J. KOMÍNEK, M. POHANKA: ESTIMATION OF THE NUMBER OF FORWARD TIME STEPS ...
207–210
ESTIMATION OF THE NUMBER OF FORWARD TIME STEPS FOR
THE SEQUENTIAL BECK APPROACH USED FOR SOLVING
INVERSE HEAT-CONDUCTION PROBLEMS
UGOTAVLJANJE [TEVILA VNAPREJ[NJIH ^ASOVNIH KORAKOV
ZA SEKVEN^NI BECKOV PRIBLI@EK PRI RE[EVANJU
PROBLEMOV INVERZNE TOPLOTNE PREVODNOSTI
Jan Komínek, Michal Pohanka
Brno University of Technology, Faculty of Mechanical Engineering, Technická 2, 616 69 Brno, Czech Republic
kominek@lptap.fme.vutbr.cz
Prejem rokopisa – received: 2014-08-13; sprejem za objavo – accepted for publication: 2015-04-08
doi:10.17222/mit.2014.192
Direct heat-conduction problems are those whose boundary conditions, initial state and material properties are known and the
entire temperature field in a model can be computed. In contrast, an inverse problem is defined as the determination of the
unknown causes based on the observation of their effects. The inverse heat-conduction method is often used for problems where
the boundary conditions cannot be measured directly but are computed from the recorded temperature history inside the model.
A very effective method for solving this difficult problem is the sequential Beck approach. To stabilize this inverse problem, a
proper regularization parameter must be used. For this method, the regularization parameter is the number of the forward time
steps that stabilize the inverse computation. This paper describes two methods for computing the number of the recommended
forward time steps for nonlinear heat-conduction models with temperature-dependent material properties. The first method is
based on tracking the sensitivity (at the interior point of a measurement) to the Dirac heat-flux pulse on the surface. The second
method determines the number of the forward time steps from the residual function computed from the heat fluxes obtained
from the inverse computation. The stability and noise (in the results) of several variants of these methods are compared. The
results showed that the first method is much less computationally intensive and gives a slightly higher value of the number of
forward time steps than the second method.
Keywords: inverse heat-conduction problem, Beck approach, number of forward time steps
Neposredni problemi prevajanja toplote so tisti pri katerih so poznani robni pogoji, za~etno stanje in lastnosti materiala ter
mo`nost izra~una temperaturnega polja znotraj modela. Nasprotno pa je inverzni problem definiran kot dolo~anje nepoznanih
vzrokov na osnovi opazovanja njihovih vplivov. Metoda inverznega prevajanja toplote se pogosto uporabi pri problemih, kjer se
robni pogoji ne morejo neposredno izmeriti, temve~ se jih izra~una iz zabele`enega poteka temperature znotraj modela. Zelo
u~inkovita metoda za re{evanje tovrstnega problema je sekven~ni Beckov pribli`ek. Za stabilizacijo tak{nega inverznega
problema se mora uporabiti ustrezen regulirni parameter. Pri tej metodi je regulirni parameter {tevilo priporo~enih ~asovnih
korakov, ki stabilizirajo inverzni izra~un. ^lanek opisuje dve metodi za izra~un {tevila priporo~enih ~asovnih korakov za
nelinearni model prenosa toplote, s temperaturno odvisnimi lastnostmi materiala. Prva metoda temelji na iskanju ob~utljivosti,
na notranji to~ki merjenja, do Dirac utripa toplotnega toka na povr{ini. Druga metoda dolo~a {tevilo vnaprej{njih ~asovnih
korakov iz preostale funkcije izra~unane iz toplotnih tokov, ki so dobljeni z inverznim izra~unom. V rezultatih je primerjana
stabilnost {uma pri ve~ variantah teh metod. Rezultati so pokazali, da je prva metoda mnogo manj ra~unsko intenzivna in daje
rahlo ve~jo vrednost {tevila predhodnih ~asovnih korakov kot druga metoda.
Klju~ne besede: problem inverzne toplotne prevodnosti, Beckov pribli`ek, {tevilo vnaprej{njih ~asovnih korakov
1 INTRODUCTION
Heat-conduction problems are often solved in engi-
neering applications during simulations. The problem is
well known as a direct task. The effect (the temperature
field in time) is computed from the causes (the known
initial and boundary conditions). Complex direct prob-
lems can be solved using many numerical methods such
as FDM,1 FVM,2 FEM.3 The situation is opposite for an
inverse heat-conduction problem and it is a much more
complicated problem. The causes (e.g., the boundary
conditions) are determined from the observation of the
effects (the temperature record in several points). There
are some computational methods dealing with this in-
verse problem, including the Beck approach,4 Tikhonov
regularization5 and neural networks.6 We focus on the
sequential Beck approach in this paper.
The basic idea of the sequential approach is to solve
the entire task step by step in time. The measured
temperature at an interior point at times tn, tn+1,..,tn+Nf is
used to compute the heat flux on the boundary at time tn,
where Nf is the number of forward time steps (the regu-
larization parameter). A computation of Nf temperature
fields using a direct task is performed for two different
values of constant heat fluxes in each time step. The
temperature responses from these two direct tasks are
compared to the measured temperature. The new value
of the heat flux at time tn is computed. The choice of an
appropriate value for Nf is essential for practical compu-
tations. A small value leads to instability and a large
Materiali in tehnologije / Materials and technology 50 (2016) 2, 207–210 207
UDK 519.61/64:543.58:536.2 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 50(2)207(2016)
value smoothes sudden changes in the boundary condi-
tions. Thus, the appropriate value of this parameter is
essential.
2 IMPACT OF THE NUMBER OF FORWARD
TIME STEPS ON THE COMPUTED RESULTS
The main function of parameter Nf is to guarantee the
stability of the computation of this difficult problem. The
stability increases with an increasing value of Nf. In Fig-
ure 1, three results for Nf = 13, 20, 30 are compared to
the correct heat-flux record, which was used to generate
the input temperature record for the inverse task. The
noise was also added to this temperature record (a stan-
dard deviation of 0.05 °C). A large oscillation of the
computed heat flux for a low value of Nf is obvious. This
is mainly due to the added noise in the input data. The
noise reduction in the input data is more effective for
larger values of Nf (Figure 1). This effect indicates that
the use of a large Nf is recommended. Unfortunately,
increasing Nf has two effects. First, the computation cost
is proportional to Nf. A higher value of Nf results in a
longer computational time. Second, a large Nf value
smoothes the computed results. Abrupt changes as well
as the maximum values of the ideal heat flux (Figure 1)
are significantly reduced when Nf increases. For Nf =30,
the computed maximum heat flux is less than 50 % of
the ideal heat flux for this test case.
3 METHODS
The appropriate value of forward time steps is diffe-
rent for each computational model. Two types of me-
thods are described in this article to determine its
amount. The first, newly proposed, method is based on
the temperature response. The idea is to compare two
temperature responses at an interior location (usually a
thermocouple position) to two Dirac pulses of heat flux
that are the same but shifted in time by one time step
(Figures 2 and 3). The first temperature response is com-
puted for the Dirac pulse applied from time step zero to
time step one and the second temperature response is
computed for the Dirac pulse applied from time step one
to time step two. The computed difference between these
two temperature responses is shown in Figure 4. The
computed curve provides an idea of how the information
about the changes in the boundary condition is delayed
from time step zero to time step two and spread over the
time. This curve shows the distribution of the informa-
tion about the temperature response. This is the informa-
tion about what happened at the beginning of the
J. KOMÍNEK, M. POHANKA: ESTIMATION OF THE NUMBER OF FORWARD TIME STEPS ...
208 Materiali in tehnologije / Materials and technology 50 (2016) 2, 207–210
Figure 3: Temperature response
Slika 3: Temperaturni odziv
Figure 1: Influence of Nf on the inverse heat-conduction problem
Slika 1: Vpliv Nf na problem inverzne toplotne prevodnosti
Figure 2: Heat-flux pulses in the subsequent time steps
Slika 2: Sunki toplotnega toka v poznej{ih ~asovnih korakih
Figure 4: Difference between temperature responses
Slika 4: Razlika med temperaturnimi odzivi
simulation (from time step zero to time step two) at the
boundary of the computational model.
For practical computations, it should be noted that
both temperature responses are the same except for the
time shift, which is one time step. In addition, the tem-
perature response difference Tn = Tn – Tn–1 corresponds
to the numerical derivation except for the multiplication
by constant c (Equation (1)):
T T c T c
T T
tn n
n n− = ⋅ = ⋅
−
−
−
1
1
d
Δ
(1)
where c = t.
The shape of the temperature response to the Dirac
pulse depends on many parameters. The most important
are the material properties (density, thermal conductivity
and thermal capacity), the distance of the thermocouple
from the boundary, the thermocouple type, the material,
and the thermal resistance between the thermocouple and
the material.
The number of forward time steps (forward time t,
respectively) is taken from the derivation of the tempe-
rature response D(t) so that D(t) meets a certain criterion.
For example, tDmax is the time when D(tmax) is maxi-
mal. tDmax,p%,1 and tDmax,p%,2 are the times when the deri-
vation of the temperature response reaches p % of its
maximum. An example for p = 60 % is shown in
Figure 5.
The second estimation method for determining the
number of forward time steps can be done with a
repeated computation of the inverse heat-conduction
problem by changing Nf. The sum of the residuals
R = 
(Q’i – Qi)2 is evaluated from each inverse task
where Qi is the computed heat flux and Q’i is the correct
heat flux from the test task. An example of how R is
dependent on Nf is shown in Figure 6 and the Nf,min value
(Nf when R is minimal) can be found here. The value of
Nf (slightly larger than Nf,min) is taken as an estimate for
the number of forward time steps. The Nf,min value is not
used due to the risk that a small shift of the estimated Nf
value to the left (to a smaller value) can rapidly increase
the R value (Figure 6). An analogical application of the
search for the optimum regularization parameter in a
Tikhonov digital filter is described by Woodbury.7
4 DISCUSSION
The first method described is much less computa-
tionally intensive than the second one because the first
method needs only one direct computation instead of
many inverse (and therefore much more complicated)
computations. Each method provides a different value of
forward time steps Nf. It is not easy to say which value is
better. Generally, this depends on what is more essential
for each application. The larger value of Nf smoothes the
results but the average values for certain time intervals
are correct. A small value of Nf can result in heat fluxes
that better fit true values, but the results include more
oscillation than would be expected in reality. The choice
of the appropriate testing function in the second method
also significantly influences the computed value of Nf.
Two examples of the testing functions and the obtained
Nf value are shown in Figure 7.
The comparison of the inverse computations per-
formed with Nf,Dmax = 37, Nf,Dmax,60%,2 = 65 (from the first
method) and Nf,1 = 24, Nf,2 = 18 (from the second
method) is shown in Figure 8. The curve for
Nf,Dmax,60%,1 = 23 is not plotted because it is almost the
same as that for Nf,1 = 24. These inverse computations
J. KOMÍNEK, M. POHANKA: ESTIMATION OF THE NUMBER OF FORWARD TIME STEPS ...
Materiali in tehnologije / Materials and technology 50 (2016) 2, 207–210 209
Figure 7: Two examples of testing functions and obtained Nf,min,
using the second method
Slika 7: Dva primera preizkusnih funkcij in dobljen Nf,min pri uporabi
druge metode
Figure 5: Example of tDmax and tDmax,60%,1–2
Slika 5: Primer za tDmax in tDmax,60%,1–2
Figure 6: Residual chart for Nf values
Slika 6: Grafikon ostankov za vrednosti Nf
were made for the 1D inverse heat-conduction problem
with thermally dependent material properties. The tem-
perature record from the real measurements was used.
Therefore, the correct heat-flux function is unknown.
The heat-conduction problem is described with diffe-
rential Equation (2):
∂
∂
∂
∂
2
2
1T
x
T
t
=

(2)
where T is the temperature, t is the time and x is the
coordinate. The boundary conditions for (Equation (3))
cooled and insulated surfaces are:4
− = − =
= =
k
T
x
q t k
T
xx x l
∂
∂
∂
∂0
0( ); (3)
The test sample was made from a thick stainless-steel
plate (L = 10 mm). One side (x = 0) of the sample was
cooled down by water and the other side (x = L) was
insulated. A thermocouple was placed under the cooled
surface (x = 2 mm).
The curves for Nf = 37 and Nf = 24 (Figure 8) appear
to be acceptable. The curve for Nf = 65 is too smooth.
The curve for Nf = 18 begins to be unstable and the com-
puted heat flux is less than zero for some points, which is
physically impossible in this experiment.
5 CONCLUSION
Two methods for determining the number of forward
time steps Nf for the sequential Beck approach were
described. The first method (based on the derivation of
the temperature response to the Dirac heat-flux pulse) is
computationally much less intensive. The choice of
Nf = Nf,Dmax is acceptable for most applications. For some
similar tasks, it may be better to use Nf,Dmax,p% with the
same suitable value of p.
The second method, which is computationally very
intensive, can be useful when the shape of the heat-flux
curve is known and the appropriate testing function can
be used. The obtained values of Nf were smaller than
those computed using the first method and the computed
heat fluxes showed more oscillation.
Acknowledgement
This work is an output of the research and scientific
activities of project LO1202, financially supported by
the MEYS under programme NPU I.
6 REFERENCES
1 G. D. Smith, Numerical solution of partial differential equations,
University Press, Oxford, UK 1978
2 S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere
Publishing Corporation, GB 1980
3 O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method, Volume
I: Basis, Butterworth-Heinemann, Oxford 2000
4 J. V. Beck, B. Blackwell, R. C. Charles, Inverse Heat Conduction:
Ill-posed Problems, Wiley, New York 1985
5 A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-posed problems, V. H.
Winston & Sons, New York 1977
6 M. Raudenský, J. Horský, J. Krejsa, Usage of neural network for
coupled parameter and function specification inverse heat conduction
problem, Int. Comm. Heat Mass Transfer, 22 (1995) 5, 661–670,
doi:10.1016/0735-1933(95)00052-Z
7 K. A. Woodbury, J. V. Beck, Estimation metrics and optimal regu-
larization in a Tikhonov digital filter for the inverse heat conduction
problem, International Journal of Heat and Mass Transfer, 62 (2013),
31–39, doi:10.1016/j.ijheatmasstransfer.2013.02.052
J. KOMÍNEK, M. POHANKA: ESTIMATION OF THE NUMBER OF FORWARD TIME STEPS ...
210 Materiali in tehnologije / Materials and technology 50 (2016) 2, 207–210
Figure 8: Results of the heat flux for four different values of Nf and
the measured temperature
Slika 8: Rezultati toplotnega toka za {tiri razli~ne vrednosti Nf in
izmerjena temperatura