|Journal||Journal of Mechanical Engineering|
|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)|
|Cited by||J. of Mech. Eng., 2019, vol. 22, no. 3, pp. 4-13|
Volodymyr M. Sirenko, Yuzhnoye State Design Office (3, Krivorizka Str., Dnipro, 49008, Ukraine), e-mail: firstname.lastname@example.org
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.
Full text: Download in PDF
- Beck, J., Blakuell, B., & Sent-Kler, Ch. Jnr. (1989). Nekorrektnyye obratnyye zadachi teploprovodnosti [Ill-conditioned inverse heat conduction problems]. Moscow: Mir, 312 p. (in Russian).
- Matsevityy, Yu. M. (2003). Obratnyye zadachi teploprovodnosti: v 2 t. T. 1. Metodologiya. T. 2. Prilozheniya [Inverse problems of heat conduction: in 2 vols. Vol. 1. Меtodologiya. Vol. 2. Applications]. Kiyev: Naukova dumka, 408 p. (vol. 1), 392 p. (vol. 2) (in Russian).
- Kozdoba, L. A. & Krukovskiy, P. G. (1982). Metody resheniya obratnykh zadach teploperenosa [Methods for solving inverse heat transfer problems]. Kiyev: Naukova Dumka, 360 p. (in Russian).
- Alifanov, O. M., Artyukhin, Ye. A., & Rumyantsev S. V. (1988). Ekstremalnyye metody resheniya nekorrektnykh zadach [Extreme methods for solving ill-conditioned problems]. Мoscow: Nauka, 288 p. (in Russian).
- Tikhonov, A. N. & Arsenin, V. Ya. (1975). Metody resheniya nekorrektnykh zadach [Methods for solving ill-conditioned problems]. Moscow: Nauka, 288 p. (in Russian).
- Matsevityy, Yu. M. & Slesarenko A. P. (2014). Nekorrektnyye mnogoparametricheskiye zadachi teploprovodnosti i regionalno-strukturnaya regulyarizatsiya ikh resheniy [Ill-conditioned multi-parameter heat conduction problems and regional-structural regularization of their solutions]. Kiyiv: Naukova dumka, 292 p. (in Russian).
- Matsevityy, Yu. M., Slesarenko, A. P., & Ganchin, V. V. (1999). Regionalno-analiticheskoye modelirovaniye i identifikatsiya teplovykh potokov s ispolzovaniyem metoda regulyarizatsii A. N. Tikhonova [Regional analytical modeling and identification of heat fluxes using the A. N. Tikhonov regularization method]. Problemy mashinostroyeniya – Journal of Mechanical Engineering, vol. 2, no. 1–2, pp. 34–42 (in Russian).
- Matsevityy, Yu. M., Safonov, N. A., & Ganchin, V. V. (2016). K resheniyu nelineynykh obratnykh granichnykh zadach teploprovodnosti [On the solution of nonlinear inverse boundary value problems of heat conduction]. Problemy mashinostroyeniya – Journal of Mechanical Engineering, vol. 19, no. 1, pp. 28–36 (in Russian). https://doi.org/10.15407/pmach2016.01.028
- Graham, N. Y. (1983). Smoothing with Periodic Cubic Splines. Bell System Techn. J., vol. 62, pp. 101–110. https://doi.org/10.1002/j.1538-7305.1983.tb04381.x
- Reinsch, C. H. J. (1967). Smoothing by Spline Function. Numerische Mathematik, vol. 10, pp. 77–183. https://doi.org/10.1007/BF02162161
Received: 29 May 2019
Received: 29 May 2019