Solution to Non-stationary Inverse Heat Conduction Problems for Multi-layer Bodies, Based on Effective Search for the Regularization Parameter

Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 22, no. 3, 2019 (September)
Pages 4-13
Cited by J. of Mech. Eng., 2019, vol. 22, no. 3, pp. 4-13



Yurii M. Matsevytyi, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi Str., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0002-6127-0341

Volodymyr M. Sirenko, Yuzhnoye State Design Office (3, Krivorizka Str., Dnipro, 49008, Ukraine), e-mail:

Andrii O. Kostikov, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi Str., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0001-6076-1942

Mykola O. Safonov, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi Str., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0002-3951-4805

Valerii V. Hanchyn, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi Str., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0001-9242-6460



To obtain a stable solution to the inverse heat conduction problem (IHCP), the article uses A. N. Tikhonov’s method with an effective algorithm for finding the regularization parameter. The required heat flux at the boundary and the thermal contact resistance in the time coordinate are approximated by Schoenberg splines of the third degree, with the sum of the squares of the desired value, its first and second derivatives, being used as a stabilizing functional. The object of this study is multilayer plates or shells, such as solid-fuel rocket engine bodies. To a first approximation, the problem is considered in a one-dimensional non-stationary linear formulation. The shell thickness-to-radius ratio will be considered such that in the heat equation, the curvature of the shell can be neglected and considered as a flat plate. This assumption was chosen to simplify the presentation of the material, and it does not limit the applicability of the methodology under consideration for the case of axially symmetrical shells, as well as for the case when a mathematical model is converted from the rectangular coordinate system to the cylindrical one. Three inverse problems are considered. In the first two, heat fluxes in a composite body with the ideal and real thermal contacts are determined. In the third IHCP, with the real thermal contact, thermal contact resistance is determined. Heat fluxes in multi-layer bodies are represented as linear combinations of Schoenberg splines of the third degree with unknown coefficients, which are calculated by solving a system of linear algebraic equations. This system is a consequence of the necessary condition for the minimum functional based on the principle of the least squares of the deviation of the temperature being simulated from the one obtained as a result of a thermophysical experiment. To regularize the solutions to the IHCP, in this functional, the stabilizing functional with the regularization parameter, as a multiplicative factor, is used as the summand to the sum of squares. This functional is the sum of the squares of heat fluxes, their first and second derivatives with the corresponding multipliers. These multipliers are selected according to the previously known properties of the desired solution. The search for the regularization parameter is carried out using the algorithm similar to the one for searching for the root of a nonlinear equation.


Keywords: inverse heat conduction problem, heat flux, thermal contact resistance, A. N. Tikhonov regularization method, functional, stabilizer, regularization parameter, identification, approximation, Schoenberg spline of the third degree.


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Received: 29 May 2019

Published 30 September 2019