very fast chip-level thermal analysis keiji nakabayashi†, tamiyo nakabayashi‡, and kazuo...
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Very Fast Chip-level Thermal Analysis
Keiji Nakabayashi†, Tamiyo Nakabayashi‡, and Kazuo Nakajima*
†Graduate School of Information Science, Nara Institute of Science and TechnologyKeihanna Science City, Nara, Japan, [email protected]
‡Graduate School of Humanities and Sciences, Nara Women’s UniversityKitauoyahigashi-machi, Nara, Japan, [email protected]
*Dept. of Electrical and Computer Engineering, University of MarylandCollege Park, MD 20742, USA, [email protected]
Budapest, Hungary, 17-19 September 2007
2
Abstarct
We present a new technique of VLSI chip-level thermal analysis. We extend a newly developed method of solving two dimensional Laplace equations to thermal analysis of four adjacent materials on a mother board. We implement our technique in C and compare its performance to that of a commercial CAD tool. Our experimental results show that our program runs 5.8 and 8.9 times faster while keeping smaller residuals by 5 and 1 order of magnitude, respectively.
3
1. Introduction
• Thermal phenomena is very important factor in VLSI and board design in the post-90 nm era.
• Thermal analysis is modeling of thermal conduction by a Laplace equation and its solution by finite difference method (FDM).
• We developed a new, very fast chip-level thermal analysis technique.
4
2. Problem• Consider a multi-layer structure, where four
layers of materials p, q, r, and s of thermal conductivities kp, kq, kr, and ks, respectively, are stacked together.
• Heat travels through a heat transfer pass consisting of the chip die, the adhesive, the heat spreader, and the heat sink, and goes out to the ambient air.
• Our problem is to find temperature distribution through two dimensional steady-state thermal conduction analysis.
5
3. Method • Recently, a new efficient direct method, called
Symbolic Partial Solution Method (S-PSM) was developed in the area of computational fluid dynamics.
• S-PSM-based solution process goes through many levels of repeated operations of decomposition and merging.
• We extend this S-PSM-based Laplace equation solver to a multi-layer structure.
6
h
h : number of Y-axis grid pointsg : number of X-axis grid points where g = (np+2)+(nq+2)+(nr+2)+(ns+2)Cell size : = = 2048(um) / h
Tleft=20( )℃(Reference)
X
Tright=20( )℃(Reference)
128um
256um
64um
Y
Chip die ks= 100 [W/(m ・K)]
Heak sink kp= 260 [W/(m ・K)]
Tamb=55( ) (Ambient temperature)℃
Tj=120( ) ℃（ Heat source: Si bulk temperature)
np+2
Adhesive kr= 16 [W/(m ・K)]
nq+2
nr+2
ns+2
(0,0)
2048um
g
Heak spreader kq= 230 [W/(m ・K)]
128um
Cellxy
x y
Boundary Value Problem (BVP)
7m
msnmsnsnsnsnsn
mm
mm
mm
mrnrnrnrn
wwww
ssssss
ssssss
ssssss
ssssss
rrrr
,03,02,01,0
1,1,13,12,11,10,1
1,2,23,22,21,20,2
1,1,13,12,11,10,1
1,0,03,02,01,00,0
,13,12,11,1
m
mnmnnnnn
mm
mm
mm
m
qqqq
pppppp
pppppp
pppppp
pppppp
vvvv
pppppp
,03,02,01,0
1,1,13,12,11,10,1
1,2,23,22,21,20,2
1,1,13,12,11,10,1
1,0,03,02,01,00,0
,03,02,01,0
m
mrnmrnrnrnrnrn
mm
mm
mm
mqnqnqnqn
ssss
rrrrrr
rrrrrr
rrrrrr
rrrrrr
qqqq
,03,02,01,0
1,1,13,12,11,10,1
1,2,23,22,21,20,2
1,1,13,12,11,10,1
1,0,03,02,01,00,0
,13,12,11,1
m
mqnmqnqnqnqnqn
mm
mm
mm
mpnpnpnpn
rrrr
qqqqqq
qqqqqq
qqqqqq
qqqqqq
pppp
,03,02,01,0
1,1,13,12,11,10,1
1,2,23,22,21,20,2
1,1,13,12,11,10,1
1,0,03,02,01,00,0
,13,12,11,1
The arrangement of interior grid and boundary points for the four material domains.
kp
kq
kr
ks
8
0)(2
2
2
2
y
u
x
uku
022
2
1,,1,
2
,1,,1
y
uuu
x
uuuk jijijijijijiu
yx
Laplace Equation and Finite Difference Method for each material u (=p, q, r, s) :
Finite Difference Method
Final Solution for each material :
)0(1n
)(5
)0(n
)(4
)(5
)1(1
)(0
)1(2
)1(2,
)(4
)(5
)1(2
)(0
)1(1
)1(1,
)(1
, where
u
e
u
e
eeeeeu
e
eeeeeu
e
AA
AA
uuuu
uufu
uufu
9
rq
rqrq kk
kkk
2,
thermal conductivity of the boundary between adjacent materials
0,0,1,01,0
,01,0,0,1,
jjqjjq
jjqjjnqp
qqkqqk
qqkqpkp
0,1,,,11,1
,11,1,1,
jnjorqjnjnq
jnjnqjnjnq
qqg
qgqg
qrkqqk
qqkqqk
Similarly, at its boundary with material r,
From the viewpoint of material q, the following difference equation holds at its boundary with material p (first order approximation) :
10
)0(0
)0(0
)0(1,
)1(2,
)1(1,
)0(0,
)0(1,
)1(2,
)1(1,
)0(0,
)0(1,
)1(2,
)1(1,
)0(0,
)0(1,
)1(2,
)1(1,
)0(0,
)(5
)(4
)(1
)(0
)(5
)(4
)(1
)(0
)(5
)(4
)(1
)(0
)(5
)(4
)(1
)(0
)0(2
)1(1
)1(2
)1(2
)1(1
)0(1,
,)0(
2
)1(1
)1(2
)1(2
)1(1
)0(1,
,)0(
2
)1(1
)1(2
)1(2
)1(1
)0(1,
,)0(
2
)1(1
)1(2
)1(2
)1(1
)0(1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
4
3
3
2
2
1
1
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
44
44
33
33
22
22
11
11
w
v
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
s
s
s
s
r
r
r
r
q
q
q
q
p
p
p
p
s
r
q
p
ns
es
es
ss
nrr
er
er
rr
nqq
eq
eq
npp
ep
ep
p
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
s
ee
eesssr
srrr
ee
eerrrq
rqqq
ee
eeqqqp
qppp
ee
eep
k
k
k
k
k
k
AIOO
AIOA
AOIA
OOIkAIk
IkAIkOO
AIOA
AOIA
OOIkAIk
IkAIkOO
AIOA
AOIA
OIkAIk
IkAIkOO
AIOA
AOIA
OOIA
matrix-vectors form of equations for four materials (three boundaries)
11
For material s with ks
: Merging operation for adjacent partial solutions.
{ } : Variables with known values.
Solutions: Eq. (3-24)↑
Equations: Eq. (3-20)
Final solutions: Eq. (3-35)↑
Final equations: Eq. (3-33)
Solutions: Eq. (3-31)↑
Equations: Eq. (3-28)
Solutions: Eq. (3-32)↑
Equations: Eq. (3-29)
Solutions: Eq. (3-25)↑
Equations: Eq. (3-21)
Solutions: Eq. (3-26)↑
Equations: Eq. (3-22)
Solutions: Eq. (3-27)↑
Equations: Eq. (3-23)
For material p with kp For material q with kq For material r with kr
For materials p, q, r, s with kqr
For materials p, q with kpq For materials r, s with krs
Decompose Equations Eq. (3-19) into 4 subsystems
{p0, p5, q0} {r5, s0, s5}
{q5, r0}
{p1, p4} {q1, q4} {r1, r4} {s1, s4}
Back substitution
12
)0(1,
)1(2,
)1(1,
)0(0,
)(0,
)0(0
)(5
)(4
)(1
)(0
)0(2
)1(1
)1(2
)1(2
)1(1
)0(1
1
1
21
1
1
1
11
11
pnpp
ep
ep
p
eqp
e
e
e
e
pp
ee
eep
kkAIkOO
AIOA
AOIA
OOIA
f
f
f
f
q
0
0
v
p
p
p
p
)0(0,
)12(1,
)12(2,
)0(1,
)1(5,
)3(0,
)2(0
)2(1
)2(4
)2(5
)0(1
)12(1
)12(2
)12(2
)12(1
)0(2
eq
eq
qnqq
eqp
erq
e
e
e
e
ee
eeqq
k
k
k
k
AIkOO
AIOA
AOIA
OOIkA
f
f
f
f
p
0
0
r
q
q
q
q
)0(1,
)1(2,
)1(1,
)0(0,
)(0,
)(5,
)(5
)(4
)(1
)(0
)0(2
)1(1
)1(2
)1(2
)1(1
)0(1
3
3
4
2
3
3
3
3
33
33
rnrr
er
er
rr
esr
erq
e
e
e
e
rr
ee
eerr
k
k
k
k
AIkOO
AIOA
AOIA
OOIkA
f
f
f
f
s
0
0
q
r
r
r
r
)0(0,
)1(1,
)1(2,
)0(1,
)(5,
)0(0
)(0
)(1
)(4
)(5
)0(1
)1(1
)1(2
)1(2
)1(1
)0(2
4
4
34
4
4
4
44
44
ss
es
es
ns
esr
e
e
e
e
ss
ee
ees
kkAIkOO
AIOA
AOIA
OOIAs
f
f
f
f
r
0
0
w
s
s
s
s
System Decomposition and Partial Solutions for Each Subsystem/Material
Decomposition
(3-20)
(3-21)
(3-22)
(3-23)
13
)0(1,
)1(2,
)1(1,
)0(0,
)(0,
)0(0
4,43,42,41,4
4,33,32,31,3
4,23,22,21,2
4,13,12,11,1
)(5
)(4
)(1
)(0
1
1
21
1
1
1
pnpp
ep
ep
p
eqp
pppp
pppp
pppp
pppp
e
e
e
e
kkBBBB
BBBB
BBBB
BBBB
f
f
f
f
q
0
0
v
p
p
p
p
)0(0,
)1(1,
)1(2,
)0(1,
)(5,
)(0,
4,43,42,41,4
4,33,32,31,3
4,23,22,21,2
4,13,12,11,1
)(0
)(1
)(4
)(5
2
2
1
3
2
2
2
2
eq
eq
nqq
eqp
erq
qqqq
qqqq
qqqq
qqqq
e
e
e
e
k
k
k
k
BBBB
BBBB
BBBB
BBBBq
f
f
f
f
p
0
0
r
q
q
q
q
)0(1,
)1(2,
)1(1,
)0(0,
)(0,
)(5,
4,43,42,41,4
4,33,32,31,3
4,23,22,21,2
4,13,12,11,1
)(5
)(4
)(1
)(0
3
3
4
2
3
3
3
3
rnrr
er
er
rr
esr
erq
rrrr
rrrr
rrrr
rrrr
e
e
e
e
k
k
k
k
BBBB
BBBB
BBBB
BBBB
f
f
f
f
s
0
0
q
r
r
r
r
)0(0,
)1(1,
)1(2,
)0(1,
)(5,
)0(0
4,43,42,41,4
4,33,32,31,3
4,23,22,21,2
4,13,12,11,1
)(0
)(1
)(4
)(5
4
4
34
4
4
4
ss
es
es
ns
esr
ssss
ssss
ssss
ssss
e
e
e
e
kkBBBB
BBBB
BBBB
BBBBs
f
f
f
f
r
0
0
w
s
s
s
s
Partial solution
(3-24)
(3-25)
(3-26)
(3-27)
14
5
0
5
0
)(01,1,
)(01,4,
)0(01,4
)0(01,1
)(5
)(0
)(5
)(0
4,1,
4,4,
4,4,
4,1,
3
3
2
2
1
1
q
q
p
p
eqrq
eqrq
p
p
e
e
e
e
qqp
qqp
pqp
pqp
Bk
Bk
B
B
IOBkO
OIBkO
OBkIO
OBkOI
ff
ff
ff
ff
r
r
v
v
q
q
p
p
5
0
5
0
)0(01,1
)0(01,4
)(51,4,
)(51,1,
)(5
)(0
)(5
)(0
4,1,
4,4,
4,4,
4,1,
2
2
4
4
3
3
s
s
r
r
s
s
errq
errq
e
e
e
e
ssr
ssr
rsr
rsr
B
B
Bk
Bk
IOBkO
OIBkO
OBkIO
OBkOI
ff
ff
ff
ff
w
w
q
q
s
s
r
r
)0(0,4,1
)14(1,3,1
)14(2,2,1
)0(1,1,15
)0(0,4,4
)14(1,3,4
)14(2,2,4
)0(1,1,40
)0(1,4,4
)13(2,3,4
)13(1,2,4
)0(0,1,45
)0(1,4,1
)13(2,3,1
)13(1,2,1
)0(0,1,10
)0(0,4,1
)12(1,3,1
)12(2,2,1
)0(1,1,15
)0(0,4,4
)12(1,3,4
)12(2,2,4
)0(1,1,40
)0(1,4,4
)11(2,3,4
)11(1,2,4
)0(0,1,45
)0(1,4,1
)11(2,3,1
)11(1,2,1
)0(0,1,10
ss
ses
ses
s
snss
s
ss
ses
ses
s
snss
s
rnrr
rer
rer
rr
rrr
rnrr
rer
rer
rr
rrr
qeq
qeq
q
qnqq
qeq
qeq
q
qnqq
pnpp
pep
pep
pp
pp
pnpp
pep
pep
pp
pp
BkBBB
BkBBB
BkBBBk
BkBBBk
BkBBBk
BkBBBk
BkBBB
BkBBB
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffff
ffffffwhere
(3-28)
(3-29)
Merge
15
5
0
5
0
)(01,1,
)(01,4,
)0(01,4
)0(01,1
4,24,1
3,23,1
2,22,1
1,21,1
)(5
)(0
)(5
)(0
3
3
2
2
1
1
q
q
p
p
eqrq
eqrq
p
p
pqpq
pqpq
pqpq
pqpq
e
e
e
e
Bk
Bk
B
B
IBBO
OBBO
OBBO
OBBI
ff
ff
ff
ff
r
r
v
v
q
q
p
p
5
0
5
0
)0(01,1
)0(01,4
)(51,4,
)(51,1,
4,24,1
3,23,1
2,22,1
1,21,1
)(5
)(0
)(5
)(0
2
2
4
4
3
3
s
s
r
r
s
s
errq
errq
rsrs
rsrs
rsrs
rsrs
e
e
e
e
B
B
Bk
Bk
IBBO
OBBO
OBBO
OBBI
ff
ff
ff
ff
w
w
q
q
s
s
r
r
rs
pq
srs
ppq
e
e
rrrsrq
qqpqrq
BB
BB
IBBBk
BBBkI
ff
ff
w
v
r
q)0(
01,41,2
)0(01,44,1
)3(0
)2(5
1,11,41,1,
1,11,44,2,
rspqsrsppqe
rspqsrsppqe
DDBBDBBD
DDBBDBBD
ffffwvr
ffffwvq
43)0(
01,41,24)0(
01,44,13)(
0
21)0(
01,41,22)0(
01,44,11)(
5
))(())((
))(())((3
2
Partial solution
Merge
Final solution
(3-31)
(3-32)
(3-33)
(3-35)
16
Temperature distributions of steady-state heat conduction for four layers of materials :
our program vs. commercial tool Raphael
1: Silicon bulk (heat source) 2: Chip die 3: Adhesive 4: Heat spreader 5: Heat sink
0 5 10 15 20 25 30 35 40 45 50
0 20 40 60 80 100 120 140
20
40
60
80
100
120
T"mat_all_data.txt" u 1:2:3
120 110 100 90 80 70 60 50 40 30
XY
T
Y
Ｔ(℃
)
X1
2 3 4 51: Silicon bulk (heat source) 2: Chip die 3: Adhesive 4: Heat spreader 5: Heat sink
0 5 10 15 20 25 30 35 40 45 50
0 20 40 60 80 100 120 140
20
40
60
80
100
120
T"mat_all_data.txt" u 1:2:3
120 110 100 90 80 70 60 50 40 30
XY
T
Y
Ｔ(℃
)
X1
2 3 4 5
X1
2 3 4 5
0 5 10 15 20 25 30 35 40 45 50
0 20 40 60 80 100 120 140
20
40
60
80
100
120
T"mat_all_data.txt" u 1:2:3
120 110 100 90 80 70 60 50 40 30
XY
T
1: Silicon bulk (heat source) 2: Chip die 3: Adhesive 4: Heat spreader 5: Heat sink
X1
2 3 4 5
Y
Ｔ(℃
)
0 5 10 15 20 25 30 35 40 45 50
0 20 40 60 80 100 120 140
20
40
60
80
100
120
T"mat_all_data.txt" u 1:2:3
120 110 100 90 80 70 60 50 40 30
XY
T
1: Silicon bulk (heat source) 2: Chip die 3: Adhesive 4: Heat spreader 5: Heat sink
X1
2 3 4 5
X1
2 3 4 5
Y
Ｔ(℃
)
Our ProgramSolver: S-PSM
Raphael [7]Solver : Iteration method
17
4. Results and Discussion• Table I shows the CPU times required and the
residuals produced by our program and Raphael. • The results demonstrate that for the largest grid, our
program ran 5.8 and 8.9 times faster while keeping smaller residuals by 5 and 1 order of magnitudes, than Raphael [7]
h g n p n q n r n s CPU(sec) Residual CPU(sec) Residual CPU(sec) Residual
128 44 8 8 4 16 0.05 1.3E-13 0.26 9.5E-07 0.32 9.3E-11
256 80 16 16 8 32 0.18 5.1E-13 1.98 9.9E-07 2.78 9.4E-11
512 152 32 32 16 64 1.7 2.1E-12 13.2 9.9E-07 19.5 9.9E-11
1,024 296 64 64 32 128 15.4 8.9E-12 89.9 9.7E-07 136.6 9.9E-11
Number of grid points S-PSMRaphael
(EPS=1.0E-6)Raphael
(EPS=1.0E-10)
18
5. Conclusions • We have proposed a new technique of solving
two dimensional Laplace equations to thermal analysis for multi-layer VLSI chips.
• Our program is superior to a commercial CAD tool, Raphael (iteration method) [7].
• Further Work : extension to Poisson equation (heat generation), three dimensional, transient heat conduction analysis, and the case of complex shapes and boundary conditions of materials.