Download Multipole-Accelerated 3-D Capacitance Extraction Algorithms for Structures with Conformal Dielectrics

Multipole-Accelerated
3-D
Structures
Capacitance
with
K.
Conformal
Nabors
Massachusetts
Dept.
MA
Abstract
Dielectrics
and
Computer
new
paper extends
program
three-dimensional
FASTCAP2
gram,
is described.
FASTCAP,
accelerated
capacitance
FASTCAP2
algorithm
low
three-dimensional
part
of an iterative
that
Like
the original
eficient
capacitance
design
from
FASTCAP
in that
with
multiple
dielectrics,
bility
of the
multipole-accele
class
of integrated
FASTCA
is able
thus
to
packaging
problems
the
approach
packagtng
The
self and
integrated
coupling
innately
tric
materials
fashion.
surrounding
layers
rated
by
conformal
and
involve
or
or
off-chip
connectors
ceramic
for three-dimensional
form
iterative
analysis
dielectric
However,
conductors
for
the
equivalent-
solution
method.
The
method’s
space-filling
have
dielectrics.
Also,
interconnection
problems
passing
several
through
capacitance
recent
computationally
structures
inexpensive
This work
search Projects
of-
methods
sur-
and the rest
conductor
i
capacitance.
charge is nu-
of its coupling
ing all the dielectric
regions to free
in this equivalent
free-space problem,
in a uni-
[5]. This
Science
Foundation
contract
(MIP-8S58764
A02),
from I.B.M.
and Digital
Equipment
Corporation.
capacitances
the conductor
capaci-
Given the conductor
potentials,
the conductor
the
surface charges can be computed
by replacing
conductor-dielectric
and dielectric-dielectric
interfaces
with a surface charge layer of density U(Z) and chang-
plas-
has made
was supported
by the Defense
Advanced
Agency
contract
NOOO1 4-91- J-I 69S, the
m conductors,
merically equal to the negative
i.
tance to conductor
development
extraction
complex
mulsepa-
all the self and coupling
with
i is raised to unit potential
conductor
are grounded,
then the total charge on
is numerically
equal to conductor
i’s self
Furthermore,
any other conductor’s
total
dielec-
in a complicated
boundary-element
of very
accu-
conductors,
Formulation
face charges must be computed
m times, with m different sets of conductor
potentials.
In particular,
if
analyz-
with
example,
metal
The
To determine
mined
matches
tional
grants
describes
Equivalent-Charge
of a structure
final
involves
structures
dielectrics.
rnultipole-accelerated
accurate
reliability.
or
with
are be-
determining
capacitances
circuits,
of polysilicon
associated
packaging
in
three-dimensional
tiple
tic
and
of these
Integrated
packaging
and
important
performance
estimation
ing
capacitances
interconnect
increasingly
circuit
of
2
circuit
coming
ten
section
eral examples.
Introduction
rate
following
utility
is demonstrated
in Section 4 by applying
our
implementation
of the algorithm,
FASTCAP2,
to sev-
problems.
1
where
dielectric
problems.
accelerated
applica-
to a wzder
and
to problems
by multiple
charge approach to analyzing
structures
with dielectric interfaces.
Section 3 then shows how this formulation can be used as a framework
for a multipole-
be
P2 d~ffers
extending
interconnect
The
al-
to
to analyze
rated
circuit
enough
calculations
process.
ii
pro-
is based on a multipoleis
method
are surrounded
regions of arbitrary
shape, thus allowing the analysis
of more realistic
integrated
circuit
interconnect
and
calculation
the earlier
Science
USA
02139
the conductors
The
for
of Technology
Engineering
Cambridge,
Algorithms
J. White
Institute
of Electrical
Extraction
ReNa-
by insisting
conductor
that
it produce
potentials
space.
Then
a(x) is deter-
a potential
which
at conductor-dielectric
interfaces, and sat isfies normal electric-field
conditions
at the dielectric-dielectric
interfaces [2, 6].
and
To numerically
compute
u, the conductor
surfaces
0738 -KWQ92
1992 IEEE
29th ACM/lEEE Design Automation Conference@
710
Psper 42.3
$3.00@
field difference
terface
at the center of that
i-th
dielectric
panel to the sum of the contributions
displacement
in-
to that
field due to the n charge distributions
on
if panel i lies on the interall n panels. In particular,
face between dielectrics with permittivities
C. and cb,
then the Gauss’s Law condition
(4)
must hold.
Here ni is a normal
the expression
to panel i. Substituting
(2) breaks (4) into a superposition
over
all the panels,
Dilql
+Dizqz+
““”+
~ijqj
+
““”+
~infln
=
0,
(5)
where
Careful
Figure
The
1:
panels
dielectric-coated
used
bus-crossing
to
discretize
problem.
The
evaluation
important
a 2x 2
of the
special
derivative
in (6) leads
to the
ca3e [3]
lower
(7)
conductors’
surfaces are discretized
in the same way
as the two upper conductors’
surfaces.
Collecting
all n equations
leads to the dense linear
of the form
(2) and (5)
system
and dielectric interfaces are discretized into n = np +
nd small panels or tiles, with np panels oll conductor
surfaces and nd panels on dielectric
interfaces a-s in
Figure 1. It is then assumed that on each panel i, a
charge, qi, is uniformly
distributed.
For each conductor surface panel, an equation is written which relates
the potential
at the center of that
i-th
panel, denoted
pi, to the sum of the contributions
to that potential
from the n charge distributions
on all n panels. For
example, the contribution
of the charge on panel j to
the
potential
superposition
at the
center
i is given
of panel
or
[mq’=m
by the
integral
where
P
● RnPxn
is the matrix
cients,
D
6 Rndxn
1
“ s the matrix
(9)
of potential
coeffi-
representing
the di-
q E Rn is the
electric interface boundary
conditions,
vector of panel charges, and p c RnP is the vector of
where
~i is the center
of panel
i, aj
is the area of
conductor-panel
panel j, co is the permittivity
of free space, and the
constant charge density qj /aj has been factored o~lt of
the integral.
The total potential
at xi, is the sum of
the contributions
from all n panels,
center-point
‘s[:l
potentials.
Using
WI)
’10)
gives
(11)
Aq=b
pi(~i)
=
~ilql+~i2!?2+<
“ “+~ijqj
+“
“ “+pinqn,
(2)
as the
where
linear
system
charge densities.
Pij ~
L
LZj
!
linear
1
~ane, j 47fCollZi –
da’.
(3)
2’[1
Similarly,
for each dielectric interface panel, an equation is written
that relates the normal displacement-
system
to solve
to for
In the standard
(11)
is solved
using
the n x n
elimina-
algorithm
at a cost
Our
algorithm
uses the multipole-accelerated
mn
of the
next
section
which
conductor
a Gaussian
tion
method
of order
the
approach
n3 operations
requires
[8, 6].
iterative
only
order
operations.
711
Pspe.r 42~3
3
The
Multipole
Approach
~
d
evacuation
d
The dense linear system of (11) can be solved to
compute panel charges from a given set of panel potentials,
ming
the
<
and the capacitances can be derived by sumIf Gaussian elimination
is
panel charges.
used to solve
(11),
der n3. Clearly,
ally intractable
the number
this approach
of operations
is or-
‘“4
becomes computation-
if the number
of panels exceeds several
Figure 2: The direct evaluation
to d panel charges at d points.
d
1: GMRES
Make an initial
>
m
hundred.
Instead, consider solving the linear system
(11) using a conjugate-residual
style iterative
method
like GMRES
[9]. Such methods have the form given
below:
Algorithm
points
paneIs
algorithm
for solving
guess to the solution,
evaluation
of the potential
due
points
\
(11)
q“.
Set k = 0.
do {
Compute
the residual,
rk = b – Aqk.
if Ilrll < tol, return qk as the solution.
else {
Choose
a’s and ~ in
‘+’
= ~$=o
to linimize
\
o!jq~ + brk
Figure 3: The evaluation
panel charges at d points
llrk+lll.
Setk=k+l.
of a the potential
due to d
using a multipole
expansion.
}
}
Forming
Here
this
(3)
II. II is the Euclidean
strategy
and
(6)
norm.
are calculating
before
the
operations
The
the
n2 entries
iterations
begin,
forming
nz
product
Aqk
on each
forming
most
of A and to substantially
of computing
tolerated
to
dominant
compute
iteration.
the
It
then
to Agk
to calculating
panel center points,
mensions
of electric
displacement,
in terms
of potential
evaluations,
the cost
uation
with
3.1
Potent
the multipole
it can be expressed
leading
to fast eval-
algorithm.
can be
[7].
cost of forming
Aqk
does not necessarily
to order
imply
n operations
that
[1].
each iteration
Aqk
=
[1
Pq~
~qk
.
Evaluations
employed
in the multipole
algorithm
is the evaluation
of potentials
using multipole expansions.
Consider evaluating
the potential
at
d panel centers due to charges on another d panels as
of the
algorithm
can be computed with order n opIf the number of GMRES iterations
required
RES iterations
required to achieve convergence
below n for large problems [4].
The produat Agk is, using (9),
ial
The key approximation
This
in
to achieve convergence approaches n, then the minimization
in each GMRES iteration
requires order n2
operations.
This problem is avoided through the use
of a preconditioned
which reduces the number of GM-
Figure
2.
center-point
equation
In
a direct
potential
like
(2).
ucts are computed
evaluation,
is calculated
a single
using
panel’s
an explicit
prodthe Figure
2 case, d Pijqj
and added at a cost of d operations.
In
Repeating
the process for all d evaluation
quires d2 operations.
to well
points
re-
When the charge panels are well-separated
from the
evaluation
points the potentials
can be approximated
at much lower cost using a multipole
expansion.
A
multipole
expansion is a truncated
series expansion of
(12)
712
Psper 42.3
is equivalent
to avoid
The approximation
used here is the multipole
algorithm, an efficient way of calculating
the potential
due
to a charge distribution
in free space that reduces the
GMRES
erations,
Pqk
at all the conductor
making it possible to approximate
it directly with the
Dqk has the difast multipole
algorithm.
Although
per-
matrix-vector
reduce
the product
the potential
of A using
and
is possible
if an approximation
Aqk
costs of
the form
where (ri, @i, Oi) are the spherical
i’s center point (the evaluation
tive to the origin of the multipole
coordinates
of panel
point) measured relaexpansion, Y{ (~i, Oi)
are the surface spherical harmonics, 1 is the expansion
order, and A4~ are the multipole
coefficients which are
determined
ing
from
the well-separated
The use of the expansion
A single multipole
panel charges us-
is illustrated
expansion
in Figure
for the potential
4’
3.
due to
Xa
the d panel charges is evaluated at the d evaluation
points.
Since the expansion is used many times, the
cost of computing
it may be neglected.
tipole evaluation of the same d potentials
Figure 4: The electric fields on both sides of the dielectric panel are approximated
with divided differences.
Thus the lmulrequires
only
d operations.
The
reduction
gregation
which
in complexity
of distant
can
be
panels
used
to
resulting
into
from
multipole
evaluate
potentials
at
panel centers is the source of the multipole
efficiency.
Maintaining
tributions
this efficiency
of panels while controlling
hierarchical
A detailed
multipole
description
the
ag-
many
Electric
for general
The evaluation
dis-
error leads to the
left-hand
alently,
imated
ences
ure
amounts
side of rzd equations
(4).
The
left-hand
by replacing
constructed
pacitances
the
near
to calculating
of (4)
derivatives
panel
product
gorithm
the n multiply-adds
of
ing material
the
ducting
(5) or, equiv-
can
in DqL
potential
are replaced
evaluations
differin
The
Fig-
P=(~~)
accuracy
of
parameterized
1 are calculated.
The
has permittivity
~h = 3.96..
x lpm
All
the con-
cross-sections,
and all
and inter-conductor
spacings are lpm.
dielectric is nominally
0.25pm thick.
accuracy
attained
by
our
program,
The
FAST-
CAP2, is investigated
using the Figure 1 problem.
The
smallest coupling
and self capacitances
in the problem are calculated using FASTCAP2
and by Gaussian
elimination
applied directly
to (11), the standard direct method. The entries in Table 1 represent the capacitances associated with the top, rear conductor
in
to find one inner
by three multipole
alfollowed by the divided
Figure
1. By default
duce capacitances
difference calculation
(15). In this way the bulk of the
displacement
field calculation
is performed
using the
efficient multipole
algorithm
to compute the potentials
Pi(~i),
the easily
of Figure
bars have lpm
overhang
conformal
be approx-
by divided
i as illustrated
required
with
structure
and
capacitance extraction
algomultiple
dielectrics,
the ca-
Figure 1 problem is called the 2 x 2 bus-crossing problem and is representative
of the 1 x 1 through 5 x 5
bus-crossings examined here. In all these problems the
4, yielding
Thus
efficiency
lower bus is covered with a layer of conformal
dielectric with permittivity
c. = 7.5c0 while the surround-
of the form
side
the
associated
bus-crossing
Evaluations
of Dqk
demonstrate
the multipole-accelerated
rithm for problems with
algorithm
used in FASTCAP2.
of the complete multipole
al-
Field
Results
To
algorithm’s
gorithm
is given in [1] and its use in the context
capacitance extraction
is described in [5, 4].
3.2
4
expansions
FASTCAP2
within
is configured
to pro-
1% of those calculated
using
direct
factorization,
as is clearly the case here. Thus
any error in the FASTCAP2
capacitances
is dominated by discretization
error rather than multipole
approximation
effects.
and pb(z~).
713
Pspe.r42.3
C34
2x2
3x3
-0.2112
-0.2112
0.9854
–0.3200
Panels
664
1984
3976
6640
FASTCAP2
–0.2113
–0.2112
0.9886
–0.3212
Direct
1.4
41
(320)
(1400)
0.44
2.4
8.6
C32
C31
Problem
1X1
4X4
C33
Direct
FASTCAP2
Table 1: Comparison
Figure 1 problem.
of capacitances
(in fF)
20
for the
Table
2:
Comparison
RS600/540
CPU
of
execution
minutes.
times
Values
in
30
40
in
I.B.M.
parenthesis
are
extrapolated.
20
18 16 14 12 10 864 20
10
0
20
(number
Figure
Figure
5: The
pacitance
Some
used
of a dielectric-coated
of
been
discretization
the
outer
removed
to
to compute
sphere
dielectric-bou?~
show
the
in
dary
inner
the
free
ity
ca-
cution
space.
panels
conductor
of
6:
of cmdwtms,
Demonstration
times.
of the
using
FASTCAP2
m)*(tiousamds
Times
the four
in CPU
IsJl
50
of panels, n)
order
mn
complex-
bus crossing
minutes
exe-
on an I.B.M.
RS6000/540.
have
surface
panels.
crate
Using
the
dielectric-coated
FASTCAP2’S
accuracy
comparing
its
conducting
sphere
result
to
layer
rounding
region
structure
has capacitance
lated
using
further
with
lrn
relative
is free
FASTCAP2
By
applied
5,
The
to
the
l%
Finally,
thick
Law,
such
calcu-
a
discretization
of the
The
program’s
execution
problems
is
standard
direct,
method
parenthesis
indicate
responding
to problenls
the
standard
method
time
requirements.
leads
to
much
speed
compared
in
to
Table
extrapolated
that
due
execution
2.
not
The
values
in
permittivity
corusing
memory
and
ter.
42.3
is shown
to
made pos-
Connectors
and
are
of this
spaced
type
have 0.65mmx0.65mm
3.25mm
must
center
be analyzed
bus connections
fully when used for high-speed
discretization,
FASTCAP2
Using a 9524-panel
nlod-
714
Paper
time
of the kind of analysis
of 3.5; the pins
cross-sections
complexity
even
execution
sible by FASTCAP2,
consider the connector
in Figure 7. The U-shaped polyester
body has a relative
the
be solved
for
As a final example
busof
times
lower
times
four
speed
excessive
FASTCAP2’S
lower
the
the
execution
could
to
for
FASTCAP2’S
the execution time data points closely follow the bestfit straight line, indicating
order mn complexity.
analytic
value.
crossing
In
grow roughly linearly with mn, where m is the number
of conductors and n is the number of panels. Figure 6
plots the FASTCAP2
execution times in Table 2 verses
mrz for the bus-crossing
problems.
As is illustrated,
sur-
value
like the 2 x 2 bus crossing.
calculations.
inner
2. The
The
within
by
by a lm
Gauss’s
148.35pF.
well
Figure
value.
is coated
permittivity
space.
5 is 148.5pF,
of
demonstrated
an analytic
of radius
dielectric
of Figure
sphere
is
sized problems
particular,
in the time required to compute the capacitance of the 4 x 4 bus crossing problem using standard
direct methods, FASTCAP2
can perform seventy such
to
cencare-
[10].
com-
earlier
draft
of this paper,
References
[1] L. Greengard.
Fields
in
bridge,
The
Particle
Rapid
Evaluation
Massachusetts,
in
Academic
Verlag,
Potential
Linear
Berlin,
Integral
Theory
Press, London,
[3] R. Kress.
Press, Cam-
1988.
[2] M. A. Jaswon and G. T. Symm.
Methods
of Potential
M.I.T.
Systems.
and
Equation
Elastosiaiics.
1977.
Integral
Springer-
Equations.
1989.
[4] K. Nabors,
S. Kim,
J. White,
and S. Senturia.
Fast capacitance
extraction
of general
of
In Proceedings
three-dimensional
structures.
Figure
7: The backplane
connector
example.
the 1991 IEEE
puter
all the self and coupling
pins in thirty
CPU
An identical
ination
same
the
est self
four
analysis
algorithms
on the
have
minutes
using
standard
requires
machine.
The
capacitances
on an IBM
pins
highest
self capacitances:
capacitance
is 0.481pF
corner
Gaussian
roughly
four
pins. The strongest
[5] K. Nabors
CPU
in the
the
coupling
pins on a diagonal
coupling
two-pin
capacitance
the maximum
is reduced
to around
signal
0.065pF.
[7] V. Rokhlin.
of classical
multipole-accelerated
algorithm
haa been
capacitance
extended
shaped,
multiple-dielectric
tended
algorithm
as implemented
with
regions.
ar-
and
-1448,
Rapid
Systems,
November
solution
potential
Physics,
ticonductor
The ex-
Design
10(11): 1447–
1984.
of integral
theory.
equations
Journal
of
Com-
60(2): 187–207, September
in FASTCAP2
has
15,
entific
process.
many help-
[10]
ful discussions with Prof. Stephen Senturia and Brian
Johnson.
We are also grateful
to Songmin Kim for
his help in implementing
FASTCAP2,
and to several
reviewers
for their
careful
readings
and
IEEE
Ikansactions
Techniques,
on
21(2):76-82,
1973.
[9] Y. Saad and M. H. Schultz. GMRES: A generalized minimal
residual algorithm
for solving nonJournal
on Scisymmetric
linear systems. SIAM
July
like to acknowledge
Theory
February
lar, FASTCAP2
is fast enough to allow capacitance
extraction
of complex
three-dimensional,
multipledielectric
geometries to be part of an iterative
design
would
systems.
Microwave
the same 170 accuracy and reduced time and memory
requirements
of the original
algorithm.
In particu-
anonymous
program.
Computer-Aided
[8] A. E. Ruehli and P. A. Brennan.
Efficient
capacitance calculations
for three-dimensional
mul-
extraction
to problems
bitrarily
The authors
A multipole
1985.
Conclusion
The
Fastcap:
1991.
MTT-32(11):1441
pin
putational
5
Circuits
Proces-
1991.
[6] S. M. Rae, T. K. Sarkar, and R. F. Barrington.
The electrostatic
field of conducting
bodIEEE
Transies in multiple
dielectric
media.
Theory and Techniques,
actions
on Microwave
capacitances,
and the four pins on the remaining
diagonals,
on
on Com-
and
by the
slightly more than 0.2pF, occur between pin pairs next
to the sides of the connector body. By grounding
four
parallel
October
Transactions
1459, November
low-
Computers
3-D capacit ante extraction
of Integrated
center
The
is attained
IEEE
days
Conference
in
and J. White.
accelerated
elim-
eight
0.547pF.
and
for the
RS6000/540.
VLSI
pages 479-484,
sors,
putes
International
Design:
and
Statistical
Computing,
7(3):856–869,
1986.
W. Whitehead.
The role of computer
models
in connector
standardization.
In .%l#th Annual
Connector
posium,
and
Interconnection
pages 103-110,
Technology
San Diego,
CA,
Sym-
1991.
of an
715
Paper 42.3