This work presents the combined use of the integral transform method, for the direct problem solution, and of Bayesian inference, for the inverse problem analysis, in the simultaneous estimation of spatially variable thermal conductivity and thermal capacity for one-dimensional heat conduction within heterogeneous media. The direct problem solution is analytically obtained via integral transforms and the related eigenvalue problem is solved by the generalized integral transform technique (GITT), offering a fast, precise, and robust solution for the transient temperature field. The inverse problem analysis employs a Markov chain Monte Carlo (MCMC) method, through the implementation of the Metropolis-Hastings sampling algorithm. Instead of seeking the functions estimation in the form of local values for the thermal conductivity and capacity, an alternative approach is employed based on the eigenfunction expansion of the thermophysical properties themselves. Then, the unknown parameters become the corresponding series coefficients for the properties eigenfunction expansions. Simulated temperatures obtained via integral transforms are used in the inverse analysis, for a prescribed concentration distribution of the dispersed phase in a heterogeneous media such as particle filled composites. Available correlations for the thermal conductivity and theory of mixtures relations for the thermal capacity are employed to produce the simulated results with high precision in the direct problem solution, while eigenfunction expansions with reduced number of terms are employed in the inverse analysis itself, in order to avoid the inverse crime. Gaussian distributions were used as priors for the parameter estimation procedure. In addition, simulated results with different randomly generated errors were employed in order to test the inverse analysis robustness.

## References

References
1.
Beck
,
J. V.
, and
Arnold
,
K.
, 1977,
Parameter Estimation in Engineering and Science
,
Wiley
,
New York
.
2.
Alifanov
,
O. M.
, 1994,
Inverse HeatTransfer Problems
,
Springer-Verlag
,
New York
.
3.
Ozisik
,
M. N.
, and
Orlande
,
H. R. B.
, 2000,
Inverse Heat Transfer: Fundamentals and Applications
,
Taylor and Francis
,
New York
.
4.
Zabaras
,
N.
, 2006, “
Inverse Problems in Heat Transfer
,”
Handbook of Numerical Heat Transfer
,
2nd ed.
,
W. J.
Minkowycz
,
E. M.
Sparrow
, and
J. Y.
Murthy
, eds.,
Wiley
,
New York
, pp.
525
557
.
5.
Flach
,
G. P.
, and
Ozisik
,
M. N.
, 1989, “
Inverse Heat Conduction Problem of Simultaneously Estimating Spatially Varying Thermal Conduction and Heat Capacity Per Unit Volume
,”
Numer. Heat Transfer, Part A
,
16
, pp.
249
266
.
6.
Huang
,
C. H.
, and
Ozisik
,
M. N.
, 1990, “
A Direct Integration Approach for Simultaneously Estimating Spatially Varying Thermal Conductivity and Heat Capacity
,”
Int. J. Heat Fluid Flow
,
11
(
3
), pp.
262
268
.
7.
Lesnic
,
D.
,
Elliot
,
L.
,
Ingham
,
D. B.
,
Clennell
,
B.
, and
Knioe
,
R. J.
, 1999, “
The Identification of the Piecewise Homogeneous Thermal Conductivity of Conductors Subjected to a Heat Flow Test
,”
Int. J. Heat Mass Transfer
,
42
, pp.
143
152
.
8.
Divo
,
E.
,
Kassab
,
A.
, and
Rodriguez
,
F.
, 2000, “
Characterization of Space Dependent Thermal Conductivity With a BEM-Based Genetic Algorithm
,”
Numer. Heat Transfer, Part A
,
37
, pp.
845
875
.
9.
Huang
,
C. H.
, and
Chin
,
S. C.
, 2000, “
A Two-Dimensional Inverse Problem in Imaging the Thermal Conductivity of a Non-homogeneous Medium
,”
Int. J. Heat Mass Transfer
,
43
, pp.
4061
4071
.
10.
Rodrigues
,
F. A.
,
Orlande
,
H. R. B.
, and
Dulikravich
,
G. S.
, 2004, “
Simultaneous Estimation of Spatially Dependent Diffusion Coefficient and Source Term in a Nonlinear 1D Diffusion Problem
,”
Math. Comput. Simul.
,
66
, pp.
409
424
.
11.
Huttunen
,
J. M. J.
,
Huttunen
,
T.
,
Malinen
,
M.
, and
Kaipio
,
J.
, 2006, “
Determination of Heterogeneous Thermal Parameters Using Ultrasound Induced Heating and MR Thermal Mapping
,”
Phys. Med. Biol.
,
51
, pp.
1011
1032
.
12.
Huang
,
C. H.
, and
Huang
,
C. Y.
, 2007, “
An Inverse Problem in Estimating Simultaneously the Effective Thermal Conductivity and Volumetric Heat Capacity of Biological Tissue
,”
Appl. Math. Model.
,
31
, pp.
1785
1797
.
13.
Liu
,
C. S.
, 2008, “
An LSGSM to Identify Nonhomogeneous Heat Conductivity Functions by an Extra Measurement of Temperature
,”
Int. J. Heat Mass Transfer
,
51
, pp.
2603
2613
.
14.
Naveira-
Cotta
,
C. P.
,
Cotta
,
R. M.
,
Orlande
,
H. R. B.
, and
Fudym
,
O.
, 2009, “
Eigenfunction Expansions for Transient Diffusion in Heterogeneous Media
,”
Int. J. Heat Mass Transfer
,
52
, pp.
5029
5039
.
15.
Naveira-
Cotta
,
C. P.
,
Orlande
,
H. R. B.
, and
Cotta
,
R. M.
, 2010, “
Integral Transforms and Bayesian Inference in the Identification of Variable Thermal Conductivity in Two-Phase Dispersed Systems
,”
Num. Heat Transfer Part B
,
57
(
3
), pp.
1
30
.
16.
Cotta
,
R. M.
, 1993,
Integral Transforms in Computational Heat and Fluid Flow
,
CRC Press
,
Boca Raton, FL
.
17.
Cotta
,
R. M.
, and
Mikhailov
,
M. D.
, 1997,
Heat Conduction: Lumped Analysis, Integral Transforms, Symbolic Computation
,
Wiley
,
New York
.
18.
Cotta
,
R. M.
, 1998,
The Integral Transform Method in Thermal and Fluids Sciences and Engineering
,
Begell House
,
New York
.
19.
Cotta
,
R. M.
, and
Mikhailov
,
M. D.
, 2006, “
Hybrid Methods and Symbolic Computations
”,
Handbook of Numerical Heat Transfer
,
2nd ed.
,
W. J.
Minkowycz
,
E. M.
Sparrow
, and
J. Y.
Murthy
, eds.,
Wiley
,
New York
, pp.
493
522
.
20.
Mikhailov
,
M. D.
, and
Cotta
,
R. M.
, 1994, “
Integral Transform Method for Eigenvalue Problems
,”
Commun. Numer. Methods Eng.
,
10
, pp.
827
835
.
21.
Sphaier
,
L. A.
, and
Cotta
,
R. M.
, 2000, “
Integral Transform Analysis of Multidimensional Eigenvalue Problems Within Irregular Domains
,”
Numer. Heat Transfer, Part B
,
38
, pp.
157
175
.
22.
Wolfram
,
S.
, 2005,
The Mathematica Book
, version 5.2,
Cambridge-Wolfram Media
.
23.
Kaipio,
J.
and
Somersalo
,
E.
, 2004,
Statistical and Computational Inverse Problems
,
Springer-Verlag
,
New York
.
24.
Gamerman
,
D.
, and
Lopes
,
H. F.
, 2006,
Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference
,
2nd ed.
,
Chapman & Hall/CRC
,
Boca Raton, FL
.
25.
Migon
,
H. S.
, and
Gamerman
,
D.
, 1999,
Statistical Inference: An Integrated Approach
,
Arnold/Oxford
,
London/New York
.
26.
Orlande
,
H. R. B.
, Colaço, M. J., and
Dulikravich
,
G. S.
, 2008, “
Approximation of the Likelihood Function in the Bayesian Technique for the Solution of Inverse Problems
,”
Inverse Probl. Sci. Eng.
,
16
, pp.
677
692
.
27.
Fudym
,
O.
,
Orlande
,
H. R. B.
,
Bamford
,
M.
, and
Batsale
,
J. C.
, 2008, “
Bayesian Approach for Thermal Diffusivity Mapping from Infrared Images Processing with Spatially Random Heat Pulse Heating
,”
J. Phys. Conf. Ser.
,
135
, pp.
12
42
.
28.
Wang
,
J.
, and
Zabaras
,
N.
, 2004, “
A Bayesian Inference Approach to the Inverse Heat Conduction Problem
”,
Int. J. Heat Mass Transfer
,
47
, pp.
3927
3941
.
29.
Parthasarathya
,
S.
and
Balaji
,
C.
, 2008, “
Estimation of Parameters in Multi-mode Heat Transfer Problems using Bayesian Inference—Effect of Noise and a Priori
”,
Int. J. Heat Mass Transfer
,
51
, pp.
2313
2334
.
30.
Tarantola
,
A.
, 2005,
Inverse Problem Theory and Methods for Model Parameter Estimation
,
Society for Industrial and Applied Mathematics, SIAM
,
.
31.
Metropolis
,
N.
,
Rosenbluth
,
A. W.
,
Rosenbluth
,
M. N.
,
Teller
,
A. H.
, and
Teller
,
E.
, 1953, “
Equations of State Calculations by Fast Computating Machines
,”
J. Chem. Phys.
,
21
, pp.
1087
1092
.
32.
Hastings
,
W. K.
, 1970, “
Monte Carlo Sampling Methods Using Markov Chains and their Applications
,” Biometrika, 57, pp. 97–109.
33.
Barker
,
A. A.
, 1965, “
Monte Carlo Calculation of the Radial Distribution Functions for a Proton-Electron Plasma
,”
Aust. J. Phys.
,
18
, pp.
119
133
.
34.
Peskun
,
P. H.
, 1973, “
Optimum Monte Carlo Sampling using Markov Chain
,” Biometrika, 60, pp. 607–612.
35.
Tavman
,
I. H.
, 1996, “
Thermal and Mechanical Properties of Aluminum Powder-Filled High-Density Polyethylene Composites
,”
J. Appl. Polym. Sci.
,
62
, pp.
2161
2167
.
36.
Kumlutas
,
D.
,
Tavman
,
I. H.
, and C¸
oban
,
M. T.
, 2003, “
Thermal Conductivity of Particle Filled Polyethylene Composite Materials
,”
Compos. Sci. Technol.
,
63
(
1
), pp.
113
117
.
37.
Lewis
,
T.
, and
Nielsen
,
L.
, 1970, “
Dynamic Mechanical Properties of Particulate-Filled Polymers
”,
J. Appl. Polym. Sci.
,
14
(
6
), pp.
1449
1471
.