Hierarchical Divergence Conforming Bases for Tip Singularities in Quadrilateral Cells

Electromagnetic scattering from targets such as thin conducting plates induce singular currents and charges at sharp edges and sharp tips. In this article, a hierarchical family of divergence-conforming singular basis functions are presented for modeling the singularities associated with current and charge density at tips. These new basis functions are used to increment existing edge-singular bases so that on cells that contain a singular tip where two singular edges join together, the final base combines a hierarchical polynomial representation with linearly independent singular terms that incorporate general exponents that may be adjusted for the specific wedge angle of interest and for the specific angle at the tip. Several variations on the tip functions are proposed.


Hierarchical Divergence Conforming Bases for Tip Singularities in Quadrilateral Cells
Roberto D. Graglia , Fellow, IEEE, Andrew F. Peterson , Fellow, IEEE, and Paolo Petrini , Member, IEEE Abstract-Electromagnetic scattering from targets such as thin conducting plates induce singular currents and charges at sharp edges and sharp tips. In this article, a hierarchical family of divergence-conforming singular basis functions are presented for modeling the singularities associated with current and charge density at tips. These new basis functions are used to increment existing edge-singular bases so that on cells that contain a singular tip where two singular edges join together, the final base combines a hierarchical polynomial representation with linearly independent singular terms that incorporate general exponents that may be adjusted for the specific wedge angle of interest and for the specific angle at the tip. Several variations on the tip functions are proposed.
Index Terms-Basis functions, hierarchical basis functions, method of moments, singular basis functions, tips, wedges.

I. INTRODUCTION
T HE electromagnetic modeling of complex 3-D structures requires numerical techniques. In recent years, higher-order representations have been developed to improve accuracy and efficiency of these approaches [1]. 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]- [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]. Throughout this article we use the term "edge singularity" to describe a singularity occurring along a line segment within the domain, such as the boundary of a conducting plate on which the current is expanded. An edge singularity takes up one side Manuscript  of a cell. A "tip singularity" occurs at a single point in a domain, typically the corner of a cell, and involves a different behavior as explained below (see Fig. 1). A tip singularity may be modeled by the apex of a cone. A conducting cone that has been flattened in the limiting case is known as a plane angular sector (PAS), and can be used to determine the field behavior near the tips of plates or apertures [7]- [9]. The PAS was studied by Satterwhite and Kouyoumjian [7] who proposed an exact solution in sphero-conal coordinates (r, θ, ψ), where r is the distance from the tip and the angle ψ ranges from 0 • to 180 • across one side of the sector (note that the polar frame (ρ, ς) can describe a PAS with a common distance from the tip, but the angles ς and ψ are not simply related).
The application of interest is electromagnetic scattering, involving an incident uniform plane wave of monochromatic frequency that induces currents on the conducting target, which in turn produce scattered fields. 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] with a corresponding expression for the charge density in the form The exponents in (1)-(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 current-components are associated with even values of m = 2 n. The odd or tangential current-components are associated with odd values of m = 2n − 1. The even/odd designation originates with the type of boundary conditions imposed on the Lame equations [7]. Note that (2) involves only odd exponents. Note also that the terms in the sums (1)-(3) may or may not be excited from a particular source, and therefore may or may not be present in a particular solution even if a vertex is present. Brinkley [23] used the results of [7] to improve the accuracy of the far-field results obtained with high-frequency techniques and current-based approaches (i.e., physical optics and its extensions) in many nonspecular regions around a flat plate. In fact, by considering a finite (but not too small) number of coefficients ν oq and ν eq , one gets better approximations of "the diffracted vertex current" that yield results that compare better with numerical results and measurements for most cases.
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), and (2) is not directly related to that of the charge density (3). In fact, (1) and (2) and Fig. 2 show that for salient sectors the leading order singularity of the current is ν o − 1 while that for the charge density is ν e − 1.
Throughout this article, as in (1)-(3), we denote terms related to even components with the subscript "e," while we use the subscript "o" to denote terms related to odd components. In the following the dependence on q is understood and the (integer) subscript q may be omitted. In addition, we sometimes specify the singularity order by the values of q and n since in the exponents we report the correct value 2n and 2n − 1 for the even and odd components, respectively.
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 • sector), and in [6] and [9] for q = 1. Sectors having aperture angle higher than 180 • are termed "reentrant"; these are out of the scope of this article and ought to be numerically modeled in a manner different from the one reported in this article.
Previous work on this topic was published by Andersson, who carried out a method of moments (MoM) analysis of the rectangular plate by developing special basis functions that provide the leading order current and charge singularities on rectangular cells within the electric field integral equation (EFIE) [5]. However, most MoM approaches for the EFIE are based on divergence-conforming basis functions, and Andersson's functions for the square plate are only approximately so. A second attempt at the corner problem was proposed by Ozturk et al. [24], who employed triangular cells, but included only one exponent at the tip.
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.
Section II discusses why numerical models capable of modeling edge singularities are inadequate for modeling tip singularities. Section III introduces the coordinates and variables used to describe tip cells. The new functions to model the odd and even current component near a tip are presented and discussed in Section IV. 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 article [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.

II. WHAT THE OTHER HIGH-ORDER BASES ARE MISSING
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. Table I reports the Andersson bases in the Cartesian reference system with origin at the 90 • sector tip [5]. The bottom  Table I reports the "reconstructed" currents and divergences obtained from the hierarchical basis of [3, Table II] consisting of edge-singular functions associated with the two edges that form the sector tip. The fact that the charge density (3) does not contain a term proportional to r ν o −2 led Andersson to model the "odd" current-component proportional to r ν o −1 with a basis function having zero divergence over its entire definition domain [5].
By rewriting the Andersson bases from Table I in polar (ρ, ς) coordinates, one obtains equations that agree with (1)-(3); for instance, Although the Andersson basis functions and the reconstructed functions of Table I obtained from the hierarchical  basis of [3, Table II] look similar, the odd current of Andersson is unbounded at the tip whereas the reconstructed J o approx. is exactly equal to zero on the sector tip to satisfy the boundary conditions along the edges. Also, the divergence of J o approx. vanishes for x and y going to zero with the same speed, i.e., for x = y → 0. A similar consideration is repeated for the even current-components. At the tip, the singularity of the divergence of Andersson's function is completely different from the reconstructed one. Approaching the tip, both ∇ · J e and ∇ · J e approx. diverge, but clearly with different rates since, from Table I This suggests that the models must be improved in the immediate vicinity of the tip itself. In Sections III-V, we introduce vector sets able to mimic the asymptotic expressions (1)-(3) and then extract basis functions of singularity order q and n.
In the end, we will retain only the functions that are unbounded at the tip. This implies that we will retain only the functions with singularity order n = 0 because the other functions for n ≥ 1 yield a zero contribution to the current and its divergence around the tip.
III. PARENT VARIABLES AND SINGULAR FUNCTIONS ON TIP CELLS The tip basis functions are piecewise defined on three adjacent quadrilateral cells, locally numbered counterclockwise from 1 to 3 in Fig. 3. The tip of the sector coincides with the free vertex of cell 2, not adjacent to cell 1 nor to cell 3.
The cell edges are also locally numbered from 1 to 4 in counterclockwise order, and four edge-vectors that are functions of the child-variable r are then defined, with [1] β + β±2 = 0.
To avoid confusion, we identify with different subscripts the variables and the edge-vectors of cell 2 with respect to those used for cells 1 and 3, so as to ensure the following. 1) Cell 1 has its ξ i+1 = 0 edge in common with the ξ j = 0 edge of cell 2. 2) Cell 3 has its ξ i−1 = 0 edge in common with the ξ j −1 = 0 edge of cell 2. Hence, the dummy variables used to describe cells 1 and 3 are {ξ i , ξ i±1 , ξ i+2 }, while cell 2 is described by the variables . Cell 2 has its vertex ξ j +1 = ξ j +2 = 0 on the sector tip, while the singular edges of cells 1 and 3 are along the local coordinate-line ξ i = 0, as in [3]. Fig. 3 shows the edge vectors j +1 and j +2 of cell 2, and the edge-vector i of the cells 1 and 3. Table II shows the expression of the tip singular functions of singularity order n. T and are tangential functions, R is an even radial function. The functions T and R are piecewise defined on three adjacent cells locally numbered counterclockwise from 1 to 3, as shown in Fig. 3. j +1 , j +2 are bubble tangential functions that span only the number 2 cell. In the Table, J (with no subscript) indicates the Jacobian of the transformation from parent to child space of each cell. Demonstration of the continuity of the normal component of T and R along the edge in common to a cell pair is straightforward, and it is left to the interested reader. The value of the two constants K T and K R of Table II does not affect the conformity to the divergence of the functions and therefore, in applications, one can safely use K T = K R = 0.

IV. BASIS FUNCTIONS FOR TIP SINGULARITIES
By replacing T n (ξ i , 1) with (νξ ν−1 i − 1) in the T 1 and T 3 parts of T on cells 1 and 3, one obtains the edge-singular of [3, Table I], apart a sign adjustment. The same applies to functions R 1 and R 3 .
The piece T 2 of T has zero divergence and is obtained from the potential 2 as follows: wheren is the unit normal to the plane angular sector. Fig. 4 shows the behavior of T and R for n = 0 and K T = K R = 0 on a 90 • sector described by three equal square cells of unit side. Fig. 5 shows the behavior of the bubble functions j +1 and j +2 for n = 0 on the same 90 • sector. Outside a small circular area around the tip, the normal component of the tip functions along the ξ j +1 = 0 edge (and the ξ j +2 = 0 edge) must go to zero as the square root of the distance from the edge itself [27], therefore proportional to ξ j +1 (or ξ j +2 ). Since the functions we consider are even or odd, the behavior is shown only for the components  orthogonal to the ξ j +1 = 0 edge, that is, the components along the gradient of ξ j +1 . In fact, in the limit of small ξ j +1 , and for 0 ≤ ξ j +1 < ξ j +2 and K T = K R = 0, one obtains ⎡ with A T = −(2 n + ν o − 1/2). Table III clarifies the modeling capabilities of the tip functions on a 90 • corner-cell with unitary Jacobian and side length. In this case, by introducing a local polar reference frame (ρ, ς) centered on the 90 • tip, the tangential functions T 2 and R 2 , and the bubble functions simplify as in Table III, in agreement with Andersson [5] for n = 0, and with cos ς and sin ς given as in Table IV. Also, by considering the variation in ρ of these functions it is quite evident that, in the vicinity of the tip, these are in full agreement with the asymptotic expressions (1)-(3). Sufficiently far away from the tip and from the edge(s) opposite the tip, however, the singularity goes progressively over to the "edge" type. The T , R, and functions can be normalized, for example, by setting As can be seen from Table III, the order m = 2n − 1 of the T function is lower than the order m = 2n + 1 of the bubble functions . Therefore, functions of order n should only be used with the T function of order n + 1. At this point, it is apparent that the bubble functions do not contribute to the currents near the tip. Thus, in applications it may be sufficient to use only the T and R functions of order n = 0. As a matter of fact, the bubble functions were introduced and discussed only to highlight that the function family of Table II models the asymptotic behavior of the current given in (1)- (3). A user may instead decide to use additional functions in the q series higher than order q = 1. However, it may be quite challenging to use functions of order q > 1 in an MoM code because these functions are not very linearly independent of each other. Given that as q grows the functions are essentially zero near the tip (ρ = 0), the singular tip behavior is primarily provided by the functions obtained with q = 1 (and n = 0).

V. MODIFICATION OF T AND R INTO BUBBLE FUNCTIONS
The divergence-conforming T and R functions of Section IV span three cells; cell 2 containing the tip and adjacent cells 1 and 3. However, it is simple to modify these functions so that they span only cell 2 by adding some corrective terms to T 2 and R 2 that make them vanish along the edges in common with cells 1 and 3. The corrective terms shall not modify the value of the function and its divergence at the tip. In this way, one obtains functions of the type shown in Fig. 6 (we suggest the reader compare this figure with Fig. 4).
For example, the T 2 function of Table II obtained for K T = 0 can be modified to obtain the tangential function The divergence of T is no longer zero but vanishes at the tip. Likewise, one can add a corrective term to the R 2 function of Table II obtained for K R = 0 to produce the radial function The last two terms on the right-hand side of (21) ensure that the divergence does not change sign along the two edges that form the tip. The modified tip functions T and R of singularity order n = 0 are shown in Fig. 6. The preceding equations constitute an additive correction technique, but alternatively one can use a multiplicative correction technique to replace T 2 and R 2 with The divergence of (24) and (25) is with Once again, the last two terms on the right-hand side of (21) should be added to (25) to guarantee that the divergence of R does not change sign along the two edges that form the tip. We leave as an exercise for the interested reader to derive the analytical expression for the divergence of these latter functions.

VI. NUMERICAL RESULTS
This section reports results for infinitely thin metal plates obtained by solving the EFIE with and without the tip-singular functions presented in this article; the representation includes the edge-singular quadrilateral basis of order [ p, s, m] illustrated in [3]. The edge-singular background base is described by an integer p denoting the background polynomial order, an integer s giving the number of fractional exponents from [3, eq. (4)] included in the edge-singular representation, and an integer m providing the order of the edge-singular basis subset in the longitudinal direction. For the tip-singular functions, we report results obtained as follows.
1) The tip-functions described in Section IV that span three rectangular cells (tip-functions A).
2) The multiplicative corrected tip-functions given in (24) and (25) that span only the cell at the corner (tipfunctions B). Fig. 7 shows results for the current induced by a monochromatic plane wave normally incident on a square plate of size 1λ×1λ meshed with 25 square cells of equal size λ/5 × λ/5. The plate is in the x, y-plane with sides parallel to x and y, and the incident E-field is polarized in the negativê y-direction. The figure shows results only for the real part of the current; the imaginary part which is smaller than the Imaginary part of the induced current at grazing incidence on two plates 0.2λ × 0.4λ spaced 0.05λ obtained using the [1,1,0] order base augmented with the type B tip-singular functions. For the far-zone fields, the results with and without the tip-singular functions are similar. Nonetheless, near-field differences in the current density are clearly visible in Fig. 7 at top right. The order of the system matrix of the problem treated with the edge-singular base of [3] (without the tip functions) is 216, with CN = 440. With the tip functions, the order of the system matrix is 224, with CN = 600.
Figs. 8 and 9 consider the case of two 0.2λ × 0.4λ plates located in the Cartesian plane x, y with sides parallel to x and y, and separated by a gap of 0.05λ. Each plate is meshed identically by using 32 square cells of sidelength  λ/20. The plates are illuminated by a grazing-incident plane wave propagating along the positiveŷ-axis with a unitary E-field polarized along the positivex-direction. This structure has been studied numerically by using the [1,1,0] order base augmented or not with the type B tip-singular functions. While the real part of the induced current shows little difference after incorporating the tip-functions, some differences are evident in the imaginary component of the induced current shown in Fig. 8. Fig. 9 shows the difference in the imaginary currents of the two-plate problem of Fig. 8 obtained with and without the tip-singular functions, magnified by a factor of 20. It is evident that the dominant difference is due to the radial (even) component of the current at the two adjacent tips along the trailing edges (top of figure) of the plates. Quadrilateral cells can model tips for a fairly wide range of salient angles; for example, let us consider the triangular plate of Fig. 10 located in the first quadrant of the Cartesian plane x, y, with a side parallel to the x-axis, and on which impinges a grazing plane-wave propagating along the positiveŷ-axis with unitary E-field polarized along the positivex-direction. Fig. 10 shows the real (top row) and the imaginary part (bottom row) of the current on the triangular plate (with sides equal to 1λ) obtained with the [1,1,0] base augmented by the tip-basis functions of the B type. Exponents for a 60 • corner are ν o = 0.91904 and ν e = 0.24010. It appears that in this case the even tip singularity prevails over the odd one for the two corners at bottom; for the top corner it is the other way around.
The preceding plots show differences between the results obtained with and without the tip singular functions in the near-field region (for example, for the induced currents), but differences can also be observed for far-field results. Fig. 11 shows the backscatterer radar-cross section (RCS) obtained with bases of different orders for the square plate of size 1λ×1λ considered in Fig. 7 (left-hand column), and for the equilateral triangular plate with sides equal to 1λ of Fig. 10 (right-hand column). Normal incidence results are reported at top, grazing incidence results are given at bottom. The results obtained with a base augmented by the tip singular functions B are those marked with the red diamonds. From these results, it is quite evident that relative to a pure polynomial representation, the edge-singular functions dramatically improve the convergence of these results, while the tip-singular functions provide a more minor, secondary level of improvement that is more noticeable for grazing incidence than for normal incidence.

VII. CONCLUSION
New families of divergence-conforming basis functions for modeling the singularities associated with current and charge density at tips are presented. The new basis functions are used to increment existing edge-singular bases so that on cells that contain a singular tip where two singular edges join together, the final base combines a hierarchical polynomial representation with linearly independent singular terms that incorporate general exponents that may be adjusted for the specific wedge angle of interest along edges and for the specific corner angle at the tip. Results for the current induced by plane waves incident on infinitely thin rectangular or triangular plates are presented. From the few test cases examined in this article, all of which involved problems where large edge currents were induced by the excitations, one may conclude that the addition of tip-singular functions to the existing edge-singular basis sets generally provides second-order improvements in the numerical results. In situations where edge effects are less pronounced, however, tip singularities may take on greater importance.