A NUMERICAL METHOD FOR THE SOLUTION OF TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEMS

A numerical method for the solution of inverse heat conduction problems in two-dimensional rectangular domains is established and its performance is demonstrated by computational results. The present method extends Beck's8 method to two spatial dimensions and also utilizes future times in order to stabilize the ill-posedness of the underlying problems. The approach relies on a line approximation of the elliptic part of the parabolic differential equation leading to a system of one-dimensional problems which can be decoupled.

[1]  T. Yoshimura,et al.  Inverse heat-conduction problem by finite-element formulation , 1985 .

[2]  P. Monk.  Error estimates for a numerical method for an ill-posed Cauchy problem for the heat equation , 1986 .

[3]  L. Rosenhead.  Conduction of Heat in Solids , 1947, Nature.

[4]  D. Colton.  Solution of boundary value problems by the method of integral operators , 1976 .

[5]  James V. Beck,et al.  Methodology for Comparison of Inverse Heat Conduction Methods , 1988 .

[6]  J. Beck,et al.  EFFICIENT SEQUENTIAL SOLUTION OF THE NONLINEAR INVERSE HEAT CONDUCTION PROBLEM , 1982 .

[7]  J. V. Beck,et al.  Combined function specification-regularization procedure for solution of inverse heat conduction problem , 1984 .

[8]  J. Taler.  Ein numerisches Verfahren zur experimentellen Ermittlung des Wärmeübergangskoeffizienten in zylindrischen Bauteilen , 1986 .

[9]  P. Knabner,et al.  The optimal stability estimate for some ill‐posed Cauchy problems for a parabolic equation , 1988 .

[10]  Hans-Jürgen Reinhardt.  Analysis of Approximation Methods for Differential and Integral Equations , 1985 .

[11]  Charles F. Weber,et al.  Analysis and solution of the ill-posed inverse heat conduction problem , 1981 .

[12]  P. Sjölin.  On the convergence almost everywhere of double Fourier series , 1976 .

[13]  K. Miller,et al.  Calculation of the surface temperature and heat flux on one side of a wall from measurements on the opposite side , 1980 .

[14]  Heinz W. Engl,et al.  Numerical solution of an inverse problem connected with continuous casting of steel , 1985, Z. Oper. Research.

[15]  J. Baumeister.  Stable solution of inverse problems , 1987 .

[16]  A. Tikhonov.  Inverse heat conduction problems , 1975 .

[17]  Lars Eldén,et al.  The Numerical Solution of a Non-Characteristic Cauchy Probelm for a Parabolic Equation , 1983 .

[18]  P. Knabner.  Regularization of the cauchy problem for the heat equation by norm bounds , 1984 .

[19]  Nichtkorrekte Aufgaben in der Rheometrie , 1983 .

[20]  M. N. Özişik,et al.  A direct analytic approach for solving two-dimensional linear inverse heat conduction problems , 1986 .

[21]  O. Burggraf.  An Exact Solution of the Inverse Problem in Heat Conduction Theory and Applications , 1964 .

[22]  Alfred S. Carasso,et al.  Infinitely divisible pulses, continuous deconvolution, and the characterization of linear time invariant systems , 1987 .

[23]  G. Weir.  Surface mounted heat flux sensors , 1986, The Journal of the Australian Mathematical Society Series B Applied Mathematics.

[24]  R. Ewing,et al.  A direct numerical procedure for the Cauchy problem for the heat equation , 1976 .

[25]  Rak-Joong Kim.  NON-CHARACTERISTIC CAUCHY PROBLEM FOR A PARABOLIC EQUATION , 1993 .

[26]  G. Stolz.  Numerical Solutions to an Inverse Problem of Heat Conduction for Simple Shapes , 1960 .

[27]  H. R. Busby,et al.  Numerical solution to a two‐dimensional inverse heat conduction problem , 1985 .

[28]  D. Murio.  On the estimation of the boundary temperature on a sphere from measurements at its center , 1982 .

[29]  F. John,et al.  Differential equations with approximate and improper date , 1955 .

[30]  Diego A. Murio,et al.  The mollification method and the numerical solution of the inverse heat conduction problem by finite differences , 1989 .

[31]  J. Beck.  Criteria for comparison of methods of solution of the inverse heat conduction problem , 1975 .

[32]  Peter Knabner,et al.  Stabilization of ill-posed Cauchy problems for parabolic equations , 1987 .

[33]  R. Gorenflo,et al.  A noncharacteristic cauchy problem for the heat equation , 1991, Acta Applicandae Mathematicae.