Chapter 1
Moment Method solution of Maxwell’s equations
Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. Its differential and integral forms are reviewed in this Chapter along with the surface equivalence principle introducing their numerical evaluation through the Method of Moments. In particular, the numerical solution of the EFIE and the CFIE type of integro-differential equations through the MoM is presented.
1.1. Maxwell’s equations
Consider a canonical scattering problem comprising a source-free, linear, isotropic region of space in which a three-dimensional body characterized by relative permittivity ε and r
permeability µ is immersed (Fig. 1.1). When the scatterer is illuminated by an r
electromagnetic wave, in the absence of external sources, the produced fields must satisfy Maxwell’s equations: 0 0 0 0 ( ) 0 ( ) 0 r r r r E j H H j E E H ωµ µ ωε ε ε ε µ µ ∇× = − ∇× = ∇ ⋅ = ∇ ⋅ = (1.1)
where E and H are complex representations of time harmonic electric and magnetic fields measured in units of [V/m] and [A/m] respectively.
The relative permittivity and permeability can in general be complex to account for the electromagnetic properties of a given structure to be described. In particular, in the limit of infinite conductivity for a given material, its surface is denoted as a Perfect Electric Conductor (PEC).
In the process of numerically solving Maxwell’s equations applied for deriving the scattered, internal or total fields, boundary conditions are of fundamental relevance in order to determine a unique solution to the problem.
On the boundary of a PEC, the electric and magnetic field vectors in the absence of external sources must satisfy:
0 ˆ 0 ˆ ˆ ˆ 0 s s r n E n H J n E n H ρ ε ε × = × = ⋅ = ⋅ = (1.2) where Js
denotes the surface current density, ρ the surface charge density and s nˆ represents the outward unit normal on surface S, as shown in Fig. 1.2.
The conditions to be satisfied for the electromagnetic fields between two homogeneous penetrable materials characterized by dielectric permittivity ε ,r1 εr2 and permeability µr1,
2 r µ are:
(
)
(
)
(
)
1 2 1 2 1 1 2 2 1 1 2 2 ˆ 0 ˆ ( ) ˆ ˆ ˆ 0 s r r s r r n E E n H H J n n E E n H H ε ε ρ µ µ × − = × − = × ⋅ − = ⋅ − = (1.3)where nˆ is a unit vector normal to the boundary pointing from medium 2 into medium 1 (Fig. 1.2(b)).
In words, the tangential components of the E-field always results continuous while the normal components present a discontinuity across the interface between two dielectrics. The tangential components of the H-field between the two media must be discontinuous to account for a given surface current density (if the material presents a non-zero conductivity).
1.2. Description of a scattering problem: source-field relationship 3
(a) (b)
Figure 1.2. Field conditions to be satisfied across a PEC (a) and fields directions at the boundary
between two dielectrics (b).
1.2. Description of a scattering problem: source-field relationships
A possible way to schematize the electromagnetic scattering mechanism is through a simple scheme that helps to fictitiously separate the field contributions for a given problem. Suppose the scatterer in Fig. 1.1 is illuminated by an incident field produced by a primary source located outside its region. This incident radiation will induce currents and charges at the scatterer location which in turn will reradiate.The produced fields can be split up in a contribution from the primary sources in the absence of the scatterer, named as the incident fields and a contribution from the induced sources to which we refer to as the scattered fields. The superposition of these contributions yields the total original fields which are established in the presence of the scatterer:
inc scat inc scat E E E H H H = + = + (1.4)
The incident fields in the vicinity of the scatterer (away from the primary source) satisfy the vector Helmholtz equations:
2 2 2 2 0 0 inc inc inc inc E k E H k H ∇ + = ∇ + = (1.5)
while the scattered fields are solution to the equations: 2 2 0 0 2 2 0 0 s s s s J E k E j J M j M H k H J j M j ωµ ωε ωε ωµ ∇∇ ⋅ ∇ + = − + ∇× ∇∇ ⋅ ∇ + = −∇× + − (1.6)
where J and M represent mathematical equivalent sources which radiate in an homogeneous medium and are function of the total fields. An alternative version of Maxwell’s equations including equivalent sources can be written, as we will focus on later, to simplify the solution of a problem involving inhomogeneities.
A combination of Maxwell’s equations for the incident fields with primary sources and for the scattered fields with mathematical sources yields (1.1). Radiation conditions ensure then that the fields satisfying (1.6) propagate away from the scatterer.
There exist a number of strategies to find a solution to equation (1.6) in an homogeneous, infinite space, which can be considered more or less efficient from a numerical point of view. A classical approach consists in expressing the fields in terms of the magnetic A and the electric F vector potentials as follows:
2 0 2 0 s s A k A E F j F k F H A j ωε ωµ ∇∇ ⋅ + = − ∇× ∇∇ ⋅ + = ∇× + (1.7)
A solution verifying the radiation conditions can be written in the form:
A J G F M G = ⊗ = ⊗ (1.8)
where G represents the three-dimensional Green’s Function which yields the potential after an operation of convolution with the source:
( ) ( ) . 4 jk r r e A r J r dr r r π ′ − − ′ ′ = ′ −
∫∫∫
(1.9)1.2. Description of a scattering problem: source-field relationships 5
The integration → differentiation procedure to construct the potentials → fields is not the best suited for numerical implementation. In fact, the integrals arising from source-field relationships seldom can be expressed in closed-form and a numerical implementation of second order derivatives (1.7) can result inaccurate and inefficient. A differentiation →
integration scheme can result in a more efficient answer given that for observation points
outside the source region, derivatives can be brought inside the integrals and the Green’s function can be easily differentiated analytically.
Unluckily, the singular behaviour of the Green’s function in the region containing the sources make impossible (unless the integral can be carried out in closed-form in the source region) to interchange the two operators without violating Leibnitz’s rule.
An alternative to the pure vector potential source-field relationship is a mixed-potential formalism whose solution for the scattered fields reads:
0 0 s e s m E j A F H A j F ωµ ωε = − − ∇Φ − ∇× = ∇× − − ∇Φ (1.10)
with Φe, Φm scalar potential functions. While the derivation of the vector potentials is equivalent to that in (1.8), the scalar potentials are defined as:
0 0 e e m m G G ρ ε ρ µ Φ = ⊗ Φ = ⊗ (1.11)
This particular solution isolates the contribution from electric and magnetic current and charge densities from that of the scattered fields. After integration of the Green’s functions over the sources to find the potentials, as an advantage over the pure vector-potential solution, a single differentiation is required in equations (1.10) to get the scattered fields. Other solutions involving the Dyadic Green’s functions can be formulated; of course all possible different source-field relationships are equivalent and produce identical results.
1.3. The surface equivalence principle
To simplify the solution of integral equations that involve an heterogeneous environment, the original scattering problem can be simplified replacing the inhomogeneity with equivalent mathematical sources which radiate in free space [39]. Both mathematical volume sources (polarization currents and charges) and sources distributed on the problem boundary surface can be fictitiously introduced, in the case we want to apply a volume or a surface equivalence principle. The first type of equivalence principle is used in the formulation of volume integral equations. Since our attention will be mainly focused on solving surface integral equations, we will mainly delve into the surface equivalence principle.
Consider two regions of space separated by a mathematical surface S (Fig. 1.3). Region 1 is homogeneous with parameters ε , 1 µ whereas region 2 presents inhomogeneities that may 1 include perfectly conducting materials. We postulate that a source (J2,M2) located in region 2, radiating in the presence of inhomogeneities, will produce fields E2 and H2
throughout region 1.
1.3. The surface equivalence principle 7
We postulate a second source ( ,J M1 1)
, located in region 1 and radiating fields E1
and H1 in a homogeneous ( ,ε µ1 1) space. Both produced fields satisfy the radiation condition on the boundary at infinity (S∞).
Throughout region 1, Maxwell’s curl equations hold:
1 1 1 1 1 1 1 1 2 1 2 2 1 2 E j H M H j E J E j H H j E ωµ ωε ωµ ωε ∇× = − − ∇× = + ∇× = − ∇× = (1.12)
Therefore, in region 1 the following equations can be constructed:
2 1 1 2 1 2 1 2 1 1 2 1 2 1 1 2 1 1 2 1 2 1 1 2 H E j H H H M E H j E E E J H E j H H E H j E E ωµ ωε ωµ ωε ⋅∇× = − ⋅ − ⋅ ⋅∇× = ⋅ + ⋅ ⋅∇× = − ⋅ ⋅∇× = ⋅ (1.13)
By subtracting (1.13) (3) left hand side from (1) and (4) from (2):
2 1 1 2 1 2 1 2 2 1 2 1
H ⋅∇× −E H ⋅∇×E +H ⋅∇×E −E ⋅∇×H =E ⋅ −J H ⋅M (1.14) which is equivalent to:
(
E1 H2 E2 H1)
E2 J1 H2 M1∇ ⋅ × − × = ⋅ − ⋅ (1.15) Equation (1.15) is a form of the Lorentz reciprocity theorem [18]. Integrating both sides of (1.15) over region 1 and applying the divergence theorem:
1 ˆ ˆ S S W dv W n dS W n dS ∞ Γ ∇ ⋅ = ⋅ + ⋅
∫∫∫
∫∫
∫∫
(1.16)where ˆn is the unit vector normal to the surface pointing out of region 1, yields:
(
)
(
)
1 1 2 2 1 ˆ 2 1 2 1 S E H E H n dS E J H M dv Γ × − × ⋅ = ⋅ − ⋅∫∫
∫∫∫
(1.17)considering that the integral over S∞vanishes as an effect of the radiation condition. Vector identities dictate that:
(
)
1 2 ˆ 1 ˆ 2 E ×H ⋅ = − ⋅ ×n E n H (1.18) and(
)
2 1 ˆ 1 2 ˆ E ×H n ⋅ = − ⋅H E ×n (1.19) As a consequence, equation (1.17) becomes:(
)
(
)
(
)
1 1 ˆ 2 1 2 ˆ 2 1 2 1 S E n H H E n dS E J H M dv Γ ⋅ − × − ⋅ − × = ⋅ − ⋅ ∫∫
∫∫∫
(1.20)which represents a generalized statement of reciprocity. By supposing the sources in region 1 are:
(
)
1 1 0 J r r M δ ′ = − = (1.21)For this sources, equation (1.20) can be written as:
(
)
(
)
2 1 2 1 2 ˆ ˆ ˆ r S u E⋅ =∫∫
E ⋅ − ×n H −H ⋅ − ×E n (1.22)where E1 and H1 are the fields produced at location r′ in an infinite homogeneous space by sources J1 and M1 residing at r. By expressing the fields through (1.7), after some algebraic manipulation (App. A) the following expression can be written:
(
)
(
)
2 2 2 1 2 ˆ ˆ ˆ 4 ˆ ˆ 4 jk r r r S jk r r S k e u E u n H dS j r r e u E n dS r r ωε π π ′ − − ′ − − ∇∇ ⋅ + ′ ⋅ = ⋅ − × ′ − ′ − ⋅∇× − × ′ −∫∫
∫∫
(1.23)1.3. The surface equivalence principle 9
Equation (1.23) is a statement that the fields produced by (J2,M2)
at some location outside region 2 can be expressed as an integration over tangential fields on surface of region 2. In fact, by comparing (1.23) with (1.7) it is apparent that the field is equivalent to that produced by surface current densities
2 2 ˆ ˆ s s J n H M E n = − × = − × (1.24)
located on S and radiating in an homogeneous space having constitutive parameters ε1 and 1
µ . This property represents the fundamental idea of the Huygens’ surface equivalence principle.
Let’s now consider, as a practical application of what stated above, a source located in region 1, which radiates in the presence of some inhomogeneities (Fig. 1.4).
Figure 1.4. The electromagnetic source located in Region1 radiates in the presence of some inhomogeneities located in Region2. The two regions are separated by surface S.
Now consider mathematical sources Js
and Ms
residing on S and satisfying:
1 1 ˆ ˆ s s J n H M E n = × = × (1.25)
where ˆn represents the outgoing normal. According to Huygens’ principle, the
combination of the original and the equivalent sources leads to the same original fields in region 1 (Fig. 1.5). The fields in region 2 are not the same as those generated in the original problem, in fact, the combination of the original and equivalent sources generates a null field in region 2 (this result is known as the extinction theorem).
Figure 1.5. The equivalent representation involving mathematical sources on S produces exactly the
same original fields in region1 and null fields in region2. For the purpose of solving the exterior problem most conveniently, the eventual inhomogeneities that are present in region2 are replaced with same medium of region1 in the equivalent representation.
Since the fields produced in region 2 through the equivalent representation vanish, the inhomogeneities in region 2 can be replaced with anything without affecting the result for the fields in region 1. By replacing them with an homogeneous medium presenting same characteristics of region 1, the original exterior problem can be solved through a more convenient equivalent one whose sources radiate in an homogeneous environment [20].
1.4. The Method of Moments 11
1.4. The Method of Moments
The Moment Method is a mathematical instrument able to reduce functional equations to matrix equations [40], [41]. If an inhomogeneous equation is considered:
( )
L f =g (1.26)
where L is a linear operator and g is known; if it is not possible to determine the inverse operator 1
L− in a straightforward manner, it is common to seek a numerical representation. An approximate solution to f can be obtained by expanding it in a series of functions in the L domain: 1 N n n n f α B = ≅
∑
(1.27)where scalars
{ }
α are unknown coefficients to be determined and n{ }
Bn
are known expansion functions or basis functions and must form a complete set for a correct representation.
Substituting (1.27) in (1.26) and using the linearity of L , g can be written as follows:
1 ( ) N n n n L B g α = =
∑
(1.28)Assuming that a suitable inner product 〈B g, 〉 [App. B] has been defined for the problem, a set of weighting functions, or testing functions
{
T T 1, 2,...,TN}
can be defined in the range ofL . By forcing the residual
1 1 N N n n n n n n L α B g α LB g = = − = −
∑
∑
(1.29)to be orthogonal to the set of testing functions
{ }
TN , produces:, , n m n m n T LB T g
α
〈 〉 = 〈 〉∑
(1.30)This set of equations can be written in matrix form:
{ }
Lm n,{ } { }
αn = gm (1.31) having entries:{ }
1 1 1 2 1 3 , 2 1 2 2 2 3 , , , , , , m n T LB T LB T LB L T LB T LB T LB 〈 〉 〈 〉 〈 〉 = 〈 〉 〈 〉 〈 〉 ⋅⋅ (1.32) and{ }
{ }
1 1 2 2 , , , n m N N T g T g g T g α α α α 〈 〉 〈 〉 = = 〈 〉 (1.33)If matrix
{ }
L is non-singular, its inverse{ }
1L− exists and coefficients
{ }
α are given by: n{ }
{ }
1{ }
, n Ln m gm
α = −
(1.34)
Therefore the solution for f , given by (1.27), can be rearranged as follows:
{ }
{ }
{ }
{ }
1{ }
, n n n m n m f = B α = B l− g (1.35) with{ }
Bn ={
B1, B2,B3, ,BN}
. This solution may be more or less accurate depending upon the choice of
{ }
Bn
and
{ }
Tn
. The particular choice
{ } { }
Bn = Tn
is known as the
Galerkin’s method [42], [43].
If the sets
{ }
Bn and{ }
Tn are finite, the resulting matrix is of finite order and can be inverted by known methods. If{ }
L is of infinite order, it can only be inverted if it is diagonal. The main task for a given problem is represented by the choice of the{ }
Bn and{ }
Tn .
1.4. The Method of Moments 13
The
{ }
Bn
should be linearly independent and chosen in a way such that their superposition results in a reasonable approximation of f . The
{ }
Tn should also be linearly independent and selected in such a way that products 〈T gn, 〉
depend on relatively independent properties of g .
Furthermore, a number of factors as: (1) the accuracy of the desired solution, (2) the ease in the evaluation of the matrix elements, (3) the maximum size of the matrix that can be inverted and (4) the well conditioning of the matrix should be taken into account while selecting
{ }
Bn
and
{ }
Tn .
There are infinite possible combinations of testing and basis functions that can be chosen to represent a given type of problem. Some sets may give faster convergence, matrices easier to evaluate or generate acceptable results through smaller matrices with respect to other sets for a given problem.
A simple way to get approximate solutions, known as the point-matching technique, is to require (1.20) to be satisfied at discrete points in the region of interest.
Another type of approximation is the method of subsections in which basis functions only defined over subsections of f are employed. This way, each coefficient generated through matrix inversion affects the reconstruction of f only in a subsection of it. This procedure often simplifies the evaluation and/or the form of the matrix. It can also be used in conjunction with the point matching method.
For single-dimension scalar quantities, the simplest basis functions in use are illustrated in Fig. 1.6. These include the Dirac delta function:
0( ) ( 0)
B x =
δ
x−x (1.36) A function which is defined only over one sub-interval is the pulse function, or piecewiseconstant function: 1 2 1 1 2 1 ( ) ( ; , ) 0 x x x B x p x x x otherwise < < = = (1.37)
However, for L= −∂ ∂ the operation LP does not yield a function in the range of L . 2 x2 Therefore pulse functions cannot be used as sub-sectional bases for this case unless the operator is extended or approximated.
Figure 1.6. Definition of basis and testing functions
A better-behaved function results to be the triangle function defined as follows:
3 3 4 4 3 2 3 4 5 5 4 5 5 4 ( ) ( ; , , ) x x x x x x x B x t x x x x x x x x x x x − < < − = = − < < − (1.38)
A linear combination of Bn=p x( ) according to (1.28) gives a step approximation to f , as represented in Fig. 1.7(a).
A linear combination of triangle functions:
1 ( ) N n n n f α T x x = =
∑
− (1.39)1.4.1. Numerical solution to the EFIE 15
Figure 1.7. Functional approximations: step (a) and piecewise linear (b).
1.4.1. Numerical solution to the EFIE
Among several existing Moment Method formulations, some have been found to be better suited for particular scattering problems based upon their geometry and material characteristics [44], [45]:
i Conducting scatterers (homogeneous, isotropic):
a. Electric Field Integral Equation formulation (EFIE) for closed and open bodies
b. Magnetic Field Integral Equation formulation (MFIE) for closed bodies
ii Dielectric scatterers (homogeneous, isotropic):
Combined Field Integral Equation formulation
iii Anisotropic scatterers (homogeneous):
Combined Field Integral Equation (CFIE) formulation modified for material characteristics.
In the following section, the numerical solution of the EFIE with particular sets of basis functions is introduced. Let S be the surface of a PEC object with unit normal nˆ. An electric field i
E , defined as the field propagating in the absence of the scatterer, is incident on and induces surface currents J on S.
The produced electric fields derived from the surface currents are defined as follows:
scat
E = −j A
ω
− ∇φ
(1.40) where the magnetic vector potential is defined as:( ) 4 jkR S e A r J dS R µ π − ′ =
∫
(1.41)and the scalar potential:
1 ( ) 4 jkR S e r dS R φ ρ πε − ′ =
∫
(1.42)where ε and µ represent the dielectric parameters of the surrounding medium and
| |
R= r−r′ is the distance between an arbitrary located source point r′ on S and observation point r. The surface divergence of the current is related to the charge through the continuity equation:
s J j
ωρ
∇ ⋅ = − (1.43)
By enforcing the continuity of the tangential components of the electric fields, reading
(
)
ˆ i s 0
n× E +E = , on S , an integro-differential equation for J is derived which reads:
(
)
i t t E j Aω φ r on S − = − + ∇ (1.44)1.4.1. Numerical solution to the EFIE 17
Set of equations (1.41)-(1.44) is well-known as the electric field integral equation. The derivatives on the current and on the scalar potential involved in the process, suggest that particular care should be taken while selecting the expansion functions and the testing procedure in the development of the numerical solution.
Suppose a set of expansion functions, suitable for use with the EFIE and triangular patch modeling (Fig. 1.8) [19], is selected such that the current on S can be approximated in terms of
{ }
Bn as follows: 1 ( ) N n n n J I B r = ≅∑
(1.45)where N represents the number of interior (no boundary) triangle edges used to represent the structure. Since a basis function is associated with each non-boundary edge of a triangulated object, N represents the number of unknowns for the problem. Due to particular characteristics [19], [46], a weighted, linear superposition of the basis functions
{ }
Bn
is able to accurately represent the current flow in an arbitrary direction within the structure.
(a) (b)
The next step in the Moment Method consists in selecting a testing procedure. For the purpose of presenting a numerical solution to the EFIE, the Galerkin’s testing will be employed, namely the same expansion functions
{ }
Bn will be used as testing functions. By making use of a symmetric product defined as follows:,
S
f g f g dS
〈 〉 =
∫
⋅ (1.46)The (1.44) is tested with
{ }
Bn , yielding:, , , .
i
m m m
E B j
ω
A Bφ
B〈 〉 = 〈 〉 + 〈∇ 〉 (1.47) It is now possible to take advantage of a surface vector calculus identity [47] and the properties of the selected bases at the edges of S to find, for the last term of (1.47):
, m s m
S
B B dS
φ φ
〈∇ 〉 = −
∫
∇ ⋅ (1.48)Thanks to the expansion functions properties, the integral can be easily approximated as a summation of the value of the scalar potential at triangle’s centroids rmc+, rmc−:
( ) ( )
1 1 m m c c s m m m m m m m S T T B dS l dS dS l r r A A φ φ φ φ φ + − + − + − ∇ ⋅ = − ≅ − ∫
∫
∫
(1.49)Through similar approximations, the remaining two terms in (1.47) may be written as:
1 1 , 2 2 ( ) ( ) 2 ( ) ( ) m m i i i m m m m mT mT i c i c m c m c m m m c c m m E E E B l dS dS A A A A A E r E r l A r A r ρ ρ ρ ρ + − + − + − + − + − + − = ⋅ + ⋅ ≅ ⋅ + ⋅
∫
∫
(1.50)where the integral over each triangle is eliminated by approximating A or E ( i) by its value at the triangle centroid.
1.4.1. Numerical solution to the EFIE 19
Through (1.48)-(1.50), (1.47) enforced at each triangle edge becomes:
( )
( )
( )
( )
( ) ( )
1, 2, , 2 2 2 2 c c i c m i c m m m m c c c m c m c c m m m m m m l E r E r m N j l A r A r l r r ρ ρ ρ ρ ω φ φ + − + − + − + − + − ⋅ + ⋅ = = = ⋅ + ⋅ + + (1.51)The testing procedure is aimed at reducing the differentiability requirement on φ in (1.44) by integrating ∇ first. φ
Approximations (1.49) and (1.50) are aimed at eliminating surface integration of the potential quantities, allowing a double surface integral to be replaced by a quantity involving a single surface integral in the numerical computation of the moment matrix elements.
The approximations are justified by the locally smooth nature of the potentials and the source within each sub-domain.
By substituting the current expansion (1.45) into (1.51), a N× system of linear equations N
is obtained, which can be written in matrix form as follows:
Z I = V (1.52)
where Z= Zm n, is a N×N matrix and I=
[ ]
In and V=[ ]
Vn are N×1 column vectors. The matrix element entries are defined as:, , , , , 2 2 c c m m m n m m n m n m n m n Z l jω A ρ A ρ φ φ + − + − + − = ⋅ + ⋅ + + (1.53)
while right-hand side column vector is defined as:
2 2 c c m m m m m m V l E ρ E ρ + − + − = ⋅ + ⋅ (1.54)
where potentials are defined as: , , ( ) , 4 1 ( ) , 4 m m jkR m n m m S jkR m n s m m S c m m e A B r dS R e B r dS j R R r r µ π φ π ωε ± ± − ± ± − ± ± ± ± ′ ′ = ′ ′ ′ = − ∇ ⋅ ′ = −
∫
∫
(1.55)and the incident field:
( )
i c
m m
E± =E r ± (1.56)
Suppose a plane wave is incident upon a given PEC structure; in this case, the right hand side vector is built as:
(
ˆ0 ˆ0)
( ) i jk r E r = Eθθ +Eφφ e− ⋅ (1.57)where ( ,θ φˆ0 ˆ0) define the angle of arrival of the incident wave and k represents the propagation vector.
Once the elements of the MoM matrix and the right hand side vector which represents the incident field are determined, the resulting matrix equation can be solved for the weights of the unknown current vector.
The integrals in (1.55) are evaluated by numerical quadrature techniques tailored for triangular sub domains. For the terms in which m=n, the integrands are singular, therefore the singular portion of each integrand must be removed and integrated analytically [48].
1.4.2. Numerical solution to the CFIE 21
1.4.2. Numerical solution to the CFIE
The numerical solution of the EFIE and the MFIE for conducting bodies can be combined [44], [49] and extended for numerically describe any arbitrary shaped dielectric object, for any given excitation. In the process of numerically solving a dielectric problem both equivalent electric and magnetic surface currents are expanded in terms of known basis functions.
Referring to Fig. 1.9, S denotes the surface which bounds a homogeneous dielectric scatterer having volume V located in region 2.
Figure 1.9. A homogeneous dielectric scatterer immersed in an isotropic free space medium.
The scatterer is immersed in region 1, representing an isotropic, lossless free space medium. Let
(
E H1s,1s)
be the electric and magnetic fields scattered in region 1 and(
2, 2)
s s
Referring to the surface equivalence principle (see Section 1.3), the electric and magnetic fields scattered in regions 1 and 2 can be written as:
1 1 1 1 1 1 1 1 1 1 1 ( ) ( ) ( ) 1 ( ) ( ) ( ), s s E j A r V r F r H j F r U r A r ω ε ω µ = − − ∇ − ∇× ′ = − − ∇ + ∇× (1.58) 2 2 2 2 2 2 2 2 2 2 1 ( ) ( ) ( ) 1 ( ) ( ) ( ), s s E j A r V r F r H j F r U r A r ω ε ω µ = + ∇ + ∇× ′ = + ∇ − ∇× (1.59)
for r on or outside S ; where the various vector (Ai
, Fi) and scalar potentials (Vi, Ui), for the different regions of interest i=1, 2 are given by:
( ) ( ) ( , ) ( ) 4 ( ) ( ) ( , ) ( ) 4 1 ( ) ( ) ( , ) ( ) 4 1 ( ) ( ) ( , ) ( ) 4 i i i S i i i S i e i i S i m i i S A r J r G r r dS r F r M r G r r dS r V r r G r r dS r U r r G r r dS r µ π ε π ρ πε ρ πµ ′ ′ ′ = ′ ′ ′ = ′ ′ ′ = ′ ′ ′ =
∫∫
∫∫
∫∫
∫∫
(1.60)the dielectric constant in the two different regions is defined as:
1 i i i i j σ ε ε ωε ′ = − (1.61) and 1 ( ) ( ) 1 ( ) ( ) e s m s r J r j r M r j ρ ω ρ ω ′ = − ∇ ⋅′ ′ ′ = − ∇ ⋅′ ′ (1.62)
1.4.2. Numerical solution to the CFIE 23
The Green’s functions defined in (1.60) for i=1, 2 is given by:
( , ) i jk R i e G r r R R r r − ′ = ′ = − (1.63)
and the propagation constant also needs to vary to account for the propagation in two different media:
i i i
k =
ω µ ε
′ (1.64)In particular, for the above expressions, ( ,ε µ σ1 1, 1= and 0) ( ,ε µ σ2 2, 2) represent the permittivity, permeability and conductivity for regions 1 and 2. Note that J represents the equivalent electric current and M the equivalent magnetic current on the surface of the dielectric scatterer. The equivalent electric and magnetic currents are related to the total tangential fields on the surface S through:
ˆ ( ) ( ) ˆ ( ) ( ) , J r n H r M r E r n ′ = × ′ ′ = ′ × (1.65)
where r′ lies on surface S, nˆ represents an outward unit normal on S as reported in Fig. 1.9.
On enforcing the continuity of the total tangential electric and magnetic fields across the surface of the arbitrary dielectric scatterer, the following combined field integral equations can be obtained in terms of the unknown surface equivalent electric and magnetic currents:
[
]
[
]
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i t t i t t F r F r E r j A r A r V r V r A r A r H r j F r F r U r U r ω ε ε ω ε ε = + + ∇ + ∇ + ∇× + ′ ′ = + + ∇ + ∇ − ∇× ′ + ′ (1.66) where i E and iH denote the incident electric and magnetic fields in th
i region and
Through the method of moment technique the above coupled integro-differential equations (1.66) can be reduced to a corresponding partitioned matrix equation for the unknown electric and magnetic currents on the surface of the scatterer.
For the purpose of numerically describing arbitrarily shaped bodies, the surface needs to be efficiently modeled by triangular or square surface patches (see Fig. 1.8). Some vector basis functions
{ }
Bn
suited for the representation of J and M on the discretized surface S , presenting some particular mathematical properties [46] need to be selected.
Referring to the dielectric scatterer shown in Fig. 1.9, the surface electric and magnetic current distributions are expanded in terms of selected vector basis functions. Denoting with N the total number of bases, then:
1 1 ( ) ( ) ( ) ( ) N n n n N n n n J r I B r M r M B r = = ′ = ′ ′ = ′
∑
∑
(1.67)where I and n M are weights yet to be determined. As pointed out in basis vectors n development [46], the weighted superposition of basis functions will conveniently represent a current flowing on the structure.
In order to obtain the current coefficients, the combined field integral equations are tested with respect to testing functions (Galerkin’s testing is employed as in Section 1.4.1). Hence, testing equation (1.66) yields:
1 2 1 2 1 2 1 2 , ( , ) ( , ) , ) i m m m m F F E B jω A A B V V B B ε ε 〈 〉 = 〈 + 〉 + 〈 ∇ +∇ 〉 + 〈∇× ′+ ′ 〉 (1.68) 1 2 1 2 1 2 1 2 , ( , ) ( , ) , ) i m m m m A A H B jω F F B U U B B ε ε 〈 〉 = 〈 + 〉 + 〈 ∇ +∇ 〉 + 〈∇× + 〉 ′ ′ (1.69)
on surface S . Due to some mathematical properties of Rao Wilton Glisson (RWG) type of basis functions, the first terms in (1.68) and (1.69) can be simplified by evaluating the vector potentials at centroids of respective triangles (as previously shown in (1.50) for the EFIE testing).
1.4.2. Numerical solution to the CFIE 25
Moreover, the second terms with gradient in (1.68) and (1.69) can be simplified as follows:
(
)
, m s m S V B V B dS 〈∇ 〉 = −∫∫
∇ ⋅ (1.70)( ) ( )
1 1 m m c c m m m m mT mT l V dS V dS l V r V r A − A + − + − + = − ≅ − ∫
∫
(1.71)Note that the integrals in (1.71) have been approximated by evaluating the scalar potentials at the centroids of the two triangles Tn± respectively. Similarly, the third terms with curl operator in (1.68) and (1.69) can be simplified as:
, m ( ) m S A B A B dS 〈∇× 〉 = −
∫∫
∇× ⋅ (1.72) 1 1 ( ) ( ) 2 2 m m m m m m T m T l A dS A dS A −ρ A + ρ − − + + − + = ⋅ ∇× + ⋅ ∇× ∫∫
∫∫
(1.73)On substituting relations (1.70)-(1.73) into tested (1.66), the functional form of combined integral equations are obtained. In particular for electric fields:
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ( ( ) ( )) ( ( ) ( )) 2 2 ( ( ) ( )) ( ( ) ( )) ( ) ( ) ( ) ( ) c c c c c c m m m m m m m c c c c m m m m m c c c c m m m m j l A r A r A r A r l V r V r V r V r P r P r P r P r ρ ρ ω ε ε ε ε + − + + − − − − + + + + − − ⋅ + + ⋅ + + + + − + + + + + ′ ′ ′ ′ ( ) ( ) , 2 2 c c i c i c m m m m m l ρ E r ρ E r + − + − = ⋅ + ⋅ (1.74)
on scatterer surface S , m=1, 2, 3,,N edges. A similar expression can be obtained for magnetic fields which reads:
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ( ( ) ( )) ( ( ) ( )) 2 2 ( ( ) ( )) ( ( ) ( )) ( ) ( ) ( ) ( ) c c c c c c m m m m m m m c c c c m m m m m c c c c m m m m j l F r F r F r F r l U r U r U r U r Q r Q r Q r Q r l ρ ρ ω µ µ µ µ + − + + − − − − + + + + − − ⋅ + + ⋅ + + + + − + − + + + = ( ) ( ) , 2 2 c c i c i c m m m H rm H rm ρ + ρ − + − ⋅ + ⋅ (1.75)
on scatterer surface S , m=1, 2, 3,,N edges. In the above tested equations:
1, 2 1, 2 1, 2 1, 2 ( ) ( ) 2 ( ) ( ) 2 m m m m m m m T m m m m m T l P r F r dS A l Q r A r dS A ρ ρ ± ± ± ± ± ± ± ± ± ± = ⋅ ∇× = ⋅ ∇×
∫∫
∫∫
(1.76)In functional equations (1.74), (1.75) the vector and scalar potentials Ai, Fi
, Vi, Uiare given by (1.60). The parts in Pi
and Qi which contain the curl operator can be further simplified as: ( ) 4 i i S F r ε M π ′ ∇× =
∫∫
( ) ( , ) ( ) ( ) 4 i i i S r G r r dS r A r µ J π ′ ×∇′ ′ ′ ∇× = ∫∫
( )r′ ×∇′G r r dS ri( , ′) ( )′ (1.77)Where the symbol
∫∫
represents Cauchy principal value of the integral. In usual numerical development techniques, integrals (1.76) are evaluated numerically.By substituting the expanded electric and magnetic currents (1.67) into the tested CFIE (1.68), (1.69), its functional form is reduced to the corresponding partitioned matrix equation [44]:
1.4.2. Numerical solution to the CFIE 27
[ ]
[ ]
[ ]
[ ]
, , , , JJ JM m n m n n m MJ MM n m m n m n Z C I V M H D Y = (1.78)For m=1, 2, 3,,M edges and n=1, 2, 3,,N M = N edges.
Elements of diagonal sub-matrices for electric and magnetic currents are given by:
2 2 2 , , , , , 1 1 1 ( ) ( ) ( ) 2 2 c c JJ m m i m n m i i i m n i i i m n i m n i m n i i i i Z l jk A jk A j k ρ + η ρ − η η φ φ + − − + = = = = ⋅ + ⋅ + −
∑
∑
∑
(1.79) and 2 2 2 , , , , , 1 1 1 1 ( ) 2 2 c c MM m i m i m n m i m n i m n i m n i m n i i i i i i i jk jk Z l F F j k ρ ρ ψ ψ η η η + − + − − + = = = = ⋅ + ⋅ + − ∑
∑
∑
(1.80)Elements of off-diagonal sub-matrices representing the cross contributions are given by:
2 2 , , , 1 1 J M m n i m n i m n i i C P+ P− = = = +
∑
∑
(1.81) and 2 2 , , , 1 1 M J m n i m n i m n i i D Q+ Q− = = = + ∑
∑
(1.82)Elements of electric and magnetic field excitation:
( ) ( ) 2 2 c c i c i c m m m m m m V l ρ E r ρ E r + − + + − − = ⋅ + ⋅ (1.83) and ( ) ( ) 2 2 c c i c i c m m m m m m H l ρ H r ρ H r + − + + − − = ⋅ + ⋅ (1.84)
and the vector and scalar potential integrals take the following form [44], [46] for i=1, 2: , , ( ) 1 ( ) ( , ) ( ) 4 n n c i m n n i m i m n T T A B r G r r dS r F π + − ± ± ± + ′ ′ ′ =
∫∫
= (1.85) , , ( ) 1 ( ) ( , ) ( ) 4 n n c i m n s n i m i m n T T B r G r r dS r φ ψ π + − ± ± ± + ′ ′ ′ ′ =∫∫
∇ ⋅ = (1.86) , , ( ) 1 ( ) ( , ) ( ) ( ) 4 2 m n n m i m n m n i m i m n m T T T l P B r G r r dS r dS r Q A ±ρ π + − ± ± ± ± ± + ′ ′ ′ ′ ′ = ⋅ ×∇ = ∫∫
∫∫
(1.87) and ( , ) i jk R i m e G r r R ± − ± ± ′ = (1.88) m R± = r±−r′ (1.89) 3 ( , ) ( )(1 ) ( ) i jk R i m m i e G r r r r jk R R ± − ± ± ± ± ′ ′ ′ ∇ = − + (1.90)Integrals (1.85)-(1.87) are in a convenient form for numerical evaluation [46]. Matrix equation (1.78) is then solved for the electric and magnetic current coefficients I and n M . n It is crucial to underline that for an efficient numerical algorithm development, only Zm n, elements can be generated since elements Zm n, and Ym n, are similar except for floating constants. Then Ym n, can be conveniently obtained from Zm n, by changing multiplying constants while filling the impedance matrix. Same is true for off-diagonal sub-blocks Cm n, and Dm n, .
Furthermore, by referring to expressions (1.79)-(1.82), it can be noted that the elements of sub matrices contain terms belonging to both region 1 and 2. The potential integral expressions for regions 1 and 2 are in fact identical except for the dielectric parameters which appear in the propagation constants and in some multiplying factors. Therefore, same subroutines are simultaneously employed for deriving region 1 and region 2 integral terms. To obtain the electric and magnetic current distribution coefficients, the matrix equation can be directly inverted or rearranged so as to eliminate one unknown and resubstituted back to get the second unknown. Another possible option is the application of iterative methods [50].