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An asymptotic approach to inverse scattering problems on weakly nonlinear elastic rods. (English) Zbl 1029.74028

Summary: Elastic wave propagation in weakly nonlinear elastic rods is considered in 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 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 a nonlinear parameter in stress-strain relation for a homogeneous elastic rod, (ii) recovery of cross-sectional area for a homogeneous elastic rod, (iii) recovery of elastic modulus for an inhomogeneous elastic rod.

MSC:

74J25 Inverse problems for waves in solid mechanics
74J20 Wave scattering in solid mechanics
74H10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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