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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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