Formulas for the discrepancy gradient in the iterative solution of inverse heat-conduction problems. II. Determining the gradient in terms of a conjugate variable

1987 ◽  
Vol 52 (4) ◽  
pp. 489-495 ◽  
Author(s):  
O. M. Alifanov ◽  
S. V. Rumyantsev
Keyword(s):  
Heat Conduction ◽  
Iterative Solution ◽  
Inverse Heat Conduction ◽  
Conjugate Variable ◽  
Inverse Heat Conduction Problems

Efficient Numerical Solution of Inverse Heat Conduction Problems

1995 ◽  
Author(s):  
Hans-Jürgen Reinhardt ◽  
Dinh Nho Hao
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Numerical Methods ◽  
Stationary Point ◽  
Forthcoming Paper ◽  
Inverse Heat Conduction ◽  
Piecewise Constant ◽  
Numerical Tests ◽  
Inverse Heat Conduction Problems ◽  
Special Case

Abstract In this contribution we propose new numerical methods for solving inverse heat conduction problems. The methods are constructed by considering the desired heat flux at the boundary as piecewise constant (in time) and then deriving an explicit expression for the solution of the equation for a stationary point of the minimizing functional. In a very special case the well-known Beck method is obtained. For the time being, numerical tests could not be included in this contribution but will be presented in a forthcoming paper.

scholarly journals To the Solution of Geometric Inverse Heat Conduction Problems

2021 ◽  
Vol 24 (1) ◽  
pp. 6-12
Author(s):  
Yurii M. Matsevytyi ◽  
◽  
Valerii V. Hanchyn ◽  
Keyword(s):  
Heat Conduction ◽  
Iterative Process ◽  
Regularization Parameter ◽  
Outer Boundary ◽  
Linear Equations ◽  
The Body ◽  
Two Dimensional ◽  
Inverse Heat Conduction ◽  
System Of Linear Equations ◽  
Inverse Heat Conduction Problems

On the basis of A. N. Tikhonov’s regularization theory, a method is developed for solving inverse heat conduction problems of identifying a smooth outer boundary of a two-dimensional region with a known boundary condition. For this, the smooth boundary to be identified is approximated by Schoenberg’s cubic splines, as a result of which its identification is reduced to determining the unknown approximation coefficients. With known boundary and initial conditions, the body temperature will depend only on these coefficients. With the temperature expressed using the Taylor formula for two series terms and substituted into the Tikhonov functional, the problem of determining the increments of the coefficients can be reduced to solving a system of linear equations with respect to these increments. Having chosen a certain regularization parameter and a certain function describing the shape of the outer boundary as an initial approximation, one can implement an iterative process. In this process, the vector of unknown coefficients for the current iteration will be equal to the sum of the vector of coefficients in the previous iteration and the vector of the increments of these coefficients, obtained as a result of solving a system of linear equations. Having obtained a vector of coefficients as a result of a converging iterative process, it is possible to determine the root-mean-square discrepancy between the temperature obtained and the temperature measured as a result of the experiment. It remains to select the regularization parameter in such a way that this discrepancy is within the measurement error. The method itself and the ways of its implementation are the novelty of the material presented in this paper in comparison with other authors’ approaches to the solution of geometric inverse heat conduction problems. When checking the effectiveness of using the method proposed, a number of two-dimensional test problems for bodies with a known location of the outer boundary were solved. An analysis of the influence of random measurement errors on the error in identifying the outer boundary shape is carried out.

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