Yu.M. Matsevytyi, M.O. Safonov, I.V. Hroza
An approach to solving the internal inverse problem of heat conduction (OCT) is proposed, using the principle of regularization of A. N. Tikhonov and the method of influence functions. In this work, the power of the energy source is presented as a linear combination of the first-order Schoenberg splines, and the temperature is presented as a linear combination of influence functions. The influence function method makes it possible to use the same vector of unknown coefficients in both representations. The unknown coefficients are found as a result of solving the system of equations, which follows from the necessary condition for the minimum of the functional of A.N. Tikhonov with an effective algorithm for finding the regularization parameter, the use of which makes it possible to obtain a stable solution to the GST. For the regularization of the OST solutions, this functional also uses a stabilizing functional with the regularization parameter as a multiplicative factor. The article presents the numerical results of identifying the power of thermal energy by temperature, measured with an error characterized by a random variable distributed according to the normal law.
inverse problem, influence functions, identification, regularization, functional.
- Beck, J.V., Blackwell, B. and St. Clair, C.R., jr. (1985), Inverse heat conduction. Ill-posed problems, J Wiley & Sons, NY, USA. URL: https://doi.org/10.1002/ 19870670331.
- Matsevytyi, Yu.M. (2002), Inverse heat conduction problems: in 2 vols. Vol. 1. Metodologiya, Naukova dumka, Kyiv, Ukraine.
- Kozdoba, L.A. and Krukovskyi, P.G. (1982), Metody resheniya obratnykh zadach teploperenosa [Methods for solving inverse heat transfer problems], Naukova dumka, Kyiv, Ukraine.
- Alifanov, O.M., Artyukhin, Ye.A. and Rumyantsev, S.V. (1988), Ekstremalnyye metody resheniya nekorrektnykh zadach [Extreme methods for solving ill-posed problems], Nauka, Moscow, Russia.
- Tikhonov, A.N. and Arsenin, V.Ya. (1979), Metody resheniya nekorrektnykh zadach. [Methods for solving ill-posed problems], Nauka, Moscow, Russia.
- Matsevytyi, Yu.M. and Slesarenko, A.P. (2014), Nekorrektnyye mnogoparametricheskiye zadachi teploprovodnosti i regionalno-strukturnaya regulyarizatsiya ikh resheniy [Incorrect Incorrect multi-parameter heat conduction problems and regional structural regularization their solutions], Naukova dumka, Kyiv, Ukraine.
- Khamzaev, Kh.M. (2020), “Algorithm for determining the trajectory of the heat source along the heated homogeneous rod”, Elektronnoye modelirovaniye, Vol. 42, no. 1, pp. 25–32.
- Guseynzade, S.O. (2018), “Pressure recovery at the reservoir boundary based on the solution of the inverse problem”, Elektronnoye modelirovaniye, 40, no. 4, pp. 19–28.
- Ivanov, V.K., Vasin V.V. and Tanaka V.P. (1978), Teoriya lineynykh nekorrektnykh zadach i yeye prilozhtniya [Theory of linear ill-posed problems and its applications], Nauka, Moscow, Russia.
- Vatulyan, A.O. (2007), Obratnye zadachi v mekhanike deformiruemogo tvyordogo tela [Inverse problems in mechanics of deformable solids], Fizmatlit, Moscow, Rusiia.
- Sergienko, I.V. and Deyneka, V.S. (2009), Sistemnyi analiz mnogokompanentnykh raspredelyonnykh system [System analysis of multicomponent distributed systems], Naukova dumka, Kyiv, Ukraine.
- Denisov, A.M. (1994), Vvedeniye v teoriyu obratnykh zadach [Introduction to the theory of inverse problems], Izdatelstvo MGU, Moscow, Russia.
- Romanov, V.G. (1984), Obratnye zadachi matematicheskoi fiziki [Inverse problems of mathematical physics], Nauka, Moscow, Russia.
- Kabanikhin, S.I. (2009), Obratnye i nekorrektnye zadachi [Inverse and ill-posed problems], Sibirskoye nauchnoye izdatelstvo, Novosibirsk, Russia.
- Vikulov, A.G. and Nenarokomov, A.V. (2019), “Identification of mathematical models of heat transfer in spacecraft”, Inzhenerno-fizicheskiy zhurnal, 92, no. 1, pp. 32–44.
- Golovin, D.Yu., Divin, A.G., Samodurov, A.A., Turin, A.I. and Golovin, Yu.I. (2020), “New express method for determining the thermal diffusivity of materials and finished products”, Inzhenerno-fizicheskiy zhurnal, 93, no. 1, pp. 240–247.
- Nenarokomov, A.V., Chebakov, E.V., Krainova, I.V., Morzhukhina, A.V., Reviznikov, D.L. and Titov, D.M. (2019), “Geometric inverse problems of radiation heat transfer as applied to the development of backup attitude control systems for spacecraft”, Inzhenerno-fizicheskiy zhurnal, 92, no. 4, pp. 979–987.
- Machanyek, A.A., Goranov, V.A. and Dedyu, V.A. (2019), “Determination of the thickness of the protein layer on the surface of polydisperse nanoparticles from the distribution of their concentration along the measuring channel”, Inzhenerno-fizicheskiy zhurnal, 92, no. 1, pp. 21–32.
- Ahlberg, J.H., Nilson, E.N. and Walsh, J.L. (1967), The theory of splines and their applications, Academic Press. URL: https://doi.org/10.1002/19700500646.
- Matsevytyi, Yu.M. and Hanchin, V.V. (2020), “Multiparametric identification of several thermophysical characteristics by solving the internal inverse heat conduction problem”, Problemy mashinostroeniya, 23, no 2, pp. 14–20.
- Matsevytyi, Yu.M. and Lushpenko, S.F. (1990), Identificatsiya teplofizicheskikh cvoystv tverdykh materialov [Identification of thermophysical properties of solid materials], Naukova dumka, Kyiv, Ukraine.
- Matsevytyi, Yu.M., Safonov, N.A. and Hanchin, V.V. (2016), “On the solution of nonlinear inverse boundary problem of heat conduction”, Problemy mashinostroeniya, 19, no. 1, pp. 28–36.
- Matsevytyi, Yu.M., Sirenko, V.N., Kostikov, A.O., Safonov, N.A. and Hanchin, V.V. (2020), “Method for identification of non-stationary thermal processes in multilayer structures”, Kosmicheskaya nauka i nechnologiya, Vol. 26, no. 1(122), pp. 79–89.
- Matsevytyi, Yu.M., Kostikov, A.O., Safonov, N.A. and Hanchin, V.V. (2017), “To the solution of nonstationary nonlinear boundary value problems of heat conduction”, Problemy mashinostroeniya, Vol. 20, no. 1–2, pp. 34–45. URL: https://doi.org/10.15407/ pmach2017.02. 022.