Efficient Numerical Methods of Inverse Coefficient Problem Solution for One Inhomogeneous Body

In the present paper, the problems of longitudinal and flexural vibrations of an inhomogeneous rod are considered. The Young's modulus and density are variable in longitudinal coordinate. Vibrations are caused by a load applied at the right end. The proposed method allows us to consider a wider class of inhomogeneity laws in comparison with other numerical solutions. Sensitivity analysis is carried out. A new inverse problem related to the simultaneous identification of the variation laws of Young's modulus and density from amplitude-frequency data, which are measured in given frequency ranges, is considered. Its solution is based on an iterative process: at each step, a system of two Fredholm integral equations of the first kind with smooth kernels is solved numerically. The analysis of the kernels is carried out for different frequency values. To find the initial approximation, several approaches are proposed: a genetic algorithm, minimization of the residual functional on a compact set, and additional information about the values of the sought-for functions at the ends of the rod. The Tikhonov regularization and the LSQR method are proposed. Examples of reconstruction of monotonic and non-monotonic functions are presented.

Automated summary

This paper focuses on the longitudinal and flexural vibrations of an inhomogeneous rod, considering the variable Young's modulus and density along the longitudinal coordinate. The vibrations are caused by a load applied at the right end. The proposed method enables the consideration of a wider range of inhomogeneity laws compared to other numerical solutions. Sensitivity analysis is performed and a new inverse problem related to the simultaneous identification of the variation laws of Young's modulus and density is addressed. The solution involves an iterative process and the analysis of Fredholm integral equations.