Hierarchical Divergence Conforming Bases for Edge Singularities in Quadrilateral Cells

Abstract

Singular divergence-conforming bases have been proposed for the solution of integral equations although they have seen only occasional use in practical applications. The existing singular bases are not hierarchical, which prevents their use in adaptive p-refinement applications. In this paper, a new family of singular hierarchical basis functions is proposed for quadrilateral cells. These functions model the singularities associated with current and charge density at edges and are more convenient for modeling such singularities than triangular bases of the same kind. The basis functions are of the additive kind and combine a hierarchical polynomial representation on quadrilaterals with linearly independent singular terms that incorporate general exponents that may be adjusted for the specific wedge angle of interest. Moreover, the added singular basis functions are computed on the fly. On the basis of various reported numerical results, this paper also illustrates the difficulties, the advantages, the accuracy, and the cost of using such bases in the method of moment solutions of integral equations.Index Terms-Basis functions, hierarchical basis functions, method of moments (MoM), singular basis functions, wedges.

Hierarchical Divergence Conforming Bases for Tip Singularities in Quadrilateral Cells

“…Higher order representations on smooth parts of an object can be effective, but the presence of wedges, sharp edges and tips can negate the improvement in efficiency due to the unbounded nature of current and charge densities in these locations. 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 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 tangential current component is unbounded along the two edges forming the tip, although the even components vanish at the tip. Of interest is the fact that the leading order singularity of the current (1, 2) is not directly related to that of the charge density (3). In fact, (1, 2) and Fig.…”

“…In the following, improved divergence-conforming basis functions are proposed for modeling tip singularities of salient sectors of arbitrary aperture angle. These new functions form a hierarchical set whose lowest-order members are motivated by the Andersson bases; members of the set may be added to the edge-singular bases of [3] in cells connected to tips to more properly model the current and charge density singularities. We present results for planar conducting plates.…”

“…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.…”