Skip to main content
SpringerLink
Log in

Identification of the Thermal Conductivity Coefficient in the Three-Dimensional Case by Solving a Corresponding Optimization Problem

  • OPTIMAL CONTROL
  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

The inverse problem of determining a temperature-dependent thermal conductivity coefficient in a parallelepiped is considered and investigated. The consideration is based on the Dirichlet boundary value problem for the three-dimensional nonstationary heat equation. The coefficient inverse problem is reduced to an optimization problem, which is solved numerically by applying gradient methods for functional minimization. The performance and efficiency of the proposed approach are demonstrated by solving several nonlinear problems with temperature-dependent coefficients.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. L. A. Kozdoba and P. G. Krukovskii, Methods for Solving Inverse Heat Transfer Problems (Naukova Dumka, Kiev, 1982) [in Russian].

    Google Scholar 

  2. O. M. Alifanov, Inverse Heat Transfer Problems (Mashinostroenie, Moscow, 1988; Springer, Berlin, 2011).

  3. P. N. Vabishchevich and A. Yu. Denisenko, “Numerical methods for solving coefficient inverse problems,” in Methods of Mathematical Modeling and Computational Diagnostics (Mosk. Gos. Univ., Moscow, 1990), pp. 35–45 [in Russian].

    Google Scholar 

  4. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics (Nauka, Moscow, 1997; de Gruyter, Berlin, 2007).

  5. G. I. Marchuk, Adjoint Equations and Analysis of Complex Systems (Nauka, Moscow, 1992; Kluwer Academic, Dordrecht, 1995).

  6. B. Czél and G. Gróf, “Inverse identification of temperature-dependent thermal conductivity via genetic algorithm with cost function-based rearrangement of genes,” Int. J. Heat Mass Transfer 55 (15), 4254–4263 (2012).

    Article  Google Scholar 

  7. Miao Cui, Kai Yang, Xiao-liang Xu, Sheng-dong Wang, and Xiao-wei Gao, “A modified Levenberg–Marquardt algorithm for simultaneous estimation of multi-parameters of boundary heat flux by solving transient nonlinear inverse heat conduction problems,” Int. J. Heat Mass Transfer 97, 908–916 (2016).

    Article  Google Scholar 

  8. Yu. M. Matsevityi, S. V. Alekhina, V. T. Borukhov, G. M. Zayats, and A. O. Kostikova, “Identification of the thermal conductivity coefficient for quasi-stationary two-dimensional heat conduction equations,” J. Eng. Phys. Thermophys. 90 (6), 1295–1301 (2017).

    Article  Google Scholar 

  9. Yu. G. Evtushenko, Optimization and Fast Automatic Differentiation (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 2013) [in Russian].

    Google Scholar 

  10. A. F. Albu, Yu. G. Evtushenko, and V. I. Zubov, “Choice of finite-difference schemes in solving coefficient inverse problems,” Comput. Math. Math. Phys. 60 (10), 1589–1600 (2020).

    Article  MathSciNet  Google Scholar 

  11. A. A. Samarskii and P. N. Vabishchevich, Computational Heat Transfer (Wiley, New York, 1996; Editorial URSS, Moscow, 2003).

  12. J. Douglas and H. H. Rachford, “On the numerical solution of heat conduction problems in two and three space variables,” Trans. Am. Math. Soc. 82, 421–439 (1956).

    Article  MathSciNet  Google Scholar 

  13. D. W. Peaceman and H. H. Rachford, “The numerical solution of parabolic and elliptic differential equations,” J. Soc. Ind. Appl. Math. 3 (1), 28–41 (1955).

    Article  MathSciNet  Google Scholar 

  14. Ch. Gao and Y. Wang, “A general formulation of Peaceman and Rachford ADI method for the N-dimensional heat diffusion equation,” Int. Commun. Heat Mass Transfer 23 (6), 845–854 (1996).

    Article  Google Scholar 

  15. A. F. Albu, Yu. G. Evtushenko, and V. I. Zubov, “Application of the fast automatic differentiation technique for solving inverse coefficient problems,” Comput. Math. Math. Phys. 60 (1), 15–25 (2020).

    Article  MathSciNet  Google Scholar 

  16. A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1977; Marcel Dekker, New York, 2001).

  17. A. F. Albu and V. I. Zubov, “Identification of thermal conductivity coefficient using a given temperature field,” Comput. Math. Math. Phys. 58 (10), 1585–1599 (2018).

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation, project no. 075-15-2020-799.

Author information

Authors and Affiliations

  1. Federal Research Center “Computer Science and Control,” Russian Academy of Sciences, 119333, Moscow, Russia

    A. F. Albu & V. I. Zubov

Authors
  1. A. F. Albu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. I. Zubov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. I. Zubov.

Additional information

Translated by I. Ruzanova

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Albu, A.F., Zubov, V.I. Identification of the Thermal Conductivity Coefficient in the Three-Dimensional Case by Solving a Corresponding Optimization Problem. Comput. Math. and Math. Phys. 61, 1416–1431 (2021). https://doi.org/10.1134/S0965542521090037

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542521090037

Keywords:

Search

Navigation