Upload
briana-richards
View
219
Download
2
Embed Size (px)
Citation preview
April 4 2001 1
Thermal-ADI: a Linear-Time Chip-Level Dynamic Thermal Simulation Algorithm
Based on Alternating-Direction-Implicit(ADI) Method
Ting-Yuan Wang
Charlie Chung-Ping Chen
Electrical and Computer Engineering
University of Wisconsin-Madison
April 4 2001 2University of Wisconsin-
Madison
Motivation
1999 International Technology Roadmap for Semiconductor (ITRS) Maximum power Number of metal layers Wire current density
April 4 2001 3University of Wisconsin-
Madison
Year 1999 2000 2001 2002 2003 2004 2005 2008 2011 2014
Technology Node (nm)
180 - - 130 - - 100 70 50 35
Maximum Power (W)
90 100 115 130 140 150 160 170 174 183
On-chip,
across-chip clock
(MHz)
1200 1321 1454 1600 1724 1857 2000 2500 3004 3600
Maximum number of metal level
7 7 7 8 8 8 9 9 10 10
5.8e5 7.1e5 8.0e5 9.5e5 1.1e6 1.3e6 1.4e6 2.1e6 3.7e6 4.6e6
)105 (
)/( 2max
Catwire
cmAJo
1999 ITRS
April 4 2001 4University of Wisconsin-
Madison
Existing Thermal Simulation Methods
Finite Difference Method Easy, good for regular geometry, fast
Finite Element Method More complicated, good for irregular
geometry
Equivalent RC Model (S.M. Kang) Compatible with SPICE model, need
to solve large scale matrix
April 4 2001 5University of Wisconsin-
Madison
X
n
ji,Tn
j1,-iT n
j1,iT
n
1ji,T
n
1ji,T
(0,0) (i,0)…
(0,j)
x
y
(I,0)
(0,J)
…
…
…
Finite-Difference Formulation of the Heat Conduction on a
Chip
Space DomainTime Domain
April 4 2001 6University of Wisconsin-
Madison
Heat Conduction Equation
where : Temperature: Material density: Specific heat: Heat generation rate: Time: Thermal conductivity
),()],(),([),(
trgtrTTrt
trTcp
Tpc
tg
April 4 2001 7University of Wisconsin-
Madison
A xq | xxq |pκ,ρ,c
Δx) g(A
Δx
Δy
Δzdt
dE
1-D Flow of Thermal EnergyUnit Element
G
xAg
xq|x
TA
xxq|x
TA
dtdE
t
TcxA xxxp
][][)(
Δx
Energy Conservation
Increasing rate of stored energy which causestemperature increase
Net rate of energy transferring into
the volume
Heat generation ratein the volume
April 4 2001 8University of Wisconsin-
Madison
Space Domain Discretization
Heat Conduction Equation
Central-Finite-Difference Approximation
pc
tyxg
y
tyxT
x
tyxT
t
tyxT
),,(),,(),,(),,(
2
2
2
2
2
2
2
,1,,1
22
,1,,1,2
2
2
)(2
x
T
x
TTT
xOx
TTT
x
T
nx
nji
nji
nji
nji
nji
njin
ji
cc
April 4 2001 9University of Wisconsin-
Madison
Time domain discretization
Heat Conduction Equation
Simple Explicit Method Simple Implicit Method Crank-Nicolson Method
gcy
T
x
T
t
TT
p
yxnn
1
)()( 2
?2
2
?21
nji
nji
nji
ny
nji
nji
nji
nx
TTTT
TTTT
1,,1,2
,1,,12
2
2
April 4 2001 10University of Wisconsin-
Madison
Accuracy:Stability Constraint:
No matrix inversion but time steps are limited by space discretization
gcy
T
x
T
t
TT
p
ny
nx
nn
1
)()( 2
2
2
21
],, [ 22 yxt
2
11122
yxt
Simple Explicit Method
nji
nji
nji
ny
nji
nji
nji
nx
TTTT
TTTT
1,,1,2
,1,,12
2
2
April 4 2001 11University of Wisconsin-
Madison
Accuracy: Unconditionally StableNo limits on time step but involves with large scale matrix inversion
gcy
T
x
T
t
TT
p
ny
nx
nn
1
)()( 2
12
2
121
],, [ 22 yxt
Simple Implicit Method
11,
1,
11,
12
1,1
1,
1,1
12
2
2nji
nji
nji
ny
nji
nji
nji
nx
TTTT
TTTT
April 4 2001 12University of Wisconsin-
Madison
Accuracy:Unconditionally stableNo limits on time step but involves with large scale matrix inversion
gcy
TT
x
TT
t
TT
p
ny
ny
nx
nx
nn
1
)(2)(2 2
212
2
2121
],, [ 222 yxt
Crank-Nicolson Method
nji
nji
nji
ny
nji
nji
nji
nx
TTTT
TTTT
1,,1,2
,1,,12
2
2
11,
1,
11,
12
1,1
1,
1,1
12
2
2nji
nji
nji
ny
nji
nji
nji
nx
TTTT
TTTT
April 4 2001 13University of Wisconsin-
Madison
n
m1616***
****
****
****
*****
*****
*****
*****
*****
*****
*****
*****
****
****
****
***
e.x. m=4,n=4
Total node number N = mn
Matrix size = NxN
Analysis of Crank-Nicolson Method
April 4 2001 14University of Wisconsin-
Madison
Alternating Direction Implicit Method
Solves higher dimension problem by successive Lower dimension methodsAccuracy:Unconditionally stableNo limits on time step and no large scale matrix inversion
],, [ 222 yxt
April 4 2001 15University of Wisconsin-
Madison
Step I: x-direction implicit y-direction explicitStep II: x-direction explicit y-direction implicit
n
Alternating Direction Implicit Method
• Peaceman-Rachford Algorithm• Douglas-Gunn Algorithm
April 4 2001 16University of Wisconsin-
Madison
Step I
Step II
gc
tT
rrT
rr
p
ny
yx
xny
yx
x
)2
1)(2
1()2
1)(2
1( 22122
gc
tT
rT
r
p
ny
yn
xx
2)
21()
21( 22
12
gc
tT
rT
r
p
n
xxn
yy
2)
21()
21( 2
1212
Peaceman-Rachford Algorithm
April 4 2001 17University of Wisconsin-
Madison
Douglas-Gunn Algorithm
Step I
Step II
gc
tTT
rTT
rTT
p
nnyynnxxnn
12
12
1
22
gc
tTrTT
rTT
p
nyy
nnxxnn
22
122
1
)(2
gc
tTT
rTT
rTT
p
nnyynnxxnn
12
2
121
2)(
2
April 4 2001 18University of Wisconsin-
Madison
Illustration for ADI
X-direction implicit Y-direction implicit
i = 1 2 … m
1,njiT
Step I Step II
j = 1
n
2
…
1 2 … m
11,
njiT
11,
njiT
j = 1
n
2…
21
,njiT
21
,1
njiT
21
,1
njiT
April 4 2001 19University of Wisconsin-
Madison
Analysis of ADI Method
mm
**
***
***
***
**
Tridiagonal Matrix
Time complexity: O(N)
2xnxm = 2nm =2N
2 stepsn matrices tridaigonal matrix
X-direction implicit
21
,njiT
i = 1 2 … mj = 1
n
2
…
21
,1
njiT
21
,1
njiT
April 4 2001 24University of Wisconsin-
Madison
Si Heat Source Heat Source Heat Source
(i,j) (i+1,j)
(i,j+1)
(i,j-1)
(i,j)(i-1,j) (i+1,j)
(i,j+1)
(i,j-1)
(i,j)(i-1,j)
(i,j+1)
(i,j-1)
Si
(case I) (case II) (case III)
Three Different Locations of Node
(i-1,j) (i+1,j)
April 4 2001 25University of Wisconsin-
Madison
CN v.s. ADI_DG v.s. ADI_PR
0
10
20
30
40
50
60
70
80
90
100
0 100 200 300 400 500 600
Number of Time Increment
Tem
per
atu
re
CN
ADI_DGADI_PR
Results comparison
April 4 2001 26University of Wisconsin-
Madison
R u n T im e o f CN , D G , a nd P R v.s. N um b e r o f N o de s
1E+00
1E+01
1E+02
1E+03
1E+04
1E+05
1E+06
0 0 .2 0 .4 0 .6 0 .8 1
N u mbe r o f No d es (M illio n s)
Ru
n T
ime
(S
eco
nd
s)
Pe a ce ma n -R ac hfor d
Do u gla s -G un n
C ra nk -N ic o lso n
Result – Run Time Comparison I
5000X
April 4 2001 27University of Wisconsin-
Madison
Run Time o f D G a nd PR v.s.Number of Nod es
0
30
60
90
120
150
0 10 20 30 40 50 60 70 80 90 100
Thousa nds
N umber o f N ode s (M illions)
Ru
n T
ime
(Se
co
nd
s)
Peac ema n-R achfo rd
D o ugla s-G unn
Result – Run Time Comparison II
April 4 2001 28University of Wisconsin-
Madison
Memory Usages of CN, DG, and PR
0
200
400
600
800
1000
0 1 2 3 4
Number of Nodes (Millions)
Mem
ory
Siz
e (M
B)
D ouglas -G unn Peaceman-Rachford
Crank-Nico lso n
Results – Memory Usages I
April 4 2001 29University of Wisconsin-
Madison
Memory Usages of CN, DG, and PR
0
500
1000
1500
2000
0 20 40 60 80 100
Number of Nodes (Millions)
Mem
ory
Siz
e (M
B)
Douglas-Gunn
Peaceman-Rachford
Crank-Nicolson
Results – Memory Usages II
April 4 2001 30University of Wisconsin-
Madison
Results – Stability Constraint
Douglas-Gunn Algorithm v.s. Gamma
0
10
20
30
40
50
60
70
80
90
0 500 1000 1500 2000 2500
Time Interval (ns)
Tem
pera
ture
(C)
0.40.811.21.641020
gamma = 20
gamma = 0.4
Gamma is the stability limit for simple explicit method
April 4 2001 32University of Wisconsin-
Madison
Thank you for your attention!