Identification of two-dimensional prestress fields in~inhomogeneous~plates

I. V. Bogachev ; R. D. Nedin

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 4, pp. 456-471

Résumé

Based on the model of in-plane oscillations of inhomogeneous prestressed plates, the new inverse problems of identifying the components of the prestress tensor via acoustic response probing are considered for the plates with and without holes and inclusions; the prestress components are assumed to be functions of two coordinates. Prestresses were set as a result of solving auxiliary problems of static loading of plates by some initial mechanical load. To solve the main and auxiliary problems of calculating the plates’ displacement functions, a finite element (FE) scheme was developed based on the derived corresponding weak problem statements, implemented in the form of software systems in the FE package FreeFem++. Rectangular plates clamped along one face, both solid and having a hole or a rigid insert, were considered. Inverse problems of identification of three prestress functions depending on two coordinates are formulated on the basis of additional data about the acoustic response on the non-clamped edges of the plates as a result of considering several sets of probing loads at several frequencies. In view of the nonlinearity of the inverse problems under study, an iterative approach was developed to solve them, which combines solving the direct problems for current approximations of the desired functions and the determination of the corresponding corrections from the operator equation built at each iteration. To solve the operator equation, a projection method has been employed that allows one to present the corrections in the form of expansions in terms of some smooth given functions and reduce the problem solution to the study of ill-conditioned SLAEs with respect to sets of the expansion coefficients using the A. N. Tikhonov method. The results of computational experiments on the simultaneous identification of two-dimensional prestress fields corresponding to various types of initial actions on the considered plates are discussed.