On Detecting the Shape of an Unknown Object in an Electric Field

Jukka-Pekka Humaloja, Timo Hämäläinen, Seppo Pohjolainen

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    Abstract

    The problem discussed in this paper is detecting the shape of an unknown object in a 2-dimensional static electric field. For simplicity, the problem is defined in a partially rectangular domain, where on a part of the boundary the potential and/or its normal derivative are known. On the other part of the boundary the boundary curve is unknown, and this curve is to be determined. The unknown part of the boundary curve describes the shape of the unknown object.
    The problem is defined in the complex plane by an analytic function w=f(z) = u(x,y)+iv(x,y) with the potential u as its real part. Then the inverse function is given as f^{-1}(w) = x(u,v)+iy(u,v), where the functions x and y are harmonic in a rectangle with an unknown boundary condition on one boundary. The alternating-field technique is used to solve the unknown boundary condition.
    Original languageEnglish
    Title of host publicationProgress in Industrial Mathematics at ECMI 2014
    EditorsG. Russo, V. Capasso, G. Nicosia, V. Romano
    PublisherSpringer International Publishing
    ISBN (Electronic)978-3-319-23413-7
    ISBN (Print)978-3-319-23412-0
    DOIs
    Publication statusPublished - 2016
    Publication typeA4 Article in conference proceedings
    EventEUROPEAN CONFERENCE ON MATHEMATICS FOR INDUSTRY -
    Duration: 1 Jan 2000 → …

    Publication series

    NameMathematics in Industry
    PublisherSpringer-Verlag
    Volume22
    ISSN (Electronic)1612-3956

    Conference

    ConferenceEUROPEAN CONFERENCE ON MATHEMATICS FOR INDUSTRY
    Abbreviated titleECMI
    Period1/01/00 → …

    Keywords

    • free boundary problem
    • industrial mathematics

    Publication forum classification

    • Publication forum level 0

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