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Cavity identification in linear elasticity and thermoelasticity
Author(s) -
Ameur Hend Ben,
Burger Martin,
Hackl Benjamin
Publication year - 2007
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.772
Subject(s) - overdetermined system , mathematics , inverse problem , regularization (linguistics) , elasticity (physics) , identifiability , linear elasticity , mathematical analysis , stability (learning theory) , iterative method , boundary (topology) , mathematical optimization , finite element method , computer science , statistics , materials science , physics , artificial intelligence , machine learning , composite material , thermodynamics
Abstract The aim of this paper is an analysis of geometric inverse problems in linear elasticity and thermoelasticity related to the identification of cavities in two and three spatial dimensions. The overdetermined boundary data used for the reconstruction are the displacement and temperature on a part of the boundary. We derive identifiability results and directional stability estimates, the latter using the concept of shape derivatives, whose form is known in elasticity and newly derived for thermoelasticity. For numerical reconstructions we use a least‐squares formulation and a geometric gradient descent based on the associated shape derivatives. The directional stability estimates guarantee the stability of the gradient descent approach, so that an iterative regularization is obtained. This iterative scheme is then regularized by a level set approach allowing the reconstruction of multiply connected shapes. Copyright © 2007 John Wiley & Sons, Ltd.