Two-Parameter Topological Expansion of Helmholtz Problems with Inhomogeneity

  • Victor A. KovtunenkoEmail author
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 26)


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.


Forward and inverse Helmholtz problem Inhomogeneity Singular perturbation Two-parameter asymptotic analysis High-order asymptotic expansion Variational method Shape and topology optimization Topological derivative 



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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© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Institute for Mathematics and Scientific ComputingKarl-Franzens University of Graz, NAWI GrazGrazAustria
  2. 2.Lavrentyev Institute of HydrodynamicsSiberian Division of the Russian Academy of SciencesNovosibirskRussia

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