Facet-Oriented Discretization of the Electric-Magnetic Field Integral Equation for the accurate scattering analysis of perfectly conducting sharp-edged objects Eduard Ubeda *, José M. Tamayo and Juan M. Rius Signal Theory and Communications Department Universitat Politècnica de Catalunya (UPC) Barcelona, Spain
[email protected],
[email protected],
[email protected] arising at the transition from triangular and quadrangular regions [5].
Abstract— In this paper, we present the discretization in Method of Moments of the Electric-Magnetic Field Equation with the divergence-Taylor-Orthogonal basis functions, a facetoriented set of basis functions. We show for a sharp-edged object that the computed RCS with this discretization offers better accuracy than the Loop-Star discretization.
The MoM-discretizations of the MFIE with RWG basis functions show some deviation in the computed RCS when compared with the same discretization of the Electric-Field Integral Equation (EFIE) [6]. The monopolar-RWG basis functions [7] mitigate to some extent such error. Just like the div-TO basis functions, the monopolar-RWG also represent a triangle-oriented set, derived from breaking the normalcontinuity constraint in the conventional RWG basis functions. Both the div-TO and the monopolar-RWG basis functions expand the same discretized current space in triangular discretizations [3].
Keywords - Integral Equations; Method of Moments; Basis functions; Second-Kind Integral Equations
I.
INTRODUCTION
In this paper, we show that the divergence-TaylorOrthogonal basis functions (div-TO), a facet-oriented set of basis functions, is well suited for the accurate discretization in Method of Moments (MoM) of the Electric-Magnetic Field Integral Equation (EMFIE) [1]. We name these basis functions divergence-Taylor-Orthogonal because: (i) they are orthogonal and therefore lead to a diagonal Gram-matrix, which is advantageous in the MoM-discretization of Magnetic-Field Integral Equation (MFIE) [2][3]; (ii) they are derived from the uniform terms and from the linear, divergence-conforming, contributions in the 2D Taylor’s expansion of the current at a reference surface point. We place these reference points inside each facet of the discretization. The adoption as reference points of the midpoints of the facets –barycenters in triangulations– ensures the property of orthogonality. Therefore, the div-TO basis functions are both point- and facet-oriented. The definition of the div-TO basis functions does not depend on the type of facet adopted [3]. In particular, their application is interesting in mixed discretizations, resulting from combining triangular and quadrangular meshings. This is not the case for other basis functions sets, such as the zeroth order divergenceconforming RWG or rooftop [4], which become only applicable, respectively, to triangular or rectangular facets, and rely on rather different formal definitions. Since these basis functions are edge-oriented, they even require for mixed meshings other basis functions across the edges
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The MFIE and the EMFIE stand for examples of second kind Integral Equations. We have observed that the LoopStar discretization [8] of the EMFIE produces some inaccuracy in the computed RCS in similar terms as the RWG-discretization of the MFIE. The Loop basis functions, a vertex-oriented set, and the Star basis functions, a facet oriented set, represent a rearrangement of the RWG-space. This rearrangement is required for the discretization of the EMFIE [1]. In this paper, we show how the div-TO discretization of the EMFIE improves the accuracy of the computed RCS. The div-TO discretization of the EMFIE necessarily requires the computation of the electric radiation due to the div-TO functions, which by definition leak out of the facet. The MoM-discretization of the EFIE has been traditionally carried out with divergence-conforming basis functions [4][6], which do not allow current leakage outwards from the function subdomain. In this paper, we show the definition for the scalar potential in the discretized space of the div-TO basis functions. II.
DIVERGENCE-TAYLOR BASIS FUNCTIONS
G
We approximate the current in the vicinity of a point r0 of the surface of the body under analysis in terms of the firstorder Taylor’s expansion, as
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AP-S/URSI 2011
G J r0 | x, y | > J x @ 0,0 xˆ ª¬ J y º¼
0,0
ª wJ º yˆ « x » xxˆ ¬ wx ¼ 0,0
ª wJ º ª wJ º ª wJ º « y» xyˆ « x » yxˆ « y » yyˆ w w x y ¬ ¼ 0,0 ¬ ¼ 0,0 ¬ wy ¼ 0,0
vˆn
G rcn
where (x, y) stand for the local cartesian planar coordinates
> @ 0,0 , ª¬ J y º¼ 0,0 denote
z
around the point. The quantities J x
G
the current components at r0 . In view of (1), the divergence
y
G and the normal component of the curl of the current at r0 become
x
G J r0 | x, y
ª wJ y º ª wJ x º « » « wx » ¬ ¼ 0,0 ¬ wy ¼ 0,0
> nˆ u J @ 0,0
ª wJ y º ª wJ º « x » « » w x ¬ ¼ 0,0 ¬ wy ¼ 0,0
Figure 1. Definition of the div-TO basis function for a triangular facet
The introduction of these quantities in (1) results in G J r0 | x, y | > J x @ 0,0 xˆ ª¬ J y º¼ yˆ 0,0 1 1 ˆ ˆ ˆ > J @ 0,0 UU > n u J @ 0,0 UI 2 2
III.
>
contributions depending on J
nˆ u H i H s
`
s
J
V K 0H 0
J jk
where the superscripts -i and -s denote the incident and scattered fields and nˆ , J , V stand for the unit normal vector, the electric current and the charge density. The wavenumber and the free-space impedance are, respectively, k and K0 . The Galerkin-discretization of the equation (6) with the zeroth-order basis functions in (5) results in the following matrix elements
The approximation in (4) shows four degrees of freedom that we can capture by means of four basis functions: two zeroth order basis functions, x- or y-oriented [2], and two first-order basis functions ȡ- orҏ I-oriented. Since for the EMFIE we require a correct expansion of the charge density, we define the set of divergence-Taylor basis functions as
^
i
K0
are relevant in the expansion of the Electric or the Magnetic scattered fields, respectively. The remaining linear current contributions in (1) are ignored in (4) because they provide null divergence and normal curl.
^on `
nˆ E E
@ 0,0 or > nˆ u J @ 0,0
G G uˆn / An , vˆn / An , r rcn /(2 An )
DIVERGENCE-TAYLOR-ORTHOGONAL DISCRETIZATION OF THE EMFIE
The Electric-Magnetic Field Integral Equation (EMFIE) is based on imposing the following magnetic- and electricboundary conditions over the surface S of the conductor
where (ȡ, I ) stand for the local polar coordinates. The linear
uˆn
H Z mn
§ · 1 om on ds ³³ om ¨ nˆ u ³³ G u on ds ' ¸ds ³³ ¨ F ,CPV ¸ 2 Fm Fm © ¹ n
where G stands for the free-space Green’s function and Fm, Fn represent, respectively, the field- and source-facets. Similarly, the testing of (7) with piecewise constant pulses on each facet leads to
where uˆn , vˆn denote two perpendicular unitary vectors, An
E Z mn
Gn
represents the nth-facet area and rc stands for the reference point of the nth-facet of the discretization (see Fig. 1). The choice of the reference point is free as long as it lies inside the facet subdomain. In this paper, we adopt as reference points the mid-points of the discretization because they lead to a diagonal Gram-matrix, which makes the MoMdiscretization of the MFIE easier [3]. Accordingly, we name these basis functions div-Taylor-orthogonal (div-TO).
1 1 3m on ds 2 jk ³³ K0 Fm
³³ 3 Fm
§ · jk ³³ 3 m nˆ ¨ ³³ Gon ds ' ¸ds ¨F ¸ Fm © n ¹
m
nˆ ) >nCPV @, D on ds
Therefore the div-TO discretization in Method of Moments of the EMFIE leads to a matrix with 3 times the number of facets. Indeed, the expansion of the current is carried out with three unknowns per facet. Moreover, we test the tangential component of the magnetic field in (8) with two functions and the normal component of the electric field in (9) with one function. The first impedance elements in the right-hand side of equations (8) and (9) lead to diagonal matrices. We call ) D
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the discrete scalar potential because it is defined consistently with the Gauge-Lorentz condition in the discretized space. The Gauge-Lorentz condition relates, prior to discretization, the vector and scalar potentials, A and ) , as
)
ª º K 1 A = 0 « ³³ GJds' » jk jk P0H 0 ¬S ¼
CONCLUSIONS The accuracy of the computed RCS with the discretization of the EMFIE with the divergence-TaylorOrthogonal basis functions, a facet-oriented set of basis functions, is better than with the Loop-Star basis functions, which represent a rearrangement of the RWG basis functions, an edge-oriented set.
However, after discretization, the scalar potential term )>nCPV @, D in (9) results in
) nD ,>CPV @ =
K0 ª
º « v³ G on nˆc dl ' ³³ G ' on ds' » jk ¬« wFn Fn ¼»
ACKNOWLEDGMENT This work was supported by the Spanish Interministerial Commission on Science and Technology (CICYT) under Projects TEC2010-20841-C04-02, TEC2007-66698-C0401/TCM and CONSOLIDER CSD2008-00068.
where wFn and nˆc denote, respectively, the closed contour around the source facet Fn and the unit-normal vector to this contour. Note that the line-integral in (11) disappears if divergence-conforming basis functions are adopted because these basis functions ensure normal-continuity of the current across the edge. IV.
REFERENCES [1]
[2]
RESULTS
In Fig.2, we show the yz-cut of the bistatic RCS for a tetrahedron with side 0.2m and Ȝ=1m discretized with 144 triangles under an impinging z-directed x-polarized planewave. We see clearly how the div-TO discretization of the EMFIE, EMFIE[div-TO], outperforms the Loop-Star discretization of the EMFIE, EMFIE[Loop;Star], and approaches the RWG-discretization of the EFIE.
[3]
[4]
[5] Bistatic RCS, yz-plane -22
[6]
-23 z
-24
y
-25 dBsm
[7]
x
-26
[8]
-27 -28 EFIE[RWG] EMFIE[Loop;Star] EMFIE[div-TO]
-29 -30
0
20
40
60
80
100
120
140
160
180
\
Figure 2. Computed RCS for a tetrahedron with side=0.2m meshed with 144 triangles under a z-directed x-polarized planewave with the MoMdiscretizations EMFIE[div-TO] , EMFIE[Loop-Star] and EFIE[RWG]
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