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.