Download Application of POD-RBF technique for retrieving thermal diffusivity of

Application of POD-RBF technique for retrieving thermal
diffusivity of anisotropic material
W.P. Adamczyk, Z. Ostrowski, Z. Buliński, A. Ryfa
Silesian University of Technology, Institute of Thermal Technology, Gliwice, Poland
[email protected]
Abstract. The knowledge about thermal diffusivity is essential in wide range of applications. It
is often used as the parameters, which describe the quality of the materials or plays crucial
parameter in numerical simulations where heat transfer process need to be evaluated. The
method presented in this paper was developed based on the classical Parker flash method
which is classified as the transient measurement technique. The idea of measurement procedure
is to locally heat up a small portion of a sample surface by a laser flash and record the resulting
transient temperature field by an IR camera. In contrary to classical flash method in proposed
paper the laser and IR camera are located in the same said of the sample. The
measurement/inverse procedure presented in this work treats the thermal conductivity
components as decisive variables which are retrieved by minimization of the difference
between experimental and numerical data. In presented work as the low dimensional model of
the considered system based on proper orthogonal decomposition technique (POD) extended
by radial base function (RBF) for finding relations between snapshots was used as the direct
solver. The numerical simulation results were validated with experiments conducted by the use
of commercially available Parker’s flash method.
1. The first section in your paper
Robustness, relatively low cost and rapidness made numerical simulations a primary technique of
engineering analysis. Numerical models can replace expensive and time consuming experimental
methods. The reliability and accuracy of the numerical results depends strongly on the input data. In
case of simulation of heat transfer within a solid body, the exact definition of the material properties,
for instance heat diffusivity is extremely important. There is a vast and constantly increasing literature
on the available experimental techniques of determining TC [5]. Detail information can also be found
in position [6]. The most popular techniques are those which fits to the transient one. Extensive study
covering the transient techniques can be found in [9,11,10] where advantages and disadvantages of
different measurement concepts have been widely discussed. One of the most popular, and widely
used 1D transient technique for measuring of the TD is the laser flash method [1,17]. This method has
found application in commercial Laser Flash Apparatus (LFA). Other examples of transient method
application can be found in [4,12]. The method presented in this work also belongs to the transient
laser flash technique and should be seen as continuation of works [2,3,13] which ultimately should
lead to development of the procedure capable for rapid measurement of the TC of the body with
arbitrary shape and with anisotropic structure.
Reference [13] presents the development process of the analytical model capable to retrieve thermal
conductivity of the large solid body with anisotropic structure. Presented model allows to calculate
thermal conductivity of the cuboid large carbon blocks in couple of minutes based on the in situ
collected experimental data. Analytical model assumes that the carbon block due to its large size in
contrary to laser emission spot and duration of the experiment (less than 3s) can be treated as semiinfinity space which considerable simplified the model formulation. In [2,3] the simple analytical
model was replaced by numerical solver which allows also for taking into account heat losses from
observation surface on the retrieved thermal conductivity. The possible application of numerical
method combined with inverse technique for retrieving thermal properties of the solid body has been
discussed. The major drawback in case of using numerical models is long simulation time which
considerable limits it’s practical application. To speed-up the simulation an application of a reduced
order technique described in this paper can be applied. In presented work, the Proper Orthogonal
Decomposition (POD) method is proposed. This approach uses POD empirical vectors as
approximation basis, while dependency of reduced model on input parameters is obtained by means of
Radial Basis Functions (RBF). This technique, introduced by co-author of current research [14] is
known as truncated POD-RBF approximation [15]. The core of this technique is the expression of the
temperature field in a form of a linear combination of space dependent basis vectors (modes), being
vectors of truncated POD base. The amplitudes of this approximation are then expressed as linear
combination of RBFs, being a function the parameters to be retrieved by the inverse technique. The
usage of POD modes as approximation basis makes this approximation optimal, i.e. for given
approximation order, POD base produces minimal error. This comes from the fact that the
approximation vector basis is not chosen arbitrary, but is derived from approximated data. Thanks to
proposed approach, the inverse technique which is used to retrieve the TD can be accelerated by
several orders of magnitude. The results presented in this work should be seen as the preliminary study
of the POD-RBF technique for retrieving components of the TD tensor.
2. Experimental setup
The initial tests were performed using steel cuboid samples. The developed in-house apparatus for
collecting experimental data is shown in Fig. 1. As the heat source the IPG Photonics laser is used. It
can operate in power range from 20 W to 200 W with adjusted emission time period. Both parameters
ensure that the power of the laser pulse can be appropriately adjusted to the expected material
properties. The spatial and temporal temperature distribution after laser emission is recorded by the
Infrared (IR) camera (FLIR A325, Flir Systems, Inc., USA) working with 60 frames per second
frequency. The measurement and data acquisition processes are controlled using an in-house PC
application written in LabVIEW environment (National Instruments Corp., USA). To reduce the
geometrical distortion of the spot of the laser ray and the image of the temperature field, the optical
axis of both devices are orthogonal to the heated spot. To achieve this effect the laser ray impinges the
probe in the direction of the local normal surface of the sample, and then the probe is rotated back to
the optical axis of the camera. The construction of experimental rig allows also smooth regulation of
the leaser head distance from emission surface. Base on that feature the size of the laser spot can be
also adjusted for specific condition, for instance to prevent measured material overheating. The
measurement procedure was organized as fallow
• positioning the sample horizontally to camera lens,
• recording initial temperature field,
• rotating the sample to bring the local normal to a position parallel to the laser optical axis ,
• heating the sample by laser pulse,
• moving sample back to previous position with sample normal cantered at the camera lens,
• recording spatial and temporal distribution of the temperature on the heated surface (see Fig. 2
where selected IR diagrams are shown),
•
•
converting the IR intensity diagrams to temperature one,
retrieving the thermal conductivity components by fitting the measured data to calculated
temperature field using the generated POD base.
Figure 1: Experimental test rig
Figure 2: Recorded temperature distribution at the observation surface in three selected times
The thermal diffusivities (TD) of the tested materials were measured using the Netzsch Laser Flash
Apparatus (LFA) 457 (NETZSCH-Geratebau GmbH). The measurements were conducted within
temperature range 30 − 40oC using cylindrical sample with thicknesses 3 mm and diameter of 12.66
mm. The measured thermal diffusivities for samples C16 and C20, were equal to 4.28 ∙ 10*+ m2/s and
6.25 ∙ 10*+ m2/s, respectively. The accuracy of the LFA device is ±3 − 5%. The material thermal
conductivity was calculated using the Cowan model (Cowan 1963), where the heat capacities of
material was measured using the Netzsch STA 409 PG (NETZSCH-Geratebau GmbH, Germany)
device. The STA records mass changes of a sample (Thermogravimetry (TG)) and the heat flow rate
between a sample and furnace (Differential Scanning Calorimetry (DSC)). Based on this information,
𝑐0 𝑇 can be calculated. The temperature measurement range was set between 25oC and 130oC.
Measured specific heats for sample C16 and C20 are illustrated in Fig. 3. The density of both samples
C16 and C20 were 7668 kg/m3 and 7886 kg/m3, respectively.
Figure 3: Measured specific heat 𝑐0 (𝑇) for samples C16 and C20 using STA device
3. Numerical model
The computational geometry used for producing set of snapshots for POD model is shown in Fig. 4.
To reduce cost of numerical simulations only the one-quarter of the model was used during
simulations. The heat transfer within sample and movement of gas above the observation surface were
modelled using computational fluid dynamics (CFD) applying the Ansys Fluent solver extended by
user defined function implemented into the solution procedure. To run the numerical summation,
several parameters need to be defined i.e. specific heat, density, initial temperature, emission time and
amount of heat provided to the sample. To give answer which of mentioned parameter has the biggest
influence on evaluated data the uncertainty quantification should be carried out, which is the subject of
future research. Here to minimize the influence of exact definition of the amount of absorbed energy
by the sample on the evaluated TC a tricky definition of the objective function Eqs. (1,2,3) was
proposed. Taking the ratio of two temperature fields Θ in two times instance 𝜏 the surface emissivity 𝜖
and amount of absorbed energy 𝑞by the body are simplified. Detailed description of the objective
function definition can be found in earlier author [3], where all heat losses from observation plane
were neglected by using simple numerical model limited to solid body and treating the observation
surface as perfectly insulated. The POD-RBF base was generated using external procedure which
controls the snapshots calculation strategy, collects the numerical data and builds the snapshot base.
Once the POD base was generated for given material it can be used as the direct solver for retrieving
TC of various samples made from the same material. The objective function used in this study was
defined as
D
min
Θ<=>,@ − ΘAB<,@
C
1
@EF
where 𝑁 stands for the total number of sampling points used by inverse analysis to retrieve TC of the
material. The Θ<=> and ΘAB< are defined as
Θ<=>,H =
𝑇 𝑥@ , 𝑦@ , 𝜏F , 𝑘MM , 𝑘NN , 𝑘OO , 𝑞, 𝜖 − 𝑇P (𝑥@ , 𝑦@ , 𝜖)
𝑇 𝑥@ , 𝑦@ , 𝜏C , 𝑘MM , 𝑘NN , 𝑘OO , 𝑞, 𝜖 − 𝑇P (𝑥@ , 𝑦@ , 𝜖)
2
ΘAB<,H =
𝑇 𝑥@ , 𝑦@ , 𝑞, 𝜏F , 𝜖 − 𝑇P (𝑥@ , 𝑦@ , 𝜖)
𝑇 𝑥@ , 𝑦@ , 𝑞, 𝜏C , 𝜖 − 𝑇P (𝑥@ , 𝑦@ , 𝜖)
3
where 𝑇P (𝑥@ , 𝑦@ ) is the initial temperature within sampling point 𝑖 located at the observation surface,
𝑘MM , 𝑘NN , 𝑘OO are the thermal conductivity components, 𝑥@ , 𝑦@ stands for the coordinates of sampling
point, 𝑞 is the heat flux, 𝜖 defines the surface emissivity. The objective function is minimized using
the Levenberg-Marquardt [16] optimization which is implemented within MatLAB.
Figure 4: Numerical model
4. POD-RBF
The idea is to solve a sequence of direct problems within the body under consideration by changing
input data upon which the field depends on. In current research, numerical technique is used to obtain
the output fields, further referred to as snapshots. The solutions are sampled at a predefined set of
points (e.g. nodes of numerical grid). The snapshots are stored as subsequent columns of snapshots
matrix 𝑈. The aim of POD analysis is to construct a small set of orthonormal vectors 𝜙 resembling the
original matrix 𝑈 (all snapshots 𝑢) in an optimal way [14.15], i.e.
4
∅=𝑈∙𝐴
where 𝐴 is matrix of amplitudes of expansion and ∅ stands for truncated POD base. The base ∅ is
orthonormal, so the amplitudes matrix can be immediately evaluated as
𝐴 = ∅W ∙ 𝑈
5
The amplitudes 𝐴 are then expressed as a linear combination of RBFs. First, all the input parameters
used for generation of snapshot matrix 𝑈 are chosen as the nodes of RBF network, i.e.
6
𝐴 =𝐵∙𝐺
where 𝐵 is matrix storing the unknown coefficients of the combination and 𝐺 stands for known matrix
whose entries are defined as
𝐺
Z
@
= fH { p − p ^ }
7
with 𝑓@ being the j-th function (thin plate spline RBF function)
𝑓@ 𝑝 = 𝑓@ 𝑝 − 𝑝 @
C
= 𝑝 − 𝑝 @ ln |𝑝 − 𝑝 @ |
8
Having in mind that 𝐴 matrix is known (5), the sought for 𝐵 matrix can be evaluated by a solution of
linear set (6). The experience of using the POD-RBF method shows that when the number of the
RBF’s becomes large, the accuracy of the network sometimes decays (being opposite to expected
trend). The investigations show that the reason for this behaviour is the ill-posedness (or nearly illposedeness, in numerical sense) of the 𝐺 matrix. To alleviate this, standard Gaussian solver is replaced
by a Singular Value Decomposition (SVD) [8] procedure used to solve the set of equations defined in
eq. (6). Using this approach, the resulting matrix of coefficient is expressed as
𝐵 W = 𝑅 ∙ 𝑊 *F ∙ 𝐿W ∙ 𝐴
9
where orthogonal 𝑅, 𝐿 and diagonal 𝑊 matrices are the result of a SVD of 𝐺 W matrix. By setting to
zero small entries of the diagonal matrix 𝑊 (i.e. singular values) the influence of the superfluous
nodes of the RBF network is eliminated. After the coefficient matrix 𝐵 is evaluated, a low dimensional
model of the temperature field can be set as
𝑢(𝑝) = 𝜙 ∙ 𝐵 ∙ 𝑓(𝑝)
10
where 𝑢(𝑝)stands for temperature field (snapshot) for arbitrary parameter vector 𝑝 and 𝑓(𝑝) stands
for the vector of above defined 𝑓@ functions (8). This approximation, hereafter referred to as the PODRBF network approximation, is capable of reproducing temperature fields that correspond to an
arbitrary set of parameters 𝑝. For transient cases, the parameters vector p contains time, thus the time
variable is built into the approximation formula. Hence, both the temporal and spatial variation of the
temperature field can be reproduced by formula (10).
5. Sample results
To validate proposed calculation algorithm using the POD-RBF technique, the steel isotropic samples
were used. To ensure stable and accurate solution appropriate times τF , τC have to be selected as the
input for objective function definition Eq. (1). Other important issue is to select pixels where the
objective function can be calculated. The possible answer for both formulated questions can be
obtained by using an analytic model developed in work [13]. This model quickly calculates the TCs
for different times ratios and defines the position of pixels based on the isotherm shapes and
sensitivity, saying in which time ratio the TC values do not significantly changes. Figure 5 illustrates
changes of TCs calculated for different time ratios using analytical model. It can be seen that for late
times between 0.6s and 0.85s the calculated TC assume almost uniform value. The same tendency is
expected when the generated POD-RBF base is used as the direct solver. Moreover, Fig. 6 shows the
plot of the sensitivity functions evaluated for several combinations of time ratios. The region where
the sensitivity function attains large values is the external circular ring located at a certain distance
from the heated area (cf. Fig. 6). Thus, it is natural to use locations within this ring in the definition of
the objective function (1).
Figure 5: Calculated thermal conductivity for different time ratios 𝜏F /𝜏C
Pixelsusedforobjective
functiondefinition
Rangewheretemperature
excessisverysmall
Figure 6: Calculates sensitivity function for several time ratios
Created POD-RBF bases consist temperature fields (snapshots) generated in thermal conductivity
ranges 10 − 20 W/mK, 20 − 30 W/(mK) for the C16 and C20 samples, respectively. Defined TC
ranges were divided into 20 subregions. First set of calculations was performed for sample C16 using
for several combinations of time ratios. Calculated data is listed in Table 1.
𝜏F /𝜏C
𝜏F /𝜏C
𝜏F /𝜏C
𝜏F /𝜏C
𝜏F /𝜏C
: 0.683/0.700
: 0.683/0.716
: 0.683/0.733
: 0.683/0.750
: 0.683/0.766
𝑘NN ,
𝑘MM ,
𝑘OO ,
𝑘ijk ,
W/(mK) W/(mK) W/(mK) W/(mK)
15.99
15.99
15.99
15.99
15.78
15.64
15.46
15.62
15.46
15.46
16.46
15.46
15.46
15.46
15.46
15.46
15.39
15.03
15.88
15.43
Table 1: Calculated TC for sample C16
𝑎,
m2/s
4.10E-06
4.00E-06
3.96E-06
3.96E-06
3.96E-06
PODRBF/LFA, %
4.2
6.4
7.4
7.4
7.5
For sample C20 different strategy was used. The simulations were run using measurement data
collected during three experiments. The set of retrieved TCs for different time ratios (𝜏F /𝜏C ) is
illustrated in Fig. 7. It can be seen that for tests 1 and 3 in time range 0.675 − 0.775s the solutions
fluctuate around some average value, while for test 2 such tendency was not observed. Using the
retrieved TCs from specified time range, the average TCs can be calculated. For tests 1 and 2 the
average TCs were equal to 27.0 (4.41%) and 27.1 (4.53%), respectively. Is worth to mention here that
these calculations were run for case without analysing the optimal position of pixels. It can be seen
that in such case it was very difficult to find stable solution. To overcome this problem, always the
solution of analytical model, together with sensitivity function study need to be used to determine
optimal pixels position and time ratios.
Figure 7: Calculated thermal conductivity for different time ratios using the POD-RBF base as the
direct solver for C20 sample
6. Summary
A non-destructive technique for measuring TC was presented in this work. The TC component was
calculated applying combination of experimental work, numerical modelling and advance inverse
technique based on the POD-RBF low order approximation. The numerical model used for generation
POD base was built using Ansys code, where the calculation algorithm was fully controlled by
external procedure. The developed measurement procedure accommodates the finite dimension of
laser spot diameter, heat loses due to the convection and radiation and emission time. Furthermore,
main advantage of the developed measurement procedure is its possibility of future application for fast
evaluation of the TC of the material within isotropic and orthotropic structure without necessity of
specified sample preparation. Presented results should be seen as the preliminary one where the
calculation procedure still need to be improved to ensure more reliable and stable solution.
Nevertheless, presented method and its application has high application potential and should be deeply
studied and developed. It is also planned to run the uncertainty and quantification analysis using the
Latin Hypercube sampling technique to determine influence of input date on accuracy of the numerical
model. Moreover, the simulation for material with real anisotropic structure need to be run.
Acknowledgments
The research has been supported by National Science Centre within SONATA scheme under contract
Nr. 2014/15/D/ST8/02620.
References
[1] ASTM 1461 – 13, Standard Test Method for Thermal Diffusivity by the Flash Method, 2013
[2] W.P. Adamczyk, R.A. Bialecki, T. Kruczek, Retrieving thermal conductivity of isotropic and
orthotropic materials, Applied Mathematical Modelling 40(4):1-12, 2015
[3] W.P. Adamczyk, R.A. Bialecki, T. Kruczek, Measuring thermal conductivity tensor of orthotropic
solid bodies, Measurement, 101:93-102, 2017
[4] N. Bianco and O. Manca. Theoretical comparison of two-dimensional transient analysis between
back and front laser treatment of thin multilayer films. International Journal of Thermal Sciences,
43:611–621, 2004
[5] F. Cernuschi, P.G. Bison, A. Figari, S. Marinetti, and E. Grinzato. Thermal diffusivity
measurements by photothermal and thermographic techniques. International Journal of
Thermophysics, 25(2):439–457, 2004
[6] A. Cezairliyan, K.D. Maglic, and V.E. Peletsky. Compendium of Thermo- physical Property [1]
Measurement Methods. Volume 2: Recommended mea- surement Techniques and Practices. New
York, 1992
[7] R.D. Cowan. Pulse method of measuring thermal diffusivity at high temperatures. Journal of
Applied Physics, 34:926–927, 1963. [8] G.H. Golub and C.F. Van Loan, Matrix computations, 1989
[9] S.E. Gustafsson, E. Karawacki, and M.N. Khan. Transient hot-strip method for simultaneous
measuring thermal conductivity, thermal diffusivity of solids, liquids. Journal of Applied Physics,
12:1411–1421, 1979
[10] Germany NETZSCH-Gerätebau GmbH. www.netzsch-thermal-analysis.com
[11] U. Hammerschmidt and W. Sabuga. Transient hot strip (ths) method: Uncertainty assessment.
International Journal of Thermophysics, 21:217– 248, 2000
[12] N.D. Milosevic, M. Raynaud, and K.D. Maglic. Simultaneous estimation of the thermal
diffusivity and thermal contact resistance of thin solid films and coatings using the two-dimensional
flash method. International Journal of Thermophysics, 24:799–819, 2003
[13] T. Kruczek, W.P. Adamczyk, and R.A. Bialecki. In situ measurement of thermal diffusivity in
anisotropic media. International Journal of Thermophysics, 34(3):467–485, 2013
[14] Z. Ostrowski, R.A. Bialecki, and A.J. Kassab. Estimation of constant thermal conductivity by use
of Proper Orthogonal Decomposition. Computa- tional Mechanics, 37(1):52–59, 2005
[15] Z. Ostrowski, R.A. Bialecki, and A.J. Kassab. Solving inverse heat conduction problems using
trained POD-RBF network inverse method. Inverse Problems in Science and Engineering, 16(1):39–
54, 2008
[16] W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes, Cambridge University
Press, Cambridge, 2007
[17] J.W. Parker, R.J. Jenkins, C.P. Butler, and G.L. Abbott. Flash method of determining thermal
diffusivity, heat capacity and thermal conductivity. Journal of Applied Physics, 32(9):1679–1684,
1961
[18] L. Vozar, W. Hohenauer, Flash method of measuring the thermal diffusivity. High TemperaturesHigh Pressures, 35/36: 253-264, 2003/2004
[19] H. Watanabe. Further examination of the transient hot-wire method for the simultaneous
measurement of thermal conductivity, thermal diffusivity. Metrologia, 39:65–81, 2002