Methods for the Evaluation of Regular, Weakly Singular and Strongly SingularSurface Reaction Integrals Arising in Method of Moments
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Methods for the Evaluation of Regular, Weakly Singular and Strongly SingularSurface Reaction Integrals Arising in Method of MomentsAbstract
The accurate and fast evaluation of surface reaction integrals for Method of Moments computations is presented. Starting at the classification of the integrals into regular and weakly, strongly and nearly singular integrals, appropriate methods are presented that handle each. A Gauss-Legendre quadrature rule evaluates regular integrals. For singular integrals, the singularity is lifted or weakened by an extraction of the singularity, a transform to polar coordinates or a domain transform. The resulting regular integral is in turn solved by a quadrature rule. The different methods are finally applied to an example, and the resulting accuracy tested against the analytical result. The presented methods are general enough to be used as integration methods for integrands with various degrees of singularity and is not limited to Method of Moments.
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References
[Abr70] M. Abramowitz, I.A. Stegun, Handbook of
Mathematical Functions, Dover Publications,
New York, 1970.
[Dom93] J. Dominguez, Boundary Elements in Dynamics,
Computational Mechanics Publications,
Southhampton, Co-published with Elsevier
Applied Science, London, 1993.
[Duf82] "Quadrature over a Pyramid or Cube of Integrands
with a singularity at the Vertex", SIAM J.
Numer. Anal., Vol. 19, No 6, Dec. 1982.
[GC87] M. Guiggiani, P. Casalini, "Direct Computation
of Cauchy Principal Value Integrals in Advanced
Boundary Elements", International Journal for
Numerical Methods in Engineering, Vol. 24, pp.
-1720, 1987.
-1
0.5 1 1.5 2
n = 4, regular
n = 32, regular
n = 4, nearly sing.
n = 32, nearly sing.
error (%)
zP (m)
Figure 14 Nearly singular integrand. Error curves for the
methods in 4.1 and 4.4 and an increasing number
of nodes with observation point at various
distances to the surface element.
[GG90] M. Guiggiani, A. Gigante, "A General Algorithm
for Multidimensional Cauchy Principal Value
Integrals in the Boundary Element Method",
Transactions of the ASME, Vol. 57, pp. 906-
, December 1990.
[HvHW00] A. Herschlein, J. v. Hagen, W. Wiesbeck,
"Electromagnetic scattering by Arbitrary Surfaces
Modelled by Linear Triangles and Biquadratic
Quadrangles", IEEE Symposium on Antennas and
Propagation 2000, Salt Lake City, UT, USA,
, Vol. 4, pp. 2290-2293.
[Hay92] K. Hayami, A Projection Transformation Method
for Nearly Singular Surface Boundary Element
Integrals, Springer-Verlag, 1992.
[HRHR97] C.J. Huber, W. Rieger, M. Haas, W.M.
Rucker, "The numerical treatment of singular
integrals in boundary element calculations",
ACES Journal, vol. 12, nu. 2, pp. 121-126,
[PG89] P. Parreira, M. Guiggiani, "On the
Implementation of the Galerkin Approach in the
Boundary Element Method", Computers &
Structures, Vol. 33, pp. 269-279, 1989.
[PM73] A. J. Poggio, E. K. Miller, "Integral Equation
Solutions of Three-Dimensional Scattering
Problems" in Computer Techniques for
Electromagnetics, Volume 7, edited by R. Mittra,
Pergamon Press, New-York, NY, USA, 1973.
[SC95] J.M. Song, W.C. Chew, "Moment Method
Solutions using Parametric Geometry", Journal
of Electromagnetic Waves and Appl., Vol. 9,
No. 1/2, pp. 71-83, January-February, 1995.
[Sch93] H.R. Schwarz, Numerische Mathematik, B.G.
Teubner, Stuttgart, 1993.
[Tel87] J.C.F. Telles, "A self-adaptive coordinate
Transformation for efficient numerical Evaluation
of general Boundary Element Integrals",
International Journal for Numerical Methods in
Engineering, Vol. 24, pp. 959-973, 1987.
