Iterative algorithms for solving inverse problems of heat conduction in multiply connected domains

• Published in: International journal of heat and mass transfer Vol. 55; no. 4; pp. 744 - 751
• Main Authors: Frckowiak, A., Ciakowski, M., Wrblewska, A.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 31.01.2012; Elsevier
• Subjects: Algorithms; Analytical and numerical techniques; Boundaries; Cauchy problem; Criteria; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Heat conduction; Heat transfer; Inverse problem; Inverse problems; Iterative algorithm; Iterative algorithms; Mathematical analysis; Mathematical models; Physics; Iterative algorithm; Inverse problem; Cauchy problem; Laplace equations; Algorithms; Inverse problems; Digital simulation; Modelling; Iterative methods; Thermal conduction
• DOI: 10.1016/j.ijheatmasstransfer.2011.10.035
• ISSN: 0017-9310

The presented paper displays a method of solving the inverse problems of heat transfer in multi-connected regions, consisting in iterative solving of convergent series of the direct problems. For known temperature and flux values at the outer boundary of the region the temperature and flux values at the inner boundaries are sought (the cauchy problem for the Laplace equation). In case of such a formulation of the problem, the solution does not always exist, one of the conditions is met in the mean-square sense, providing the optimization criterion. The idea of the process consists in solving the direct problem in which the boundary condition is subject to iterative changes so as to attain minimum of the optimization criterion (the square functional). Two algorithms have been formulated. In the first of them the heat flux at the inner boundaries of the region, while in the other the temperature were subject to changes. Convergence of both the algorithms have been compared. The numerical calculation has been made for selected examples, for which an analytical solution is known. The effect of random disturbance of the boundary conditions on the solution obtained with iterative algorithms has been checked. Moreover, a function was defined, serving as convergence measure of the solution of the inverse problem solved with the algorithms proposed in the paper. The properties of the function give evidence that it tends to the value exceeding unity.