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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References
Atkinson, K. E.: Solving integral equations on surfaces in space. In: Constructive methods for the practical treatment of integral equations (Hämmerlin, G., Hoffmann, K. eds.), pp. 20–43. Birkhäuser: ISNM, 1985.
Atkinson, K. E., Chien, D.: Piecewise polynomial collocation for boundary integral equations. Report No. 29, University of Iowa, 1992 (submitted to Siam J. SSC).
Buthmann, U.: Zur praktischen Berechnung fast singulärer Integrale über ebenen Dreiecken. Diploma Thesis, Universität zu Kiel, 1991.
Ciarlet, Ph. G.: The finite element method for elliptic problems. Amsterdam: North-Holland 1987.
Davis, P. J., Rabinowitz, P.: Methods of numerical integrations. London: Academic Press 1980.
Friedman, A.: Partial differential equations. New York: Holt, Rinehart and Winston 1969.
Gröbner, W., Hofreiter, N.: Integraltafel, erster Teil: unbestimmte Integrale. Wien New York: Springer 1957.
Guiggiani, M., Gigante, A.: A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. ASME J. Appl. Mech.57, 907–915 (1900).
Guiggiani, M.: Direct evaluation of hypersingular integrals in 2D BEM. In: Proc. 7th GAMM Seminar on Numerical Techniques for BEM, Kiel 1991. Notes on Numerical Fluid Mechanics, vol. 33 (Hackbusch, W. ed.), pp. 23–34. Braunschweig: Vieweg 1991.
Hackbusch, W., Sauter, S. A.: On the efficient use of the Galerkin method to solve Fredholm integral equations. Report Nr. 92-18, DFG-Schwerpunktreihe: “Randelementmethoden”, Universität zu Kiel, 1992.
Hansen, E. R.: A table of series and products. Englewood Cliffs: Prentice Hall 1975.
John, F.: Plane waves and spherical means. New York: Springer 1955.
Johnson, C. G. L., Scott, L. R.: An analysis of quadrature errors in second-kind boundary integral methods. SIAM J. Numer. Anal.26, 1356–1382 (1989).
Kieser, R., Schwab, C., Wendland, W. L.: Numerical evaluation of singular and finite part integrals on curved surfaces using symbolic manipulation. Computing49, 279–301 (1992).
Nedelec, J. C.: Curved finite element methods for the solution of singular integral equations on surfaces inR 3. Comput. Meth. Appl. Mech. Eng.8, 61–80 (1976).
Rabinowitz, P., Richter, N.: New error coefficients for estimating quadrature errors for analytic functions. Math. Comp.24, 561–570 (1970).
Schwab, C., Wendland, W. L.: Kernel properties and representations of boundary integral operators. Math. Nach.156, 187–218 (1992).
Schwab, C., Wendland, W. L.: On numerical cubatures of singular surface integrals in boundary element methods. Num. Math. Vol. 343–369 (1992).
Telles, J. C. F.: A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals. Int. J. Num. Meth. Eng.24, 959–973 (1987).
Wendland, W. L.: Boundary element methods and their asymptotic convergence. In: Theoretical acoustics and numerical treatments (Filippi, P. ed.), pp. 289–313. London: Pentech Press 1981.
Wendland, W. L.: Asymptotic accuracy and convergence for point collocation methods. In: Topics in boundary element research, Vol. 2 (Brebbia, C. A. ed.), pp. 230–258. Berlin Heidelberg New York Tokyo: Springer 1984.
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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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DOI: https://doi.org/10.1007/BF02238073