Super‐hyper singularity treatment for solving 3D electric field integral equations

The super-hyper singularity treatment is developed for solving three-dimensional (3D) electric field integral equations (EFIE). EFIE usually takes two forms: one of which includes a super-hyper singular kernel generated by the double gradient of the Green's function. So far, there is no way to evaluate these super-hyper singular integrals needed for constructing the pertinent matrix equation. We apply the series expansion of the Green's function to the super-hyper singular kernel and derive closed-form expressions for their evaluations in the Cauchy principal-value sense. The derivation is based on the constant current approximation over a flat triangle patch, but it can be extended easily to higher-order approximations of the current. The scheme can be used to accurately calculate the near and self interaction terms in the impedance matrix for the method of moments (MoM), Nystrom method or boundary element method (BEM). © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 1383–1388, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI.10.1002/mop.22443

[1]  D. Wilton,et al.  Electromagnetic scattering by surfaces of arbitrary shape , 1980 .

[2]  H. B. Dwight,et al.  Tables of Integrals and Other Mathematical Data , 1934 .

[3]  D. Wilton,et al.  Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces , 1980 .

[4]  M. G. Duffy,et al.  Quadrature Over a Pyramid or Cube of Integrands with a Singularity at a Vertex , 1982 .

[5]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[6]  Jian-Ming Jin,et al.  Fast and Efficient Algorithms in Computational Electromagnetics , 2001 .

[7]  Roberto D. Graglia,et al.  Static and dynamic potential integrals for linearly varying source distributions in two- and three-dimensional problems , 1987 .

[8]  P. Yla-Oijala,et al.  Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions , 2003 .

[9]  John J. Ottusch,et al.  Numerical Solution of the Helmholtz Equation in 2D and 3D Using a High-Order Nyström Discretization , 1998 .

[10]  Seppo Järvenpää,et al.  Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra , 2003 .

[11]  D. Wilton,et al.  Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains , 1984 .

[12]  Roberto D. Graglia,et al.  On the numerical integration of the linear shape functions times the 3-D Green's function or its gradient on a plane triangle , 1993 .

[13]  S. D. Gedney.  On deriving a locally corrected Nystrom scheme from a quadrature sampled moment method , 2003 .