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Reconstruction in the inverse crack problem by variational methods

Published online by Cambridge University Press:  01 December 2008

LUCA RONDI
LUCA RONDI*
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, via Valerio 12/1, I-34127 Trieste, Italy email: rondi@units.it
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Abstract

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We deal with a variational approach to the inverse crack problem, that is the detection and reconstruction of cracks, and other defects, inside a conducting body by performing boundary measurements of current and voltage type. We formulate such an inverse problem in a free-discontinuity problems framework and propose a novel method for the numerical reconstruction of the cracks by the available boundary data. The proposed method is amenable to numerical computations and it is justified by a convergence analysis, as the error on the measurements goes to zero. We further notice that we use the Γ-convergence approximation of the Mumford–Shah functional due to Ambrosio and Tortorelli as the required regularization term.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 6 , December 2008 , pp. 635 - 660
Copyright
Copyright © Cambridge University Press 2008

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