taylor-orthogonal basis functions for the … · 86 ubeda, tamayo, and rius 1. introduction the...

Progress In Electromagnetics Research, Vol. 119, 85–105, 2011

TAYLOR-ORTHOGONAL BASIS FUNCTIONS FOR THEDISCRETIZATION IN METHOD OF MOMENTS OF SEC-OND KIND INTEGRAL EQUATIONS IN THE SCATTER-ING ANALYSIS OF PERFECTLY CONDUCTING OR DI-ELECTRIC OBJECTS

E. Ubeda, J. M. Tamayo, and J. M. Rius*

AntennaLab, Department of Signal Theory and Communications(TSC), Universitat Politecnica de Catalunya (UPC), Campus NordUPC, D-3, C/Jordi Girona, 1-3, Barcelona 08034, Spain

Abstract—We present new implementations in Method of Momentsof two types of second kind integral equations: (i) the recently proposedElectric-Magnetic Field Integral Equation (EMFIE), for perfectlyconducting objects, and (ii) the Muller formulation, for homogeneousor piecewise homogeneous dielectric objects. We adopt the Taylor-orthogonal basis functions, a recently presented set of facet-orientedbasis functions, which, as we show in this paper, arise from the Taylor’sexpansion of the current at the centroid of the discretization triangles.We show that the Taylor-orthogonal discretization of the EMFIEmitigates the discrepancy in the computed Radar Cross Sectionobserved in conventional divergence-conforming implementationsfor moderately small, perfectly conducting, sharp-edged objects.Furthermore, we show that the Taylor-discretization of the Muller-formulation represents a valid option for the analysis of sharp-edged homogenous dielectrics, especially with low dielectric contrasts,when compared with other RWG-discretized implementations fordielectrics. Since the divergence-Taylor Orthogonal basis functionsare facet-oriented, they appear better suited than other, edge-oriented,discretization schemes for the analysis of piecewise homogenous objectssince they simplify notably the discretization at the junctions arisingfrom the intersection of several dielectric regions.

Received 17 May 2011, Accepted 22 June 2011, Scheduled 21 July 2011* Corresponding author: Juan Manuel Rius ([email protected]).

86 Ubeda, Tamayo, and Rius

1. INTRODUCTION

The discretization in Method of Moments (MoM) with RWG basisfunctions, a zeroth-order example of divergence-conforming set, of theMagnetic-Field Integral Equation (MFIE) [1, 2], a second kind surfaceintegral equation, for perfectly conducting (PeC) objects shows somediscrepancy [3] in the computed Radar Cross Section (RCS) withrespect to the EFIE [4], a first kind surface integral equation [5].This discrepancy is especially evident in the analysis of electricallymoderately small perfectly conducting sharp-edged objects. Over thelast years, other sets of basis functions have been proposed for theMoM-discretization of the MFIE in the scattering analysis of PeC-objects that reduce the observed discrepancy [6–10].

The Muller formulation [11–13] stands for a second kind IntegralEquation in the scattering analysis of homogeneous or piecewisehomogeneous dielectric objects. The MoM-implementation of theMuller-formulation is normally carried out with the RWG basisfunctions [13, 14]. Since the Muller formulation combines magnetic-field and electric-field contributions from the PeC-case in such a waythat the gram-matrix contributions are enhanced, one may expectthat this implementation of the Muller-formulation may reproduce,to some extent, the observed performance for the RWG-discretizationof the MFIE in conductors. In our experience, as we show inthis paper, the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT)formulation [15–17], a first kind Integral Equation for dielectric objects,provides a stable trend for convergence for electrically coarser degreesof meshing than the Muller-formulation. Moreover, we observe thatmoderately small sharp-edged objects with low dielectric contrast showin general much smaller RCS deviation with the RWG-implementationof the Muller-formulation than in the PeC-case. However, forhigh dielectric contrasts, the observed degree of deviation increases,although it depends greatly on the shape of the object under analysis.In view of our tests, the RCS deviation becomes especially significantfor cubes, as it has been also reported by others [13, 14].

In this paper, we use the recently presented facet-orientedorthogonal basis functions [18], well-suited for the discretization ofSecond-kind Integral Equations. We now decide to rename the firstorder orthogonal basis functions presented in [18] as divergence-Taylor-Orthogonal basis functions (div-TO) because, as we showin this paper, they are derived from the uniform terms and fromthe linear, divergence-conforming, contributions in the 2D Taylor’sexpansion of the current at a reference surface point inside afacet arising from the discretization. We present two new MoM-

Progress In Electromagnetics Research, Vol. 119, 2011 87

implementations with the div-TO basis functions of two secondkind Integral Equations: (i) The Electric-Magnetic Field IntegralEquation (EMFIE), recently presented in [19] under a Loop-Stardiscretization [20, 21], for conductors and (ii) the Muller formulation,for homogeneous or piecewise homogeneous dielectric objects. Thedevelopment of the div-TO discretization of the EMFIE and theMuller formulation, unlike the same implementation for the MFIEin [18], requires the discretization of the scalar potentials. The sourcecontributions of the scalar potentials are usually discretized in thecontext of the EFIE or PMCHWT formulations with divergence-conforming basis functions [4, 22], which preserve normal continuityof the current across the edges. In this paper, though, we providea more general definition for the discrete electric and magnetic scalarpotentials because the normal component of the current expanded withthe div-TO basis functions is not confined in the facets arising fromthe discretization.

2. TAYLOR-ORTHOGONAL BASIS FUNCTIONS

Interestingly, the imposition of the tangential continuity of the currentacross edges appears better-suited than the imposition of the normalcontinuity for the MoM-discretization of the MFIE for sharp-edgedobjects [6]. The normal-continuity constraint in RWG is applied in thediscretizations of the EFIE but it is not clear, in our opinion, that itmust be imposed for second kind Integral Equations like the MFIE orthe EMFIE.

The current may become singular exactly on the tips of sharpedges [23] but is bounded and continuous at any surface point ~r0 outof sharp edges or corners. In general, we can approximate the currentin the vicinity of the reference point ~r0, J(~r0|u, v), in terms of thefirst-order 2D Taylor’s expansion, as

J(~r0|u, v) ≈ J(0,0) +[∂J∂u

]

(0,0)

u +[∂J∂v

]

(0,0)

v

= [Ju](0,0)u + [Jv](0,0)v +[∂Ju

∂u

]

(0,0)

uu

+[∂Jv

∂u

]

(0,0)

uv +[∂Ju

∂v

]

(0,0)

vu +[∂Jv

∂v

]

(0,0)

vv (1)

where (u, v) stand for the local cartesian planar coordinates around~r0, and J(0,0), [∂J

∂u ](0,0), [∂J∂v ](0,0) denote the current and the partial

derivatives of the current at ~r0. The definition of the following


quantities A, B, C, D

A =[∂Ju

∂u

]

(0,0)

+[∂Jv

∂v

]

(0,0)

B =[∂Ju

∂u

]

(0,0)

−[∂Jv

∂v

]

(0,0)

(2)

C =[∂Jv

∂u

]

(0,0)

+[∂Ju

∂v

]

(0,0)

D =[∂Jv

∂u

]

(0,0)

−[∂Ju

∂v

]

(0,0)

(3)

allows the expression of (1) equivalently as

J (~r0|u, v) ≈ [Ju](0,0)u + [Jv](0,0)v +12(A + B)uu

+12(C + D)uv +

12(C −D)vu +

12(A−B)vv (4)

and

J (~r0|u, v) ≈ [Ju](0,0)u + [Jv](0,0)v +A

2(uu + vv)

+B

2(uu− vv) +

C

2(uv + vu) +

D

2(uv − vu) (5)

where the local polar coordinates (ρ, φ) and the local cartesiancoordinates (u, v) are related as (see Figure 1)

ρρ = uu + vv ρφ = uv − vu (6)

x

y

z cr

nc

n

v

u φρρ

Figure 1. Local polar and cartesian coordinates for the definition ofthe Taylor-Orthogonal basis functions.


In view of (2) and (3), the definitions for A and D become

A = [∇ · J](0,0) D = [n · ∇ × J](0,0) (7)

The coefficients A, D are important in the linear approximation ofJ at ~r0 because [∇ · J](0,0) and [n · ∇ × J](0,0) are relevant in theexpansion of the normal components of the Electric and Magneticscattered fields, respectively. However, the remaining terms in (5),associated to the coefficients C, B, provide null divergence and normalcurl, whereby they may be ignored. The expression (5) can be thenrewritten accordingly as

J (~r0|u, v) ≈ [Ju](0,0)u+[Jv](0,0)v+12[∇·J](0,0)ρρ+

12[n·∇×J](0,0)ρφ (8)

which represents a local linear approximation of the current in thevicinity of the reference point ~r0 with four degrees of freedom: [Ju](0,0),[Jv](0,0), [∇ · J](0,0) and [n · ∇ × J](0,0). Four basis functions are hencerequired to capture each of the unknowns. We establish as domains forthese basis functions each of the facets arising from the discretization.

The zeroth order Taylor-orthogonal basis functions [25] are

b0,u =1√Ar

u b0,v =1√Ar

v (9)

where Ar represents the area of the facet. This definition allows∫

F

b0,u · b0,vds = 0 (10)

∫

F

b0,u · b0,uds =∫

F

b0,v · b0,vds = 1 (11)

where F denotes the facet. We define the remaining first-order basisfunctions as

b1,ρ =1

2Ar(~r − ~rc) b1,n×ρ =

12Ar

n× (~r − ~rc) (12)

where ~rc stands for the geometrical center of the facet, the centroid ina triangle (see Figure 1). From this definition it is accomplished

∇ · b1,ρ = n · (∇× b1,n×ρ) = 1/Ar (13)∫

F

b0,u · b1,ρds =∫

F

b0,v · b1,ρds =∫

F

b1,ρ · b1,n×ρds = 0 (14)

The properties (10) and (14) show that these basis functionsare orthogonal, whereby we name them Taylor-orthogonal basis


functions. We define two sets of Taylor-orthogonal basis functions:the divergence-Taylor-orthogonal basis functions (div-TO) and thecurl-Taylor-orthogonal basis functions (curl-TO). Both sets of basisfunctions group three Taylor-orthogonal basis functions per facet.Whereas the div-TO set retains b0,u, b0,v and b1,ρ, the curl-TO setconsiders b0,u, b0,v and b1,n×ρ.

The monopolar RWG and nxRWG sets [10, 24] stand for facet-oriented sets of basis functions too. They arise from breaking,respectively, the normal or tangential continuity constraints enforcedby the RWG or nxRWG sets across the edges. The div-TO and thecurl-TO sets expand the same spaces of current as, respectively, themonopolar RWG and the monopolar nxRWG sets [18]. Therefore, themonopolar sets and the div-TO or curl-TO basis functions require thesame number of unknowns. Indeed, twice the number of edges is equal,for closed objects, to three times the number of facets. This representstwice the number of unknowns of the RWG basis functions.

3. TAYLOR-ORTHOGONAL DISCRETIZATION OF THEELECTRIC-MAGNETIC FIELD INTEGRAL EQUATION

The Electric-Magnetic Field Integral Equation (EMFIE) is based onimposing at the same time the tangential magnetic and the normalelectric field boundary conditions over the surface S embracing thescatterer; that is,

n×Hi =J2−

(1µ0

)n× [∇×A]CPV (15)

n ·Ei

η0=

∇ · J2(−jk)

+(

1η0

)n · [∇Φ]CPV +

(jk

µ0

)n ·A (16)

which are second-kind integral equations because the sourcemagnitudes, current and divergence of the current, respectively, J and∇ · J, come out from the source-integrals in the potentials. Hi, Ei

stand for the incident magnetic and electric fields and the quantities k,n, ε0, µ0, η0 represent, respectively, the wavenumber, the unit vectornormal to the surface, the free-space dielectric permittivity, the free-space magnetic permeability and the free-space impedance. The MFIEis obtained by imposing only the tangential continuity of the magneticfield in (15).

The limiting values of the singular Kernel contributions in thesource integrals depend on the current for the magnetic field in (15)and on the divergence of the current for the normal component of thescattered electric field in (16). The definition for the magnetic vector


potential A is

A = µ0

∫∫

S

GJds′ (17)

The definition of the potential related quantities in (15) and (16)capturing the Cauchy Principal value of the surface integrals are

[∇×A]CPV = µ0

∫∫

S,CPV

∇G× Jds′ (18)

[∇Φ]CPV =η0

(−jk)

∫∫

S,CPV

∇G∇′ · Jds′ (19)

where G = e−jk|~r−~r′|4π|~r−~r′| stands for the free-space Green’s function.

In view of (15) and (16), the source magnitudes appearpredominantly in the range spaces of the integral equations derived.It makes then sense to undertake the MoM-testing of (15) with thebasis functions adopted for the expansion of the current. Similarly,in the MoM-testing of (16) we need to employ the divergence ofthe basis functions expanding the current. It is then clear thata successful MoM-implementation of the EMFIE must provide forthe proper expansion for both the current and the charge density.The solenoidal contribution of the current has zero divergence andcannot therefore expand properly the range space of the normal-electricintegral equation in (16). The nonsolenoidal component of the current,with non-null divergence, is well suited for the expansion of the chargedensity, which is mandatory for the proper accomplishment of (16).Therefore, the MoM-implementation of the EMFIE with the Loop,solenoidal, and the Star, nonsolenoidal, basis functions follows theseguidelines accordingly [19].

The RWG discretization of the MFIE provides some deviation inthe computed RCS of sharp-edged perfectly conducting objects withrespect to the EFIE. The EMFIE, based to some extent on the MFIE,also shows this RCS discrepancy with the Loop-Star discretization,which represents a solenoidal-nonsolenoidal rearrangement of the RWGcurrent space. In the div-TO discretization, the solenoidal subspace ofcurrent must be captured by the divergence-free zeroth-order terms,b0,u, b0,v. The b1,ρ term, in contrast, with non-zero divergence, mustprovide for the expansion of the nonsolenoidal subspace of current.Furthermore, the monopolar RWG discretization of the MFIE reducessignificantly the RCS deviation observed with the RWG discretizationfor moderately small sharp-edged objects [10]. Since the div-TO basisfunctions expand the same space as the monopolar RWG set, it is


reasonable to expect some improvement in the div-TO discretizationof the EMFIE too with respect to the Loop-Star discretization.

We arrange the divergence-Taylor-Orthogonal basis functions inthree subsets B0,u = {b1

0,u . . . bNf

0,u}, B0,v = {b10,v . . . bNf

0,v} and

B1,ρ = {b11,ρ . . . bNf

1,ρ} gathering, respectively, the u-, v- and ρ-contributions in the Nf facets arising from the discretization. Thesesubsets are introduced consecutively in the definition of {on}, thewhole set of divergence-Taylor-Orthogonal basis functions, so that{on} = {B0,u,B0,v,B1,ρ}. The expansion of the electric current withthe div-TO basis functions then yields

J ≈3Nf∑

n=1

Jnon (20)

where Jn denote the current coefficients. The div-TO discretization ofthe EMFIE results in the following matrix system

H im =

3Nf∑

n=1

ZHmnJn m = 1 . . . 2Nf (21)

Eim =

3Nf∑

n=1

ZEmnJn m = (2Nf + 1) . . . 3Nf (22)

The magnetic-field quantities ZHmn and H i

m denote, respectively, theimpedance matrix elements and the tested incident magnetic field orexcitation vector, defined as

H im =

[⟨om, nm ×Hi

⟩]=

∫∫

Fm

(om × nm) ·Hids (23)

ZHmn =

12

∫∫

Fm

om · onds− 1µ0

∫∫

Fm

om · (nm × [∇×A]nCPV) ds (24)

where Fm stands for the m-th facet arising from the discretization andnm denotes the unit vector normal normal to Fm.

The first term in the right-hand side in (24) is related withthe Gram-matrix, which is diagonal because of the property oforthogonality of the div-TO basis functions. The testing in (23)and (24) is carried out with the zeroth-order Taylor-orthogonal basisfunctions [25]. In view of (18), the contribution of the n-th div-TObasis function in the expansion of [∇×A]CPV becomes

[∇×A]nCPV = µ0

∫∫

Fn,CPV

∇G× onds′ (25)


The electric-field quantities ZEmn and Ei

m in (22) denote, respectively,the normal-electric impedance elements and the normally testedincident electric field, which are defined as

Eim =

[⟨Πmnm,

Ei

η0

⟩]=

(1η0

)∫∫

Fm

Πmnm ·Eids (26)

ZEmn =

12(−jk)

∫∫

Fm

Πm∇ · onds +1η0

∫∫

Fm

Πmnm · [∇Φ]nCPVds

+(

jk

µ0

)∫∫

Fm

Πmnm ·Ands (27)

where Πm stands for a constant pulse defined over the testing facet.The definition of this testing-pulse is consistent with the uniformityof the divergence of the term b1,ρ in the div-TO set. In view of (17),the n-th div-TO contribution in the discrete expansion of the magneticvector potential in (27) results in

An = µ0

∫∫

Fn

onGds′ (28)

Furthermore, [∇Φ]nCPV, the other potential magnitude in (27), denotesthe contribution of the n-th div-TO basis function to the CauchyPrincipal value of the gradient of the electric scalar potential.

The use of the potentials in the analysis of PeC-scattering prob-lems with Integral Equations is associated with the accomplishmentof the Gauge-Lorentz condition, which, prior to discretization, estab-lishes the definition of the electric scalar potential Φ in terms of themagnetic vector potential A as

Φ = − 1jk√

µ0ε0∇ ·A =− η0

jk∇ ·

∫∫

S

GJds′

(29)

After discretization, the space of current is determined by theexpansion in terms of the adopted divergence-Taylor-Orthogonal basisn-th div-TO basis function in the expansion of the electric scalarpotential is

Φn = − η0

jk∇ ·

∫∫

Fn

Gonds′

(30)

which becomes equivalently


Φn = − η0

jk

∫∫

Fn

∇G · onds =η0

jk

∫∫

Fn

∇′G · onds′

=η0

jk

∫∫

Fn

∇′ · (Gon)ds′ −∫∫

Fn

G∇′ · (on)ds′

=η0

jk

∮

∂Fn

G (on · nc) dl′ −∫∫

Fn

G∇′ · (on)ds′

(31)

where ∂Fn and nc denote, respectively, the closed contour aroundthe source facet Fn and the unit-normal vector to this contour (seeFigure 1). Note that the line-integral in (31) becomes zero fordivergence-conforming discretizations, like RWG or rooftop, becausethey ensure, by definition, normal continuity of the current across theedges. As a matter of fact, this leads to the widely used expression for[∇Φ]nCPV arising for example in divergence-conforming discretizationsof the EFIE [4, 22]. However, the current expanded by the divergence-Taylor-Orthogonal basis functions leaks out from the facet domain andtherefore the full expression in (31) needs to be considered. Finally,the expression for [∇Φ]nCPV required in (27) must be

[∇Φ]nCPV =1

jωε0

∮

∂Fn

∇G (on · nc) dl′−∫∫

Fn,CPV

∇G∇′ · (on)ds′

(32)

4. TAYLOR-ORTHOGONAL DISCRETIZATION OF THEMULLER FORMULATION

The Muller formulation arises in the scattering analysis of homoge-neous or piecewise homogeneous dielectric objects. In general, the EMscattering from a penetrable object is solved through the definition ofan equivalent problem resulting from the combination of two homo-geneous problems associated to the outer and inner dielectric regions,respectively R1 and R2, in general with different pairs of dielectric per-mittivities and magnetic permeabilities, (ε1, µ1) and (ε2, µ2). Bothregions have accordingly different wavenumbers and impedances, (k1,η1) and (k2, η2). In particular, the Muller formulation results from im-posing the following magnetic-field and electric-field conditions across


the surface S embracing the penetrable object so that

µ1n1 ×Hi =(µ1 + µ2)

2J1 − µ1n1 × [Hs

1]CPV+µ2n2×[Hs2]CPV (33)

ε1n1×Ei = −(ε1+ε2)2

M1−ε1n1 × [Es1]CPV+ε2n2×[Es

2]CPV (34)

which are second-kind integral equations because the sourcemagnitudes, the electric and magnetic currents, come out from thesource radiating integrals of the scattered fields. The terms Ei, Hi

denote the incident electric and magnetic fields. The vectors n1, n2

denote the unit vectors normal to the surface pointing into the outerand inner regions, respectively, and accomplish n2 = −n1. The electricand magnetic currents at both sides of the surface, J1, J2 and M1,M2, are also related so that J2 = −J1 and M2 = −M1. The scatteredmagnetic and electric fields at the outer and inner sides of the surfacestand for Hs

1, Hs2 and Es

1, Es2. The extraction of the limiting value of

the singular Kernel contribution in the source integrals of the scatteredfields results in

ni ×Hsi =

Ji

2+ ni × [Hs

i ]CPV (35)

ni ×Esi = −Mi

2+ ni × [Es

i ]CPV (36)

where i = 1, 2 denotes the dielectric region involved.The contributions to the scattered fields in (33) and (34) in terms

of the vector and scalar potentials in each region become

µini×Hsi =

µiJi

2+ni×[∇×Ai]CPV−jkiηini × Fi−µini ×∇Ψi(37)

εini×Esi =−εiMi

2−ni × [∇×Fi]CPV−

jki

ηini×Ai−εini×∇Φi(38)

The integral expressions for the magnetic vector and the electric scalarpotentials, Ai and Φi, related with the electric source currents, yield

Ai = µi

∫∫

S

GiJids′ Φi =ηi

(−jki)

∫∫

S

Gi∇′ · Jids′ (39)

Similarly, the integral expressions for the electric vector and magneticscalar potentials, related with the magnetic source currents, become

Fi = εi

∫∫

S

GiMids′ Ψi =1

(−jkiηi)

∫∫

S

Gi∇′ ·Mids′ (40)

where Gi = e−jki|~r−~r′|4π|~r−~r′| stands for the Green’s function in the region Ri.


The Galerkin-discretization of the Muller-formulation with thediv-TO basis functions is formally similar to the procedure describedin [13] for the RWG basis functions. Now the expansion of the electricand magnetic currents with the div-TO basis functions is

J1 ≈3Nf∑

n=1

J1non M1 ≈

3Nf∑

n=1

M1non (41)

where J1n and M1

n denote the electric and magnetic current coefficients,respectively, in region R1. The current coefficients at the other side ofthe surface, in region R2 are related so that J2

n = −J1n and M2

n = −M1n.

The div-TO discretization of the Muller-formulation results in thefollowing matrix system

H im =

3Nf∑

n=1

ZHJmn J1

n +3Nf∑

n=1

ZHMmn M1

n m = 1 . . . 3Nf (42)

Eim =

3Nf∑

n=1

ZEJmnJ1

n +3Nf∑

n=1

ZEMmn M1

n m = 1 . . . 3Nf (43)

The quantities H im and Ei

m represent the excitation vectors for,respectively, the incident magnetic and incident electric fields,

H im =

[⟨om, nm

1 ×Hi⟩]

=∫∫

Fm

(om × nm1 ) ·Hids (44)

Eim =

[⟨om, nm

1 ×Ei⟩]

=∫∫

Fm

(om × nm1 ) ·Eids (45)

where Fm stands for the m-th facet arising from the discretizationof the surface and nm

1 denotes the unit vector normal to Fm pointingtowards the region R1. The impedance elements in the submatricesalong the main diagonal of the system in (42) and (43), ZHJ

mn andZEM

mn , are defined as

ZHJmn =

(µ1 + µ2)2

∫∫

Fm

om · onds

−∫∫

Fm

om · [nm1 × ([∇×A1]

nCPV − [∇×A2]

nCPV)] ds (46)


ZEMmn = −(ε1 + ε2)

2

∫∫

Fm

om · onds

+∫∫

Fm

om · [nm1 × ([∇× F1]

nCPV − [∇× F2]

nCPV)] ds (47)

where the first term in the right-hand side is related with the Gram-matrix, which is diagonal because of the orthogonality of the div-TObasis functions. The contribution of the n-th div-TO basis functionin the expansion of the Cauchy principal value of the source integralsin (46) and (47) yields

[∇×A1]nCPV − [∇×A2]nCPV =∫∫

S,CPV

(µ1∇G1 − µ2∇G2)× onds′ (48)

[∇× F1]nCPV − [∇× F2]nCPV =∫∫

S,CPV

(ε1∇G1 − ε2∇G2)× onds′ (49)

The definition for the off-diagonal submatrices in (42) and (43), ZHMmn

and ZEJmn, is

ZHMmn = −

∫∫

Fm

om · [nm1 × (jk1η1Fn

1 − jk2η2Fn2 )] ds

−∫∫

Fm

om · [nm1 × (µ1∇Ψn

1 − µ2∇Ψn2 )] ds (50)

ZEJmn = −

∫∫

Fm

om ·[nm

1 ×(

jk1

η1An

1 −jk2

η2An

2

)]ds

−∫∫

Fm

om · [nm1 × (ε1∇Φn

1 − ε2∇Φn2 )] ds (51)

where the contribution of the n-th div-TO basis function in theexpansion of the vector electric and magnetic potentials result in

(jk1η1Fn1 − jk2η2Fn

2 ) = jω

∫∫

S

(µ1ε1G1 − µ2ε2G2)onds′ (52)

(jk1

η1An

1 −jk2

η2An

2

)= jω

∫∫

S

(µ1ε1G1 − µ2ε2G2)onds′ (53)

and, in view of the discrete expansion of the gradient of the scalarelectric potential in the perfectly conducting case in (32), in the


dielectric case the discrete contribution in (51) becomes

(ε1∇Φn1 − ε2∇Φn

2 ) =1jω

[∮

∂Fn

(∇G1 −∇G2) (on · nc) dl′

−∫∫

Fn

(∇G1 −∇G2)∇′ · (on)ds′

(54)

and, in an analogous manner, for the expansion of the magnetic scalarpotential in (50), we can write

(µ1∇Ψn1 − µ2∇Ψn

2 ) =1jω

[∮

∂Fn

(∇G1 −∇G2) (on · nc) dl′

−∫∫

Fn

(∇G1 −∇G2)∇′ · (on)ds′

(55)

5. RESULTS

We compute the impedance elements in the resulting impedancematrices with great accuracy. We carry out the analytical extractionof the quasi-singular R−3 and R−1 contributions of the Kernels withtraditional integrating schemes over triangles [1, 26, 27]. The remainingsource contributions and the field testing integrals are computednumerically with a 9-point Gaussian quadrature rule. Interestingly,the Muller-formulation results in the advantageous cancelation ofsome R−3-contributions. For example, the expression (∇G1 − ∇G2)in (54) and (55) shows a milder R−1-dependence after the subtractionprocess [13]. Similarly, since µ1 = µ2 = µ0 in all tested cases, thesubtraction (µ1∇G1 − µ2∇G2) arising in (48) turns out also R−1-dependent.

In our tests of RCS accuracy for the div-TO MoM-implementations we adopt sharp-edged objects with moderately smallelectrical dimensions (with electrical dimensions of wavelength frac-tions) for the following reasons: (i) the presence of sharp-edges is dom-inant in the scattering process and therefore their influence becomesrelevant for any direction of observation; (ii) it is possible to assess theconvergence of the computed RCS against the number of unknownsbecause the required computational requirements for very fine mesh-ings stay within the available resources. In all the tested cases, weexcite the bodies with a +z-propagating x-polarized plane wave. Theadopted wavelength and free-space wavelength, respectively, for theperfectly conducting or dielectric bodies is 1 m.


0 20 40 60 80 120 140 180-33

-32

-31

-30

-29

-28

-27

-26

-25

-24

100 160ψ

dB

sm

Bistatic RCS, yz plane

x

yz

EFIE [RWG]

MFIE [RWG]

EMFIE [Loop-Star]

MFIE [div-TO]

EMFIE [div-TO]

Figure 2. yz plane cut ofthe RCS computed with MFIE[div-TO], EMFIE [div-TO], EFIE[RWG], MFIE [RWG] and EMFIE[Loop-Star] for a PeC-cube withside 0.1m meshed with 108 trian-gular facets and wavelength 1 m.

EFIE [RWG]

MFIE [RWG]

EMFIE [Loop-Star]

MFIE [div-TO]

EMFIE [div-TO]

x

yz

Number of unknowns10

3

-58.8

-59

-59.2

-59.4

-59.6

-59.8

-60

-60.2

dB

sm

Backward RCS: PeC-Tetrahedron, side = 0.05 m

Figure 3. Backward RCS ofa regular PeC-tetrahedron withside 0.05 m computed with MFIE[div-TO], EMFIE [div-TO], EFIE[RWG], MFIE [RWG] and EMFIE[Loop-Star]. The wavelength is1m.

In Figure 2, we present the computed RCS for a perfectlyconducting cube with side 0.1 m discretized with 108 triangular facets.We see clearly how for this sharp-edged object the computed RCSpatterns with the div-TO discretizations of the Second Kind IntegralEquations, MFIE and EMFIE, MFIE [div-TO] and EMFIE [div-TO],respectively, reduce the observed discrepancy of the RWG and Loop-Star discretizations of the MFIE and the EMFIE, MFIE [RWG] andEMFIE [Loop-Star], with respect to the RWG discretization of theEFIE, EFIE [RWG].

In Figures 3 and 4, we show, respectively, the backwardand forward RCS against the number of unknowns of the MoM-implementations EFIE [RWG], MFIE [RWG], MFIE [div-TO] andEMFIE [div-TO] for a perfectly conducting regular tetrahedron withside 0.05 m. The gain in RCS accuracy by adopting a div-TOdiscretization of MFIE and EMFIE under a given mesh is clearly betterthan increasing the number of unknowns for a RWG discretizationby making the mesh finer. Also, we see that the improvement inthe div-TO discretizations is in relative terms much better for lessfine meshings, when the influence of the sharp-edges becomes moreimportant. The same is observed in Figures 5 and 6 for a regulartetrahedron with bigger electrical dimensions, with side 1 m (onewavelength).


EFIE [RWG]

MFIE [RWG]

EMFIE [Loop-Star]

MFIE [div-TO]

EMFIE [div-TO]


3

dB

sm

Fordward RCS: PeC-Tetrahedron, side = 0.05 m-65.4

-65.6

-65.8

-66

-66.2

-66.4

-66.6

-66.8

-67

Figure 4. Forward RCS ofa regular PeC-tetrahedron withside 0.05 m computed with MFIE[div-TO], EMFIE [div-TO], EFIE[RWG], MFIE [RWG] and EMFIE[Loop-Star]. The wavelength is1m.

x

yz

dB

sm

Backward RCS: PeC-Tetrahedron, side = 1 m

EFIE [RWG]

MFIE [RWG]

MFIE [div-TO]

EMFIE [div-TO]


3

4.2

4

3.8

3.6

3.4

3.2

3

Figure 5. Backward RCS ofa regular PeC-tetrahedron withside 1 m computed with MFIE[div-TO], EMFIE [div-TO], EFIE[RWG] and MFIE [RWG]. Thewavelength is 1m.

dB

sm

Forward RCS: PeC-Tetrahedron, side = 1 m

EFIE [RWG]

MFIE [RWG]

MFIE [div-TO]

EMFIE [div-TO]


3

5

4.9

4.8

4.7

4.6

4.5

4.4

4.3

4.2

Figure 6. Forward RCS ofa regular PeC-tetrahedron withside 1m computed with MFIE[div-TO], EMFIE [div-TO], EFIE[RWG] and MFIE [RWG]. Thewavelength is 1 m.

x

yz

PMCHWT [RWG]

Muller [RWG]

Muller [div-TO]

Number of unknowns

0 1000 2000 3000 4000 5000 6000

0.038

0.036

0.034

0.032

0.03

0.028

0.026

0.024

0.022

sq

-me

ter

Backward RCS: Cube, side = 1/3 m, ε = 4r

Figure 7. Backward RCS ofa dielectric cube with εr = 4with side 0.33 m computed withPMCHWT [RWG], Muller [RWG]and Muller [div-TO]. The free-space wavelength is 1 m.

In Figure 7, we validate our MoM-implementations for dielectricsthrough the backward RCS plot against the number of unknowns ofa cube with side 0.33m and relative permittivity of 4 (see Figure 3of [14]). In Figures 8 and 9, we show the forward scattered RCS againstthe number of unknowns for two different sharp-edged moderately


small dielectric regular polyhedra, a cube and a tetrahedron, with lowdielectric contrast (relative permittivities of 3 and 2) and side 0.1 m.Note how the observed RCS deviation now, in the dielectric case, forthe RWG discretization of the Muller-formulation, Muller [RWG], ismuch lower, 0.3 dB at most for very coarse meshings, than in theperfectly conducting case for the RWG discretizations of the MFIE

r

x

yz

Fordward RCS: Cube, side = 0.1 m, ε = 3


3

PMCHWT [RWG]

Muller [RWG]

Muller [div-TO]

-36.68

-36.7

-36.72

-36.74

-36.76

-36.78

-36.8

-36.82

-36.84

-36.86

Figure 8. Forward RCS ofa dielectric cube with εr = 3and side 0.1 m computed withPMCHWT [RWG], Muller [RWG]and Muller [div-TO]. The free-space wavelength is 1m.

PMCHWT [RWG]

Muller [RWG]

Muller [div-TO]

Number of unknowns

103

dB

sm

rFordward RCS: Tetrahedron, side = 0.1 m, ε = 2

z

y

x

-59.75

-59.8

-59.85

-59.9

-59.95

-60

-60.05

-60.1

-60.15

Figure 9. Forward RCS of a di-electric regular tetrahedron withεr = 2 and side 0.1 m computedwith PMCHWT [RWG], Muller[RWG] and Muller [div-TO]. Thefree-space wavelength is 1 m.

PMCHWT [RWG]

Muller [RWG]

Muller [div-TO]


3

Backward RCS: Cube, side = 0.1 m, ε = 50

zy

x

dB

sm

r

-35.5

-36

-36.5

-37

-37.5

-38

-38.5

-39

-39.5

Figure 10. Backward RCS ofa dielectric cube with εr = 50and side 0.1 m computed withPMCHWT [RWG], Muller [RWG]and Muller [div-TO]. The free-space wavelength is 1m.

dB

sm

PMCHWT [RWG]

Muller [RWG]

Muller [div-TO]


3

Fordward RCS: Cube, side = 0.1 m, ε = 50r

-22.5

-22.6

-22.7

-22.8

-22.9

-23

-23.1

-23.2

-23.3

-23.4

Figure 11. Forward RCS ofa dielectric cube with εr = 50and side 0.1m computed withPMCHWT [RWG], Muller [RWG]and Muller [div-TO]. The free-space wavelength is 1 m.


or the EMFIE. For these two cases, the div-TO discretization of theMuller-formulation, Muller [div-TO], follows the RCS computed withthe PMCHWT-formulation, PMCHWT [RWG], more closely than theRWG-discretization of the Muller-formulation, Muller [RWG].

In Figures 10 and 11, we show the backward and forward RCSagainst the number of unknowns for a cube with side 0.1m and ahigh dielectric contrast (εr = 50). Now the observed RCS deviationfor the discretizations of the Muller-formulation is bigger than forlower dielectric contrasts. In view of these figures, the RWG orthe div-TO sets do not appear as competitive options with respectto the PMCHWT-formulation, which shows a more stable trend forconvergence.

6. CONCLUSIONS

We present the Taylor-Orthogonal basis functions as a set of basisfunctions suitable for the discretization in Method of Moments oftwo types of Second Kind Integral Equations: (i) the recentlyintroduced Electric-Magnetic Field Integral Equation (EMFIE), forconductors, and (ii) the Muller formulation, for homogeneous orpiecewise homogeneous dielectrics. These basis functions are derivedfrom the 2D-linear Taylor’s expansion of the current around thegeometric centers of the facets arising from the discretization.We define the divergence-Taylor-Orthogonal [div-TO] and the curl-Taylor-Orthogonal [curl-TO] sets as the basis functions capturing,respectively, the divergence of the current and the normal componentof the curl of the current at the centroids of triangles. We showthat the div-TO implementation of the EMFIE reduces the observedRCS discrepancy of the RWG and Loop-Star discretizations of theMFIE and EMFIE, respectively, for moderately small sharp-edgedconductors. Moreover, in view of our tests with moderately smallsharp-edged dielectrics, we observe that the div-TO discretization ofthe Muller-formulation better approaches the RCS computed with thePMCHWT-formulation for sharp-edged objects with low contrasts.For a cube with high-dielectric contrast, both RWG and div-TOdiscretizations of the Muller-formulation show a bigger deviation withrespect to PMCHWT, which shows a stable trend of convergence inany case. The facet-oriented discretization of the Muller-formulationpresented in this paper allows a straightforward implementation ofthe Muller-formulation for composite piecewise homogeneous dielectricobjects. Indeed, assigning unknowns to facets avoids imposing thenormal continuity of the expanded current across the junctions arisingfrom the intersection of several dielectric regions, as it is required whenthe edge-oriented RWG basis functions are adopted [28].


ACKNOWLEDGMENT

This work was supported by the Spanish Interministerial Commissionon Science and Technology (CICYT) under Projects TEC2010-20841-C04-02, TEC2007-66698-C04-01/TCM, TEC2009-13897-C03-01 andCONSOLIDER CSD2008-00068.

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