Inverse Heat Conduction Linear Algebra Model for
Metal Heating by Ultra-Short Laser Pulses
Benxin Wu
Abstract: The heat transfer process of metal heating by ultra-short laser pulses, where the pulse duration is
less than 1 nanosecond, is governed by parabolic two-step (PTS) heat conduction equations. In this paper,
we proposed, for the first time to our knowledge, an inverse heat conduction linear algebra model for the
one-dimensional metal heating by ultra-short laser pulses through an approximate explicit finite difference
method. The simulation results by this model are compared with experimental data and a reasonably good
agreement has been obtained.
The heat transfer process of metal heating by ultra-short laser pulses, where the pulse
duration is less than 1 nanosecond, is governed by parabolic two-step (PTS) heat
conduction equations, whose one-dimensional form is as follows [1]:
Ce
Te
 2Te
T
 ke
 G (Te  Tl )  S , (1); Cl l  G (Te  Tl ) , (2); S  I 0 exp( x) , (3);
2
t
t
x
Please see appendix 1 for the nomenclature. We assume, without causing obvious errors,
that the thermal properties are temperature independent. If we neglect the heat loss at the
boundary, the initial conditions and boundary conditions are:
Te ( x, t initial )  Tl ( x, t initial )  T0 ,and
Te Tl

 0 at x  0 and x  L
x
x
(4)
By applying an approximate explicit finite difference method to equations (1) to (4),
we set up a concise linear algebra model as follows [2]:
Tenew=BTeold+a3Tlold+S
(5)
Tlnew= a5Teold+a4Tlold
(6)
See reference [2] for the nomenclature and the detailed derivation of the linear algebra
model. Here, we will derive an inverse heat conduction linear algebra model based on our
previous work in reference [2].
From equations (5) and (6), we can get:
(
a
1
1
1
1
B  5 I )Teold  ( S  Tenew  Tln ew )
a3
a4
a3
a3
a4
If the matrix M  (
(7)
a
1
B  5 I ) is invertible, then from equations (5) to (7), we can
a3
a4
get the inverse heat conduction linear algebra model, which is expressed by:
Teold  M 1 (
Tlo 
1
1
1
S  Tenew  Tln ew )
a3
a3
a4
a
1
Tln ew  5 Teold
a4
a4
(8)
(9)
From equations (8) and (9), we can the temperature field at previous time step, Teold
and Tlo , based on the current temperature field, Tenew and Tln ew , if we know all the other
necessary information. Whether the model is valid or not depends on if the matrix M is
invertible. M is a tri-diagonal matrix and there is no easy analytical expression for its
determinant, so we should check by numerical experiment if it is invertible or not.
We can see that the entries for M are functions of a1 to a5, and for a given material, a1
to a5 are functions of x and t chosen. Even though M is invertible, it may not give us
convergent results. Therefore practically, in our numerical simulations, we should first
choose some reasonable value for x and t by estimation, then after several numerical
experiments and parameter modifications, we can get a combination of the values for
x and t , which can give us a convergent result.
This inverse heat conduction linear algebra model is very concise, and also very
useful. If we know the laser parameters and materials properties, and the temperature
field for the material after being heated by a laser pulse, we can apply this model to
calculate the temperature field at previous time step, and even the initial temperature.
This situation is very common in real research projects.
A comparison of experimental data [1] with the simulation result by this model is
shown in figure 1 in appendix 2. See appendix 3 for the program code. Please notice that
the simulation results are obtained by the following method: we get the temperature field
at the last time step by the experimentally verified linear algebra model in reference [2],
and then we use this temperature field and our inverse heat conduction linear algebra
model to calculate back the temperature field at previous time steps. Then we compare
the results with the experimental data, we can see that the agreement is reasonably good.
Reference:
1. T.Q. Qiu, T. Juhasz, C. Suarez, W.E. Bron, and C.L. Tien, Femtosecond Laser
Heating of Multi-Layer Metals-II Experiments, Int. J. Heat Mass Transfer 37, pp.
2799-2808, 1994.
2. B.X. Wu, A Concise Linear Algebra Model for Metal Heating by Ultra-Short
Laser Pulses, project 1 for the Linear Algebra course (Math 208), University of
Missouri – Rolla, July 2003.
Appendix 1 Nomenclature
a1, a2, a3, a4, a5.
Constants in finite difference equations
Ce
Volumetric heat capacity of electrons
Cl
Volumetric heat capacity of the lattice
G
Electron-phonon coupling factor
i
Grid point index in x direction
I0
Absorbed laser power density
ke
Effective electron thermal conductivity
L
Length of heated metal
n
Time step index
N0
Maximum grid point index in x direction
S
Laser heating source
t
Time
t
Temporal step size
tinitial
Pulse starting time
T0
Initial temperature
Te
Temperature of electrons
Tl
Temperature of the lattice
n
Temperature of electrons at x  (i  1)x and t  (n  1)t ,
Tl (i)
n
Temperature of the lattice at x  (i  1)x and t  (n  1)t ,
x
Spatial step size ( grid size )

Absorption coefficient
Te (i)
Appendix 2: Comparison of experimental data [1] with our linear algebra model
Our inverse model
Experimental data
Fig. 1 Comparison of experimental data with our linear algebra model simulation result
( Pulse duration: tp = 100 fs ; Pulse starts at t = -200 fs; Pulse energy density: 10 J/m2; L
= 100 nm. Material: gold. The normalized electron temperature change is at the front
surface, and is defined as: (Te - T0 )/(Te,max – T0). See reference 1 for materials properties
and the temporal shape of the laser pulse).
Appendix 3: MatLab Program Code
G=2.6e016;
Ce=21000;
ke=315;
Clt=2.5e006;
tp=100e-015;
tst=-2.*tp;
tend=2000e-015;
tN=44001;
deltt=(tend-tst)./(tN-1);
J=10;
afa=0.065e009;
L=0.1e-006;
N0=21;
deltx=L./(N0-1);
a1=deltt./(Ce.*deltx.*deltx).*ke;
a3=G.*deltt./Ce;
a2=1-2.*a1-a3;
a5=G.*deltt./Clt;
a4=1-a5;
for i=1:N0
for j=1:N0
B(i,j)=0;
end
end
B(1,2)=2.*a1;
B(N0,N0-1)=2.*a1;
for i=1:N0
B(i,i)=a2;
end
for i=2:N0-1
B(i,i+1)=a1;
B(i,i-1)=a1;
end
for i=1:N0
T0(i)=300;
end
Tenew=T0';
Teold=T0';
Tltnew=T0';
Tltold=T0';
Tefront(1)=300;
Terear(1)=300;
for n=2:tN
t=tst+(n-1).*deltt;
%*******************
I0=0.94.*J./tp.*exp(-2.77.*(t./tp).^2);
SS(1)=I0.*2.*deltt./(Ce.*deltx).*(1-exp(-afa.*deltx./2));
for i=2:N0-1
SS(i)=I0.*deltt./(Ce.*deltx).*( exp(-afa.*(i-0.5).*deltx)-exp(-afa.*(i+0.5).*deltx) );
end
SS(N0)=I0.*2.*deltt./(Ce.*deltx).*( exp(-afa.*(N0-0.5).*deltx)-exp(-afa.*N0.*deltx) );
S=SS';
%*******************
Tenew=B*Teold+a3.*Tltold+S;
Tltnew=a5.*Teold+a4.*Tltold;
Tefront(n)=Tenew(1,1);
Terear(n)=Tenew(N0,1);
Teold=Tenew;
Tltold=Tltnew;
end
n=1:tN;
figure(1);
plot((tst+(n-1).*deltt).*10.^12, (Tefront(n)-300)./(max(Tefront(n))-300) );
hold on;
figure(2);
plot((tst+(n-1).*deltt).*10.^12, (Terear(n)-300)./(max(Terear(n))-300) );
hold on;
expt=[0,0.1,0.15,0.30,0.41,0.59,0.70,0.87,1.00,1.15,1.30,1.41,1.71,2];
expTe=[0.5,1.0,0.95,0.75,0.60,0.50,0.40,0.35,0.30,0.25,0.24,0.22,0.20,0.17];
for i=1:14
figure(1);
plot(expt(i)', expTe(i)','o');
hold on;
end
M=(1./a3).*B-(a5./a4).*eye(N0);
detM=det(M)
for n=tN:-1:1
t=tst+(n-1).*deltt;
%*******************
I0=0.94.*J./tp.*exp(-2.77.*(t./tp).^2);
SS(1)=I0.*2.*deltt./(Ce.*deltx).*(1-exp(-afa.*deltx./2));
for i=2:N0-1
SS(i)=I0.*deltt./(Ce.*deltx).*( exp(-afa.*(i-0.5).*deltx)-exp(-afa.*(i+0.5).*deltx) );
end
SS(N0)=I0.*2.*deltt./(Ce.*deltx).*( exp(-afa.*(N0-0.5).*deltx)-exp(-afa.*N0.*deltx) );
S=SS';
%*******************
Teold= ( M^(-1) )*( -1./a3.*S+1./a3.*Tenew-1./a4.*Tltnew );
Tltold=1./a4.*Tltnew-a5./a4.*Teold;
Tefront(n)=Teold(1,1);
Terear(n)=Teold(N0,1);
Tenew=Teold;
Tltnew=Tltold;
end
n=1:tN;