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Thermal characterization of materialsusing Karhunen–Loève decompositiontechniques – Part II. HeterogeneousmaterialsElena Palomo Del Barrio a , Jean-Luc Dauvergne b & ChristophePradere ba Université Bordeaux 1, Laboratoire TREFLE , Talence , Franceb CNRS, Laboratoire TREFLE , Talence , FrancePublished online: 15 Feb 2012.
To cite this article: Elena Palomo Del Barrio , Jean-Luc Dauvergne & Christophe Pradere (2012)Thermal characterization of materials using Karhunen–Loève decomposition techniques – Part II.Heterogeneous materials, Inverse Problems in Science and Engineering, 20:8, 1145-1174, DOI:10.1080/17415977.2012.658517
To link to this article: http://dx.doi.org/10.1080/17415977.2012.658517
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Inverse Problems in Science and EngineeringVol. 20, No. 8, December 2012, 1145–1174
Thermal characterization of materials using Karhunen–Loeve
decomposition techniques – Part II. Heterogeneous materials
Elena Palomo Del Barrioa*, Jean-Luc Dauvergneb and Christophe Pradereb
aUniversite Bordeaux 1, Laboratoire TREFLE, Talence, France; bCNRS, LaboratoireTREFLE, Talence, France
(Received 30 November 2010; final version received 13 January 2012)
A new method for thermal characterization of heterogeneous materials has beenproposed. As for homogeneous materials in the first part of the paper, the methodis based on the use of Karhunen–Loeve decomposition (KLD) techniques inassociation with infrared thermography experiments or any other experimentaldevice providing dense data in the spatial coordinate. Orthogonal properties ofKLD eigenfunctions and states are used for achieving simple estimates of thermaldiffusivities. It has been proven that diffusivities can be estimated without explicitknowledge of variables and parameters related to heat exchanges at the interfaces(i.e. thermal conductivities, thermal contact resistances). Indeed, the diffusivitiesestimates only depend on some few KLD eigenelements. As a result, a significantamplification of the signal/noise ratios is reached. Moreover, it is shown thatspatially uncorrelated noise has no effect on KLD eigenfunctions, the noise beingentirely reported on states (time-dependent projection coefficients). This isparticularly interesting because thermal diffusivities estimates involve spatialderivatives of the eigenfunctions. Consequently, the proposed method results inan attractive combination of parsimony and robustness to noise. The effectivenessof the method is illustrated through some simulated experimental applications.
Keywords: thermal characterization; infrared thermography; Karhunen–Loevedecomposition; singular values decomposition
Nomenclature
Roman letters
k Thermal conductivityh Inverse of the thermal resistance
Tðx, tÞ Temperature fieldTðtÞ Vector of temperature
t TimeVmðxÞ Eigenfunctions of W
V Matrix of eigenfunctions
*Corresponding author. Email: [email protected]
ISSN 1741–5977 print/ISSN 1741–5985 online
� 2012 Taylor & Francis
http://dx.doi.org/10.1080/17415977.2012.658517
http://www.tandfonline.com
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Wðx, x0Þ Energy functionW Energy matrixx Coordinates
zmðtÞ StatesZðtÞ Vector of states
Greek letters
� Thermal diffusivity� Thermal loss coefficient
"ðx, tÞ Noise fieldeðtÞ Noise vector�2m Eigenvalues of W�2" Noise energy
Symbols
�k k Unitarily invariant norm,h i Scalar product� Time derivative� Noisy variable
Mean value^ Estimated value
Abbreviations
KLD Karhunen–Loeve decompositionPCA Principal components analysisSVD Singular values decomposition
1. Introduction
Heterogeneous media involve spatial variations of the thermophysical properties in
different ways, such as large-scale variations in functionally graded materials, abrupt
variations in composites and random variations due to local concentration fluctuations in
dispersed phase systems. This paper deals with the thermal characterization of composite
materials formulated by physical or chemical bounding of different homogeneous phases.
Such kind of materials, with abrupt changes of the physical properties at the interfaces,
have been providing engineers with increased opportunities for tailoring structures to meet
a variety of property and performance requirements.There are three main experimental devices of particular interest for thermal
characterization of non-homogeneous materials which are able to provide thermal
imaging of the samples surfaces: scanning thermal microscopy (SThM), photoreflectance
microscopy (PhRM) and infrared tomography (IRT).The SThM [1,2] is based on an atomic force microscope equipped with a thermal probe
to carry out thermal images while simultaneously obtaining contact mode topography
images. It allows thermal imaging at spatial resolution around 100 nm. However, the main
issue of this device is to establish suitable models to take advantage of the experimental
measurements. Two simplified models of the probe have been recently proposed [3]: a model
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out of contact which enables calibration of the probe, and a model in contact to extractthermal conductivity from the sample under study.
PhRM is based on the measurement and analysis of the periodic temperature increaseinduced by the absorption of an intensity modulated laser beam (pump beam) [4]. Bydetecting the thermally induced reflection coefficient variations with the help of asecondary continuous laser beam (probe beam), the temperature increase at the samplesurface can be measured at the micrometric scale. To image the temperature on a samplesurface, both beams (probe and pump) are simultaneously scanned with constant offset.The phase signal at each position is characteristic of the local material properties, a simplemodel relaying the phase lag with the thermal diffusivity of the sample over the distancebetween beams. This technique has been applied to characterize both functionally gradedmaterials and composites [5–7].
IRT is widely used to measure thermal diffusivities. An experiment usually consists inapplying a heat flux on the front face of a sample by a laser beam and detecting the samplethermal response on either its front or its rear face using an infrared camera with a focalplane array of infrared detectors. Compared to SThM and PhRM, IRT is simpler and lessexpensive. Indeed, because scanning is not required, experiments are really short. An IRTexperiment typically takes less than 1min, whereas scanning a small sample (�100 mm2) byPhRM takes several hours. However, maximum spatial resolution of IRT is around 20 mm.Concerning materials thermal characterization by IRT, most part of the work carried outrelates to homogeneous materials [8–13] as well as to inverse methods based on simpleanalytical solutions of the heat conduction problem [8–11]. Thermal characterization ofheterogeneous materials is obviously more complicated because involving estimation of asmany thermal diffusivities as homogeneous phases (composites) or continuous spatialvariations of thermal properties (functionally graded materials). A fairly commonapproach for estimating thermal properties of non-homogeneous media is related to theminimization of an objective function that usually involves the quadratic differencebetween measured and estimated dependent variables, such as the least squares norm, orits modified versions with the addition of regularization terms [14–19]. Bayesianapproaches are rarely used because it is computationally expensive. To overcome suchdifficulty, Bayesian inference combined with the integral transform method for solving thedirect problem has been recently proposed [20]. Although popular and useful in manysituations, all these techniques are generally rather sensitive to the initial guess. Besides,they have been numerically tested with moderate noise-corrupted temperature data, withadded noise usually around 1% of either the maximum or the mean temperature, whichcorrespond to rather high-quality data in the framework of IRT. To reduce computingefforts when handling the large amount of data provided by IRT, the use of point-by-pointleast squares estimation approaches has been also proposed [21–23]. However, thesensitivity to measurement noise of such approaches is very high because involving timeand space derivates of the temperature data.
This paper proposes a new method for estimating the thermal diffusivities of thedistinct phases of composite materials with arbitrary microstructure. As for homogeneousmaterials in the first part of the paper [13], the method is based on the use of Karhunen–Loeve (KLD) techniques in association with one single infrared thermography experiment.Compared to the inverse techniques discussed above, the proposed KLD-based methodallows dealing efficiently with large amount of noise-corrupted temperature data, as thoseusually provided by IRT experiments. Indeed, the method does require neither initial guessnor knowledge of the thermal parameters defining heat exchanges at the interfaces
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between the phases of the composite material (i.e. thermal conductivities, thermal contact
resistances). Although the objective consist in identifying the thermal diffusivity of a fixed
number of homogeneous phases, the method proposed does not reduce to just identifying
a fixed number of homogeneous materials because it is based on one single experiment and
performs simultaneous identification of the whole set of thermal diffusivities.KLD techniques are widely used for multivariate data reduction in many areas of
application. A low-dimensional approximate description of the whole set of data is
obtained by projecting the initial high-dimensional set on the dominant KLD
eigenfunctions. When dealing with regular signals, KLD yields optimal low-dimensional
descriptions. This means that it provides the lowest dimension for a given approximation
precision or, alternatively, the best precision for a given dimension. Such a method has
been developed about 100 years ago by Pearson [24] as a tool for graphical data analysis
and re-developed several times since them in different areas of application [25–27], so that
it goes under many names as principal components analysis (PCA), Karhunen–Loeve
decomposition (KLD), singular value decomposition (SVD), etc. PCA/KLD/SVD is a
very common tool today in image processing and signal processing for compression,
noise reduction, signals classification, data clustering and information retrieval problems
[28–31]. In thermal analysis, SVD-based methods have been developed for efficient
reduction of linear and non-linear heat transfer problems [32–36], as well as for solving
inverse problems dealing with unknown heat sources [37–39].The content of the paper is as follows. The problem is stated in section 2. The method
for estimating thermal diffusivities is presented in section 3. The effectiveness of the
method is illustrated through simulated experimental applications in section 4.
2. Problem statement
Let us consider a heterogeneous material coming from physical aggregation of p different
phases, as well as a thin sample (plate) of this material. Let �i (i ¼ 1, 2, . . . , p) denote the
region of the space occupied by the i-phase, so that � ¼ �1 [�2 [ � � � [�p represents the
indoor domain of the plate. The boundary separating the medium from its environment is
@�, and the interface between the i-phase and j-phase is referred as @�ij. To simplify
notation, the phases are supposed to be isotropic, although theoretical analysis and
methods in this paper could be applied to anisotropic cases too.For time t4 0 and points belonging to �i (i ¼ 1, 2, . . . , p), the energy equation can be
written as
@Tðx, tÞ
@t¼ �ir
2Tðx, tÞ � �iTðx, tÞ ð1Þ
where x ¼ ðx, yÞ represents point coordinates. Tðx, tÞ is the excess of temperature with
regard to the surrounding, which is assumed to remain at uniform and constant
temperature during the experiment. Parameter �i represents the thermal diffusivity of the
i-phase. Parameter �i is defined as �i ¼ h=ð�ciLeÞ, where h represents the effective heat
transfer coefficient between the plate and its surrounding, �ci is the thermal capacity of the
i-phase and Le is the thickness of the plate. Last one is assumed to be small enough for the
thermal gradient in the thickness direction to be negligible. The Biot number (Bi ¼ hLe=ki;ki¼ thermal conductivity) hence has to be small, let us say less than 0.1.
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Initial condition is Tðx, 0Þ ¼ ToðxÞ. The plate dimensions are as large as required towarrant the border is not reached by thermal perturbations. Hence, adiabatic boundaryconditions on @� can be assumed
rTðx, tÞn ¼ 0 8t, 8x 2 @� ð2Þ
n represents the unit outward-drawn vector normal to @� at point x. If there is a perfectthermal contact among phases, the equations verified at the interfaces @�ij are(8t, 8x 2 @�ij):
Tðx, tÞ��i¼ Tðx, tÞ
��j ð3Þ
kirTðx, tÞ��inij þ kjrTðx, tÞ
��jnij ¼ 0 ð4Þ
Otherwise, Robin-type conditions are assumed (8t, 8x 2 @�ij):
�kirTðx, tÞ��inij ¼ �hij Tðx, tÞ
��i�Tðx, tÞ��j� �ð5aÞ
�kjrTðx, tÞ��jnij ¼ �hij Tðx, tÞ
��j�Tðx, tÞ��i� �ð5bÞ
where ki and kj are, respectively, the thermal conductivity of i-phase and j-phase. hijrepresents the inverse of the thermal contact resistance at the interface (if any). nij is theunit normal vector to the interface directed from �i to �j at point x.
To be fully in line with the model above, infrared thermography experiments must becarried out on thin samples (plates) located in an environment at constant and uniformtemperature. Starting from a plate in thermal equilibrium with its environment, the initialcondition ToðxÞ can be established using, i.e. a laser beam with almost arbitrary spatial andtime patterns. Thermal relaxation of the plate is thus observed using an infrared camera.At each sampling time, a plate temperature map is recorded and stored. We suppose thelateral resolution of the camera is high enough for temperature maps provide a goodenough approximation of Tðx, tÞ.
Thermal characterization aims at determining �i (i ¼ 1, 2, . . . , p) values from therecorded temperature data. The microstructure of the sample is assumed to be known. Onthe contrary, coefficients �i (i ¼ 1, 2, . . . , p), thermal conductivities ki (i ¼ 1, 2, . . . , p) andthermal contact resistances are not.
3. Thermal diffusivities estimation method
The method we are proposing for estimating thermal diffusivities �i (i ¼ 1, 2, . . . , p) is heredescribed. Section 3.1 introduces the definition of the KLD as well as its main properties.In section 3.2., orthogonal properties of KLD eigenfunctions and states are intensivelyused for getting some fundamental equations on which thermal parameters estimationswill be based. They prove that diffusivities �i, as well as coefficients �i, can be estimatedwithout explicit knowledge of variables and parameters related to heat exchanges at theinterfaces (i.e. thermal conductivities, thermal contact resistances). Estimates for param-eters �i and �i are presented in section 3.3. It is shown that the information required forthermal diffusivities estimation reduces to eigenfunctions ViðxÞ(i ¼ 1, 2, . . . , p), and to theassociated states ziðtÞ (i ¼ 1, 2, . . . , p). In other words, we prove that the p-dimensional
Inverse Problems in Science and Engineering 1149
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KLD approximation of the temperature field provides information enough for estimationpurposes. As shown later, this leads to an exciting combination of parsimony and
robustness to noise.
3.1. Karhunen–Loeve decomposition of the thermal field
The energy function associated to Tðx, tÞ is defined as
Wðx, x0Þ �
Zt
Tðx, tÞTðx0, tÞdt ð6Þ
It can be proven that the eigenfunctions of Wðx, x0Þ, noted as VmðxÞ� �
m¼1...1in the
following, define a complete orthogonal set in the Hilbert space associated to the problem.
Indeed, the spectrum of Wðx,x0Þ consists of 0 (zero) together with a countable infinite setof real and positive eigenvalues: �21 � �
22 � � � � � 0. The problem defining eigenvalues and
eigenfunctions of Wðx, x0Þ is
Wðx, x0Þ ¼X1m¼1
VmðxÞ�2mVmðx
0Þ ð7Þ
with orthogonal condition:
Vk,Vmh i��
Z�
VkðxÞVmðxÞdx ¼ �km ð8Þ
The Karhunen–Loeve decomposition of the temperature field, also called singularvalues decomposition, results from Tðx, tÞ expansion on Wðx, x0Þ eigenfunctions:
8t, Tðx, tÞ ¼X1m¼1
VmðxÞzmðtÞ ð9Þ
where the projection coefficients (states in the following) are given by
zmðtÞ ¼ Tðx, tÞ,VmðxÞ� �
��
Z�
Tðx, tÞVmðxÞdx ð10Þ
Taking into account Equations (6)–(8), it can be easily proven that the states are
orthogonal, they verify
zmðtÞ, zkðtÞ� �
t�
Zt
zmðtÞzkðtÞdt ¼ �mk�2m ð11Þ
Let us now consider noise-corrupted observations:
~Tðx, tÞ ¼ Tðx, tÞ þ "ðx, tÞ ð12Þ
with
8x, x0Zt
"ðx, tÞTðx0, tÞdt ¼ 0 ð13Þ
8x, x0 W"ðx, x0Þ �
Zt
"ðx, tÞ"ðx0, tÞdt ¼ �2" �ðx� x0Þ ð14Þ
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In such conditions (spatially uncorrelated noise), the KLD of ~Tðx, tÞ is [13]:
~Tðx, tÞ ¼X1m¼1
VmðxÞ ~zmðtÞ ð15Þ
Equation (15) proves that the noise has no effect on the KLD eigenfunctions. On thecontrary, it is entirely reported on the states:
~zmðtÞ ¼
Z�
~Tðx, tÞVmðxÞdx ¼
Z�
Tðx, tÞVmðxÞdxþ
Z�
"ðx, tÞVmðxÞdx ¼ zmðtÞ þ �mðtÞ ð16Þ
�mðtÞ represents the orthogonal projection of the noise on eigenfunction VmðxÞ. Itverifies [13]
8m, �2",m �
Zt
�2mðtÞdt ¼ �2" ð17Þ
The signal/noise ratio of the states is hence given by ð�2m=�2" Þ. As �21 � �
22 � � � � � 0, the
effect of noise on the states is as much significant as the energy of the state is lower.More detailed information on the KLD of the temperature field, as well as proofs of its
properties, is provided in the first part of this paper [13].
3.2. Fundamental equations for estimation purposes
The equations that will be used for parameters estimation are derived here. MultiplyingEquation (1) by VkðxÞ and integrating over �i (i ¼ 1, 2), leads to
@Tðx, tÞ
@t,VkðxÞ
�i
¼ �i r2Tðx, tÞ,VkðxÞ
� ��i��i Tðx, tÞ,VkðxÞ
� ��i
ð18Þ
The second theorem of Green, with adiabatic conditions on @�, allows writing:
r2Tðx, tÞ,VkðxÞ� �
�i¼ r2VkðxÞ,Tðx, tÞ� �
�iþIðiÞk ðtÞ ð19Þ
with
IðiÞk ðtÞ ¼
Z@�
½VkðxÞrTðx, tÞnij � Tðx, tÞrVkðxÞnij�dx ð20Þ
It can be demonstrated that hIðiÞk ðtÞ, zkðtÞit ¼ 0. Introducing KLD of Tðx, tÞ into
Equation (20) yields:
IðiÞk ðtÞ ¼
Z@�
VkðxÞX1m¼1
zmðtÞrVmðxÞnij � rVkðxÞnijX1m¼1
zmðtÞVmðxÞ
" #dx ð21Þ
Multiplying this equation by zkðtÞ, integrating over time and taking into accountorthogonal property of the KLD states, we obtain
IðiÞk ðtÞ, zkðtÞ
D Et¼
Z@�
�2kVkðxÞrVkðxÞnij � �2kVkðxÞrVkðxÞnij
� �dx ¼ 0 ð22Þ
This equation applies both for continuity conditions at the interfaces (Equations (3)and (4)) and for Robin-type conditions (Equations (5a) and (5b)).
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Reporting now Equation (19) into Equation (18) and adding resulting equations for
i ¼ 1, 2, . . . , p yields
@Tðx, tÞ
@t,VkðxÞ
�
¼Xi¼1...p
�i Tðx, tÞ,r2VkðxÞ
� ��iþ IkðtÞ �
Xi¼1���p
�i Tðx, tÞ,VkðxÞ� �
�ið23Þ
with IkðtÞ ¼ Ið1Þk ðtÞ þ � � � þ I
ð pÞk ðtÞ.
We now replace Tðx, tÞ in Equation (23) by its KLD, we multiply by zkðtÞ and we
integrate over time. Taken into account hzmðtÞ, zkðtÞit ¼ �mk�2m and hIkðtÞ, zkðtÞit ¼ 0, we
obtain
1
�2k_zkðtÞ, zkðtÞ� �
t¼Xi¼1...p
�i r2VkðxÞ,VkðxÞ
� ��i�Xi¼1...p
�i VkðxÞ,VkðxÞ� �
�ið24Þ
or
z2kðtÞ
2�2k
�����tf
t¼0
¼Xi¼1...p
�i r2VkðxÞ,VkðxÞ
� ��i�Xi¼1...p
�i VkðxÞ,VkðxÞ� �
�ið25Þ
The estimation of the diffusivities will be based on these equations. It must be noticed
that variables and parameters related to heat exchanges at the interfaces @�ij do not appear
in Equation (25). This is an advantage for reliable estimation of the diffusivities because the
total number of physical parameters to be estimated is hence significantly reduced
(i.e. thermal conductivities and thermal resistances at the interfaces have not to be
estimated).At last, it can be easily proven that integration over � of Equation (1) leads to
�1�1
d �T1ðtÞ
dtþ � � � þ
�p�p
d �TpðtÞ
dt¼ � �TðtÞ ð26Þ
�TiðtÞ ¼ hTðx, tÞi�i(i ¼ 1, . . . , p) and �i is the fraction of the plate surface which is occupied
by the i-phase (surface of �i / surface of �). Integrating from 0 to t the equation above,
yields:
�1�1
D �T1ðtÞ þ � � � þ�p�p
D �TpðtÞ ¼ �
Z t
¼0
�TðÞd ð27Þ
with D �TiðtÞ ¼ �TiðtÞ � �Tið0Þ. Estimation of parameters �i (i ¼ 1, . . . , p) will be based on this
equation.
3.3. Parameters estimates
For free-noise observations, parameters �i and �i (i ¼ 1, . . . , p) can be easily calculated
from any 2� p arbitrarily chosen Equation (25). On the contrary, for noise-corrupted
observations, it is convenient to do otherwise.Taking into account the efficiency of the spatial-mean operator, as well as time
cumulative integrations, for noise reduction, the best estimate of parameters �i(i ¼ 1, . . . , p) is achieved applying the linear least squared method to Equation (27).
1152 E. Palomo Del Barrio et al.
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This leads to
1=�1
..
.
1=�p
2664
3775 ¼ ðM0MÞ�1ðM0yÞ ð28Þ
with
ðM0MÞ ¼
Zt�1D �T1ðtÞ � � � �pD
_�TpðtÞ
h i0�1D �T1ðtÞ � � � �pD
_�TpðtÞ
h idt
ðM0yÞ ¼ �
Zt�1D �T1ðtÞ � � � �pD
_�TpðtÞ
h i0 Z t
¼0
�TðÞd
�dt
ð29Þ
On the contrary, the estimation of the diffusivities �i must be based on Equation (25).
It is evidence that at least p of these equations are required because the problem involves p
unknown diffusivities. However, keeping all of them will be a wrong strategy because there
are terms in the KLD of ~Tðx, tÞ which are not significant compared to the noise. As shown
in section 3.1, signal/noise rate for states ~zmðtÞ is �2m=�
2" , so that the effect of noise is as
much significant as the energy of the state is low. Besides, eigenvalues �2m usually decrease
quickly: �21 �22 �23 � � � Hence, Equation (25) that will be preferred for diffusivities
estimation are those involving the states with largest eigenvalues.First step towards diffusivities estimation consists in verifying that the experiment
carried out is informative enough. States showing high enough signal/noise ratio,
z1ðtÞ, z2ðtÞ, . . . , zrðtÞ, are hence identified. If r5 p, the experiment must be rejected. If r � p,
Equation (25) for k ¼ 1, . . . , r is then written in the matrix form:
~y ¼M
�1
..
.
�p
2664
3775 ð30Þ
with
~y ¼
D ~z21= ~�21 þP2i¼1
�i V1ðxÞ,V1ðxÞ� �
�i
D ~z22= ~�22 þP2i¼1
�i V2ðxÞ,V2ðxÞ� �
�i
� � �
D ~z2r= ~�2r þP2i¼1
�i VrðxÞ,VrðxÞ� �
�i
26666666664
37777777775
D ~z2i ¼1
2~z2i��tft¼0
ð31Þ
and
M ¼
r2V1ðxÞ,V1ðxÞ� �
�1� � � r2V1ðxÞ,V1ðxÞ
� ��p
r2V2ðxÞ,V2ðxÞ� �
�1� � � r2V2ðxÞ,V2ðxÞ
� ��p
� � � � � � � � �
r2VrðxÞ,VrðxÞ� �
�1� � � r2VrðxÞ,VrðxÞ
� ��p
266664
377775 ð32Þ
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The solution in the least squares sense of Equation (30) is
�1
..
.
�p
2664
3775 ¼M#~y with M# ¼ ðM0MÞ�1M0 ð33Þ
For r ¼ p, the sensitivity matrix M# becomes M# ¼M�1. In section 4.3 we show thatthe best results are achieved when r ¼ p. The reason is that increasing the number of KLDeigenelements to be used implies reducing signal/noise ratios: KLD truncation acts as asignal/noise ratio amplifier, the amplification being as much significant as r is small.
Provided that �2" �2p , the statistical properties of the a-estimates defined byEquation (33) can be assumed to be as follows (see Appendix for proof):
E½a� ¼ a a0 ¼ ½�1 �2 � � � �p � ð34Þ
covðaÞ � E½ða� aÞða� aÞ0� ¼M#½Dvarð"Þ þ AcovðbÞA0�ðM#Þ0 ð35Þ
Dð p� pÞ is a diagonal matrix with elements dmm ¼ ðzmðtf Þ2þ zmð0Þ
2Þ=�4m. Að p� pÞ is the
matrix whose elements are ami ¼ VmðxÞ,VmðxÞ� �
�i. E represents the expectation operator
and covðbÞ if the covariance matrix of the of parameters �i.
3.4. Estimation in practice
Infrared thermography experiments provide a plate temperature map at each samplingtime; that is, a finite-dimensional approximation of the infinite-dimensional thermal field.
Let ~TðtÞ ¼ TðtÞ þ eðtÞ be the vector including temperature measurements (platetemperature map) at time t. TðtÞ represents useful information while eðtÞ is themeasurement noise. The dimension of vector ~TðtÞ (n� 1) is equal to the number ofpixels of the infrared image. We suppose the lateral resolution of the camera is highenough for TðtÞ provides a good enough approximation of Tðx, tÞ. Moreover, measure-ments noise eðtÞ is assumed to be spatially uncorrelated. The noise energy matrix is hence:W" � heðtÞetðtÞit ¼ �
2" I.
First of all, parameters �i (i ¼ 1, . . . , p) are estimated using Equations (28) and (29).Let ~TiðtÞ (i ¼ 1, . . . , p) be the vector including the elements of ~TðtÞ belonging to �i. Meanvalues �TðtÞ and �TiðtÞ (i ¼ 1, . . . , p) at time t are assumed to be equal to the mean values of~TðtÞ and ~TiðtÞ, respectively.
Since parameters �i (i ¼ 1, . . . , p) become known, thermal diffusivities estimation canbe carried out. First step towards parameters �i (i ¼ 1, . . . , p) estimation is KLD of ~TðtÞ.This involves
– Calculation of the energy matrix of ~TðtÞ
~W �
Zt
~TðtÞ~TtðtÞdt ð36Þ
If ~TðtÞ provides a good enough approximation of ~Tðx, tÞ, then ~W is expected to bea good enough approximation of ~Wðx, x0Þ.
– Calculation of the eigenvalues and the eigenvectors of ~W. We remind that ~W
(n� n) is a symmetric, definite positive matrix. Accordingly, spectral
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decomposition of ~W leads to a n-dimensional set of orthonormal eigenvectors,~V ¼ ~v1 ~v2 � � � ~vn
� �with ~V
t ~V ¼ I, and associated eigenvalues verifying~�21 � ~�22 � � � � � ~�2n � 0. The energy matrix can hence be written as
~W ¼ ~V~D~Vt
with ~D ¼ diag ½ ~�21 ~�22 � � � ~�2n � ð37Þ
If ~W provides a good enough approximation of ~Wðx, x0Þ, then ~vi is expected to bea good enough approximation of ~ViðxÞ (i ¼ 1, . . . , p). Moreover, for spatiallyuncorrelated noise ~ViðxÞ ¼ ViðxÞ and hence ~V ¼ V.
– Calculation of the states: ~ZðtÞ ¼ ~Vt~TðtÞ.
Next step is parameters �i (i ¼ 1, . . . , p) estimation using Equation (33). For discreteapproximations of eigenfunctions, as those coming from KLD of ~TðtÞ, the inner productsin Equations (31) and (32) become
~VkðxÞ, ~VkðxÞ� �
�i� vtkPivk & r2 ~VkðxÞ, ~VkðxÞ
� ��i� vtkPiLvk ð38Þ
where L is the numerical approximation of r2 and Pi is a 1/0 diagonal matrix which selectsthe elements of the eigenvector vk associated to the pixels belonging to �i.
As previously noted, the microstructure of the sample is assumed to be known. Hence,it has to be determined before starting with thermal parameters estimations. An efficientmethod based on KLD techniques and infrared thermography experiments has beenrecently proposed for retrieving the microstructure of composite materials [40]. Aconstant, uniform heat flux is applied on the rear face of a thin sample whilesimultaneously the thermal response on the front face is recorded with an infraredcamera. The experiment is short enough (typically less than 1 s) for heat exchanges throughthe interfaces to be almost negligible. The number of phases of the composite is proven tobe equal to the rank of the energy matrix of the thermal field. Phases are thusdiscriminated by simple analysis of the sign of the KLD eigenfunctions. Compared totechniques based on optical or electronic microscopes, the method proposed in [40] allowsretrieving the sample microstructure at the same spatial resolution and with exactly thesame pixel-grid than in later thermal characterization experiment because based on thesame experimental device.
It must be noticed too that the thermal capacity of the phases should be obtained bysome other independent method in case the thermal conductivity and/or the effective heattransfer coefficient h are to be known.
4. Numerical examples
Some numerical examples are being used for illustrating the appropriateness of ourdevelopments. The materials and the experiments are described in sections 4.1 and 4.2.Main KLD eigenelements are analysed in section 4.3. Average temperatures required forestimating �i parameters are examined in section 4.4. The last section includes thermalparameters estimations and discussion of the results.
4.1. Description of the plates microstructure
Square plates (L� L, L ¼ 6mm) with realistic two-phase random microstructures havebeen considered. The concept of random morphology description functions (RMDF) has
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been used to create the microstructures [41–43]. In this approach, the morphology of a
random two-phase material at the microsctructural level is defined through a level cut of a
random field. The RMDF is defined as the sum of N 2D Gaussian functions:
f ðxÞ ¼XNi¼1
ci exp �x� xi
wi
� �2" #
ð39Þ
The magnitudes ci 2 ½�1, 1� and the centres xi of the Gaussian functions are randomly
chosen. The spatial widths of the individual Gaussian functions are wi ¼ L=ffiffiffiffiNp
, so that
increasingly complex morphological features are achieved as more and more Gaussian
functions are included in RMDF. For convenience, the RMDF f ðxÞ and the cut-off value
fo are normalized to lie in the range ½0, 1� as follows: f ðxÞ ð f ðxÞ � fminÞ=ð fmax � fminÞ,
where fmin and fmax are the minimum and the maximum value of f ðxÞ. A two-phase
random microstructure is thus obtained by applying the cut-off value fo to f ðxÞ: if
f ðxÞ4 fo ) x 2 �1; otherwise, x 2 �2.The microstructures generated for further analysis are represented in Figure 1. They
are referred as ‘PxxNyyy’, where ‘xx’ denotes the volume fraction of phase 2 (20, 50, 70%)
and ‘yyy’ is the number of Gaussian functions that are used to define the RMDF
(N ¼ 100, 500, 1000). As previously noted, the complexity of the morphology increases as
the number of Gaussian functions is increased.
Figure 1. Two-phase heterogeneous plates (6mm� 6mm).
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Images in Figure 1 include 60� 60 pixels (pixel size: 100 mm� 100 mm). Connectedcomponents in each image have been identified. Just for comparisons purposes, the‘effective’ length of one component is defined as l ¼
ffiffiffiffiffinpp
, where np is the number of pixelswithin the component. l is hence the length of the square of size equal to that of thecomponent. Table 1 provides a short statistical description of the ‘effective’ lengths ofconnected components in the studied microstructures. Median values range froml ¼ 4:2 ðpixelsÞ to l ¼ 29:1 ðpixelsÞ for phase 1, whereas they vary from l ¼ 3:1 ðpixelsÞ tol ¼ 32:8 ðpixelsÞ for phase 2.
4.2. Numerical experiments
Let us consider the plates in Figure 1 (L� L, L ¼ 6mm) that exchange heat byconvection/radiation with an environment at uniform and constant temperature, say at0�C. As in actual experiments, the plate is assumed to be in thermal equilibrium with theenvironment at the beginning of the experiment. Thus, a heat flux is applied on the centreof the plate during a short time using either a laser or a lamp with a mask:
qðx, y, tÞ ¼ qo for ðx, yÞ 2 �spot and 05 t to
qðx, y, tÞ ¼ 0 otherwise
ð40Þ
�spot represents the laser/lamp spot, which is assumed to be circular with diameterspot ¼ L=8. The equation governing the thermal evolution of points belonging to �i
(i ¼ 1, 2) is
@Tðx, y, tÞ
@t¼ �ir
2Tðx, y, tÞ � �iTðx, y, tÞ þ ’iðx, y, tÞ ð41Þ
with ’iðx, y, tÞ ¼ qðx, y, tÞ=ð�ciLeÞ. At the interfaces between phases, continuity of bothtemperature and heat flux is assumed. At points on the plate boundaries (x ¼ 0,x ¼ L, y ¼ 0, y ¼ L), adiabatic conditions are applied. Thermal parameters are�1 ¼ 1:5152� 10�7 m2 s�1, �2 ¼ 3:0303� 10�7 m2 s�1, �1 ¼ 0:0152 s�1, and�2 ¼ 0:0303 s�1. The effective heat transfer coefficient is assumed to be h ¼ 5Wm�2 s�1
Table 1. Analysis of the plates microstructure. Results of the statistical analysis carried out on thesize of connected components.
‘Effective’ length of connected components
Maximum value Mean value Minimum value Median value
Phase 1 Phase 2 Phase 1 Phase 2 Phase 1 Phase 2 Phase 1 Phase 2
P20N100 53.0 53.6 34.6 20.0 8.1 2.0 26.8 9.5P50N100 42.4 42.4 30.0 30.0 10.2 7.6 29.1 29.5P70N100 50.2 47.6 24.5 34.6 4.3 15.6 10.1 32.8P20N500 53.6 53.6 34.6 12.0 2.2 1.0 26.8 3.1P50N500 42.4 42.4 17.3 21.2 1.0 1.0 6.7 13.9P70N500 50.2 50.1 13.4 34.6 1.0 1.0 4.8 32.8P20N1000 53.6 53.6 34.6 8.6 1.0 1.0 26.8 3.1P50N1000 42.4 42.4 13.1 18.9 1.0 1.0 4.2 3.6P70N1000 50.2 50.0 10.0 30.0 1.0 1.7 4.2 23.4
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and the thickness of the plates is Le ¼ 0:1mm. In the most unfavourable case (Phase 1),
the Biot number is less than 0.001, so that that Equation (41) applies.The finite volume method has been applied on an equally spaced n� n (n ¼ 60, pixel
size: 100 mm� 100 mm) grid for discretization of Equation (41). This leads to the state-
space model:
_TðtÞ ¼ ATðtÞ þ fðtÞ ð42Þ
Temperature data are thus generated by time integration of Equation (42). The
experiments have been generated with qo ¼ 4000Wm�2, to ¼ 2s, tend ¼ 12 s (final time)
and Dt ¼ 0:5� 10�2 s (sampling time). It must be noticed that only data for t4 to, those
describing the thermal relaxation of the temperature field established by the applied heat
flux, are being used for estimation purposes. In the following, we note t ðt� toÞ and
Tð0Þ TðtoÞ is referred as initial temperature field.For estimation purposes, the plates thermal behaviours are corrupted with additive
noise: ~TðtÞ ¼ TðtÞ þ eðtÞ (n� 1, n ¼ 3600), with W" ¼ �2" I. Three different values of noise
amplitude have been considered: �0:5�C (bad quality data), �0:1�C (medium quality data)
and �0:02�C (good quality data). The quality of the experiments can be appreciated
through the following index evaluating signal/noise ratio:
SN ¼XnTi¼1
var½TiðtÞ�
! XnTi¼1
var½"iðtÞ�
!�1ð43Þ
TiðtÞ and "iðtÞ represent the elements of vectors TðtÞ and eðtÞ, respectively, and var is the
variance. SN index for the different experiments that have been generated are reported in
Table 2. Higher quality temperature data (SN ¼ 9123:3) correspond to the plate P20N500
with added noise amplitude equal to �0:02�C, whereas worst temperature data (SN ¼ 1)
are those generated with P50N100 and noise amplitude equal to �0:5�C. The thermal
response (relaxation period) of the plates P20N500 (noise: �0:02�C) and P50N100
(noise: �0:5�C) is depicted in Figure 2. The respective initial temperature fields are
represented on the left side of this figure, while the time evolution of the pixels temperature
is depicted on the right side.
Table 2. Evaluation of the quality of the experiments. Signal/noise ratio (Equation (43)) of thetemperature data.
Noise: �0.02�C Noise: �0.10�C Noise: �0.50�C
Phase 1 Phase 2 Whole Phase 1 Phase 2 Whole Phase 1 Phase 2 Whole
P20N100 3678.3 3970.4 3736.7 147.1 158.8 149.4 5.8 6.3 6.0P50N100 766.7 556.8 661.8 30.7 22.3 26.5 1.2 0.9 1.0P70N100 3213.9 1999.2 2363.8 128.4 79.9 94.5 5.1 3.2 3.7P20N500 9764.6 6554.2 9123.3 390.3 262.0 364.6 15.6 10.5 14.6P50N500 8029.5 4682.3 6357.1 321.1 187.1 254.1 12.8 7.5 10.1P70N500 3688.7 2171.9 2627.6 147.6 86.8 105.1 5.9 3.4 4.2P20N1000 9177.2 7213.3 8785.0 367.1 288.3 351.3 14.6 11.5 14.0P50N1000 1332.9 1246.1 1289.5 53.2 49.8 51.5 2.1 1.9 2.0P70N1000 2030.0 1522.2 1674.7 81.2 60.9 67.0 3.2 2.4 2.6
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4.3. Main eigenelements analysis
As previously noted, the information required for estimating the thermal diffusivity of thedistinct phases reduces to some few eigenelements, namely the two first eigenfunctions andthe two first states for a two-phase medium. The purpose of this section is to highlight theadvantages of using eigenelements instead of employing measured temperature data.
Let us consider the experiment generated using the microstructure P50N100 and addednoise with amplitude equal to �0:5�C (worst case in terms of signal/noise ratio). KLD ofthe set of free-noise signals has been carried out as described in section 3.4: TðtÞ ¼ VZðtÞ.Table 3 includes eigenvalues �21 , . . . , �26 as well as the corresponding contribution to thetotal energy of TðtÞ signals [13]: �2i =ð�m¼1,...,n�
2mÞ. It can be noticed that most part of the
TðtÞ signals energy is captured by the two first KLD components. First and secondeigenfunctions are depicted in Figure 3(a) and (c), while the time evolution of thecorresponding states zmðtÞ
� �m¼1,2
is represented in Figure 4 (continuous line).KLD of noise-corrupted data ~TðtÞ has been also carried out: ~TðtÞ ¼ ~V~ZðtÞ. Eigenvalues
~�2i (i ¼ 1 . . . 6) as well as relative errors ð ~�2i � �2i Þ=�
2i
�� �� are reported in Table 3. Next row ofthis table includes calculated signal/noise ratio for states (�2i =�
2" ). As expected, relative
Figure 2. Simulated temperature data: (a) initial temperature field on the plate P20N500 with noiseamplitude equal to �0.02�C; (b) temperature time behaviour within the plate P20N500 (�0.02�Cnoisy data), one curve by pixel; (c) initial temperature field on the plate P50N100 (�0.5�C noisydata); (d) temperature time behaviour within the plate P50N100 (�0.5�C noisy data), one curve bypixel.
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Figure 3. Eigenfunctions associated to the largest eigenvalues (plate P50N100): (a) Firsteigenfunction calculated from free-noise data, V1ðx, yÞ; (b) residuals V1ðx, yÞ � ~V1ðx, yÞ (�0.5
�Cnoisy data); (c) second eigenfunction calculated from free-noise data, V2ðx, yÞ; (d) residualsV2ðx, yÞ � ~V2ðx, yÞ (�0.5
�C noisy data).
Table 3. KLD eigenvalues analysis for free-noise and for noise-corrupted temperature data(E¼ total energy of the whole set of temperature data; SN¼ signal/noise ratio; Tr¼KLDr-dimensional approximation of the thermal field).
P50N100 Free-noise data 1 2 3 4 5 6
Eigenvalues 3650.9 473.3 45.7 4.0 0.5 0.06Contribution to E (%) 87.45 11.33 1.09 0.09 0.012 0.0016
P50N100 Noise-corrupted data 1 2 3 4 5 6Eigenvalues 3650.9 473.6 45.9 4.3 0.9 0.06Relative error (%) 0.0017 0.0763 0.4507 7.30 70.45 801.35SN of the states 1812.6 1747.5 190.6 16.2 1.6 1SN of Tr 1812.9 1780.6 1253.0 944.3 762.8 636.3
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error increases and signal/noise ratio reduce as the state energy �2i decreases. It can be seen
that for i � 4 signal/noise ratios start to be close or less than one. This means that useful
information in signals ~ziðtÞ with i � 4 is completely bogged down in noise. On the contrary,
signal/noise ratios for the two first states, those that will be used for parameters
estimation, are very high (>1700). Figure 4 shows the time behaviour of ~z1ðtÞ and ~z2ðtÞ
(symbols).If thermal diffusivities can be estimated using only some few KLD eigenelements,
namely the first r ones, this means that the r-dimensional approximation of the thermal
field
~Trðx, tÞ ¼Xrm¼1
VmðxÞ ~zmðtÞ r ¼ 1, 2, . . . , 6 ð44Þ
provides information enough for estimation purposes. The signal/noise of such approx-
imations (SN index, Equation (43)) has been also analysed. As shown in Table 3 (last row),
SN index for approximations up to r ¼ 6 is much higher than that of the raw temperature
data (SN¼ 1). Indeed, a huge amplification of the signal/noise ratio is achieved when
reducing primary signals to their 2D or 3D KLD approximation. This explains the
robustness to noise of the method we are proposing for diffusivities estimation. Indeed,
using the 2D KLD approximation instead of the 3D one will provide additional advantage
with regard to noise rejection. The SN index of the 2D approximation is 1780 times higher
than that of primary temperature data, while the SN index of the 3D approximation
reduces to 1253.The maps in Figure 3(b) and (d) represent the difference between the eigenfunctions
(first and second one) coming from KLD of TðtÞ and those from KLD of ~TðtÞ. These
differences (residuals in the following) are due to a not strictly diagonal W" matrix. As
shown in [13], the statistical correlation of an arbitrary element of eðtÞ, say "iðtÞ, withelements "j¼1,...,nðtÞ ( j 6¼ i) is almost zero, but no zero. Consequently, energy matrix W" is
diagonally dominant, but not strictly diagonal. As a result, measurement noise affects
Figure 4. States associated to the largest eigenvalues (plate P50N100): ~z1ðtÞ and ~z2ðtÞ for �0.5�C
noisy data (symbols); z1ðtÞ and z2ðtÞ for free-noise data (continuous lines).
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KLD eigenfunctions too. Figure 3(b) and (d) shows that the residuals in eigenfunctionsincrease as eigenvalues diminish. However, it remains very low for the two firsteigenfunctions even for highly noised temperature data. Indeed, the statistical analysiscarried out shows that the residuals can be described as Gaussian, spatially uncorrelatednoise. The histogram of the residuals for the second eigenfunction is depicted inFigure 5(a), whereas Figure 5(b) shows the spatial-autocorrelation. It can be seen that theresiduals lie in the �10�3 interval and are symmetrically distributed around zero. Besides,the autocorrelation function does not show significant values for spatial lags differentthan zero.
4.4. Averaged temperatures
As explained in section 3.3, data required for parameters �i (i ¼ 1, 2) estimation are theaverage temperatures of phases 1 and 2, namely �TiðtÞ (i ¼ 1, 2). Figure 6(a) represents the
Figure 5. Statistical analysis of the second eigenfunction residuals (plate P50N100, noiseamplitude¼�0.5�C): (a) histogram of the residuals; (b) spatial autocorrelation function.
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average temperatures corresponding to the worst experiment, the one generated using the
microstructure P50N100 and measurements noise amplitude equal to �0:5�C. The
differences between the average temperatures calculated from noise-corrupted temperature
data and those obtained from free-noise data are depicted in Figure 6(b). Compared to the
noise added to temperature data (�0:5�C), the noise amplitude in average temperatures is
much lower (�0:015�C). The signal/noise ratio is also improved, SN index (Equation (43))
is SN ¼ 9:7 for the average temperature of the phase 1 and SN¼ 30 for the phase 2.
However, the amplification of the signal/noise ratio supplied by averaging is negligible
compared to that provide by KLD. The SN values of the signals required for diffusivities
estimation are much higher (see Table 3). As shown in next section, the quality of the
estimations will be hence controlled by the quality of the average temperatures.
Consequently, it should be convenient to apply some filtering process to reduce their
noise amplitude.
Figure 6. Time evolution of the average temperature of the phases 1 and 2 (a) and residuals (b).Plate P50N100, experiment with added noise amplitude equal to �0.5�C.
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In the framework of KLD techniques, a common filter consists in replacing raw
temperature data ~Tðx, tÞ by a low-dimensional KLD approximation ~Trðx, tÞ (Equation
(44)). Average temperatures are thus calculated using ~Trðx, tÞ instead of raw data.
However, if the dimension r of the approximation is too high, the filtering effect becomes
negligible; on the contrary, if the dimension is too low, significant bias can be induced. In
the studied experiment (P50N100, noise amplitude¼�0:5�C), a 3D approximation is
required to observe interesting reduction of the noise amplitude. However, it leads to
highly biased average temperatures.Another filtering method is hence proposed. It consists in (a) identifying significant
states with regard to noise: ~z1ðtÞ to ~zrðtÞ; (b) filtering them as explained below: zf1ðtÞ
to zfrðtÞ; (c) replacing ~Tðx, tÞ by the r-dimensional approximation Tf ðx, tÞ ¼Pm¼1,...,r VmðxÞzfmðtÞ; and (d) calculating average temperatures from filtered temperature
data Tf ðx, tÞ.Applying transformation TðtÞ ¼ VZðtÞ to Equation (42) with fðtÞ ¼ 0 (relaxation/
observation period), and taking into account that V0V ¼ I, it follows that the states
behaviour is governed by a model of the form: _ZðtÞ ¼ FZðtÞ. The following r-dimensional
black-box state model is hence proposed for representing and filtering noise-corrupted
states:
_Zf ðtÞ ¼ BZf ðtÞ þ KeðtÞ
~ZrðtÞ ¼ Zf ðtÞ þ eðtÞð45Þ
where ~ZrðtÞ ¼ ½ ~z1ðtÞ ~z2ðtÞ � � � ~zrðtÞ �0 and eðtÞ represents added white noise. Filtered
states Zf ðtÞ can be obtained by fitting model (45) on ~z1ðtÞ to ~zrðtÞ data. The well-known
iterative prediction-error minimization algorithm [44] can be used for fitting.In the studied experiment (P50N100, noise amplitude¼�0:5�C), only states ~z1ðtÞ to
~z5ðtÞ are significant with regard to noise (see Table 3). They are represented in Figure 7(a)
and (b). Filtered states zf1ðtÞ to zf5ðtÞ (symbols), as well as free-noise states z1ðtÞ to z5ðtÞ
(continuous lines), are depicted in Figure 7(c) and (d). Average temperatures calculated
from filtered temperature data Tf ðx, tÞ are represented in Figure 8(a), while Figure 8(b)
depicts the differences between theoretical �TiðtÞ (i ¼ 1, 2) values (calculated from free-noise
primary temperature data) and average temperatures calculated from Tf ðx, tÞ. The
efficiency of the applied filtering process can be appreciated when comparing such results
with data in Figure 6(a) and (b). Residuals are now less than �0:6� 10�3�C (Figure 8b),
while the noise amplitude of average temperatures before filtering is �0:015�C (Figure 6b).
However, residuals do not behave this time as a noise. Average temperatures calculated
from Tf ðx, tÞ are hence slightly biased.
4.5. Thermal properties estimation
The estimation of the parameters �i and �i (i ¼ 1, 2) has been carried out for the whole set
of experiments (9 microstructures� 3 level of noise¼ 27 experiments) by the method
described in section 3. Equations (28) and (29) with p ¼ 2 allow estimating �i parameters,
while Equations (31)–(33) with r ¼ p ¼ 2 provide �i estimates. Parameters �i have been
estimated two times: the first one using raw temperature data for calculating average
temperatures, and the second one using the filtering process described in section 4.4.
1164 E. Palomo Del Barrio et al.
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Results achieved are summarized in Figures 9 and 10, where bias and/or 95% confidence
intervals for estimated parameters are represented:
bias ð%Þ ¼ ðp� pÞ=p� 100,
uncertainity ð%Þ ¼ 1:96ffiffiffiffiffiffiffiffiffiffiffiffiffivarðpÞ
p=p� 100, p ¼ �i,�i
ð46Þ
Squares represent the results achieved when parameters �i are estimated using average
temperatures coming directly from raw temperature data (without filtering), whereas filled
circles correspond to the results obtained when parameters �i are estimated on filtered
average temperatures. It can be seen that
– Both, the bias and the uncertainty of the estimated diffusivities are strongly
correlated with, respectively, the bias and the uncertainty of the estimated �iparameters (Figure 9a and b). The quality of the estimated diffusivities is mainly
controlled by the quality of �i estimates, the morphology of the tested plates
appearing as a secondary parameter.– The uncertainty of the estimated parameters reduces when the quality of primary
temperature data increases. Figure 9(c) represents the uncertainty of the estimated
diffusivities against the signal/noise ratio of the temperature data (SN index,
Equation (43)).
Figure 7. Time behaviour of the states (plate P50N100, noise amplitude¼�0.5�C): (a–b) ~z1ðtÞ to~z5ðtÞ calculated from the raw temperature data; (c–d) Filtered states zf1ðtÞ to zf5ðtÞ (symbols) andtheoretical states z1ðtÞ to z5ðtÞ (continuous lines).
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– As shown in Figure 9(c), a significant reduction of the uncertainty of theestimated diffusivities is achieved when parameters �i are estimated using filteredaverage temperatures.
– For estimations carried out with raw average temperatures, the bias of theparameters tends to be reduced when increasing the quality of the primarytemperature data (Figure 9d, squares). On the contrary, for estimations carriedout with filtered average temperatures, the bias seems to be independent of thetemperature data quality (Figure 9d, filled circles).
– For bad quality temperature data (SN< 50), there is no significant difference interms of bias between the estimations carried out with either raw or filteredaverage temperatures (Figure 9d). On the contrary, when the quality of thetemperature data improves (SN> 50), the bias of the estimations carried out usingfiltered average temperatures is globally higher.
– Because the bias is as important as the uncertainty to judge about the quality ofthe results, Figure 10 shows the maximum between bias and uncertainty in the
Figure 8. Average temperature of phases 1 and 2 (plate P50N100, experiment with added noiseamplitude equal to �0.5�C): (a) calculated from the filtered states and the corresponding 5Dapproximation of the thermal field; (b) residuals.
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Figure 9. Summary of the estimations results: (a) parameters �i uncertainty vs. �i parametersuncertainty; (b) bias of �i estimations vs. bias of estimated �i parameters; (c) uncertainty of �iestimations vs. signal/noise of raw temperature data; (d) bias of �i estimations vs. signal/noise of rawtemperature data.
Figure 10. Summary of the estimations results. Maximum (�i-bias, �i-uncertainty) against signal/noise of raw temperature data.
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estimated diffusivities as a function of the quality of the temperature data. It canbe seen that using filtered average temperatures is not advantageous whenprimary temperature data are of very high quality (SN> 1000, noiseamplitude¼�0.02�C).
Tables 4–6 include estimated values for parameters �i and �i, as well as thecorresponding bias and uncertainty (Equation (46)). For high-quality temperaturedata (SN> 1000, noise amplitude¼�0.02�C) the estimations have been carried outwithout filtering average temperatures (Table 4), while filtering has been appliedfor medium and bad quality temperature data (Tables 5 and 6). Bold characters havebeen used to identify results with either �i-bias or �i-uncertainty greater than 2%: onecase in Table 4 and four cases in Table 5. It must be noticed that the worst results achieved(plates P50N100 and P50N1000 with noise amplitude equal to �0.5�C) correspond tothe experiments with lowest quality temperature data, those with SN index rangingfrom 0.9 to 2.
Table 4. Estimated values for the plates thermal parameters using temperature data of high quality(noise amplitude¼�0.02�C). True values are �1 ¼ 1:5152� 10�7 m2 s�1, �2 ¼ 3:0303� 10�7 m2 s�1,�1 ¼ 0:0152 s�1, and �2 ¼ 0:0303 s�1.
Geometry
�1 (s–1)Bias (%)
Uncertainty (%)
�2 (s–1)Bias (%)
Uncertainty(%)
�1 ð�10�6 m2 s�1Þ
Bias (%)Uncertainty
(%)
�2 ð�10�6 m2 s�1Þ
Bias (%)Uncertainty
(%)
P20N100 0.0152 0.0303 0.1515 0.30270.0078 0.0607 0.0261 0.09540.1150 1.8602 0.2784 1.7373
P50N100 0.0152 0.0303 0.1515 0.30310.0007 0.0115 0.0210 0.03441.1418 1.1667 0.7207 1.6333
P70N100 0.0151 0.0303 0.1514 0.30290.0339 0.0303 0.0928 0.02850.3155 0.2539 0.2400 0.3430
P20N500 0.0152 0.0303 0.1515 0.30320.0009 0.0470 0.0020 0.04560.0471 0.9833 0.1480 0.9045
P50N500 0.0152 0.0303 0.1514 0.30290.0011 0.0213 0.0779 0.05790.0076 0.4304 0.1472 0.2999
P70N500 0.0151 0.0303 0.1514 0.30290.0112 0.0043 0.0485 0.03290.3156 0.0613 0.1483 0.2381
P20N1000 0.0151 0.0303 0.1515 0.30320.0110 0.0666 0.0078 0.07000.2038 1.0537 0.1730 1.1111
P50N1000 0.0151 0.0303 0.1514 0.30340.1036 0.0776 0.0648 0.12372.0016 1.3945 1.2460 2.2260
P70N1000 0.0151 0.0303 0.1514 0.30310.0759 0.0670 0.1037 0.03910.6293 0.5142 0.6458 0.5238
1168 E. Palomo Del Barrio et al.
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5. Conclusion
A new method for estimating the thermal diffusivity of the whole set of homogeneous
phases of a composite media has been proposed. The method is based on the use of KLD
techniques in association with one single infrared thermography experiment. Orthogonal
properties of KLD eigenfunctions and states allow achieving simple estimates of thermal
diffusivities. It is proven that diffusivities can be estimated without explicit knowledge of
variables and parameters describing heat transfer at the interfaces (i.e. thermal
conductivities, thermal contact resistances). Indeed, diffusivity estimates only depend on
some few KLD eigenelements. As a result, the proposed method is an attractive
combination of parsimony and robustness to noise. Last feature comes from the ability of
KLD to amplify signal/noise ratios when truncated. Moreover, it is shown that free-noise
KLD eigenfunctions are achieved when spatially uncorrelated measurement noise applies.
This is also interesting because thermal diffusivity estimates involve spatial derivatives of
the eigenfunctions.
Table 5. Estimated values for the plates thermal parameters using temperature data ofmedium quality (noise amplitude¼�0.1�C). True values are �1 ¼ 1:5152� 10�7 m2 s�1,�2 ¼ 3:0303� 10�7 m2 s�1, �1 ¼ 0:0152 s�1, and �2 ¼ 0:0303 s�1.
Geometry
�1 (s–1)Bias (%)
Uncertainty(%)
�2 (s–1)Bias (%)
Uncertainty(%)
�1 ð�10�6m2s�1Þ
Bias (%)Uncertainty
(%)
�2 ð�10�6m2s�1Þ
Bias (%)Uncertainty
(%)
P20N100 0.0152 0.0297 0.1520 0.29810.1069 1.8351 0.3206 1.62630.1314 1.6678 0.2816 1.5399
P50N100 0.0150 0.0305 0.1510 0.30650.7686 0.6889 0.3277 1.13840.9407 0.8497 0.5178 1.3272
P70N100 0.0149 0.0306 0.1502 0.30761.3827 0.9855 0.8602 1.52070.3316 0.2524 0.2249 0.3771
P20N500 0.0151 0.0304 0.1516 0.30390.0258 0.2304 0.0291 0.28520.0800 1.0629 0.1477 1.0150
P50N500 0.0151 0.0308 0.1509 0.30640.0679 1.4978 0.4349 1.12490.0425 0.4631 0.1483 0.3610
P70N500 0.0150 0.0303 0.1511 0.30540.9165 0.1231 0.2660 0.78060.3107 0.0559 0.1032 0.2759
P20N1000 0.0151 0.0307 0.1513 0.30790.3105 1.4562 0.1658 1.60360.2063 0.9854 0.1452 1.0782
P50N1000 0.0148 0.0308 0.1495 0.31152.3856 1.6728 1.3005 2.8100
1.9545 1.4090 1.2287 2.2824P70N1000 0.0150 0.0306 0.1500 0.3067
1.2844 0.9519 1.0194 1.22341.0089 0.7667 0.9126 0.9112
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Compared to other existing estimation approaches, the proposed KLD-based method
allows dealing efficiently with large amount of rather noised temperature data, as those
usually provided by IRT experiments. Indeed, the method does require neither initial guess
for thermal diffusivities nor explicit knowledge of the thermal parameters defining heat
exchanges at the interfaces.The numerical tests carried out show the effectiveness of the proposed method.
However, some further research is still required to complete the development and to
establish limits of application. Open topics to be mentioned are those related to the
maximum uncertainty allowed in the knowledge of the sample microstructure, the
maximum number of phases that can be simultaneously identified, the minimum spatial
resolution required, the parametric variations allowed, etc. Actual experiments have to be
carried out too. Moreover, some extensions of the method can be envisioned as well, as its
extension for estimating thermal resistances at the interfaces or its extension for
functionally graded materials characterization.
Table 6. Estimated values for the plates thermal parameters using temperature data of bad quality(noise amplitude¼�0.5�C). True values are �1 ¼ 1:5152� 10�7m2s�1, �2 ¼ 3:0303� 10�7m2s�1,�1 ¼ 0:0152s�1, and �2 ¼ 0:0303s�1.
Geometry
�1 (s–1)Bias (%)
Uncertainty(%)
�2 (s–1)Bias (%)
Uncertainty(%)
�1 ð�10�6 m2 s�1Þ
Bias (%)Uncertainty
(%)
�2 ð�10�6 m2 s�1Þ
Bias (%)Uncertainty
(%)
P20N100 0.0152 0.0298 0.1519 0.29860.0502 1.6725 0.2782 1.45800.4285 6.3805 0.9517 5.9673
P50N100 0.0151 0.0305 0.1513 0.30570.4741 0.4903 0.1389 0.88714.9815 4.9273 2.8108 7.4119
P70N100 0.0149 0.0306 0.1503 0.30801.4274 1.0169 0.8085 1.65381.4652 1.3220 1.1581 1.7388
P20N500 0.0152 0.0303 0.1516 0.30350.0073 0.0573 0.0781 0.13980.2119 3.7944 0.4973 3.5828
P50N500 0.0151 0.0307 0.1509 0.30630.0764 1.3851 0.3737 1.08050.2005 2.1840 0.6528 1.6983
P70N500 0.0150 0.0303 0.1513 0.30520.7193 0.0937 0.1222 0.70011.4652 0.3295 0.5282 1.3025
P20N1000 0.0151 0.0307 0.1514 0.30720.2083 1.2768 0.1027 1.38961.1131 5.5136 0.8028 5.9665
P50N1000 0.0148 0.0310 0.1491 0.31352.5875 2.1786 1.5631 3.445910.1518 7.6746 6.3741 12.1658
P70N1000 0.0150 0.0305 0.1509 0.30550.6748 0.5498 0.4153 0.81734.1677 3.4269 4.0315 3.7592
1170 E. Palomo Del Barrio et al.
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Appendix
Let us consider the observer ~ym:
~ym ¼ D ~z2m=ð2 ~�2mÞ þXpi¼1
ami�i ðA:1Þ
with
D ~z2m ¼ ðzmðtf Þ þ �mðtf ÞÞ2� ðzið0Þ þ �mð0ÞÞ
2
�mðtÞ ¼
Z�
"ðx, tÞVmðxÞdx
ami ¼ VmðxÞ,VmðxÞ� �
�i
~�2m ¼ �2m þ �
2"
8>>>>>>>><>>>>>>>>:
ðA:2Þ
As already mentioned, the measurements noise "ðx, tÞ is assumed to be spatially uncorrelated.Indeed, we consider "ðx, tÞ to be stationary and white. Hence
8x, 8t E½"ðx, tÞ� ¼ 0 & E½"2ðx, tÞ� ¼ varð"Þ
8ðx1, x2Þ, 8ðt1, t2Þ E½"ðx1, t1Þ"ðx2, t2Þ� ¼ �ðx1 � x2Þ�ðt1 � t2Þvarð"Þ
ðA:3Þ
where E represents the expectation operator and varð"Þ is the variance of the noise. Taking intoaccount the statistical properties of "ðx, tÞ, as well as the orthonormality of eigenfunctions, it follows:
8m E½�mðtf Þ� ¼ E½�mð0Þ� ¼ 0
8m E½�2mðtf Þ� ¼ E½�2mð0Þ� ¼ varð"Þ
8m E½�mðtf Þ�mð0Þ� ¼ 0
8 ðm, kÞ E½�mðtf Þ�kðtf Þ� ¼ E½�mð0Þ�kð0Þ� ¼ 0
8 ðm, kÞ E½�mðtf Þ�kð0Þ� ¼ E½�mð0Þ�kðtf Þ� ¼ 0
8>>>>>>>><>>>>>>>>:
ðA:4Þ
We suppose that the statistical moments of higher order of the variables �mðtÞ, �kðtÞ, . . . can beneglected. Besides, we assume that �i is unbiased, so that E½�i� ¼ �ið8iÞ. We note covðbÞ thecovariance matrix of parameters �i.
From equations above, it can be easily proven that
E½ ~ym� ¼Dz2m
2�2m½1þ ð�2" =�
2mÞ�þXpi¼1
ami�i ðA:5Þ
varð ~ymÞ ¼zmðtf Þ
2þ zmð0Þ
2
�4m½1þ ð�2" =�
2m�
2varð"Þ þ amcovðbÞa
0m ðA:6Þ
covð ~ym ~ykÞ ¼ amcovðbÞa0k
ðA:7Þ
with am ¼ ½ am1 am2 � � � amp � and ak ¼ ½ ak1 ak2 � � � akp �.Equation (a.5) shows that ~ym provides a biased approximation of ym that could lead to biased
estimations of parameters �i. The necessary condition for the bias becomes negligible is �2" �2m. Insuch a case, the mean value and the variance of the observers can be approached by
E½ ~ym� ¼Dz2m2�2mþXpi¼1
ami�i ¼ ym ðA:8Þ
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ovem
ber
2014
varð ~ymÞ ¼zmðtf Þ
2þ zmð0Þ
2
�4mvarð"Þ þ amcovðbÞa
0m
ðA:9Þ
Provided that �2" �2p (almost unbiased observers), the statistical properties of the diffusivitiesestimates defined by Equation (33) can be hence assumed to be as follows:
E½a� ¼ a a0 ¼ ½�1 �2 � � � �p � ðA:10Þ
E½ða� aÞ2� ¼M#½Dvarð"Þ þ AcovðbÞA0�ðM#Þ0 ðA:11Þ
where Dð p� pÞ is a diagonal matrix whose elements are dmm ¼ ðzmðtf Þ2þ zmð0Þ
2Þ=�4m, and Að p� pÞ
is the matrix whose elements are ami ¼ VmðxÞ,VmðxÞ� �
�i.
1174 E. Palomo Del Barrio et al.
Dow
nloa
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by [
Que
ensl
and
Uni
vers
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f T
echn
olog
y] a
t 05:
50 2
1 N
ovem
ber
2014