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On Simultaneous Determination of Thermal Conductivity and Volume Heat Capacity of Substance

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Abstract

The study of nonlinear problems associated with heat transfer in substance is important for practice. Earlier, the authors proposed an efficient algorithm for determining the thermal conductivity from experimental observations of the dynamics of the temperature field in an object. In this work, we explore the possibility of extending the algorithm to the numerical solution of the problem of simultaneous identification of the temperature-dependent volume heat capacity and the thermal conductivity of the substance under study. The consideration is based on the Dirichlet boundary value problem for the one-dimensional nonstationary heat equation. The coefficient inverse problem in question is reduced to a variational problem, which is solved by applying gradient methods based on the fast automatic differentiation technique. The uniqueness of the solution to the inverse problem is analyzed.

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REFERENCES

  1. V. G. Zverev, V. D. Gol’din, and V. A. Nazarenko, “Radiation–conduction heat transfer in fibrous heat-resistant insulation under thermal effect,” High Temp. 46 (1), 108–114 (2008).

    Article  Google Scholar 

  2. O. M. Alifanov and V. V. Cherepanov, “Mathematical simulation of high-porosity fibrous materials and determination of their physical properties,” High Temp. 47 (3), 438–447 (2009).

    Article  Google Scholar 

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

  4. W. K. Yeung and T. T. Lam, “Second-order finite difference approximation for inverse determination of thermal conductivity,” Int. J. Heat Mass Transfer 39, 3685–3693 (1996).

    Article  Google Scholar 

  5. H. T. Chen, J. Y. Lin, C. H. Wu, and C. H. Huang, “Numerical algorithm for estimating temperature-dependent thermal conductivity,” Numer. Heat Transfer B 29, 509–522 (1996).

    Article  Google Scholar 

  6. A. A. Samarskii and P. N. Vabishchevich, Computational Heat Transfer (Wiley, New York, 1996).

    Google Scholar 

  7. S. Kim, M. C. Kim, and K. Y. Kim, “Non-iterative estimation of temperature dependent thermal conductivity without internal measurements,” Int. J. Heat Mass Transfer 46, 1891–1810 (2003).

    Article  MATH  Google Scholar 

  8. E. Majchrzak, K. Freus, and S. Freus, “Identification of temperature dependent thermal conductivity using the gradient method,” J. Appl. Math. Comput. Mech. 5 (1), 114–123 (2006).

    MATH  Google Scholar 

  9. 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 

  10. 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 

  11. Y. Evtushenko, V. Zubov, and A. Albu, “Inverse coefficient problems and fast automatic differentiation,” J. Inverse Ill-Posed Probl. 30 (3), 447–460 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  12. C. H. Huang and J. Y. Yan, “An inverse problem in simultaneously measuring temperature-dependent thermal conductivity and heat capacity,” Int. J. Heat Mass Transfer 38, 3433–3441 (1995).

    Article  Google Scholar 

  13. A. Imani, A. A. Ranjbar, and M. Esmkhani, “Simultaneous estimation of temperature-dependent thermal conductivity and heat capacity based on modified genetic algorithm,” Inverse Probl. Sci. Eng. 14 (7), 767–783 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

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

    Google Scholar 

  16. Yu. G. Evtushenko and V. I. Zubov, “Generalized fast automatic differentiation technique,” Comput. Math. Math. Phys. 56 (11), 1819–1833 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  17. Yu. G. Evtushenko, E. S. Zasukhina, and V. I. Zubov, “Numerical optimization of solutions to Burgers’ problems by means of boundary conditions,” Comput. Math. Math. Phys. 37 (12), 1406–1414 (1997).

    MathSciNet  MATH  Google Scholar 

  18. A. F. Albu and V. I. Zubov, “Investigation of the optimal control of metal solidification for a complex-geometry object in a new formulation,” Comput. Math. Math. Phys. 54 (12), 1804–1816 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  19. Yu. G. Evtushenko, V. I. Zubov, and A. F. Albu, Optimal Control of Thermal Processes with Phase Transitions (Maks Press, Moscow, 2021) [in Russian]. https://doi.org/10.29003/m2449.978-5-317-06677-2

    Book  Google Scholar 

  20. A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).

  21. A. F. Albu and V. I. Zubov, “On the efficient solution of optimal control problems using the fast automatic differentiation approach,” Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 21 (4), 20–29 (2015).

    Google Scholar 

  22. L. Hascoet and V. Pascual, “The Tapenade automatic differentiation tool: principles, model, and specification,” ACM Trans. Math. Software 39 (3), 1–43 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  23. R. J. Hogan, “Fast reverse-mode automatic differentiation using expression templates in C++,” ACM Trans. Math. Software 40 (4), 26–42 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Yu. Gorchakov, “On software packages for fast automatic differentiation,” Inf. Tekhnol. Vychisl. Sist., No. 1, 30–36 (2018).

  25. A. Albu, A. Gorchakov, and V. Zubov, “On the effectiveness of the fast automatic differentiation methodology,” Commun. Comput. Inf. Sci. 974, 264–276 (2019).

    MathSciNet  MATH  Google Scholar 

  26. Yixuan Qiu, L-BFGS++ (2021). https://github.com/yixuan/LBFGSpp/

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Funding

This work was carried out using the resources of the shared facility center “High-Performance Computations and Big Data” (Informatics) of the Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences (Moscow).

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Authors and Affiliations

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

    A. Yu. Gorchakov & V. I. Zubov

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  1. A. Yu. Gorchakov
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  2. V. I. Zubov
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Correspondence to V. I. Zubov.

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The authors declare that they have no conflicts of interest.

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Translated by I. Ruzanova

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Gorchakov, A.Y., Zubov, V.I. On Simultaneous Determination of Thermal Conductivity and Volume Heat Capacity of Substance. Comput. Math. and Math. Phys. 63, 1408–1423 (2023). https://doi.org/10.1134/S0965542523080079

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  • DOI: https://doi.org/10.1134/S0965542523080079

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