To the solution of non-stationary nonlinear reverse problems of thermal conductivity
Keywords:
inverse boundary-value heat conduction problem, heat flux, Tikhonov's regularization method, functional, stabilizer, regularization parameter, identification, approximation, Schoenberg splinesAbstract
To solve the nonlinear boundary inverse heat conduction problem, two approaches are used with the regularizing method of A. N. Tikhonov, for which an effective algorithm for finding the regularizing parameter has been developed. The required functions with respect to the time coordinate are approximated by Schoenberg splines and the boundary inverse problem is reduced to the determination of the approximation coefficients. In the first approach, the temperature function is replaced by two terms of the Taylor series, depending on the approximation parameters. In this case, one must calculate the partial derivatives of the temperature function with respect to all the approximation parameters. Because of the very complicated dependence of the temperature function on the approximation parameters, the partial derivatives must be calculated using the finite difference method, which ultimately leads to the need to solve for each parameter an additional direct problem at each step of the iteration process. This leads to additional computational costs. The second approach uses the influence function method for the linearized mathematical model of the thermal process. This approach allows us to significantly reduce the time of the solution of the problem, but at the first steps of the iterative process it is necessary to take into account that the temperature field is still far from the true state and the nonlinear thermophysical characteristics that depend on this state are still far from the true values. In conclusion, it should be noted that the first approach is more universal, but for a large number of parameters leads to large computational costs. For the second approach, computational costs do not increase as much as for the first approach, but it can be used only for solving boundary inverse heat conduction problems. From this it can be concluded that for the solution of multidimensional inverse problems these two approaches are desirable to be combined.References
Beck, J. Nekotorye obratnye zadachi teploprovodnosti / J. Beck, B. Blakuell, Ch. Sent-Kler (ml.) – M.: Mir, 1989. – 312 s.
Маtsevitiy, Ju. М. Оbratnye zadachi teploprovodnosti: V 2-h т. / Ju. М. Маtsevitiy. – Кiev: Nauk. dumka, 2002-2003. Т. 1: Меtodologiya. – 408 s.; Т. 2: Prilozheniya. – 392 s.
Коzdoba, L. А. Меtody resheniya obratnykh zadach teploperenosa / L. А. Коzdoba., P. G. Кrukovskiy. – Кiev: Nauk. dumka, 1982. – 360 s.
Аlifanov, О. М. Ekstremalnye меtody resheniya nekorrektnykh zadach / О. М. Аlifanov, Е. А. Аrtuchin, S. V. Rumyantsev. – М.: Nauka, 1988. – 288 s.
Тikhonov, А. N. Меtody resheniya nekorrektnyh zadach / А. N. Тikhonov, V. Ja. Аrsenin. – М.: Nauka, 1979. – 288 s.
Маtsevitiy, Ju. М., Slesarenko. А. P. Nekorrektnye mnogoparametricheskie zadachi teploprovodnosti i regionalno-strukturnaya regulyarizatsiya ikh resheniy / Ju. М. Маtsevitiy, A. P. Slesarenko / – К.: Nauk. dumka, 2014. – 292 s.
Shlykovв Ju .P. Kontaktnoe termicheskoe soprotivlenie / Ju. P. Shlykov, E. А. Ganin, S. N. Tsаrevskiy. – М.: Energia, 1977. – 328 s.
Кrukovskiy P. G. Оbratnye zadachi teploperenosa (оbschii inzhenernyi podkhod) / P. G. Кrukovskiy. – К.: In-t tehn. Тeplofiziki NAN Ukrainy, 1998. – 224 s.
Jakovleva R. А. Novye оgnezaschitnye pokrytiya pо мetallu i identifikatsiya ikh teplofizicheskikh svoystv / R. A. Jakovleva, S. L. Fomin, N. A. Safonov, A. M Bezuglyi // Nauk. visnyk budivnytstva, vyp. 48, 2008, s. 250-268. Kharkiv, Kharkivskii derzhavnyi universitet budivnytstva ta arkhitektury , KhОТV АBU 2008.
Маtsevitiy Ju. М. Rеgionalno-analiticheskoe моdelirovanie i identifikatsiya teplovykh potokov s ispol‘zovaniem меtoda regulyarizatsii А. N. Тikhonova / Ju. М. Маtsevitiy, А. P. Slesarenko, V. V. Ganchin // Probl. маshinostroeniya. – 1999. – Т. 2, № 1-2. – S. 34-42.
Маtsevitiy Ju. М. К resheniyu nelineynykh оbratnykh granichnykh zadach teploprovodnosti / Ju М. Маtsevitiy, N. А. Safonov, V. V. Ganchin // Probl. маshinostroeniya. – 2016. – Т. 19, № 1. – S. 28-36.
Graham N. Y. Smoothing with Periodic Cubic Splines / N. Y. Graham // Bell System Tech. J. – 1983. – Vol. 62. – P. 101-110.
Reinsch C. H. J. Smoothing by Spline Function / C. H. J. Reinsch // Numerische Mathematik. – 1967. – Vol. 10. – P. 177-183.
Kartashov E. М. Analiticheskie мetody resheniya krayevykh zadach uravneniya teploprovodnosti v oblasti s dvizhuschimisya granizami: Obzor. / E. М. Kartashov, B. Ya. Lubov // Izv. АN SSSR Energetika i transport. – 1974. №6. – S. 83-111.
Kartashov E. М. Analiticheskie мetody v teorii teploprovodnosti tverdykh tel. – М.: Vysshaya shkola, 2001. – 553 s.
Tikhonov А. N. Uravneniya matematicheskoi fiziki. Ucheb. posobie. / А. N. Tikhonov А. А. Samarskii – М.: Izd-vо МGU, 1999. – 799 s.
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