Divergence-conforming discretization of second-kind integral equations for the RCS computation in the Rayleigh frequency region

Authors


Abstract

[1] The divergence-conforming discretization in method of moments of the magnetic field integral equation (MFIE) shows huge inaccuracies in the computed radar cross section (RCS) at the very low or extremely low frequency regime. We justify theoretically that the presence of this error in the computed RCS can be mitigated to some extent for some sharp-edged objects, where this discrepancy is most evident, through the adoption of uniformly triangular or quadrangular meshings. Moreover, we present two discretizations in method of moments of second-kind integral equations that are free from these huge RCS inaccuracies: (1) a modified implementation of the conventional Rao-Wilton-Glisson (RWG) discretization of the MFIE that provides in two steps the static and dynamic current components so that the nonsolenoidal static contribution can be discarded in the far-field computation; and (2) the loop-star discretization in method of moments of the novel electric-magnetic field integral equation (EMFIE) which provides inherently a null static nonsolenoidal current contribution whereby a static-dynamic decomposition of the current is not required.

1. Introduction

[2] The conventional discretization in method of moments (MOM) of the magnetic field integral equation (MFIE) shows in general huge inaccuracies in the computed radar cross section (RCS) at very low frequencies [Zhang et al., 2003]. As shown by Zhang et al. [2003], these huge RCS inaccuracies come from the nonnull radiating nonsolenoidal static component of the current arising in the MFIE. The irruption of these huge inaccuracies in the computed RCS depends ultimately on the degree of cancellation of the contributions from the nonnull nonsolenoidal static current in the far-field computation. For a curved object like a sphere, the error appears for extremely low electrical dimensions, below 10−15λ [Zhang et al., 2003]. However, this inaccuracy in the computed RCS for a sharp-edged object is first observed at electrical dimensions around 10−3λ.

[3] Zhang et al. [2003] propose an implementation of the MFIE free from the observed inaccuracy at very low frequencies by discarding the static nonnull nonsolenoidal contribution of the current in the computed RCS after the iterative computation of the solenoidal and nonsolenoidal components of the current through the jk-Taylor's expansion around k = 0, the static limit. The whole procedure becomes rather time consuming because it is required the inversion of the resulting impedance matrix at each iterative step, although it is true that in the very low frequency regime few terms are required.

[4] In section 2, we discuss with examples how the appearance of the error in the computed RCS with a conventional discretization of the MFIE for sharp-edged objects can be shifted to much smaller electrical dimensions with the adoption of uniform meshes. We justify that in this case the overall cancellation in the computed far-field from the nonsolenoidal static contributions is much better reached than with a nonuniform discretization. This represents a simple strategy to extend the validity of the conventional Rao-Wilton-Glisson (RWG) discretization of the MFIE deeper inside of the Rayleigh frequency region, down to the extremely low frequency regime, with no need to implement other more complicated and time-consuming schemes. This strategy, though, can only be applied to objects that are amenable to uniform discretization.

[5] We then provide two other approaches relying on the MOM discretization of integral equations of the second kind that become free from this observed inaccuracy and lead to the right λ−4-asymptotic RCS scaling deep inside the Rayleigh frequency region for any object under analysis:

[6] In section 3, we present a modification of the MOM-MFIE implementation at very low frequencies provided by Zhang et al. [2003]. We propose to simplify notably the process described by Zhang et al. [2003] in two steps. We first compute the static component of the current, and second, the remaining contributions. This strategy requires two matrix inversions for a solution valid at any frequency.

[7] In section 4, we present a new integral equation, the electric magnetic field integral equation (EMFIE), which we implement under a MOM discretization with the loop-star basis functions [Wu et al., 1995]. The EMFIE, just like the MFIE, represents a second-kind integral equation and inherits the advantageous properties of the matrices resulting from MOM-MFIE implementations regarding stable and good condition numbers. The loop-star discretization of the EMFIE provides a zero static nonsolenoidal component of the current which allows the λ−2-asymptotic scaling of the real part of the nonsolenoidal current observed in the electric field integral equation (EFIE) at very low frequencies.

2. Mitigation of the Observed Error With Uniform Meshings

[8] At low frequencies, the discrete expansion of the radiation vector equation image, from which the discrete expansion of the scattered far fields are obtained, can be approximated as

equation image

where S denotes the closed surface embracing the discretized object, k stands for the wave number and equation image denotes the discrete divergence-conforming expansion of the current in terms of the solenoidal (s), loop, {Ln}, and nonsolenoidal (ns), star, {Sn}, sets

equation image

where Nv and Nf denote, respectively, the number of vertices and facets arising from the discretization. Jns and Jnns represent the set of coefficients in the expansion of the current.

[9] The first term in (1) can be expressed for each Cartesian component in a different manner by taking into consideration that

equation image

where the first term becomes null because of the normal continuity across the edges arising from the discretization with divergence-conforming basis functions. Therefore, (1), equivalently, becomes

equation image

As shown by Zhang et al. [2003], the jk-Taylor's expansion around the static limit k = 0, yields

equation image

where the zeroth-order term represents the static component of the current. We call the other components dyn, dynamic, because they gather the remaining terms as defined by Zhang et al. [2003]; that is,

equation image

[10] At very low or extremely low frequency region, the zeroth- and first-order loop-star contributions in the MFIE computed current in (4) lead to the dominant, zeroth-, and first-order, terms in the jk-Taylor's series of the computed radiation vector equation image; that is,

equation image

In view of (2), the development of the first term in (7) becomes

equation image

where Dn stands for the surface subdomain of the nth star basis function. As reasoned by Zhang et al. [2003], this term is responsible for the huge RCS inaccuracies arising in the conventional MFIE implementations in the low frequency regime because the nonsolenoidal static component of the current Jns(0) is nonzero. However, it is possible to mitigate to some extent the irruption of this inaccuracy in the RCS patterns by adopting discretizations schemes that favor the cancellation of (8).

[11] In our experience, for a smooth-shaped object like a sphere, the discrepancy appears in the RCS computation for electrical dimensions below 10−15λ. However, a sharp-edged object shows in general a major deviation in the computation of the normalized far-field (we define the normalized scattered electric far-field as equation image) already for electrical dimensions of 10−3λ (see Figures 1, 3, and 4). These dimensions are even bigger than the observed electrical dimensions where the EFIE blows up due to the low-frequency breakdown (10−8λ) [Qian and Chew, 2008]. Interestingly, the adoption of uniform meshings, equilateral triangular or quadrangular, in sharp-edged objects shifts the irruption of the huge RCS inaccuracies to much smaller electrical dimensions. Note, for example, in Figure 2, for a regular tetrahedron uniformly meshed, with equilateral triangles, how the far-field is accurately computed up to electrical dimensions of 10−8λ, which allows the computed RCS to scale correctly up to 10−15λ (see Figure 3). The same is observed for a cube meshed with squares or rectangles in comparison with a triangular meshing, which becomes necessarily nonuniform for a cube (see Figure 4). See also in Figure 3 how the introduction of a subtle irregularity breaking the original uniformity of the mesh ruins the right scaling of the RCS at bigger electrical dimensions.

Figure 1.

Contribution in the normalized backscattered far-field from the nonsolenoidal component of the current computed with the loop-star discretization of the electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) of a pyramid with square basis and side 1 m meshed with 200 triangles.

Figure 2.

Contribution in the normalized backscattered far-field from the nonsolenoidal component of the current computed with the loop-star discretization of the EFIE and the MFIE of a regular tetrahedron with side 1 m meshed with 100 equilateral triangles under and impinging x-polarized z-directed plane wave.

Figure 3.

Backscattered radar cross section (RCS) computed with the loop-star discretization of the EFIE and the MFIE of a regular tetrahedron with side 1 m meshed with 100 triangles under an impinging x-polarized z-directed plane wave. Different types of triangles for the MFIE are adopted: all equilateral or a few slightly irregular. The parameter dev stands for the relative % shift (referenced to the mesh size with uniform meshing) of the limiting vertices position after introducing the irregularity with respect of their position before introducing the irregularity.

Figure 4.

Backscattered RCS computed with the loop-star discretization of the EFIE and the MFIE of a cube with side 1 m meshed with 192 triangles, 150 squares, or 94 rectangles under an impinging x-polarized z-directed plane wave.

[12] A MOM discretization can reach by construction the minimization of (8) through the cancellation of equation image for some basis functions. The facet-oriented star basis function Sn is defined formally in the same manner for triangulations [Wu et al., 1995; Lee et al., 2003] and for quadrangular meshings [Vecchi, 1999] as

equation image

where fequation image denote the unnormalized RWG or rooftop basis functions, defined over the edges shaping the central triangular (ns = 3) or rectangular (ns = 4) facet so that the surface divergence in the central facet is positive. The unnormalized RWG or rooftop sets are obtained by dividing the conventional basis functions RWG [Rao et al., 1982] or rooftop [Glisson and Wilton, 1980] by the edge length [Graglia et al., 1997]. Equivalently, the nth star basis function in (9) can be defined as

equation image

where fequation image represent the contributions with negative surface divergence of fequation image, which embrace the facets around the central facet. The star contribution in the central facet, fcn, becomes (see Figure 5)

equation image

where Acn and equation image denote, respectively, the area and the midpoint of the central facet. The contribution in the source of error in (8) due to a single arbitrary star basis function arising from a triangular or a quadrangular discretization, becomes (see Figures 6 and 7)

equation image

Note that only the contributions from the facets around the central facet are relevant in the computation of (12) because the integration of (11) is zero.

Figure 5.

Star basis function over a triangular or quadrangular meshing.

Figure 6.

Domain for the nth star basis function defined over a triangulation.

Figure 7.

Domain for the nth star basis functions defined over a quadrangular meshing.

[13] In general, the source of error in (12) becomes nonzero for an arbitrary mesh. Interestingly, the particularization of (12) for a coplanar meshing with equilateral triangles or with squares or rectangular facets makes it zero (see Figures 6 and 7). Although a sharp-edged object provides flat sides that may be amenable to a uniform discretization, there exist also, at any degree of meshing, nonfavorable star basis functions with local subdomains embracing sharp edges that eventually prevent the right scaling. In any case, the adoption of uniform meshings in sharp-edged objects shifts the inaccurate computation of the RCS from the very low frequency regime (electrical sizes about 10−3λ) to the extremely low frequency regime (electrical sizes about 10−15λ). This may be convenient to extend the validity of the conventional MFIE schemes deeper inside the Rayleigh region with no extra cost (see, for example, Figure 5 of Wildman and Weile [2004] where it is analyzed an octahedron with side 4.7e-9λ and uniformly meshed).

[14] We present two approaches relying on the MOM discretization of Integral Equations of the second kind that are free from the source of error described by Zhang et al. [2003]. In section 3, we present a modification of the scheme presented by Zhang et al. [2003] for the MFIE, where the RCS is computed by discarding Jns(0). In section 4, we present the loop-star discretization of the recently proposed electric-magnetic field integral equation (EMFIE) [Ubeda and Rius, 2010], which enforces the tangential magnetic and the normal electric boundary conditions so that the property Jns(0) = 0 arises automatically, like in the loop-star discretization of the EFIE. Both schemes rely on the expansion of the discretized subspace of solenoidal current in terms of the vertex-oriented loop basis functions [Wu et al., 1995], whereby the abovementioned MOM implementations are only valid for simply connected closed surfaces [Vecchi, 1999; Lee et al., 2003].

3. Two-Step Implementation of the Magnetic Field Integral Equation Free From the Nonsolenoidal Static Current

[15] The magnetic field integral equation (MFIE) is based on the imposition of the tangential magnetic field boundary condition on the closed surface S enclosing the perfectly conducting body under analysis.

equation image

where Hi and Hs denote the incident and scattered magnetic fields and equation image and J represent, respectively, the normal unit vector and the current distribution.

[16] The kernel in equation image × HS relies on the gradient of the free-space Green's function, ∇G, as

equation image

where CPV stands for the Cauchy principal value of the surface integral. The implementation in method of moments of the MFIE results in the following matrix system

equation image

where N denotes the number edges arising from the discretization. ZmnH and Hmi represent, respectively, the impedance elements and the tested incident magnetic field or excitation vector, which are defined as

equation image

where Dm stands for the local region in S spanned by the adopted zero-order divergence-conforming set {bm}. As shown in (2), the current coefficients Jn determine the discrete expansion of the current equation image in terms of the loop-star basis functions.

[17] The expression for the magnetic incident field as propagating plane wave over an arbitrary direction equation image becomes

equation image

where H0 denotes the magnetic vector amplitude. We carry out a decomposition of the impinging magnetic field as

equation image

In accordance with the Taylor's expansion around k = 0 of Hi, the terms in the right-hand side of (18) become

equation image
equation image

Hi(0) thus represents the zeroth-order, static, component of the incident magnetic field and Hidyn, dynamic, gathers all the remaining components in the Taylor's expansion.

[18] Therefore, in view of (5) and (18), the matrix equation for the MFIE can be expressed equivalently as

equation image

where Hmi,(0), Hmi,dyn denote the excitation vectors derived from testing the incident magnetic field components (19), (20) and the matrices [ZmnH,(0)], [ZmnH,dyn] stand for the static and dynamic contributions in the MOM-MFIE impedance matrix. This involves adopting the static and dynamic components in the (k = 0)-centered Taylor's expansion of ∇G

equation image
equation image

with R = ∣equation image∣.

[19] The matrix system in (15) can be applied exclusively to the static components so that

equation image

This matrix system is only solved for the solenoidal component of Js(0), expanded with the loop set, and Jns(0) is set to zero (see equations (20) and (24) of Zhang et al. [2003]). The solution of (24) allows to establish another matrix system from (21) as

equation image

where now the dynamic current becomes the unknown. The dynamic contribution to the current is expanded with the loop-star set since both solenoidal and nonsolenoidal contributions need to be considered. Note that the loop-star MFIE impedance elements ZmnH in (25) result from the summation of their two previously computed static and dynamic contributions, respectively, ZmnH,(0) and ZmnH,dyn,

equation image

We then obtain a two-step MOM procedure that reduces the computational effort in the analysis of the MFIE presented in the paper by Zhang et al. [2003]. First, in (24) we solve the matrix equation for the static component of the current. The nonstatic matrix equations presented by Zhang et al. [2003] are included altogether in the dynamic system equation in (25). Therefore, now only two MOM matrices, static and dynamic, need to be generated. Also, as shown in (20) and (23), the dynamic contributions in the excitation vector or in ∇G can be computed in closed form.

[20] This procedure results in a computed RCS free from the observed inaccuracy. However, two systems, (24) and (25), still need to be handled and two matrices, ZmnH,(0) and ZmnH,dyn, need to be generated. In the next section, we present a new integral equation of second kind that provides in one step a computed RCS free from the inaccuracy described in this paper.

4. Loop-Star Discretization of the Electric-Magnetic Field Integral Equation

[21] We present a new integral equation of second kind, like the MFIE, that provides a null static nonsolenoidal component of the current, like in the EFIE. The MFIE provides a solution for the current but in the formulation there is no explicit hint to the charge density. In view of (8), the observed error depends exclusively on the right cancellation of a static far-field quantity relying on the divergence of the static current. It makes then sense to try to insert in the integral equations a charge condition that provides explicitly an expansion for the surface divergence too.

[22] The electric-magnetic field integral equation (EMFIE) is defined by the imposition of the tangential magnetic field boundary condition in (13) together with the normal electric field boundary condition, which is

equation image

where Ei and Es denote the incident and scattered electric fields and σ stands for the electric charge density. Just like equation image × HS, equation image · ES depends on ∇G so that

equation image

Note how both integral equations in (14) and (28) stand for second-kind integral equations because of the nonzero limiting value of the integrals of the strongly singular terms of the Kernel.

[23] It is convenient to decompose the electric current into its solenoidal, loop, and nonsolenoidal, star, components. The electric field equation in (28) is mainly dependent at very low frequencies on the nonsolenoidal contribution of the current because the solenoidal component has by definition zero divergence. For objects with electrically very small dimensions, the nonsolenoidal component of the current tends to vanish as Jns = (−jk)Jns, which lets this field condition stable at very low frequencies. For the same reason, the magnetic field equation in (14) becomes mainly dependent at very low frequency regime on the solenoidal component of the current. The expression of the EMFIE in terms of the solenoidal current Js and the k-normalized nonsolenoidal current Jns becomes

equation image
equation image

[24] In the implementation in method of moments of the EMFIE it makes then sense (1) to test the magnetic field equation (29) with the solenoidal loop basis functions, because they represent the hegemonic current component, solenoidal, in the magnetic field computation, at very low frequencies; and (2) to test the electric field equation (30) with normally oriented piecewise constant vector pulses because they expand the hegemonic charge density component, nonsolenoidal, in the electric field computation, in the very low frequency regime. Similarly, as shown in (2), the expansion of the solenoidal and the k-normalized nonsolenoidal components of the current needs to be carried with the loop {Ln}and star {Sn} basis functions. The corresponding set of coefficients of current now become, respectively, Jns and Jnns.

[25] The MOM discretization of the EMFIE with loop-star basis functions results in the following matrix system

equation image
equation image

where ZmnsH, ZmnsE and ZmnnsH, ZmnnsE denote, respectively, the solenoidal and nonsolenoidal contributions in the magnetic field or electric field impedance elements. The electric field and magnetic field excitation vectors are Emi and Hmi. The magnetic quantities are defined as

equation image

and the electric matrices and vectors become

equation image

where Πm represents a constant pulse over the testing facet and Dn, Dm denote the local subdomains embraced by the expanding or testing function. Note that the EMFIE requires the explicit definition of the nonsolenoidal component of the current as vanishing for very small frequencies, which automatically lets the RCS computation free from the source of error described by Zhang et al. [2003]. There exist other integral equation implementations [Taskinen and Ylä-Oijala, 2006] that consider the surface normal electric field boundary condition but they assign different unknowns to the current and to the charge density and do not provide for the solenoidal-nonsolenoidal decomposition of the current.

5. Results

[26] In Figures 8, 9 and 10, we present the normalized backscattered far-field contributions due to the real part of the nonsolenoidal component of the computed current for triangulations of, respectively, the tetrahedron with uniform discretization, the square pyramid and the cube. We compute these quantities with the loop-star discretization of the EFIE, the EMFIE and the free MFIE; that is, the loop-star discretization of the MFIE free from the static nonsolenoidal current contribution in the computed RCS (see section 3). Now, with the adopted MOM implementations, the far-field quantities scale perfectly, which is in agreement with the expected physical far-field performance in the Rayleigh frequency region.

Figure 8.

Contribution in the normalized backscattered far-field from the nonsolenoidal component of the current computed with the loop-star discretization of the EFIE, the MFIE free from the static nonsolenoidal component, and the EMFIE for a regular tetrahedron with side 1 m meshed with 100 triangles, a few of which are nonuniform (with 10% deviation from uniformity), under an impinging x-polarized z-directed plane wave.

Figure 9.

Contribution in the normalized backscattered far-field from the nonsolenoidal component of the current computed with the loop-star discretization of the EFIE, the MFIE free from the static nonsolenoidal component, and the EMFIE for a pyramid with square basis and side 1 m meshed with 200 triangles, under an impinging x-polarized z-directed plane wave.

Figure 10.

Contribution in the normalized backscattered far-field from the nonsolenoidal component of the current computed with the loop-star discretization of the EFIE, the MFIE free from the static nonsolenoidal component, and the EMFIE for a cube with side 1 m meshed with 192 triangles, under an impinging x-polarized z-directed plane wave.

[27] In Figures 11, 12 and 13, we show RCS cuts for the abovementioned loop-star discretizations of the EFIE, EMFIE and free MFIE for, respectively, a tetrahedron, a cube and a square pyramid. They all are analyzed in the Rayleigh region, with wavelengths of, respectively, 106 m, 107 m and 108 m. The tetrahedron is discretized with a nonuniform meshing for the proposed EMFIE and free MFIE. In Figure 11, the conventional MFIE requires an equilateral triangulation for the sake of accurate performance in the very low frequency regime (see Figure 3). We adopt an exciting plane wave x-polarized with +z-propagation. In view of these figures, we have two second-kind integral equations, free MFIE and EMFIE that work all over the frequency range. Moreover, we see that both approaches produce very similar accuracy in the RCS computation of these sharp-edged objects.

Figure 11.

RCS over the yz plane computed with the loop-star discretization of the EFIE, the conventional MFIE, the MFIE free from the nonsolenoidal static component, and the EMFIE for a regular tetrahedron with side 0.25 m meshed with 196 triangles under an impinging x-polarized z-directed plane wave and λ = 106 m. The triangles adopted are all equilateral for the conventional MFIE and a few slightly nonuniform for the MFIE free of the static nonsolenoidal component and the EMFIE. Ψ denotes the observation angle over the yz plane, Ψ = 0 being the backscattering direction.

Figure 12.

RCS over the yz plane computed with the loop-star discretization of the EFIE, the conventional MFIE, the MFIE free from the nonsolenoidal static component, and the EMFIE for a cube with side 1 m meshed with 300 triangles under an impinging x-polarized z-directed plane wave and λ = 107 m. Ψ denotes the observation angle over the yz plane, Ψ = 0 being the backscattering direction.

Figure 13.

RCS over the xz plane computed with the loop-star discretization of the EFIE, the conventional MFIE, the MFIE free from the nonsolenoidal static component, and the EMFIE for a pyramid with square basis and side 1 m meshed with 392 triangles under an impinging x-polarized z-directed plane wave and λ = 108 m. Ψ denotes the observation angle over the xz plane, Ψ = 0 being the backscattering direction.

[28] It is interesting to remark that some deviation of the computed RCS with the free MFIE and EMFIE divergence-conforming schemes with respect to the EFIE is still observed in Figure 11, Figure 12, and Figure 13. This discrepancy becomes inherent in the computed RCS with the conventional RWG discretization of the MFIE and is especially evident for electrically moderately small sharp-edged objects [Rius et al., 2001; Ergül and Gürel, 2004]. See, for example, that the deviation appears more clearly for the tetrahedron (see Figure 11) than for the cube (see Figure 12), whereby it appears to be associated with the degree of sharpness of the object. This discrepancy is independent from the inaccuracy described in this paper and leads to a slower accuracy rate of convergence against the number of unknowns with respect to the RWG-discretized EFIE. Peterson [2008] shows a rigorous description of the convergence rates of the RCS computed with different implementations of the MFIE for smooth objects. In view of Figures 11, 12 and 13, the deviation exists at very low frequency regime too but it can only be noticed after removing the huge inaccuracy addressed by Zhang et al. [2003]. In recent years several authors have mitigated such deviation with the adoption of other basis functions sets in the discretization of the MFIE [Ubeda and Rius, 2005, 2006; Ergül and Gürel, 2006]. Similarly, an improved implementation in method of moments for the EMFIE with a facet-oriented discretization has been released recently [Ubeda and Rius, 2011; Ubeda et al., 2011].

[29] In Figure 14, we show the total computational time versus the number of unknowns required by the EMFIE and free MFIE implementations compared with the loop-star discretization of the EFIE, a successful MOM scheme in the very low frequency regime. The overall computational time in Figure 14 is mainly dependent on the time required to generate the impedance matrix. The matrix systems are solved by direct inversion methods because the electrically small objects under analysis require a moderate number of unknowns that can be handled with the available computational resources. In consequence, the extra time required in solving the two matrix systems of the free MFIE (see equations (24) and (25)) becomes still irrelevant for the degrees of meshing in Figure 14. In view of Figure 14, the free MFIE and the loop-star discretization of the EMFIE involve, as expected, more computational effort than the loop-star implementation of the EFIE. The reasons for this are the following: (1) in the two-step implementation of the MFIE two static and dynamic matrices need to be generated; (2) the loop-star discretization of the EMFIE relies internally on the construction of two discrete integral electric field and magnetic field implementations.

Figure 14.

Computational time required to solve the loop-star implementations of the EFIE, the MFIE free from the nonsolenoidal static component and the EMFIE for a cube with side 1 m meshed with 192, 300, 432, 588, 768, 972 and 1200 triangular facets and λ = 107 m.

[30] The loop-star discretization of the EFIE solves the low-frequency breakdown for the EFIE leading to poorly conditioned impedance matrices but bearable within the machine precision. The loop-star EFIE matrix system can then be solved by direct inversion methods but when the matrix equation is solved iteratively the iteration count is usually large [Zhao and Chew, 2000]. Since the free MFIE and the loop-star implementation of the EMFIE represent discretizations of second-kind integral equations, they become much better conditioned than the loop-star EFIE for any degree of meshing (see Figure 2 of Ubeda and Rius [2010]). Therefore, in an iterative search of the solution the free MFIE and EMFIE become especially advantageous in front of the loop-star EFIE.

6. Conclusions

[31] We present two MOM approaches relying on second-kind integral equations that are free of the misbehavior associated to the incapacity of the conventional MFIE implementations at very low frequency of expanding a nonradiating static nonsolenoidal component of the current. The first implementation stands for a modification of the approach regarding the loop-star discretization of the MFIE presented by Zhang et al. [2003] to compute the static and dynamic components of the current separately in order to discard the static nonsolenoidal contribution in the RCS computation (free MFIE). Alternatively, we present another new second-kind integral equation, the electric-magnetic field integral equation (EMFIE), which we discretize with the loop-star basis functions. The EMFIE establishes at the same time the tangential magnetic and the normal electric field boundary conditions at the surface of the scatterer and provides inherently a null static nonsolenoidal contribution of the current, like the loop-star discretization of the EFIE. We show that the loop-star discretization of both the EMFIE and the free MFIE produce very similar RCS results. These free MFIE and EMFIE implementations are very robust to the type of meshing adopted, which otherwise, with a conventional divergence-conforming discretization of the MFIE leads in general to huge inaccuracies in the computed RCS in the Rayleigh frequency region.

Acknowledgments

[32] This work was supported by the Spanish Interministerial Commission on Science and Technology (CICYT) under projects TEC2010-20841-C04-02, TEC2009-13897-C03-01, and CONSOLIDER CSD2008-00068.

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