Exact integration of surface and volume potentials

Michael Carley, Stefano Angioni

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A method for exact analytical integration of potentials from sources
distributed on planar and volume elements is presented. The method
is based on reduction of the surface integrals to a function similar
to an incomplete elliptic integral, giving the integrals in closed
form as functions of geometric properties of the surface or volume
element. Explicit formulae and recursions are given for the
integrals, allowing the evaluation of the potential for arbitrary
polynomial sources. Volume integrals are derived from the surface
integrals using a simple coordinate transformation which gives the
volume integral with little more effort than that required for the
surface calculation.
Original languageEnglish
Pages (from-to)93-106
Number of pages22
JournalJournal of Engineering Mathematics
Issue number1
Early online date9 Sept 2016
Publication statusPublished - Jun 2017


  • Laplace equation; potential theory; boundary element method; integral equations; quadrature; elliptic integrals


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