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On the Reconstruction of a Rigid Body in the Theory of Elasticity
Author(s) -
Dassios G.,
Rigou Z.
Publication year - 1997
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19970771208
Subject(s) - scattering , mathematics , polarization (electrochemistry) , elasticity (physics) , embedding , mathematical analysis , scattering theory , anisotropy , scalar field , inverse problem , classical mechanics , physics , optics , computer science , chemistry , artificial intelligence , thermodynamics
Abstract A method is proposed for the solution of the inverse scattering problem associated with the shape reconstruction of a 3‐D, star shaped, rigid scatterer in the theory of elasticity. The inversion procedure is based on the use of elastic Herglotz functions. A key point of the method is a basic connection formula associating the scattered field and the Herglotz function on the surface of the scatterer, with the corresponding scattering amplitudes and Herglotz kernels on the unit sphere. Analytical difficulties caused by the complexity of the spectral Navier operator were overpassed by embedding the vector elastic scattering problem into a dyadic scattering problem, which absorbs the dependence of the incident field upon the transverse polarization by considering a complete incident dyadic field. The actual elastic problem is then obtained by projecting the dyadic scattering problem into the particular polarization of the assumed incident field. The method, which for the scalar case has been developed by Colton and Monk, leads to an optimization scheme which is similar, but much more complicated, to the corresponding scheme in acoustics.