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 clas...

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Bibliographic Details
Published in:Axioms Vol. 12; no. 10; p. 912
Main Authors: Vatulyan, Alexandr, Uglich, Pavel, Dudarev, Vladimir, Mnukhin, Roman
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.09.2023
Subjects:
Acoustics
Approximation theory
Density
Fredholm equations
Fredholm integral equation
Frequency ranges
Genetic algorithms
Inhomogeneity
inhomogeneous body
Integral equations
inverse problem
Inverse problems
Iterative methods
Kernels
LSQR method
Mathematical research
Methods
Modulus of elasticity
Nondestructive testing
Numerical analysis
Numerical methods
Physical properties
Regularization
Sensitivity analysis
Tikhonov regularization
Russia
ISSN:2075-1680, 2075-1680
Online Access:Get full text
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Summary: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.
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ISSN:2075-1680
2075-1680
DOI:10.3390/axioms12100912