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

A. O. Vatulyan ; S. A. Nesterov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 22 (2022) no. 2, pp. 180-195.

Résumé

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.

Bibliographie

[1] Birman V., Byrd L. W., “Modeling and analysis of functionally graded materials and structures”, Applied Mechanics Reviews, 60:5 (2007), 195–216 | DOI

[2] Lomazov V. A., Diagnostics Problems for Inhomogeneous Thermoelastic Media, OrelGTU Publ, Orel, 2002, 168 pp. (in Russian)

[3] Alifanov O. M., Artyukhin E. A., Rumyantsev S. V., Extreme Methods of Solving Ill-posed Problems, Nauka, M., 1988, 288 pp. (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, 154 (2019), 63–75 | DOI

[5] Raudensky M., Woodbary K. A., Kral J., “Genetic algorithm in solution of inverse heat conduction problems”, Numerical Heat Transfer, Part B : Fundamentals, 28:3 (1995), 293–306 | DOI

[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, 24 (2016), 181–194 | DOI

[7] Cao K., Lesnic D., “Determination of space-dependent coefficients from temperature measurements using the conjugate gradient method”, Numerical Methods for Partial Differential Equations, 34:4 (2018), 1370–1400 | DOI

[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, 153 (2020), 119638 | DOI

[9] Geymonat G., Pagano S., “Identification of mechanical properties by displacement field measurement: A variational approach”, Meccanica, 38 (2003), 535–545 | DOI

[10] Grediac M., Hild F., Pineau A., Full-Field Measurements and Identification in Solid Mechanics, Wiley-ISTE, Great Britain, 2013, 485 pp. | DOI

[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, 44:14–15 (2007), 4978–5002 | DOI

[12] Vatulyan A. O., Nesterov S. A., Coefficient Inverse Problems of Thermomechanics, Southern Federal University Publ, Rostov-on-Don–Taganrog, 2019, 146 pp. (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, 43:11 (2020), 6861–6870 | DOI

[14] Nedin R., Nesterov S., Vatulyan A., “On an inverse problem for inhomogeneous thermoelastic rod”, International Journal of Solids and Structures, 51:3 (2014), 767–773 | DOI

[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, 21:1 (2021), 35–47 (in Russian) | DOI

[16] Tikhonov A. N., Goncharskiy A. V., Stepanov V. V., Yagola A. G., Numerical Methods for Solving Ill-Posed Problems, Nauka, M., 1990, 230 pp. (in Russian)