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Identification of Cavities in a Three-Dimensional Layer by Minimization of an Optimal Cost Functional Expansion

A.E. Martínez-Castro1, I.H. Faris1, R. Gallego1
1 University of Granada, Granada, Spain

Computer Modeling in Engineering & Sciences 2012, 87(3), 177-206.


In this paper, the identification of hidden defects inside a three-dimen -sional layer is set as an Identification Inverse Problem. This problem is solved by minimizing a cost functional which is linearized with respect to the volume defects, leading to a procedure that requires only computations at the host domain free of defects. The cost functional is stated as the misfit between experimental and computed displacements and spherical and/or ellipsoidal cavities are the defects to locate. The identification of these cavities is based on the measured displacements at a set of points due to time-harmonic point loads at an array of source points. The topological expansion of the displacement field due to the presence of a small cavity provides the topological expansion of the cost functional. This expansion, called the Cost Functional Expansion, depends quadratically on the cavity volumes. Therefore, considering that the cavity center coordinates are fixed, the optimum volumes of the defects are easily computed by a closed-form formula. The evaluation of the Cost Functional Expansion for the optimum cavity volumes defines the Optimal Cost Functional Expansion, which depends only on the cavity center coordinates. The evaluation of the Optimal Cost Functional Expansion is very fast since it depends only on information computed at the non-damaged layer. Finally, a zero-order algorithm, such as Genetic Algorithms is proposed to find the optimal positions of the cavity centers. A set of numerical tests have been carried out, in order to test the main properties of the proposed procedure. It is shown to be a very effective technique to find hidden cavities in problems in which no a-priori information is known with respect to the number, position and size of defects.


Topological Derivative, Identification Inverse Problems, Genetic Algorithms, Experimental Noise, Layered media, Elastodynamics, Boundary Elements

Cite This Article

Martínez-Castro, A., Faris, I., Gallego, R. (2012). Identification of Cavities in a Three-Dimensional Layer by Minimization of an Optimal Cost Functional Expansion. CMES-Computer Modeling in Engineering & Sciences, 87(3), 177–206.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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