IEEE
Antennas and Propagation Society Newsletter, April 1988
FiniteDifference TimeDomain (FDTD) Modeling
of
Electromagnetic Wave Scattering and Interaction Problems
bY Allen Taflove
EECS
Department Northwestern University Evanston,
IL
60201
Abstract
This rticle, based pon invited apers
at he
XXII URSI
General Assembly (Tel viv, ugust 1987) nd the
URSI
National adio cience eeting
(Boulder, anuary 19881, reviews ecent pplications of he initedifference timedomain (FDTO) method for umerical odeling f lectromagnetic wave
scattering and interaction problems. One of he goals of this article is to demonstrate that recent advances in FDTD modeling oncepts and software mplementa tion, combined with advances in computer echnology, have xpanded the scope, ccuracy, nd peed of FDTD modeling o he oint where
it
may be the referred choice or structures hat cannot be easily reated by
conventional ntegral quation and asymptotic approaches. As a class, such structures are electric ally arge and ave omplex hapes, material composi
ti
ons, apertures, and
i
teri or cavi ties
1. INTRODUCTION
Contemporary ighfrequency lectromagnetic engineering roblems can involve wave interactions with complex, electricallylarge hreedimensional structures. These structures an ave shapes, material ompositions, pertures, r avities hich
produce ear fields hat annot be resolved nto
finite sets
of
modes or ays. Proper numerical model
ing f such ear fields equires ampling t sub
wavelength esolution o void liasing f magnitude ani phase information. The goal is o provide a self consistent model of he utual oupling f he
electrical
ly
smal 1 cell
s
corfipri sing he structure.
A
candidate numerical modeling approach or his
purpose is he initedifference timedomain (FDTD) solution of Maxwell's url quations. his pproach
is analogous to xisting initedifference olutions of luid low problems ncountered in computational aerodynamics, in hat he umerical model
is
based upon a direct olution f he overning artial
differential quation. ursuing his nalogy, FDTD
shares he omputational equirements of he luids
codes and other imilar argescale artial iffer ential equation solvers) n erms of computer floating
point rithmetic ate, rimary random ccess memory size, and data andwidth o econdary memory. Yet,
FDTD is a nontraditional pproach o umerical
electromagnetic odeling, where requencydomain approaches have dominated. One of he goals of his article s o demonstrate that ecent advances in FDTD modeling oncepts and software mplementation, ombined ith dvances in
computer echnology, have expanded he scope, accuracy and peed of FDTD modeling to he point where
it
may be the preferred choice or certain ypes of scatter
ing and coupling roblems. With his in mind, this
article
wi
succinctly eview he ollowing ecent
FDTD modeling alidations nd esearch rontiers:
Continued
o
poge
6
Korada
R.
Umashankar
EEC4
Department University of Illinois at Chicago Chicago,
IL
60680
Introducing Feature Article Author
Allen Taflove
Allen Taflove was born in Chicago, IL on une 14, 1949. He received he
B.S.
(with ighest istinc tion),
M.S.,
and Ph.D. degrees in electrical engineer ing rom orthwestern niversity, vanston, L,
in
1971, 1972, and 1975, respectively. From 975 to 1984, e was a staff member at 111 7esearch Institute n Chicago, L, olding he osi tions f ssociate ngineer, esearch ngineer, anc Senior ngineer. here, in ddition o esearch
i
2lectromagnetic wave scattering and penetration, hc xovided echnical eadership or major rograms ir
joint ightofway design or 60Hz transmission ines Ind pipelines or ailroads; and extraction of oil ron shale, tar sand, nd slowly roducing ells sins
novel in itu lectromagnetic eating echnology.
H
is one of hree principal coinventors of he atter,
and has been granted
10
U.S.
patents in his area.
In
1984,
Dr.
Taflove eturned o Northwestern a: an Associate Professor in he
EECS
Department. incc then, e as eveloped everal esearch rograms
ir
analysis and umeri cal methods for el ectromagneti
(
dave interactions with arge, complex tructures.
Hi2
related nterests nclude nverse cattering targel
synthesi
s;
appl
i
ati ons of ecent ector uper. computers nd oncurrent rocessors in computationa' electromagnetics; and the new onsurface adiatior condition
OSRC)
theory or highfrequency scattering which e srcinated long with
G.
A.
Kriegsmann n(
K.
R.
Umashankar.
Dr.
Taflove s a member of Tau Beta
Pi,
Eta Kapp;
Nu,
and Sigma
Xi
He is a Senior Member of IEEE anc a member of
URSI
Commission
B.
5
IEEE
Antennas
and
Propagation Society Newsletter, Aprll
1988
Featwe ArticleContinued
from
page
5
1.
Scattering models or hreedimensional eentrant
2.
Conformal models of urved urfaces;
3.
Scattering models or wodimensional nisotropic
structures;
4.
Penetration odels or arrow lots and lapped
joints n hick screens;
5.
Coupling odels or ires and wire undles in
free space and in arbitrary metal cavities;
6.
Penetration models or he lectromagnetic ields
within detailed, inhomogeneous
ti
ssue pproxima tions f he omplete human body (at UHF
frequencies
1
;
structures spanning up to
9
wavelengths;
7.
Microstrip and microwave circuit models;
8.
Scattering models or elativistically ibrating
mirrors;
Introducing Feature Article Author
Korada
R.
Umashankar
Korada
R.
Umashankar received he
B.E.
degree rom Mysore University, ndia, n
1962;
the
M.E.
degree from he ndian nstitute f cience, angalore,
India, n
1964;
and the Ph.D. degree rom he University
of
Mississippi, niversity,
MS,
in
1974;
all n electrical engineering. From
1964
to
1969,
he was Assistant Professor and Head of he Department of lectrical ngineering,
College f ngineering, arnatak niversity, ubli,
India. uring
1974
and
1975,
he was a Postdoctoral Research ssociate, and from
1975
to
1977,
Assistant Professor of Electrical Engineering at he University
of ississippi. From
1977
to
1979,
he was the National Research Council Visiting Fellow at he
U.
S.
Air
Force Weapons Laboratory, irtland AFB,
NM.
During
1979
to
1984,
he was Senior ngineer at
IIT
Research Institute, Chicago, L. Currently,
Dr.
Umashankar is Associate rofessor of Electrical Engineering and Computer Science at he
University f llinois t Chicago. is rimary
research
is
in he development of nalytical and numerical echniques in electromagnetic heory, EMP/EMC interactions, and
EM
simulation tudies. Research opics nclude he tudy f cattering rom
bodies omprised f, r oated ith, nisotrGpic
media; extension f he method of momenti to electrically ery arge argets; and the new on
surface adiation ondition (OSRC) theory or igh
frequency scattering.
Dr.
Umashankar is a member of Eta Kappa Nu and Sigma
Xi
He is a Senior Member of
IEEE
and a member of
URSI
Commission
B.
9.
Inverse cattering econstruction f one
dimensional, patially oincident rofiles f
electrical permittivity and conductivity;
10.
Inverse cattering econstruction of wo
dimensional onducti ng, homogeneous, and
i
homo geneous dielectric argets rom inimal TM
scattered ield pulse esponse data; and
11.
Largescale computer software.
2.
GENERAL CHARACTERISTICS
OF
FDTD
As stated, FDTD is a direct solution of Maxwell’s timedependent curl quations.
It
employs o potentials. nstead,
it
applies imple, econdorder accurate centraldifference approximations
El]
for he space nd time erivatives f he lectric and
magnetic fields directly o he espective differen tial operators of he drl equations. This achieves a
sampleddata eduction f he ontinuous lectro
magnetic field n a volume of space, over a period of time. Space and time iscretizations re elected o bound errors n he sampling rocess, and to nsure numerical tability f he lgorithm
[2].
Electric and magnetic field components are nterleaved n space to ermit a natural atisfaction f angential ield continuity onditions t media nterfaces. verall, FDTD .is a marchingintime procedure which simulates
the ontinuous actual waves by ampleddata umerical analogs ropagating in a data space stored n a computer.
At
each time tep, he ystem f quations to update he ield components is ully explicit,
so
that here s no need to et up or olve a set f linear quations, and the equired omputer torage
and running ime s roportional o he lectrical
size of he volume modeled. Lottice Truncotion Plone (Invisiblb To
All
Waves)
/
x
=
1/28
Figure
1
TimeDomain WaveTracking Concept
of
the FDTD Method
Fig.
1
illustrates he timedomain wave tracking concept of he FDTD method.
A
region of space within the dashed lines s elected or ield sampling
in
space nd ime.
At
time
=
0
,
it
is assumed that all fields ithin he umerical ampling egion re
identically zero. An incident lane wave is assumed to nter he ampling egion
at
this point. Propaga tion of he ncident wave is modeled by the commence ment of ime stepping, which is simply he mplementa tion f he initedifference nalog f he url
equations. Time stepping ontinues as the umerical analog of he ncident wave strikes he modeled target embedded within he ampling egion.
A
outgoing
Continued on page 7
5
IEEE
Antennas and Propagation Society Newsletter, April
1988
Feature ArticleContinued
from
page
6
scattered wave analogs deally ropagate hrough he
1
atti e truncation pl anes with negl
i
i bl ref
1
ecti on to xit he sampling egion. Phenomena such s
induction of surface currents, scattering and multiple
scattering, enetration hrough pertures, and cavity excitation are modeled imestep y imestep by the
action of he url equations analog. Selfconsistency
of hese modeled phenomena is enerally assured
if
their patial and emporal ariations re ell
resolved by the space and time sampling process.
Time stepping is continued until he desired ate time ulse esponse r teadystate ehavior is
achieved. An important example of he atter s he sinusoidal teady tate, wherein he ncident wave is assumed to have a sinusoidal dependence, and time stepping is continued until all ields n he sampling
region xhibit inusoidal epetition. his s a
consequence of he imiting amplitude principle
C31.
Extensive umerical xperimentation ith FDTD has
shown that he number of complete ycles f he
incident wave required o be timestepped to achieve the inusoidal teady tate is approximately qual o the
Q
factor f he tructure r phenomenon being model ed.
EV
'Y
X
Figure 2. Positions
of
the Field Components about a Unit Cell
of
the YEE Lattice
111
Fig.
2
illustrates he positions of he electric
and magnetic ield components about a unit cell of he FDTD lattice n Cartesian coordinates
C11.
Note that each magnetic field vector component is surrounded y four circulating electric ield vector components, and vice ersa. his rrangement ermits ot nly a
centereddifference analog o he space derivatives of the url quations, ut lso a natural eometry or implementing he ntegral orm f araday's Law and
Ampere's Law at he pacecell evel. his ntegral
interpretation permits a simple but effective modeling of he hysics of moothly urved arget urfaces,
penetration hrough narrow slots having subcell gaps,
and coupling
to
thin wires having ubcell diameters,
as
w
be een later. Fig.
3
illustrates how an arbitrary hree dimensional catterer is embedded in an FDTD space lattice comprised of he nit ells f ig.
2.
Simply, desired alues of electrical permittivity and
conductivity re ssigned o ach lectric ield
component of he attice. orrespondingly, esired
values f agnetic erineabi
1
i
y cna quivalent conductivity re ssigned o ach agnetic ield
component of he attice. The media arameters re
interpreted by the FDTD program as local coefficients for he imestepping lgorithm. pecification f
media properties in this componentbycomponent manner
FD
TD
UNIT
CELL

Y
Figure
3.
Arbitrary ThreeDimensional Scatterer Embedded n a FDTD Lattice
results n a steppededge, or taircase, pproxima
tion f urved urfaces. ontinuity f angential
fields s assured at he nterface of dissimilar media with his rocedure. here s no eed for pecial
field matching t media interface oints. Stepped
edge approximation f urved urfaces has een found to be adequate in he FDTD modeling problems tudied in he
1970's
and early
1980's,
including wave inter actions with biological issues
C41,
penetration nto cavities
[SI,
[SI,
and electromagnetic ulse
EMPI
interactions ith complex structures
[7]

C91.
However, recent nterest n wide dynamic ange models of cattering by curved argets has rompted the
development of urfaceconforming FDTD approaches which eliminate taircasing. These
wi
be ummarized later n his article.
1
1
Reqion
I
:
Toial Fields Region
2
:
Scattered Fields Source Lattice Truncation (a
1
Region
I
:
Totai ields
E
Region
2
:
Scottered Flelds
x
jo
+...
..
~~...t..,~...t...~...t.,.~
,,.
,.._._

HY
e
&

H
iI
(b)
Figure
4.
Division
of
FDTD Lattice into TotalField and Scattered Field Regions. a) Lattice division;
(b)
Field component geometry
at
connecting plane y
=
joS
[lo] 1111
Fig. 4 illustrates he ivision f he FDTD
lattice nto otalfield and scatteredfield egions. This division has een ound to be ery seful ince
it
permits he fficient imulation f an incident plane wave in he otalfield egion with rbitrary
Continued on poge
8
7

IEEE
Antennas and Propagation Society Newsletter. April
1988
Feature ArticleContinued
from
page
7
angle f ncidence, olarization, imedomain
waveform, nd duration
[lo],
C111.
Three dditional important enefits rise rom his attice ivision. a. b.
C.
A
large nearfield computational dynamic ange is achieved, ince he catterer f nterest is embedded in he otalfield egion. Thus, low ield evels n shadow regions or within shielding nclosures re computed directly without uffering ubtraction noise as would be he ase
if
scattered ields n such regions were imestepped ia FDTD, and then added to a cancel ing ncident ield o obtail, the ow otalfield evels). Embedding the catterer n he otal field
region ermits a natural atisfaction f
tangential ield ontinuity cross media
interfaces, as iscussed arlier, ithout
having o compute the ncident ield t
possibly numerous points along a complex locus that s unique o each scatterer. The zoning arrangement of Fig.
4
requires computatjon of the ncident ield only along he ectangblar connecting surface between the otalfield and scatteredfield egions. his urface s
.fixed, .e., ndependent f he shape
or
composition f he nclosed catterer eing
modeled. The provision f a welldefined cattered field egion n he FDTD lattice permits he neartofar ield ransformation llustrated
in Fig.
5.
The dashed virtual urface shown in ig.
5
can e ocated long onvenient
lattice lanes n he catteredfield egion
of
Fig.
4.
Tangential cattered
E
and
H
fields computed via FDTD at his irtual surface can hen be weighted by he reespace
Green's unction and then ntegrated (summed) to rovide he arfield esponse and radar
cross ection full bistatic esponse or he assumed illumination ngle)
[lll

C131.
The nearfield ntegration urface has
a
fixed rectangular shape, nd thus is independent of the shape or omDosition f he nclosed

I
I
I
I
I
I
I
I
scattered being modeled.
(
P,
?is)
d
I
NO
SOURCES
8
ZERO
FIELDS
I
I
I
I
I
I
I
(a)
(bJ
Figure
5.
NeartoFar Field Transformation Geometry (a) Original problem; (b) Equivalent problem external
o
the
virtual surface, Sa
[ll]
Fig.
4
uses the erm lattice runcation o
designate he utermost 1 attice planes n he cat teredfield egion. The fields at hese planes cannot
be omputed sing he entereddifferencing pproach discussed arlier ecause f he assumed absence of known field ata t oints utside f he attice truncation. These data are needed to orm he central differences. herefore, n uxiliary attice runca tion ondition s necessary. his ondition must be
consistent with Maxwell's auations n hat n ut
going catteredwave umerical nalog triking he
lattice runcation must exit he attice ithout
appreciable onphysical eflection, ust as
if
the lattice runcation was invisible.
It
has een shown that he equired attice runcation ondition s
really a radiation ondition n he near ield
[lo],
C141

C171.
Further,
it
has een shown that convenient ocal approximations of he exact adiation
condition an e enerated and applied ith good
results
[IO]

[171.
Based upon this esearch, he procedure or constructing more precise ocal approxi mations of he exact adiation condition is reasonably
well nderstood. hese pproximations re urrently
under tudy for numerical mplementation in he FDTD computer programs C181.
3.
THREEDIMENSIONAL FDTD SCATERINC MODELS
Analytical and experimental Val
i
ati ons have been
obtai ne6 relative o FDTD modeling of anonical threedimensional conducting argets spanning
1/3
to
9
wavelengths
C121, C131, C191, C201.
For revity, nly one such validation wi1,l be eviewed here.
Fin
E2
f
ky
HX
0.37
m
Corner Reflector
Looks
/
o'<
0'.
1800
Fin Center
I
I
I
I
I
I
I
L
0'
=
90'
r__________________
/
LATTICE
EDGE
9 .
00
Fig.
6
Geometry
of
CrossedPlate Scatterer and Illumination
13],
1191,
1201
Fig.
6
depicts he geometry of a crossedplate scatterer omprised f wo flat plate:, Zlectrically bonded together o orm he shape of a
T
.
The main plate has the dimensions
30
cm x
10
cm x 0.33 cm, and the bisecting in has the dimensions
10
cm x
10
cm
x
0.33
cm. The illumination s a plane wave at
0
elevation ngle and TE polarization elative o he main late, and at he requency
9.0
GHz. Thus, the main plate spans
9.0
wavoelengths. ote that ook angle zimuths etween
90
and
180
provide ubstan tial corner eflector physics, n addition o he edge diffraction, orner iffraction, and other ffects
found or an isolated lat plate. For he 9GHz DTD model, the attice ell ize is
0.3125
cm, approximately
1/11
wavelength. The main plate s formed y
32
x
96
x
1
cells; he bisecting fin s formed by
32
x
32
x
1
cells; and the overall lattice s comprised of
48
x
112
x
48
cells
(1,548,288
unknown field components) ontaining
212.6
cubic
Continued
on
page
9