IEEE Antennas and Propagation Society Newsletter, April 1988
Finite-Difference Time-Domain (FD-TD) Modeling
of Electromagnetic Wave Scattering and Interaction Problems
bY
Korada R. Umashankar
EEC4 Department
University of Illinois at Chicago
Chicago, IL 60680
Allen Taflove
EECS Department
Northwestern University
Evanston, IL 60201
AbstractT h iasr t i c l e ,
based
upon
i n v i t epda p e r s
X X I I U R S I General Assembly (Tel
Aviv,
August
athe
1987)
and
t h e URSI National
Radio
Science
Meeting
(Boulder,
January
19881, r e v i e w sr e c e nat p p l i c a t i o n s
otfh ef i n i t e - d i f f e r e n c e
time-domain (FD-TO) method
numerical
for modeling
electromagnetic
of
wave
s c a t t e r i n g and i n t e r a c t i o n problems. One ofthegoals
of t h i s a r t i c l e i s t o demonstrate t h a t r e c e n t advances
i n FD-TD modeling
concepts
and software
implementat i o n , combined w i t h advances i n computertechnology,
haveexpanded
t h e scope,accuracy,andspeed
o f FD-TD
m o d e l i n gt ot h ep o i n t
where it may be t h ep r e f e r r e d
c h o i c ef o rs t r u c t u r e st h a tc a n n o t
be e a s i l y t r e a t e d b y
c o n v e n tiinotneaqglur aalt i o n
and
asymptotic
approaches. As a class, such s t r u c t u r e sa r ee l e c t r i c a l l y l a r g e andhavecomplexshapes,
m a t e r i a l compositions, apertures, and in t e r i o r c a v i t i e s
Introducing Feature Article Author
.
1.
INTRODUCTION
Allen Taflove
Contemporary
high-frequency
electromagnetic
engineering
problems
can i n v o l v e wave i n t e r a c t i o n s
w i t h complex,
e l e c t r i c a l l y - l a rtgher e e - d i m e n s i o n a l
structures.
These structures
have
can
shapes,
material
compositions,
apertures,
or
cavities
which
produce
near
f i e l dt sh ac ta n n o t
be r e s o l v ei dn t o
f i n i t e s e t s o f modes orrays.Propernumericalmodeli n go f
such
near
f i e l d rse q u i r e s a m p l i n ag t
subw a v e l e n g t hr e s o l u t i o nt oa v o i da l i a s i n go fm a g n i t u d e
a n i phase i n f o r m a t i o n . The goal i s t o p r o v i d e a s e l f c o n s i s t e n t model tohnfeu t ucaol u p l itnohgfe
e l e c t r i c a l l y smal 1 c e l l s corfipri s i n gt h es t r u c t u r e .
A l l e nT a f l o v e was born i n Chicago, I L onJune
14,
1949.
He r e c e i v etdh e
B.S.
( w i thhi g h e sdti s t i n c t i o n ) , M.S.,
and Ph.D. degrees i n e l e c t r i c a l e n g i n e e r ing
from
Northwestern
University,
Evanston,
IL,
in
1971, 1972, and 1975, r e s p e c t i v e l y .
From
1975
t o 1984,
he
was a s t a f f member a t 111
7esearch I n s t i t u t ei n C h i c a g oI,Lh, o l d i n gt h ep o s i tions
of
Associate
Engineer,
Research
Engineer,
anc
Senior
Engineer.
There,
i na d d i t i o nt or e s e a r c h
ir
2lectromagnetic wave s c a t t e r i n g and p e n e t r a t i o n , hc
x - o v i d e tde c h n i c al el a d e r s h i fpom
r a j opr r o g r a m s
ir
j o i n tr i g h t - o f - w a yd e s i g nf o r
60-Hz t r a n s m i s s i o n l i n e s
I n dp i p e l i n e so rr a i l r o a d s ;
and e x t r a c t i o n o f o i l f r o n
shale, t a r sand,
and
s l o w pl yr o d u c i nwge l ul ss i n s
HE
novel i ns i t ue l e c t r o m a g n e t i ch e a t i n gt e c h n o l o g y .
i s one o ft h r e ep r i n c i p a lc o - i n v e n t o r so ft h el a t t e r ,
and hasbeen granted 10 U.S. p a t e n t s i n t h i s area.
A candidatenumericalmodelingapproachforthis
purpose i st h ef i n i t e - d i f f e r e n c e
time-domain (FD-TD)
s o l u t i o n o f Maxwell's
curl
equations.
This
approach
i s analogous t oe x i s t i n gf i n i t e - d i f f e r e n c es o l u t i o n s
o f l u i df l o w
problems
encountered
i n computational
aerodynamics, i tnh atth ne u m e r i c a l
model i s based
upon a d i r e cs to l u t i ootnhfgeo v e r n i npga r t i a l
d i f f e r e n t i aelq u a t i o nP. u r s u i ntgh iasn a l o g y ,
FD-TD
shares
the
computational
requirements
o tf h ef l u i d s
codes
(and
o t h e rs i m i l a rl a r g e - s c a l ep a r t i a dl i f f e r e n t i a le q u a t i o ns o l v e r s )i nt e r m so f
computer f l o a t i n g
p o i nat r i t h m e t i cr a t ep, r i m a r y
random
access
memory
size, and data
bandwidth
to
secondary
memory.
Yet,
FD-TD i s a n o n - t r a d i t i o naapl p r o antcouhm e r i c a l
electromagnetic
modeling,
where
frequency-domain
approacheshavedominated.
I n 1984, Dr. T a f l o v er e t u r n e dt oN o r t h w e s t e r n
a:
an AssociateProfessor
i nt h e EECS Department.
Sincc
then,
he
has
developed
several
research
programs
ir
a n a l y s i s andnumeri
c a l methods f o re le c t r o m a g n e t i (
dave i n t e r a c t i o n sw i t hl a r g e ,
complexstructures.
Hi2
r e l a t e di n t e r e s t si n c l u d ei n v e r s es c a t t e r i n g/ t a r g e l
appl ic a t i ons orfe c e nvt e c t osr u p e r .
synthesi s;
computersandconcurrentprocessors
i n computationa'
electromagnetics;
and t h e new o n - s u r f a crea d i a t i o r
c o n d i t i o n (OSRC) t h e o r yf o rh i g h - f r e q u e n c ys c a t t e r i n g
which
he
o r i g i n a t e da l o n gw i t h
G. A. Kriegsmann
an(
K. R. Umashankar.
One o f t h e g o a l s o f t h i s a r t i c l e i s t o d e m o n s t r a t e
t h a tr e c e n t
advances i n FD-TD modelingconcepts
and
software
implementation,
combined
with
advances
in
computertechnology,haveexpandedthe
scope, accuracy
andspeed
o f FD-TD modeling t o t h e p o i n t where it may
be t h ep r e f e r r e dc h o i c ef o rc e r t a i nt y p e so fs c a t t e r i n g and c o u p l i n pg r o b l e m sW
. i t thh i s
i n mind, t h i s
i
l s u c c i n c t l yr e v i e wt h ef o l l o w i n gr e c e n t
article w
FD-TD m o d e l i nvga l i d a t i o nasnrde s e a r cfhr o n t i e r s :
Continued on poge 6
D r . T a f l o v e i s a member o f Tau Beta P i , E t a Kapp;
Nu, and Sigma X
.
i He i s a Senior Member o f IEEE anc
a member o f U R S I Commission B.
5
IEEE Antennas and Propagation Society Newsletter, Aprll 1988
F e a t w e Article-Continued
I n v es rcsaet t errei ncgo n s t r u cot ifo n
oned i m e n s i o n as p
l , a t i a cl loy i n c i d epnrto f i l oe fs
e l e c t r i c a l p e r m i t t i v i t y and c o n d u c t i v i t y ;
10. I n v esrcsaet t e r iencgo n s t r u c t i o n
twoof
dimensionalconducti ng, homogeneous, and inhomogeneous d i e l e c t a
r ircgferm
tosm
inimal
TM
s c a t t e r e df i e l dp u l s er e s p o n s ed a t a ;
and
from page 5
9.
1.
S c a t t e r i n gm o d e l sf otrh r e e - d i m e n s i o n arle e n t r a n t
structures spanning up t o 9 wavelengths;
2.
Conformalmodels
3.
S c a t t e r i n gm o d e l sf otrw o - d i m e n s i o n aal n i s o t r o p i c
structures;
4.
P e n e t r a t i om
n o d e lf sonra r r oswl o t s
j o i n t s i n t h i c k screens;
5.
Coupling
models
for
wires
and wire
bundles
f r e e space and i n a r b i t r a r y m e t a l c a v i t i e s ;
6.
P e n e t r a t i o nm o d e l sf o trh ee l e c t r o m a g n e t i cf i e l d s
w i t h i nd e t a i l e d ,
inhomogeneous ti ssueapproximat itcohoonefm
s plete
human body ( a t UHF
frequencies 1 ;
7.
M i c r o s t r i p andmicrowave
8.
Scattering
m o d e l fsorre l a t i v i s t i c a l l vy i b r a t i n g
mirrors;
ofcurvedsurfaces;
11. Large-scalecomputersoftware.
and lapped
in
2.
GENERAL CHARACTERISTICS OF FD-TD
As s t a t e d , FD-TD i s a d i r e c t s o l u t i o n o f M a x w e l l ’ s
It employs
no
time-dependent ecquur al t i o n s .
it applies
simple,
second-order
p o t e n t i a l sI.n s t e a d ,
a c c u r a t ec e n t r a l - d i f f e r e n c ea p p r o x i m a t i o n s
El] f o r t h e
space
and
t i mdee r i v a t i v etohsef el e c t r i c
and
magnetic f i e l d sd i r e c t l yt ot h er e s p e c t i v ed i f f e r e n t i a lo p e r a t o r so ft h ec d r le q u a t i o n s .T h i sa c h i e v e s
a
s a m p l e d - d art ea d u c t i o
tohcnfeo n t i n u o uesl e c t r o magnetic f i e l d i n a volume o f space, over a p e r i o d o f
time.
Space and t i m ed i s c r e t i z a t i o n sa r es e l e c t e dt o
bound e r r o r si nt h e
sampling
process,
and t oi n s u r e
n u m e r i c aslt a b i l i toyt fh ae l g o r i t h m
[2].
Electric
and magnetic f i e l d components a r e i n t e r l e a v e d i n space
t op e r m i t
a n a t u r a sl a t i s f a c t i o no ft a n g e n t i a fl i e l d
c o n t i n u i t cy o n d i t i o n as t
media
interfaces.
Overall,
FD-TD . i s a marching-in-timeprocedurewhichsimulates
t h ec o n t i n u o u sa c t u a l
waves bysampled-datanumerical
analogs
propagating
i n a data space s t o r eidn
a
computer.
A t each timestep,thesystemofequations
t o u p d a t et h ef i e l d
components i s f u l l y e x p l i c i t ,
so
t h a tt h e r ei s
no need t os e t
up o sr o l v e
a s e to f
l i n e aer q u a t i o n s ,
and the
required
computer
storage
and r u n n i n tgi m ieps r o p o r t i o n at ltoh e l e c t r i c a l
s i z e o f t h e volumemodeled.
c i r c u i t models;
Introducing Feature Article Author
LotticeTruncotionPlone
(InvisiblbTo All W a v e s )
Korada R. Umashankar
/
Korada R. Umashankar r e c e i v e dt h e B.E. degreefrom
1962;
t h e M.E. degree
Mysore U n i v e r s i t yI ,n d i ai ,n
f r otm
hI ned i aI nns t i t uSot ecf i e n cBe a, n g a l o r e ,
I n d i ian,
1964;
and t h e Ph.D. degree
from
the
U n i v e r s i t y o f M i s s i s s i p p iU, n i v e r s i t y ,
MS, i n 1974;
a l li ne l e c t r i c a le n g i n e e r i n g .
From 1964 t o 1969, he was A s s i s t a n tP r o f e s s o r and
Head ot hf e
Department oEfl e c t r i c aEln g i n e e r i n g ,
College
of
Engineering,
Karnatak
University,
Hubli,
I n d i aD. u r i n g
1974 and 1975, he was a P o s t d o c t o r a l
ResearchAssociate,
and from 1975 t o 1977, A s s i s t a n t
P r o f e s s o ro fE l e c t r i c a lE n g i n e e r i n ga tt h eU n i v e r s i t y
M
o fi s s i s s i p p i .
From 1977 t o 1979, he was t h e
N a t i o n a lR e s e a r c hC o u n c i lV i s i t i n gF e l l o wa tt h e
U. S.
Air Force Weapons L a b o r a t o rKyi ,r t l a n d
AFB, NM.
D u r i n g 1979 t o 1984, he was Senior
Engineer
at I
I
T
Research I n s t i t u t e , Chicago,IL.
x = 1/28
Figure 1. Time-Domain Wave-Tracking Concept of the FD-TD
Method
C u r r e n t l y , Dr. Umashankar i s Associate
Professor
o fE l e c t r i c a lE n g i n e e r i n g
andComputer
Science a t t h e
U n i v e r s Iiolt lfyi n oai ts
Chicago.
primary
His
research i s i tnh e
development oafn a l y t i c a l
and
numerical
techniques
i n electromagnetic
theory,
EMP/EMC i n t e r a c t i o n s , and EM s i m u l a t i os nt u d i e s .
R e s e a r c ht o p i c si n c l u d et h es t u d yo sf c a t t e r i n gf r o m
bodies
comprised
coated
of,
or with,
anisotrGpic
media;
e x t e n st hiooef n
method o f momenti t o
e l e c t r i c a l lvye rl ya r gtea r g e t s ;
and t h e new ons u r f a c er a d i a t i o nc o n d i t i o n
(OSRC) t h e o r fyohr i g h f r e q u e n c ys c a t t e r i n g .
F i g . 1 i l l u s t r a t e st h e
time-domain wave t r a c k i n g
c o n c e p to ft h e FD-TD method. A r e g i o n o f space w i t h i n
t h e dashed l i n e si ss e l e c t e df o rf i e l d
sampling i n
spaceandtime.
A t t i m e = 0 , it i s assumed t h a t a l l
f i e lw
d si t ht nh
i nue m e r i csaalm p l i rnegg i ao rne
i d e n t i c a l l y zero.
An i n c i d e npt l a n e
wave i s assumed
t oe n t e rt h es a m p l i n gr e g i o n
a t t h i sp o i n t .
Propagationoftheincident
wave i s modeled by t h e commencement o ft i m es t e p p i n g ,w h i c h
i s simplytheimplementat i ootnhf fei n i t e - d i f f e r e n caen a l ootghf ceu r l
equations.
Time stepping
continues
as the
numerical
analogoftheincident
wave s t r i k e s t h e modeled t a r g e t
embedded w i t h ti nhsea m p l i nr ge g i o n .
A
ll outgoing
D r . Umashankar i s a member o E
f ta
Kappa Nu and
Sigma X
.
i He i s a Senior Member o f IEEE and a member
o f U R S I Commission B.
Continued on page 7
5
IEEE Antennas and Propagation Society Newsletter, April 1988
Feature Article-Continued
from page 6
s c a t t e r e d wave a n a l o g si d e a l l yp r o p a g a t et h r o u g ht h e
1a t t i c e t r u n c a t i o n p l anes w i t h n e g l ig i bel r e f 1e c t i on
teoxtiht e
sampling
region.
Phenomena such
as
inductionofsurfacecurrents,scattering
and m u l t i p l e
s c a t t e r i n g ,p e n e t r a t i o nt h r o u g ha p e r t u r e s ,
and c a v i t y
e x c i t a t i o na r e
modeledtime-stepbytime-step
by t h e
a c t i o no ft h ec u r le q u a t i o n sa n a l o g .S e l f - c o n s i s t e n c y
otfh e s e
modeled phenomena i sg e n e r a l l y
assured i f
t h es ipra t i a l
and
temporal
variations
are
well
r e s o l v e d by t h e spaceand timesamplingprocess.
Time s t e p p i n g i s c o n t i n u e d u n t i l t h e d e s i r e d l a t e time
pulse
response
steady-state
or
behavior
is
achieved.
An i m p o r t a n t example o ft h el a t t e ri st h e
s i n u s o i d a ls t e a d ys t a t e ,w h e r e i nt h ei n c i d e n t
wave i s
assumed t o have a s i n u s o i d a l dependence, and t i m e
stepping i s c o n t i n u e d u n t i l a l l f i e l d s i n t h e s a m p l i n g
r e g i oe nx h i bs i tn u s o i draelp e t i t i oTnh.i iss
a
consequence o ft h el i m i t i n ga m p l i t u d ep r i n c i p l e
C31.
Extensive
numerical
experimentation
with
FD-TD has
shown t h at ht e
number o f c o m p l e tcey c l eot shf e
i n c i d e n t wave r e q u i r e dt o
be time-stepped t oa c h i e v e
t h es i n u s o i d a ls t e a d ys t a t e
i s approximatelyequalto
t h e Q f a c t oortfh es t r u c t u r eo r
phenomenon being
model ed.
EV
FD
-
- TD -
UNIT
CELL
Y
Figure 3. Arbitrary Three-Dimensional ScattererEmbeddedina
FD-TD Lattice
r e s u l t si n
a stepped-edge, or
staircase,
approximat i oocnfu r v esdu r f a c e sC. o n t i n u i totyaf n g e n t i a l
f i e l d s i s assured a t t h e i n t e r f a c e o f d i s s i m i l a r
media
w i tthh ipsr o c e d u r eT. h e ries
no
need
f osrp e c i a l
fielm
d atchina
gt
media i n t e r f a c pe o i n t s .
Steppededge approximationofcurvedsurfaces
has
been
found
t o beadequate i n t h e FD-TD modelingproblemsstudied
i n t h e 1970's and e a r l y 1980's, i n c l u d i n g wave i n t e r a c t i o n sw i t hb i o l o g i c a lt i s s u e s
C41, p e n e t r a t i o ni n t o
pulse
(EMPI
c a v i t i e s [ S I , [SI, and electromagnetic
interactiow
n si t h
complex
structures
[ 7 ] - C91.
However, r e c e n t i n t e r e s t i n widedynamicrangemodels
os fc a t t e r i n g
by c u r v et ad r g e t s
has
prompted
the
development surface-conforming
of
FD-TD approaches
w h i c he l i m i n a t es t a i r c a s i n g .
These w
i
l besummarized
later in this article.
1
1 Reqion I :
Toial
Fields
'Y
X
Region 2 :
Scattered
Fields
Figure 2. Positions of the Field Components about a Unit Cell of
the YEE Lattice 111
F i g . 2 i l l u s t r a t e st h ep o s i t i o n so ft h ee l e c t r i c
and m a g n e t i c f i e l d componentsabout a u n i t c e l l o f t h e
FD-TD l a t t i c ei nC a r t e s i a nc o o r d i n a t e s
C11. Note t h a t
eachmagnetic
f i e l d v e c t o r component i s surroundedby
four circulating electric field vector
components,and
vice
versa.
This
arrangement
permits
not
only
a
c e n t e r e d - d i f f e r e n c ea n a l o gt ot h e
space d e r i v a t i v e s o f
t h ec u rel q u a t i o n s b, u at l s o
a n a t u r agl e o m e t r yf o r
implementingtheintegralformofFaraday's
Law and
Ampere's Law atth es p a c e - c e llle v e lT. h i si n t e g r a l
interpretation permits a simplebuteffectivemodeling
otfh ep h y s i c s
of
smoothly
curved
target
surfaces,
p e n e t r a t i o nt h r o u g hn a r r o ws l o t sh a v i n gs u b - c e l l
gaps,
and c o u p l i n g t o t h i nw i r e sh a v i n gs u b - c e l ld i a m e t e r s ,
as w
l
i beseen l a t e r .
Source
(a
Region I :
FTi eo ltdasi
1
- - - -
"x
j o +...
Region 2 :
Scottered
Flelds
Fig.
3 i l l u s t r a t e s how an a r b i t r at hr yr e e d i m e n s i o n asl c a t t e r e r
i s embedded i n an FD-TD space
l a t t i c e comprised ot hf uen ci te l loFsfi g .
2.
S i m p l y ,d e s i r e dv a l u e so fe l e c t r i c a lp e r m i t t i v i t y
and
c o n d u c t i v i at yraes s i g n ee
to
da cehl e c t rfi ice l d
component ot hf lea t t i c eC. o r r e s p o n d i n g l yd,e s i r e d
values
of
magnetic
perineabi
1it y
c neaq u i v a l e n t
c o n d u c t i v iat yraes s i g n eet oda cmha g n e tfiice l d
component o tf h el a t t i c e .
The media
parameters
are
i n t e r p r e t e d by t h e FD-TD program as l o c a l c o e f f i c i e n t s
f tohtriem e - s t e p p i nagl g o r i t h m
S .p e c i f i c a t i oonf
m e d i a p r o p e r t i e s i n t h i s component-by-componentmanner
Lattice
Truncation
E
.. ~ ~ . . . t . . , ~ . . . t . . . ~ ..,,.
. . t ,.._._
.,.~
-HY
e
&
-
:H
iI
(b)
Figure 4. Division of FD-TD Lattice into Total-Field and ScatteredField Regions. (a) Lattice division; (b) Field component
geometry at connecting plane y = joS [lo], 1111
F i g . 4 i l l u s t r a t etshdei v i s i oont fh e
FD-TD
l a t t i c ei n t ot o t a l - f i e l d
and s c a t t e r e d - f i e l dr e g i o n s .
T h i sd i v i s i o n
hasbeenfound
t o beveryusefulsince
it p e r m i t st h ee f f i c i e n st i m u l a t i o no f
an i n c i d e n t
p l a n e wave i nt h et o t a l - f i e l dr e g i o nw i t ha r b i t r a r y
Continued on poge 8
7
-
-
IEEE Antennas and Propagation Society Newsletter. April 1988
Feature Article-Continued
from page 7
incidence,
angle
of polarization,
time-domain
waveform,
and
d u r a t i o n [lo], C111. T h r eaed d i t i o n a l
i m p o r t a n tb e n e f i t sa r i s ef r o mt h i sl a t t i c ed i v i s i o n .
a.
b.
C.
1
I
-"
(
A l a r g en e a r - f i e l dc o m p u t a t i o n a l
dynamicrange
i s a c h i e v e d s, i n c et h es c a t t e r e ro if n t e r e s t
i s embedded i nt h et o t a l - f i e l dr e g i o n .
Thus,
l o wf i e l dl e v e l si n
shadow r e g i o n so rw i t h i n
s h i e l d i negn c l o s u r easr e
computed d i r e c t l y
w i t h o u ts u f f e r i n gs u b t r a c t i o nn o i s e( a sw o u l d
the
be
case
i f s c a t t e r ef di e l di ns
such
r e g i o n s weretime-steppedvia
FD-TD, and then
added t o a c a n c e l l i n g i n c i d e n t f i e l d t o o b t a i l ,
thelowtotal-fieldlevels).
Embedding t h es c a t t e r e ir nt h et o t a -l f i e l d
r e g ipoenr m i t s
a n a t u rsaal t i s f a c t i o fn
t a n g e n tf i ace lodn t i n uai ct yr o s s
media
i n t e r f a c e s , dai s c u s seeadr l i w
e ri t, h o u t
h a v i nt og
compute t hi ne c i d e fni te al dt
p o s s i b l y numerous p o i n t s a l o n g a complex l o c u s
t h a t i s u n i q u et o
each s c a t t e r e r . The zoning
arrangement o fF i g .
4 r e q u i r e sc o m p u t a t j o no f
t h ei n c i d e n tf i e l do n l ya l o n gt h er e c t a n g b l a r
connectingsurface between t h e t o t a l - f i e l d and
s c a t t e r e d - f i reel dg i o T
n sh u
.i sr f ai sc e
.fixed,
i.e.,
independent
the
of
shape or
c o m p o s i t i o ont fh e n c l o s e sdc a t t e r ebr e i n g
modeled.
The p r o v i s i oonf
a w e l l - d e f i n esdc a t t e r e d f i e l dr e g i o ni nt h e
FD-TD l a t t i c ep e r m i t st h e
n e a r - t o - f afri e l tdr a n s f o r m a t i o inl l u s t r a t e d
i nF i g .
5.
The dashed v i r t u asl u r f a c e
shown
i nF i g .
5 can
be
located
along
convenient
l a t t i c ep l a n e si nt h es c a t t e r e d - f i e l dr e g i o n
o f F i g . 4.
T a n g e n t isacla t t e r e d
E and H
f i e l d s computed v i a FD-TD at th i vs i r t u a l
surfacecanthenbeweightedbythefree-space
G r e e n ' sf u n c t i o n
and t h e ni n t e g r a t e d
(summed)
t op r o v i d et h ef a r - f i e l dr e s p o n s e
and r a d a r
c r o s ss e c t i o n( f u l lb i s t a t i cr e s p o n s ef o rt h e
assumed i l l u m i n a t i o na n g l e )
[lll - C131. The
n e a r - f i e lidn t e g r a t i osnu r f a c e
has a f i x e d
r e c t a n g u l a r shape,and
thus i s independent o f
t h e shape comDosition
or
the
of
enclosed
s c a t t e r e d b e i n g modeled.
I
I
I
I
NO SOURCES
8
ZERO FIELDS
!
If
ky
HX
Corner Reflector Looks
1800
/ go'< 0 ' .
I
I
I
I
I
L
-
r__-________________'
0' =
90'
/
I
I
I
I
I
I
(a)
Fin
I
I
!
A n a l y t i c a l and experimental Val id a t i onshavebeen
o b t a i ne6 r e l a t i v et o
FD-TD modeling ocfa n o n i c a l
three-dimensionalconductingtargetsspanning
1/3 t o 9
wavelengths C121,
C131,
C191,
C201.
F o rb r e v i t y ,o n l y
one suchvalidationwi1,lbereviewedhere.
I
I
I
d
3. THREE-DIMENSIONAL FD-TD SCATERINC MODELS
Fin Center
I
I
I
-
0.37cm
--I
I
-
E2
P,?is)
I
going
scattered-wave
numerical
analog
striking
the
l a t t i cter u n c a t i o n
must e x ti ht l ea t t i cwe i t h o u t
appreciable
n o n - p h y s i c arle f l e c t i o nj ,u s t
as i f t h e
It has
been
shown
l a t t i c et r u n c a t i o n
was i n v i s i b l e .
t h at three q u i r elda t t i cter u n c a t i ocno n d i t i oins
r e a l l y a r a d i a t i o nc o n d i t i o ni nt h en e a rf i e l d
[lo],
C141
C171.
Further,
it been
has
shown t h a t
convenientlocalapproximationsoftheexactradiation
condition
can
be
generated
and a p p l i ewd i t h
good
r e s u l t s [IO] [171.
Based upon t h i rse s e a r c ht,h e
p r o c e d u r ef o rc o n s t r u c t i n g
more p r e c i s e l o c a l a p p r o x i mationsoftheexactradiationcondition
i s reasonably
well
understood.
These
approximations
are
currently
understudy
fornumericalimplementation
i n t h e FD-TD
computerprograms C181.
(bJ
Figure 5 . Near-to-Far Field Transformation Geometry
(a) Original problem; (b) Equivalent problem external
to
the virtual surface, Sa [ll]
F i g . 4 uses t htee r "ml a t t i cter u n c a t i o nt o"
designatetheoutermost
1a t t i c ep l a n e si nt h es c a t t e r e d - f i e l dr e g i o n .
The f i e l d sa tt h e s ep l a n e sc a n n o t
becomputedusingthecentered-differencingapproach
d i s c u s s e de a r l i e br e c a u s eo tf h e
assumed absence o f
known f i e l d a t a pt o i n t so u t s i d eotfh el a t t i c e
t r u n c a t i o n . These d a t aa r e needed t of o r mt h ec e n t r a l
d i f f e r e n c e sT. h e r e f o r ea, na u x i l i a r yl a t t i c et r u n c a t i o nc o n d i t i o ni s
necessary.
This
condition
must
be
c o n s i s t e nw
t i t hM a x w e l l ' se a u a t i o n si nt h a at no u t -
LATTICE
EDGE
9'. 00
Fig. 6 . Geometry of Crossed-Plate Scatterer and Illumination
[13],
1191, 1201
F i g . 6 d e p i c t tsh e
geometry o f a crossed-plate
scatterercomprisedoftwo
f l a t plate:, Z l e c t r i c a l l y
bonded t o g e t h e rt of o r mt h e
shape o f a T
The main
p l a t e has thedimensions 30 cm x 10 cm x 0.33 cm, and
10 cm x 10 cm x
t h eb i s e c t i n gf i n
has thedimensions
0.33 cm.
The i l l u m i n a t i o ins
a plane wave a t 0"
e l e v a t i o na n g l e
and TE p o l a r i z a t i o nr e l a t i v et ot h e
m a i np l a t e ,
and a t h ef r e q u e n c y
9.0 GHz. Thus, t h e
Note
t h al ot o k
main p l a t e spans 9.0 wavoelengths.
angle
azimuths
between
90 and 180" providesubstant i a l cornerreflectorphysics,inadditiontothe
edge
d i f f r a c t i o nc ,o r n edri f f r a c t i o n ,
and o t h eerf f e c t s
f o u n d f o r an i s o l a t e d f l a t p l a t e .
.
F o rt h e
9-GHz
FD-TD
model, t h el a t t i c ec e l ls i z e
i s 0.3125 cm, approximately 1/11 wavelength. The main
32 x 96 x 1 c e l l s ;t h eb i s e c t i n g
p l a t e i s formedby
f i n i s formed by 32 x 32 x 1 c e l l s ; and t h eo v e r a l l
l a t t i c e i s comprised o f 48 x 112 x 48 c e l l s ( 1 , 5 4 8 , 2 8 8
unknown f i e l d components)
containing
212.6
cubic
Continued on page 9
IEEE Antennas and Propagation Society Newsletter, April 1988
12
e
1
5
-n
m
C
.o
c
4
o
-4
U
W
* -e
v)
v)
0
5
-I 2
L
40
-16
CK
I'
0 -20
.-c
0
c
0
C
-24
@ FD-TO
I
0
r
-28
Modeling
Results
( 3 min single-processor Cray-2 time/ point)
S R I Meosurements(Range of doto
ot eoch look ongle)
-32
- 36
r
0"
I
20"
I
4Oo
1
I
1
I
I
80"
loo"
1204
#J' ( Look Angle , Degrees From Broadside 1
60"
140"
I
160"
1
180"
Fig. 7. Comparison of FD-TD Modeling with SRI Measurementsof Monostatic Radar Cross Section for the Crossed-Plate
Scattererat 9 GHZ (maximum scatterer size = 9 wavelengths) [13], [19],[20]
w a v e l e n g t h sN. o tteh at ht lea t t i cter u n c a t i o nasr e
o n l y 8 c e l l s (0.75 wavelength)fromthetarget'smain
p l a t e and f i n edges. The s l i g h t l ye c c e n t r i cp o s i t i o n ing of the bisecting fin is
accountedfor i n t h e FD-TD
model
S t a r t i n gw i t hz e r o - f ie l
d i n i t i a lc o n d i t i o n s ,
661 t i m es t e p sa r e
used, e q u i v a l e n t o3 1c y c l e so f
t h e i n c i d e n t wave a t 9 GHz.
resonantresponses
by 1%t o 2% f o r Q f a c t o r so f 30 t o
80, and c a np o s s i b l yi n t r o d u c es p u r i o u sn u l l s .I nt h e
a r esoacf a t t e r i n g
and RCS, the
use
stepped
of
surfaces has p r e v e n t e d a p p l i c a t i o n o f FD-TD f o r modeli ntghi em p o r t a nc tl a sotsaf r g e t s
where s u r f a c e
roughness,
exact
curvature,
and
dieleco
t rri c
RCS.
permeableloading i s c r u c i a l i n d e t e r m i n i n g
Measurements o f t h e m o n o s t a t i c r a d a r c r o s s s e c t i o n
(RCS) v s . look
angle
azimuth
were
performed
i nt h e
anechoic chamber f a c i l i t y o p e r a t e d by S R I I n t e r n a t i o n a l , MenloPark,
CA. F i g . 7 compares t h e FD,-TD p r e d i c t i o n sw i t ht h e
S R I measurements a t 32 key lookangles
RCS response.
w h i c hd e f i n et h em a j o rf e a t u r e so ft h e
It i s seen t h a t h e
agreement i sw i t h i na b o u t
1 dB
over a t o t a l RCS-pattern
dynamic
range
o f 40 dB.
L o c a t i o n s o f peaks
and
n u l lotshf pea t t e ranr e
a c c u r a t e l py r e d i c t e d
to
within
1" o f azimuth.
Note
e s p e c i a l ltyheex c e l l e n t
agreement f ol or oakn g l e
azimuthsgreaterthan
90",where
t h e r e i s a pronounced
corner-reflece
t of fr e c t .
As s t a t iend
[13],
it
appears t h a t t h i s case(andsimilarthree-dimensional
9-wavelengthtargetsstudied
i n [ 2 0 ] r) e p r e s e n t st h e
l a r g e s dt e t a i l e dt h r e e - d i m e n s i o n anl u m e r i c asl c a t t e r i n g model s o f any t y peev evre r i f i ewd h e r e i n
a
u n i f o r m l yf i n es p a t i a lr e s o l u t i o n
and t h ea b i l i t yt o
t r e a tn o n m e t a l l i cc o m p o s i t i o ni si n c o r p o r a t e di nt h e
model.
R e c e n t l yt,w od i f f e r e nt y p e s
o f FD-TD conformal
surface
models
have
been
proposed and
examined
for
two-dimensionalproblems:
.
4.
a.
Faraday's Law contour
path
models
c211.
These p r e s e rtbvhaeeC
s iac r t e s igarni d
arrangement o ff i e l d
components a t a l l space
c e l lesx c e pt ht o saed j a c e nttothtea r g e t
surface.
Space c e l l sa d j a c e n t ot h es u r f a c e
are
deformed
ct oo n f o rwmit hseu r f a c e
S l i g hm
t l yo d i f itei m
d e-stepping
1ocus
e x p r e s s i o n sf o rt h em a g n e t i cf i e l d
components
a d j a c e n t ot h es u r f a c ea r ed e r i v e df r o mt h e
i n t e g r a l f o r m oFfa r a d a y ' s
Law implemented
aroundtheperimetersofthedeformedcells.
.
b.
TWO-DIMENSIONAL CONFORMAL MODELS
OF CURVED SURFACES
Stretched,
conforminq
mesh models
[221,[231.
These emDlov a v a i l a b l en u m e r i c a l
mesh senerat i o n schemes t oc o n s t r u c tn o n - C a r t e s i a ng r i d s
w h i cahrceo n t i n u o u s lsyt r e t c h etdcoo n f o r m
with
smoothly
shaped targets.
Time-stepping
e x p r e s s i oaenriset haedra p tf er todhme
C a r t e s i a n FD-TD c a s[ e2 2o1rb t a i n evdi a
a n a l o gtytohceo m p u t a t i o n af l u id y n a m i c s
(CFD) case C231.
Research i s ongoing
for
each
these
of types
of
conformal
surface
models.
Key q u e s t i oin sc l u d e :
ease o f mesh generation;
suppression
on fu m e r i c a l
a r t i f a c t s such
as
i n s t a b i l i t yd, i s p e r s i o n ,
and
nonp h y s i c a l wave r e f l e c t i o ncso; d i nc og m p l e x i t y ;
and
modeling execution time.
A key f l a w i n p r e v i o u s
FD-TD models o fc o n d u c t i n g
s t r u c t u r e sw i t h
smooth
curved
surfaces
has
been
the
need t o use stepped-edge(staircase)approximations
of
t h ea c t u a sl t r u c t u r es u r f a c e A
. l t h o u g hn o t
a serious
problemforcomputing
wave p e n e t r a t i o n and c o u p l i n g
i n t o l o w - Qm e t acl a v i t i e s r, e c e n t
FD-TD studieshave
shown t h a ts t e p p e da p p r o x i m a t i o n so fc u r v e dw a l l sa n d
a p e r t u rseu r f a c ecsasnh icf te n t ef rre q u e n c i eosf
Continued on page 10
9
lEEE Antennas and Propagation Society Newrletier, April 1988
using FD-TD t o model materialtargets having diagonalt e n s o er l e c t r i c
and magnetic properties. No a l t e r a FD-TD algorithm i s required. The
tion of thebasic
more complicated
behavior
associated
with
offdiagonal
tensor
components can also be modeled, in
principal, with some algorithm
complications
[ZO].
from page 9
Feature Article-Continued
E,,
=2
koS=S
"."
hw = 4
___ CFIE solutlon
o o
+i=900
0
FD-TD (12cycles)
(Frequency = 150 MHz)
-.,-I--_L-.-L
0'
pxx= 2
30-
60'
90'
120'
150'
180'
R
Fig. 8. Comparison of FD-TD and Exact Solution for Azimuthal
Surface Electric Current on a ka = 5 Circular Conducting
Cylinder, TM Case (0.05 wavelength grid cell size) [21]
S
ID
>-
DI.20
0'.40
0'.60
D'.BO
NORMALIZED CONTOUR LENGTH
ll.00
.'=B
0.0
/--.L--L-L~l-L
0'
1--
30-
A_._
bo'
L
I...._
I
-
90'
120'
150-
180'
Fig. 10.Comparison of FD-TD and CFIE Solutions for Longitudinal Surface Electric Current on A Kos = 5 Square
Anisotropic Cylinder, TM Case [24]
B
Fig. 9. Comparison of FD-TD.and Exact Solution for Longitudinal
Surface Electric Current on a Ka = 5 Circular Conducting
Cylinder, TM Case (0.05 wavelength grid cell size) [21]
Recent development of combined-field, coupled
surfaceintegralequationsfor
modeling scattering by
a r b i t r a r y shaped two-dimensional anisotropic
targets
[24] has permitteddetailedtests
of theaccuracy of
FD-TD anisotropic models.
Fig. 10 i l l u s t r a t e s one
equivalent
such t e s t .
Here, the magnitude of the
s u r f a c e e l e c t r i c c u r r e n t induced by TM illumination o f
a squareanisotropiccylinder
i s graphed as a function
of positionalongthecylindersurfacefor
b o t h the
FD-TD and combined-field
integral
equation
(CFIE)
models.
The incident wave propagates i n the +y
direction and has a + z - d i r e c t e de l e c t r i cf i e l d .
The
cylinder has an e l e c t r i c a l s i z e k s = 5 , permittivity
E
= 2 , and diagonal
permeabilqty
tensor
!-I
= 2
a66 uyy = 4 . From F i g . 10, we seethatthe
FD-% and
The accuracy of the
Faraday's
Law contour p a t h
models f o r smoothly curved targetssubjected t o TE ar;d
TM illumination iisl l u s t r a t e d
i n Figs. 8 and 9 ,
respectively.
Here, a moderate-resolution
Cartesian
FD-TD grid(having
1/20 wavelength c e l ls i z e ]i s
used
t o compute the azimuthal
longitudinal
or
current
distribution on the
surface
of a ka = 5 c i r c u l a r
metal cylinder.
For b o t h polarizations,
the
contour
path FD-TD model achieves an accuracy o f 1.5% or
b e t t e ra t most surfacepointsrelative
t o theexact
o n l y 3.5%,
s e r i esso l u t i o n .
The worst-case
error,
occursforthe
TE case a t a point i n t h ec e n t e r - l i t
region where contourdeformation
i s maximum. Running
timeforthe
conformal FD-TD model i s essentiallythe
same asfortheoldstaircase
FD-TD model, since o n l y
a few H components immediately adjacenttothetarget
surface
require
a s l i g h t l y modified
time-stepping
re1 ation.
5.
CFIE results agree very well overalmost everywhere on
the
cylinder
surface.
Disagreement i s noted a t h e
cylindercorners where CFIE predictssharplocal
peaks
local
nul 1 s.
Studies
are
whereas FD-TD predicts
continuingtoresolvethecornerphysicsissue.
SCATTERING MODELS FOR TWO-DIMENSIONAL
ANISOTROPIC STRUCTURES
6.
The a b i l i t yt o
independently
specify
electrical
permittivity and conductivity
for
each E vector
FD-TD l a t t i c e , and magnetic permeacomponent inthe
b i l i t y and equivalentconductivityfor
each H vector
of
component, leads immediately t o t hpeo s s i b i l i t y
PENETRATION MODELS FOR NARROW SLOTS
AND LAPPED JOINTSI N MICK SCREENS
The physics of electromagnetic wave transmission
t h r o u g h narrow s l o t s and lapped joints in
shielded
understood
to
permit
enclosures must be accurately
Continued on page I I
10
IEEE Antennas and Propagation Society
Feature Article-Continued
c.
from page 10
good engineeringdesignofequipmenttomeetspecifications
for
performance
concerning
electromagnetic
p u l s e (EMP), l i g h t n i n g , high-power
microwaves
(HPM),
e l e c t r o m a g n e t i nc t e r f e r e n c e
and c o m p a t i b i l i t y (EM1
In
and EMC), u n d e s i r erda d i a t esdi g n a l s ,
and RCS.
many cases, s l o t s and j o i n t s may haveverynarrowgaps
f i l l e d by a i r ,o x i d a t i o nf i l m s ,o rl a y e r so fa n o d i z a t i o no rp a i n t .J o i n t s
canbesimple(say,twometal
s h e e tbsu t t etdo g e t h e r ) ;
more complex (laa p p eodr
" f u r n i t u r e "j o i n t ) ;o r
even more complex (athreaded
screw-typeconnectionwith
random p o i n t so fm e t a l - t o metal
contact,
depending
upon t h et i g h t e n i n g ) E
. xtra
c o m p l i c a t i o n sa r i s ef r o mt h ep o s s i b i l i t y
o f electrom a g n e t i cr e s o n a n c e sw i t h i nt h ej o i n te, i t h e irnt h e
t r a n s v e r s eo rl o n g i t u d i n a l( d e p t h )d i r e c t i o n .
i c :
D
E a u i v a l esnlltoota d i n q
C251.
Here, r u l e s
a r es e t od e f i n e
an e q u i v a l e n tp e r m i t t i v i t y
and p e r m e a b i l i t y i n a s l o t formedby a s i n g l e c e l l gap t o e f f e c t i v e l y narrowthe
gap t ot h e
desi red degree.
b.
Subqridding C261. Here, t h er e g i o nw i t h i nt h e
s l o to rj o i n ti sp r o v i d e dw i t h
a sufficiently
f i n eg r i d .T h i sg r i di sp r o p e r l y
connected t o
thecoarsergridoutsideoftheslot.
A
Observation
locus
A
( 0 )
50.
0'
-100'
0
FD-TD(XJ40
3A
B
Continued on page 12
100'
-
I
The accuracy
of
the
Faraday's
Law contour
path
model f o rn a r r o ws l o t sa n dj o i n t si si l l u s t r a t e di n
F i g s . 11 and12by
d i r e c t comparison o ft h e computed
gape1 e c t r i c f i e l d a g a i n s t h i g h - r e s o l u t i o n n u m e r i c a l
thick
benchmarks.
F i g . 11 models a 0.1 wavelength
conductingscreenwhichextends
0.5 w a v e l e n g t ht o each
s i d e o f a s t r a i g h t s l o t which has a gap o f 0.025 wavelength.
Broadside
TE i l l u m i n a t i o ni s
assumed. Three
t y p e sopf r e d i c t i v ed a t aa r e
compared:
( 1 ) The lowhe
r e s o l u t i o n ( 0 . 1 wavelength) FD-TD model u s i nt g
contourpathapproach
t o treattheslot
as a 1 / 4 - c e l l
gap;
( 2 1 A high-resolution
(0.025
wavelength)
FD-TD
model t ot r e a t h es l o t
as a 1 - c e l l gap;
and (3) A
h i g h - r e s o l u t i o n method o f moments (MOM) model ( h a v i n g
0.0025 wavelengthsampling
i nt h es l o t )w h i c ht r e a t s
t hsel o t t esdc r e e n
as a p u rsec a t t e r i ngge o m e t r y .
From F i g . 11, we see t h a t t h e r e i s e x c e l l e n t
agreement
between a lt lh r e se e t ospf r e d i c t i v d
e a t iab
noth
magnitude and
phase.
O f p a r t i c u l a irn t e r e s itst h e
FD-TD model, u s i n g t h e
a b i 1it y o f t h e l o w - r e s o l u t i o n
c o n t o u rp a t ha p p r o a c h ,t oa c c u r a t e l y
compute t h e peak
e l e c t r i c f i e l d i n t h es l o t .
FRONT OF SCREEN k S L O T d
B
BACK OF SCREEN
- 50.
A'
E
Fig. 12a. Geometry of U-Shaped Lapped JointFor TE Illumination,
Shown to Scale [27]
1
4,
C'
.025A
Fig. 12(a)
Three d i f f e r e ntty p e os f
FD-TD sub-cell
models
have
been
proposed and
examined
formodelingnarrow
s l o t s and j o i n t s :
a.
Faraday's Law contour
path
model C271. Here,
space c e l l s a d j a c e n t t o and w i t h i n t h e s l o t o r
j o i n t a r e deformed t o c o n f o r m w i t h t h e s u r f a c e
l o c u s( i n
a manner s i m i l a trot h ec o n f o r m a l
curved
surface
model 1.
S l i g h tml yo d i f i e d
time-stepping
expressions
for
the
magnetic
f i e l d components i n t h e s ec e l l sa r ed e r i v e d
f r o mt h ei n t e g r a lf o r mo fF a r a d a y ' s
Law implemented
around
t h ep e r i m e t e r so tf h ed e f o r m e d
cells.
*I--
C l e a r l y , t o make any headway w i t h t h i s c o m p l i c a t e d
group o f problems
using
the
FD-TD approach, it i s
necessary t od e v e l o p
andVal i d a t e FD-TD modelswhich
c a ns i m u l a t et h eg e o m e t r i cf e a t u r e so fg e n e r i cs l o t s
and j o i n t s S
. ince
a keygeometricfeature
i sl i k e l y
t o be thenarrow gap o f t h e s l o t o r j o i n t r e l a t i v e t o
one FD-TD space c e l l , it i s i m p o r t a nttou n d e r s t a n d
how s u b - c e l l gapscanbe
e f f i c i e n t l y modeled.
2.5
Newsletter, April 1988
resolution1
FD-TOOko/10 resolution
Contour model
FRONT OF SCREEN
A
( b)
Fig. 11. Comparison of FD-TD and MOM Solutions for the Gap
Electric Field Distribution, Straight Slot Case: (a)
Magnitude; (b) Phase 1271
11
IEEE Antennas ond Propagation Society Newsletter, April 1988
Feature Article-Continued
s i m i l a r t o conformal
the
curved
surface
model 1.
l / r s i n g u l a r i t i e ostfh ae z i m u t h a l
magnetic f i e l d and r a d i a le l e c t r i cf i e l da r e
assumed t oe x i s tw i t h i nt h e
deformed c e l l s .
S l i g h tm
l yo d i f i et idm e - s t e p p i negx p r e s s i o n s
f o rt h ea z i m u t h a lm a g n e t i cf i e l d
components i n
these cell s are derived from the integral form
o f Faraday's Law implemented
around
the
perimeterofthedeformedcells.
from page I 1
I -
-
- 120"
\9
-0-
0
FD-TD(0.025A. resolulioni
FD-TD (0.09X. resolution I
contour model
aJ
u)
a
P
/
-240°
/"
/
SERIES SOLUTION (EXACT)
FD-TD
0
(b)
Fig. 12b. FD-TD Computed Gap Electric Field Within the
0.45-Wavelength Path Length, U-Shaped Lapped Joint
'
(First Transmission Resonance):
(a) I Egap/Einc I ; (b) LEgap/Hz(A) [27l
0.8
-
0.6
F i g . 12a shows thegeometry
o f a U-shaped lapped
j o i n t which was s e l e c t e d f o r d e t a i l e ds t u d yo fp a t h l e n g t (hd e p t h )
power transmission
resonances.
The U
shape o ft h ej o i n tp e r m i t sa d j u s t m e n to ft h eo v e r a l l
j o i n tp a t hl e n g t hw i t h o u td i s t u r b i n gt h ep o s i t i o n so f
t h ei n p u t and o u t p u tp o r t sa t
A andF.
A u n i f o r m gap
of 0.025 wavelength i s assumed, as i s a s c r e e nt h i c k ness o f 0.3 wavelength and w i d t ho f
3 wavelengths.
F i g . 12b
compares
t h e gap e l e c t r i cf i e l dw i t h i nt h e
contour
j o i n t as computed by: ( 1 ) A low-resolution,
p a t h FD-TD model having 0.09 wavelength c e l l s i z e and
and ( 2 ) A h i g h t r e a t i ntgh e
gap as 0.28 c e l l ;
r e s o l u t i o n FD-TO model having 0.025 wavelength c e l l
s i z e and t r e a t i n gt h e
gap as 1 c e l l .
The t o t a pl a t h
l e n g t hw i t h i nt h el a p p e dj o i n it sa d j u s t e dt oe q u a l
0.45 wavelength,whichprovides
a sharp power t r a n s mission peak t ot h e
shadow s i d eo tf h es c r e e n .
From
F i g . 12b, we see a very goodagreementbetween
thelow
and h i g hr e s o l u t i o n FD-TD models,eventhough
thisis
a n u m e r i c a l lsyt r e s s f u rl ,e s o n a npte n e t r a t i ocna s e .
An i m p l i c a t i o n o f t h e s er e s u l t si st h a ct o a r s e
(0.1
wavelength) FD-TD g r i d d i n gc a n
be e f f e c t i v e l y used t o
model t hfei n e - g r a i n epdh y s i cosf
wave p e n e t r a t i o n
t h r o u g hs l o t s
and j o i n t s , i f s i m p l ea l g o r i t h mm o d i f i c a t i o n sa r e
made i n accordancewiththecontourpath
approach.
This
can
substantially
reduce
computer
resourcerequirements
and c o d i n gc o m p l e x i t yf o r
FD-TD
models o f complex s t r u c t u r ew
s ,i t h o suat c r i f i c i n g
appreciable accuracy i n t h e m o d e l i n g r e s u l t s .
7.
(X. /IO resolulion. contour
integral model of w i r e l
I.o
0.4
0.2
P
01
3,000
30,000
I
I
300
30
Wire radius, in wavelengths
Fig. 13. Comparison of FD-TD and Exact Solution for the Scattered
Azimuthal Magnetic Field at a Point 1/20 Wavelength
From the Center of an Infinite Wire [29]
-
2.0 r
MOM (A./60 resolution1
0 FD-TO (?,/IO
resolulion,
contour integral
model of wire)
1.5 -
1.0 -
0.5 -
COUPLING MODELS FOR WIRES AND WIRE BUNDLES
Center
End
Position along half dipole
I n equipmentdesignfor
EMP, HPM, and EM1 / EMC,
understandingelectromagnetic
wave c o u p l i n gt ow i r e s
and c a b l eb u n d l e sl o c a t e dw i t h i ns h i e l d i n ge n c l o s u r e s
i s a p r o b l e mt h a t
i s complementary t ot h a to f
wave
p e n e t r a t i o nt h r o u g ha p e r t u r e so tf h es h i e l d( s u c h
as
n a r r o ws l o t s
and j o i n t s ) .S i m i l a rt ot h en a r r o ws l o t
problem, a keydimension o f t h e i n t e r a c t i n g s t r u c t u r e ,
i n t h i s casethewireorbundlediameter,
may besmall
r e l a t i vteo
one FD-TD space c e l l .
Thus, it i s
i m p o r t a n tt ou n d e r s t a n d
how t h i n ,s u b - c e l l ,w i r e s
and
b u n d l e sc a nb ee f f i c i e n t l y
modeled i f FD-TO i s t o have
much a p p l i c a t i o n t o c o u p l i n g p r o b l e m s .
Two d i f f e r e n tt y p e so f
been proposed
and
examined
Fig.14.
The accuracy
of
the
Faraday's
Law contour
path
model f o rt h i nw i r e si nf r e e
space i s i l l u s t r a t e d i n
F i g s . and
14.
13
F i g . graphs
13 the
scattered
azimuthalmagnetic
f i e l d a t a f i x e dd i s t a n c eo f1 / 2 0
wavelengthfromthecenterof
an i n f i n i t e l y l o n g w i r e
having a radius
ranging
between
1/30,000
and
1/30
wavelength.
TM i l l u m i n a t i o ni s assumed. We see t h a t
t h e r ei se x c e l l e n t
agreementbetween
t h ee x a c ts e r i e s
s o l u t i o n and t h e l o w - r e s o l u t i o n (0.1 wavelength) FD-TD
contourpath
model o v e rt h ee n t i r e
3-decade r a n g eo f
w i r er a d i u sF. i g .
14
graphs
the
scattered
azimuthal
m a g n e t ifci e lddi s t r i b u t i oanl o n g
a 2.0-wavelength
( a n t i r e s o n aw
n ti )r e
o f radius
1/300
wavelength.
Broadside TM i . l l u m i n a t i o n i s assumed, and t h e f i e l d i s
o f 1/20wavelengthfrom
observed a t a f i x e dd i s t a n c e
FO-TD sub-cell modelshave
f om
r o d e l i n gt h i nw i r e s :
a.
Equivalent
inductance
C281.
Here, an equivalent inductance is defined for
a wire within a
space c e l l p e r m i t t i n g a l u m p e d - c i r c u i t model
of the wire to
be s e t up andcomputed.
I
b.
Comparison of FD-TD and MOM Solutions for the
Scattered Azimuthal Magnetic Field Distribution Along a
2.0-Wavelength Wire ofRadius 1/300 Wavelength (Broadside TM illumination) [29]
Faraday's Law contour
path
model L291. Here,
space c e l l s a d j a c e n t t o t h e w i r e a r e
deformed
t oc o n f o r mw i t ht h es u r f a c el o c u s
( i n a manner
Continued on poge 13
12
tEEE Antennas and Propagation Society Newsletter. April 1988
-
I
Cylindrical
shielding
enclosure
I 1
I
Single FD-TD c e l l
virtualsuriace
::
LANE
YBRID FD-TD/MOM
aperture
Normalized posftion
-
SYSTEM
Fig. 15. Comparison of Hybrid FD-TD / MOM Modeling Predictions withDirect EFIE for Induced on a WireBundle
Illuminated Broadside bya Plane Wave in Free Space [29]
Feature Article-Continued
from page 12
t hw
e i rcee n t e r .
We see t h atth e r iees x c el le n t
agreement
between
a MOM s o l u t i o sna m p l i n tgh w
e ire
currentat1/60wavelengthincrements,
and thelowr e s o l u t i o n (0.1 wavelength) FD-TD contourpathmodel.
Fig. 16. Geometry of the Cylindrical Shielding Enclosure and
Internal Wire or Wire-Pair [29]
Wave p e n e t r a t i o n i n t o t h e i n t e r i o r o f t h e e n c l o s u r e i s
through a c i r c u m f e r e n t i a ls l o ta p e r t u r e
(0.125 m a r c
l e n g t h , 0.0125 m gap) a t h e
ground
plane.
For
the
cases
studied,
an i n t e r n asl h o r t i n gp l u gi sl o c a t e d
0.40 m above t h e
ground
p l a n eF. otrh e
single-wire
t e s t , a w i r e o f l e n g t h 0.30 m and r a d i u s 0.000495 m i s
c e n t e r ew
d i t h itnhien t e r i o r
and connected t toh e
groundplanewith
a lumped 50-ohm l o a d .F o rt h ew i r e p atier spt ,a r a l l ewli r eotshf e sdei m e n s i o nasr e
l o c a t e d 0.01 m a p a r tw
, ith
one w i r es h o r t e dt ot h e
ground
plane
and t h e
o t h ecr o n n e c t e d
to
the
ground
p l a nwei t h
a lumped 50-ohm load.
A
ll r e s u l tasr e
n o r m a l i z e dt o a 1 v/m i n c i d e n t wave e l e c t r i c f i e l d .
The FD-TD contourpath
model can be extended t o
t r e atth i w
n irb
eundles,
as w e l l as s i n g l w
e ires.
F i g : 15 shows t h ea n a l y t i c avl a l i d a t i o nr e s u l t sf o r
t h ei n d u c e dc u r r e n t s
on a bundlecomprised o f 4 wires,
where 3 a r eo ef q u alle n g t h .
Here, a w i r eo fl e n g t h
0.6 m (2.0wavelengths)
i s assumed a t h ec e n t e r
of
thebundle,
and t h r e ep a r a l l e lw i r e so fl e n g t h
:0.3 m
(1.0 wavelength)
are
assumed t o be l o c a t e da t
120"
separations on a c o n c e n t r i cc i r c l eo fr a d i u s
0.005 m
( 1 / 6 0 wavelength).
The r a d o
i ai f w
l l i r e istnh e
0.001 m (1/300 wavebundle
are
equal
and s etto
length).
The assumed e x c i t a t i o n
i isn
free
space,
p r o v i d e d by a 1-GHz broadside TM plane wave. Followi n gt h et e c h n i q u eo f
C291, thebundle i s r e p l a c e d by a
s i n g l ew i r eh a v i n gv a r y i n ge q u i v a l e nrt a d i u sc o r r e sponding t ot h et h r e es e c t i o n sa l o n gt h eb u n d l ea x i s .
The p h y s i c so ft h es i n g l ew i r eo fv a r y i n ge q u i v a l e n t
r a d i u si si n c o r p o r a t e di n
a l o w - r e s o l u t i o n (0.1 wavel e n g t h ) FD-TD contourpath
model, as discussed above.
The FD-TD model i s t h e nr u nt oo b t a i nt h et a n g e n t i a l
E and H f i e l d a
st
a v i r t u as lu r f a cceo n v e n i e n t l y
locatedatthecell
b o u n d a r yc o n t a i n i n gt h ee q u i v a l e n t
w i r e (shown
as
a dashed l i nieFn i g1.5 ) .
These
f i e l d sa r et h e nu t i l i z e d
as e x c i t a t i o nt oo b t a i nt h e
c u r r e n t si n d u c e d on t h e i n d i v i d u a l w i r e s o f t h e o r i g i n a lb u n d l e .T h i sl a s ts t e pi sp e r f o r m e db ys e t t i n g
up
an e l e c t r i c f i e l d i n t e g r a l e q u a t i o n ( E F I E )
and s o l v i n g
v i a MOM.
F i g . 15 shows an excellent
correspondence
between t h e r e s u l t s o f t h e h y b r i d
FD-TD/MOM procedure
described aboveand t h e u s u a l d i r e c t
EFIE s o l u t i o n f o r
t h ei n d u c e dc u r r e n td i s t r i b u t i o n
oneach
w i r eo ft h e
bund1 e.
The h y b r i d FD-TD / MOM p r o c e d u r e f o r m o d e l i n g t h i n
wirebundles i s mostuseful when thebundle i s l o c a t e d
w i t h i n a s h i e l d i ne gn c l o s u rFei.g .
17 shows t h e
e x p e r i m e n t a vl a l i d a t i o nr e s u l t sf o rt h ev a r i a t i o no f
i n d u c e dl o a dc u r r e n tw i t hf r e q u e n c yf o r
a s i n g l ew i r e
and a w i r e - p a i r l o c a t e d a t t h e c e n t e r o f t h e c y l i n d r i The enclosure
calenclosuredepicted
i n F i g . 16 [291.
a
i s 1.0 m high, 0.2 m i n diameter,andreferencedto
large
metal
ground
plane.
Approximate
plane
wave
e x c i t a t i oinpsr o v i d e d
an
by
e l e c t r i c al yl -a1r g e
c o n i c a l monopole r e f e r e n c e dt ot h e
same groundplane.
-.-
Hybrid FD-TD /MOM w i t h
VI
0.6
'
DX = 1.2144 cm:
d a t aa t
15 cyclesbelow
1.1 GHz,
above
30 cycles
s
. .
1.1 GHz
I
Frequency IGHz)
Fig. 17(a)
From F i g . 17, we see t h a t t h e r e i s
a good c o r r e spondencebetween t h e measuredand numerically modeled
w i r el o a dc u r r e n ft o rb o t ht e s t
cases.
The two-wire
t e spt r o v e dt o
b e s p e c i a l l cy h a l l e n g i n gs i n c et h e ,
coupling
response
(center
observed Q f a c t oortfh e
Continued on page 14
13
0.8
,
IEEE Antennas and Propagation Society Newsletter, April 1988
r
h
1 mW/cm* plane wavepower
density
Hybrid FD-TD / MOM with
OX = 1.23 cm;
@
d a t aa t
30 cycles except a t
140,
1.141 GHr and 1.15 GHz.
180
1
B @
i..
J"
0.8
0.3
Y "
1.0
&,Jh
1.2
1.1
Frequency (GHz)
Fig. 17. Comparison of Hybrid FD-TD / MOM Modeling Predictions With Experimental Data For Induced Load Current:
(a) Single Wire in Shielding
Enclosure; (b) Wire-Pair in
Shielding Enclosure [29]
Feature Article-Continued
from page 13 .
Fig. 19a. FD-TD Computed Contour Map of the Specific Absorption Rate (SAR) Distribution Along a Horizontal Cut
Through the Head of the 3-D Inhomogeneous Man Model
(350 MHz) [32]
f r e q u e n cdyi v i d e d
by the
half-power
bandwidth)
is
quith
e i g ha, b o u t
75.
Indeed, it i s f o u n dt h atth e
FD-TD code
has
t o be stepped
through
as
many as 80
c y c l etsao p p r o x i m a t e lrye a c thh sei n u s o i d aslt e a d y
s t a tfeoerx c i t a t i ofnr e q u e n c i ense at rhree s o n a n t
peak.
However, s u b s t a n t i a lfl ey w ec ry c l eo
t isfm e
s t e p p i na gr e
needed away from
the
resonance,
as
indicatedinthefigure.
I nf a c t ,
one o tf h ee a r l i e s at p p l i c a t i o n so f
FD-TD
i n v o l v e dt h ec o n s t r u c t i o nodf e t a i l e d
inhomogeneous
t i s s u em o d e l so tf h e
human eye t oo b t a i np r e d i c t i v e
d a t af o r
UHF / m i c r o w a v ep e n e t r a t i o n
and h e a t i n g E41.
The emergence o f supercomputers
has
recently
p e r m i t t e d FD-TD t o be s e r i o u s l y a p p l i e d t o a number o f
it was
i m p o r t a n tb i o - e l e c t r o m a g n e t i cp r o b l e m s .F i r s t ,
shown t h a t FD-TD p r o v i d e es x c e l l e n t
agreement w i t h
s e r i e ss o l u t i o n sf o rt h ep e n e t r a t i n gf i e l dd i s t r i b u t i o n s w i t h i n homogeneous and l a y e r e dt i s s u ec y 1i n d e r s
and
spheres
C301, E311.
F i g . 18, taken
from
C301,
shows t h
ae
n a l y t i cvaal l i d a t i o
r ens u lfttohsre
p e n e t r a t i n ge l e c t r i cf i e l dv e c t o r
components w i t h i n a
The i n n e r
0.15 m r a d i u sm u s c l e - f a tl a y e r e dc y l i n d e r .
l a y e r( r a d i u s
= 0.079 m) i s assumed t o be
comprised
o f muscle
having
a r e l a t i v ep e r m i t t i v i t yo f
72 and
c o n d u c t i v i t yo f
0 . 9 Slm.
The o u t e lra y e r
i s assumed
t o becompr-ised
of fat having a re1 ative permittivity
o f 7.5and
c o n d u c t i v i t yo f 0.048 Slm. TE i l l u m i n a t i o n
a t a f r e q u e n c yo f
100 MHz i s modeled.
From
F i g . 18,
we see t h a t t h e FD-TD s o l u t i o n f o r t h e i n t e r n a l f i e l d s
agreesvery
we1 1 w i t ht h ee x a c ts o l u t i o n ,d e s p i t et h e
f a c tt h a t a stepped-edge(staircase)approximationof
t h ec i r c u l a rl a y e rb o u n d a r i e si s
used.
8. PENETRATION MODELS FOR BIOLOGICAL TISSUES
Two c h a r a c t e r i s t i c so f FD-TD cause it t o be v e r y
p r o m is i n gf o r
model ing electromagnetic wave i n t e r a c t i o n sw i t hb i o l o g i c a tl i s s u e s :
( 1 ) E l e c t r i c a l media
f o r each v e c t o rf i e l d
can
be
s p e c i f i e di n d e p e n d e n t l y
component, so t i s s u e so f
enormous complexity can
be
s p e c i f i e di np r i n c i p l e ;
and ( 2 ) The requiredcomputer
resources f o r t h i s t y p e o f d e t a i l e d v o l u m e t r i c
model i n ga r ed i m e n s i o n a l l yl o w o, n l yo of r d e r
N, where N
i st h e
number o f space c e l l s i n t h e FD-TD l a t t i c e .
I
0 30
'
h '
I
A f t e rv a l i d a t i o no f
FD-TD models o f p e n e t r a t i n g
f i e l d sf o rc a n o n i c a lb i o l o g i c a lt i s s u e
shapes, a t t e n t i o nt u r n e dt o w a r dm o d e l i n gh i g h l yr e a l i s t i c
inhomogeneous t i s s uaep p r o x i m a t i o n s
o f t h e human body.
S p e c i f i ce l e c t r i c apl a r a m e t e r sw e r ea s s i g n e dt o
each
o ft h ee l e c t r i cf i e l dv e c t o r
components a t t h e 16,000
t o 40,000 space c e l l s comprising
the
body
model.
Assignmentswerebaseduponcross-sectiontissue
maps
o f t h e body ( a t s p a c i n g s o f a b o u t
one inch, as o b t a i n ed v i a c a d a v es tr u d i e a
s )v a i l a b l e
itnhm
eedical
of
tissue
1it e r a t u r e , and cataloged measurements
d i e l e c t r i cp r o p e r t i e s .
Space r e s o l u t i o n s as f i n e as
0.013 m t h r o u g h o tuhe
t en t i r e
human body
proved
l 9 b , takenfrom
p o s s i b l eu s i n g
FD-TD.
F i g s . 19a
and
E321,
show
t h e computed contour maps o ft h es p e c i f i c
a b s o r p t i orna t e
(SARI d i s t r i b u t i o an l o n hg o r i z o n t a l
c u t st h r o u g ht h e
headand
1iv e r , r e s p e c t i v e l y , o f t h e
In Fig
three-dimensional inhomogeneous man model
19a, t h ei n c i d e n tp l a n e
wave has a power d e n s i t y of
1 mW/cm2 a t 350 MHz, and
each
contour
i s 20 rnW1Kg.
+ Y
I
018m
0.12
O.
0.00
O'
E
0.00
0.
.
Fig. 18. Comparison of FD-TD and Exact Solution for Penetrating
Electric Field Vector Components Within
a Circular Muscle
- Fat Layered Cylinder, TE Polarization, 100 MHZ I301
.
Continued on page 1.
14
IEEE Antennas and Propagation Society Newsletter, April 1988
1 mW/cm2 plane wave power d e n s i t y
I
Fig. 19b. FD-TD Computed Contour Map of the Specific Absorption Rate (SAR) Distribution Along a Horizontal Cut
Through the Liver of the 3-D Inhomogeneous Man Model (100 MHz) [32]
Feature Article-Confinued from page 14
I nF i g .
19b, t h ei n c i d e n t
wave has t h e same power
100 MHz;
contours i nt h e arms a r e
d e n s i t yb u ti sa t
a t 20 mW/kg i n t e r v a l s ,w h i l ec o n t o u r si nt h e
body a r e
a t 10 mW/kg i n t e r v a l s .
These contour maps i l l u s t r a t e
t h eh i g h 1eve1 o f d e t a i l o f l o c a l f e a t u r e s o f t h e
SAR
d i s t r i b u t i o n t h a t i s p o s s i b l e v i a FD-TD modeling.
I n C353, FO-TD i s f i r s t
used t oo b t a i nr e s o n a n t
f r e q u e n cs o
ieefvsetrharle e - d i m e n s i o nc a vl i t i e s
loaded
by
d i e l e c t br ilco c kNse. txrhte,es o n a n t
f r e q u e n c yo f
a f i n l i n ec a v i t yi s
computed. L a s t t, h e
resonantfrequenciesof
a m i c r o s t r i pc a v i t y on anisot r o p si cu b s t r a taeroeb t a i n e d ,
and t hdei s p e r s i o n
c h a r a c t e r i s t i c so ft h em i c r o s t r i p
used i nt h ec a v i t y
a r ec a l c u l a t e d .
FD-TD m o d e l i n gr e s u l t a
s re
compared
p r i m a r i ltytoh o soeb t a i n euds i ntghter a n s m i s s i o n
l i n em a t r i x
(TLM) approach,
and
thetwo
methods a r e
foundtogivepracticallythe
same r e s u l t s .
More r e c e n t workhas
d e p a r t e df r o ms i m u l a t i o n so f
plane wave i l l u m i n a t i o n o f t h e
human body. C u r r e n t l y ,
FD-TD i s b e i n gu s e dt o
model annular phased a r r a y so f
a p e r t u r e and dipole
antennas
used f ohry p e r t h e r m i a
C331. A 17,363 c e l l , 0.013 m r e s o l u t i o n ,a n a t o m i c a l l y
basedmodel
o ft h e human torsosurrounded
by p b o l u s
o fd e i o n i z e dw a t e r
i s used f o rc a l c u l a t i o n so f
SARs.
Testruns
on t h ec a l c u l a t i o no ff i e l d si nt h ew a t e r f i l l e d i n t e r a c t i o n spaceand w i t h homogeneous c i r c u l a r
and e l l i p t i c a lc y l i n d e r
phantoms c o r r e l a t ew e l w
l ith
the
experimental
data
ti hnl iet e r a t u r lee, n d i n g
s u p p o r tt ot h ea c c u r a c yo ft h e
FD-TD method f o rn e a r f i e l d exposureconditions C331.
9.
Continued on poge 16
MICROSTRIP AND MICROWAVE CIRCUIT MODELS
Recently, FD-TD modeling
has
been extended
p r o v i ddee t a i l ecdh a r a c t e r i z a t i o nm
osfi c r o s t r i p s ,
r e s o n a t o r fsi,n l i n e s ,
and
two-dimensional
microwave
c i r c u i t sI.n
C341,
FD-TD
i s used t co a l c u l a t teh e
d i s p e r s i v ec h a r a c t e r i s t i c so f
a t y p i c a lm i c r o s t r i p
a g a l l i ua m
r s e n isdueb s t r a t e .
A Gaussian
pulse
e x c i t a t i oi ns
used, and t heef f e c t i vdei e l e c t r i c
c o n s t a n t and c h a r a c t e r i s t i c impedancevs.frequency
e f f i c i e n t l yo b t a i n e do v e r
a broadfrequencyrangevia
F o u r i e rt r a n s f o r mo ft h et i m e - d o m a i nf i e l dr e s p o n s e .
to
on
is
Fig. 2O(a)
1s
I E I E Antennas and Propagation Society Newsletter, April 1988
g i v edni r e c ttilh
nyl ae b o r a t o r y
frame.
This
is
accomplishedbyimplementingtheproper
relativistic
b o u n d a r yc o n d i t i o n sf o rt h ef i e l d sa t h es u r f a c e
of
themovingconductor.
Scattered fie1
Total fields
Vibrating mirror
E
2
6
4
8
le
(b)
12
Lattice truncation
f (GHzl
Fig. 20. Comparison of FD-TD Modeling Predictions With
Measurements of 1 S21 I for a Two-Port Microstrip Ring
Circuit: (a) Geometry and Griddingof Microstrip Circuit;
(b) Comparative Results Over 2 - 12 GHZ [36]
Feature Article-Continued
- Analytical
z/d=-50
2.0
from poge 15
results
/9=0.2, kd= 1
I n [36], a m o d i f i e dv e r s i o no f
FD-TD i s presented
wphr oi cvchiednetsr a l - d i f f e rtei m
n cee- s t e p p i n g
e x p r e s s i o n sf o rd i s t r i b u t i o n so vf o l t a g e
and s u r f a c e
current density along arbitrary-shaped two-dimensional
microwave c i r c u i t s .T h i s
approach i sq u i t ed i f f e r e n t
f r o mt h a to f[ 3 4 1
and C351, which u t i l i z e t h e o r i g i n a l
v o l u m e t r ifci e lsda m p l i ncgo n c e pf o
tr
FD-TD.
As a
r e s u l t ,t h e
method o f[ 3 6 ]r e q u i r e sf e w e r
unknowns t o
be solved, and avoidsthe
need f o r a r a d i a t i o n bounda cr yo n d i t i o n .
However, an a u x i l i a rcyo n d i t i oi ns
r e q u i r edt ode s c r i bt hel oe a d i negf f e c t s
o f the
f r i n g i nf ige l dtahst e
edges tohm
f ei c r o s t r i p
conductingpaths.Fig.
20, takenfrom C361, shows t h e
I S 2 1 1 , as a f u n c t i o no f
FD-TD computed S parameter,
f r e q u e n c yf o r a t w o - p o r tm i c r o s t r i pr i n gc i r c u i t .
The
r i n gc i r c u i t ,g r i d d e d
as shown i nt h ef i g u r e ,
hasan
7 mm, s u b s t r a t e
i n n e rr a d i u so f
4 mm, o u t e rr a d i u so f
10 and r e l a t i v ep e r m e a b i l i t y
r e l a t i v ep e r m i t t i v i t yo f
and i s connected t o two
o f 0.93 ( s i m u l a t i n gd u r o i d ) ,
50-ohm 1ines
making
a 90" angle.
The broadband
response o tf h ec i r c u i it so b t a i n e du s i n g
a single
FD-TD r uf onr
an a p p r o p r i aptuelesxec i t a t i o n ,
f o l l o w e d by F o u r i et rra n s f o r m a t i ootnhfdee s i r e d
20, we see
response
time-domain
waveform.
From
Fig.
goodagreement
o ft h ep r e d i c t e d
and measured c i r c u i t
responseoverthefrequencyrange
2 - 12 GHz and a
dynamicrange o f about 30 dB. E361 c o n c l u d e st h a tt h e
a p p l i c a t i o oni tf s
FD-TD approach t ao r b i t r a r i l y shaped m i c r o s t r i pc i r c u i t si s
encouraging,
but
more
work i s needed t o d e t e r m i n et h em o d e l i n gl i m i t a t i o n s ,
e s p e c i a l l ya th i g h e rf r e q u e n c i e s
wheremedia
d i spersioncan become i m p o r t a n t .
IE,I
1 .o
H
FD-TO
0.0
0
2n
71
Rt
(b)'
Fig. 21. Comparison of FD-TD and Analytical ResultsForthe
Envelope of the Scattered E Field VS. Time For a
Monochromatic Plane WaveIlluminating a Vibrating
Mirror at 30" [37]
F i g . 21 shows r e s u l t s f o r one of t h e more i n t e r e s t i n g problems o ft h i st y p e
modeled so f a r ,t h a to f
o b l i q u ep l a n e wave i n c i d e n c e onan
infinitevibrating
m i r r o rT. h i sc a s e
i s much more complicatedthanthe
normalincidence
case, i n t h a t it hasnoclosed-form
solution.
An analysis
presented
i n t h e 1it e r a t u r e
C381 w r i t e st h es o l u t i o ni n
an i n f i n i t es e r i e sf o r m
usingplane-waveexpansions,wherethe
unknown coeff i c i e n t si nt h es e r i e sa r es o l v e dn u m e r i c a l l y T. h i s
a n a l y s is e r v e s
as t h eb a s i o
scf o m p a r i s o nf otrh e
FD-TD model r e s u l t fsotrh et i m ev a r i a t i o notfh e
s c a t t e r e df i e l de n v e l o p ea pt o i n t sn e a tr h em i r r o r .
10. SCATTERING MODELS FOR RELATIVISTICALLY
MOVING SURFACES I N ONE ANDTU0 DIMENSIONS
Since it idsi f f i c u tl to
model e x a c t l y an
i n f i n i t pe l a n m
e i r r oi rn
a f i n i t e two-dimensional
g r i d , a l o n gt,h i nr,e c t a n g u l apr e r f e c t l y - c o n d u c t i n g
s l a bi s
used
as
t h em i r r o r
model, as shown i nF i g .
21a.
R e l a t i v i s t i c b o u n d a r yc o n d i t i o n sf o rt h ef i e l d s
are
implemented
on t h ef r o n t
and
back
s i d e so tf h e
slab.
The o t h e tr w os i d e s p, a r a 1l e 1t ot h ev e l o c i t y
v e c t o r ,a r ei n s e n s i t i v et ot h em o t i o no ft h es l a b ,
and
t h e r e f o r e no r e l a t i v i s t i c boundary
conditions
are
r e q u i r e dt h e r e .
To m i n i m i z et h ei m p a c ot f
edge d i f f r a c t i o nt,h e
slab
length
i cs a r e f u l l sy e l e c t e d
so
A n a l y t i c a l v a l i d a t i o n s havebeen r e c e n t l y o b t a i n a monochromatic
ed f o r FD-TD model s o f r e f l e c t i o n o f
p l a n e wave by a p e r f e c t l yc o n d u c t i n gs u r f a c ee i t h e r
moving a t a u n i f o r m r e l a t i v i s t i c v e l o c i t y o r v i b r a t i n g
a t a frequency and a m p l i t u d el a r g e enough so t h a t t h e
s u r f a c ea t t a i n sr e l a t i v i s t i c
speeds C371.
The
FD-TD
approach o f C371 i s novel i n t h a t i t does not r e q u i r e
a s y s t e mt r a n s f o r m a t i o n
where theconductingsurface
i sa tr e s t .I n s t e a d ,t h e
FD-TD g r i d i s a t r e s t i n t h e
l a b o r a t o r y frame,
and
t h e computed f i e l ds o l u t i o ni s
Continued on poge 17
16
IEEE Antennas and Propagation Society Newsletter, April 1988
Feature Article-Continued
from page 16
Original
profile
Reconstructed
values
\
t h atth es l a ba p p e a r st o
be i n f i n i t ei ne x t e n ta t
o b s e r v a t i o pn o i n t ,
P , d u r i n g a w e l l - d e f i n e ed a r l y timeresponse when t h e edge e f f e c t has n o t y e t propag a t etdo
P
Since
the
TM case
does
n optr o v i d e
substantially different results than the
TE case C381,
o n l yt h e TE case i s considered.FromFig.
21b, we see
good agreement
between
the
FD-TD and a n a l y t i c a l
r e s u l t so b t a i n e df r o m
C381 f o rt h e
envelope o ft h e
sc:ttered
E f i e l d vs.time,for
an i n c i d e n ta n g l e o f
30 , peak m i r r o r speed 20% t h a t o f l i g h t , andobservat i o np o i n t sz / d
= -5 andz/d
= -50 , where kd = 1
S i m i l a r agreement i s f o u n df o r
an
even
more
oblique
angle, 60" C371.
T h i s agreement i ss a t i s f y i n gs i n c e
is
the action of the relativistically vibrating mirror
socomplicated,generating
a r e f l e c t e d wave having a
spreadboth i n f r e q u e n c ya n ds p a t i a lr e f l e c t i o na n g l e ,
as w e l l asevanescent modes.
.
.
11.
INVERSE SCATTERING RECONSTRUCTIONS
IN ONE AND MO DIMENSIONS
I n i t i a l workhasdemonstratxedthe
possibility of
a c c u r a t e l yr e c o n s t r u c t i n go n e - d i m e n s i o n a p
l r o f i l e so f
p e r m i t t i v i t y and c o n d u c t i v i t y C391, and t h e shapeand
d i e l e c t cr iocm p o s i t i o n
o f two-dimensional
targets
C401,C411
f r o mm i n i m a ls c a t t e r e df i e l dp u l s er e s p o n s e
data.
The general
approach
involves
setting
up a
numerical
feedback
loop
which
uses
a one- o r twodimensional
FD-TD
code
as
a forward-scattering
element,
and
a s p e c i aclol yn s t r u c nt eodn l i n e a r
o p t i m i z a t i o n code
as
the
feedback
element.
FD-TD
generates a t e s tp u l s er e s p o n s ef o r
a t r i a ll a y e r i n g
The t e spt u l s ies
ot ra r g e t
shape / composition.
compared t o t h e measuredpulse,andan
errorsignalis
developed.
Working
on
t h iesr r osri g n a tl ,h e
nonl i n e a ro p t i m i z a t i o ne l e m e n tp e r t u r b st h et r i a ll a y e r i n go rt a r g e ts h a p e / c o m p o s i t i o ni n
a manner t o d r i v e
down t h e r r o r ,
Upon r e p e a t e idt e r a t i o n st ,h pe r o posed l a y e r i n g or t a r g ei dt e a l lcyo n v e r g e s
t o the
a c t u a l one, a s t r a t e g y s i m i l a r t o t h a t o f
C421.
The advantage o fw o r k i n gi nt h et i m e
domain i s
t h a t a l a y e r e d medium o rt a r g e t
shape
can
be
recons t r u c t e ds e q u e n t i a l l y i n t i m e as t h ew a v e f r o n to ft h e
i n c i d e npt u l s e
sweeps through,
taking
advantage
of
causality.
This
reduces
the
complexity
reconof
s t r u c t i o ns i n c eo n l y
a p o r t i o notfh el a y e r i n g
or
t a r g e t shape i s being
generated
a t each i t e r a t i o n .
Advanced s t r a t e g i e s f o r r e c o n s t r u c t i o n i n t h e p r e s e n c e
o f a d d i t i v e n o i s e may i n v o l v e t h e u s e o f p r e d i c t i o n
/
c o r r e c t i o n , where t h ter i al a
l y eortra r g e t
shape
i s c o n s i d e r e dt ob e
a p r e d i c t o r o f t h ea c t i a l
case,
which i s subsequentlycorrected by o p t i m i z a t i o n o f t h e
e n t i r el a y e r e d
medium otra r g eut s i n gt h ec o m p l e t e
scattered pulse.
F i g . 22 shows t h ea p p l i c a t i o no ft h eb a s i c
FD-TC
feedbackstrategy
t o a one-dimensionallayered
medium
i n t h e absence o fn o i s e .B o t ht h ee l e c t r i c a p
l ermitt i v it y
and c o n d u c t i v i t oyt fh e
medium vary i n a
"sawtooth" manner wdiet hp t h .
The curves show
simulated measured d afttoahrre f l e c t epdu l s e
f otrh r e e
cases
defined
by
the
peak values of t h e
c o n d u c t i v i t y (0.001 S/m, 0.01 S/m, and 0.1 S/m) and
t h ec o r r e s p o n d i n gs p a t i a l l yc o i n c i d e n t
peak v a l u e so f
medium.
r e l a t i v ep e r m i t t i v i t y
(3, 2, and 4 ) o tf h e
I n eachcase,
t h ei n c i d e n tp u l s e
i s assumed t ob e a
50 cm betweenzerocrossings.
half-sinusoidspanning
N o t itnhtdghaae
dt rosktusp e r i m p o s e d
on t h e
" s a w t o o trhe"p r e s etrh
nee
t c o n s t r u c tveadl uoefs
p e r m i t t i v i t y and c o n d u c t i v i t y , we see t h a tt h eb a s i c
i s quite
s u c c e s s f ui ln
the
FD-TD feedback
strategy
absence o f n o i s e C391.
Continued on page I8
Fig. 22. Application of theFD-TD
/ FeedbackStrategy
to
Reconstruct A I-D Sawtooth Variation of Electrical Permittivity and Conductivity in the Absence of Noise [39]
4.0
0.01 S f m
2.0
0.01
Exact Profiles
Reconstructed Profiies
- 40 dB SIN
ReCOnStrMCted Profiles
- 20 dB SIN
Fig. 23. Applications of the FD-TD / FeedbackStrategy to
Reconstruct a 2-D Lossy Dielectric Target in the Presence
of Noise [41]
IEEE Antennas and Propagation Society Newsletter, April 1988
Table 1. Computation Times
F i g . 23 shows t h ea p p l i c a t i o no ft h e
FD-TD feedback s t r a t e g yt or e c o n s t r u c t
a two-dimensionallossy
10-WavelengthModel
d i e l e c t r i ct a r g e t .
The t a r g e ti s
a 30 cm x 30 cm
Present FD-TDCode
Machi ne
squarecylinderhaving
a u n i f o r mc o n d u c t i v i t yo f
0.01
S/m,
and a t e n t - l i k er e l a t i v ep e r m i t t i v i t yp r o f i l e
VAX 111780 ( n o f l o a t i n g p o i n t
'which s t a r t s a t 2.0 a t t h e f r o n t
and l e f ts i d e s
and
40.0 h
accelerator1
i n c r e a s e sl i n e a r l yt o
a peak v a l u e o f 4.0 a t t h e back
Cray-2 ( single processor, us? ng
;corner
on
t h er i g h ts i d e .
These p r o f i l e sa r ei l l u s rnin
t h e VAX F1 o2 r. t0r a n )
t r a t e d i n a p e r s p e c t i v e manner a t t h e t o p o f F i g .
23.
Cray-2
(
s
i
n
g
l
e
p
r
o
c
e
s
s
o
r
,
some
The t a r g e t i s assumed t o b e i l l u m i n a t e d by a TM p o l a r code o p t i m i z a t i o n )
3.0 min
i z e dp l a n e wave t h a t i s d i r e c t e dt o w a r dt h ef r o n to f
t h et a r g e t( a sv i s u a l i z e da tt h et o po fF i g .2 3 ) .
The
Cray-2 ( f o u rp r o c e s s o r s , some
i n c i d e n t waveform i s a 3 - c y c l es i n u s o i d a tl o n eb u r s t
code o p t i m i z a t i o n )
( e s1t . m
) in
having a 60-MHz carrier
frequency.
For
the
reconmachine
Gflop10 True
2 sec ( e s t . )
s t r u c t i o n ,t h eo n l yd a t au t i l i z e di st h e
time-domain
waveform o ft h es c a t t e r e de l e c t r i cf i e l d
asobserved
-*
a t twopoints.
These p o i n t sa r el o c a t e d
1 m f r o mt h e
1.55 106 unknown f i e l d components,661
timesteps
f r o nottfh et a r g e t ,
and a rpeo s i t i o n e1d5
cm t o
e i t h e sr i d eotfh et a r g ect e n t e lri n e .
To s i m u l a t e
a s s o c i a t e d memories
arranged
i n a h i g h l ye f f i c i e n t
measureddata,thecomputedscatteredfield
waveforms
manner for
processor-to-procesor
communication.
With
a r ec o n t a m i n a t e dw i t ha d d i t i v eG a u s s i a nn o i s e .I n
a1 1
t h e CM, a s i n g l ep r o c e s s o rc a nb ea s s i g n e dt os t o r e
o ft h er e c o n s t r u c t i o n s ,t h et a r g e t
shapeand l o c a t i o n
and time-step a s i n g l e r o w o f v e c t o r f i e l d
components
i s assumed t o be known.
i n a three-dimensional FD-TD space l a t t i cFeo. r
example, 1.5 * l o 6 p r o c e s s o r sw o u l db es u f f i c i e n t o
From F i g . 23, we see t h afto r
a signal/noise
s t o r et h e 6 C a r t e s i a n components o f E end tI f o r each
r a t i o o f 40 dB, t h e a v e r a g e e r r o r i n t h e p e r m i t t i v i t y
o tf h e
500 x 500 rows o f a c u b i cl a t t i c es p a n n i n g
and c o n d u c t i v i t yp r o f i l e si s
1.5% and
2.3%,
respect50 wave1 engths(assuming
10 c e l l d w a v e l e n g t h r e s o l ui v e l y . I f t h es i g n a l / n o i s er a t i oi sr e d u c e dt o
20 dB,
t i o n ) . FD-TD timesteppingwouldbeperformedviarow
ta
h ve e r a e
g rer oi rnsc r e a tsoe
6.9%
and
10.4%,
CM processors..
o p e r a t i o n s mapped o n t ot h ei n d i v i d u a l
r e s p e c t i v e l y C411.
Research i s ongoing t do e t e r m i n e
These rowoperationswouldbeperformedconcurrently.
means oifm p r o v i n gt h en o i s ep e r f o r m a n c ee, s p e c i a l l y
Thus, f o r a f i x e d number otfi m se t e p st,h teo t a l
u s i n gp r e d i c t o r / c o r r e c t o rt e c h n i q u e sb r i e f l yd i s c u s s e d
r u n n i n gt i m ew o u l db ep r o p o r t i o n a lt ot h et i m e
needed
e a r l i e rG
. i v e nt h er e l a t i v e l ys m a l l
amount of scattpoe r f o r m
a s i n g l e row
operation,
which
i nt u r n
teredfielddatautilized,the
FD-TD feedbackstrategy
w o u l db ep r o p o r t i o n atlot h e
number o fv e c t o rf i e l d
appearspromisingforfuturedevelopment.
components i nt h e
row, o r O ( N ' / ' 1
For
the
50w a v e l e n g t hc u b i cl a t t i c en o t e d
above, t h i s wouldimply
a dimensional
reduction
of
the
computational
burden
12. LARGE-SCALE
COMPUTER
SOFTMARE
from O(500'
t o O(5001, a tremendous b e n e f i t . As a
r e s u l t , it i s c o n c e i v a b l et h a t
a s u i t a b l ys c a l e d
CM
The FD-TD method i s n a t u r a l l ys u i t e df o rl a r g e c o u l d modelone
l o o ka n g l eo f
a 50-wavelengththreescale
processing
by
state-of-the-art
vector
superd i m e n s i o n a ls c a t t e r e ri no n l y
a fewseconds,achieving
computersandconcurrentprocessors.This
i s because
e f f e c t i vf leo a t i npgo i rnat t tei hnoser d oe fr
e s s e n t i a l l ya l lo ft h ea r i t h m e t i co p e r a t i o n si n v o l v e d
100 Gflops.Forthisreason,
FD-TD a l g o r i t h md e v e l o p i n a t y p i c a l FD-TD r u nc a nb ev e c t o r i z e do rc a s ti n t o
ment f o r t h e CM i s a p r o m i s i n g a r e a o f r e s e a r c h .
O(N)
a h i gchol yn c u r r feonrtmFaut .r t htehre,
demand forcomputer
memory and clcckcycles(where
N
i s t h e number o f l a t t i c e space c e l i s ) i s d i m e n s i o n a l l y
13. CONCLUSION
low,and
permitsthree-dimensional
FD-TD mode7s span100wavelengths t o b e a n t i c i p a t e d by1990.
n i n g 50
Recent
advances
i n FD-TD modeling
concepts
and
software
implementation,
combined w i t h advances i n
L e t us now considercomputationtimes
o f present
computer
technology,
have
expanded
the
scope,
FD-TD codes. Table 1 l i s t s computationtimes(derived
accuracy, and
speed
o f FD-TD m o d e l i n gt ot h ep o i n t
e i t h e r frombenchmark r u n s o r based on a n a l y s t s ' e s t i where i t may be t h r e
preferred
choice
f ocr e r t a i n
mates) f o rm o d e l i n g one l o o ka n g l eo f
a 10-wavelength
t y p e so fe l e c t r o m a g n e t i c
wave s c a t t e r i n g and c o u p l f n g
three-dimensional
scatterer
using
the
present
FD-TD
p r o b l e m sT.h iasr t i c l e
has
attempted
tpor o v i d e
a
'code.Fourcomputingsystemsare
l i s t e di nt h et a b l e .
s u c c i n c ts t a t e - o f - t h e - a r tr e v i e wo f
FD-TD model ing
!The f i r s t i s t h e D i g i t a l Equipment VAX 11/780,without
a p p l i c a t i o n s . The
reader
is
r e f et hrte
roe d
; f l o a t i n gp o i n ta c c e l e r a t o r .
The secondand t h i r d a r e ,
journalpapers
and r e p o r t s l i s t e d below f o rd e t a i 1s
r e s p e c t i vseilnyg, l e - p r o c e s s o r
and four-processor
o f t h e FD-TD a l g o r i t h m s and a p p l i c a t i o n s .
v e r s i o n so ft h e
Cray-2.
The f o u r t hi s a h y p o t h e t i c a l
n e x t - g e n e r a t i o nm a c h i n eo p e r a t i n ga ta na v e r a q er a t e
o f 10 G f l o p s ( 1 0 - b i i l i o n f l o a t i n g p o i n t o p e r a t i o n s p e r
14. REFERENCES
second).Thislastcomputer
i sg e n e r a l l ye x p e c t e dt o
b ea v a i l a b l ea b o u t
1990
1992.
c11 K. S . Yee, "Numericalsolution o f i n i t i a l boundar y Valueproblems
in v o l v i ngMaxwell s equations
From Table 1, it if sa i r lcyl e at hr as tt e a d i l y
IEEE Trans.
i si no t r o p i c
media,"
Antennas
\advancingsupercomputertechnology
wil p e r m i t r o u t i n e
Propaqat.,vol.
AP-14, pp. 302-307, May 1966.
e n g i n e e r i n g usage o f FD-TD f o rm o d e l i n ge l e c t r i c a l l y largeproblems by t h e e a r l y 1 9 9 0 ' s .
[2] A. T a f l o v e and M. E. Brodwin,"Numericalsolution
Feature Article-Continued
from page 17
-
.
-
-
An i n t e r e s t i n gp r o s p e c tt h a t
has r e c e n t l ya r i s e n
i st h er e d u c t i o no ft h e
O ( N ) computationalburden
of
T h i sp o s s i b i l i t yi s
a consequence
FD-TD t o O ( N 1 ' 3 ) .
of
the
appearance
of
the
Connection
Machine
(CM),
whichhastens
o f thousands o fs i m p l ep r o c e s s o r s
and
o fs t e a d y - s t a t ee l e c t r o m a g n e t i cs c a t t e r i n gp r o b I s equalems using
the
time-dependent
Maxwell
tions,"
IEEE Trans.
Microwave
Theory
Tech.,
v01. MTT-23, pp. 623-630, Aug. 1975.
Continued on page 7 9
ILEE Antennas and PropagatiaIn Society Newsletter, Aprll 1988
Feature Article-Continued
[3]
[4]
J. Comp. Physics,
from poge I8
G. A. Kriegsmann, " E x p l o i t i n gt h el i m i t i n ga m p l i t u d ep r i y i p l et on u m e r i c a l l ys o l v es c a t t e r i n g
problems,
Wave Motion,
vol.
4, 371-380,
pp.
1982.
A. Taflove and M. E. Brodwin,"COmpUtatiOn
Of t h e
e le c t r o m a g n e t i c f i e l d s and inducedtemperatures
w i t h i n a model o ft h em i c r o w a v e - i r r a d i a t e d
human
eye,"
IEEE Trans.MicrowaveTheory
Tech. , v o l
MTT-23, pp. 888-896, N ~ v . 1975.
.
[5]
A. T a f l o v e ," A p p l i c a t i o no ft h ef i n i t e - d i f f e r e n c e
time-domain
method
tsoi n u s o i d a l
stead!
state
electromagnetic
penetration
problems,
IEEE
Trans.
Electromaqn.
Compat.,
vol
EMCz
pp. 191-202, Aug. 1980.
.
[6]
A. Tafloveand K. R: Umashankar, " A h y b r i d moment
method / f i n i t e - d i f f e r e n c e t i m e - d o m a i n a p p r o a c h t o
electromagnetic coupling and a p e r t u r e p e n e t r a t i o n
i n t o complex
geometries,"
IEEE Trans.
Antennas
Propaqat.,vol
AP-30, pp. 617-627, J u l y 1982.
.
[ 7 1 R. Holland,
Threde:
A free-field
EMP c o u p l i n g
and s c a t t e r i n g code," IEEE Trans.
Nuclear
Sci.,
v o l . NS-24, pp. 2416-2421, Dec. 1977.
i n press.
[19] A. T a f l o v e and K. R. Umashankar, "The f i n i t e method
for
difference
time-domain
(FD-TD)
interaction
e l e c t r o m a qsnceat itct e r i n g
and
problems,'
J. Electromaqn. Waves and
Appls.,
V O ~ . 1, pp. 243-267,
1987.
[20] A. T a f l o v e and K. R. Umashankar, " A n a l y t i c a l
models f eo lre c t r o m a g n e tsi c a t t e r i n gF, "i n a l
Report RADC-TR-85-87
on Air F o rCcoen t r a c t
F19628-82-C-0140,
Electromagn.
Sci.
Division,
Rome Air Development
Center,
Hanscom AFB, MA,
May 1985.
[21] T. G. Jurgens, A. Taflove, and K. R. Umashankar,
"FD-TD conformal m o d e l oi nf g
smooth curved
1987. U R S I Radio
Science
surfaces,"
Proc.
VA, June1987,
p. 227.
Meeting,Blacksburg,
[22]
M.
Fusco, "FDTD a l g o r i t h mi nc u r v i l i n e a rc o o r d i nates,"
submitted
to
IEEE Trans.
Antennas
Propaqat.
[23] V. Shankarand
W. H a l l , "A t i m e domain d i f f e r e n st o
i ealflvoeercrt r o m a g n es ct iac t t e r i n g
problems,"
Proc.
1988 U R S I National
Radio
ScienceMeetinq,Boulder,
CO, Jan.
1988,
p. 103.
[8]
K.
S. Kunz and K:
M. Lee, " A three-dimensional
f i n it e - d if f e r e n c es o lu t i
on o ft h ee x t e r n a l
response o f an a i r c r a f t t o a complex t r a n s i e n t EM
environmen::
P a r t I-The method
and
i t s implementation,
IEEE Trans.
Electromaqn.
Cornpat.,
v o l . EMC-20, pp. 328-333, May 1978.
[24] B. Beker, K . K. Umashankar, and A. Taflove,
"Numerical
analysis
and v a l i d at thoi eof n
combined f i e lsdu r f a ci en t e g r ae lq u a t i o nf so r
e l e c t r o m a g n e t i cs c a t t e r i n g
by a r bti z a r y
shaped
two-dimensional
anisotropic
objects,
submitted
t o IEEE Trans.AntennasPropaqat.
[9]
D. E. Meriwether, R. F i s h e r , and F. W.
C251
[lo]
Smith, "On
implementing a numericHuygens'source
scheme i n
a f i n i t e - d i f f e r e n c e program t oi l l u m i n a t es c a t t e r i nbgo d i e s , "
IEEE Trans.
Nuclear
Sci.,
vol.
NS-27, pp.1819-1833,
Dee. 1980.
G. Mur,
"Absorbingboundaryconditionsforthe
f i n i t e - d i f f e r e nac pe p r o x i m a t ti tohi fe
m
n edomai n electromagneti c f i e l de q u a t i o n s ," IEEE
Trans,
Electromaqn.
Compat., v o l
EMC-23,
377-382, Nov. 1981.
.
x
[ll]
K. R. Umashankar and A. Taflove, "A novelmethod
t oa n a l y z ee l e c t r o m a g n e t i cs c a t t e r i n go f
complex,
IEEE Trans.Electromaqn.
Compat., v o l .
objects,"
EMC-24, pp. 397-405, N ~ v . 1982.
cros:
[12] A. T a f l o v e and K. R. Umashankar, "Radar
s e c t i o n ofaeneralthree-dimensionalscatterers,
IEEE Trans, Electromagn. Compat., v o l . EMC-25,
pp.433-440,
NO^. 1983.
[13] A. Taflove, K. R. Umashankar, and T. G. Jurgens,
" V a l i d a t i o n o f FD-TD m o d e l i n go ft h er a d a rc r o s s
s e c t i o notfh r e e - d i m e n s i f n asl t r u c t u r e ss p a n n i n g
IEEE Trans.
Antennas
up nine
to wavelengths,
Propaqat.,vol.
AP-33, pp.662-666,June1985.
C141 B. Engqui s t and A. Majda,
"Absorbing
boundary
c o n d i t i ofnosr
nt huem e r isciaml u l a toi of n
waves,"
Math. Comp.,
v o l . 31, pp. 629-651,
J u l y 1977.
and
C. Morawetz, " S o l v i n gt h e
[15] G. A. Kriegsmann
H e l m h o l tezq u a t i ofnoerx t e r i o r p, ,r o b l e mws i t h
I, S I A M J. Sci.
v a r i a b l ien d eoxr fe f r a c t i o n :
1980.
S t a t . Comput., v o l . 1, pp. 371-385, Sept.
C161 A . B a y l i s s and E. T u r k e l",R a d i a t i y
c o n d i t i o nfso r
wave-1 ike
equations,
Pure Appl. Math., v o l . 33, pp. 707-725,
[17]
L.
boundary
Commun.
1980.
N. T r e f e t h e n and L. Halpern,
"Well-posedness
o f one-way wave equationsandabsorbingboundary
c o n d i t i o n s ,I"n s t .
Comput. Appl.
Sci.
and Eng.
(ICASE), NASA Langley Res. Center, Hampton, VA,
Report 85-30, June 1985.
C181 J. G. Blaschak and G. A. Kriegsmann, "A comparat i vset u doayfb s o r b i nbgo u n d a rcyo n d i t i o n s , "
J. G i l b e r t and
R. Holland,"Implementationofthe
t h i n - s l o tf o r m a l i s mi nt h ef i n i t e - d i f f e r e n c e
EMP
code THREDII ," IEEE Trans.NuclearSci
, vol
NS-28, pp.
4269-4274,
Dec. 1981.
.
.
E261 K.
S.
Yee,
"A numerical
method
soof l v i n g
Maxwell'sequationswith
a c o a r s eg r i db o r d e r i n g
a f i n eg r i d , "
SGEMP Note
#9,
Document
D-DV-860008, D D i v i s i o n , Lawrence Livermore
National
Laboratory,July
1986.
C271 A . Taflove, K. R. Umashankar, B. Beker,
and
F.
Harfoush,
"Detailed
FD-TD a n a l y s i osef l e c t r o m a g n e tfiice l dpse n e t r a t i nnga r r oswl o t s
and
l a p p ej odi ntithnsi c ko n d u c t i nsqc r e e n s ,
"
I E E E Trans.
Antennas
Propcqat.,
Gol. AP-36,
Feb. 1988 ( i n p r e s s ) .
C281 R. H o l l a n d and L. Simpson, " F i n i t e - d i f f e r e n c e
EMP c o u p l i nttgoh isnt r u t s
and
a n a l y s iosf
wires,"
IEEE Trans.
Electromaqn.
Compat., v o l .
EMC-23, pp.88-97,
May 1981.
C291 K. R. Umashankar, A. Taflove, and B. Beker,
" C acl ual t i o n
and experimental Val id a t i o n o f
i n d u c e d c u r r e n t s on couoledwires i n an a r b i t r a- r v*
shaped c a v i t y , "
IEEE i r a n s . Antennas
Propagat.,
~ 0 1 . AP-35, pp.1248-1257,
NOV. 1987.
[301 D. T. Borup, D. M. S u l l i v a n , and 0. P. Gandhi,
"Comparison o f t h e FFT c o n j u g a t eg r a d i e n t method
a n dt h ef i n i t e - d i f f e r e n c et i & - d o m a i n
method f o r
t h e 2-0
absorption
problem,"
IEEE Trans.
MicrowaveTheory Tech., v o l . MTT-35,pp:
383-395,
A p r i 1 1987.
C311 D. M. S u l l i v a n , D. T. Borup,
and
0. P. Gandhi,
"Use o f t h e f in i t e - d i f f e r e n c e ti me-domai n method
i nc a l c u l a t i n g EM a b s o r p t i o ni n human t i s s u e s , "
BME-34, pp.148IEEE Trans. Bfomed. Enqrq.,vol
157, Feb. 1987.
.
0. P. Gandhi , and A. Taflove,
E321 D. M. S u lilv a n ,
"Use o f t h e f i n i t e - d i f f e r e n c e
time-domainmethod
i cn a l c u l a t i n g
EM a b s o r p t i o n i n man models,"
IEEE Trans.
Biomed.
Engrq.,
v o l . BME-35, 1988
(inpress).
Continued on poge 20
19
IEEE Antennas and Propagation Society Newdetter. April 1988
Feature Article-Continued
De Z u t "t eRre, f l e c t i of r1no.sm
inearly
v i b r a t ionbgj e c pt sl a: m
n ei rorabotlri q u e
IEEE Trans.
Antennas
Propaqat.,
incidence,"
v o l . AP-30, pp. 898-903, Sept. 1982.
C381 D.
from poge 19
Wang and 0. P. Gandhi, "Numericalsimulat i o no fa n n u l a r
phased a r r a y sf o r, , a n a t o m i c a l l y basedmodelsusingthe
FDTD method, submitted t o
IEEE Trans.MicrowaveTheoryTech.
[33] C.-Q.
R.
Umashankar, S.
K.
Chaudhuri, and A.
Taflove,"Finite-differencetime-domainformulat i o no f
an i n v e r s es c a t t e r i n g
scheme f o r remot:
s e n s i nogf
inhomogeneous l o s slya y e r e d
media,
s u b m i t t e dt o IEEE Trans.AntennasPropaqat.
C391 K.
Zhang, J . Fang, K.K.
Mei,
and
Y. L i u ,
" C a l c u l a t i o n sotfh ed i s p e r s i v ec h a r a c t e r i s t i c s
o fm i c r o s t r i p s by thetime-domain
finitedifferIEEE Trans.
Microwave
Theory
'ence
method,"
Tech., v o l . 36, pp. 263-267, Feb. 1988.
C341 X.
[40j M. A. S t r i c k e l , A. Taflove, and K. R. Umashankar,
"Accurate
reconstruction
two-dimensional
of
conducting and homogeneous d i e l e c t rtiacr g e t
shapes from a s i n g l e - p o i n t TM s c a t t e r e df i e l d
pulse
response,"
submitted
to
IEEE Trans.
AntennasPropaqat.
H. Choi
and
W. J . R. Hoefer, "The f i n i t e differencetime-domain methodand i t s a p. p. l i c a t i o n
t o e i genval ue problems, "
IEEE Trans.Microwave
Theory
Tech.,
v o l . MTT-34, pp. 1464-1470, Dec.
[35] D.
E411 M. A. S t r i c k e l and A. Taflove,"Reconstructionof
1986.
[36] W. K.
one- andtwo-dimensionalinhomogeneous
diele7,tric
t a r g e t us s i n tgh e
FD-TD / feedback
method,
in
preparation.
Gwarek, " A n a l y s i o
saf r btir a r i l y - s h a p e d
two-dimensionalmicrowave
c i r c u i t s by t h e f i n i t e difference
time-domain
method,"
IEEE Trans.
MicrowaveTheoryTech.,
v o l . 36, 1988 ( i n p r e s s ) .
L. Bennett and G. F. Ross, "Time-domain
&
electromagnetics and i at sp p l i c a t i o n s , "
IEEE, v o l . 66, pp. 299-318, March 1978.
C421 C.
[37] F. Harfoush, A.
Taflove, and G. A. Kriegsmann,
"A n u m e r i c tael c h n i q uf e
oa rn a l y z i negl e c t r o magnetic wave scatteringirommovingsurfaces
in
one
dimensions,
and
two
submitted
to
Trans.AntennasPropaqat.
IEEE
Editor's Comments-Continued
from page 4
I Feature Articles Solicited
managed, i t h a sn or e s e r v ec a p a c i t y .I fe v e no n e
of a
r a t h e rl a r g e number ofemployees
is i l l o r q u i t s ,t h e
effect.
The
r e s u l t is a d i s p r o p o r t i o n a t e l ayr g e
c o n t i n g e n c yp l a n se i t h e rd o n ' t
work or d o n ' te x i s t .
One o f t h e more f r u s t r a t i n g e x p e r i e n c e s o c c u r r e d w i t h
tF
h e b r u airsys uBe e. c a u s e
of t hJ ae n u a r y
URSI
m e e t i n g ,e v e r y o n ew o r k e dv e r yh a r dt og e tt h e
material
o f f t o N e w Yorkby
t h eu s u a ld e a d l i n e .I t
"sat", w i t h
at least a week a n do n e - h a l f
n o t h i n gb e i n gd o n e ,f o r
after i t a r r i v e d .T h i si s s u e
w i l l s e r v e as s o m e t h i n g
o f a test.
If New York c a nb er e s p o n s i v e g, r e a t .
If
n o t , we may have t o e x a m i n o
e t h e ar l t e r n a t i v e s
for
g e t t i n gt h e Newsletter o u to nt i m e .I ' l lb em a k i n g
a
f u l l report at t h eJ u n e
AdCom m e e t i n g ". S t a yt u n e d " ,
a n d I welcome a n y s u g g e s t i o n s o r comments.
Ercument Arvos
Dept. of Electricol ond
Computer Engineering
Syracuse University
Syracuse, NY 13244-1240
(315) 423-2655
Richard Compton
Dept. of Electrical Engineering
Phillips Hall
CornellUniversity
Ithoco, NY 14853
(607)255-9231
T h e AP-S Newsletter continuestoactivelysolicitfeature
articleswhichdescribeengineeringactivitiestakingplace
i ni n d u s t r y g, o v e r n m e n t a, n du n i v e r s i t i e s E
. m p h a s i si s
p l a c e d o n p r o v i d i n gt h er e a d e rw i t h
a general
understandingofeither
a particulartechnicalarea,or
of
t h et e c h n i c a lp r o b l e m sb e i n ga d d r e s s e db yv a r i o u s
engineeringorganizationsaswellastheircapabilitiesto
c o p ew i t ht h e s ep r o b l e m s I. f
y o u a r ei n t e r e s t e di n
submitting an article, please contact either Ercument Arvas
or RichardComptontodiscusstheappropriateness
of the
topic.
On a much b r i g h t enro t ey,o u r
Newsletter has
turned 30!
A c t u a l l yt,h eF e b r u a r yi s s u e
was i s s u e 1
tothhfei r t i evt oh l u mbeu, t
I u n d e r s t atnhda t
p u b l i c a t i o n was notbegun
a t t h e start o f a y e a r .
I
h a v eh a dt h ef e e l i n gt h a ts o m e t h i n gs h o u l da p p e a r
in
t h e spea g etso
celebrate since I s t a r t e tdh i n k i n g
last F a l l ,b u t
I can't
a b o u tt h es i g n i f i c a n c eo ft h i s ,
come u pw i t h
a good i d e a( a n ys u g g e s t i o n s ? ) .P e r h a p s
t h a t is as i t s h o u l db e .
We'll s i m p l yk e e pd o i n gt h e
b e s tj o b
we can, a n db ep l e a s e dw i t h
the growthand
v i t a l i t yo fo u rs o c i e t y .P l e a s en o t et h a t
I n e v e ru s e
t h es o - c a l l e d" e d i t o r i a l
e".
The " w e " refers t ot h e
A s s o c iEa tdei t oorfsf,i c earus t, h oaorntshd, e r
members a n ds u p p o r t e r s of AP-S -- a n do u rr e a d e r s
-who p u t a great deal o f time a n de f f o r it n t ot h i s
Newsletter. Thankyouvery
much, one and a l l . I hope
t o see as many o f y o u as p o s s i b l e at S y r a c u s e i n J u n e .
Best wishes u n t i l t h e n .
20