Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Vatulyan A. O., Nesterov S. A. On the Peculiarities of Solving the Coefficient Inverse Problem of Heat Conduction for a Two-Part Layer. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 4, pp. 409-423. DOI: 10.18500/1816-9791-2019-19-4-409-423, EDN: FWBUCB

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
02.12.2019
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Language: 
Russian
Heading: 
Mechanics
Article type: 
Article
UDC: 
536.24
DOI: 
10.18500/1816-9791-2019-19-4-409-423
EDN: 
FWBUCB

On the Peculiarities of Solving the Coefficient Inverse Problem of Heat Conduction for a Two-Part Layer

Autors: 
Vatulyan Alexander Ovanesovitsch, Southern Federal University
Nesterov Sergey A., Southern Mathematical Institute — the Affiliate of Vladikavkaz Scientific Centre of Russian Academy of Sciences
Abstract: 

The coefficient inverse problem of thermal conductivity about the determination of the thermophysical characteristics of the functional-gradient part of a two-component layer is posed. The input information is the temperature measurement data on the top face of the layer. After the Laplace transform and dimensioning, the direct problem of heat conduction is solved on the basis of Galerkin projection method. Conversion of transformant on the basis of the theory of residues is carried out. The influence of various laws of changes in the thermophysical characteristics and thickness of the functional-gradient part on the input information was studied. To solve the inverse problem, two approaches are used. The first approach is based on the algebraization of the direct problem using Galerkin projection method. The second approach is a development of the previously developed iterative approach, at each step of which the Fredholm integral equation of the first kind is solved. Computational experiments were carried out to restore various laws of change in thermophysical characteristics. Practical advice on the choice of a time interval for additional information is given. A comparison of two approaches to solving the coefficient inverse problem of heat conduction is made. 

Key words: 
thermal conductivity coefficient
specific heat capacity
layered materials
functional-gradient materials
inverse problem
identification
algebraization
iterative scheme
integral equation
Galerkin projection method
References: 
  1. Wetherhold R. C., Seelman S., Wang S. The use of functionally graded materials to eliminated or control thermal deformation. Compos. Science Techn., 1996, vol. 56, iss. 9, pp. 1099–1104. DOI: https://doi.org/10.1016/0266-3538(96)00075-9
  2. Birman V., Byrd L. W. Modeling and analysis of functionally graded materials and structures Appl. Mech. Rev., 2007, vol. 60, iss. 5, pp. 195–216. DOI: https://doi.org/10.1115/1.2777164
  3. Agisheva D. K., Shapovalov V. M. Engineering Analysis of Non-Steady-State Heat Conduction of Multi-Layer Plate. Vestnik TGTU, 2002. vol. 8, no. 4, pp. 612–617 (in Russian).
  4. Kudinov V. A., Kuznetsova A. E., Eremin A. V., Kotova E. V. Analytical solutions of thermoelasticity problems for multilayer structures with variable properties. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2013, iss. 1 (30), pp. 215–221 (in Russian). DOI: https://doi.org/10.14498/vsgtu1128
  5. Alifanov O. M., Artyukhin E. A., Rumyantsev S. V. Ekstremal’nye metody resheniya nekorrektnykh zadach [Extreme Methods of Solving Ill-Posed Problems]. Moscow, Nauka, 1988. 288 p. (in Russian).
  6. Lesnic D., Elliot L., Ingham D. B., Clennell B., Knioe R. J. The identification of the piecewise homogeneous thermal conductivity of conductors subjected to a heat flow test. International Journal of Heat and Mass Transfer, 1999, vol. 42, iss. 1, pp. 143–152. DOI: https://doi.org/10.1016/S0017-9310(98)00132-X
  7. Penenko A. V. Discrete-analytic schemes for solving an inverse coefficient heat conduction problem in a layered medium with gradient methods. Numerical Analysis and Applications, 2012, vol. 5, iss. 4, pp. 326–341. DOI: https://doi.org/10.1134/S1995423912040052
  8. Lukasievicz S. A., Babaei R., Qian R. E. Detection of material properties in a layered body by means of thermal effects. J. Thermal Stresses, 2003, vol. 26, no. 1, pp. 13–23. DOI: https://doi.org/10.1080/713855763
  9. Pobedrya B. E., Kravchuk A. S., Arizpe P. A. Identification of the coefficients in a nonstationary heat conductivity equation. Computational Continuum Mechanics, 2008, vol. 1, no. 4, pp. 78–87 (in Russian). DOI: https://doi.org/10.7242/1999-6691/2008.1.4.41
  10. Denisov A. M. Vvedeniye v teoriyu obratnykh zadach [Introduction to the theory of inverse problems]. Moscow, Moscow Univ. Press, 1994. 206 p. (in Russian).
  11. Kravaris C., Seinfeld J. H. Identification of spatially varying parameters in distributed parameters systems by discrete regularization. J. Math. Analys. Appl., 1986, vol. 119, pp. 128–152. DOI: https://doi.org/10.1137/0323017
  12. Chen W. L., Chou H. M., Yang Y. C. An inverse problem in estimating the space — dependent thermal conductivity of a functionally graded hollow cylinder. Composites Part B: Engineering, 2013, vol. 50, pp. 112–119. DOI: https://doi.org/10.1016/j.compositesb.2013.02.010
  13. Kabanikhin S. I., Hasanov A., Penenko A. V. A gradient descent method for solving an inverse coefficient heat conduction problem. Numerical Analysis and Applications, 2008, no. 1, pp. 34–45. DOI: https://doi.org/10.1134/S1995423908010047
  14. Hao D. N. Methods for inverse heat conduction problems. Frankfurt/Main, Peter Lang Publ. Inc, 1998. 249 p.
  15. Isakov V., Kindermann S. Identification of the diffusion coefficient in a one dimensional parabolic equation. Inverse Problems, 2000, vol. 16, no. 3, pp. 665–680. DOI: https://doi.org/10.1088/0266-5611/16/3/309
  16. Raudensky M., Woodbary K. A., Kral J. Genetic algorithm in solution of inverse heat conduction problems. Numerical Heat Transfer, Part B: Fundamentals, 1995, vol. 28, pp. 293–306. DOI: https://doi.org/10.1080/10407799508928835
  17. Xu M. H., Cheng J. C., Chang S. Y. Reconstruction theory of the thermal conductivity depth profiles by the modulated photo reflectance technique. J. Appl. Phys., 2004, vol. 84, no. 2, pp. 675–682. DOI: https://doi.org/10.1063/1.368122
  18. Vatul’yan A. O., Nesterov S. A. A Method of Identifying Thermoelastic Characteristics for Inhomogeneous Bodies. Journal of Engineering Physics and Thermophysics, 2014, vol. 87, iss. 1, pp. 225–232. DOI: https://doi.org/10.1007/s10891-014-1004-6
  19. Nedin R., Nesterov S., Vatulyan A. On an inverse problem for inhomogeneous thermoelastic rod. International Journal of Solids and Structures, 2014, vol. 51, iss. 3–4, pp. 767–773. DOI: https://doi.org/10.1016/j.ijsolstr.2013.11.003
  20. Nedin R., Nesterov S., Vatulyan A. On reconstruction of thermalphysic characteristics of functionally graded hollow cylinder. Appl. Math. Modell., 2016, vol. 40, iss. 4, pp. 2711– 2719. DOI: https://doi.org/10.1016/j.apm.2015.09.078
  21. Nedin R., Nesterov S., Vatulyan A. Identification of thermal conductivity coefficient and volumetric heat capacity of functionally graded materials. International Journal of Heat and Mass Transfer, 2016, vol. 102, pp. 213–218. DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2016.06.027
  22. Vatulyan A. O., Nesterov S. A. On an Approach to the Solution of the Coefficient Inverse Heat Conduction Problem. Ekologicheskiy vestnik nauchnykh tsentrov CHES [Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation], 2018, vol. 15, no. 1, pp. 50–60 (in Russian). DOI: https://doi.org/10.31429/vestnik-15-1-50-60
  23. Vatulyan A. O. Koeffitsyentnye obratnye zadachi mekhaniki [Coefficient inverse problems of mechanics]. Moscow, Fizmatlit, 2019. 272 p. (in Russian).
  24. Danilaev P. G. Coefficient inverse problems for parabolic type equations and their applications. Utrecht, Boston, Koln, Tokyo, VSP, 2001. 115 p.
  25. Lam T. T., Yeung W. K. Inverse determination of thermal conductivity for onedimensional problems. J. Themophys. Heat Transf., 1995, vol. 9, no. 2, pp. 335–344. DOI: https://doi.org/10.2514/3.665
  26. Yeung W. K., Lam T. T. Second-order finite difference approximation for inverse determination of thermal conductivity. International Journal of Heat and Mass Transfer. 1996, vol. 39, iss. 17, pp. 3685–3693. DOI: https://doi.org/10.1016/0017-9310(96)00028-2
  27. Marple S. L. Tsifrovoi spektral’nyi analiz i ego prilozheniya [Digital spectral analysis and its applications]. Moscow, Mir, 1990. 584 p.(in Russian).
  28. Tikhonov A. N., Goncharskiy A. V., Stepanov V V., Yagola A. G. Chislennye metody resheniya nekorrektnykh zadach [Numerical methods for solving ill-posed problems]. Moscow, Nauka, 1990. 230 p. (in Russian).
Received: 
23.05.2019
Accepted: 
30.06.2019
Published: 
02.12.2019
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