On identification of two-dimensional density of an elastic inhomogeneous cylinder

Within the framework of the linear theory of elasticity, using the model of an isotropic body, the problem of steady-state vibrations of an inhomogeneous hollow cylinder is formulated. The vibrations of the cylinder are caused by a load applied to the side surface, and the conditions of sliding embedding are implemented at the ends. The variable material properties are described by the Lamé parameters and density, which change along the radial and longitudinal coordinates. The direct problem solution of the cylinder vibrations is constructed using the finite element method implemented in the FlexPDE package, its main advantages are noted. To study the influence of variable properties on the values of resonant frequencies and components of the displacement field, the laws of these properties are considered in the general form used in modern works for modeling functionally graded materials. On the basis of the performed numerical calculations, the degree of influence of the amplitude values of each of the Lamé parameters and density on the first resonant frequency and the displacement field are studied. Graphs are also presented that demonstrate the influence of the type of the law of density change on the values of the displacement field components. A new inverse coefficient problem is formulated to determine the density distribution function in the cylinder wall from the displacement field data measured at a finite set of points inside the considered area for a fixed frequency. The main difficulties in the implementation of the reconstruction procedure in practice are noted. To increase the accuracy of calculating the first and second derivatives of the two-dimensional functions calculated in the finite element package, which are used in solving the inverse problem, an approach based on the locally weighted regression algorithm is proposed. The results of computational experiments on solving the inverse problem are presented, which demonstrate the possibility of using the proposed method to restore various types of two-dimensional laws of density. Practical recommendations are given for the implementation of the most accurate reconstruction procedure.