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Mathematics. Mechanics. Informatics

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

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Vatulyan A. O., Nesterov S. A. Solution of the inverse problem of two thermomechanical characteristics identification of a functionally graded rod. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 2, pp. 180-195. DOI: 10.18500/1816-9791-2022-22-2-180-195, EDN: OTAJWA

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Solution of the inverse problem of two thermomechanical characteristics identification of a functionally graded rod

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

An approach to solving the inverse problem of the simultaneous identification of two thermomechanical characteristics of a functionally graded rod is presented. Two problems of thermoelasticity with different heat loads at the ends of the rod are considered. The input information is the temperature measurement data at the end of the rod over a finite time interval. Direct problems after applying the Laplace transform are solved on the basis of the shooting method and inversion of transformants based on the  expanding the original in a series in the shifted Legendre polynomials. The analysis of the influence of the variable characteristics change laws on the values of the input information taken in the experiment is carried out. The solution to the nonlinear inverse problem is based on an iterative process. The initial approximation for the iterative process is in the class of linear functions, the coefficients of which are determined from the condition of the minimum value of the residual functional. To find corrections to the laws of change in thermomechanical characteristics on the basis of a weak statement of each direct problem and the linearization method, a system of Fredholm integral equations of the 1st kind is obtained. The system of integral equations is regularized based on the method of  A. N. Tikhonov. Computational experiments on the simultaneous reconstruction of two thermophysical characteristics with known laws of change in other characteristics are carried out. Pairs of both monotonically increasing and monotonically decreasing functions were reconstructed.

This work was supported by the Southern Mathematical Institute, a branch of the Vladikavkaz Scientific Center of the Russian Academy of Sciences.
  1. Birman V., Byrd L. W. Modeling and analysis of functionally graded materials and structures. Applied Mechanics Reviews, 2007, vol. 60, iss. 5, pp. 195–216. https://doi.org/10.1115/1.2777164
  2. Lomazov V. A. Zadachi diagnostiki neodnorodnykh termouprugikh sred [Diagnostics Problems for Inhomogeneous Thermoelastic Media]. Orel, OrelGTU Publ., 2002. 168 p. (in Russian).
  3. 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).
  4. Razzaghi H., Kowsary F., Ashjaee M. Derivation and application of the adjoint method for estimation of both spatially and temporally varying convective heat transfer coefficient. Applied Thermal Engineering, 2019, vol. 154, pp. 63–75. https://doi.org/10.1016/j.applthermaleng.2019.03.068
  5. 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, iss. 3, pp. 293–306. https://doi.org/10.1080/10407799508928835
  6. Dulikravich G. S., Reddy S. R., Pasqualette M. A., Colaco M. J., Orlande H. R., Coverston J. Inverse determination of spatially varying material coefficients in solid objects. Journal of Inverse and Ill-posed Problems, 2016, vol. 24, pp. 181–194. https://doi.org/10.1515/jiip-2015-0057  
  7. Cao K., Lesnic D. Determination of space-dependent coefficients from temperature measure[1]ments using the conjugate gradient method. Numerical Methods for Partial Differential Equations, 2018, vol. 34, iss. 4, pp. 1370–1400. https://doi.org/10.1002/num.22262
  8. Helmig T., Al-Sibai F., Kneer R. Estimating sensor number and spacing for inverse calculation of thermal boundary conditions using the conjugate gradient method. International Journal of Heat and Mass Transfer, 2020, vol. 153, Art. 119638. https://doi.org/10.1016/j.ijheatmasstransfer.2020.119638
  9. Geymonat G., Pagano S. Identification of mechanical properties by displacement field measurement: A variational approach. Meccanica, 2003, vol. 38, pp. 535–545. https://doi.org/10.1023/A:1024766911435
  10. Grediac M., Hild F., Pineau A. Full-Field Measurements and Identification in Solid Mechanics. Great Britain, Wiley-ISTE, 2013. 485 p. https://doi.org/10.1002/9781118578469
  11. Avril S., Pierron F. General framework for the identification of constitutive parameters from full-field measurements in linear elasticity. International Journal of Solids and Structures, 2007, vol. 44, iss. 14–15, pp. 4978–5002. https://doi.org/10.1016/j.ijsolstr.2006.12.018
  12. Vatulyan A. O., Nesterov S. A. Koeffitsiyentnye obratnye zadachi termomekhaniki [Coefficient Inverse Problems of Thermomechanics]. Rostov-on-Don, Taganrog, Southern Federal University Publ., 2019. 146 p. (in Russian).
  13. Dudarev V. V., Vatulyan A. O., Mnukhin R. M., Nedin R. D. Concerning an approach to identifying the Lame parameters of an elastic functionally graded cylinder. Mathematical Methods in the Applied Sciences, 2020, vol. 43, iss. 11, pp. 6861–6870. https://doi.org/10.1002/mma.6428
  14. 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, pp. 767–773. https://doi.org/10.1016/j.ijsolstr.2013.11.003
  15. Vatulyan A. O., Nesterov S. A. On the identification problem of the thermomechanical characteristics of the finite functionally graded cylinder. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 1, pp. 35–47 (in Russian). https://doi.org/10.18500/1816-9791-2021-21-1-35-47
  16. 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).