An Efficient Framework for Using Higher-Order Curved Div

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 8, AUGUST 2012

An Efficient Framework for Using Higher-Order Curved Div-Conforming Elements With Cancellation of Weak Singularities in Local Space for Surface Integral Equation Gil José Ma, Member, IEEE

Abstract—A reliable scheme for using higher-order curved divconforming elements is presented. This procedure takes advantage of tangential basis functions developed for finite-element applications. Throughout the theorem of trace, the Helmholtz decomposition into solenoidal and nonsolenoidal subspaces is converted into a Hodge decomposition for the surface currents. Furthermore, a cancellation of the singular term appearing within the integrals is carried out acting in the local space, avoiding the difficulties associated to the curved domains, when coordinate transformations are used with this propose. Index Terms—Div-conforming elements, Helmholtz decomposition, singular cancellation.

I. INTRODUCTION

T

HE method of moments (MoM) transforms a functional operator equation into a matrix equation. When the operator is the surface integral, its solution is useful for the study of antenna and scattering problems involving homogeneous objects. In recent years, the developed codes usually model the geometry of the structure by means of a set of subdomains or elements of simple shapes. The basis functions are defined on rectilinear elements in a local space, which are then transformed into curvilinear real space. Since the lower-order basis by Rao, Wilton, and Glisson, higher order basis have been proposed by several authors, including interpolatory basis functions in both curved triangle and quadrilateral patches [1] or hierarchical basis functions in curved triangles [2] and quadrilaterals [3]. It is known the bad behavior of the numerical solution of Maxwell’s equations when the frequency tends towards zero or when the size of the elements is small compared to the wavelength. This is the result of the decoupling of electric and magnetic fields. The separation of the bases into solenoidal and nonsolenoidal functions is of importance when addressing this issue. Basis functions dealing with the issue have been called as Manuscript received May 05, 2011; revised November 14, 2011, January 12, 2012; accepted March 09, 2012. Date of publication May 23, 2012; date of current version July 31, 2012. This work was supported in part by the Spanish Ministry of Science and Innovation under reference TEC2010-20249-C02-01 and by Program Consolider Ingenio under the project Terasense CSD2008-00068. The author is with the Department of Electromagnetic and Circuit Theory, UPM, 28040 Madrid, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2012.2201084

“Loop-Tree” [4] and “Loop-Star” [5], [6] and with curved elements [7] in the literature. A broad band approach should be compatible with a solenoidal-non solenoidal decomposition and higher-order basis functions for curved domains. All of these requirements are hard to find simultaneously, and they are the main goal of the present work. Furthermore, in mixed potential integral equation formulations, a singular integration must be solved. Conventional methods of extraction [8], [9] or cancellation of the singularity [10], [11] can be algebraically complicated when both higher order basis functions and curved elements are used simultaneously. In this work, we obtain higher order div-conforming basis functions directly from the curl-conforming finite-element families by rotation. In this way, we take advantage of the wide and fine work developed by the finite-elements community. Finiteelement method (FEM) and MoM are similar; a mathematical operator in an infinite dimension space is projected into a discrete subspace ( dimension) spanned by a finite base of functions ( functions). The obtained div-conforming elements are curved triangles, hierarchical, nearly orthogonal and isotropic in relation to the vertex. All the integrals are calculated on a reference straight triangle. The obtained basis functions can be of any order and the Helmholtz decomposition in the curl-conforming elements, i.e., the decomposition of the basis functions space into two subspaces Gradient and Rotational [12] is transformed by means of the surface rotation into a Hodge decomposition for surface currents consisting of a subspace of solenoidal functions and a complement of nonsolenoidal functions. These two subspaces are called “Loop” and “Star”, for analogy with the Loop-Star decomposition although the approach is here different. We work with a known family of elements, quite popular among the finite-element designers [13], although any other finite-element basis might be useful as well. The curl-conforming and div-conforming bases share the same components in local space; they are the coefficients of curvilinear vectors which give the properties of continuity of the vector basis functions across the boundaries of the elements. The decomposition of the bases to avoid the low frequency breakdown is suitably kept throughout the surface rotation. In addition, for addressing the problem of the singular term in the integral equation, we use the coordinate transformation described in [14] and similar to that used in [15] from local space

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JOSÉ MA: EFFICIENT FRAMEWORK FOR USING HIGHER-ORDER CURVED DIV-CONFORMING ELEMENTS

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to a new space, which, by means of the Jacobean, introduces a term that cancels the singular behavior. The singularity cancellation is carried out directly on the local domain. The procedure is compatible with any order of curvature of the elements and invariant with the order of the basis functions. In order to test the good behavior of the elements some canonical scattering PEC problems are analyzed.

being the local matrix which relates the interaction between the m and n elements.

II. FORMULATION

Some excellent descriptions of the instability of the low frequency solutions in EFIE can be found in the literature; see, , the mage.g., [17]. In summary, it is well known that if neto-static and electrostatic equations are uncoupled. the surface curIn magneto-static, as a result of . The rent must be solenoidal (div-free) because solenoidal current, induced by the low frequency (LF) magnetic field governs the inductive phenomena. In electrostatic, and this current is the result of the charge (LF electric field), , and which must be nonsolenoidal (curl-free) because it governs the capacitive phenomena. The decoupling of the very low frequency magnetic and electric fields, known as Helmholtz decomposition, is projected into the currents in a decomposition of two components, solenoidal and nonsolenoidal. The numerical method must provide this decomposition in the basis functions. This has been addressed by means of the “Loop-Tree” and “Loop-Star” decomposition. We are going to maintain the nomenclature by calling the loop current (div-free) , and the star current (curl-free) . Hence, when

Consider a perfect electric conductor (PEC) of surface in free space and excited by an incident plane wave . The condition that the tangential total electric field must be zero on the surface of the object leads to the electric field integral equation (EFIE)

(1) where is the distance from the source point to an arbitrary observation point located on ; and are the permittivity and permeability of the medium, respectively, , and is the unit external normal vector to . The surface current density is represented by a superposition of tangential to vector basis functions

III. HIGHER-ORDER DIV-CONFORMING BASIS FUNCTIONS FOR CURVED TRIANGLES A. Low Frequency Instability of EFIE

(7)

(2) A Galerkin projection procedure yield to an algebraic system as (3) where are the unknowns; an element of the impedance matrix is given by

(4) (5) The object to be studied is meshed by means of curved triangles which are transformed into a straight reference element in (p, q) space. We solve the integrals in the local space, element by element and, after an assembling procedure, we conclude the discretization process of the EFIE. The Z matrix is the sum of local element matrices (FEM philosophy [16]) (6)

Moreover, the EFIE, after Galerkin, when test function Nj, is reduced to

and for a

(8)

and the solution is formed by solenoidal currents, i.e., stationary currents. They constitute the null space of the divergence operator, N (div). Therefore, the accuracy of the LF solution depends on the approximation of the N (div). Moreover, LF agrees with the near field, and a good representation of N (div) improves the solution for the near field. For high frequency (HF), or far field, the potential vector term is the relevant; it is convenient to use higher order basis functions to approximate this current. In addition, the inconvenient of the frequency scaling inherent to the EFIE can be compensated by making use of the decomposition of the current into loop and star (7). The [Z] matrix, , is made up of blocks with a different frequency when dependence: (9) To improve the condition number of the [Z] matrix and balance the equation, a frequency normalization might be carried out; this can be seen as a post and preconditioning that makes the final matrix diagonally dominant and, then, stable at low frequencies. Numerical results can be found in literature (see [17]).

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B. Div-Conforming Basis Functions Space Once the convenience of the decomposition of the current into solenoidal and nonsolenoidal components is seen, we will use a space of functions as follows (10)

are the coordinates of the vertices and order polynomials, with

are second(14)

being shown in the equation at the bottom of the page and the unit external normal vector to is

The space is divided into a “Loop” subspace made up of solenoidal (zero div) functions of degree and a “Star” subspace made up of nonsolenoidal (nonzero div) functions of degree ; this nomenclature is borrowed from the literature for analogy with the Loop-Star decomposition:

(11) The subspace constitutes the null space of the divergence is made up of nonzero div funcoperator, N (div). The space tions to complete the vector space. Each subspace can be further decomposed into three subspaces associated with the edges of the triangle and one subspace of functions associated with the face of the triangle. Functions in an edge subspace will have a nonzero normal component along one edge and they will be tangential (or zero) along the rest of the edges, and the functions in a face subspace will be tangential (or zero) to all the edges of the triangle. These high-order vector elements have two orders (m, n); when both orders are the same, the element is complete to order m; if not, the elements are mixed-order (see [13]). As it has often been claimed by the FEM community ([13], [18]), these mixed-order elements remove degrees of freedom associated to the N(div) or space Loop and they do not affect the approximations of the scalar potential term (divergence term) and they do not improve the order of convergence. C. Mapping of Curved Elements The surface current is approximated in the element by means of a base of functions as described before. (12) being the position vector and the number of basis functions, respectively, in the element. The mapping is determined by three parametric functions

(16)

D. Rotation of Curl-Conforming Basis Functions We obtain the div-conforming elements throughout the trace theorem, i.e., by rotating the surface trace of curl-conforming basis functions. This is carried out by maintaining the advantages of the basis developed within the FEM framework; for this work and referencing to the FEM elements described in [13] they are hierarchic, quasi-orthogonal until order 2; the face functions are isotropic in relation to the numeration of vertex, and more important, the Helmholtz decomposition for the curl-conforming space is converted into a Hodge decomposition for the surface currents. Moreover, curl-conforming mixed or complete (m, n) order elements are going to share the same components of the basis functions with the div-conforming ones. The curl-conforming functions are tangential to the surface and they have tangential continuity across the elements allowing normal discontinuity. A tangential function is (17) are the covariant components and are the reciprocal base vectors. If we rotate it on the surface

(18) We define two vectors as a consequence of the rotation

(13)

(15)


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TABLE I DIMENSION OF LOOP/STAR/OVERALL FUNCTION SPACES FOR SOME ELEMENTS

(19) It can be seen that (20) are the base vectors. The obtained div-conforming functions are TABLE II LOCAL COMPONENTS OF BASIS FUNCTIONS FOR THE (1, 2) ELEMENT

(21) The surface divergence becomes (22) As shown, we obtain the div-conforming basis functions by means of the surface rotation of curl-conforming tangential basis functions which are decomposed into two subspaces; we and rotational , in order to achieve the call gradient Helmholtz decomposition for the electromagnetic fields. The overall function space is of dimension (m, n)

The Gradient space:

(23) (24)

is the null space of the curl operator, and the Rotational space (25) is the complement to complete the overall function space. soThe Gradient subspace approximates the static lutions, which generates spurious modes, and the quasi-static singular modes in the vicinity of the sharp edges. If we rotate the tangential vectors to the surface belonging to the Gradient space we obtain nonsolenoidal tangential functions, i.e., these functions belong to the defined “Loop” subspace. Looking at the following property (26) when the function is irrotational, the divergence of the rois zero. Then, the Loop subspace could be tated function identified as the rotated Gradient subspace. When is not irrotais not zero. tional, the divergence of the rotated function Then, the Star subspace could be identified as the rotated Rotational subspace as well. In this way, we have the overall space of functions, for the surface current, decomposed into two subspaces, div-free, and another subspace containing, at least, some irrotational (curl-free) functions. This decomposition achieves a broad band and stable discretization of the surface integral equation [5]. In terms of the Sobolev vector spaces, the curl-conforming function space is defined by (27)

and this is where the and fields reside in a natural way. If we apply the twisted tangential trace operator (28) we obtain the surface trace of a div-conforming space (trace theorem) where the surface current resides. Formal descriptions of these spaces can be found in [19]. This space admits a Hodge decomposition (29) being the null space (kernel) of the surface divergence operator, where the solenoidal currents reside and the complement subspace of functions which has some components in the curl-free space. If we apply the twisted tangential trace (30) The dimensions (number of independent functions) of the subspaces are shown in Table I. The components of the basis functions in local space (p, q), for a (1, 2) element are given in Table II. Some of them are associated to edges and the others are associated to the face of the element. They are the same for both curl-conforming and div-conforming elements, as derived from (17) and (21) (see [12]). It can be useful to point out that, when the element is in the vicinity of a sharp edge, we can use the singular functions developed in the literature as in [14] to model singular magnetic and electric fields to approximate the correct behavior of the current density near the edge of a wedge. This fact simplifies and

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reduces the need to develop new families of basis functions to model singular current densities. IV. COORDINATES TRANSFORMATION IN LOCAL SPACE FOR CANCELLATION OF THE WEAK SINGULARITY OF EFIE It is derived from the EFIE (1) that, when source and field elements coincide, a singular kernel of integrals arises. This is the self-term case. Furthermore, if both elements are very close to each other, but they do not overlap, the integrals are nearly singular. This kind of singular kernel is the result of the factor 1/R within the Green function

Fig. 1. Mapping to unity square.

(31) We use a cancellation method based on a geometrical transformation whose Jacobean cancels out the singular term of the integral kernel. The cancellation is carried out directly in the local space. This is an additional advantage when curvilinear domains are involved, because the equations governing the geometrical transformation are linear and independent of the real curvilinear domain. The method uses an approach similar to used by Duffy, dividing the domain, in local space, into three rectilinear subtriangles, sharing the singular point or its projection as a common vertex. In order to cancel the singularity we use a different geometrical transformation which is a conversion to triangular polar coordinates of the transformation due to Wait [20], which has been employed in fracture mechanic problems and applied to singular electromagnetic in [21]. The singular integral is of the form (32) represents the nonsingular integrand; it can contain basis functions of any order. The 1/R factor is the singular one. A geometrical transformation, dealing with curvilinear elements (order equal or greater than two), is applied to each element. That is described with the expressions (13), (14), and (15). The integral becomes

(33) In order to cancel the singularity, a second coordinate transformation (Wait modified) between rectilinear elements is made. The p-q local triangle is transformed into the unity square in the space (see Fig. 1). The vertex 1, which coincides with the point where the singularity is located, is . transformed into the edge The coordinate transformation is given by the following expression:

Fig. 2. Partition of the triangle into three subtriangles sharing the singular point.

The “m” factor adds some flexibility to the capacity to model singularities with different orders, useful to nearly singularities, that is, when the source point is very close to the field point. It generalizes the cancellation of the 1/R term into the integrand to . The possibility to a singular integrand with any order handle nearly singularities for integrands in the Integral Equation is not explored in the present paper. This study demands the definition of a criteria to assign the nearly singularity in relation to the proximity between elements, and relative to its electrical size. The “m” parameter must be chosen to suit the level of near singularity; the election of the “m” parameter requires a study, even an optimization process to smooth the near singularity. , because our goal is to In this work, we are going to set cancel the 1/R divergence in the self-term. Then, the following equation of proportionality is produced (35) In this way, the Jacobean cancels out the 1/r type divergence in the integral. The singularity on the vertex 1 of each subtriangle is “diluted” along the edge 1 of the unity square. This as present in method also softens a stronger singularity the magnetic field integral equation (MFIE). For each Gauss integration point for the outer integral in the (p, q) space, a singularity occurs within the inner integral. The triangle is divided into three subtriangles , and sharing the singular point as a common vertex in the integral I is then calculated as the sum of three integrals extended to each subtriangle (Fig. 2) (36)

(34) with being the vertex of the where element. The Jacobean of the transformation is (35)

The three integrals are integrated on the unity square by means of a Gauss quadrature. The current approach has the effect of clustering Gaussian points toward the singular point (see Fig. 3). Analogous expressions can be obtained for quadrilateral or tetrahedral elements.


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Fig. 3. Clustering of integration points towards the singular point.

Fig. 5. Radiation pattern of a =2 dipole antenna.

of the field components related to the vector potentials this region. For a PEC object

in

(39)

Fig. 4. Current density of a =2 dipole antenna.

(40) Finally, the integral is where is the number of elements in the mesh, is the number of basis functions in the element and are the basis functions, with

(37) The geometrical transformation is different for each subtriangle. The integration points and the Jacobeans change for each of those. If we apply the expression (34), with , we obtain the vertex of the three subtriangles, , with , and the Jacobeans

The integral is carried out, element by element, and all the contributions are summed. This is done for each space direction. Moreover, the radar cross section can be calculated as (42)

VI. SOME NUMERICAL APPLICATIONS

(38) These geometrical transformations can deal with a singular point located outside the local domain, or on its border. If it is outside, at least one coordinate is negative, so the Jacobean would be negative as well. V. FAR FIELD COMPUTATION Once the surface current density on the object has been calculated, we can compute the far field by using the expressions

In order to show the good behavior of the basis functions obtained by the present approach, some standard scattering problems are explored. They check the performance of the basis when expand surface current with smooth variation (dipole), divergent currents nearby to a metallic edge (2-D plate), and smooth currents in curved 3-D domains (conducting sphere). A. Dipole Antenna A simple dipole antenna was simulated by means of a thin long strip of 2 cm long and 0.05 cm wide. The antenna is excited by a plane wave whose incidence direction is along the dipole axis. The frequency of 7.5 GHz gives a dipole whose length is half the wavelength of the incident wave. Two types of curved

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Fig. 7. Scattered field along a '

Fig. 6. Current density of a

2 metallic plate.

= 0 cut for a metallic plate.

Fig. 8. Surface current density ('-component) versus , with '

= =2.

elements were implemented with orders (0, 1) and (1, 2). Because of the simplicity of this case, almost identical results with both elements are obtained. In Fig. 4, the module of the density current along the axis of the dipole is shown together with the mesh used; this consists of 114 elements. Computation of current density was carried out by using (1, 2) elements. The results are compared with those shown in [22]. From (39) and (40), the known bidirectional radiation pattern is calculated and can be seen in Fig. 5. B. Thin Square Metallic Plate This target, located on the plane, is excited by a plane wave incident along the direction. Current distributions along the two principal cuts are shown in Fig. 6, for a mesh of 60 elements and the use of both (0, 1) and (1, 2) elements. The results are compared with [23], obtaining a good agreement in both cases. Finally, the far electric field is computed from the calculated current coefficients by using the expressions (39) and (49). The results for a by metallic plate, with (1, 2) elements, can be seen in Fig. 7. They are compared with those by [24], showing a good concordance.

Fig. 9. Surface current density ( -component) versus , with '

= 0.

C. Conducting Sphere

component of the surface current distribution along two cuts, obtained with 572 elements of order (1, 2). A good behavior of the elements is concluded from the comparison with theoretical calculations. Finally, we calculate the scattered field again by using these basis functions. In Fig. 10, we can see the expected variation of cut. the amplitude of the far field along a

The conducting sphere is another commonly used benchmark for checking the discretization of the surface integral equation in a 3-D domain. We meshed a sphere with a diameter of . The frequency of the incident wave is 30 GHz, which is -traveling and -polarized. We show in Figs. 8 and 9, the module of the

In this work, a reliable framework for using higher-order curved div-conforming elements in surface integral equation is presented. This approach incorporates the advantage of using

VII. CONCLUSION


Fig. 10. Scattered field for the sphere of diameter =

(cut ' = 0).

hierarchical tangential basis functions, developed for finite-element applications, to the integral equation. That includes the Helmholtz decomposition for the surface density currents. Moreover, the singular terms within the integrals is cancelled out from the local space. In this way, the difficulties associated with curved domains when coordinate transformations are used, are avoided.

ACKNOWLEDGMENT The author is grateful to Prof. J. Zapata for providing some useful software for this work.

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