Handling the Low-frequency Breakdown of the
PMCHWT Integral Equation with the
quasi-Helmholtz Projectors
Yves Beghein∗
Rajendra Mitharwal
†
Abstract — This contribution presents a quasiHelmholtz projectors based regularization of the
low frequency breakdown of the PMCHWT integral equation. The PMCHWT equation in the lowfrequency regime shows an ill-conditioned behavior
inherited from the Electric Field Integral Operators
it contains. The stabilization via quasi-Helmholtz
projectors, differently from the use of standard
Loop-Star/Tree decompositions, does not introduce
an additional mesh-size-related ill-conditioning and
it applies smoothly to both simply and non-simply
connected geometries. The presentation of the main
formulation will be complemented by numerical results demonstrating the effectiveness and accuracy
of the proposed scheme.
1
INTRODUCTION
The
Poggio-Miller-Chan-Harrington-Wu-Tsai
(PMCHWT) integral equation is one of the main
formulations for solving radiation and scattering
problems for penetrable objects in the frequency
domain [1].
The equation is of the first kind and, similarly to the electric field integral equation (EFIE)
[2], it suffers from severe ill-conditioning problems
when the simulation frequency is low. This is
the well-known “low-frequency” breakdown phenomenon [3, 4]. Historically, the breakdown can
be solved by using quasi-Helmholtz decompositions
such as Loop-Star and Loop-Tree basis functions [5,
6, 7, 8]. These decompositions however, although
solving the frequency-related ill-conditioning, introduce a parasitic and mesh-dependent form of illconditioning [9] since they can be interpreted as
graph counterparts of differential operators [10].
Recently, a solution to this problem has been introduced for the EFIE based on newly introduced
quasi-Helmholtz projectors [11]. These projectors
realize a quasi-Helmholtz decomposition, but in an
implicit and well–conditioned way, so that the un-
Kristof Cools
‡
Francesco P. Andriulli
†
desired mesh-dependent ill-conditioning generated
by standard approaches does not occur. The approach has been also successfully exploited in time
domain [12].
This work generalizes this strategy to the treatment of the low-frequency breakdown of the PMCHWT integral equation. The generalization is
challenging since, differently from the EFIE, the offdiagonal blocks of the PMCHWT requires a suitable analysis and treatment to ensure low-frequency
stability.
Numerical results corroborates the theory and
shows the practical applicability of the new approach.
2
BACKGROUND AND NOTATION
Consider a dielectric region Ω of intrinsic
impedance ηk with its external region Ω/R3 of intrinsic impedance ηk . We denote with Γ = ∂Ω its
boundary surface with outward directed normal n̂.
The Electric Field Integral Operator (EFIO) and
Magnetic Field Integral Operator (MFIO) read
eiα|r−r |
Tα (f (r)) = n̂ × iα
f (r )dΓ
Γ 4π|r − r |
1
eiα|r−r | ∇ · f (r )dΓ,
− n̂ × ∇
iα
Γ 4π|r − r |
(1)
Kα (f (r)) = −n̂ ×
Γ
∇
eiα|r−r |
× f (r )dΓ, (2)
4π|r − r |
where the subscript α represents the wave number k or k for the region Ω/R3 or Ω respectively.
The PMCHWT equation models scattering from
the penetrable region Ω when illuminated by an
incident electromagnetic field (E i (r), H i (r)). It
reads
∗ Department
of
Information
Technology
(INM (r)
Tk /ηk + Tk /ηk
−(Kk + Kk )
TEC), Ghent University, Ghent, Belgium;
e-mail:
(Kk + Kk )
(ηk Tk + ηk Tk ) J(r)
yves.beghein@ugent.be
† CERL
Laboratory,
Télécom Bretagne / In−n̂ × H i (r)
=
, (3)
stitut
Mines-Télécom,
Brest,
France;
e-mail:
−n̂ × E i (r)
francesco.andriulli@mines-telecom.fr
‡ Electrical Systems and Optics Research Division,
University of Nottingham, Nottingham, U.K.; e-mail:
kristof.cools@nottingham.ac.uk
where J (r) and M (r) represent the electric and
magnetic currents, respectively.
978-1-4799-7806-9/15/$31.00 ©2015 IEEE
1534
definition of these matrices for space limitations,
the reader is invited to refer to [11] for an exhaustive definition) as PΣ = Σ(ΣΣT )+ ΣT and
PΛH = I − Σ(ΣΣT )+ ΣT on the surface mesh
and its dual counterparts PΛ = Λ(ΛΛT )+ ΛT and
n=1
N
PΣH = I − Λ(ΛΛT )+ ΛT on the dual mesh [11].
βn f (r). The overall discretized PMCHWT The decomposition operator matrix is defined as
√
n=1
M = PΛH √1k + iPΣ k. Similarly, the associequation is given by
matrix is de ated dual decomposition operator
√
Λ √1
ΣH
h
β
−(Kk + Kk )
Tk /ηk + Tk /ηk
fined
as
M
=
P
+
iP
k.
The
associated
=
k
e
(Kk + Kk )
(ηk Tk + ηk Tk ) α
Gram matrix linking the basis functions on the pri(4) mal and the dual mesh is defined as [G ]
mix mn =
where,
element-wise,
(Tk )mn
= BC
BC
f m (r), n̂r × f n (r) where {f n (r)}N
n=1 is the
n̂ × f m (r), Tk (f n (r))Γ using the notation
Γ
a · bχ = χ a · b dχ (Tk , Kk and Kk are set of Buffa-Christiansen (BC) basis functions [13]
similarly defined). The right hand side (RHS) defined on the dual mesh.
The proposed regularization based on the quasivectors are given by hm = n̂ × f m (r), H i (r)Γ
Helmholtz
projectors for the EFIO blocks in the
and em = n̂ × f m (r), E i (r)Γ . The EFIO blocks
equation
(7)
reads
on the diagonal of the system matrix in equation
(4) suffer from the low-frequency breakdown, a
x
Λ −1
x
ΛH
x Σ
MT G−1
− PΣH G−1
mix T M = (P Gmix TA P
mix Tφ P )
problem which is consequently transferred to the
x
Σ
ΣH −1
+ ik(PΛ G−1
Gmix TxA PΛH )
entire PMCHWT equation.
mix TA P + P
On each edge of the triangular surface mesh approximating the surface Γ, an RWG basis function f n (r) is defined [2]. This set approximates
N
the currents as J (r) =
αn f n (r) and M (r) =
3
PROPOSED REGULARIZATION
x
Σ
− k 2 PΣH G−1
mix TA P
(8)
To properly regularize the effect of low frequency
where the superscript x can be either a or b debreakdown in equation (4), it is useful to separate
pending on the diagonal block matrix. In order
the different contributions in the EFIO
to simplify the above expression, we have used
x
1 A
Txφ PΛH = 0 and PΛ G−1
a
A
mix Tφ = 0. Along the same
TA = (Tk /ηk + Tk /ηk )
lines, the proposed regularization for the MFIO
k
blocks is
Taφ = k(Tφk /ηk + Tφk /ηk )
(5)
1
1 Λ −1
Σ
ηk + TA
TbA = (TA
MT G−1
P Gmix Ke PΛH + i(PΛ G−1
k ηk )
mix KM =
mix Kz P
k k
k
Σ
ΣH −1
+ PΛ G−1
Gmix Kz PΛH
Tbφ = k(Tφk ηk + Tφk ηk )
mix Ke P + P
where the subscripts φ and A represent the scalar
and vector potential term respectively (the first and
second term on the RHS of equation (1)). The
MFIO matrix should also be decomposed as
ΛH
Σ
+ PΣH G−1
) − k(PΣH G−1
mix Ke P
mix Ke P
Σ
+ PΣH G−1
mix Kz P )
(9)
where we have used the property of static MFIE
ΛH
= 0.
operator matrix as PΛ G−1
mix Kz P
(6)
The
proposed
quasi-Helmholtz
projected
PM
Ke = Kk + Kk − Kz
CHWT system, where the new unknown vectors
where K0 is the static MFIO matrix corresponding are α̃ = M−1 α and β̃ = M−1 β, reads
to k = 0. Using equation (5) and (6), the system
h
Z11 Z12 α̃
matrix in the equation (4) can be written as
= p
(10)
Z
Z
ep
β̃
21
22
a
1 a
kTA + k Tφ −(Ke + Kz )
.
(7)
1 a
a
where Z11 = MT G−1
kTbA + k1 Tbφ
Ke + Kz
mix (kTA + k Tφ )M, Z12 =
−1
T
−M Gmix (Ke + Kz ))M, Z21 = MT G−1
mix (Ke +
The objective behind these matrix decompositions K )M and
−1
1 b
T
b
Z
=
M
G
(kT
+
T
)M.
z
22
mix
A
k φ
is to have a low-frequency-stable behavior of the
conditioning of the overall system matrix after ap- 4 NUMERICAL RESULTS
plying the quasi-Helmholtz projectors. These projectors can be defined using the loop/star-to-RWGs The proposed regularization is applied to a scenario
transformation matrices Σ and Λ (we omit the involving scattering due to a homogeneous dielecKz = 2K0
1535
Condition number
θ
r|E |
tric sphere of the radius 1m with relative permittivity as r = 3. The electromagnetic source is a
plane wave travelling in the z direction and is polarf=1.0e−40Hz, εr=3.0
ized along the x axis with an electric field intensity
−95
10
of 1V/m. The condition number of the regularized system matrix stays constant when compared
−100
10
to the original PMCHWT matrix as shown in Figure 1 when the frequency decreases from 1MHz to
−105
10−40 Hz. The far field computed using the pro10
This work
posed regularization shows a good agreement with
Mie Series
the Mie series in Figure 2 and 3 for the frequency
−110
10
−40
0
50
100
150
200
1MHz and 10 Hz respectively.
Angles (Degrees)
In a second example, a torus model with the centerline radius of 3m and the cross-section radius of Figure 3: Sphere: Far field calculated at frequency
0.5m with relative permittivity as r = 3 is chosen. of 10−40 Hz.
The parameters for the electromagnetic source are
kept equal to that of the sphere. The condition
number of the regularized system matrix stays constant as seen in Figure 4 when the frequency is decreased till 10−40 Hz. The far field due to both the
original and regularized formulation shows a perfect
matching in Figure 5 for a frequency of 10MHz.
PMCHWT
This work
20
10
PMCHWT
This work
20
10
10
10
10
10
−40
10
−40
−20
10
0
10
10
Frequency (Hz)
20
10
20
Figure 4: Torus: Condition number w.r.t to frequency.
f=1.0e+06Hz, εr=3.0
0
10
This work
Mie Series
f=1.0e+07Hz, εr=3.0
0
10
−5
10
θ
r|E |
θ
0
10
Figure 1: Sphere: Condition number w.r.t to frequency.
r|E |
−20
10
10
Frequency (Hz)
−10
10
0
50
100
150
Angles (Degrees)
200
−5
Figure 2: Sphere: Far field calculated at frequency
of 1MHz.
5
CONCLUSION
10
0
This work
PMCHWT
50
100
150
Angles (Degrees)
200
Figure 5: Torus: Far field calculated at frequency
of 10MHz.
We have presented a quasi-Helmholtz projectors
based regularization for the system matrix arising
1536
from the discretization of the PMCHWT integral
equations. The proposed regularization ensures a
frequency independent behavior for the condition
number of the system matrix. Numerical results
demonstrate the accuracy and applicability of the
new scheme to both simply and non-simply connected geometries.
[8] G. Vecchi, “Loop-star decomposition of basis functions in the discretization of the efie,”
IEEE TAP, vol. 47, no. 2, pp. 339–346, 1999.
[9] F. P. Andriulli, “Loop-star and loop-tree decompositions: Analysis and efficient algorithms,” IEEE TAP, vol. 60, no. 5, 2012.
[10] F. P. Andriulli, A. Tabacco, and G. Vecchi,
“Solving the efie at low frequencies with a conditioning that grows only logarithmically with
This work was supported in part by the Agence Nathe number of unknowns,” IEEE TAP, vol. 58,
tionale de la Recherche under Project FASTEEGno. 5, 2010.
ANR-12-JS09-0010, in part by the European Union
Marie Curie Project NEUROIMAGEEG, in part by [11] F. Andriulli, K. Cools, I. Bogaert, and
E. Michielssen, “On a well-conditioned electric
the Brittany Region, France, in part by the HPC
field integral operator for multiply connected
resources through the GENCI-TGCC Project ungeometries,” IEEE TAP, vol. 61, no. 4, 2013.
der Grant 2015-gen6944, and in part by a doctoral
grant from the Agency for Innovation by Science
[12] Y. Beghein, K. Cools, and F. Andriulli, “A
and Technology in Flanders (IWT).
dc stable and large time step well-balanced
td-efie based on quasi-helmholtz projectors,”
References
IEEE TAP, 2015.
Acknowledgments
[1] W. C. Chew, E. Michielssen, J. Song, and [13] F. P. Andriulli, K. Cools, H. Bagci,
J. Jin, Fast and efficient algorithms in comF. Olyslager, A. Buffa, S. Christiansen, and
putational electromagnetics. Artech House,
E. Michielssen, “A multiplicative calderon preInc., 2001.
conditioner for the electric field integral equation,” IEEE TAP, vol. 56, no. 8, 2008.
[2] S. M. Rao, D. Wilton, and A. W. Glisson,
“Electromagnetic scattering by surfaces of arbitrary shape,” IEEE TAP, vol. 30, no. 3,
1982.
[3] J.-S. Zhao and W. C. Chew, “Integral equation
solution of maxwell’s equations from zero frequency to microwave frequencies,” IEEE TAP,
vol. 48, no. 10, pp. 1635–1645, 2000.
[4] J.-F. Lee, J.-F. Lee, and R. J. Burkholder,
“Loop star basis functions and a robust
preconditioner for efie scattering problems,”
IEEE TAP, vol. 51, no. 8, pp. 1855–1863, 2003.
[5] D. Wilton and A. Glisson, “On improving
the electric field integral equation at low frequencies,” Proc. URSI Radio Sci. Meet. Dig,
vol. 24, 1981.
[6] M. Burton and S. Kashyap, “A study of a recent, moment-method algorithm that is accurate to very low frequencies,” ACES Journal,
vol. 10, pp. 58–68, 1995.
[7] W.-L. Wu, A. W. Glisson, and D. Kajfez, “A
study of two numerical solution procedures for
the electric field integral equation at low frequency,” ACES Journal, vol. 10, no. 3, 1995.
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