To the Solution of Geometric Inverse Heat Conduction Problems

Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Pidhornyi Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 24, no. 1, 2021 (March)
Pages 6-12
Cited by J. of Mech. Eng., 2021, vol. 24, no. 1, pp. 6-12



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

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



On the basis of A. N. Tikhonov’s regularization theory, a method is developed for solving inverse heat conduction problems of identifying a smooth outer boundary of a two-dimensional region with a known boundary condition. For this, the smooth boundary to be identified is approximated by Schoenberg’s cubic splines, as a result of which its identification is reduced to determining the unknown approximation coefficients. With known boundary and initial conditions, the body temperature will depend only on these coefficients. With the temperature expressed using the Taylor formula for two series terms and substituted into the Tikhonov functional, the problem of determining the increments of the coefficients can be reduced to solving a system of linear equations with respect to these increments. Having chosen a certain regularization parameter and a certain function describing the shape of the outer boundary as an initial approximation, one can implement an iterative process. In this process, the vector of unknown coefficients for the current iteration will be equal to the sum of the vector of coefficients in the previous iteration and the vector of the increments of these coefficients, obtained as a result of solving a system of linear equations. Having obtained a vector of coefficients as a result of a converging iterative process, it is possible to determine the root-mean-square discrepancy between the temperature obtained and the temperature measured as a result of the experiment. It remains to select the regularization parameter in such a way that this discrepancy is within the measurement error. The method itself and the ways of its implementation are the novelty of the material presented in this paper in comparison with other authors’ approaches to the solution of geometric inverse heat conduction problems. When checking the effectiveness of using the method proposed, a number of two-dimensional test problems for bodies with a known location of the outer boundary were solved. An analysis of the influence of random measurement errors on the error in identifying the outer boundary shape is carried out.

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


Full text: Download in PDF



  1. Matsevytyy, Yu. M. & Kostikov, A. O. (2014). Geometricheskiye obratnyye zadachi teploobmena [Geometric inverse problems of heat transfer]. Kiyev: Naukova dumka. 223 p. (in Russian).
  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. 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).
  4. Tikhonov, A. N. & Arsenin, V. Ya. (1979). Metody resheniya nekorrektnykh zadach [Methods for solving ill-posed problems]. Moscow: Nauka, 288 p. (in Russian).
  5. 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.
  6. Kostikov, A. O. (2004) Yedinyy metodologicheskiy podkhod k postanovke i resheniyu geometricheskikh obratnykh zadach teploprovodnosti [A unified methodological approach to the formulation and solution of geometric inverse problems of heat conduction]. Problemy Mashinostroyeniya Journal of Mechanical Engineering, vol. 7, no. 4, pp. 52-60 (in Russian).
  7. 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).
  8. Lavrentyev, M. M. (1962). O nekotorykh neofitsialnykh zadachakh matematicheskoy fiziki [On some ill-posed problems of mathematical physics]. Novosibirsk: Publishing house of the Siberian Branch of the USSR Academy of Sciences, 68 p. (in Russian).
  9. Mаtsevitiyy, Yu. M. & Ganchin, V. V. (2020). Multiparametric identification of several thermophysical characteristics by solving the internal inverse heat conduction problem. Journal of Mechanical Engineering, vol. 23, no. 2, pp. 14-20.
  10. Tikhonov, A. N. & Samarskiy, A. A. (1966). Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow: Nauka, 596 p. (in Russian).
  11. 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 mashinostroyeniyaJournal of Mechanical Engineering, vol. 2, no. 1–2, pp. 34–42 (in Russian).
  12. 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 mashinostroyeniyaJournal of Mechanical Engineering, vol. 19, no. 1, pp. 28–36 (in Russian).



Received 09 December 2020

Published 30 March 2021