Efficient Representation of Multilevel QR Algorithm for Analysis of
Microstrip Structures
Zhaoneng Jiang
1,3
and Ting Wan
2,3
1
Hefei University of Technology, Hefei 230009
jiangzhaoneng@hfut.edu.cn
2
Nanjing University of Posts and Telecommunications, Nanjing 210003
want@njupt.edu.cn
3
State Key Laboratory of Millimeter Waves, Nanjing 210096
Abstract ─ This paper presents a novel approach for the
efficient solution of large-scale microstrip structures
with the mixed potential integral equation (MPIE) in
conjunction with the method of moments (MoM) based
on the conventional Rao-Wilton-Glisson (RWG) basis
functions. Although multilevel QR (MLQR) is efficient
compared with direct method, it consumes more
computation time and storage memory. A novel matrix
compression technique is presented to recompress the
sub-matrices of MLQR algorithm. The advantages of
applying the novel recompression technique are illustrated
by numerical results, the computation time as well as the
memory requirements are compared to the conventional
MLQR algorithm and the matrix decomposition
algorithm-singular value decomposition (MDA-SVD). It
is demonstrated that the use of proposed method can
result in significant savings in computation time and
memory requirements, with little or no compromise in
the accuracy of the solution.
Index Terms ─ Compression techniques, microstrip
structure, Multilevel QR Algorithm (MLQR).
I. INTRODUCTION
The method of moments (MoM) [1-5] has been
widely used for the analysis of microstrip structures. The
first kind is the spectral domain MoM which has been
used to analyze the electromagnetic problems in [3]. The
second kind is the spatial domain MoM which is
proposed by Michalski and Hsu in [4] for scattering by
microstrip patch antennas in a multilayered medium.
However, the conventional MoM using subsectional
basis functions and λ/10 discretization becomes highly
inefficient for the analysis of large-scale microstrip
structures. This is because the size of the associated
MoM matrix grows very rapidly as the dimensions
become large in terms of the wavelength, or a fine mesh
is used to model a complex structure to guarantee good
solution accuracy, and this in turn places an inordinately
heavy burden on the CPU in terms of both memory
requirement and computational complexity, which
increase with O(N
2
)and O(N
3
), where N is the number of
unknowns. This difficulty can be circumvented by using
the Krylov iterative method, which can reduce the
operation count to O(N
2
).
To make the iterative methods more efficient,
many fast algorithms are developed to speed up the
matrix-vector product operation, such as the fast multipole
method (FMM) [6-7] and the matrix decomposition
techniques [8-22], etc. The multilevel fast multi-pole
algorithm (MLFMA) [23] has been successfully applied
to the free space problem since the memory cost is
reduced to the order of NlogN. To be noticed, though the
FMM is successfully applied to the microstrip problems,
the procession is always difficult because of its
dependence on the Green’s function. At the beginning,
FMM is tried to combine with discrete complex image
method (DCIM) to solve the static and two-dimensional
problems [6]. Unfortunately, it will be lack of accuracy
when the frequency is high. Though FMM is employed in
[7] for full wave analysis, the implementation is very
complicated because the surface-wave poles are extracted
in DCIM. The FMM also has been applied to thin layer
structures as the thin stratified medium fast multipole
algorithm [24] which is adaptive to thin-stratified
media. In contrast with FMM, the matrix decomposition
technique is purely algebraic and, therefore, independent
of the problem of Green’s function. It can be easily
interfaced to existing MOM codes. QR is a popular matrix
decomposition technique, which has been successfully
applied in [18-22] to electromagnetic problems.
The aim of this paper is to present a novel
representation of MLQR algorithm for analyzing the
electromagnetic scattering and radiation problems of
1054-4887 © 2016 ACES
Submitted On: November 22, 2015
Accepted On: February 20, 2016
777ACES JOURNAL, Vol. 31, No. 7, July 2016
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