, that is the maximal number

of interconnected triangles [4]. As a consequence, AGMG may

fail for structures containing an increasing number of sheets

per junction, while performance does not deteriorate when this

number remains constant.

10 210 310 410 510 610 710 8

Frequency (Hz)

10 0

10 4

10 8

10 12

Condition Number

This work

Loop-Star

Standard EFIE

1m

Fig. 1. Comparison of the condition number of the EFIE matrix with or

without preconditioning from medium to low frequency.

1m

This work

Loop-Star

Standard EFIE

0/4 /2 3 /4

(rad)

-220

-190

-160

-130

RCS (dBsm)

Fig. 2. Radar Cross Section (RCS) obtained with and without preconditioning

with a frequency of 1·104Hz. The structure is irradiated by a plane wave.

V. NUMERICAL RESULTS

In our numerical results, we have used two models of

structures containing different types and numbers of junctions.

Results obtained by solving the EFIE with and without pre-

conditioning from medium to low frequencies are compared

along with the loop-star decomposed EFIE [1]. The condition

number of the linear systems as a function of the frequency

is reported in Fig. 1. The preconditioned matrices clearly

remain well-conditioned when the frequency decreases, while

the condition number drastically increases for the standard for-

mulation. In addition, the quasi-Helmholtz projectors show a

better conditioning than the loop-star decomposition thanks to

their ﬂat spectrum. Fig. 2 shows that the proposed formulation

and the loop-star formulation yield comparable results at low-

frequency, unlike the standard formulation.

ACKNOWLEDGMENT

This work was supported in part by the European Research

Council (ERC) through the European Union’s Horizon 2020

Research and Innovation Programme under Grant 724846

(Project 321) and in part by the H2020-MSCA-ITN-EID

project COMPETE GA No 955476.

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