Multiparametric Identification of Several Thermophysical Characteristics by Solving the Internal Inverse Heat Conduction Problem

Authors

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

Valerii V. Hanchyn, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi St., Kharkiv, 61046, Ukraine), e-mail: gan4ingw@gmail.com, ORCID: 0000-0001-9242-6460

Abstract

Approaches to the identification of thermophysical characteristics, using methods for solving inverse heat conduction problems and A. N. Tikhonov’s regularization method, are developed. According to the results of the experiment, temperature-dependent coefficients of heat conductivity, heat capacity, and internal heat sources are determined. In this case, the thermophysical characteristics are approximated by Schoenberg’s cubic splines, as a result of which their identification reduces to determining unknown coefficients in the approximated dependencies. Therefore, the temperature in the body will depend on these coefficients, and it can be represented using two members of the Taylor series as a linear combination of its partial derivatives with respect to the unknown coefficients, multiplied by the increments of these coefficients. Substituting this expression into the Tikhonov functional and using the minimum property of the quadratic functional, we can reduce the solution of the problem to the solution of a system of linear equations with respect to the increments of unknown coefficients. By choosing a certain regularization parameter and some functions as an initial approximation, we can implement an iterative process in which the vector of unknown coefficients for the current iteration will be equal to the sum of the vector of the coefficients obtained in the previous iteration and the coefficient increment vector as a result of solving a system of linear equations. Such an iterative process of identifying the thermophysical characteristics for each regularization parameter makes it possible to determine the mean-square discrepancy between the resulting temperature and the temperature measured as a result of the experiment. It remains to choose the regularization parameter so that this discrepancy is within the root-mean-square measurement error. Such a search, for example, is identical to algorithms for searching roots of nonlinear equations. When checking the efficiency of using the proposed method, a number of test problems were solved for bodies with known thermophysical characteristics. An analysis of the influence of random measurement errors on the error of the identifiable thermophysical characteristics of the body being studied was carried out.

Keywords: inverse heat conduction problem, Tikhonov’s regularization method, stabilization functional, regularization parameter, identification, approximation, Schoenberg’s cubic splines.

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References

1. 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.
2. 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).
3. 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).
4. 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).
5. Tikhonov, A. N. & Arsenin, V. Ya. (1979). Metody resheniya nekorrektnykh zadach [Methods for solving ill-posed problems]. Moscow: Nauka, 288 p. (in Russian).
6. Matsevityy, Yu. M. & Slesarenko, A. P. (2014). Nekorrektnyye mnogoparametricheskiye zadachi teploprovodnosti i regionalno-strukturnaya regulyarizatsiya ikh resheniy [Incorrect multi-parameter heat conduction problems and regional structural regularization of their solutions]. Kiyev: Naukova dumka, 292 p. (in Russian).
7. Matsevityy, Yu. M. & Multanovskiy, A. V. (1990). Odnovremennaya identifikatsiya teplofizicheskikh kharakteristik sverkhtverdykh materialov [Simultaneous identification of the thermophysical characteristics of superhard materials]. Teplofizika vysokikh temperatur – High Temperature, vol. 5, pp. 924–929 (in Russian).
8. 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).
9. Lavrentyev M. M. (1962). O nekotorykh nekorrektnykh zadachakh matematicheskoy fiziki [About some incorrect problems of mathematical physics]. Novosibirsk: Izdatelstvo Sibirskogo otdeleniya AN SSSR, 68 p. (in Russian).
10. Ivanov, V. K., Vasin, V. V., & Tanaka, V. P. (1978). Teoriya lineynykh nekorrektnykh zadach i yeye prilozheniya [The theory of linear ill-posed problems and its applications]. Moscow: Nauka, 208 p. (in Russian).
11. Tikhonov, A. N. & Samarskiy, A. A. (1966). Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow: Nauka, 596 p. (in Russian).
12. 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).
13. 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, vol. 19, no. 1, pp. 28–36 (in Russian). https://doi.org/10.15407/pmach2016.01.028.

Received 02 March 2020

Published 30 June 2020