Proceedings of IMECE2005
2005 ASME International Mechanical Engineering Congress and Exposition
Orlando, Florida, USA, November 5-11, 2005
IMECE2005-79542
PERFORMANCE COMPARISON OF NUMERICAL PROCEDURES FOR EFFICIENTLY
SOLVING A MICROSCALE HEAT TRANSPORT EQUATION DURING
FEMTOSECOND LASER HEATING OF NANOSCALE METAL FILMS
Ravi Ranjan Kumar∗
J. M. McDonough
M. P. Mengüç
Department of Mechanical Engineering
University of Kentucky
Lexington, Kentucky, 40506
Illayathambi Kunadian
Center for Applied Energy Research
University of Kentucky
Lexington, Kentucky 40511
Email: ikuna0@engr.uky.edu
ABSTRACT
An alternative discretization and solution procedure for implicitly solving a 3-D microscale heat transport equation during
femtosecond laser heating of nanoscale metal films has been developed (Kunadian et al. [1]). The proposed numerical technique
directly solves a single partial differential equation, unlike other
techniques available in the literature which splits the equation
into a system of two equations and then apply discretization. The
present paper investigates performance of its split and unsplit
methods of solution via numerical experiments using Gauss–
Seidel, conjugate gradient, generalized minimal residual and δform Douglas–Gunn time-splitting methods to compare the computational cost involved in these methods. The comparison suggests that the unsplit method [1] employing δ-form Douglas–
Gunn spatial time-splitting is the most efficient way in terms of
CPU time taken to complete the simulation of solving the 3-D
time dependent microscale heat transport equation.
relativity). In order to ensure finite propagation of thermal disturbances a hyperbolic heat conduction equation (HHCE) was
proposed (Vernotte, Cattaneo [2, 3]). However, since the classical and hyperbolic models neglect the thermalization time (time
for electrons and lattice to reach thermal equilibrium) and relaxation time of the electrons, their applicability to very short pulse
laser heating (Qui,Qui [5, 6]) becomes questionable.
Anisimov et al. [7] proposed a two-step model to describe
the electron temperature Te and the lattice temperature Tl during short-pulse laser heating of metals. Later, Qiu and Tien [5, 6]
rigorously derived the hyperbolic two-step model from the Boltzmann transport equation for electrons. They numerically solved
the equations of this model in the context of a 96 f s duration laser
pulse irradiating a thin film of thickness 0.1µm. Predicted temperature change of the electron gas during the picosecond transient agreed well with experimental data, supporting the validity of the hyperbolic two-step model for describing heat transfer
mechanisms during short-pulse laser heating of metals.
Tzou proposed the dual phase-lag (DPL) model [8–11] that
reduces to diffusion, thermal wave, phonon-electron interaction
[5, 6], and pure phonon scattering (Guyer et al. [12]) models
under special values of timescale parameters τq and τT . Over
the years, various numerical methods have been investigated for
solving the DPL equation. Most early numerical studies involved
only the 1-D equation, often using explicit discretization [21].
Recent studies have considered 2-D and 3-D DPL equations, em-
INTRODUCTION
Fourier’s law predicts thermal disturbances propagating at
infinite velocities, implying that a thermal disturbance applied
at a certain location in a solid medium can be sensed immediately anywhere else in the medium (violating precepts of special
∗ Address
all correspondence to this author.
1
c 2005 by ASME
Copyright heating of the sample is expressed as
ploying implicit discretizations [16, 17, 19, 20]. Dai and Nassar [13–17] have developed an implicit finite-difference scheme
in which the DPL equation is split into a system of two equations with the individual equations discretized using the CrankNicolson scheme and solved sequentially. Dai showed by the
discrete energy method [13–18] that the scheme was unconditionally stable.
Zhang and Zhao [19, 20] employed the iterative techniques
Gauss–Seidel, successive overrelaxation (SOR), conjugate gradient (CG), and preconditioned conjugate gradient (PCG) to solve
a Dirichlet problem for the 3-D DPL equation. Kunadian et
al. [1] developed an alternative discretization and solution procedure for implicitly solving a 3-D microscale heat transport equation and applied to simulation of this and related femtosecond
laser heating of nanoscale metal films. They found that the δform Douglas–Gunn time-splitting method outperformed other
numerical techniques available in the literature in terms of computational time taken to complete the simulation.
The purpose of this paper is to more thoroughly investigate
the computational cost involved in the split and the unsplit methods by employing Gauss–Seidel, conjugate gradient, generalized
minimal residual (GMRES) and δ-form Douglas–Gunn timesplitting methods. Performance comparison of this sort over a
wide range of currently-used numerical methods has high applicability in making a wise choice of an efficient numerical method
for solution of large scale time-dependent problems.
First, we present the discretization of the split and unsplit 3D DPL equations. Next, we outline a numerical solution strategy
for the solution of discretized equations and compare the numerical results with the experimental results of Brorson et al. [23]
and Qiu and Tien [5, 6] corresponding to the transient temperature distribution obtained during short pulse laser heating of thin
metal films. Finally we present the performance data for both
split and the unsplit methods in terms of computational time requirements observed for the above-noted methods.
x 1.992 | t − 2t p |
1−R
,
S(x,t) = 0.94J
exp − −
tpδ
δ
tp
(2)
where J is the laser fluence, t p describes laser heating of the
electron-phonon system from a thermalization state; δ is laser
penetration depth, and R is reflectivity [21]. Dai and Nassar
[13–17] split Eq. (1) by introducing an intermediate function
u(T,t), discretized the individual equations using the Crank–
Nicolson scheme and solved them sequentially. The intermediate
function is defined as
u(T,t) = T + τq
∂T
,
∂t
(3)
which upon substitution into Eq. (1) and discretization we obtain
n+1
n+1
n+1
n+1
C4 Ti,n+1
j,k +C5 Ti+1, j,k + Ti−1, j,k +C6 Ti, j+1,k + Ti, j−1,k
n
n+1
+
T
+C7 Ti,n+1
j,k+1
i, j,k−1 = F ,
(4)
where
n
n
F n = C8 Ti,nj,k +C9 Ti+1,
+
T
j,k
i−1, j,k
n
n
+C10 Ti, j+1,k + Ti, j−1,k +C11 Ti,nj,k+1 + Ti,nj,k−1
+2uni, j,k + ∆tG∗ .
(5)
The various constants are defined as follows:
DISCRETIZATION OF SPLIT AND UNSPLIT 3-D DPL
EQUATION
The governing transport equation used to describe the thermal behavior of microstructures in 3D is expressed as [8]
τq
1 ∂T
+
− τT
2
α ∂t
α ∂t
∂2 T
∂(∇2 T )
∂t
= ∇2 T +
∂S
1
S + τq
k
∂t
∆t
1
C5 = − τT +
,
2 ∆x2
1
∆t
,
C6 = − τT +
2 ∆y2
∆t
1
C7 = − τT +
,
2 ∆z2
1 2τq
+
− 2(C5 +C6 +C7 )
C4 =
α α∆t
∆t
1
C9 = −τT +
,
2 ∆x2
(1)
for (x, y, z) ∈ Ω ⊂ R3 and t ∈ (t0 ,t f ]. Here k, α, τT and τq are conductivity, diffusivity, phase lag temperature gradient and phase
lag of heat flux, respectively. In one dimension the volumetric
2
(6a)
(6b)
(6c)
(6d)
(6e)
c 2005 by ASME
Copyright 1
∆t
,
2 ∆y2
∆t
1
C11 = −τT +
,
2 ∆z2
1 2τq
− 2(C9 +C10 +C11 ),
C8 = − +
α α∆t
" n+1
#
n
Si, j,k + Si,n j,k
Si,n+1
j,k − Si, j,k
∗
G =
+ τq
2
2
2τq
2τq
n+1
n+1
Ti, j,k + 1 −
Ti,nj,k − uni, j,k .
ui, j,k = 1 +
∆t
∆t
C10 =
−τT +
NUMERICAL SOLUTION STRATEGY
To test performance of the numerical methods listed above,
consider use of Neumann boundary conditions,
(7a)
∂T
=0
∂n
(7b)
(10)
(7c)
on all of ∂Ω, where n = x, y, z, and initial conditions
(7d)
T (x, y, z) = To
and
(7e)
∂T
(x, y, z, 0) = 0.
∂t
(11)
It is convenient to rearrange Equation (4) to the form
In the case of the unsplit method, Eq. (1) is solved directly using Trapezoidal integration [1]. The final form of the discretized
equation remains the same as the split method Eq. (4) except
for details of the coefficients and the RHS term F n . In this case
coefficients are
1
∆t
C5 = − τT +
,
2 ∆x2
1
∆t
,
C6 = − τT +
2 ∆y2
∆t
1
C7 = − τT +
,
2 ∆z2
τq
1
+
− 2(C5 +C6 +C7 )
C4 =
α α∆t
∆t
1
C9 = −τT +
,
2 ∆x2
1
∆t
C10 = −τT +
,
2 ∆y2
1
∆t
,
C11 = −τT +
2 ∆z2
1 2τq
C8 =
− 2(C9 +C10 +C11 ),
+
α α∆t
" n+1
#
n+1
n
n
S
−
S
S
+
S
i,
j,k
i,
j,k
i,
j,k
i,
j,k
G∗ =
+ τq
.
2
2
n+1
n+1
n+1
n+1
C7 Ti,n+1
j,k−1 +C6 Ti, j−1,k +C5 Ti−1, j,k +C4 Ti, j,k +C5 Ti+1, j,k +
n+1
n
C6 Ti,n+1
j+1,k +C7 Ti, j,k−1 = F ,
which can be written in matrix form as AT = F. The matrix A is
a sparse 7-band matrix, which can be stored in a N ×7 array, with
N (= Nx × Ny × Nz ) being number of grid points in the physical
domain. Thus Eq. (12) can be grouped in matrix form as follows
(8a)
(8b)
(8c)
 


 T n+1
Fi,nj,k−1
j,k−1
2C7  i,n+1

Fi,nj−1,k 
 T j−1,k  
C7 

 
  i,n+1


n
 T j,k   Fi−1,
. 
j,k 

  i−1,

n+1
n


Fi, j,k 
. 

  Ti, j,k  = 


n
 n+1 
. 

Fi+1,
  Ti+1, j,k  
j,k


  Fn
. 


 Ti,n+1
i, j+1,k
j+1,k
n
0
n+1
F
Ti, j,k+1
i, j,k+1
(13)
where T is the solution vector, and F is the known RHS vector.
Hence the system of linear equations (13) is strictly diagonally
dominant. The coefficient matrix A is a nonsymmetric positive
semi-definite seven banded matrix.
The system of linear equations AT = F arising from finite
difference discretization for both split and unsplit procedures has
been solved using Gauss–Seidel, conjugate gradient, generalized
minimal residual and δ-form Douglas–Gunn time-splitting methods.
Equation (4) is three-level in time, and it can be efficiently
solved using a Douglas–Gunn time splitting method [22]. By
dividing both sides of Eq. (4) by

0
 C7

 .

 .

 .

 .
2C7
(8d)
(8e)
(8f)
(8g)
(8h)
(8i)
and the right-hand side function is
n
n
F n = C8 Ti,nj,k +C9 Ti+1,
j,k + Ti−1, j,k
+C10 Ti,nj+1,k + Ti,nj−1,k +C11 Ti,nj,k+1 + Ti,nj,k−1
−
τq n−1
T
+ ∆tG∗.
α∆t i, j,k
(12)
0
C6
.
.
.
.
2C6
0 C4
C5 C4
. C4
. C4
. C4
. C4
2C5 C4
2C5
C5
.
.
.
.
0
2C6
C6
.
.
.
.
0
τq
1
+
α α∆t
(9)
3
(14)
c 2005 by ASME
Copyright we obtain
1.0
Analytical, Chiu
Experimental, Qui & Tien
n+1
n+1
0
1 − 2C50 − 2C60 − 2C70 Ti,n+1
j,k +C5 Ti+1, j,k + Ti−1, j,k
n+1
0
n+1
n+1
n
+C60 Ti,n+1
+
T
+C
T
+
T
7
j+1,k
i, j−1,k
i, j,k+1
i, j,k−1 = S , (15)
Normalized Temperature Change
Experimental, Brorson et al.
where,
Sn =
C50 =
C60
=
C70 =
Fn
τq ,
1
α + α∆t
C5
τq ,
1
α + α∆t
C6
τq 1
+
α
α∆t
C7
τq .
1
α + α∆t
(16a)
(I + Ay )T
(I + Az )T
(3)
=T
=T
(2)
− Ay T
Figure 1.
(16d)
(17b)
n
(17c)
(1)
= δT
2.5
+T ,
0.1µm thick gold film.
NUMERICAL EXPERIMENTS
We conducted numerical experiments on a 3.4GHz Xeon
em64t processors@2GB machine using time step ∆t = 0.001ps
and t = 2.5ps to test performance of the split and unsplit methods. Convergence tests were performed using L2 (Ω) norm of the
difference between successive iterates with a convergence tolerance of 10−7 . Figure 1 shows the comparison between the numerical (explicit and implicit scheme), analytical [21] and the
experimental results of Brorson et al. [23] and Qiu and Tien
[5, 6] corresponding to the front surface transient response for
a 0.1µm thick gold film. The laser heat source term given by
Eq. (2) is used for this purpose. The thermal properties (α =
1.2 × 10−4m2 s−1 , k = 315W m−1 K −1 , τT = 90ps, τq = 8.5ps)
are assumed to be constant. The temperature change is normalized by the maximum value that occurs during the short-time
transient. The results from the present numerical scheme compare well with experimental and analytical results. The HHCE
and the parabolic heat conduction models neglect the microstructural interaction effect in the short-time transient, rendering an
over estimated temperature in the transient response as seen in
the figure.
Tables 1 and 2 show the CPU time required for entire 3-D
simulation using split and unsplit methods for the explicit procedure and iterative schemes, and the direct δ-form Douglas–
(19a)
(19c)
n
Front surface transient response for a
We remark that this form is the most efficient of the forms
found in the literature. Again, the implicit part of the above equations (19d) is tridiagonal, and is thus easily solved using LU decomposition.
(19b)
(I + Az )δT (3) = δT (2)
T
2.0
315Wm−1 K −1 , τT = 90ps, τq = 8.5ps.
(17a)
n
(I + Ax )δT (1) = Sn − (I + A)T n
(3)
1.5
Comparison between numerical (explicit and implicit schemes), analytical [21] and experimental results [6, 23]. α = 1.2 × 10−4m2 s−1 , k =
T (1) , T (2) and T (3) denote intermediate estimates of T n+1 with
T n+1 = T (3) . The implicit part T n+1 of the above equations (17c)
is tridiagonal, and is thus easily solved using LU decomposition.
By applying δ-form Douglas–Gunn time-splitting [22] we
can represent Eq. (15) as follows
n+1
1.0
(16c)
− Az T ,
= δT
0.5
Time (ps)
(1)
(1)
(1)
(I + Ax )T (1) = 1 − 2C50 Ti, j,k +C50 Ti+1, j,k + Ti−1, j,k (18a)
(2)
(2)
(2)
(I + Ay )T (2) = 1 − 2C60 Ti, j,k +C60 Ti, j+1,k + Ti, j−1,k (18b)
(3)
(3)
(3)
(I + Az )T (3) = 1 − 2C70 Ti, j,k +C70 Ti, j,k+1 + Ti, j,k−1 (18c)
(I + Ay )δT
Numerical, HHCE
0.4
0.0
0.0
where
(2)
0.6
Numerical, DPL
(16b)
(I + Ax )T (1) = Sn − Ay T n − AzT n
(1)
Numerical, Parabolic
0.2
This equation is in the general form required for construction
of a multilevel Douglas and Gunn time splitting [22]. Applying
Douglas–Gunn time-splitting technique to Eq. (15) we have
(2)
0.8
(19d)
4
c 2005 by ASME
Copyright Numerical techniques
N=21
N=41
N=51
N=101
Explicit Scheme
4.88
147.62
450.26
7920.00
Gauss–Seidel
13.46
175.1
415.86
7800
GMRES
13.56
133.98
267.8
2873.2
Conjugate gradient
12.12
110.5
233.96
2733.4
δ-form D-G
9.15
75.22
153.3
1638
Table 1.
split method [1] employing δ-form Douglas–Gunn time-splitting
is the most efficient way of solving the 3-D time dependent microscale heat transport equation in terms of CPU time taken to
complete the simulation.
Total CPU time taken in seconds
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Performance comparison of different numerical methods for
solving the discretized 3-D DPL equation using split method.
Numerical techniques
Total CPU time taken in seconds
N=21
N=41
N=51
N=101
Explicit Scheme
4.88
147.62
450.26
7920.00
Gauss–Seidel
14.14
253.42
627.03
11343.06
Conjugate Gradient
12.33
124.83
270.3
3614.69
GMRES
13.86
145.00
270.8
2996.0
δ-form D-G
8.54
70.5
140.92
1344.4
Table 2.
Performance comparison of different numerical methods for
solving the discretized 3-D DPL equation using unsplit method.
Gunn time-splitting method for different values of the spatial discretization parameter N. Here N = Nx = Ny = Nz . We can make
several observations from Tables 1 and 2. When N = 21 the explicit method consumes significantly less CPU time than the rest
of the numerical techniques, but for the spatial discretization parameter N > 21, all implicit methods except the GS procedure
perform better than the explicit method employed in this numerical experiment. The poor performance of the iterative methods namely GS, CG and GMRES in the unsplit approach owes
to the fact that the coefficient matrix A (Eq. (13)) is less diagonally dominant than that of the split method. Clearly, for highresolution calculations on grids having greater than a million grid
points, the splitting method will be significantly more efficient
for time-dependent problems than any iterative technique.
SUMMARY AND CONCLUSIONS
We have investigated performance of split and unsplit techniques for solving the DPL equation by conducting numerical
experiments using Gauss–Seidel, conjugate gradient, generalized
minimal residual, and δ-form Douglas–Gunn methods and determined the computational cost involved in each approach. The un5
c 2005 by ASME
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6
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Copyright