1

A High Efficiency Full-Chip Thermal Simulation AlgorithmYong Zhan and Sachin S. Sapatnekar

Department of Electrical and Computer EngineeringUniversity of Minnesota

Abstract— Thermal simulation has become increasingly importantin chip design especially in the nanometer regime, where theon-chip hot spots severely degrade the performance and reliability ofthe circuit and increase the leakage power. In this paper, wepresenta highly efficient and accurate thermal simulation algorithm that iscapable of performing full-chip temperature calculations at the celllevel. The algorithm is a combination of several important numer-ical techniques including the Green function method, the discretecosine transform (DCT), and the frequency domain computations.Experimental results show that our algorithm can achieve orders ofmagnitude speedup compared with previous Green function basedalgorithms while maintaining the same accuracy.

I. INTRODUCTION

Thermal simulation algorithms in chip design can be roughlydividedinto two categories based on whether the meshing of the entire substrate isnecessary during the simulation process. The generic thermal simulationalgorithms such as the finite difference method (FDM) and thefiniteelement method (FEM) used in [1] and [2] enjoy the advantagesof highflexibility in handling different kinds of boundary conditions in thermalproblems and the capability of achieving high accuracy. However, therequirement of meshing the entire substrate and later solving a large systemof linear equations in the simulations using these methods makes themrelatively inefficient. In [3], the thermal-ADI algorithm was proposed toefficiently solve the transient thermal problems using a meshing schemesimilar to that used in the FDM. However, for steady-state analysis, theADI algorithm can also become slow if the initial guess of thetemperaturedistribution is far from the final solution.

The boundary element method (BEM) constitutes another class of ther-mal simulation algorithms in which the volume meshing of thesubstrate iscompletely avoided. An important underlying concept in theBEM is theGreen function which describes the temperature distribution in the chipwhen a unit point power source is present. In [4] and [5], the analyticalforms of the Green function were derived by assuming that thechip wasinfinitely large horizontally. One significant advantage ofthe analyticalforms of the Green function is that they are very cheap to evaluate andhence can be easily incorporated into optimization procedures where theGreen function needs to be evaluated many times. However, byassumingthat the chip is infinitely large horizontally, the derived Green functionstend to severely underestimate the temperature, although they can correctlyidentify the locations of the hot spots as shown in [4]. In [6], a Greenfunction that is suitable for the rectangular shaped chip geometry waspresented and look-up tables were established based on the Green functionto assist the efficient evaluation of the temperature field. Nevertheless,the cost of this algorithm can become prohibitive for cell level full-chipthermal simulations where both the number of heat sources and the numberof field regions are large.

In [7], an efficient thermal simulation algorithm based on the solution ofthe finite difference equations using the multigrid approach was proposed,and it has the capability of performing the full-chip thermal analysis.In this paper, we present another highly efficient and accurate full-chip thermal simulation algorithm that is based on the Greenfunctionmethod, the discrete cosine transform (DCT), and the frequency domaincomputations. Since the temperature field can be obtained byconvolvingthe power distribution with the underlying Green function,using thefrequency domain computations in conjunction with the DCT will result ina significant improvement in efficiency as can be achieved in many signalprocessing works where the time or space domain convolutionis replacedby the frequency domain analysis. The functional eigen-decompositionapproach used in the substrate parasitic extraction work in[8] is also aspecific implementation of the frequency domain analysis. Our algorithmtakes a piece-wise constant power density map as the input and generatesa piece-wise constant temperature map as the output. The primary stepsof the algorithm include

1) Obtain the frequency domain representation of the power densitymap using the 2D DCT.

This work was supported in part by DARPA under grant N66001-04-1-8909 and NSF underaward CCR-0205227.

The authors thank the University of Minnesota Supercomputing Institute for providing thecomputing facilities.

2) Calculate the frequency domain representation of the temperaturemap by multiplying each frequency component of the power densitymap by the corresponding frequency response of the linear systemdetermined by the Green function.

3) Use a 2D inverse discrete cosine transform (IDCT) to obtain thetemperature map from its frequency domain representation.

Both the 2D DCT and the 2D IDCT can be calculated efficiently using the2D fast Fourier transform (FFT) inO((M ·N)×log(M ·N)) time, whereM ·N is the total number of grid cells in the power density map, which isalso the total number of grid cells in the resulting temperature map. Thisis a significant improvement over theO((M ·N)2) time complexity of thealgorithm presented in [6]. Experimental results show thatthe algorithmproposed in this paper can achieve orders of magnitude speedup over thealgorithm in [6], while still maintaining the same accuracy.

II. PROBLEM FORMULATION

Chip

Packaging

(a) (b)

a

b

Heat spreaderHeat sink

Chip

z = 0z = −d1z = −dN−1z = −dN

Fig. 1. Schematic of a VLSI chip with packaging (a) IC chip andthe packagingstructure (b) simplified model of the chip and packaging.

Fig. 1(a) shows an IC chip with the associated packaging, andFig. 1(b)shows a schematic of the structure in Fig. 1(a) where the packagingincluding the heat spreader and the heat sink has been simplified but themultilayered structure of the chip is explicitly shown. Thesteady statetemperature distribution inside the chip are governed by the Poisson’sequation

∇2T (r) = −

g(r)

kl(r)

(1)

where r = (x, y, z), T (r) is the temperature (°C) distribution insidethe chip, g(r) is the volume power density (W/m3), and kl(r) is thethermal conductivity (W/(m·°C)) of the layer where pointr is located [9].The vertical surfaces and the top surface of the chip are assumed tobe adiabatic [10], and the bottom surface of the chip is assumed to beconvective, with an effective heat transfer coefficienth (W/(m2·°C)) [11].In mathematical forms, these boundary conditions can be expressed as

∂T (r)

∂x

˛˛x=0,a

=∂T (r)

∂y

˛˛y=0,b

= 0 (2)

∂T (r)

∂z

˛˛z=0

= 0 (3)

kN∂T (r)

∂z

˛˛z=−dN

= h(T (r)|z=−dN− Ta) (4)

whereTa is the ambient temperature, andkN is the thermal conductivityof the bottom layer of the chip. In addition, we enforce the continuityconditions at the interface between adjacent layers withinthe multilayeredchip, i.e.,

T (r)|z=−di+ε = T (r)|z=−di−ε (5)

ki∂T (r)

∂z

˛˛z=−di+ε

= ki+1∂T (r)

∂z

˛˛z=−di−ε

(6)

whereε is an infinitely small quantity andki is the thermal conductivityof the ith material layer in the multilayered chip structure.

III. FULL -CHIP THERMAL SIMULATION ALGORITHM

A. The Green function of the rectangular-shaped multilayeredstructureLet G(r, r′), with r = (x, y, z) andr

′ = (x′, y′, z′), be the distributionof temperature aboveTa in the multilayer when a unit point power sourceof 1W is placed at positionr′. ThenG(r, r′) satisfies the equation

∇2G(r, r

′) = −δ(r − r

′)

kl(r)

(7)

and the boundary conditions

∂G(r, r′)

∂x

˛˛x=0,a

=∂G(r, r′)

∂y

˛˛y=0,b

= 0 (8)

∂G(r, r′)

∂z

˛˛z=0

= 0 (9)

kN∂G(r, r′)

∂z

˛˛z=−dN

= hG(r, r′)|z=−dN

(10)

G(r, r′)|z=−di+ε = G(r, r

′)|z=−di−ε (11)

ki∂G(r, r′)

∂z

˛˛z=−di+ε

= ki+1∂G(r, r′)

∂z

˛˛z=−di−ε

(12)

where δ(r, r′) = δ(x − x′)δ(y − y′)δ(z − z′) is the three-dimensionalDirac delta function, andG(r, r′) is the Green function. The temperaturefield under an arbitrary power density distribution can be obtained easilyas

T (r) = Ta +

Za

0

dx′

Zb

0

dy′

Z0

−dN

dz′G(r, r

′)g(r′) (13)

Using a derivation similar to that presented in [12], the Green functioncan be written in the form

G(r, r′) =∞X

m=0

∞X

n=0

cos

„mπx

a

«cos

„nπy

b

«cos

mπx′

a

!cos

nπy′

b

!Z′

mn(z, z′) (14)

whereZ′

mn(z, z′)′s are functions of only thez coordinates of the sourceand field points.

B. Full-chip thermal simulation algorithmIn the following analysis, we assume that both the heat sources and the

field regions are located on discrete horizontal planes. Since the verticaldimensions of the devices are much smaller than that of the silicon chip,this assumption is reasonable for most practical purposes.For a particularpair of source and field planes, i.e., for a particularz and z′, the Greenfunction can be written as

G(x, y, x′, y

′) =

∞X

m=0

∞X

n=0

Cmncos

„mπx

a

«cos

„nπy

b

«cos

mπx′

a

!cos

nπy′

b

!(15)

The temperature distribution on the field plane due to the heat sources onthe source plane is given by

T (x, y) = Ta +

Za

0

dx′

Zb

0

dy′G(x, y, x

′, y

′)Pd(x′, y

′) (16)

wherePd(x′, y′) is the power density distribution on the source plane. Theconvolution integral in (16) can be considered as the governing equationof a linear system determined by the Green functionG(x, y, x′, y′).

As stated previously, the first step of our algorithm is to obtain thefrequency domain representation of the power density map inthe form

Pd(x′, y

′) =

∞X

i=0

∞X

j=0

aijφij(x′, y

′) (17)

where

φij(x, y) = cos

„iπx

a

«cos

„jπy

b

«(18)

It is easy to show thatφij(x, y) satisfies the equation

λijφij(x, y) =

Z a

0

dx′

Z b

0

dy′G(x, y, x

′, y

′)φij(x′, y

′) (19)

where

λij =

8<:

abCij if i = j = 012 abCij if i = 0, j 6=0 or i6=0, j = 014 abCij if i6=0, j 6=0

(20)

is the response of the linear system to the frequency component φij(x, y).After the frequency domain representation of the power density distributionin the source plane is obtained, the temperature distribution in the fieldplane can be calculated easily by

T (x, y) = Ta +

∞X

i=0

∞X

j=0

λijaijφij(x, y) (21)

As will be shown next, both the frequency decomposition in (17) and thedouble-summation in (21) can be calculated efficiently using the DCT andIDCT through the FFT.

M grid cells

N g

rid

cel

ls

y’

x’

p

p

p

p

p p

p

p

p

M−1,0

M−1,1

10

1101

00

0,N−1 1,N−1 M−1,N−1

Fig. 2. The arrangement of theM×N grid cells on the source plane.

Now we assume that both the source plane and the field plane aredivided into M×N rectangular grid cells of equal size as shown in Fig.2, and the power density in each grid cell on the source plane is uniform,i.e., the power density distribution can be written in the piece-wise constantform

Pd(x′, y

′) =

M−1X

m=0

N−1X

n=0

PmnΘ(x′ − (m +1

2)∆x, y

′ − (n +1

2)∆y) (22)

where

Θ(x′, y

′) =

1 if |x′|≤ 1

2∆x and |y′|≤ 12∆y

0 otherwise (23)

and ∆x = aM

, ∆y = bN

. Pmn is the power density of themnth gridcell. Substituting (22) into (17) and using the orthogonality property ofthe cosine functions in the integral sense, we obtain

aij = Aij

M−1X

m=0

N−1X

n=0

Pmncos

„iπ(2m + 1)

2M

«cos

„jπ(2n + 1)

2N

«(24)

where

Aij =

8>>>><>>>>:

1MN if i = j = 0

4iNπ sin

`iπ2M

´if i6=0, j = 0

4Mjπ

sin“

jπ2N

”if i = 0, j 6=0

16ijπ2 sin

`iπ2M

´sin“

jπ2N

”if i6=0, j 6=0

(25)

Note that to accurately represent the power density distributionPd(x

′, y′) using (17), the theoretical upper limit of the double summationshould be infinity. In practical implementations, however,the summationmust be truncated to ensure a reasonable runtime. Since (17)is essentiallythe Fourier expansion ofPd(x

′, y′), a natural criterion for determining thetruncation point is that enough “energy” contained inPd(x

′, y′) is coveredby the truncated Fourier expansion. Mathematically, we have

Za

0

dx′

Zb

0

dy′P

2d (x

′, y

′) = ab

∞X

i=0

∞X

j=0

sija2ij (26)

where

sij =

8<:

1 if i = j = 012 if i = 0, j 6=0 or i6=0, j = 014 if i6=0, j 6=0

(27)

Substituting (22) into the left hand side of (26), we obtain

1

MN

M−1X

m=0

N−1X

n=0

P2mn =

∞X

i=0

∞X

j=0

sija2ij (28)

which can be considered as a form of the Parseval’s theorem. Thetruncation pointsM ′ andN ′ are then determined by

M′−1X

i=0

N′−1X

j=0

sija2ij ≥ η

1

MN

M−1X

m=0

N−1X

n=0

P2mn

!(29)

where η is the proportion of the “energy” of the space domain signalPd(x

′, y′) that must be covered by the truncated Fourier expansion. Inpractice, we found that settingη to 90% will usually be enough to obtainvery accurate results in temperature calculations.

Note that for0≤i<M and0≤j<N , the double summation in (24) canbe considered as a term in the 2D type-II DCT [13] of the power densitymatrix P . For i≥M or j≥N , we can always find integerss1 ands2 suchthat i = 2s1M±i andj = 2s2N±j where0≤i<M and0≤j<N . Hence,for any i and j, we always have

aij = ±AijePij (30)

where

ePij =

M−1X

m=0

N−1X

n=0

Pmncos

iπ(2m + 1)

2M

!cos

jπ(2n + 1)

2N

!(31)

with 0≤i<M and 0≤j<N is the 2D type-II DCT of theP matrix andthe sign of (30) is determined by whethers1 and s2 are even or oddnumbers. Equation (31) can be calculated efficiently using the 2D FFT inO((M ·N)×log(M ·N)) time. After the 2D DCT matrixeP is obtained,the calculation ofaij simply involves computing the coefficientAij andfinding the corresponding termePij .

From (18) and (21), the temperature distributionT (x, y) can now bewritten as

T (x, y) = Ta +

M′−1X

i=0

N′−1X

j=0

λijaijcos

„iπx

a

«cos

„jπy

b

«(32)

and the average temperature of themnth grid cell can be obtained by

Tmn =1

∆x∆y

Z(m+1)∆x

m∆x

dx

Z(n+1)∆y

n∆y

dyT (x, y)

= Ta +MN

ab

M′−1X

i=0

N′−1X

j=0

Bijcos

„iπ(2m + 1)

2M

«cos

„jπ(2n + 1)

2N

«(33)

where

Bij =

8>>>><>>>>:

λijaijab

MN if i = j = 02λijaij

abiNπ sin

`iπ2M

´if i6=0, j = 0

2λijaijab

Mjπ sin“

jπ2N

”if i = 0, j 6=0

4λijaijab

ijπ2 sin`

iπ2M

´sin“

jπ2N

”if i6=0, j 6=0

(34)

As stated previously, anyi≥M andj≥N can be written asi = 2s1M±iand j = 2s2N±j such that0≤i<M , 0≤j<N and s1,s2 are integers.Using the periodicity of the cosine function, we can finally castTmn intothe form

Tmn = Ta +

M−1X

i=0

N−1X

j=0

Lijcos

iπ(2m + 1)

2M

!cos

jπ(2n + 1)

2N

!(35)

whereLij = KijePij andKij can be calculated as follows

1) if i = j = 0

Kij =λ00

MN(36)

2) if i6=0, j = 0

Kij =8M

Nπ2sin2

iπ

2M

!X

i < M′

i = 2s1M±i

λi0

i2(37)

3) if i = 0, j 6=0

Kij =8N

Mπ2sin2

jπ

2N

!X

j < N′

j = 2s2N±j

λ0j

j2(38)

4) if i6=0, j 6=0

Kij =64MN

π4sin

2

iπ

2M

!sin

2

jπ

2N

!X

i < M′

i = 2s1M±i

X

j < N′

j = 2s2N±j

λij

i2j2

(39)

Input:

• Chip geometry and physical properties of the material layers.• Power density map - matrixP .

Output: Temperature distribution map - matrixT .Algorithm:

1) Calculate the Green function coefficientsCij′s;

2) Calculate the frequency responses of the systemλij′s;

3) Calculate the type-II 2D DCT of the power density matrixeP = 2DDCT(P );4) TSE = 1

MN

PM−1m=0

PN−1n=0 P 2

mn;5) M ′ = M, N ′ = N ;

ASE =PM′−1

i=0

PN′−1j=0 sija2

ij ;while ( ASE < η×TSE )

M ′ = M ′ + M, N ′ = N ′ + N ;UpdateASE;

end while;6) Calculate the matrixK;7) Calculate the matrixL with Lij = Kij

ePij ;8) Calculate the temperature distribution map using the type-II 2D IDCT T =

Ta + 2DIDCT(L);

Fig. 3. Thermal simulation algorithm using the Green function method, the DCT,and the frequency domain computations.

After the coefficientsK′

ijs are calculated, the matrixL can be easily

obtained by point-wise multiplying the matricesK and eP . The doublesummation in (35) can then be calculated efficiently using the 2D IDCT.

The complete thermal simulation algorithm using the Green functionmethod, the DCT, and the frequency domain computations is shown inFig. 3. The asymptotic time complexity of the algorithm isO((M ·N) ×log(M ·N)) where M ·N is the total number of grid cells. This is asignificant improvement over theO((M ·N)2) complexity of the algorithmgiven in [6]. Note that up to now, we have focused on the effectof one source layer on the temperature distribution of the field layer.When multiple source layers are present, such as in the emerging 3DIC technology, their effects can be calculated individually and summedup. The ambient temperatureTa should only appear once in the finalsummation.

IV. EXPERIMENTAL RESULTS

A. Accuracy and Efficiency of the Algorithm

01

23

4

0

2

40

50

100

150

200

250

x(mm)y(mm)

Pow

erD

ensi

ty(W

/cm

2)

(a)

01

23

4

0

2

470

75

80

85

90

95

x(mm)y(mm)

T−

Ta

(°C

)

(b)

Fig. 4. Power density and temperature distribution of an example chip (a) powerdensity distribution (b) temperature distribution.

Because the method presented in [6] is accurate except for the very smalltruncation error of the Green function, we use the result in [6] as the bench-mark to test the accuracy and efficiency of our algorithm. Fig. 4 shows thepower distribution of an example chip and the calculated temperature mapusing the algorithm proposed in this paper. The heat sourcesare assumedto be located on the top surface of the chip and the size of the temperaturemap is 64×64. The chip has dimensions of 3.3mm×3.3mm×0.5mm. Thethermal conductivity of silicon is 148W/(m·°C) and the bottom surfaceof the chip has an effective heat transfer coefficient of 8700W/(m2·°C).We require our algorithm to achieve a similar accuracy as that in [6] bychoosingη in (29) appropriately, i.e., within 1% error compared with theresults from commercial computational fluid dynamic softwares, and theruntimes of the two algorithms are compared. Each runtime isdividedinto two parts, i.e., the time spent on the steps that are independent ofthe input power density matrix and hence can be pre-calculated outsidethe optimization loop in thermal-aware designs, and the time spent onthe steps that depend on the input power density matrix and hence mustbe executed within the optimization loop. For both algorithms, the Greenfunction coefficientsCij

′s can always be pre-calculated and stored. Inaddition, the look-up tables in the algorithm in [6] can be pre-calculatedwhile the frequency responsesλij

′s in our algorithm can be pre-calculated.For the pre-calculated steps, the runtime is dominated by the computation

of the coefficientsCij′s in both algorithms, which may take about 95sec

to obtain a 2048×2048C matrix. However, since these steps only need tobe executed once for each die geometry and then used many times in laterthermal simulations, the amortized cost is usually extremely small andwe will ignore this part of the runtime in further analysis. Experimentalresults show that for the steps that are not pre-calculated,the runtime of ouralgorithm using MATLAB is 0.09sec while that of the algorithm in [6] is128sec. Note that the runtime of the algorithm in [6] is linear with respectto the number of heat sources and there are only 13 heat sources in theexample given here. For cell level full-chip simulations where the numberof heat sources is significantly larger, the advantages of our algorithm willbecome even more obvious.

B. Cell Level Full-Chip Thermal Simulation

0

0.5

1

0

0.5

10

50

100

150

x(cm)y(cm)

Pow

erD

ensi

ty(W

/cm

2)

(a)

0

0.5

1

0

0.5

120

30

40

50

60

70

x(cm)y(cm)

T−

Ta

(°C

)

(b)

Fig. 5. Cell level power density and temperature distribution of a 1cm×1cm chip(a) power density distribution (b) temperature distribution.

In this subsection, we show an example of cell level full-chip thermalsimulations. We consider a chip with dimensions of 1cm×1cm×0.5mmand the same physical properties as the chip used in the previous example.There are 1024×1024 square grid cells of equal size located on the topsurface of the chip and a 1024×1024 temperature distribution map of thecell layer is calculated. Fig. 5 shows the input power density map andthe resulting temperature map. The time it takes for MATLAB to obtainthis temperature map containing 1.05M grid cells is only 6.4sec excludingthe time for the pre-calculations while the runtime of the algorithm in [6]becomes intractable.

V. DISCUSSIONS- STRATEGIES FORPERFORMING THETHERMALSIMULATIONS WITH LOCAL HIGH ACCURACY REQUIREMENTS

The situation frequently arises in real design environments where theaccuracy requirements on the thermal simulation differ from place to placeon the same chip. For example, in mixed signal designs where the analogcircuits are fabricated on the same chip as the digital circuits, the analogblocks often have more stringent accuracy requirements on the thermalsimulation because the operations of the analog circuits are more sensitiveto temperature. For these kinds of problems, a better strategy can beadopted to accelerate the runtime of the algorithm further.The key ideais to use a coarse grid to divide the source and field layers such that eachgrid cell can contain several logic gates or analog functional units. Thepower density of each grid cell is calculated by summing up the powerdissipations of all the logic gates and analog functional units located in itand divide the sum by the area of the grid cell. A coarse temperature mapis then obtained from the coarse power density map using the algorithmpresented in section III and is used for the digital blocks onthe chip. Notethat we must ensure that the coarse grid is fine enough for the digitalblocks but they may not achieve the accuracy requirements ofthe analogblocks.

M grid cells

N g

rid c

ells

Fig. 6. A mixed signal chip where the analog block has higher requirement onthe accuracy of the thermal simulation.

Fig. 6 shows a chip that is divided intoM×N coarse grid cells eachof which contains several logic gates or analog functional units, and letthe shaded area represent the analog block. AnM×N temperature mapis first obtained. The inaccuracies in the temperature calculations, besidesthat due to the truncation of the eigen-decomposition of thepower densitymap, will come from two sources which include

• Assuming that the power density in each grid cell is uniform.• Only the average temperature of each grid cell is calculated, i.e., all

the logic gates and analog functional units inside the same grid cellobtain the same calculated temperature.

Now assume that we need to calculate the temperature of the analogfunctional unit located in theijth grid cell and represented by the blackrectangle more accurately. LetTij andTij,kl be the average temperature ofthe ijth grid cell and the contribution of the average power density of theklth grid cell to the average temperature of theijth grid cell in the coarsegrid temperature calculations respectively, and letTa.c. represent the moreaccurate average temperature of the analog functional unit. We divide thegrid cells into two categories, i.e., those with close interactions with theijth grid cell (denoted byCIij) and those without close interactions withtheijth grid cell. The effects of the logic gates and analog functional unitsthat are contained in the grid cells belonging toCIij are re-calculated forincreased accuracy. For example, we can put theijth grid cell and all thegrid cells surrounding it into the first category and all the other grid cellsinto the second category. If higher accuracy is required, then more gridcells should be put into the first category. The temperatureTa.c. can thenbe calculated using

Ta.c. = Tij −X

kl∈CIij

Tij,kl +X

sT

gate and functional units (40)

where T gate and functional units is the contribution toTa.c. from the sth

logic gate or analog functional unit in the grid cells that have closeinteractions with theijth grid cell. BothTij,kl andT gate and functional unit

s

can be calculated efficiently using the table look-up approach given in [6]and will not be reiterated here.

VI. CONCLUSIONS

In this paper, we presented a cell level full-chip thermal simulationalgorithm that is a combination of the Green function method, the DCT,and the frequency domain computations. Experimental results show thatour algorithm can achieve orders of magnitude speedup compared withprevious Green function based thermal simulation algorithms while stillmaintain the same accuracy. The simulation of a chip containing 1.05Mgrid cells only takes about 6.4sec after the pre-calculations have beenperformed. In addition, the strategies that can be used for the problemsthat have local high accuracy requirements on temperature calculations arealso discussed.

REFERENCES

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