Advances in Radio Science www.adv-radio-sci.net 1684-9965 1684-9973 4 Kleinheubacher Berichte 2005 2006 10.5194/ars-4-11-2006 http://www.adv-radio-sci.net/4/11/2006/ http://www.adv-radio-sci.net/4/11/2006/ars-4-11-2006.html http://www.adv-radio-sci.net/4/11/2006/ars-4-11-2006.pdf 11 15 2006-09-04 Numerical quadrature for the approximation of singular oscillating integrals appearing in boundary integral equations L. O. Fichte lars-ole.fichte@hsu-hh.de S. Lange M. Clemens Professur für Theoretische Elektrotechnik und Numerische Feldberechnung, Helmut-Schmidt-Universität Universität der Bundeswehr Hamburg, PO. Box 700 822, 22 008 Hamburg, Germany Bundeswehr Hamburg, PO. Box 700 822, D-22 008 Hamburg, German Boundary Integral Equation formulations can be used to describe electromagnetic shielding problems. Yet, this approach frequently leads to integrals which contain a singularity and an oscillating part. Those integrals are difficult to handle when integrated naivly using standard integration techniques, and in some cases even a very high number of integration nodes will not lead to precise results. We present a method for the numerical quadrature of an integral with a logarithmic singularity and a cosine oscillator: a modified Filon-Lobatto quadrature for the oscillating parts and an integral transformation based on the error function for the singularity. Since this integral can be solved analytically, we are in a position to verify the results of our investigations, with a focus on precision and computation time. Abramowitz,~M. and Stegun,~I A.: Handbook of Mathematical Functions, New York 1970. Ehrich,~M., Fichte,~L O., and Lüer,~M.: Contribution to Boundary Integrals by the Singularity of Kernels satisfying Helmholtz' Equation, CJMW'2000 China-Japan Joint Meeting on Microwaves, Nanjing, PR China, CD-ROM, 2000. Ehrich,~M., Kuhlmann,~J., and Netzler,~D.: High accuracy integration of boundary integral equations describing axisymmetric field problems, Asia-Pacific Microwave Conf., Hong Kong, Microwave Conf. Proc., CD-ROM, 1997. Fichte,~L O.: Berechnung der Stromverteilung in einem System rechteckiger Massivleiter bei Wechselstrom mit Hilfe der Randintegralgleichungsmethode, PhD-Thesis, to be published. Fichte,~L O., Ehrich,~M., and Kurz,~S.: An Analytical Solution to the Eddy Current Problem of a Conducting Bar, EMC 2004 Intern. Symposium on Electromagnetic Compatibility, Sendai Conf. Proc., CD-ROM, 2004. Fichte,~L O., Lange,~S., Steinmetz,~T., Clemens,~M.: Shielding Properties of a Conducting Bar calculated with a Boundary Integral Equation Method, Adv. in Radio Sci., 3, 119–123, 2005. Hanson,~G W. and Yakovlev,~A B.: Operator Theory for Electromagnetics, Springer, New York, 2002. Iserles,~A.: On the numerical quadrature of highly-oscillating integrals, IMA J. of Numerical Anal., 24, 365–391, 2004.