Theoretical development of elliptic cross-sectional hyperboloidal harmonics and their application to electrostatics

J. C.-E. Sten, G. Fragoyiannis, P. Vafeas, P. K. Koivisto, G. Dassios

    Research output: Contribution to journalArticleScientificpeer-review

    Abstract

    The analytic computation of electric and magnetic fields near corners and edges is important in many applications related to science and engineering. However, such complicated situations are hard to deal with, since they accumulate charges and consequently they mathematically represent singularities. In order to model this singular behavior, we introduce a novel method, which is related to the geometry and the analysis of the ellipsoidal coordinate system. Indeed, adopting the benefits of the corresponding coordinate surfaces, we use a general non-circular double cone, being the asymptote of a two-sided hyperboloid of two sheets with elliptic cross section, which matches almost perfectly the particular physics and captures the corresponding essential features in a fully three-dimensional fashion. To this end, our analytical technique employs the ellipsoidal geometry and adapts the ellipsoidal functions (solutions of the well-known Lamé equation) so as to construct a new set of the so-called elliptic cross-sectional hyperboloidal harmonics, supplemented by the appropriate orthogonality rules on every constant coordinate surface. By first recollecting the key results of the coordinate system and the related potential functions, including the indispensable orthogonality results, we demonstrate our method to the solution of two boundary value problems in electrostatics. Both refer to a non-penetrable two-hyperboloid of elliptic cross section and its double-cone limit, the first one being charged and the second one scattering off a plane wave. Closed form expressions are derived for the related fields, while the already known formulae from the literature are readily recovered, all cases being followed by the appropriate numerical implementation.
    Original languageEnglish
    Article number053505
    JournalJournal of Mathematical Physics
    Volume58
    Issue number5
    DOIs
    Publication statusPublished - 1 May 2017
    MoE publication typeA1 Journal article-refereed

    Fingerprint

    Electrostatics
    Harmonic
    electrostatics
    Orthogonality
    harmonics
    orthogonality
    Cone
    Cross section
    Asymptote
    cones
    Accumulate
    Potential Function
    Plane Wave
    asymptotes
    Electric Field
    cross sections
    Closed-form
    geometry
    Magnetic Field
    Boundary Value Problem

    Cite this

    Sten, J. C.-E. ; Fragoyiannis, G. ; Vafeas, P. ; Koivisto, P. K. ; Dassios, G. / Theoretical development of elliptic cross-sectional hyperboloidal harmonics and their application to electrostatics. In: Journal of Mathematical Physics. 2017 ; Vol. 58, No. 5.
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    abstract = "The analytic computation of electric and magnetic fields near corners and edges is important in many applications related to science and engineering. However, such complicated situations are hard to deal with, since they accumulate charges and consequently they mathematically represent singularities. In order to model this singular behavior, we introduce a novel method, which is related to the geometry and the analysis of the ellipsoidal coordinate system. Indeed, adopting the benefits of the corresponding coordinate surfaces, we use a general non-circular double cone, being the asymptote of a two-sided hyperboloid of two sheets with elliptic cross section, which matches almost perfectly the particular physics and captures the corresponding essential features in a fully three-dimensional fashion. To this end, our analytical technique employs the ellipsoidal geometry and adapts the ellipsoidal functions (solutions of the well-known Lam{\'e} equation) so as to construct a new set of the so-called elliptic cross-sectional hyperboloidal harmonics, supplemented by the appropriate orthogonality rules on every constant coordinate surface. By first recollecting the key results of the coordinate system and the related potential functions, including the indispensable orthogonality results, we demonstrate our method to the solution of two boundary value problems in electrostatics. Both refer to a non-penetrable two-hyperboloid of elliptic cross section and its double-cone limit, the first one being charged and the second one scattering off a plane wave. Closed form expressions are derived for the related fields, while the already known formulae from the literature are readily recovered, all cases being followed by the appropriate numerical implementation.",
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    Theoretical development of elliptic cross-sectional hyperboloidal harmonics and their application to electrostatics. / Sten, J. C.-E.; Fragoyiannis, G.; Vafeas, P.; Koivisto, P. K.; Dassios, G.

    In: Journal of Mathematical Physics, Vol. 58, No. 5, 053505, 01.05.2017.

    Research output: Contribution to journalArticleScientificpeer-review

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