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Obstacles reconstruction from partial boundary measurements based on the topological derivative concept

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Abstract

In this work a new method for obstacles reconstruction from partial boundary measurements is proposed. For a given boundary excitation, we want to determine the quantity, locations and sizes of a number of holes embedded within a geometrical domain, from partial boundary measurements related to such an excitation. The resulting inverse problem is written in the form of an ill-posed and over-determined boundary value problem. The idea therefore is to rewrite it as an optimization problem where a shape functional measuring the misfit between the boundary measurement and the solution to an auxiliary boundary value problem is minimized with respect to a set of ball-shaped holes. The topological derivative concept is used for solving the associated topology optimization problem, leading to a second-order reconstruction algorithm. The resulting algorithm is non-iterative – and thus very robust with respect to noisy data – and also free of initial guess. Finally, some numerical results are presented in order to demonstrate the effectiveness of the proposed reconstruction algorithm.

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Acknowledgments

This research was partly supported by CNPq (Brazilian Research Council), CAPES (Brazilian Higher Education Staff Training Agency) and FAPERJ (Research Foundation of the State of Rio de Janeiro). These supports are gratefully acknowledged.

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  1. Laboratório Nacional de Computação Científica LNCC/MCT, Coordenação de Matemática Aplicada e Computacional, Av. Getúlio Vargas 333, 25651-075, Petrópolis, RJ, Brasil

    S. S. Rocha & A. A. Novotny

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  1. S. S. Rocha
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  2. A. A. Novotny
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Correspondence to A. A. Novotny.

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Rocha, S.S., Novotny, A.A. Obstacles reconstruction from partial boundary measurements based on the topological derivative concept. Struct Multidisc Optim 55, 2131–2141 (2017). https://doi.org/10.1007/s00158-016-1632-x

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  • DOI: https://doi.org/10.1007/s00158-016-1632-x

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