Computation of EFIE Matrix Entries With Singular Basis Functions

Abstract

The use of singular basis functions enhances the convergence of method-of-moments (MoM) solutions for structures containing edges. While standard algorithms for computing the MoM matrix entries treat Green's function singularities, these are not well-suited for integrating the singular basis functions: conventional quadrature routines exhibit slow convergence and may produce inaccurate results. In this paper, new algorithms are proposed for handling the combination of edge singularities and Green's function singularities on quadrilateral cells.

Integral Transformations to Handle Corner Function Singularities

“…The Green's singularity may be cancelled as in [2] by subdividing the grandparent cell into four triangles sharing a common vertex at (u g , v g ). With reference to Fig.…”

“…The mapping formulas from parent to child coordinates are usually of the polynomial type so that it is simple to turn parent into child coordinates. When the structures to be modeled contain edges or corners it is possible to augment the set of bounded basis functions (normally vector polynomials) with an appropriate set of unbounded or singular basis functions that incorporate the singular behavior of the fields in the neighborhood of the edges [1], [2] and corners [3]. In these cases a second non-linear mapping from the parent to a grandparent ζ-space becomes necessary to perform the required numerical integrals [2].…”

“…When the structures to be modeled contain edges or corners it is possible to augment the set of bounded basis functions (normally vector polynomials) with an appropriate set of unbounded or singular basis functions that incorporate the singular behavior of the fields in the neighborhood of the edges [1], [2] and corners [3]. In these cases a second non-linear mapping from the parent to a grandparent ζ-space becomes necessary to perform the required numerical integrals [2]. In addition to these two mappings, integral equation MoM applications such as those discussed in [3], [4], [5], [6] require further transformations of variables to cancel the Green's function singularity from the self and near-self integrals [7]- [25].…”

“…for u or v negative. Therefore, we use (5) only to deal with the case of an observer in the first quadrant (i.e., for u ≥ 0 and v ≥ 0) while, as previously discussed in [2], we use transformations other than (5) to deal with observer cases in one of the remaining three quadrants, thus working in grandparent spaces other than (5).…”

Hierarchical Divergence Conforming Bases for Tip Singularities in Quadrilateral Cells

“…The functions of Section IV span three rectangular cells; Section V introduces corrective techniques that reduce the domain of the tip singular functions to a single quadrilateral cell. The numerical technique used to compute the MoM integrals associated with singular tip-functions is described in a companion paper [25], while [26] describes the integration of the edge-singular basis functions given in [3]. Numerical results in the frequency domain are reported and discussed in Section VI.…”