The application of inverse problems to thermal diagnostics of homogeneous structures is outlined, variational formulations of the problems under study are considered, their solvability is proven, and an iterative solution algorithm is described. The results are applied to the thermal diagnostics of thermoelastic structures, a theorem of existence and uniqueness of the solution to the problem of determining the inhomogeneous thermal characteristics of the medium, which is used to restore the physical and technical image of the diagnosed object, is proved, and the necessary condition for the extremum is formulated in the form of a variational inequality. An example of a numerical solution to the problem under consideration is given and an analysis of the calculation results on the application of the proposed diagnostic technique is given.
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References
A. N. Tikhonov and V. Ya. Arsenin, Methods for Solving Ill-Posed Problems [in Russian], Nauka, Moscow (1979).
A. N. Tikhonov, V. Ya. Arsenin, and A. A. Timonov, Mathematical Problems of Computed Tomography [in Russian], Nauka, Moscow (1987).
A. N. Tikhonov, A. S. Leonov, and A. G. Yagola, Nonlinear Ill-Posed Problems [in Russian], KURS, Moscow (2017).
F. P. Vasil'ev, Optimization Methods [in Russian], Faktorial Press, Moscow (2011).
A. D. Iskenderov, Variational formulation of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 274, No. 3, 531–533 (1983).
A. D. Iskenderov, T. B. Gardashov, and T. M. Ibragimov, Solution of inverse problems for a system of quasilinear equations of heat conduction in a self-similar regime, J. Eng. Phys. Thermophys., 58, No. 2, 102–107 (1989).
A. D. Iskenderov, G. Ya. Yagubov, and M. A. Musaeva, Identification of Quantum Potentials [in Russian], Çarşi oğlu, Baku (2012).
M. A. Musaeva, Variational Methods for Determining Quantum Potentials [in Russian], Elm-Takşil, Baku (2018).
M. A. Musaeva, Variational method for determining complex coefficients of nonlinear and nonstationary Shrödingertype equation, Zh. Vychisl. Mat. Fiz., 60, No. 11, 1985–1997 (2020).
D. A. Nesteruk and V. P. Vavilov, Thermal Monitoring and Diagnostics [in Russian], TGU, Tomsk (2007).
O. F. Moshoshin, Diagnostics of Aviation Equipment [in Russian], MGTU im. N. É. Baumana, Moscow (2007).
O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).
M. Goebel, On existence of optimal control, Math. Nachr., 93, 67–73 (1979).
A. V. Lapin, Iterative Methods for Solving Grid Variational Inequalities [in Russian], KGU, Kazan, (2008).
V. K. Ivanov, V. V. Vasin, and V. A. Tanana, Theory of Linear Ill-Posed Problems and Its Applications [in Russian], Nauka, Moscow (1978).
V. A. Morozov, Methods for Regularizing Unstable Problems [in Russian], MGU, Moscow (1987).
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Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 96, No. 6, pp. 1419–1428, November–December, 2023.
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Pashaev, A.M., Iskenderov, A.D. & Musaeva, M.A. Method of Inverse Problems for Thermal Diagnostics of Thermoelastic Structures. J Eng Phys Thermophy 96, 1407–1415 (2023). https://doi.org/10.1007/s10891-023-02808-8
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DOI: https://doi.org/10.1007/s10891-023-02808-8