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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Veröffentlicht in:	Axioms Jg. 12; H. 10; S. 912
Hauptverfasser:	Vatulyan, Alexandr, Uglich, Pavel, Dudarev, Vladimir, Mnukhin, Roman
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Basel MDPI AG 01.09.2023
Schlagworte:	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
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Abstract	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.
AbstractList	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.
Audience	Academic
Author	Dudarev, Vladimir Mnukhin, Roman Vatulyan, Alexandr Uglich, Pavel
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Cites_doi	10.1016/j.compstruct.2015.08.021 10.1016/j.cam.2013.06.011 10.1115/1.2777164 10.1016/j.ijmecsci.2014.08.017 10.1002/zamm.202200430 10.1007/s42452-019-1228-3 10.1016/j.cam.2019.112677 10.1016/j.apm.2010.07.006 10.1007/978-3-319-51658-5 10.1002/mma.6428 10.3390/axioms12040326 10.1088/1402-4896/acc493 10.1016/j.ijpvp.2018.10.019 10.1134/S0037446623030175 10.1016/j.compstruct.2021.113596 10.1007/978-94-015-8480-7 10.1016/j.asej.2022.102024 10.1177/09544062221141246 10.1007/s00419-023-02406-3 10.1007/s10958-023-06449-3 10.1016/j.petrol.2020.107508 10.1016/j.mbs.2018.09.004 10.1201/9781315117638 10.1016/j.compositesb.2011.09.003 10.1016/j.compositesb.2020.108376 10.1007/978-3-319-26553-7 10.1016/j.jappmathmech.2011.01.004 10.1016/j.apmt.2016.10.001 10.1016/j.jsv.2018.02.010 10.1016/j.ijengsci.2012.08.004 10.1016/j.compstruct.2020.113309 10.1145/355984.355989
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SubjectTerms	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
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Title	Efficient Numerical Methods of Inverse Coefficient Problem Solution for One Inhomogeneous Body
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