Journal of Applied Mathematics

An asymptotic approach to inverse scattering problems on weakly nonlinear elastic rods

Shinuk Kim and Kevin L. Kreider

Full-text: Open access

Abstract

Elastic wave propagation in weakly nonlinear elastic rods is considered in the time domain. The method of wave splitting is employed to formulate a standard scattering problem, forming the mathematical basis for both direct and inverse problems. A quasi-linear version of the Wendroff scheme (FDTD) is used to solve the direct problem. To solve the inverse problem, an asymptotic expansion is used for the wave field; this linearizes the order equations, allowing the use of standard numerical techniques. Analysis and numerical results are presented for three model inverse problems: (i) recovery of the nonlinear parameter in the stress-strain relation for a homogeneous elastic rod, (ii) recovery of the cross-sectional area for a homogeneous elastic rod, (iii) recovery of the elastic modulus for an inhomogeneous elastic rod.

Article information

Source
J. Appl. Math., Volume 2, Number 8 (2002), 407-435.

Dates
First available in Project Euclid: 30 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.jam/1049074736

Digital Object Identifier
doi:10.1155/S1110757X0210903X

Mathematical Reviews number (MathSciNet)
MR1954932

Zentralblatt MATH identifier
1029.74028

Subjects
Primary: 74J30: Nonlinear waves 74B20: Nonlinear elasticity
Secondary: 74H10: Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) 74J25: Inverse problems 65M32: Inverse problems 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15]

Citation

Kim, Shinuk; Kreider, Kevin L. An asymptotic approach to inverse scattering problems on weakly nonlinear elastic rods. J. Appl. Math. 2 (2002), no. 8, 407--435. doi:10.1155/S1110757X0210903X. https://projecteuclid.org/euclid.jam/1049074736


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