On reconstruction of inhomogeneous properties of piezoelectric solids

The problem of reconstruction of the inhomogeneous laws of electroelastic body characteristics is considered. The weak statement of the problem is presented, which makes it possible to derive operator equations for the inverse problem. The proposed approach is illustrated on an example problem of reconstruction of the inhomogeneous compliance module for a longitudinally polarized electroelastic rod. The solution of the direct problem of the longitudinal vibrations of the rod is reduced to a Fredholm equation of the second kind. The inverse problem is studied using additional information on the amplitude-frequency characteristic of the free end of a cantilevered rod. The iterative process is presented. At each step of the process, corrections are determined by the Fredholm integral equation of the first kind, and its numerical solution is constructed using a Tikhonov regularization method with automatic selection of the regularization parameter. The results of computational experiments on reconstruction of the monotonic and nonmonotonic laws of the compliance module changes are presented.