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Two-Parameter Topological Expansion of Helmholtz Problems with Inhomogeneity

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Mathematical Analysis of Continuum Mechanics and Industrial Applications

Part of the book series: Mathematics for Industry ((MFI,volume 26))

Abstract

For forward and inverse Helmholtz problems with inhomogeneity in the 2-D setting, high-order topological analysis is provided based on singular perturbations and variational methods. When diminishing the inhomogeneity, the two-parameter asymptotic result is proved rigorously with respect to the size of inhomogeneity and its refractive index. In particular, for a fixed refractive index this implies the topological derivative. For identifying an unknown inhomogeneity put in a test domain, variation of a complex refractive index leads to the zero-order necessary optimality condition of minimum of the objective function. This condition is realized as an imaging function for finding center of the inhomogeneity.

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Acknowledgments

The results were obtained with the support of the Austrian Science Fund (FWF) project P26147-N26: “Object identification problems: numerical analysis” (PION), partial support of NAWI Graz, the Austrian Academy of Sciences (OeAW), and OeAD Scientific & Technological Cooperation (WTZ CZ 01/2016). The author thanks for invitation Organizers of the International Conference CoMFoS15: Mathematical Analysis of Continuum Mechanics and Industrial Applications, 16-18.11.2015, Kyushu University, Fukuoka, Japan.

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Authors and Affiliations

  1. Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstr. 36, 8010, Graz, Austria

    Victor A. Kovtunenko

  2. Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Victor A. Kovtunenko

Authors
  1. Victor A. Kovtunenko
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Correspondence to Victor A. Kovtunenko .

Editor information

Editors and Affiliations

  1. Tokyo University of Science, Tokyo, Tokyo, Japan

    Hiromichi Itou

  2. Inst of Science and Engineering, Kanazawa University, Kanazawa, Japan

    Masato Kimura

  3. Fujitsu Ltd., Tokyo, Japan

    Vladimír Chalupecký

  4. Hiroshima Kokusai Gakuin University, Hiroshima, Japan

    Kohji Ohtsuka

  5. Kyushu University, Fukuoka, Fukuoka, Japan

    Daisuke Tagami

  6. Asahi Glass Co., Ltd., Tokyo, Tokyo, Japan

    Akira Takada

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Kovtunenko, V.A. (2017). Two-Parameter Topological Expansion of Helmholtz Problems with Inhomogeneity. In: Itou, H., Kimura, M., Chalupecký, V., Ohtsuka, K., Tagami, D., Takada, A. (eds) Mathematical Analysis of Continuum Mechanics and Industrial Applications. Mathematics for Industry, vol 26. Springer, Singapore. https://doi.org/10.1007/978-981-10-2633-1_5

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  • DOI: https://doi.org/10.1007/978-981-10-2633-1_5

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-2632-4

  • Online ISBN: 978-981-10-2633-1

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