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der etc; it is advantageous to formulate FDTD in cylindrical coordinates. But, exceedingly large computer resources are required
when a high circumferential resolution imposes a small time step.
The problems associated with a short time step are particularly
pronounced at low frequencies, where time cycles are long. The
solution described interfaces two methods: the PEE and the classical FDTD in cylindrical co-ordinates, which results in a new, efficient modelling tool.
esign using genetic
algorithm
J.S. P a n
We wish to thank V, Delport for making us aware of his Letter
[l, 21. Although he uses a genetic algorithm, it differs from ours
considerably [3]. The main differences are as follows: Delport uses
the codebook indices of the training data as the coding string, the
length of the coding string is thus the number of the training data
points in the training set. In our algorithm, we use the codebook
vectors as the coding string, the length of the coding string is thus
equivalent to the number of codewords. This means that the
string is much shorter in our algorithm. Theoretically and practically, it is difficult to converge to better optimal value if the coding string is too long.
In addition, in our algorithm, the coding strings of the initial
population can be assigned randomly from the training data,
because it can be converged to optimal value easily in any initial
condition. Delport uses the binary splitting method to obtain the
better initial population to improve his algorithm.
Finally, we employ a sorting technique based on the central
value of the training data to facilitate convergence to better
optima. This is unique to our algorithm.
To sum up, Delport applies a genetic algorithm to adapt the
codebook index of points in the training data, whereas we apply a
genetic algorithm to adapt the value of the codebook vector.
31 October 1995
0 IEE 1996
Electronics Letters Online No: 19960138
J.S. Pan (Centre for Communication Interface Research, The University
of Edinburgh, 80 Soutlz Bridge, Edinburgh EH1 IHN, Scotland, United
Kingdom)
/=
FDTD
Ar
Fig. 1 Schematic diagram of method of analysis
Analysis; The geometry of the structure is shown in Fig. 1. The
PEE algorithm described in [l] is used inside the radius R,where
the field components are expanded in a Fourier series with respect
to 4.A typical component has a form
N
E~(T4
, ,z)=
(1)
e T n ( T , z)ejnd
n=l
References
1 DELPORT, v.: ‘Comment: VQ codebook design using genetic
algorithm’, Electron. Lett., 1996, 32, (3), p. 193
2 DELPORT, v., and KOSCHORRECK, M.: ‘Genetic algorithm for
codebook design in vector quantisation’, Electron. Lett., 1995, 31,
(2), pp. 84-85
3 PAN, J.s., MCINNES, F.R., and JACK, M.A.: ‘VQ codebook design using
genetic algorithms’, Electron. Lett., 1995, 31, (17), pp. 1418-1419
D method for efficient field
lindrical co-ordinates
M. Mrozowski, M. Okoniewski and M.A. Stuchly
Between R, and R,, an FDTD algorithm in cylindrical co-ordinates is used. The transition between the two algorithms (interior
PEE and exterior FDTD) is carried out as follows: fields calculated from the PEE on the surface r = R, are directly imported to
the FDTD algorithm. In the opposite direction, the fields given by
the FDTD at R,+ Ar are used to obtain the expansion coefficients
of the PEE by taking the inner product of each function [l]. In
cylindrical co-ordinates, and media independent of 4, this algorithm resembles the PEE-FDTD [2]. However, there are two
important differences between the two algorithms. The PEE computations are carried with several expansion terms simultaneously,
while the BOR-FDTD considers each mode separately. Only axisymmetric structures can be analysed with the BOR-FDTD
method, while the hybrid PEE-FDTD method can be applied to
any inhomogeneuity in the FDTD region of the computational
space.
A relative stability criterion of the hybrid method compared to
the FDTD can be derived following the method given in [3] as
Indexing terms: Numerical methods, Finite-dijference time domain
method, Electromagnetic waves
Two algorithms are combined in the cyhndrical co-ordinates for
accelerated computations of electromagnetic fields in the time
domain. The partial eigenfunction expansion (PEE) algorithm is
used close to the co-ordinate system origin, and the FDTD
algorithm is used in the remaining region. This eliminates the use
of a prohibitively short time step if high circumferentialresolution
is required away from the origin. A significant acceleration of
computations is obtained, owing to the superior stability and
higher efficiency of the PEE method.
Introduction: As time domain techniques, particularly the finite
difference time domain (FDTD), are increasingly used to simulate
electromagnetic fields in more complex and larger structures, there
is a need for new algorithms which require fewer computer
resources, and are stable. Recently, an efficient hybrid partial
eigenfunction expansion (PEE) - FDTD method was developed
for shielded structures in the Cartesian co-ordinates [l]. For some
structures; e.g. a corrugated waveguide, antenna arrays on a cylin-
194
where A is a constant dependent on b and R,, and N is the
number of terms in the expansion series. Thus, the smaller the A$
and the lower the number of terms in the expansion required, the
greater the mcrease in the tme step. Additionally, the dispersion
of the PEE algorithm is inherently lower, because of a lack of discretisation in the I$ direction, so more accurate results are to be
expected.
The increase in speed of the hybrid method compared with the
FDTD is caused by two factors: (i) as the required number of
terms in the expansion is usually small, a larger time step can be
used resulting in fewer computations, and (ii) fewer unknowns are
used to evaluate the fields close to the origin. The gain in speed
- R,)for a
caused by the second factor is proportional to R,I(R,xox
large number of discretisation points along Y and small N. The
overall acceleration is a product of both factors, and large gains in
speed are expected for large R,IRmax.The gain in speed is also
accompanied by savings in the computer memory.
ELECTRONICS LETTERS
1st February 1996
Vol. 32
No. 3
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