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Analysis and estimation of quadrature errors in weakly singular source integrals of the method of moments
Author(s) -
Botha Matthys M.,
Rylander Thomas
Publication year - 2017
Publication title -
international journal of numerical modelling: electronic networks, devices and fields
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.249
H-Index - 30
eISSN - 1099-1204
pISSN - 0894-3370
DOI - 10.1002/jnm.2269
Subject(s) - gaussian quadrature , numerical integration , singularity , mathematics , singular integral , clenshaw–curtis quadrature , quadrature (astronomy) , mathematical analysis , gauss–kronrod quadrature formula , gauss–laguerre quadrature , tanh sinh quadrature , gauss–hermite quadrature , nyström method , integral equation , physics , optics
Abstract The method of moments (MoM) is used for the numerical solution of electromagnetic field integral equations. Weakly singular integrals over surfaces in 3 dimensions (3D) are routinely evaluated for the impedance matrix setup and for post‐processing. Available numerical integration schemes range from direct application of Gauss‐Legendre product‐rule quadrature, to singularity and near‐singularity cancellation, coordinate transformation schemes. This paper presents a general, explicit, pole‐based, a priori procedure to estimate quadrature errors in the numerical evaluation of weakly singular and near‐singular, 3D surface integrals in the MoM. It is based on an error theorem for linear Gaussian quadrature, which involves the analytic extension of the integrand into the complex plane. Errors are linked to poles in the complex plane. New closed‐form estimates are presented for direct Gaussian product‐rule integration, polar‐coordinate integration, and the Radial‐Angular‐ R 1 ‐Sqrt singularity cancellation scheme, for triangle integration domains. This work can serve as a foundation/template for further, 3D MoM‐related work to identify appropriate quadrature schemes according to their error characteristics; for automatic selection of optimal schemes and quadrature orders in a computer implementation of the MoM; and for local and global estimation of MoM quadrature errors. This work can be specialized to the MoM for surfaces in 2D.