Abstract

A solution scheme based on the maximum entropy method (MEM) for the solution of two-dimensional inverse heat conduction problems is established. MEM finds the solution which maximizes the entropy functional under the given temperature measurements. The proposed method converts the inverse problem to a nonlinear constrained optimization problem. The constraint of the optimization problem is the statistical consistency between the measured temperature and the estimated temperature. Successive quadratic programming (SQP) facilitates the numerical estimation of the maximum entropy solution. The characteristic feature of the proposed method is investigated with the sample numerical results. The presented results show considerable enhancement in resolution for stringent cases in comparison with a conventional method.

Issue Section:
Technical Notes
Keywords:
Conduction, Inverse, heat conduction, inverse problems, entropy, maximum entropy methods
Topics:
Entropy, Heat conduction, Heat flux, Inverse problems, Optimization, Temperature, Impulse (Physics)
1.
Ozisik, M. N., and Orlande, H. R. B., 2000, Inverse Heat Transfer: Fundamentals and Applications, Taylor & Francis, New York.
2.
Alifanov, O. M., 1994, Inverse Heat Transfer Problems, International Series in Heat and Mass Transfer, Springer-Verlag, New York.
3.
Beck, J. V., Blackwell, B., and St. Clair, C. R., 1985, Inverse Heat Conduction: Ill-Posed Problems, Wiley, New York.
4.
Beck
,
J. V.
, and
Woodbury
,
K. A.
,
1998
, “
Inverse Problems and Parameter Estimation: Integration of Measurements and Analysis
,”
Meas. Sci. Technol.
,
9
(
6
), pp.
839
–
847
.
5.
Alifanov
,
O. M.
,
1997
, “
Mathematical and Experimental Simulation in Aerospace System Verification
,”
Acta Astronaut.
,
41
(
1
), pp.
43
–
52
.
6.
Kim
,
S. K.
,
Koo
,
B. Y.
, and
Lee
,
W. I.
,
2000
, “
The Evaluation of the Heat Transfer Coefficient During Hot-Jet Impingement Using an Inverse Method With Regularization Parameter Optimization
,”
International Journal of Transport Phenomena
,
2
(
3
), pp.
173
–
186
.
7.
Huang
,
X. C.
,
Bartsch
,
G.
, and
Schroederrichter
,
D.
,
1994
, “
Quenching Experiments With a Circular Test Section of Medium Thermal Capacity Under Forced-Convection of Water
,”
Int. J. Heat Mass Transfer
,
37
(
5
), pp.
803
–
818
.
8.
Liauh
,
C. T.
,
Hills
,
R. G.
, and
Roemer
,
R. B.
,
1993
, “
Comparison of the Adjoint and Influence Coefficient Methods for Solving the Inverse Hyperthermia Problem
,”
ASME J. Biomech. Eng.
,
115
(
1
), pp.
63
–
71
.
9.
Tikhonov, A. N., and Arsenin, V. I. A., 1977, Solutions of Ill-Posed Problems, Scripta Series in Mathematics, Washington, Winston, New York, distributed solely by Halsted Press.
10.
D’Souza
,
N.
,
1976
, “
Numerical-Solution of One-Dimensional Inverse Transient Heat-Conduction by Finite-Difference Method
,”
Mech. Eng. (Am. Soc. Mech. Eng.)
,
98
(
5
), p.
105
105
.
11.
Raynaud
,
M.
, and
Bransier
,
J.
,
1986
, “
A New Finite-Difference Method for the Nonlinear Inverse Heat-Conduction Problem
,”
Numer. Heat Transfer
,
9
(
1
), pp.
27
–
42
.
12.
Weber
,
C. F.
,
1981
, “
Analysis and Solution of the Ill-Posed Inverse Heat-Conduction Problem
,”
Int. J. Heat Mass Transfer
,
24
(
11
), pp.
1783
–
1792
.
13.
Neto
,
A. J. S.
, and
Ozisik
,
M. N.
,
1992
, “
2-Dimensional Inverse Heat-Conduction Problem of Estimating the Time-Varying Strength of a Line Heat-Source
,”
J. Appl. Phys.
,
71
(
11
), pp.
5357
–
5362
.
14.
Jarny
,
Y.
,
Ozisik
,
M. N.
, and
Bardon
,
J. P.
,
1991
, “
A General Optimization Method Using Adjoint Equation for Solving Multidimensional Inverse Heat-Conduction
,”
Int. J. Heat Mass Transfer
,
34
(
11
), pp.
2911
–
2919
.
15.
Scarpa
,
F.
, and
Milano
,
G.
,
1995
, “
Kalman Smoothing Technique Applied to the Inverse Heat-Conduction Problem
,”
Numer. Heat Transfer, Part B
,
28
(
1
), pp.
79
–
96
.
16.
Park
,
H. M.
, and
Lee
,
J. H.
,
1998
, “
A Method of Solving Inverse Convection Problems by Means of Mode Reduction
,”
Chem. Eng. Sci.
,
53
(
9
), pp.
1731
–
1744
.
17.
Battaglia
,
J. L.
,
2002
, “
A Modal Approach to Solve Inverse Heat Conduction Problems
,”
Inverse Probl. Eng.
,
10
(
1
), pp.
41
–
63
.
18.
Beck
,
J. V.
, and
Murio
,
D. A.
,
1986
, “
Combined Function Specification-Regularization Procedure for Solution of Inverse Heat-Conduction Problem
,”
AIAA J.
,
24
(
1
), pp.
180
–
185
.
19.
Dowding
,
K. J.
, and
Beck
,
J. V.
,
1999
, “
A Sequential Gradient Method for the Inverse Heat Conduction Problem (Ihcp)
,”
ASME J. Heat Transfer
,
121
(
2
), pp.
300
–
306
.
20.
Reinhart
,
H. J.
, and
Hao
,
N. H.
,
1996
, “
A Sequential Conjugate Gradient Method for the Stable Numerical Solution to Inverse Heat Conduction Problems
,”
Inverse Probl. Eng.
,
2
, pp.
263
–
272
.
21.
Kim
,
S. K.
,
Lee
,
W. I.
, and
Lee
,
J. S.
,
2002
, “
Solving a Nonlinear Inverse Convection Problem Using the Sequential Gradient Method
,”
KSME International Journal
,
16
(
5
), pp.
710
–
719
.
22.
Park
,
H. M.
, and
Jung
,
W. S.
,
2001
, “
On the Solution of Multidimensional Inverse Heat Conduction Problems Using an Efficient Sequential Method
,”
ASME J. Heat Transfer
,
123
(
6
), pp.
1021
–
1029
.
23.
Beck
,
J. V.
,
Blackwell
,
B.
, and
HajiSheikh
,
A.
,
1996
, “
Comparison of Some Inverse Heat Conduction Methods Using Experimental Data
,”
Int. J. Heat Mass Transfer
,
39
(
17
), pp.
3649
–
3657
.
24.
Wu, N., 1997, The Maximum Entropy Method, Springer Series in Information Sciences, 32, Springer, New York.
25.
Skilling
,
J.
, and
Bryan
,
R. K.
,
1984
, “
Maximum-Entropy Image-Reconstruction—General Algorithm
,”
Mon. Not. R. Astron. Soc.
,
211
(
1
), p.
111
111
.
26.
Golan
,
A.
,
Judge
,
G.
, and
Robinson
,
S.
,
1994
, “
Recovering Information From Incomplete or Partial Multisectoral Economic Data
,”
Rev. Econ. Stat.
,
76
(
3
), pp.
541
–
549
.
27.
Ramos
,
F. M.
, and
Giovannini
,
A.
,
1995
, “
Solution of a Multidimensional Heat-Conduction Inverse Problem Using the Finite Analytic Method and the Principle of Maximum-Entropy
,”
Int. J. Heat Mass Transfer
,
38
(
1
), pp.
101
–
111
.
28.
Lair
,
P.
,
Dumoulin
,
J.
, and
Millan
,
P.
,
1998
, “
Inverse Method for Flux Characterization Using Infrared Thermography in Die Forging
,”
Numer. Heat Transfer, Part A
,
33
(
3
), pp.
267
–
277
.
29.
Muniz
,
W. B.
,
Ramos
,
F. M.
, and
Velho
,
H. F. D.
,
2000
, “
Entropy and Tikhonov-Based Regularization Techniques Applied to the Backwards Heat Equation
,”
Computers & Mathematics with Applications
,
40
(
8–9
), pp.
1071
–
1084
.
30.
Kim
,
S. K.
, and
Lee
,
W. I.
,
2002
, “
Solution of Inverse Heat Conduction Problems Using Maximum Entropy Method
,”
Int. J. Heat Mass Transfer
,
45
(
2
), pp.
381
–
391
.
31.
Reklaitis, G. V., Ravindran, A., and Ragsdell, K. M., 1983, Engineering Optimization: Methods and Applications, Wiley, New York.
32.
Tan
,
Y.
, and
Deng
,
C.
,
2000
, “
Solving for a Quadratic Programming With a Quadratic Constraint Based on a Neural Network Frame
,”
Neurocomputing
,
30
(
1–4
), pp.
117
–
128
.
33.
McIntyre
,
N. S.
,
Pratt
,
A. R.
,
Piao
,
H.
,
Maybury
,
D.
, and
Splinter
,
S. J.
,
1999
, “
Resolution Enhancement of X-Ray Photoelectron Spectra by Maximum Entropy Deconvolution
,”
Appl. Surf. Sci.
,
145
, pp.
156
–
160
.
34.
Rohsenow, W. M., Hartnett, J. P., and Cho, Y. I., 1998, Handbook of Heat Transfer, 3rd ed., Mcgraw-Hill Handbooks, McGraw-Hill, New York.
35.
Lloyd, J. M., 1975, Thermal Imaging System, Plenum Press, New York.
36.
Mayer
,
R.
,
Henkes
,
R.
, and
Van Ingen
,
J. L.
,
1998
, “
Quantitative Infrared-Thermography for Wall-Shear Stress Measurement in Laminar Flow
,”
Int. J. Heat Mass Transfer
,
41
(
15
), pp.
2347
–
2360
.
37.
Ekkad
,
S. V.
, and
Han
,
J. C.
,
1996
, “
Heat Transfer Inside and Downstream of Cavities Using Transient Liquid Crystal Method
,”
J. Thermophys. Heat Transfer
,
10
(
3
), pp.
511
–
516
.
38.
Allison
,
S. W.
, and
Gillies
,
G. T.
,
1997
, “
Remote Thermometry With Thermographic Phosphors: Instrumentation and Applications
,”
Rev. Sci. Instrum.
,
68
(
7
), pp.
2615
–
2650
.
39.
Kwon
,
P.
,
Schiemann
,
T.
, and
Kountanya
,
R.
,
2001
, “
An Inverse Estimation Scheme to Measure Steady-State Tool-Chip Interface Temperatures Using an Infrared Camera
,”
Int. J. Mach. Tools Manuf.
,
41
(
7
), pp.
1015
–
1030
.
40.
Valvano
,
J. W.
,
1992
, “
Temperature Measurements
,”
Adv. Heat Transfer
,
23
, pp.
359
–
436
.
41.
Jaynes
,
E. T.
,
1957
, “
Information Theory and Statistical Mechanics
,”
Phys. Rev.
,
106
, pp.
620
–
630
.
42.
Shannon
,
C. E.
,
1948
, “
A Mathematical Theory of Communication
,”
Bell Syst. Tech. J.
,
27
, pp.
379
–
423
.
43.
Graham
,
N. Y.
,
1983
, “
Smoothing With Periodic Cubic-Splines
,”
Bell Syst. Tech. J.
,
62
(
1
), pp.
101
–
110
.
44.
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Series in Computational Methods in Mechanics and Thermal Sciences, Hemisphere Pub. Corp., New York.
45.
Raynaud
,
M.
, and
Beck
,
J. V.
,
1988
, “
Methodology for Comparison of Inverse Heat-Conduction Methods
,”
ASME J. Heat Transfer
,
110
(
1
), pp.
30
–
37
.
46.
Kim, S. K., Jung, B. S., Lee, J. S., and Lee, W. I., 2002, “An Inverse Estimation of Surface Temperature Using the Maximum Entropy Method,” in Proceedings of the Twelfth International Heat Transfer Conference, Elsevier SAS.
47.
Lawrence, C., Zhou, Z. L., and Tits, A. L., 1997, User’s Guide for Cfsqp Version 2.5, Electrical Engineering Dept. and Institute for System Research, University of Maryland, College Park, MD.
48.
IMSL Math/Library User’s Guide, 1994, Visual Numerics, http://vni.com
49.
Donoho
,
D. L.
,
Johnstone
,
I. M.
,
Hoch
,
J. C.
, and
Stern
,
A. S.
,
1992
, “
Maximum-Entropy and the Nearly Black Object
,”
J. R. Stat. Soc. Ser. B. Methodol.
,
54
(
1
), pp.
41
–
81
.
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