thermal characterization of materials using karhunen–loève decomposition techniques – part ii....

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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

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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

1146 E. Palomo Del Barrio et al.

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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

Inverse Problems in Science and Engineering 1147

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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


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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).


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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.


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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


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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


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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" =�

2mÞ�

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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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.


Dow

nloa

ded

by [

Que

ensl

and

Uni

vers

ity o

f T

echn

olog

y] a

t 05:

50 2

1 N

ovem

ber

2014

thermal characterization of materials using karhunen–loève decomposition techniques – part ii....

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