Basis Functions for Vertex, Edge, and Corner Singularities: A Review

Hierarchical Divergence Conforming Bases for Tip Singularities in Quadrilateral Cells

“…The inadequacy of polynomial or edge-singular bases to deal with tip singularities is evident by considering the behavior of the current near a 90 • sector tip; in this case the leading order singularity of the current is ν o − 1 = −0.18534 while that of the charge density is ν e − 1 = −0.70342 (see [4]). For salient sectors the singularity of the charge density of the even current-component is always stronger than that of the odd current-component.…”

“…Several types of special basis functions have been developed for the purpose of representing singular currents and charge densities near edges [1], [2], [3]. To date, less attention has been directed at the effect of tips [4], such as the tips of a square conducting plate [5]. The current density and charge density at a plate tip are known to be infinite, with a singularity that is different from that arising along the plate edge [6].…”

“…The PAS scattering problem, just like the canonical problem of the diffraction by a quarter-plane or the scattering from a cone, are classic electromagnetic problems treated by various authors using the Wiener-Hopf or other sophisticated asymptotic techniques (see for example [10]- [22] and references therein). That said, from the exact solution given in [7] one can extract an asymptotic expansion of the current in terms of sphero-conal coordinates [4]…”

“…The exponents in (1)(2)(3) are the sum of an integer m (even or odd) that depends on the outer summation index n plus an irrational number (indicated by the Greek letter ν o,q or ν e,q ) that depends on the inner summation index q. These exponents do not depend on the frequency and grow slowly with q and more rapidly with n [4]. The so-called even or radial currentcomponents are associated with even values of m = 2n.…”

“…The critical parameters are the exponents ν o and ν e of "salient" perfect electric conducting (PEC) sectors shown in Fig. 2 for q = 1; these exponents may be obtained in the manner developed in [7] and have been tabulated in [4] up to q = 4 (for a 90 degree sector), and in [6], [9] for q = 1. Sectors having aperture angle higher than 180 • are termed "reentrant;" these are out of the scope of this paper and ought to be numerically modeled in a manner different from the one reported in the present paper.…”