IEEE
Tka
nactions
on
Nucteoa
Science,
V'o.t.S24,
No.6,
Decembet
1977
THREDE:
A
FREEFIELD
EMP
COUPLING
AND
SCATTERING
CODE
Richard
Holland
Mission
Research
Corporation
PostOffice
Box
3693
Albuquerque,
New
Mexico
87108
Abstract
Equations
ToBe
Solved
THREDE
is
a
timedomain,
linear,
finitedifference,threedimensional
EMP
coupling
and
scattering
code.
In
its
present
form,
it
can
accomodate
a
problem
spacecon
sisting
of
a
30x3Ox30
mesh.
Differencing
is
linear.
Problemspace
boundaries
are
provided
with
a
radiating
condition
which
doesnot
generate
ficticious
mathemati
cal
echos
at
late
times.The
scatterermust
be
a
per
fect
conductor,
although
a
nonideal
groundplane(runway)may
be
close
by.
The
presentarticledescribes
the
mathematical
basis
ofTHREDE,and
showsthe
resultsof
applying
it
to
predict
the
responseof
an
Flll
shell
in
the
horizontallypolarized
dipole
HPD
EMP
simulator.
Amplitude
agreementbetweenexperiment
and
prediction
for
this
example
is
typically
on
the
order
of
20 ;
resonant
frequencies
are
predictedmore
closely
than
this.
THREDE
costs
about
1
second
ofCDC7600
computertime
per
program
cycle;
most
practicalanalysesrequire
500
to
1000
programcycles.
Introduction
Thispaper
describes
the
threedimensional
finite
difference
code
THREDE,
for
solving
timedomain
Mlaxwell's
equations
in
cartesian
coordinates
and
threedimensions.The
primaryapplication
ofTHREDE
is
in
calculatinq
EMP
freefieldscattering
and
surface
currents
for
a
com
plicated
strurture,such
as
an
aircraft.Scattering
structuresmay
either
be
imbedded
in
homogeneous
space
(as
an
aircraft
in
flight)
or
located
over
a
lossy
groundplane
(as
an
aircraft
on
a
runway).
The
algorithm
upon
which
THREDE
is
based
was
first
described
by
Yee1
about
ten
years
ago.
At
that
time,
computers
did
not
exist
whichcould
implement
thisscheme
practically
in
threedimensions.
This
is
no
longer
the
case.
It
is
now
possible
to
mesh
a
problem
space
ofinterest
into
a
30x30x30
grid
and
have
random
access
to
the
resulting
162,000
field
quantities.
These
advances
in
computer
technology
have
prompted
Longmire
to
credit
Yee's
algorithm
as
being
the
fastest
numeri
cal
way
of
solving
Maxwell's
equations .2
Theother
significant
cornerstoneupon
which
THREDE
is
based
is
Merewether's
radiating
boundary
condition.3
This
mathematical
techniquepermits
one
to
treat
a
finite
problem
space
without
being
troubled
at
late
times
by
numericalechos
from
the
problemspace
boun
daries.
Basically,
it
utilizes
thefact
that,
far
from
a
scatteringobject,
the
electromagnetic
scattered
fieldmust
behave
like
f(e,q)g(t
r/c)/r.
In
the
following
sections
of
this
article,
we
will
first
summarize
the
Yee
technique
for
finitedifferencing
Maxwell's
equations.
Next,
we
will
describe
theradia
ting
boundary
condition.
Finally,
we
will
illustrate
an
application
ofTHREDE
by
showing
how
it
is
used
to
predict
theEMPresponseof
an
F111
aircraft
shell
in
the
horizontallypolarized
dipole
(HPD)
simulator.
Thisfinal
section
will
include
experimentalcomparisons
withthe
THREDE
predictions.
A
more
detailed
description
of
the
THREDEcode
is
also
available,4
in
a
report
which
was
not
prepared
un
der
the
brevity
constraint
applied
to
the
present
article.
The
experimental
data
described
in
thispaper
was
ob
tained
at
the
Air
Force
Weapons
Laboratory,
Kirtland
Air
Force
Base,
New
Mexico,
under
Contract
F2960176
C0064
(EG&G,
Inc.,
prime).
Maxwell
's
equations,
in
the
presence
of
electricaland
magneticconductivity
and
current,
are
(1)
(2)
M
VxE
=
J*
*H

=
t


VAH
=
C
+
J
+
iE

at
We
will
be
dealing
withproblems
where
the
total
E
and
H
are
the
superposition
ofincident
and
scattered
fields:
T
inc
s
E
=
E
n
E
HTinc
s
=
H
+
H
(3)(4)
THREDE
is
presentlywritten
in
a
linearformwhere
u
and
o*donot
depend
on
ET
or
HT.
Subject
to
this
linearity
assumption,
the
scattered
field
alonealso
satisfiesMaxwell's
equations.
We
shallpresume
the
scatteringobject
is
perfectly
electrically
conducting
and
completely
enclosed
by
the
problem
space.
We
shall
alsopresumethereare
no
inde
pendentelectric
or
magnetic
currents
J
or
J*
in
the
problem
space.
We
donot
require
c,
ui,
a
oro*
tobe
homogeneous,
and
wo
do
not
require
the
incident
field
to
be
a
plane
wave.
Then
the
boundary
condition
at
the
surface
of
the
scattering
body
is
that
thetotal
tangential
E
field
vanish,
=Einc
=
En
:t
an
t
a
n
(5)
The
distant
boundary
onthe
scattered
field
is
that
the
field
takestheformof
an
outgoing
radial
wave
s
=
f
g,)
(t
r/c)
f(6
4)
r
(6)
At
t=
O,
the
scattered
field
is
presumed
to
be
every
where
zero.
Subject
to
the
linearity
assumption,
these
conditions
uniquely
specify
Es
and
Hs
within
the
problem
space
at
all
time.
The
logic
of
THREDE
procedes
as
follows:
Wle
assume
we
know
the
incident
field
(Einc,
Hlnc)
at
all
times
and
places;this
is
the
field
which
would
exist
in
the
absence
of
a
scatterer.
The
main
portion
of
THREDE
is
devoted
to
finitedifferencing
Maxwell
's
equations
for
the
scatteredfield
(ES,
HS).
Outputquantities
of
THREDE
are
usually
total
fields
or
total
surfacecurrents
on
the
scatterer.
Total
fields
are
given
by
(3)
or
(4).
Total
surface
currents
areT
inc
+
=
nx
(H
c+Hs
,
(7)
where
n
is
the
outwardpointing
unit
normal
on
the
scatterer.
Difference
Equations
Figure
1
illustrates
the
manner
of
imposing
the
(I,J,K)
indexspace
mesh
on
the
problem
space.
It
also
illustrates
thelocation
of
evaluation
points
forthesix
scatteredfield
components
in
a
typical
cell.
Note
thatthe
Es
and
Hs
evaluation
points
are
interlaced
al
ternatelyalong
each
spatial
axis.
This
interlacinq
is
2416
The
other
four
component
equations
may
be
identically
differenced
to
yield
corresponding
formulas.
If
the
medium
filling
the
problem
space
is
inhomogeneous,
c,
uj,
a
and
a*
will
depend
on
(I,J,K).
The
finitedifference
equations
forthe
scattered
fields
arenow
solved
as
follows:
Let
us
assume
the
grid
indices
run
I=
1
to
Nx,
J=
1
to
Ny
and
K=
1
to
Nz.
Let
us
further
assume
the
scatterer
is
modelable
as
an
assemblage
of
conductive
cells
completely
interior
to
the
problem
space.
At
t<0,
all
scattered
fields
are
zero:
EZ(
H
(I,J,K)
yvW ~~~~~~~
S
o(
I)
XO I
Xf+
z,K
Figure
1.
Convention
for
imposing
the
(I,J,K)
indices
on
the
(x,y,z)problem
space,
and
location
ofthesix
field
evaluation
points
in
a
typical
cell.also
done
in
time:
If
At
is
a
program
cycle
time,
Hs
is
evaluated
atnAt,
and
Es
is
evaluated
at
(n+12)AtY
The
THREDE
mesh
is
not
required
to
be
uniform.
However,
if
nonuniformmeshing
is
used,
we
do
not
recommend
that
adjacent
cells
differ
in
sizeby
more
than
30 .
In
componentform,
two
of
Maxwell'sequations
for
the
scatteredfields
are
3E5
3E5
az
ay=
3y
az
aHz
3Hs
zy
y
ay
az
M
s
x
3Es
x
+
s
£
t
x
(8)
(9)
The
other
four
equations
are
obtained
from
these
by
indexpermutation.
Let
us
define
E
n+(I,J,K)
to
be
Es
evaluated
at
the
point
indicated
in
Figure
1,
and
at
time
(n+'2)At.
Let
us
furtherdefine
Hn(I,J,K)
to
be
Hs
at
the
correspondinglyindicated
point,
and
at
time
nAt.
The
other
fourfield
components
may
be
analogously
spe
cified.
Subject
to
these
conventions,
Eqs.
(8)
and
(9),
when
finitedifferenced
consistently
withFigure
1,
yield
Hn(I,J_K)
Ia*/2
H

I,J
(,u/At
+
o*/2)
At+
c*/2
IJx
K
E
(I,JK+
E(Ij1K)
EE2 IJK
+
E/At
+
a/12)
1 z
,JsK
z
J
,K
10(
)
Y
E
nI,,KJK+l)
En(I9J9K)
Z(K)
Z(K1
Z
K)
(10)
0
0/
En+2
I
J
,K)
=
E/At+a/2
En
2
I,J
K
llHn
I,J,K
Hn(,,j_l
9K
+.
c/At
cI2)
Iz
Y J)
Y Jl
H(I,J,K)
Hn I,J
,K1)~
7(K
Z
K1)
E2
(I,J,K)
=
0
E
2(I,J,K)
=
0
y
E2
(I,J,K)
=
0
H
(I,J,K)=O
x
H0
I,J,
K)
=o
y
Ho
I,J
,K =
0
z
l<I<NX
;
l<J<Ny;
l<K<Nz
I
<
I
<
Nx
1
<
J
<
N
y1
;1
<
K<
N

y

z
1
<I<Nx
;
1
<J<N
y;
1
<
K<
N
1
1
<
I
<
Nx;
I
<
J
<
Ny
;
<
K
<
Nzz
1
<
I<
NX
1;
1
<
J<
N
1;
1
<
K<
N

I<N
I


y


Nz
1
2)
1
3)
(14)
1
5)
(16)
1
7)
Letthe
incident
fieldfirsttouch
the
scatterer
at
2At,
and
considerwhat
happens
then
on
a
surface
facing
x.
If
the
x
surfaceof
Cell
(I,J,K)
coincides
with
the
x
surface
ofthe
scatterer,
E2(I,J,K),
E
2(I,J,K+1),
E1 I,J,K)
and
E(I,J+1,K)
are
all
determined
by
Eq.(5).
Teese
four
quantities,
in
turn,define
Hl I,J,K)
via
Eq.
(10).
The
same
is
trueat
all
subsequent
program
cycles,
n>l.
Likewise,
on
a
surface
facing
+x,
let
Cell
(I',J',
K')
have
a
+x
surface
coincident
with
the
+x
surface
of
the
scatterer.
Then
E2
2(I+1,J,K),
En
(I
+l,J,K
+1)
E
2
12
y
Ez
i(I+l,J',K'),
and
Ez
(IV+l,J'+l,K')
are
all
determined
by
Eq.
(5)
for
n>0.
These
four
quantities
subse
quently
determine
Hx
l(I +l,JY,K')
for
n>0.
Identical
considerations
apply
on
the
scatterersurfaces
facing
±y
and
±z.
In
this
way,
the
tangential
E
and
normal
H
components
ofthe
scatteredfields
are
determinate
on
all
the
externalsurfacesof
the
cells
which
make
up
the
conducting
scatterer
model.
Mathematically,
the
difference
equations
pertaining
to
the
fields
atthe
interior
points
of
the
scatterer
thus
become
decoupled
from
the
equations
pertaining
to
the
exterior
points.
In
otherwords,
if
E
n(I,J,K)
describes
the
field
at
an
Ex
meshpoint
in
the
interior
ofthe
scatterer,
this
quantity
cannot
at
some
later
timehave
any
influence
on
any
field
at
some
mesh
point
external
tothe
scatterer.
We
can
thus
set
all
the
internal
meshpoint
fields
to
zero
every
program
cycle
without
affecting
theexternal
scatteringproblem
solution.
Knowledge
ofH
nl
(I,J,K)for
1
<
I
<
N
,
I
<J
<
N
1,
nx
x
y
1
<
K<
N
1,
of
E
2(I,J,K)for
<
I<
,
1
<
J<
N
z
zn
x
y
1
<
K<
Nzl,andEy
2(I,J,K)
for
1
<I
<
Nx,
1
<
J
<
Nyl
,
1
<
K<
Nz
now
determines
the
new
Hn(I,J,K)
for
1
<
K<
Nx,
1
<
J<
Ny.l
1
<
K<Nzl
by
means
of
Eq.
(10).
Similarly,
knowledge
of
all
the
problemspace
Hn
(1,J,K)
Ex
(I,J,K)
and
Ez
(I,J,K)
determines
all
the
new
Hn
I,J,K)
in
the
problem
space.
Finally,
all
the
Hz
(I,J,K),
Ex
(I,J,K)
and
En
I
,J,I)
may
be
used
to
find
all
the
new
Hn(I,J,K).
z
2417
YiJ
K)
1+1
,J,K)
Theelectric
fields
are
a
bitmorecomplicated
to
advance
than
the
magnetic
fieldsdue
to
the
fact
that
the
radiationboundary
condition,
Eq.
(6),
must
be
ap
plied
on
thesefields
atthe
limits
of
the
problem
space.
If
weknow
Ex
(I,J,K),
Hy(I,J,K)
and
Hz(
I,J,K)
at
all
the
meshpoints
in
the
problem
space
wecanuse
Eq.
(11)
to
find
thenew
Ex
(I,J,K)
everywhere
except
on
the
problemspace
boundary.
Thefollowing
Ex
2(I,J,K)
problemsoace
boundary
values,however,
must
be
obtained
by
meansof
the
radiationboundary
condition,
not
Eq.
(
1)
:
En
+2(I,lK)
x
En+2(IIN
,K)
x
y
En+(I,J,1)
Ex
2(I,J,N
Z)
Implementation
of
the
next
section.
1
<
I
<
N

1
,
I
<
K
<
Nz
1
I<N1,
1
<K<N
1
<
I<N
1
,1
<J<N
I

x
i
i
y
I
<
I
<
N
1
1
<
J
<
Ny
this
condition
will
be
described
in
In
a
similar
way,the
other
electric
fieldcompo
nents
Ey
2(I,J,K)
and
EZ
2(I,J,K)
are
advanced
by
finite
difference
formulas
at
interior
mesh
points,
and
bythe
radiationcondition
atthe
problemspaceboundaries.
Radiation
BoundaryCondition
The
purposeof
the
radiationboundary
condition
is
to
permit
the
volumeof
the
problem
space
to
be
finite
without
generating
nonphysicalreflections
off
the
problemspace
boundary.
This
is
achieved
in
TyREDE
by
assuming
the
fields
drop
off
asan
outgoing
r
wave
in
Eq.
(6)
if
one
is
far
enough
awayfrom
the
scatterer.
It
is,
of
course,
desirable
to
make
far
enough
as
small
as
possible,
soas
to
minimize
thenumber
ofbuf
fermeshes
which
must
be
interposed
between
thescattererandthe
problemspace
boundary.The
first
step
towards
this
goal
is
to
centerthescatterer
within
the
problem
space.
Thus,
we
shall
hereafter
presume
r=O
coincides
with
some
point
ator
near
the
center
of
thescatterer.
Subject
to
this
convention,
if
d
is
the
greatestdimensionof
thescatterer,
Merewether
has
empirically
found
that far
enough
occurs
in
a
twodimensional
program
when
r
is
about
d/2
past
thescatterer
in
everydirec
tion3
(see
Figure
2).
We
shall
presume
this
result
is
alsovalid
in
threedimensions.
Actual
construction
of
the
radiation
boundary
is
best
explained
byexample.
One
of
the
boundaryfield
quantities
we
need
to
specify
is
En
2(l1,J,K)
for
some
(JK)
such
that
1
<
J
<
Nyl,
I
<
K
<
{Iz.
(This
E
mesh
point
lies
on
the
x
facing
surface
ofthe
proNlem
space.)
In
our
scheme,
this
quantity
is
determinedfrom
old
values
of
Ey(2,J,K)
by
threepoint
interpolation
overtime.Let
Ryx(I,J,K)
be
the
value
of
r
at
which
Ey(l,J,K)
or
Ey(2,J,K)
is
calculated.
Furthermore,
define
e
(J,K,(n+')At

R
(1,J,K)/c)
=
y
yx
Ryx
(l,J,K)E
2(l
,J
,K)
y y
Equation
(18)
can
also
be
expressed
as
R
yx(l,J,K)(y
(1,J,K)
R
2,J,K)
ey(J,K,(n'2)At
+
At
y
JsK)
y
~~~~~~~~c
R
(1,J,K)

R
(2,J,K)\
R
(2,J,K)
ey
JK,(n12)At

_Y
+
ey
(J,K)At)
(18)
(19)
Figure
2.
Separationrequired
between
scatteringobject
and
problemspace
boundary
for
radiationboundary
approximation
tobe
valid.
where
R
1
yx
JK)

Ryx
(2,J,K)
o
(J,K)
=
1

yx
y
yxcAt
In
view
ofthe
radiation
assumption,
Eq.
(6),it
is
also
true
that
R
(2,J,K)E
n2
(2,J,K)
=
yx
y
c
for
any
Z.
If
we
abbreviate
the
quantity
Eq.
(21)
as
R
(2,J,K)En
12(2,J,K)
=
f
(QAt)
yx
y
yx
1
(21)
indicated
in
5
(22)
the
quantitydescribed
by
Eq.
(19)
can
be
written
R
(l,J,K)E
n2(1,J,K)
=
f
e
(J,K)At)
yxy
yx
yx
(23)
We
now
assume
that
f
x(u)
can
be
represented
by
the
firstthree
terms
in
;
Maclaurin
series,
f
(u)
=
A
+
Bu
+
Cu2
yx
(24)
where
A,
B
and
C
are
unknowns
tobe
determined.
THREDE
is
programmed
so
that
access
to
oldvalues
of
Ey(2,J,K)
is
available
forthreeselected
previouscycles.
These
old
values
are
denoted
to
be
k,l'2
and
Q3
proqram
cycles
in
thepast.
In
otherwords,
THREDE
backstores
En
2 2,J,K)
for
=Z1'Q2'Q3S
If
we
define
f
by
 ~21t3
fi
fyx
(1it)
=
Ryx
2,JyK)E
d
i(2,J,K),
i
=
13
,
(25)
we
obtainthree
equations
for
the
unknowns
A,
B
and
C
intermsofthebackstored
En12i 2,J,K):
y
f.
=
A
Bk
At
+
Ck
At2,
i
=
13
26)
2418
(20)
Simultaneous
solutionofthesethreelinear
equations
is
accomplished
bythe
usual
use
of
determinants.
Re
sultingformulas
for
A,
B
and
C
in
termsoftheback
storedfields
are
given
elsewhere,4
but
in
the
interest
of
brevity,
are
omitted
here.
It
is
now
possible
to
express
the
radiationcondition
value
forthe
desired
boundary
field
En+2 1,J,K)
entirely
in
terms
of
previouslydeterminedquantities:
9
9
E
+2(1,J,K)
=
y
A
+
BO
(J,K)At
+
Cox
(iJ,K)AtL
yx
lJ
yx
Ry
I
J,K)
Radiatingconditionboundary
fieldsmust
also
be
obtained
for
En+2(N
,J,K)
y
x
En+2
N
,J,K)
z
x
En
+2(I,N,K)
x
y
E
n+12
(I,N
,K)
z
y
l
<J
<
Ny
1
<
K<
Nz
1
<J
<
N1,
1
<
K<
Nz
1
<J
<
Ny,
1
<
K<
Nz
1
<
I
<
NX
1,
1
<
K<
Nz
I
<
I
<
Nx,
1
<
K
<
Nz
1
<
I'<
NX
1,
I
<
K<
Nz
I
<
I
<
N
,
1
<
K<
Nz
closest
interior
field,
EnM
Zi(2,J,K),
atthe
present
cycle
Q;
=
1)
andthe
two
mostrecent
past
cycles
(N
=
0
and
+1).
For
uniform
cubical
meshing,
Eq.
(28)
indicates
the
maximum
stable
timestep
to
beAt
=
AX/ v 3
c)
9
(29)
where
AX
is
the
dimension
of
any
ofthecubical
cell
edges.
Thus,
under
these
circumstances,
an
electromag
netic
signal
will
propagate
.58
AX
during
eachtimestep.At
this
point,
it
may
be
helpful
to
refer
to
Figure
3,
and
to
bear
in
mindthat
we
are
trying
to
compute
the
7)
new
E+2
(1,J,K)
from
Eyh2k
(2,J,K)
obtained
previously
Q
>
1).
As
timeprogresses,
the
field
at
I=
2
propa
gates
towards
the
I=
1
mesh
point.
At
Qi
=
1
(the
pre
sent
cycle),
this
propagation
has
just
begum.
By
i
=
0
(the
mostrecent
pastcycle),
the
propagation
has
pro
gressed
.58
AX;
by
Qi
=
1
(the
nextmostrecent
past
cycle)
the
propagation
has
progressed
1.16AX,
andthus
passed
the
I=
1
mesh
point.
Thus,
forthis
case,
taking
Qi=
1,
0
and
1
guarantees
that
the
new
boundary
field,
En+2 1,J,K),
is
interpolatively
obtained
from
previously
determined
values
which
have
propagated
to
bothsides
ofthe
relevant
I=
1
mesh
point.
Ey l,J,K)
Ey 2,J,K)
X0 l
X0(2)
0
En
2(I,J,l)
En
2(I,J,l
y
En+2(I,J,N
x
z
En
+12 I,J,N
y
z
I
<
I<
N
1
1
<J
<
N
I
<
I
<
N
1
<'J
<
Ny
I1
1
<
I
<
N
x
I,
I
J
<
Ny
I
<
I<
N
,
1
<J
<
N
y.
These
eleven
additional
cases
oftheradiation
boundary
condition
are
all
developed
by
straightforward,
but
tedious,
repetitions
of
the
derivation
of
Eq.
(27)
with
appropriatepermutations
ofthe
subscripts.
Stability
Considerations
The
algorithm
described
hereinfor
solving
Maxwell's
equations
is
unstable
if
too
large
a
value
is
assigned
the
time
step
At.
For
stability,
At
must
obey
At
<
m
in1X
I+1)

X
I))
2
1
+
(Yo(J+0
0+
(Y
(3+1)
y )
+
(z
(K+l)

Z
(K))Y2(p)
1
)
(28)
If
c
or
p
is
inhomogeneous,
their
dependence
on
(I,J,K)
must
be
included
in
evaluating
the
minimal
timestepof
Eq.
(28).
A
somewhat
moresubtle
stabilitydifficulty
canbe
generated
at
the
radiationboundary
if
one
is
careless
in
picking
values
forthe
ki.
In
particular,
onecan
inadvertantly
turn
the
radiationinterpolation
procedure
into
a
less
stable
extrapolation
procedure.
For
thecase
of
uniformcubicalmeshing,
this
difficulty
will
be
avoided
if
onetakes
k.
=l,
0
and
1
as
the
backstored
fields
used
to
determine
A,
B
and
C
from
Eq.
(26).
Thischoice
of
Li
corresponds
to
deter
mining
the
new
boundary
field
En
Nl1,J,K)
from
the
y
direction
of
propagation
.4
+t
i
1
4
.58AX
IQ
=O

{
1.
16AX
k=
1
Figure
3.
Location
ofthe
boundary
and
firstinterior
E
(I,J,K)
mesh
points.
Note
that
by
i=
=1,
the
signal
from
XO(2)
has
passed
Xo(l)
for
uniform
cubical
meshing.
Thus,
forthis
case,the
new
signal
at
Xo(l)
can
be
temporally
interpolated
from
its
Zi=
1,
0
and
+1
values
at
X
(2).
On
the
other
hand,
if
oneuses
an
expanding
mesh
near
the
radiation
boundary
in
order
to
satisfy
the
previouslydescribed
far
enouah
criterion
with
a
mini
mum
number
of
buffer
meshes,
selecting
Qi
=
1,
0
and
1
may
notbe
satisfactory.
Suppose,
for
example,
XO(l)
and
XO(2)
are
separated
by
threetimes
the
dimension
of
thesmallest
mesh.
Then
the
signal
starting
out
from
I
=2
at
Li
=
1
willhave
propagated
only
1.16/3
of
thedis
tance
to
I=
1
by
Li
=
+1.
In
this
case,
kj
=
1,0
and
1
all
correspond
to
signals
which
are
on
the
same
side
of
I=1.
In
the
case
of
nonuniform
meshing,
it
is
generally
necessary
to
make
at
least
one
ofthe
ki
greater
than
+1.
In
other
words,
generally
it
is
necessary
to
backstore
information
from
longer
ago
thanthe
two
immediately
pre
ceding
cycles.
Itis
not,
however,
necessary
to
back
storefieldsfrom
more
than
two
previous
cycles
at
any
given
time

to
do
otherwise
in
a
threedimensionalcode
is
very
extravagant
of
computermemory
resources.
The
manner
in
which
THREDE
selects
cycles
forbackstorage
and
determines
the
corresponding
Li
for
an
arbitrary,
nonuni
form
mesh
is
of
sufficient
complexity
to
preclude
pre
sentation
here.
It
has
been
documented
in
the
THREDEreference
report,
however.4
2419
I