For citation:
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. DOI: 10.18500/1816-9791-2021-21-1-35-47, EDN: BKDVPH
On the identification problem of the thermomechanical characteristics of the finite functionally graded cylinder
The problem of axisymmetric vibrations of a functionally graded finite hollow cylinder is considered. The ends of the cylinder are thermally insulated and are in a sliding fit. Zero temperature is maintained on the inner surface of the cylinder, free from stress, and a combined thermal and power load acts on the outer surface. The direct problem after applying the Laplace transform is solved based on the method of separation of variables. A set of canonical linear systems of differential equations of the 1st order is obtained, the solution of each of which is obtained numerically using the shooting method. The coefficient inverse problem of finding the thermomechanical characteristics of a finite length cylinder using additional information in Laplace transforms, given on the outer surface of the cylinder, is posed. The dimensionless thermomechanical characteristics of the cylinder were restored in two stages. At the first stage, the initial approximation was determined in the class of positive bounded functions. At the second stage, based on the solution of the corresponding Fredholm integral equations of the 1st kind, corrections of the reconstructed functions were found, and an iterative process of their refinement was constructed. In the course of computational experiments, it was found that monotonic characteristics are restored with good accuracy; the reconstruction procedure is resistant to input information noise.
- 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
- Wetherhold R. C., Seelman S., Wang S. The use of functionally graded materials to eliminated or control thermal deformation. Composites Science and Technology, 1996, vol. 56, no. 9, pp. 1099–1104. https://doi.org/10.1016/0266-3538(96)00075-9
- Lomazov V. A. Zadachi diagnostiki neodnorodnykh termouprugikh sred [Diagnostics Problems for Inhomogeneous Thermoelastic Media]. Orel, Izdatel’stvo OrelGTU, 2002. 168 p. (in Russian).
- 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. New Series. Series: Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 4, pp. 409–423 (in Russian). https://doi.org/10.18500/1816-9791-2019-19-4-409-423
- 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).
- Hao D. N. Methods for Inverse Heat Conduction Problems. Frankfurt a/M., Peter Lang Pub. Inc., 1998. 249 p.
- 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, vol. 1, iss. 1, pp. 34–45. https://doi.org/10.1134/S1995423908010047
- 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. https://doi.org/10.1016/0017-9310(96)00028-2
- Vatulyan A. O., Bogachev I. V., Nedin R. D., Yavruyan O. V. Identification of inhomogeneous elastic properties of isotropic cylinder. ZAMM — Journal of Applied Mathematics and Mechanics / Zeitschrift fur Angewandte Mathematik und Mechanik , 2016, vol. 97, iss. 3, pp. 358–364. https://doi.org/10.1002/zamm.201600179
- 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
- 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
- Grediac M., Hild F., Pineau A. Full-Field Measurements and Identification in Solid Mechanics. Great Britain, Wiley-ISTE, 2013. 485 p.
- 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
- Vatulyan A. O., Nesterov S. A. Koeffitsiyentnye obratnye zadachi termomekhaniki [Coefficient Inverse Problems of Thermomechanics]. Rostov-on-Don, Taganrog, Izdatel’stvo Yuzhnogo Federal’nogo Universieta, 2019. 146 p. (in Russian).
- Nedin R., Nesterov S., Vatulyan A. On reconstruction of thermalphysic characteristics of functionally graded hollow cylinder. Applied Mathematical Modeling, 2016, vol. 40, iss. 4, pp. 2711–2719. https://doi.org/10.1016/j.apm.2015.09.078
- Vatulyan A. O., Nesterov S. A. About the Specifics of Identification Thermomechanical Characteristics of Functionally Graded Materials. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 2014, vol. 14, iss. 3, pp. 329–335 (in Russian). https://doi.org/10.18500/1816-9791-2014-14-3-329-335
- Vatul’yan A. O., Nesterov S. A. On determination of inhomogeneous thermomechanical characteristics of a pipe. Journal of Engineering Physics and Thermophysics. 2015, vol. 88, no. 4, pp. 984–993. https://doi.org/10.1007/s10891-015-1274-7
- 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).
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