The profile and time evolution of a solid/liquid interface in a phase change process is estimated by solving an inverse heat transfer problem, using data measurements in the solid phase only. One then faces the inverse resolution of a heat equation in a variable and a priori unknown 2D domain. This ill-posed problem is solved by a regularization approach: the unknown function (position of the melting front) is obtained by minimization of a two component criterion, consisting of a distance between the output of a simulation model and the measured data, to which a penalizing function is added in order to restore the continuity of the inverse operator. A numerical study is developed to analyze the validity domain of the identification method. From simulation tests, it is shown that the minimum signal/noise ratio that can be handled depends strongly on the position of the measurement sensors.

1.
Afshari, A., 1990, “Identification de l’e´volution d’un front de fusion/solidification par re´solution inverse de l’e´quation de la chaleur dans le domaine solide,” These de l’Universite´ de Paris-Sud, Orsay, France.
2.
Afshari, A., Be´nard, C., Duhamel, C. and Guerrier, B., 1989, “On-Line Identification of the State of the Surface of a Material Undergoing Thermal Processing,” Proceedings, 5th IFAC Symposium on Control of Distributed Parameter Systems, Perpignan, France, pp. 459–463.
3.
Barbu, V., 1989, “Controlling the Free Boundary of Stefan Problem,” Proceedings, 5th IFAC Symposium on Control of Distributed Parameter Systems, Perpignan, France, pp. 255–259.
4.
Beck
J. V.
, and
Murio
D. A.
,
1986
, “
Combined Function Specification—regularization Procedure for Solution of Inverse Heat Conduction Problem
,”
AIAA Journal
, Vol.
24
, pp.
180
185
.
5.
Be´nard
C.
,
Gobin
D.
, and
Zanoli
A.
,
1986
, “
Moving Boundary Problem: Heat Conduction in the Solid Phase of a Phase Change Material during Melting Driven by Natural Convection in the Liquid
,”
International Journal of Heat and Mass Transfer
, Vol.
29
, No.
11
, pp.
1669
1681
.
6.
Be´nard
C.
, and
Afshari
A.
,
1992
, “
Inverse Stefan Problem: Tracking of the Interface Position From Measurements of the Solid Phase
,”
International Journal of Numerical Methods in Engineering
, Vol.
35
, No.
4
, pp.
835
851
.
7.
Be´nard, C., Guerrier, B., and Wang, X., 1993, “Identification de l’e´volution d’une interface liquide-solide dans une ge´ome´tric 2D,” Proceedings of 11eme Congre´s Francois de Me´canique, Lille, Vol. 3, pp. 181–184.
8.
Colton
D.
, and
Reemtsen
R.
,
1984
, “
The Numerical Solution of the Inverse Stefan Problem in Two Space Variables
,”
SIAM Journal of Applied Mathematics
, Vol.
44, 5
, pp.
996
1013
.
9.
Guerrier
B.
, and
Be´nard
C.
,
1993
, “
Two-Dimensional Linear Transient Inverse heat Conduction Problem: Boundary Condition Identification
,”
AIAA, Journal of Thermophysics and Heat Transfer
, Vol.
7
,
3
, pp.
472
478
.
10.
Hadamard, J., 1923, Lecture on the Cauchy Problem in Linear Partial Differential Equations, Yale University Press, New Haven.
11.
Hsu
H. F.
,
Rubinsky
B.
, and
Mahin
K.
,
1986
, “
An Inverse Finite-Element Method for the Analysis of Stationnarv Arc Welding Processes
,”
ASME Journal of Heat Transfer
, Vol.
108
,
737
741
.
12.
Jochum
P.
,
1980
a, “
The Numerical Solution of the Inverse Stefan Problem
,”
Numerical Mathematics
, Vol.
34
, pp.
411
42
.
13.
Jochum
P.
,
1980
b, “
The Inverse Stefan Problem as a Problem of Nonlinear Approximation Theory
,”
Journal of Approximation Theory
, Vol.
30
, pp.
81
98
.
14.
Jochum, P., 1982, “To The Numerical Solution of an Inverse Stefan Problem in Two Space Variables, Numerical Treatement of Free Boundary Value Problem,” Albrecht J. et al., eds, ISNM 58, Birkha¨luser-Verlag, Basel.
15.
Katz
M. A.
, and
Rubinsky
B.
,
1984
, “
An Inverse Finite-Element Technique to Determine the Change of Phase Interface Location in One-Dimensional Melting Problems
,”
Numerical Heat Transfer
, Vol.
7
, pp.
269
283
.
16.
Knaber
P.
,
1985
, “
Control of Stefan Problem by Means of Linear-Quadratic Defect Minimization
,”
Numerical Mathematics
, Vol.
46
, pp.
429
442
.
17.
Mannikko
P.
,
Neittaanmaki
P.
, and
Tiba
D.
,
1994
, “
A Rapid Method for the Identificadon of the Free Boundary in Two-Phase Stefan Problems
,”
IMA Journal of Numerical Analysis
, Vol.
14
,
3
, pp.
411
420
.
18.
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing, McGraw-Hill.
19.
Reemtsen
R.
, and
Kirsch
A.
,
1984
, “
A Method for the Numerical Solution of the One Dimensional Inverse Stefan Problem
,”
Numerical Mathematics
, Vol.
45
, pp.
253
273
.
20.
Tikhonov, A. N., and Arsenin, V. Y., 1977, Solutions of Ill-Posed Problems, V. H. Winston and Sons, Washington, DC.
21.
Zabaras
N.
,
Ruan
Y.
, and
Richmond
O.
,
1992
, “
Design of two Dimensional Stefan Processes with Desired Freezing Front Motion
,”
Numerical Heat Transfer
, Vol.
21, 3
, pp.
307
326
.
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