Forum for Electromagnetic Research Methods and Application Technologies (FERMAT)
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reflectors for frequency reuse systems, or for frequency
selective radomes. The conventional Method of Moments
(MoM) [8], [9] is often applied for the solution of moderatesize problems; however, it becomes numerically inefficient
for electrically large problems owing to the large sizes of the
associated system matrices. Among the direct solvers, the
Characteristic Basis Function Method (CBFM) has been
extensively used for reducing the size of the impedance
matrix system of equations both in the context of printed
antennas and scattering problems [10]-[12]. It is based on the
generation of physics-based, Characteristic Basis Functions
(CBF) to model the current distributions induced in the subdomains. The reduced version of the system matrix arising in
the CBFM consumes a relatively small CPU time compared
to the Lower Upper (LU) factorization of the conventional
MoM matrix. Another approach to ease the solution of large
scale problems is the application of hybrid methods, where
the MoM is combined with asymptotic techniques Error!
Reference source not found.. In this context, a matrix
equation is only derived for the MoM region, resulting in a
sizable reduction of the number of unknowns. However, the
accuracy of such hybrid techniques is often limited, and there
is no simple way to improve it.
In this paper, a novel method, which combines the CBFM
with the Spectral Rotation (SR) approach Error! Reference
source not found. is proposed. The technique is based on
computing the integrals pertaining to the mutual-interactions
in the spectral domain, while the self-interactions are
analyzed in the spatial domain Error! Reference source not
found.. Furthermore, the computation of the reaction
integrals in the spectral domain is speeded up via the SR
method, which circumvents the need to deal with threedimensional Fourier transforms for non-planar geometries.
The time required to fill the reduced matrix is considerably
reduced in comparison to the conventional CBFM when the
SR method is applied, but without compromising the
accuracy of the final result.
The paper is organized as follows. In Section II the CBFM
algorithm is briefly reviewed. Section III presents some of
the basic concepts and numerical considerations on the
spectral rotation approach. Section IV presents some
numerical examples which demonstrate the accuracy and the
numerical efficiency of the proposed technique. Finally,
some concluding remarks are provided in Section V.
Abstract—An efficient Characteristic Basis Function
(CBF)-based method is proposed to analyze conformal
Frequency Selective Surfaces (FSS) that are not
amenable to analysis by using conventional numerical
methods typically used to model infinite, planar and
doubly-periodic FSSs. The technique begins by
employing the CBFs to describe the currents induced on
the elements. The reaction integrals needed to derive the
reduced matrix elements are computed either in the
spatial domain, or spectral domain, depending on the
separation distance between the blocks, so as to make the
process numerically efficient. The spectral domain
integrals are evaluated by making use of the Spectral
Rotation (SR) on the spectra of the CBFs to alleviate the
computational burden to be associated with full Three
Dimensional (3-D) Fourier transform which is required
in the conventional spectral domain approach applied to
non-planar geometries. Numerical results from the new
method have good agreement with the fully-spatial
Characteristic Basis Function Method (CBFM) and the
conventional Method of Moments (MoM). However, the
required Central Processing Unit (CPU) time is greatly
reduced.
Index
Terms—Method
of
Moments
(MoM),
Characteristic Basis Function Method (CBFM), Spectral
domain analysis, Spectral Rotation (SR).
I. INTRODUCTION
Periodic structures have been widely used and investigated
in the past as they find widespread applications as
Frequency-Selective Surfaces (FSS) [1]-[3], Electromagnetic
or Photonic Bandgap (EBG or PBG) materials [4] and
metamaterials. The periodic structures are conventionally
modeled as infinite doubly periodic arrays and periodic
boundary conditions are applied to a unit cell based on the
Floquet’s theorem [5] or the periodic Green’s Function (GF)
[6]. Even if the size of the array to be analyzed may be very
large, those structures are, in practice, finite in size [7];
hence, an accurate full-wave analysis should appropriately
consider the coupling between all the elements, leading to
the solution of a large-scale problem. In this work we focus
on the analysis of conformal FSS, that are widely employed
to realize high performance frequency selective sub1
Forum for Electromagnetic Research Methods and Application Technologies (FERMAT)
algorithms, whose calculation can be relatively timeconsuming.
In the next section, we show how we can compute the
reaction between two currents which reside on two planes
whose orientation can be arbitrary, by using a 2-D technique.
II. THE CHARACTERISTIC BASIS FUNCTION METHOD
In the conventional CBFM, as a first step, the structure
under analysis is partitioned into N sub-domains, or blocks
(see Fig. 1). The induced currents in each block are solved
for separately by illuminating the block with a set of K plane
waves with both E- and H-polarizations. The solutions thus
obtained are then processed by using the Singular Value
Decomposition (SVD) algorithm, to automatically down
select and retain the M most linearly independent high-level
basis functions for that block, denoted as CBFs.
III. SPECTRAL ROTATION APPROACH
Let us consider the scenario depicted in Fig. 2, in which a
source CBF is defined over support #1 and a test CBF resides
over support #2.
Fig. 1. Partitioning of a conformal FSS into N blocks (top view).
Fig. 2. Support #1 defined over the xy-plane and support #2 defined over
the x'y'-plane. Also shown support #1 rotated by an angle r and
with
respect to the xy-plane.
The second step to be performed is the generation of the
(M·N) × (M·N) reduced-size matrix by Galerkin testing the
CBFs, which are used both as basis and testing functions.
Finally, the current distribution induced on the entire
structure is expressed as a linear combination of the CBFs,
weighted by the solution vector resulting from the inversion
of the reduced system of equations Error! Reference source
not found..
The generic element of the reduced matrix, whose
calculation comprises a reaction integral, can be computed in
the spatial domain by using:


Z iRED
  J i (r )    J j (r ') G (r  r ') d r ' d r
,j
T
S

where
To compute the fields produced by the CBFi which is
defined over support #1 at location #2, in spectral domain, we
~
first express the spectrum of J #1,i (k x , k y ) with respect to the
spectral coordinates (k x' , k y' ) of x'y'z' system:
~
~
J #1,i (k x' , k y' )  J #1,i (k x , k y )
is multiplied by an exponential term which incorporates the
projection of the vector distance d between the origins of the
two supports, over the test system. This step translates the
spectrum from the source (#1) to the test position (#2). The
final form of the reaction integral in spectral domain reads:
spatial CBF while the primed and unprimed notations refer to
the source and test coordinate system, respectively. By
invoking Parseval's theorem Error! Reference source not
found., expression (1) can be rewritten in spectral domain, as:
 
1
2 
2
 J
~


~


~




i (  k x , k y )  G ( k x , k y )  J j ( k x , k y ) dk x dk y
(3)
where r and r are the angular differences between the xyz
and x'y'z' reference systems. This first step produces a rotation
of the spectrum of the source current at #1, such that it
~
becomes parallel to test support #2 (Fig. 2). Next, J #i (k x' , k y' )
(1)
G is the spatial domain GF, Ji represents the ith
Z iRED

,j
k x  k x' cos  r cos r  k y' cos  r sinr  k z' sin r
k y  k x' sinr  k y' cos r
(2)
Z 2RED
,1 
  
where G~ ( , ) denote the spectral domain Green's function,
and J~ i is the Fourier transform of the ith CBF.
 
1
2 
2
 J
~


~


# 2 ,i (  k x ,  k y )  G ( k x , k y )
  
~
 j k  d  xˆ   k y d  yˆ   k z d  zˆ 
 J #1,i (k x , k y ) e x
dk x dk y
(4)
~
where G ( , ) denotes the spectrum of the dyadic GF and d
For planar structures, such as thin plates, two-dimensional
(2-D) Fast Fourier Transform (FFT)-based approaches can
be successfully employed in order to reduce the memory and
CPU time considerably. However, this applies only if the
source and observation surfaces are parallel to each other
Error! Reference source not found.; otherwise, it becomes
necessary to switch to three-dimensional (3-D) FFT
is the distance between the source and observation points.
The computation of the integral in (4) can be carried out
efficiently in the spectral domain for the mutual terms,
since the rotated current becomes highly attenuated in the
evanescent region of the spectrum. If we recall the
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Forum for Electromagnetic Research Methods and Application Technologies (FERMAT)
definition of the z-component of the wavenumber Error!
Reference source not found.:
In order to carry out an estimate of the accuracy of the
algorithm, we define the Normalized Error (NE), as:
  k 2  (k 2  k 2 ) if k 2  k 2  k 2 (visible spectrum)
0
x
y
x
y
0

kz = 
2
2
2
2
2
2
 j (k x  k y )  k0 if k x  k y  k0 (evanescent spectrum)


CBFM  SR
NE (%)  100 
(5)
note that kx in (3) is complex when
k x2

k y2

k 02 ,
 Fields / RCS
 ( Fields / RCS
 ( Fields / RCS )
CBFM 2
)
n
CBFM 2
n
(6)
where superscripts 'CBFM-SR' and 'CBFM ' refer to the
hybrid and the conventional CBFM, respectively. The
normalized error is 1.2% and the Relative Time (RT),
defined as the ratio between the CPU time taken by the
conventional CBFM and the CBFM-SR is 2.17.
because
kz is purely imaginary. Thus, when performing the transform
of the current, this produces attenuation in the evanescent
part of the spectrum when sin(r) ≠ 0 in (3). Additional
attenuation is introduced by the translation to be applied to
 j k  d  zˆ 
the spectrum of the field, defined by the term e z
in
(4). In the evanescent spectrum this exponential term reduces
to e
 ( k x2  k y2 )  k02 d  zˆ
producing an attenuation related to d  z  .
Let us consider a CBF distributed over a rectangular
support with length L=/4 along x and width w = L/10 at
1 GHz. We assume their distribution is sinusoidal along the
longitudinal dimension and constant along the transverse
one. The behavior of the spectrum before and after a rotation
(, ) = (3/8, /4) is reported in Fig. 3(a) and (b),
respectively.
Fig. 4. Geometry of 7×5 array of x-oriented PEC strips.
(a)
(b)
Fig. 3. (a) Spectrum of bi-dimensional sinusoidal current distribution J
and (b) Spectrum of J after a (, ) = (3/8, /4) rotation.
It is apparent that the integration interval in the spectral
domain is already significantly reduced with the use of the
rotation operation only, and this fact can be exploited to
obtain an accurate result for the integral with a relatively
small number of spectral samples. The relative advantage
also grows as the number of blocks N is increased, because
the number of self-interactions to be computed when filling
the reduced matrix is N while the number of mutualinteractions is N(N-1) Error! Reference source not found..
Fig. 5. Magnitude of scattered E-fields in the far-field region on the
elevation plane, = 0, for TE and TM polarizations.
IV. NUMERICAL RESULTS
For the second example we analyze an array of 109 patch
elements arranged as in Fig. 6. A plane wave is incident from
the top with TMx - polarization at the operating frequency of
1 GHz. A total of 19620 sub-sectional basis functions are
used to model the structure. The bistatic RCS on the
elevation plane computed by using MoM, CBFM-SR and
conventional CBFM is reported in Fig. 7. The relative time is
4.1 for this example while the normalized error is 2.1% for
normal incidence, when we use 4 CBFs. In Fig. 8 we show
the NE on the RCS when the incident  angle inc is varied
from 0 to 90o and we use 4 CBFs. The NE is below 6% over
all incident angles. Table 1 shows the NE and the RT as a
For the first example we analyze a 7×5 array of xdirected PEC strips wrapped around a cylindrical surface
(Fig. 4). The dimensions of the strip are 120mm×30mm. A
total of 1995 sub-sectional basis functions are employed to
discretize the geometry at 1 GHz. Both TEy and TMx
polarizations have been considered, with (, ) = (-, 0),
and the magnitude of the scattered x- and z- components of
the E-fields are plotted in Fig. 5 (two orders of magnitude
lower Ey component is not shown). The agreement between
CBFM-SR and CBFM is good, even for TE incidence, for
which the scattered fields are smaller than in the TM case.
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Forum for Electromagnetic Research Methods and Application Technologies (FERMAT)
function of the number of employed CBFs. As it can be
noted, the method is independent of the number of employed
CBFs (the relative error remains constant) and, as expected,
the time advantage grows as the number of CBFs is
decreased. After SVD, we find 4 CBFs to be sufficient to
model the current distribution (see Fig. 7).
The CPU time necessary to fill the reduced matrix mutualblocks becomes negligible in comparison to that required to
evaluate the self-block interactions when the problem size is
very large. This, in turn, leads to a considerable time-saving
in comparison to the conventional approach for large scale
geometries.
while retaining the accuracy of the results. The scattered
fields are found to agree well with those obtained when the
reduced matrix is generated conventionally. Furthermore, the
time-savings of the proposed method grow as a function of
the number of unknowns in the blocks.
REFERENCES
[1] B.A. Munk, Frequency selective surfaces: Theory and
design, Wiley, New York, 2000.
[2] R. Mittra, C. H. Chan and T. Cwik, “Techniques for
Analyzing Frequency Selective Surfaces - A Review”,
IEEE Proc., 76, 1593-1615 (1988).
[3] C. H. Chan and R. Mittra, “On the Analysis of
Frequency-Selective Surfaces Using Subdomain Basis
Functions,” IEEE Trans. Antennas Propag., Vol. 38,
No. 1, 40–50, Jan. 1990.
[4] Y. Rahmat-Samii and H. Mosallaei, “Electromagnetic
band-gap structures: Classification, characterization, and
applications,” in Proc. Inst. Elect. Eng. Antennas
Propagation, Manchester, U.K., Apr. 17–20, 2001.
[5] R. E. Collin, Field Theory of Guided Waves, IEEE
Press, New York, 1991.
[6] K. Yasumoto and K. Yoshitomi, “Efficient calculation
of lattice sums for free-space periodic Green’s
function,” IEEE Trans. Antennas Propagat., vol. 47, pp.
1050–1055, June 1999.
[7] T. A. Cwik and R. Mittra, "The effects of the truncation
and curvature of periodic surfaces: A strip grating,"
IEEE Trans. Antennas Propag., vol. AP-36, no. 5, pp.
612-622, May 1988.
[8] R. F. Harrington: Field Computation by Moment
Method. The Macmillan Company, 1968.
[9] A. F. Peterson, S. L. Ray and R. Mittra, Computational
Methods for Electromagnetics. New York: IEEE Press,
1998.
[10] V. Prakash and R. Mittra, “Characteristic basis function
method: A new technique for efficient solution of
method of moments matrix equations”, Microwave Opt.
Technol. Lett., vol. 36, no. 2, pp. 95-100, Jan. 2003.
[11] R. Mittra, G. Bianconi, C. Pelletti, K. Du, S. Genovesi
and A. Monorchio, “A Computationally Efficient
Technique for Prototyping Planar Antennas and Printed
Circuits for Wireless Applications”, Proceedings of the
IEEE, vol. 100, no. 7, pp. 2122-1231, July 2012.
[12] J. Wei and Z. Nie, “A wide band Scattering Analysis of
Conformal FSS by CBFM and SVD method”, AsiaPacific Microwave Conference (APMC) 2008.
Fig. 6. Geometry of 109 PEC patch elements array.
Fig. 7. Bistatic RCS on the elevation plane for TM x incidence. The
results of MoM, CBFM and hybrid CBFM (CBFM-SR) are compared.
Fig. 8. Normalized % error as a function of the incident theta angle inc.
Photo
V. CONCLUSIONS
In this paper, a technique to efficiently analyze the
problem of the curved FSS has been presented. By
computing the reduced matrix mutual-interactions in the
spectral domain, and by making use of the fully-2D spectral
rotation method, a considerable time-saving is achieved
4
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