|Journal||Journal of Mechanical Engineering – Problemy Mashynobuduvannia|
|Publisher||A. Pidhornyi Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
|ISSN||2709-2984 (Print), 2709-2992 (Online)|
|Issue||Vol. 24, no. 3, 2021 (September)|
|Cited by||J. of Mech. Eng., 2021, vol. 24, no. 3, pp. 6-13|
On the basis of A. N. Tikhonov’s regularization theory, a technique has been developed for solving inverse heat conduction problems of identifying the thermal conductivity tensor in a two-dimensional domain. Such problems are replaced by problems of identifying the principal heat conductivity coefficients and the orientation angle of the principal axes, with the principal coefficients being approximated by Schoenberg’s cubic splines. As a result, the problem is reduced to determining the unknown coefficients in these approximations and the orientation angle of the principal axes. With known boundary and initial conditions, the temperature in the domain will depend only on these coefficients and the orientation angle. If one expresses it by the Taylor formula for two terms of series and substitutes it into the Tikhonov functional, then the determination of the increments of the coefficients and the increment of the orientation angle can be reduced to solving a system of linear equations with respect to these increments. By choosing a certain regularization parameter as well as some functions for the principal thermal conductivity coefficients and the orientation angle as an initial approximation, one can implement an iterative process for determining these coefficients. After obtaining the vectors of the coefficients and the angle of orientation as a result of the 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 choose the regularization parameter in such a way that this discrepancy is within the root-mean-square discrepancy of the measurement error. When checking the efficiency of using the proposed method, a number of two-dimensional test problems for bodies with known thermal conductivity tensors were solved. The influence of random measurement errors on the error in the identification of the thermal conductivity tensor was analyzed.
Keywords: internal inverse heat conduction problem, thermal conductivity tensor, A. N. Tikhonov’s regularization method, stabilizing functional, regularization parameter, identification, approximation, Schoenberg’s cubic splines.
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- Matsevityy, Yu. M. (2002). Obratnyye zadachi teploprovodnosti. T. 1. Metodologiya [Inverse problems of thermal conductivity: In 2 vols. Vol. 1. Methodology. Kiyev: Naukova dumka, 408 p. (in Russian).
- Alifanov, O. M., Artyukhin, Ye. A., & Rumyantsev, S. V. (1988). Ekstremalnyye metody resheniya nekorrektnykh zadach [Extreme methods for solving ill-posed problems]. Moscow: Nauka, 288 p. (in Russian).
- Tikhonov, A. N. & Arsenin, V. Ya. (1979). Metody resheniya nekorrektnykh zadach [Methods for solving ill-posed problems]. Moscow: Nauka, 288 p. (in Russian).
- Beck, J. V., Blackwell B., & St. Clair, C, R. (Jr.) (1985). Inverse heat conduction. Ill-posed problems. New York etc.: J. Wiley & Sons, 308 p. https://doi.org/10.1002/zamm.19870670331.
- Formalev, V. F. (2015). Teploperenos v anizotropnykh tverdykh telakh. Chislennyye metody, teplovyye volny, obratnyye zadachi [Heat transfer in anisotropic solid bodies. Numerical methods, heat waves, inverse problems]. Moscow: Fizmatlit, 280 p. (in Russian).
- Kuznetsova, E. L. (2011). Solution of the inverse problems of heat admittance in order to derive characteristics of anisotropic materials. High Temperature, vol. 49, pp. 881–886. https://doi.org/10.1134/S0018151X11060162.
- Formalov, V. F. (2001). Teplomassoperenos v anizotropnykh telakh. Obzor [Heat and mass transfer in anisotropic bodies. Overview]. Teplofizika vysokikh temperatur – High Temperature, vol. 39, no. 5, pp. 810–832 (in Russian).
- Kolesnik, S. A. (2013). Metod chislennogo resheniya obratnykh nelineynykh zadach po vosstanovleniyu komponentov tenzora teploprovodnosti anizotropnykh materialov [Method of numerical solution of inverse nonlinear problems on the recovery of components of the heat conductivity tensor of anisotropic materials]. Vychislitelnyye tekhnologii – Computational Technologies, vol. 18, no. 1, pp. 34–44 (in Russian).
- Mаtsevytyi, Yu. M. & Hanchyn, V. V. (2020). Multiparametric identification of several thermophysical characteristics by solving the internal inverse heat conduction problem. Journal of Mechanical Engineering – Problemy Mashynobuduvannia, vol. 23, no. 2, pp. 14–20. https://doi.org/10.15407/pmach2020.02.014.
- Маtsevytyi, Yu. М. & Hanchyn, V. V. (2021). To the solution of geometric inverse heat conduction problems. Journal of Mechanical Engineering – Problemy Mashynobuduvannia, vol. 24, no. 1, pp. 6–12. https://doi.org/10.15407/pmach2021.01.006.
- Krukovskiy, P. G. (1998). Obratnyye zadachi teplomassoperenosa (obshchiy inzhenernyy podkhod) [Inverse problems of heat and mass transfer (general engineering approach)]. Kiyev: Institute of Technical Thermophysics, National Academy of Sciences of Ukraine, 224 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 – Problemy Mashynobuduvannia, 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 problems of heat conduction]. Problemy mashinostroyeniya – Journal of Mechanical Engineering – Problemy Mashynobuduvannia, vol. 19, no. 1, pp. 28–36 (in Russian). https://doi.org/10.15407/pmach2016.01.028.
Received 29 March 2021
Published 30 September 2021