On numerical cubatures of nearly singular surface integrals arising in BEM collocation

Numerische Quadratur fast singulärer Oberflächenintegrale aus der Randelementkollokation

Abstract

In this paper we present efficient methods to approximate nearly singular surface integrals arising massively when discretizing boundary integral equations via the collocation method. The idea is to introduce local polar coordinates centred at a corner of the triangle. Thus it is possible to perform the inner integration analytically, where either the corresponding formulae can be evaluated numerically stable or can be replaced by simple (rational) approximation quite efficiently. We show that the outer integration can be performed by simple Gauß-Legendre quadrature and how to adapt the order of the Gauß formulae to a required order of consistency. Numerical tests will emphasize the efficiency of our method.

Zusammenfassung

Die Arbeit präsentiert effiziente Verfahren zur Bestimmung fast singulärer Integrale, wie sie in großer Anzahl bei der Diskretisierung von Integralgleichungen durch Kollokation auftreten. Die Methode basiert auf der Einführung von lokalen Polarkoordinaten um eine Dreiecksecke. Die innere Integration läßt sich analytisch durchführen, wobei man entweder entsprechende numerisch stabile Formeln oder Funktionsapproximationen verwenden kann. Die äußere Integration kann mit gewünschter Genauigkeit mittels Gauß-Legendre-Quadratur ermittelt werden. Numerische Tests unterstreichen die Effizienz unserer Methode.

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This work was supported by the Priority Research Program “Boundary Element Methods” of the German Research Foundation (DFG).

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Hackbusch, W., Sauter, S.A. On numerical cubatures of nearly singular surface integrals arising in BEM collocation. Computing 52, 139–159 (1994). https://doi.org/10.1007/BF02238073

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AMS Subject Classification

  • 65N38
  • 45B05
  • 45E05
  • 45Z10
  • 65Y20

Key words

  • Boundary element method
  • numerical quadrature
  • collocation
  • panel method
  • cubature