Rochester Institute of Technology
RIT Scholar Works
Articles
1986
A Novel technique to the solution of transient
electromagnetic scattering from thin wires
Sadasiva Rao
Tapan Sarkar
Sohail Dianat
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Recommended Citation
IEEE Transactions on Antennas and Propagation 34N5 (1986) 630-634
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630
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL.
conducting surfaces,” ZEEE Trans. Antennas Propagat., vol. AP31, pp. 5-12, 1983.
[3] -,
“Simplifications in the stochastic Fourier transform approach to
random surface scattering,” ZEEE Trans. Antennas Propugat., vol.
AP-33, pp. 48-56, 1985.
[4] A. K. Funs and G . W. Pan, “A rough surface scattering model for the
entire frequency range,” submitted to ZEEE Trans. Antennas Propugat.
AP-34, NO. 5 , MAY 1986
[5] C. Eftimiu and P. L. Huddleston, “Scattering by a strip with a
randomly serrated edge,” Radio Sci., vol. 20, pp. 1549-1554; 1985.
Cornel Eftimiu, for a photograph and biography please see 636 of the July
1982 issue of this TRANSACTIONS.
A Novel Technique to the Solution of Transient
Electromagnetic Scattering from Thin Wires
SADASIVA M. RAO, TAPAN K. S-,
SENIOR MEMBER,
IFEE,AND SOHEIL A. DIANAT
a function of currents at previous instants. The marching-onAbstract-Previous approaches to the problem of transient scattering
by conducting bodies have utilized the well-known marching-on-in-time in-time solution procedure is simple to derive andeasily
solution procedures. However, these procedures are very dependent on
applicable to any geometrical shape. An important advantage
discretization techniques and in many cases lead to instabilities as time
of
this method, which has beenfrequently
stressed by
progresses. Moreover, the accuracy of the solntion procedure cannot be
Auckenthaler
and
Bennet
[l],
Mitzner
[2],
and
Herman
[3],
verifiedeasily and usnallythere is noerror estimation.Recently an
[4],
is
the
fact
that
no
matrix
inversion
is
required
if
one
alternate approachto thesolution of transient scatteringby thin wires was
presentedbasedon
theconjugategradient
(CG) method. In thii
carefully chooses the time and space discretizations. Unfortuprocedure, space and time arediscretized independentlyinto sobintervals
nately, this method suffers from the serious disadvantage that
and the error is minimized iteratively. Unfortunately, this procedure is
rapidly growing spurious oscillations may appear at later
very slow, not easily extendable to other geometries, and moreover, some
of the advantages of marching-on-in-time are lost.
In this paper, again the instants of time. The exact theoretical cause for this behavior
conjugate gradient methodis applied to solve the above problem, but this is not known but it is speculatedthat the accumulation of
time, reducing the error to a desired value at each time step. Since the
errors during the calculations such as round-off andtruncation
error is reduced at each time step, marching-on-in-time can still be done
errors trigger these instabilities. Although it may be possible
withouterroraccumulation
as timeprogresses.Computationally,
thii
to reduce these instabilities by employing certain smoothing
procedure is as fast as conventional marching-on-in-time. Thus, this new
procedures, these procedures are not simple. Moreover, there
method retains all the advantages of marching-on-in-time and yet does
not introduce instabilities in the late time.
It is also possible to apply this is no straightforward way of selecting the best possible
procedure to other geometries. Details of the solution procednre along
smoothing procedure’for a given geometry. AS a result the
with numerical results are also presented.
accumulation of round-off errors puts a serious limitation on
I. INTRODUCTION
REVIOUS APPROACHESto the problem of transient
utilized the
scattering by conducting
bodies
have
well-known marching-on-in-time solution procedures. In this
solution procedure spaceand time are discretized intoa
number of subintervals. A recurrence relation for the value of
’thecurrent at the present timeand space interval is obtained as
P
Manuscript received February 25, 1985; revised July 3, 1985. This work
part by the Office of NavalResearchunderContract
wassupportedin
N00014-79-C-0598.
S. M. Raoiswith
the Researchand Training Unit for Navigational
Electronics, Osmania University, Hyderabad, India.
T. K. Sarkar was with the Department of Electrical Engineering, Rochester
Institute of Technology, Rochester, NY. He is now with the Department of
Electrical Engineering, Syracuse University, Syracuse, NY 13210.
S. A. Dianat is with the Department of Electrical Engineering, Rochester
Institute of Technology, Rochester, NY 14623.
IEEE Log Number 8407332.
the applicabilityof this procedure to arbitrary geometries.
Another disadvantage of this procedure is that the accuracy of
solution cannot be easily verified and usually there is no error
estimation.
Recently an alternate procedure based on the solution of
operator equations using iterative techniques wasproposed by
Rao et al. [ 5 ] . In this method again space and time are divided
into anumberof subintervals but these diicretizations are
integrated
independent of one another. Asuitablydefined
squared error over a rectangular grid of time and space is
successively minimized at the end of each iteration by
employing either the method ofsteepest descent or the method
of conjugate gradient (CG). q e solution is obtained after the
error falls below a certain preselected value. The solution is
usuallyobtainedina
finite n m b e r of steps and since the
procedure is based on the minimization of integrated squared
error, the round-off and truncation errors are limited by the
0018-926X/86/0500-0630$01.00 0 1986 IEEE
63 1
RAO et al.: TRANSIENT ELECTROMAGNETIC SCAlTERING
last stage of iteration. The major advantage of this method is and
At
At
that one can obtain an accuracy estimate at each iteration and
for m A t - - s t < m A t + may terminate the algorithm at a desired value of the error.
2
2
(0,
otherwise.
However, the method described in [SI is computationally very
slowand also is noteasily extendible to other geometrical
structures. Thus, as claimed by the authors, this method is not We observe in (1) that t = T at z = z' . By utilizing (51471,
we can write
going to replace the marching-on-in-time procedure.
In this paper we describe a new procedure which retains all
the advantages of the marching-on-in-time method andalso of
the iterative methods. This procedure is stable with round-off
and truncation errors and is simple to apply. It is also easy to
Zt()z r Y dz'
see that this procedure is readily extendible to other geomez'= -Ad2 d ( z - z f ) 2 + a 2
tries.
In the next section we briefly describe the marching-on-intime method for completeness. In Section III, we describe the
conjugate gradient method and the new way of its application
to the present problem. A discussion on the comparison of
computational aspects is also presented. Finally, inSection
IV, the numerical results obtained with this new method for
the solution oftransient scattering of a thinwire are presented.
[
II. MAFXHING-ON-IN-TIME
SOLUTION
The primary objective of this paper is to obtain the current
distribution on a thin conducting wire as a function of time
when thewire is irradiated by a narrowelectromagneticpulse.
It is wellknown [SI thatwhen an incident electric field
impinges on a thin wire of length L and radius a, the current
distribution on the structure Z(z, t ) satisfies the following
integrodifferential equation:
where the overbar on the integral indicates thatthe effect of the
self-tern is not included. By approximating the current to be
constant in a given subdomain
at a giventime, we can write (8)
as
z(zmY tn)
1
["'i"']
4TEo
a22
c2 at2
I(z',
L
7)
d(Z- 2 ' ) ' +
dz') = -
aEyz, t)
at
(1)
Q2
where
1
c = velocity of light =-
(2)
G
C
1
c
and E & is the component ofthe incident field tangential to the
wire. In the operator notation (1) can be rewritten as
AZ= Y
(4)
where A is the integrodifferential operator and the current Z is
produced 'by the excitation Y for 0 5 z 5 L and t > 0.
Assuming the current distribution on the wire can be writtenas
Z(z, t > = C I(zrn9 tn) . Pn(z)
. Qm(t)
(5)
m.n
where
Az
(0,
Az
for nAz----Sz<nAz+2
otherwise
2
(6)
.
632
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-34, NO. 5, MAY 1986
From (9)-(12), it is clear that the current at any particular
instant is calculated by utilizing. the currents at the earlier
intervals. The method isquite efficient for computing transient
responses of objects which are of the order of the pulsewidth
of the incident field. An important advantage of this method,
which can be readily seen, is the fact that no matrix inversion
is needed if the time step is smaller than a certain upper limit
which is, among others, determined by spatial discretizationof
the object.
An important disadvantage of this procedure is the possible
Occurrenceofrapidly growing spurious oscillations at later
instants which is apparently due to the accumulation of errors
during the calculations. Althoughit may bepossible to
eliminate these instabilities by employing certain smoothing
procedures, nevertheless these spurious oscillationsputa
serious limitationon
the applicability of this technique.
Moreover, with the marching-on-in-time procedure, the accuracy of the solution cannot
be easily verified and usuallythere
is no error estimation. That is why we discuss the iterative
methods inwhich the accumulation of errors does not take
place. Also, for the iterative methodsthe time andspace
discretizations are not tiedtogether as in the case of marchingon-in-time solution. One can very wellchoose the step sizes in
time and space independently. In the following section, we
discuss briefly the method of conjugate gradient and a new
way of its application to the present problem.
For the conjugate gradient method described in [ 5 ] , the
error is defined by
1
T
0
dt
dz[AI-
YI2
0
and the method starts with an initial guess Io(z, t ) and
generates
Po= -boA*Ro= - b d * ( A I o - Y)
0
at each instant of
time. The current I(z)at the time instant ti =
iAt is computedby
starting with an initial guess and
developing(14)-(19) to minimize the error at this instant.
Again the iterations are continued until the error falls below a
preselectedvalue. It should be mentioned here that in the
present wayof applying the conjugate gradient method the
adjoint operator is the same as the operator A itself. The
conjugate gradient method described by (14)-(19) converges
for any initial guess. In our solution procedure we assume the
current to be zero on the wire at the beginning of the iterative
procedure for each instant of time.
The solution procedure starts by performing the integration
operation to obtain Q(z)at t = 0, At, and 2At, where
with the assumption thatI(z) = 0 at t
III. METHOD
OF CONJUGATE
GRADIENT
E=
certain preselected value.Once this error criterion is achieved,
we have the solution for the integrodifferentialequation for all
times. As mentioned earlier, this procedure is computationally
very slow and difficult to extend to other geometries.
In the present set-up, we define the integrated squared error
as
L
E= [ A I - Y I 2 dz
(20)
= 0 and At and with an
initial guess equal to zero at t = 2At.
We next perform the double derivative operation of Q ( z , t )
by using a finite difference approximation. There are actually
a number of ways one can approximate the double derivative
operator but the majority of these choices wouldimpose
certain conditions on the step sizes of time and space in order
to obtain a stable solution [SI. In the following, we Iist two
common forms of finite difference approximations given by
( 14)
where A* represents the adjoint operator for A which involves
advancedconvolutionin
time [5]. The conjugate gradient
method then develops
Ik+1 =Ik + a k p k
(15)
Rk+1 =Rk i-(YkAPk
(16)
and
+ Q m + l , n - 2 Q~m( ,An +ZQ)m' - I , n
where I, R , and P are two-dimensional vectors in space and
time and k denotes the iteration number. The convergence where
properties of the conjugate gradient method for electromagnetic problems are available in [6], [7]. The iterations are
continued until the integrated squared error E falls below a
633
RAO ef al.: TRANSIENT ELECTROMAGNETIC SCATITRING
approximation for the double derivative. Unfortunately, in this
case, the step size in time should be less than or equal to the
step size in space divided by the velocity of wave propagation
for stable solution. To ease the stability constraint one may
choose (23), in which no restriction is placed on the time and
space discretization. In the numerical results presented in the
next section, both forms of approximations are used, each one
performed equally well as long as certain conditions are
satisfied.
Once the double derivative operation is performed, we
evaluate the residual R at t = At by
1.2
1.0
0.8
0.6
0.4
0.2
1
ln
R=AI- Y
(25)
where we obtain Y analytically by performing the derivative
operation on the incident field. Equations (14)-(19) are
successively applied until a desired convergence is achieved.
This completes the evaluation of current distribution on the
wire scatterer at t = 2At. The current at later instantsis
obtained by repeating the solution procedure outlined here for
each time step.
The main difference in the present wayofapplying the
conjugate gradient method to the earlier work described in [5]
is that the error in the solution vector is minimized at each
instant separately rather than trying to minimize on the whole
rectangular grid of space and time. The new approach is not
only much faster with respect to computational speed butalso
requires much less core storage than the one described in [5].
In the present case we only haveto store seven columnvectors
of spatial discretization plus a rectangular matrix containing
the values of current at each instant and at the center of each
spatial subdomain. In the earlier procedure described in [ 5 ] ,
one has to store five rectangular matrices representing the time
and spatial grid. Since the error is minimized at each instant
separately, there is no error accumulation as time progresses,
and as a result, the solution is guaranteed to be stable in the
earlier as well as in late times. Moreover, it is also possible to
have a control over the error in the solution vector because
basically the method is an iterative method. Depending on the
choice of the finite difference operator, one can essentially
choose a larger time step than the spatial subdomain divided
the velocity ofEM wave. This particular choice is not allowed
in the marching-on-in-time solution withoutmatrix inversion.
Moreover, this particular choice would sometimes result in a
solution whichis computationally faster than the marching-onin-time solution. Thus we can say confidently thatthe present
application of the conjugate gradient method retains all the
advantagesofboththemarching-on-in-time
procedure and
iterative methods andyet eliminates the disadvantagesof these
methods.
IV. NUMERICAL
RESULTS
As an example consider a perfectly conductingstraight wire
of length 2.0 m and radius 0.01 m irradiated by a Gaussian
electromagnetic pulse of the form
0.0
-0.2
PIPS
-0.4
-0.6
-0.8
-1.0
-1.2
I
L
0.0
2.0
4.0
6.0
8.0
10.0
T i m in Lisht-neters
Fig. 1. Current induced at the Center of a conducting wire scatterer (length
= 2.0 m, radius = 0.01 m) by a normal incident Gaussian impulse.
from the broadside direction. In (26) 77 is the characteristic
impedance of free space, c the velocity of EM wave in free
space, a the standard deviation equal to 0 . 5 and
~ to is the time
when the incident pulse strikes the wire which is equal to 6a
and the time is measuredin light meters. The wire structure is
divided into tenequal subdomains so that each subdomain is of
0.2 m. Fig. 1 shows the transient current induced at the center
of the wire scatterer as function of time. While applying the
CG method described in this paper, we present two cases in
Fig. 1, the frst one with time step cAt = 0.2 light meters
using (22). In the second case, the time step cAt = 0.25 light
meters which is greater than Az and uses (23). It should be
mentioned here that this particular choice is not allowed inthe
conventional marching-on-in-time solution procedure without
matrix inversion. For the conjugate gradient method, the
iterations were continued until the normalized
error is less than
at each instant. For comparison, the same problem is also
solved
using
marching-on-in-time
solution
procedure by
choosing cAt = 0.2 light meters. As it is evident from the
figure, the numerical results obtained with this new method
compare very well with the results obtained by marching-onin-time solution.
In all of our computations, the conjugate gradient method
yielded
normalized
a
error of less than
at
each time
instance, in less than four iterations.
V. CONCLUSION
In this paper, a new way of applying the conjugate gradient
method to solve the problem of transient electromagnetic
scattering from conducting wires is presented. The solution
procedure is simple and efficient. The method is basically an
iterative method, thereby restricting the round-off andtruncation errors to the last stage of iteration. However, the method
may also be termed as a marching-on-in-time method since
the
current at the present time is essentially computed using the
634
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AI-34, NO. 5> MAY 1986
knowledge of the currents at previous instants. Thus, this new
methcd retains all the advantages of marching-on-in-time
procedure andyet guarantees stability as the solution progresses in time. Moreover this method is quite simple to apply
to other geometries. At present, work is in progress to apply
this method to other structures.
k E N C E S
[l] A. M.Auckenthaler and C. L. Bennet, “Computer solution of transient
time domain thin-wire antenna problems,” E E E Trans. Microwave
Theory Tech., vol. M’IT-19, pp. 892-893, 1971.
[2] K. M.Mitzner, “Numerid solution for transientscatteringfroma
hard surface by arbitrary shape-retardedpotential technique,” J.
Acousr. SOC.Am., vol. 42, pp. 391-397, 1967.
[3] G . C . Herman,“Scatteringofacousticwaves
by aninhomogeneous
obstacle,” presented at Int. URSI Symp., Munich, 1980.
[4] -,
“Scatteringoftransientacousticwaves
in fluids and solids,”
Ph.D. dissertation, Delft Inst. Tech., The Netherlands, 1981.
[5] S. M. Rao, T. K. Sarkar, and S. A. Dianat, “The application of the
conjugate gradient method to the solution of transient electromagnetic
. .
scattering from thin wires,” Radio Sci., vol.19, no. 5 , pp.13191326, Sept.-Oct. 1984.
T.K. Sarkar and S. M. Rao, “The application of the conjugategradient
method for the solution of electromagnetic scattering from arbitrarily
oriented wire antennas,” IEEE Trans. AntennasPropagat., vol. 32,
pp. 398403, Apr. 1984.
[7] T. K.Sarkar, “The application of the conjugate gradient method
for the
solution of operator equations arising in electromagneticscattering
from wire antennas,” Radio Sci., vol. 19, pp. 1156-1172, Sept.-Oct.
1984.
[8] G. F. Carrier and C. E. Pearson, Partial D$ferential Equations.
New York: Academic, 1976.
[a
Sadasiva M.Rao, for a photograph and biography please
see page 418 of the
May 1982 issue Of this TRANSACTIONS.
Tapen K. Sarkar (S’69-M’764M’81),for aphotographandbiography
please see page 539 of the May 1985 issue of this TRANSACTTONS.
Soheil A. Dianat, for a photograph and biography please
see page 663of the
July 1982 issue of this TRANSACTIONS.
. -