An approach to solving the inverse problem of thermoelasticity that employs A. N. Tikhonov′s regularization principle and the method of influence functions has been suggested. The use of A. N. Tikhonov′s regularization with an efficient algorithm of the search for the regularization parameter makes it possible to obtain a stable solution of the inverse problem of thermoelasticity. The unknown functions of displacement and temperature are approximated by the Schoenberg splines, whereas the unknown coefficients of these functions are calculated by solving the system of linear algebraic equations. This system results from the necessary condition of the functional minimum based on the principle of least squares of the deviation of the calculated stress from the stress obtained experimentally. To regularize the solutions of the inverse heat conduction problems, this functional also employs a stabilizing functional with the regularization parameter as a multiplicative multiplier. The search for the regularization parameter is effected with the aid of an algorithm analogous to the algorithm of the search for the root of a nonlinear equation, whereas the use of the influence functions makes it possible to represent temperature stresses and temperature as a function of one and the same sought vector. This article presents numerical results on temperature identification by thermal stresses measured with an error characterized by a random quantity distributed by the normal law.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu. M. Matsevityi, Yu. S. Postol′nik, and V. O. Povgorodnii, Inverse problems of thermomechanics, Probl. Mashinostr., No. 3, 30–37 (2008).
A. G. Vikulov and A. V. Nenarokomov, Identification of mathematical models of heat exchange in space vehicles, J. Eng. Phys. Thermophys., 92, No. 1, 29–42 (2019).
Kh. M. Gamzaev, On inverse problem of acoustic flow, J. Eng. Phys. Thermophys., 92, No. 1, 162–168 (2019).
D. Yu. Golovin, A. G. Divin, A. A. Samodurov, A. I. Tyurin, and Yu. I. Golovin, A new rapid method of determining the thermal diffusivity of materials and finished articles, J. Eng. Phys. Thermophys., 93, No. 1, 234–240 (2020).
S. V. Alekhina, V. T. Borukhov, G. M. Zayats, A. O. Kostikov, and Yu. M. Matsevityi, Identification of the thermal conductivity coefficient for quasi-stationary two-dimensional heat conduction equations, J. Eng. Phys. Thermophys., 90, No. 6, 1295–1301 (2017).
A. A. Makhanek, V. A. Goranov, and V. A. Dedew, Determination of the protein layer thickness on the surface of polydisperse nanoparticles from the distribution of their concentration along a measuring channel, J. Eng. Phys. Thermophys., 92, No. 1, 19–28 (2019).
A. V. Nenarokomov, E. V. Chebakov, I. V. Krainova, A. V. Morzhukhina, D. L. Reviznikov, and D. M. Titov, Geometric inverse problem of radiative heat transfer as applied to the development of alternate spacecraft orientation systems, J. Eng. Phys. Thermophys., 92, No. 4, 948–955 (2019).
S. P. Timoshenko and J. N. Goodier, Theory of Elasticity [in Russian], Nauka, Moscow (1979).
A. D. Kovalenko, Thermoelasticity [in Russian], Vishcha Shkola, Kiev (1975).
G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations [in Russian], Mir, Moscow (1980).
A. N. Tikhonov and A. N. Arsenin, Methods of Solution of Ill-Posed Problems [in Russian], Nauka, Moscow (1979).
J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications [Russian translation], Mir, Moscow (1972).
Yu. M. Matsevityi, Inverse Heat Conduction Problems, in 2 vols., Vol. 1: Methodology, Vol. 2: Appendices [in Russian], Naukova Dumka, Kiev (2002–2003).
C. A. J. Fletcher, Computational Techniques for Fluid Dynamics [Russian translation], Mir, Moscow (1988).
Yu. M. Matsevityi, N. A. Safonov, and V. V. Ganchin, Concerning the solution of nonlinear inverse boundary-value heat conduction problems, Probl. Mashinostr., 19, No. 1, 28–36 (2016).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 94, No. 5, pp. 1134–1140, September–October, 2021.
Rights and permissions
About this article
Cite this article
Matsevityi, Y.M., Strel’nikova, E.A., Povgorodnii, V.O. et al. Methodology of Solving Inverse Heat Conduction and Thermoelasticity Problems for Identification of Thermal Processes. J Eng Phys Thermophy 94, 1110–1116 (2021). https://doi.org/10.1007/s10891-021-02391-w
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10891-021-02391-w