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!