International Journal of Thermal Sciences 132 (2018) 486–497

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International Journal of Thermal Sciences

journal homepage: www.elsevier.com/locate/ijts

Detection of contact failures with the Markov chain Monte Carlo method by T
using integral transformed measurements
Luiz A.S. Abreua,b, Helcio R.B. Orlandeb,∗, Marcelo J. Colaçob, Jari Kaipioc,d, Ville Kolehmainend,
César C. Pachecob, Renato M. Cottab
a
Department of Mechanical Engineering and Energy, Polytechnic Institute/IPRJ, Rio de Janeiro State University – UERJ, Rua Bonfim 25, Nova Friburgo, RJ, 28625-570,
Brazil
b
Department of Mechanical Engineering, Politecnica/COPPE, Federal University of Rio de Janeiro – UFRJ, Cid. Universitária, Cx. Postal: 68503, Rio de Janeiro, RJ,
21941-972, Brazil
c
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland Mail Centre, Auckland, 1142, New Zealand
d
Department of Applied Physics, University of Eastern Finland, P.O. Box 1627, 70211, Kuopio, Finland

A B S T R A C T

This work deals with the solution of an inverse heat conduction problem aiming at the detection of contact
failures in layered composites through the estimation of the contact conductance between the layers. The spa-
tially varying contact conductance is estimated using a Bayesian formulation of the problem and a Markov chain
Monte Carlo method, with infrared camera measurements of the transient temperature field on the surface of the
body. The inverse analysis is formulated using a data compression scheme, where the temperature measurements
are integral transformed with respect to the spatial variable. The present approach is evaluated using synthetic
measurements and experimental data from controlled laboratory experiments. It is shown that only few trans-
formed modes of the data are required for solving the inverse problem, thus providing substantial reduction of
the computational time in the Markov chain Monte Carlo method, as well as regularization of the ill-posed
problem.

1. Introduction problems is often prohibitive. Therefore, in this paper, we extend [3,4]

to accommodate a data compression scheme. The temperatures mea-
The detection of internal failures in materials is a subject of ex- sured with an infrared camera are spatially compressed through the
tensive research due to its importance in several fields, for example, in integral transformation with eigenfunctions related to the actual phy-
structural health monitoring [1–14]. With the recent advancement and sical problem. Only a few transformed modes are then used in the in-
practical applications (e.g., in the aeronautic, space and petroleum in- verse analysis and the forward model is formulated directly in terms of
dustries) of composites consisting of layers of different materials, the transformed (compressed) temperatures. A similar data compres-
nondestructive and noninvasive methods for the detection of adhesion sion approach was used in Refs. [16,17] for the estimation of spatially
failures between the composite layers have been developed [2–15]. varying properties in a one-dimensional problem and is applied here
Heat transfer techniques can be found among these methods, by using with a two-dimensional transformation. The data compression applied
qualitative [1,2,7,13,14] as well quantitative analyses based on the in this work not only reduces the computational time required for the
solutions of inverse problems [3–12]. Markov chain Monte Carlo method, but also provides regularization for
In our previous works [3,4], the contact failures were detected the inverse problem [18]. Conceptually, the integral transform data
through the estimation of the contact conductance between the layers compression scheme falls within the broader class of orthogonal de-
of different materials from measurements of the transient temperature composition methods, such as POD – Proper Orthogonal Decomposi-
over the surface of the composite. The approach used in Refs. [3,4] was tion, Principal Component Analysis, Karhunen–Loeve decomposition
based on a Bayesian formulation of the inverse problem, with a total and Truncated Singular Value Decomposition [19–25]. The accuracy of
variation prior model for the unknowns and using the Markov chain the proposed methodology is examined with simulated measurements,
Monte Carlo (MCMC) method for the inference of the Bayesian model. as well as with actual thermographic data obtained with controlled
The computational complexity of MCMC with large dimensional laboratory experiments [9], involving samples manufactured with

∗
Corresponding author.
E-mail address: helcio@mecanica.coppe.ufrj.br (H.R.B. Orlande).

https://doi.org/10.1016/j.ijthermalsci.2018.06.006
Received 2 July 2017; Received in revised form 27 March 2018; Accepted 4 June 2018
1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.
L.A.S. Abreu et al. International Journal of Thermal Sciences 132 (2018) 486–497

Nomenclature ϕ eigenfunction given by equation (4.a)

γ eigenvalue given by equation (6.b)
a,b,c dimensions of the plate φ eigenfunction given by equation (4.b)
Bic(X,Y) dimensionless contact conductance θ dimensionless estimated temperature
D total number of measurements Θ vector of estimated dimensionless temperatures
hc(x,y) thermal contact conductance τ dimensionless time
k thermal conductivity ψ dimensionless measured temperature
kmax number of transient measurements Ψ vector of measured temperatures
q(x,y,t) heat flux imposed on the top boundary
M number of elements in the spatial grid Subscripts
P vector of parameters
T temperature 1,2 plates 1 and 2, respectively
To initial temperature in the medium i,j order of the eigenquantities in the X and Y directions,
X,Y,Z dimensionless spatial coordinates respectively
Z1 dimensionless position of the contact interface ref reference values
|A| determinant of matrix (A)
Superscripts
Greeks
∗
dimensionless thermophysical properties and heat flux
α thermal diffusivity _ transform in the X direction
β eigenvalue given by equation (6.a) ∼ transform in the Y direction

designed contact failures of different formats. ∂τ θ2 (X , Y , τ ) = α 2∗ ∇2 θ2 in 0 < X < A, 0 < Y < B, Z1 < Z < 1, for τ > 0
(2.b)
2. Physical problem and mathematical formulation ∂Z θ1 = 0 at Z = 0 in 0 < X < A, 0 < Y < B and τ > 0 (2.c)

The physical problem considered here involves heat conduction k1∗ ∂Z θ1 = k 2∗ ∂Z θ2 at Z = Z1 in 0 < X < A, 0 < Y < B and τ > 0 (2.d)
through a plate with two layers, heated through its top surface by a heat k1∗ ∂Z θ1 = Bic (X , Y )[θ2 − θ1] at Z = Z1 in 0 < X < A, 0 < Y < B and τ > 0 (2.e)
flux q (x,y,t), as illustrated by Fig. 1. The bottom surface of the plate is
thermally insulated and heat transfer is assumed negligible through its k 2∗ ∂Z θ2 = q∗ (X , Y , τ ) at Z = 1 in 0 < X < A, 0 < Y < B and τ > 0
lateral surfaces. The plate is initially at a uniform temperature, To, and (2.f)
the physical properties of each layer are assumed homogeneous and not
dependent on temperature. The length and width of the plate are a and ∂X θ1 = ∂X θ2 = 0 at X= 0, 0< Y < B , 0< Z < 1, and τ > 0 (2.g)
b, respectively, while its thickness is denoted by c. A spatially dis- ∂X θ1 = ∂X θ2 = 0 at X = A, 0 < Y < B, 0 < Z < 1, and τ > 0 (2.h)
tributed contact resistance between the two adjacent layers is modeled
by a contact conductance hc (x,y) [26]. For the inverse analysis, mea- ∂Y θ1 = ∂Y θ2 = 0 at Y = 0, 0 < X < A, 0 < Z < 1, and τ > 0 (2.i)
surements of the temperature at the top (heated) surface of the plate,
∂Y θ1 = ∂Y θ2 = 0 at Y = B, 0 < X < A, 0 < Z < 1, and τ >0 (2.j)
obtained with an infrared camera, are available.
The mathematical problem is written in dimensionless form by θ1 = θ2 = 0 at τ = 0 in 0 in 0 < X < A, 0 < Y < B, 0 < Z < 1 (2.k)
using the following variables:
where the contact interface is located at Z = Z1.
T (x , y, z , t ) − To c
θ (X , Y , Z , τ ) = q∗ (X , Y , τ ) = q (x , y, t )
To kref To 3. Forward problem
(1.a,b)
The forward (direct) problem associated with the formulation given
k α by equation (2.a-k) involves the determination of the temperature fields
k∗ = α∗ = θ1 (X , Y , Z , τ ) and θ2 (X , Y , Z , τ ) . The direct problem corresponding to
kref αref (1.c,d)
the transformed data is solved here by using a hybrid analytical-
x
X= c
y
Y= c
z
Z= c
c
Z1 = c1
a
A= c
b
B= c (1.e-j)

αref t hc (x , y ) c
τ= Bic (X , Y ) =
c2 kref (1.k,l)

and we obtain:

∂τ θ1 (X , Y , τ ) = α1∗ ∇2 θ1 in 0 < X < A, 0 < Y < B, 0 < Z < Z1, for τ > 0
(2.a)
Fig. 1. Physical problem.

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numerical approach, based on the Generalized Integral Transform θ1 (X , Y , Z , τ ) and θ2 (X , Y , Z , τ ) were within a user prescribed error
Technique (GITT) [4,27–33] and finite differences [26,34]. tolerance. The double summations in equation (7.f) were rearranged in
Problem (2) is integral transformed along the X and Y directions by the form of a single summation, by using the two-dimensional eigen-
using the following transform – inversion formulae pair: values βi2 + γ j2 in increasing order [33].
A B The numerical accuracy of the solution of the direct problem was
Transform: θ͠ 1,2 (βi , γj, Z , τ ) = ∫ ∫ ϕi φj θ1,2 (X , Y , Z , τ ) dY dX verified in our previous works [3,4]. In this study, the inverse analysis
X =0 Y =0 (3.a) in carried out with compressed data using integral transformation. In
∞ ∞
the following section, the solution of the direct problem in the trans-
Inversion formula: θ1,2 (X , Y , Z , τ ) = ∑ ∑ ϕi φj θ͠ 1,2 (βi, γj, Z , τ ) formed domain will be used for the definition of the likelihood function
i=0 j=0 (3.b) required for the solution of the inverse problem.

where the normalized eigenfunctions are obtained from Ref. [20]:

4. Inverse problem
cos(Xβi ) cos(Yγj )
ϕi = φj = for i = 0, …, ∞ and j = 0, …, ∞ The focus of this work is the detection of contact failures between
Ni Nj (4.a,b)
layers 1 and 2 (see Fig. 1) by identifying the dimensionless contact
with normalization integrals conductance Bic(X,Y). For perfect contact, Bic(X,Y) is sufficiently large
A to characterize temperature continuity at the interface, while a contact
Ni = A for i = 0, Ni = for i = 1, …, ∞ failure is detected by values of the contact conductance that tend to
2 (5.a)
zero. Simulated and real transient temperature measurements obtained
B with an infrared camera over the top surface Z = 1 will be used in an
Nj = B for j = 0, Nj = for j = 1, …, ∞
2 (5.b) inverse analysis for the identification of Bic(X,Y) (see below the section
and eigenvalues with the description of the experimental apparatus).
The inverse problem is solved through statistical inference on the
iπ posterior distribution of the model parameters, within the Bayesian
βi = for i = 0, …, ∞
A (6.a)
framework of statistics [35–38]. In the Bayesian framework, all the
jπ unknown parameters are considered as random variables and modeled
γj = for j = 0, …, ∞ in terms of statistical distributions (priors) that represent their in-
B (6.b)
formation and uncertainty before the experiment is performed. The
Hence, the following system of coupled equations for i = 0, …, ∞ priors are then combined with the information provided by the mea-
and j = 0, …, ∞ results from the transformation of problem (2): surements taken during the experiment, which is also modelled in the
∼ ∼ form of a statistical distribution (likelihood), in order to obtain the
1 ∂θ1 (βi , γj, Z , τ ) ∂2θ1 ∼
= − (βi2 + γ j2 ) θ1 for τ > 0, in 0< Z < Z1 posterior distribution of the parameters by using Bayes' theorem
α1∗ ∂τ ∂Z 2 [35–38]:
(7.a)
π (P) π (Ψ P)
∼ ∼ πposterior (P) = π (P Ψ) =
1 ∂θ2 (βi , γj, Z , τ ) ∂2θ2 ∼ π (Ψ) (9)
= − (βi2 + γ j2 ) θ2 for τ > 0, in Z1 < Z < 1
α 2∗ ∂τ ∂Z 2 where πposterior(P) is the posterior probability density, π(P) is the prior
(7.b) density, π(Ψ|P) is the likelihood function and π(Ψ) is the marginal
probability density of the measurements, which plays the role of a
∂θ͠ 1 normalizing constant.
k1∗ = 0 at Z = 0 and τ > 0
∂Z (7.c) The posterior (9) is a probability density model of the inverse pro-
blem, defined on a high dimensional space. In this study, inference on
∂θ͠ 2 ∼
k 2∗ = di . j at Z= Z1 and τ > 0 the posterior model is carried out by sampling based on Markov chain
∂Z (7.d)
Monte Carlo (MCMC) integration that is implemented through the
Metropolis-Hastings' algorithm [35–46]. In the Metropolis-Hasting al-
∂θ͠ 1 ∂θ͠
k1∗ = k 2∗ 2 at Z = Z1 and τ > 0 gorithm, a proposal distribution p (P∗,P(t−1)), which is used to draw a
∂Z ∂Z (7.e)
new candidate P∗ given the parameters in the current state P(t−1) of the
∂θ͠ 1,(i, j) ∞ ∞ Markov chain, must be selected by the user. The proposal distribution
k1∗
∂Z
= ∑ ∑ Ai,j,m,u [θ͠ 2,(m,u) − θ͠ 1,(m,u) ] at Z = Z1 and τ > 0 can be, for example, random walk processes or independent moves
m=0 u=0 (7.f)
based on the prior, but adaptive schemes are also available in the lit-
∼∗ ∼∗ erature [35–38]. Once the proposal distribution has been selected, the
θ1 = θ2 = 0 at τ = 0 in 0 < Z < 1 (7.g)
Metropolis-Hastings algorithm can be described by the following steps
where [35–46]:
A B
∼ 1. Sample a Candidate Point P∗ from the proposal distribution p
d i, j = ∫ ∫ q∗ (X , Y , τ ) ϕi φj dY dX (P∗,P(t −1)).
X =0 Y =0 (8.a)
2. Calculate the acceptance factor:
A B
Ai, j, m, u = ∫ ∫ ϕi ϕm φj φu Bic (X , Y ) dY dX R = min ⎡1 ,
π (P∗ Ψ) p (P(t − 1) , P∗) ⎤
X =0 Y =0 (8.b) ⎢ π (P(t − 1) Ψ) p (P∗, P(t − 1) ) ⎥
⎣ ⎦ (10)
The system of infinite coupled partial differential equation (7.a-g)
for the transformed fields θ͠ 1 (βi , γj, Z , τ ) and θ͠ 2 (βi , γj, Z , τ ) was then 3. Generate a random value U that is uniformly distributed on (0,1).
discretized implicitly with finite differences along the Z direction [34]. 4. If U ≤ R, set P(t) = P∗. Otherwise, set P(t) = P(t−1).
The numerical solution was obtained by truncating the infinite system 5. Return to step 1.
to a finite number of transform modes. The number of modes, as well as
the finite difference grid size, was selected so that the computed fields In this way, a sequence of random samples is generated to represent

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the posterior distribution and inference on this distribution is obtained Markov chain Monte Carlo method for the solution of the inverse pro-
from inference on the samples {P(1), P(2), …, P(n)}. We note that values blem (acquiring an adequate number of samples with the MCMC chain)
of P(i) must be ignored while the chain has not converged to equilibrium might not be feasible for general purpose computers. The use of fast
(the burn-in period). reduced mathematical models for the solution of the direct problem in
In this work the dimensionless contact conductance Bic(X,Y) is the inverse analysis, instead of the complete model that accurately re-
modeled as piecewise constant in each pixel of a grid with center points presents the physics of the problem, can be formally treated within the
(XI,YJ), where XI = IΔX, YJ = JΔY, I = 1, …, If, J = 1, …, Jf, and with Bayesian framework by modeling the approximation errors as Gaussian
grid spacing given by ΔX = A/If and ΔY = B/Jf. The total number of random variables and modifying the likelihood, such as in the
estimated points, which cover the spatial domain 0 < X < A and Approximation Error Model [35,47–52]. Sampling techniques like the
0 < Y < B , is then M = If Jf. The applied heat flux is analogously Delayed Acceptance Metropolis-Hastings algorithm [53,54] have also
modeled as piecewise constant on a similar discretization of the top been developed in order to expedite the application of Markov chain
surface. Hence, the vector of unknown parameters for the inverse Monte Carlo methods with the use of reduced models. Besides the three-
analysis is given by: dimensional nature of the heat conduction problem in this work, the
large computational times for the solution of the present inverse pro-
PT = [Bic1, Bic 2, …,BicM , q1∗, q2∗, …,qM
∗
, α1∗ , α 2∗ , k1∗ , k 2∗ ] (11) blem result from the number of measurements made available by the
The priors used for the parameters will be discussed below. The infrared camera, which can provide experimental data with high spatial
vector containing the measured temperatures is written as: resolution and high frequency. Therefore, instead of applying model
⎯→
⎯ ⎯→
⎯ ⎯→
⎯ reduction and using either the Approximation Error Model or the De-
ΨT = ( ψ1 , ψ2 , ... , ψkmax ) (12.a) layed Acceptance Metropolis-Hastings algorithm, a data compression
⎯→
⎯ approach is applied here in order to reduce the computational work
where ψk contains the measured temperatures of each of the M grid needed for the calculation of the likelihood function. We note, however,
elements at time tk, k = 1, …, kmax, that is, that with a drastic model reduction, the estimates might turn out to be
⎯⎯⎯→ misleading (in the sense of predicted posterior covariance). In such a
ψk = (ψk1 , ψk 2 , ... , ψkM ) for k = 1,..., kmax (12.b) case, the Approximation Error Models might be called for.
so that we have D = M kmax measurements in total. Data compression in this work is performed by transforming the
Temperature measurements obtained with an infrared camera have experimental data (temperatures at each pixel recorded by the infrared
errors that can be modeled as Gaussian, with zero mean and constant camera) with the same integral transform that is used in the forward
standard deviation σ [18,44]. Therefore, the likelihood function can be model, equation (3.a), that is,
expressed as [3,4,35–46]: A B
∼
1 [Ψ−Θ (P)]T [Ψ−Θ (P)]
ψ (βi , γj, τ ) = ∫ ∫ ϕi φj ψ (X , Y , τ ) dY dX
π (Ψ P) = (2πσ 2)−D /2 exp ⎧ − ⎫ X =0 Y =0 (14.a)
⎨
⎩ 2 σ2 ⎬
⎭ (13)
Such as for θ͠ 1 (βi , γj, Z , τ ) and θ͠ 2 (βi , γj, Z , τ ) , the transformed measured
where Θ (P) is the solution of the direct (forward) problem, given by ∼
equation (2.a-k) with vector P given by equation (11). data, ψ (βi , γj, τ ) , were ordered with increasing eigenvalue βi2 + γ j2 . The
The solution of the direct problem, Θ (P) , needs to be computed for number of transformed modes was selected as the same used for the
all states of the Markov chain, at each position and time that a mea- solution of problem (7. a-g), by also taking into account that the in-
surement is available. Typically, the number of states required for the version of (14. a) with
Markov chain to generate samples that appropriately represent the ∞ ∞
∼
posterior distribution is very large. Therefore, if the computational time ψ (X , Y , τ ) = ∑ ∑ ϕi φj ψ (βi, γj, τ )
for the solution of the direct problem is large, the application of the i=0 j=0 (14.b)

Fig. 2. Experimental apparatus: (a) Infrared camera, (b) Camera holder, (c) Radiating heater, (d) Sample, (e) Sample holder, (f) Data acquisition system, (g)
Microcomputer for data acquisition, (h) Plexiglass dome.

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must be represented up to desired accuracy by the truncated series. An

estimate for the number of eigenmodes can be computed based on the
related eigenvalues for any desired accuracy.
Since the transformation given by equation (14.a) is linear, and
using the fact that the standard deviation of the measurements is con-
∼
stant, the covariances of the transformed measurements ψ (βi , γj, τ ) are
then given by:
A B
σ͠ 2 (βi , γj, τ ) = σ 2 ∫ ∫ ϕi φj dY dX
X =0 Y =0 (15)
∼
With the transformed measurements obtained with equation (14.a), Ψ ,
and with the transformed estimated temperatures obtained with the
solution of the forward problem given by equation (7.a-g) at Z = 1 and
∼
each time that a measurement is available, Θ (P) , the likelihood func-
tion can be rewritten in the transformed domain as
∼ ∼ ∼ −1/2 1 ∼ ∼ T ∼−1 ∼ ∼
π (Ψ P) = (2π )−D /2 W {
exp − [Ψ −Θ (P)] W [Ψ −Θ (P)]
2 (16) }
∼ ∼
where W is the covariance matrix of Ψ , with elements given by equa-
tion (15).
∼ ∼
Vectors Ψ and Θ (P) are computed at a limited number of trans-
∼
formed modes, D , much smaller than the actual number of measure-
ments, D, thus resulting in substantial reduction of computational times
required for the solution of the inverse problem, as will be apparent
below. The integrals required for the transformation of the measure-
ments and their covariances were respectively computed numerically
(with the interpolation of ψ (X , Y , τ ) by cubic splines) and analytically.

5. Experiments

An experimental apparatus was constructed in order to validate the

approach for the detection of contact failures described above (see
Fig. 2). The experimental apparatus is composed of [9]: (a) Infrared
camera FLIR A325sc with 320 × 240pixels and measurement frequency
of 9 Hz; (b) Camera holder; (c) Radiating heater; (d) Sample; (e) Sample
holder; (f) Data acquisition system (Agilent 34970-A); (g) Micro-
computer for data acquisition; and (h) Plexiglass dome to reduce flow
over the sample from the air conditioning system. The whole system
was assembled over a table used for laser experiments, in order to ob-
tain precise positions that are maintained throughout the different ex-
periments. The radiating heater consisted of a bare electrical resistance
fixed with ceramic insulators inside an aluminum groove. The internal
and external surfaces of the lower part of the groove were coated with
graphite (Graphit 33, Kontact Chemie), in order to increase their ab-
sorptivity and emissivity, respectively, aiming at a uniform flux over
the sample. The samples where the detection of contact failures was
sought were located in a support made of plexiglas, styrofoam and
aluminum. This support was attached to the table, at the center of the
square heater, so that the sample orientation was maintained from one
experiment to another. The distances from the sample to the heater and
to the camera were kept as 100 mm and 400 mm, respectively. The
plexiglas plate that maintained the sample position and orientation also
served as thermal insulation for its lateral surfaces, while the styrofoam
block served as thermal insulation for the sample lower surface.
In order to validate the inverse problem approach of this work,
samples containing known controlled contact failures were manu-
factured by using square plates made of polymethyl methacrylate. The
plate dimensions were a = b = 0.04 m and c1 = c2 = 0.002 m. Grooves
with different shapes and depths ranging from 100 μm to 150 μm
(measured with a 3D Digital Microscope 3D Hirox model KH-8700)
were micro-machined in one of the plates. After cleaning, the plate
containing the groove was carefully attached to another plate without
Fig. 3. Manufactured plates: (a) Without contact failures, (b) With a circular any grooves by using chloroform. Hence, the contact between the plates
contact failure, (c) With a longitudinal contact failure. was ideally perfect, except at the location of the machined groove.
Samples containing a circular and a longitudinal contact failure are

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presented by Fig. 3. This figure also shows a sample made of one single actual measurements are used for the solution of the inverse problem.
plate, without any grooves, which was prepared for validation of the For this test-case, 96100 temperature measurements were available
method and for the estimation of the heat flux imposed by the radiating (experiment with duration of 10s, measurements with a frequency of
heater. The samples were all coated with a very thin graphite layer 10 Hz on a 31 × 31 grid). On the other hand, 50 modes were used for
(Graphit 33, Kontact Chemie). This coating served to increase and make the solution of the inverse problem with the likelihood function in the
more uniform the imposed heat flux over the sample surface, as well as transformed domain (see equation (16)), thus resulting on a data
to increase and control emissivity in the infrared range for accurate compression of 94.7%. Despite such large data compression through the
measurements with the infrared camera. integral transformation of the measurements, the spatial information
Temperatures measured with the infrared camera were recorded provided by the transformed modes was still sufficient to recover two
with the software FLIR ResearchIR™ at a rate of 9 frames per second. small contact failures, such as given by Fig. 4a,b.
Before the experiments were started, the equilibrium temperatures The reduction of the computation time required for the solution of
were recorded for 10 s in order to compute the standard deviation of the the inverse problem was substantial, when the likelihood function in
measurements, which was of 0.03 °C. the transformed domain was used. The solution of the inverse problem
using MCMC took 6.3 days with the regular likelihood given by equa-
6. Results and discussions tion (13), while the MCMC run using the likelihood in the transformed
domain took less than 2 h, resulting on a speedup of 79. For both cases,
Results obtained with simulated measurements, as well as with ac- the solutions were obtained with 30000 states in the Markov chains and
tual measurements in the experiment described above, are reported and by neglecting the first 10000 states (burn-in period). Computational
discussed in this section. For all cases, the prior for the dimensionless times refer to a FORTRAN code running on an Intel Core I7-2600 with
contact conductance was taken as a uniform improper distribution in 3.4 GHz clock and 16 Gb of RAM memory.
the form
6.2. Validation with actual measurements
1 , Bic (X , Y ) ≥ 0
π (Bic (X , Y )) = ⎧
⎨
⎩ 0 , Bic (X , Y ) < 0 (17) The thermophysical properties of the polymethyl methacrylate,
used for manufacturing the plates for the controlled experiments, as
where, the non-negativity constraint for the contact conductance was
described in the previous section, were measured by using the Flash
imposed. An upper bound for the prior of Bic (X , Y ) was not imposed
method (NETZSCH LFA-441). The measured thermal diffusivity and
because Bic (X , Y ) → ∞ in the perfect contact. On the other hand, from
the practical point of view, values of Bic(X,Y) larger than 10 already
characterize perfect contact for the cases examined below, because the
temperature difference across the interface becomes negligible, as given
by equation (2.e).

6.1. Verification with simulated measurements

Simulated measurements were used for the verification of the in-

verse problem procedure, based on measurements in the transformed
domain and the non-informative prior given by equation (17). Two
plates with dimensions a = b = 0.10 m and thicknesses of 0.005 m
(c = 0.01 m), representative of titanium (k = 21.9 W/mK and
α = 9.32 × 10−6 m2/s) and epoxy with graphite fibers (k = 0.87 W/
mK and α = 0.66 × 10−6 m2/s) were used for this verification test [4].
A heat flux of 25 kW/m2 was imposed uniformly over the titanium
plate. The surfaces of the plates were allowed to exchange heat by
convection with the surrounding environment, with heat transfer
coefficients taken as 10 and 100 W/m2K at the bottom and top surfaces
of the plate, respectively, such as in Ref. [4]. The initial temperature
was assumed as 25 °C. The thermophysical properties and the initial
and boundary condition parameters were supposed known for this
verification with simulated measurements. The frequency of the simu-
lated measurements was 10 Hz and the duration of the simulated ex-
periment was taken as 10s. The simulated contact failures were given in
the form of two squares with sides of 0.005 m (see Fig. 4a). The contact
conductance was discretized on a mesh with 31 × 31 pixels. The si-
mulated measurements contained additive Gaussian random noise with
zero mean and standard-deviation of 0.05 °C, therefore larger than
those of the actual measurements (see previous section).
The dimensionless contact conductance estimated with simulated
measurements and with a symmetrical proposal density [4], by using
50 transformed modes, is presented in Fig. 4b. A comparison of Fig. 4a
and 4b reveals that the contact failures were accurately detected with
the proposed approach. Furthermore, Fig. 4b shows that the estimated
contact conductance was stable with respect to the measurement errors
and did not exhibit oscillations, despite the fact that its prior is non- Fig. 4. (a) Exact dimensionless contact conductance used to generate the si-
informative. Regularization was provided by the selection of the mulated measurements, (b) Dimensionless contact conductance estimated with
number of transformed modes, as will be further discussed below when simulated measurements.

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Fig. 5. Estimated heat flux.

Fig. 6. Estimated heat flux with Kalman filter at t = 85 s.

Fig. 8. Measured temperatures at t = 90 s: (a) Plate with circular contact

failure.

in an inverse heat conduction problem in the plate without contact

failures (see Fig. 3a), namely: Markov chain Monte Carlo (MCMC)
method [41] and Kalman filter [51,52]. Temperature measurements
taken with the infrared camera were used for the estimation of the
imposed heat flux in an experiment with duration of 90 s. The Markov
chain Monte Carlo method was applied for a one-dimensional problem,
where the imposed heat flux only varies in time, but not over the sur-
face of the plate [41]. The motivation for estimating a transient heat
flux in a one-dimensional problem was the experimental result that the
temperature was practically uniform at the surface of the plate without
contact failures. The transient temperatures recorded at different pixels
Fig. 7. Temperature residuals for the estimation of the imposed heat flux. were individually used for the estimation with the MCMC method.
Moreover, the MCMC method was applied with the transient spatial
thermal conductivity were 1.31 × 10−7 m2s−1 and 0.22 Wm−1K−1, mean temperature over the surface of the plate. The prior used for the
respectively, which are in excellent agreement with the values reported heat flux was a Gaussian Markov Random field, such as in Ref. [41]. As
in the literature [55]. For the estimation procedures presented below, a validation of the heat flux estimated with the one-dimensional model
uncertainties in these parameters were taken into account through their using MCMC, the Kalman filter was also applied, such as in Refs.
priors, which were modeled as Gaussian distributions centered at these [51,52]. For the estimation of the heat flux with the Kalman filter, the
measured values, with standard deviations set to 10% of the measured more general case of a spatially varying transient function, q (x,y,t), was
values from the Flash experiment, of 1.31 × 10−8 m2s−1 and 0.022 considered with all the transient measurements taken over the sample
Wm−1K−1, respectively. surface. Details of the procedures used for the estimation of the imposed
Before addressing the identification of the contact failures of the heat flux are avoided here but can be readily found in Refs. [41,51,52].
actual samples, the heat flux imposed by the radiating heater was es- Fig. 5 compares different estimations obtained for the imposed heat
timated by inverse analyses within the Bayesian framework, as well. flux, including: (i) the transient heat flux estimated by using MCMC
Two different techniques were used for the estimation of the heat flux with the measurements of one single pixel located at x = y = 0.02 m;
(ii) the transient heat flux estimated by using MCMC with the spatial

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Fig. 9. Transformed dimensionless measured temperature modes for the plate

with circular failure: (a) First mode, (b) Other selected modes.

mean of the transient measured temperatures; (iii) the local transient

heat flux q (x,y,t) estimated with the Kalman filter at x = y = 0.02 m;
(iv) the mean of the local heat fluxes estimated with the Kalman filter at
Fig. 10. Measured temperatures recovered after transformation with 50 modes
all pixels. We note in Fig. 5 that the heat fluxes estimated with the at t = 90 s:
different approaches are in excellent agreement, indicating that the (a) Plate with circular contact failure, (b) Plate with longitudinal contact
radiating heater provides a practically uniform flux over the sample. failure.
Negative heat flux values are observed in Fig. 5 with the Kalman filter
estimates, because this is a sequential technique. Actually, the heat flux
elements taken from a Gaussian distribution with zero mean and uni-
was null during the pre-heating period, as correctly estimated with
tary standard deviation. This nonsymmetrical proposal density was
MCMC which is a whole-domain technique. Fig. 6 shows the local heat
used in order to avoid a large rejection of candidates P∗, specially in the
fluxes at t = 85 s estimated with the Kalman filter, which practically
regions of contact failure where the expected value of Bic(X,Y) is zero,
has a uniform distribution over the plate, with few oscillations that can
since this function needs to satisfy the non-negativity constraint (see
be attributed to the ill-posed character of the problem. The temperature
equation (17)). For the results presented below, which were obtained
residuals (differences between measurements and temperatures calcu-
with this proposal density, the acceptance ratio of candidate states was
lated with the estimated heat fluxes) are presented in Fig. 7; the re-
around 27%.
siduals are quite small and uncorrelated, indicating the accuracy of the
The measured temperatures at the final time (duration of the ex-
estimated heat fluxes and that there are no mismatches between the
periment) t = 90 s are presented by Fig. 8a,b, for the plates with the
physical problem and the mathematical formulations used for the es-
circular and the longitudinal contact failures, respectively (see
timation of the imposed heat flux.
Fig. 3b,c). The temperature measurements are available on a grid with
For the estimation of the contact conductance Bic(X,Y) presented
60 × 60 pixels over the top surface of the plate. Fig. 8a,b shows that, in
hereafter, the prior for the heat flux, [q1∗, q2∗, …,qM
∗
] , was given in the
the present experiments, the contact failures could be identified to some
form of Gaussian distributions, with means and covariances obtained
extent by a qualitative analysis of the surface temperature variation.
from the Kalman filter estimation described above. In addition, the
However, for the estimation of the contact conductance such qualitative
proposal density was given by the following nonsymmetrical random
information was not taken into account, in order to challenge the
walk process
method for the quantitative identification of the contact failures with a
non-informative prior. Furthermore, to be able to make the decision
P∗ = P(t − 1) + ω (δ1 + N) (18)
about the existence of a contact failure, one needs to know the posterior
where 1 is a vector with all elements equal to one and N is a vector with means together with the associated variances, which are readily

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L.A.S. Abreu et al. International Journal of Thermal Sciences 132 (2018) 486–497

Fig. 12. (a) Markov chains of the estimated dimensionless contact conductance
for the plate with circular failure obtained with 50 modes, (b) Histogram for a
point of contact failure at x = y = 0.02 m, (c) Histogram for a point of perfect
contact at x = y = 0.005 m.

Fig. 11. Dimensionless contact conductance for the plate with circular failure: modes through the application of the inverse formula given by equation
(a) Estimated means obtained with 50 modes, (b) Estimated standard deviations (14.b). These temperatures are presented by Fig. 10a,b on a grid with
obtained with 50 modes (c) Estimated means obtained with 200 modes.
21 by 21 pixels, for the circular and longitudinal failures, respectively,
at t = 90 s. A comparison of Figs. 8 and 10 reveals that the actual
provided by the MCMC samples. measurements and the temperatures recovered by using 50 modes are
The first mode of the transformed measurements is presented by in excellent agreement, despite the fact that a reduced number of pixels
Fig. 9a, while other selected modes are presented by Fig. 9b, for the was used for the results presented in Fig. 10. In fact, the temperatures
plate with the circular contact failure. These figures show that the recovered with the integral transformation/inversion were smoother
magnitude of the first mode was significantly larger than of the other than the actual measurements; hence, the measurement errors of high
modes and that the 50th mode was practically zero. Although not pre- frequency in space and time were filtered out with the integral trans-
sented for the sake of brevity, such was also the case for the plate with a formation and inversion based on the most important modes.
longitudinal failure. As a verification of the transformation process, the The estimation of the contact conductance Bic(X,Y) in the plate with
measured temperatures were then recovered by using 50 transformed the circular failure, obtained with 50 modes, is presented by Fig. 11a,b.

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L.A.S. Abreu et al. International Journal of Thermal Sciences 132 (2018) 486–497

likelihood given by equation (13), Gaussian or Total Variation priors

are required for appropriate regularization as shown in Refs. [3,4].
The Markov chains at a point of contact failure (x = y = 0.02 m)
and at a point of perfect contact (x = y = 0.005 m) obtained with 50
modes in the plate with the circular failure, are shown by Fig. 12a. After
starting at a condition of perfect contact, Bic(X,Y) = 15, the chains
reached equilibrium in about 8000 states and converged towards 12
and 0 at the pixels of perfect contact and contact failure, respectively.
The histograms of the samples of the Markov chains, between 10000
and 30000 states, at the pixels located at x = y = 0.02 m and
x = y = 0.005 m are presented by Fig. 12b,c, respectively. Notice in
Fig. 12b the skewed histogram at the point of contact failure, caused by
the non-negativity constraint for Bic(X,Y), represented by the prior
given by equation (17). The histogram at the point of perfect contact
presented by Fig. 12c shows that no samples were generated at regions
of small Bic(X,Y), at this point of perfect contact. Both histograms ex-
hibit small posterior variances, as expected from the analysis of
Fig. 11b.
Fig. 13a,b presents respectively the means and standard deviations
of samples of the Markov chains, for the contact conductance Bic(X,Y) at
each pixel at the plate with the longitudinal failure, obtained with 50
modes. As for the circular failure, the longitudinal failure could be
accurately detected with a small variance and the inverse problem so-
lution was stable. Fig. 13a also shows that the contact between the
plates was not perfect outside the manufactured failure for Y < 2 (the
bottom of the image). The lack of attachment in this region was actually
verified by a visual inspection of the target.
The temperature residuals (differences between measurements and
temperatures calculated with the estimated parameters) are presented
by Fig. 14a,b, for the plates with circular and longitudinal failures,
respectively. These figures show the residuals at points of perfect con-
tact (x = 0.015 m and y = 0.01 m) and contact failure
(x = y = 0.02 m). The residuals are small, but correlated, despite the
accurate predictions of the contact failures. Such behavior is due to the
fact that the inverse problem was solved in the transformed tempera-
ture domain with a reduced number of modes and was also previously
Fig. 13. Dimensionless contact conductance for the plate with longitudinal observed in cases involving one-dimensional data compression through
failure: (a) Estimated means obtained with 50 modes, (b) Estimated standard integral transformation [17,44].
deviations obtained with 50 modes. For the experiments with duration of 90s used in this work, the
infrared camera provided 2,916,000 measurements. On the other hand,
While Fig. 11a presents the means of the samples of the Markov chains with the use of the likelihood function in the transformed domain with
after the burn-in period, Fig. 11b presents the standard deviations of 50 modes, 40,500 transformed measurements were used in the inverse
these samples, for each pixel. Despite the non-informative prior and the analysis, thus providing a data compression of 98.6%. In terms of
small number of pixels used to recover this function, Fig. 11a,b shows computational time, the MCMC method with the likelihood function in
that the position of the contact failure could be accurately detected with the transformed domain with 50 modes took 3 h and 10 h, for the cases
small uncertainties. Note that the contact conductance is approximately with the circular and longitudinal failures, respectively.
zero within the region of the contact failure and large in the region of
perfect contact (see Fig. 11a), while the standard deviations are at least 7. Conclusions
two order of magnitudes smaller that the estimated means of the con-
tact conductance (see Fig. 11b). Hence, as for the case with simulated The detection of contact failures in layered composites was ad-
measurements, despite the fact that the measurements are compressed dressed in this work, by solving an inverse problem for the estimation of
through the integral transformation in the spatial domain, the trans- the contact conductance between the layers, with the Markov Chain
formed modes are capable of retaining the information about the spatial Monte Carlo method. In addition to simulated data, actual temperature
variation of Bic(X,Y). Furthermore, the estimated contact conductance measurements obtained with an infrared camera over the surface of
is smooth and does not exhibit oscillations, even at the edge of the plates with two layers and controlled contact failures, were used in the
contact failure. Such a result reveals that the number of modes used for inverse analysis. The estimation of the heat flux imposed by a radiant
the likelihood function in the transformed domain (equation (16)) heater to the samples was also addressed in this work, by using both the
served as regularization of the inverse ill-posed problem, by filtering MCMC method and the Kalman filter. The inverse problem of esti-
the high frequencies (noise) that would be amplified if more modes mating the interface contact conductance was solved in a transformed
would be accounted for in the inverse analysis. Indeed, the solution of domain, with measurements compressed by using the same integral
the inverse problem obtained with 200 modes was completely unstable, transformation that was applied for the solution of the forward pro-
as shown by Fig. 11c. It should be noted that the non-informative blem. The prior used for the unknown contact conductance was non-
uniform prior used here does not provide regularization, other than the informative and only dealt with the fact that the contact conductance is
non-negativity constraint, for the inverse problem solution. On the non-negative. Circular and longitudinal failures were accurately de-
other hand, if the solution of the inverse problem is considered with the tected with the proposed approach. Therefore, the transformed tem-
perature modes were capable of appropriately retaining the information

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