2 2

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________________________________________________________________________________

2

2 Method of Moments

Today, the numerical simulations play a key role in the solution of electromagnetic problem. The increase of computational power of modern computer allows us to simulate very complex problem. In the past much effort was expended to analytically manipulate solutions into forms which minimized the computational effort. It is now often more convenient to use computer to reduce the analytical effort. The method of moments [1,2] is a useful method to computing the solutions to fields problems, like radiation and scattering problem of objects with a generally shape. The solution of integral equation is related to the solution of a linear system and numerically to an inversion of matrix.

2.1 The core of method: inner product

Our aim is to solve a general integral equation, in the form of:

Lf =g (2.1)

where L is a continuous, linear operator such as the integral operators, f is the unknown to be determined (the response), and g represents a known excitation (the source). A problem of analysis involves the determination of f when g are specified, hence a problem of synthesis involves a determination of L when f and g are specified. We are interested to analysis problem. The solution of equation (2.1) is given by:

1

f

=

L g

− (2.2)

And if it exists, is unique.

L

−1 is the inverse operator and it is more difficult determinate it in practice, hence we have recourse to numerical solutions. The linear operator L maps the function f in its domain to a function g in its range. Generally, domain and space are diverse linear space. To proceed in the treatment, we introduce the notion for the inner product:

,

f g (2.3)

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________________________________________________________________________________ , , , , , *, 0 if 0 *, 0 if 0 f g f g f g w f w g w f f f f f f α β α β = + = + = = > ≠ (2.4)

where f g, and w are function, α and β are scalar and * denotes a complex conjugate. The first equation is called commutative propriety, the second and third equations define the bilinear propriety and the last two indicate that the scalar product is semi-define positive. The functions f g, are orthogonal if the inner product gives:

, 0

f g = (2.5)

If the inner product satisfies these proprieties, it can be used to define a natural norm: ,

f = f f (2.6)

and we define distance between two function as:

( , )

d f g = f −g (2.7)

A set of function

[ ]

φ

_n is indicate as a base if it is orthogonal

, 0

l m l m

φ φ

= ≠ (2.8)

and complete, namely if the zero function is the only function in the inner product space orthogonal to each member of set. A base can be used to portray any function in the inner product space, is then:

0

n n

n

f −

∑

α φ = (2.9)

where the

α

n are scalar coefficient determinate by:

, , n n n n f

φ

α

φ φ

= (2.10)

We are interested a finite-dimensional subspace of the inner product space, then we have to truncate the set of basis:

1 N N n n n f f

α φ

= ≅ =

∑

(2.11)

The scalar coefficient are decide on minimize the distance between the original function f and his clipping

f

N, therefore the approximation error

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Chapter 2: Method of Moments ________________________________________________________________________________ ________________________________________________________________________________ 29 ( , N) N d f f = f −f (2.12)

has to minimize. For this reason the coefficient

α

n are selected to make the approximation

error orthogonal to the base. This operation is called orthogonal projection:

, 0 1, 2,...,

n d n N

φ

= = (2.13)

The projection of the function f on a base can be execute also for the function g:

1 N N n n n g g

β ϕ

= ≅ =

∑

(2.14)

2.2 Method of Moments

The method of moment is a general procedure for solving linear equations. In the previous paragraph we showed in equation (2.11) how approximate a function in a projection on a finite base. The equation (2.1) is the replacement of equation (2.11) and we can obtain a system of linear equations forcing the residual:

1 1 N N n n n n n n L α φ g α φL g ε = =   − = − =   

∑



∑

(2.15)

to be orthogonal to a set of testing functions, or weighting functions,

[

ϕ ϕ

₁, ₂,...,

ϕ

_N

]

, that is

, 0

n

ϕ ε

= . The functions

{ }

φ

n are called basis function defined on the domain of L and the

scalar

{ }

α

n are unknowns coefficient to be determinate. The result obtained can be

summarized as: Lα =β (2.16) having entries:

[ ]

1, 1 1, 1 1, 2 2 2, 2, 1 2, 2 , , , mn n m n _m g L L g l L L g g ϕ α ϕ φ ϕ φ α ϕ ϕ φ ϕ φ α α _ϕ       _ _ _ _   _ _   =  = =_ _     _ _ _ _     _ _     … … ⋮ _⋮ ⋮ ⋮ ⋱ (2.17)

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________________________________________________________________________________

[ ]

1

[ ]

n l gm

α

_ − _

=   (2.18)

and the solution for f is given by equation (2.11)

The procedure described is given the name “weighted-residual method”. The “method of moments” is the name used generally in electromagnetic.

The basis and testing function used in practice are often not orthogonal sets. If the testing functions are orthogonal, the projection of the range space onto the testing function is orthogonal. Nevertheless the same rule not remains valid for the basis function. It became difficult predicts the convergence of the numerical approximation of equation (2.11) to exact solution when

N

→ ∞

.However, the result obtained from equation (2.11) is always an approximation, because

N

is a finite number for numerical calculation.

The selection of basis and testing functions is the most important issue arising within a method of moments implementation. There is several factor which affecting the choice of these functions, how the accuracy of the approximation solution, the size of the matrix that can be inverted and the relative complexity of the resulting matrix entries (well-conditioned matrix).

2.3 Basis Functions

The basis functions

φ

n chosen have to allow the reconstruction of original function f , that it

can have a general evolution. The basis function can be divided into two groups: sub-domain functions and entire domain function. The following examples concern mono dimensional discretization of dominion problems. For two-dimensional problem, the Rao Wilton Glisson (RWG) functions will be shown.

2.3.1 Sub-domain functions

These function are non-zero only over a part of the domain of original function f . These basis functions may be used without prior knowledge of the nature of the original function. The domain of interest is divided into intervals, or cells. We will show several examples of these functions. For single-dimension scalar quantities, the first and the simplest functions are the pulse or piecewise constants, shown in Figure 2.1 a). These are defined as:

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Chapter 2: Method of Moments ________________________________________________________________________________ ________________________________________________________________________________ 31

( )

1 1 , 0 otherwise. n n n x x x x φ  − ≤ ≤ =   (2.19)

Once the associated coefficients are determined, this function will produce a staircase representation of the unknown function, similar to that in Figure 2.1 b).

A different common basis set is the triangle functions, displayed in Figure 2.1 c). These are defined by: 1 1 1 1 1 1 ( ) 0 elsewhere n n n n n n n n n n n x x x x x x x x x x x x x x x φ − − − + + + −  ≤ ≤  ₋   − = ≤ ≤ −     (2.20)

The resulting representation (Figure 2.1 d) is smoother than that of the pulse constant, but at the cost of increased computational complexity.

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________________________________________________________________________________ Other common bases are piecewise sinusoidal or truncated cosine, shown in Figure 2.1e).

(

)

(

)

(

)

(

)

1 1 1 1 1 1 sin sin sin ( ) sin 0 otherwise n n n n n n n n n n n x x x x x x x x x x x x x x x β β β φ β − − − + + +  _ ₋ _ ≤ ≤  −       −      = ≤ ≤ −          (2.21)

In the equation (2.21) the parameter β regulates the slope of the basis function in the sub-domain. It highlights that the generic function

φ

_n

( )

x

not have continue derivative in

x

n.

2.3.2 High level functions

The high level functions are basis functions defined on macro domains (several wavelengths size) that include a relatively large number of conventional sub-domains. The functions are definite on a triangular or rectangular patches, that dividing the objects. This type of basis functions leads to a significant reduction in the number of unknowns. Differently entire domain basis functions, high level bases are indicate in modelling arbitrary or complicated unknown functions.

2.4 RWG Functions

For three dimensional problem, the simulating of surface can be made whit a wire-grid model, using a thick mesh to follow better the variation of the geometry. Results more accurate can be obtained using a continuous surface. The first step is divide the surface in triangular o polygonal patches. In Figure 2.2 a triangular mesh of a naval vessel model is depicted. It is important to note that the conventional method of moments requires a λ 10 or λ 20 discretization. This means that the sides of triangular patches cannot be longer than λ 10.The second one is employ the suitable basis functions. In the 1982 Rao, Wilton and Glisson [8] have presented a set of basis function, today widely used. They chose a triangular patches model, for a number of reason:

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Chapter

________________________________________________________________________________

________________________________________________________________________________ • triangular patches are capable of accurately conforming to the g

surface or boundary;

• the patches scheme is easy to pass to computer;

• the resolution can be modified varying the patch density.

Figure 2.2

Each basis function

φ

_n

( )

r

is associated on an interior model and defined non-zero only

that edge. Referring to Figure 2.3

plus or minus designation is determined by the choice of a positive current reference direction for the th

n edge. This reference is assumed to be from basis function associated with the

n n n

φ ρ

where

l

nis the length of the edge,

a generic point of T_n± and a free vertex of

Chapter 2: Method of Moments

________________________________________________________________________________

________________________________________________________________________________ triangular patches are capable of accurately conforming to the g

surface or boundary;

the patches scheme is easy to pass to computer;

the resolution can be modified varying the patch density.

A triangular mesh of a naval vessel model

is associated on an interior edge , not boundary edge

ly on this edge, that is the joint of two triangles, which have 3, we can define these two triangles as T_n+ and

plus or minus designation is determined by the choice of a positive current reference direction reference is assumed to be from Tn

+

to Tn

−

. Now, we can define t basis function associated with the th

n edge, as: in 2 ( ) in 2 0 otherwise n n n n n n n n n l r T A l r r T A ρ φ ρ + + + − − −     =     

is the length of the edge, A_n± is the area of triangle T_n±, and

ρ

_n±

is the distance between and a free vertex of T_n±.

________________________________________________________________________________

________________________________________________________________________________ triangular patches are capable of accurately conforming to the geometrical

edge, of the patch joint of two triangles, which have and T_n−, where the plus or minus designation is determined by the choice of a positive current reference direction Now, we can define the vector

(2.22)

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________________________________________________________________________________

Figure 2.3 Geometry of triangular patch

The basis function

φ

_n

( )

r

is used to approximately represent the surface current. These functions have some properties, which make it exceptionally suited for this role.

• The current has no component normal to the boundary of the surface formed by the triangles Tn

±

, and hence no line charges exist along this boundary.

• The component of current normal to the nth edge is constant and continuous across the edge. Moreover, considered the above point, all edges of both triangles are free of line charges.

• The surface divergence of

φ

_n

( )

r

, which is proportional to the surface charge density associated with the basis element, is:

in , ( ) in , 0 otherwise . n n n n S n n n l r T A l r r T A φ + + − −     ∇ ⋅ =      (2.23)

The charge density is constant in each triangle.

• The moment of

φ

_n

( )

r

is given by:

(

)

( ) 2 n n c c n n n n n n n T T l A A φ φ dS ρ ρ + − + − + − + + =

∫

= + (2.24) where

ρ

nc ±

is the vector between the free vertex and the centroid of Tn

±

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________________________________________________________________________________

________________________________________________________________________________ 35

2.5 Testing Functions

It has been shown in the paragraph 2.2 that the scalar coefficients

{ }

α

n to be determinate. The

equation (2.15) leads to an equation with N unknowns, which cannot be resolved. It is necessary to have N linearly independent equations. Previously, it was introduced inner product, which provides independent equation to solve the problem. In Figure 2.4 a series of possible choices for the testing function are shown. A widely used procedure is the Galerkin’s procedure. In the Galerkin’s method, the testing functions are chosen equal to the basis functions, that is,

ϕ

n

=

φ

n. MoM codes based on Galerkin’s method are generally more

accurate and rapidly convergent. It has been noted in the previous section that each term of matrix

[ ]

l (equation 2.18) requires two or more integrations, at least one to evaluate each

( )

n

L f , and one to perform the inner product. To carry out these integrations, a large amount of computation time can be required. A set of testing functions that reduce the number of required integrations is the set of Dirac delta weighting (testing) functions.

Figure 2.4 Different choices of testing and weighting functions in the MoM

The use of Dirac delta for both basis and testing functions is called point matching. Hence, the basic functions and the testing functions are equivalent in discrete point. Care should be taken to the point of intersection, because it must fall at a point where the function is not differentiable, giving some errors.

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________________________________________________________________________________

2.6 Solution of Integral Equation: EFIE for a Perfect Conductor

In general, the resolution of the EFIE (Electric Field Integral Equation) in the frequency domain through the method of moments allows to determine the electric field re-radiated from a perfectly conducting surface using the determination of the electric current density J induced by incident field. Once we know we can go back to the fields J re-radiated through integral relations.

We consider a surface

S

that delimits an perfectly conducting object (scatterer), open or closed. It is denoted the outward normal

n

ɵ

to this surface. Exists an electric field

E

inc in the space around the object, which strikes the surface, inducing a surface current J ( obviously this is not the current density M). If

S

is open, we regard J as the vector sum of the surface currents on the opposite sides of

S

(theorem of image). Moreover, we can define the electric scattering field, with the mixed potential formalism of equations 1.17, as

scat

E = −jωµA− ∇Φ (2.25)

where A and

Φ

are the magnetic vector potential and scalar potential, respectively, introduced in the first chapter, and defined as:

( )

( ) 4 1 4 jkR S jkR S S e A r J dS R e r dS R µ π ρ πε − − ′ = ′ Φ =

∫

(2.26)

where R= −r r′ is the distance between an arbitrarily located observation point r and a source point r′on

S

and the surface charge density

ρ

S is connected to the surface divergence

of J by the equation of continuity

S J j

ωρ

S

∇ ⋅ = − (2.27)

Following the procedure discussed in the first chapter, an integral-differential equation for J can be derived by enforcing the boundary condition nɵ×

(

Einc+Escat

)

= on 0

S

_{, obtaining}

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(

)

tan tan on inc E j A

ω

r S − = − − ∇Φ (2.28)

To solve the equation (2.29), the surface

S

of the conducting object is first discretized into triangular patches and then RWG basis functions

φ

n are used to approximate the current on

S

as follows: 1 N n n n J I

φ

= =

∑

(2.29)

where

N

is the total number of basis functions,

I

n are the unknowns coefficients to be

determined. Replacing the current expansion of equation 2.30 into equation 2.29 and then employing the testing procedures, we obtain an

N

×

N

system of linear equations, which may be written in matrix form:

[

Zmn

][ ] [ ]

In = Vm (2.30)

where

[

Zmn

]

is an

N

×

N

matrix and

[ ]

In and

[ ]

Vm are column vectors of length

N

.

Typically,

[ ]

In and

[ ]

Vm are referred to as generalized current and voltages vectors,

respectively, and

[

Zmn

]

is referred as a generalized impedance matrix. This matrix has a

physical meaning only if an EFIE formulation in the frequency domain is used. The physical interpretation of elements of matrix

[

Zmn

]

is:

• element

Z

mn, where

n m

≠

, represents the effect of cell m on cell n, which is called

coupling;

• element

Z

nn represents the self-term. It presents some problems evaluation due to the

singularity of Green’s function when the source point coincides with the observation point. In this case special arrangements are used.

The elements of

[

Zmn

]

and

[ ]

Vm are then given by:

2 2 2 2 c c m m mn m mn mn mn mn c c m m m m m m Z l j A A V l E E ρ ρ ω ρ ρ + − + − + − + − + −     =   ⋅ + ⋅ + Φ − Φ        =  ⋅ + ⋅    (2.31) where:

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________________________________________________________________________________

( )

' 4 1 4 m m jkR mn n m S jkR mn S n m S c m m e A r dS R e r dS j R R r r µ φ π φ π ωε ± ± − ± ± − + ± ± ± ′ = ′ ′ Φ = − ∇ ⋅ ′ = −

∫

(2.32)

To simplify the complexity of mathematical operation, we can approximate the averages of the scalar potential

Φ

, the vector potential A, and the incident field

E

inc over each triangle by their values at the centroids of the triangle. In this way, we can avoid the surface integrals of the potentials and reduce the double integrals over the surface to a single one. The procedure to solve via MoM the electric field integral equation expressed in a frequency domain is been shown. It is important to underline that a solution in the frequency domain obtained via factorization or inversion for a given frequency is independent of the exciting source geometry. Thus the induced currents can be obtained for any illuminating field from the product of the inverse matrix and the source vector. The time-domain formulation is, on the other hand, source geometry dependent, but can cover a broad frequency range.

2.7 Solution of Integral Equation: EFIE for an Imperfect Conductor

In the presence of a not perfect conductor object, the formulation of previous section is not longer valid. The application of the impedance boundary condition (IBC) is effective in solving electromagnetic problems involving penetrable bodies.

The IBC was first introduced by Leontovich [9] in an effort to solve the problems of dealing with the propagation of radio waves over the hearth. He showed that on the surface of a nearly perfect conductor, a scatterer with a small skin depth, the IBC boundary condition relates the tangential components of the electric to the magnetic surface fields via a surface impedance

η

s

defined by the electromagnetic properties of the scatterer. In formula:

(

)

ˆ ˆ _sˆ

E−n E n⋅ =η n H× (2.33)

Since the approximate boundary condition relates only the fields outside the scatterer, the scattered fields can be evaluated without involvement of the internal fields. The IBC represents a good approximation in such cases where the magnitude of the complex refractive index of

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________________________________________________________________________________

________________________________________________________________________________ 39

the scatterer materials is much greater than unity and if the curvature radius of the scatterer is sufficiently large.

In this section we include the impedance boundary condition in the EFIE and so extend the RWG MoM to this class of objects. The IBC has been used to model high-conductivity scatterers, absorbing coatings, plasma coatings, corrugated surfaces, rough surface and other configurations. Now, we consider a medium with permeability

µ

s and permittivity

ε

s. The

IBC can be applied if the refraction coefficient of medium, defined as N_s= ε µ_s _s , satisfies the followings conditions:

{ }

(

min

)

10 Im 2.3 / s s N N ka  ≥   ≥  (2.34)

where k is the propagation constant, and

a

_min is the minimum curvature radius of the scattering body surface. On the surface of the body, using the IBC the surface magnetic current density

M is given by:

ɵ

s

M =η J× n (2.35)

where J is the electric surface current density,

η

S is the impedance of the medium and

ɵ

n

is the unit normal to the body surface.

The integral equation can be obtained by expressing the scattering electric field in terms of magnetic vector potential F, electric vector potential A and electric scalar potential

Φ

:

1

s

E j A

ω

F

ε

= − − ∇Φ − ∇ × (2.36)

and imposing that the total electric field

E

tot

=

E

inc

+

E

scatt is equal to M =Etot×nɵ on the body surface. The integral equation is:

{

}

{

}

ˆ 1 _ˆ ( ) scatt S M n j n

J

E

A

F

η

ω

ε

− × = − − ∇Φ − ∇ × × (2.37)

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________________________________________________________________________________ 1 , , , , , i m m m m S m M E

φ

j

ω

A

φ

F

φ

η

J

φ

ε

< >= < > + < ∇Φ > + < ∇ × > + < > (2.38) Substitution of the electric current expansion

( )

1 N n n n J I

φ

r =

=

∑

into equation (2.39) yields an

N

×

N

system of linear equations which can be written in a matrix form as

ZI

=

V

.

The integration of vectorial terms can be an very complicated operation. By integrating the function with respect to the centroid of the triangle we obtain a correct approximation:

1 1 , 2 m 2 m i i i m m m m T T l dS dS F F F E E E A A A A A φ ρ ρ + − + − + − = ⋅ + ⋅ ∇ × ∇ × ∇ ×                   _     _                              

∫

(2.39)

In the same way, the scalar potentials can be evaluated at the triangle centroid:

1 1 , ( ) ( ) ( ) m m m S m m m m S T T c c m m m dS l dS dS A A l r r φ φ + − + − + − < ∇Φ >= − Φ∇ ⋅ = − Φ − Φ ≅   − _Φ − Φ _

∫

∫∫

(2.40)

At last, we obtain a

N N

×

system of linear equations:

1 1 1 1 1 1 1 2 2 , 2 2 2 2 c c N N N N m m mn mn m n n m n mn n mn n n n n c c N N N m m m mn mn n n S n n m n n n M c c m m m m m j l I A I A l I I l I F I F I l E E

ρ

ω

ρ

η

φ φ

ε

ρ

+ − + − ₊ ₋ = = = = + − + − = = = + − + −     ⋅ + ⋅ − Φ − Φ +           + _∇ × ⋅ + ∇ × ⋅ _+ < >=     ⋅ + ⋅    

∑

(2.41) where:

( )

-' ' 4 m n n jkR M mn _n m T T e A r dS R µ π

φ

± + ± ± ∪ =

∫∫

(2.42)

( )

-1 1 ' ' 4 m n n jkR mn _n M T T m e r dS jω πε

φ

R ± + ± ± ∪ Φ = −

∫∫

∇ ⋅ (2.43)

( )

-' ' ' 4 m n n jkR M mn _n _S m T T e F r r dS R

n

ε η π

φ

± + ± ± ∪ =

∫∫

× (2.44)

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Chapter ________________________________________________________________________________ ________________________________________________________________________________ m m

V

l

E

mn m mn mn m m m M Z l j A A l F F ω ε   =  ⋅ + ⋅ − Φ − Φ +     + ∇ × ⋅ + ∇ × ⋅ + < >  

2.8 Solution of Integral Equation: EFIE for a Wires and Wire

Junctions

In the previous sections the way moments for metal surfaces has

radiation problems of a thin wire attached to an arbitrarily shaped surface, it is necessary to derive integral equations that take account for the mutual coupling between wires and surface. Special vector bases are developed to

junction, the Method of Moment (MoM) is applied to obtain numerical values of the induced current on the investigated electromagnetic problem.

Figure

________________________________________________________________________________ ________________________________________________________________________________

[ ] [

Vm = Zmn

][ ]

In

2

c c m m m m m m

V

l

E

ρ

E

ρ

+ − + −





=

_

⋅

+

⋅

_





(

)

2 2 , 2 2 c c m m mn mn mn m mn mn c c m m m mn mn _S _n _m Z l j A A F F ρ ρ ω ρ ρ η φ φ + − + − ₊ ₋ + − + −     =   ⋅ + ⋅ − Φ − Φ +       + ∇ × ⋅ + ∇ × ⋅ + < >  

Solution of Integral Equation: EFIE for a Wires and Wire

the way to solve the integral equation (EFIE) with the method of s been shown. However, to treat the scattering and/or the radiation problems of a thin wire attached to an arbitrarily shaped surface, it is necessary to derive integral equations that take account for the mutual coupling between wires and surface.

pecial vector bases are developed to describe the current on the wire, the surface, and the junction, the Method of Moment (MoM) is applied to obtain numerical values of the induced current on the investigated electromagnetic problem.

Figure 2.5 PEC plate with a wire attached

________________________________________________________________________________

________________________________________________________________________________ (2.45)

(2.46)

(2.47)

Solution of Integral Equation: EFIE for a Wires and Wire-Surface

with the method of the scattering and/or the radiation problems of a thin wire attached to an arbitrarily shaped surface, it is necessary to derive integral equations that take account for the mutual coupling between wires and surface. describe the current on the wire, the surface, and the junction, the Method of Moment (MoM) is applied to obtain numerical values of the induced

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________________________________________________________________________________ Figure 2.5 shows the meshed structure of the investigated problem. The surrounding media is characterized by (ε,µ). In order to obtain the desired integral equations, the wire is supposed to be thin with respect to its total length and the working wavelength so that the thin-wire approximation can be used and any variation in the circumferential current distribution can be neglected. Moreover, the wire radius

a

may vary smoothly or discontinuously. Both the wire and the surface are made up of perfect conductor material (PEC).

To obtain the induced current over the surface we can create a triangular mesh of the surface and we can use the RWG basis functions (section 2.4) for the triangular patches. Then, we can divide the wire into N_wire straight linear segments and definite the vectors r_n ,with

0,1,...

_wire

n

=

N

, as the vector joining a local landmark

O

and end points of each linear segments along the wire axis, as shown in Figure 2.6. The same end point can be described in terms of sn , which is defined with respect to the local frame along the wire axis. In the local

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Chapter

________________________________________________________________________________

Figure

Figure 2.7 depicts the junction between the wire and the surface in node this node and the centroid of the triangle

Figure

________________________________________________________________________________

Figure 2.6 Segmentation of wire

s the junction between the wire and the surface in node i. The vector joining this node and the centroid of the triangle T_ni is defined as

ρ

n.

Figure 2.7 Wire- surface junction

________________________________________________________________________________

________________________________________________________________________________ . The vector joining

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________________________________________________________________________________

The tangential component of electric field have to vanish on PEC surface, to respect the boundary condition. For the wire, the boundary conditions are obtained forcing the tangential component of electric field to zero along the wire axis. By expressing the scattered electric fields in terms of magnetic vector potential A r

( )

and scalar potentialΦ

( )

r , we have:

(

)

(

)

tan

tan on the surface

on the wire ˆ ˆ inc inc E j A E s j A s ω ω = + ∇Φ ⋅ = + ∇Φ ⋅ (2.48) where

( )

' ' ' ( ) ( ) A = ' ' 4 4 2 1 1 1 =- '- ' j4 j4 2 ' jkR jkR S W jkR jkR S W s s e I e r J dS dS R a R e dI e r J dS dS R a ds R µ µ π π π πωε πωε π − − − − + Φ ∇ ⋅

∫

(2.49)

In equation (2.49), a s is the radius of the wire that can change along the wire axis.

( )

To represent the current induced on the wire-surface geometry, a set of basis function φ_n was introduced, as illustrated in equation (2.29). We can individuate three induced currents, the first on the body surface, the second on the wire and the last one at the junction, which will be explain below.

1) The surface current distribution ( s

J ) on the body can be expressed a superposition of the RWG basis functions as following:

( )

1 RWG surf N s s n n n J α φ r = =

∑

(2.50)

2) The linear current density _Iw

on the wire is expressed as:

( )

1 wire N w w n n n I β p s = =

∑

(2.51) where:

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( )

1 1 2 2 otherwise 1 0 n n n r r r p r =  − ≤ ≤ +  (2.52) or equivalent:

( )

1 1 2 2 otherwise 1 0 n n n s r s p s =  − ≤ ≤ +  (2.53) The points 1 2 n r± or 1 2 n

s± indicate the midpoints of wire segment in the global frame and local

frame, respectively. The derivative of the wire current also enters into the integral equation as the charge appearing in the scalar potential. According to the current continuity, the linear charge density is:

1

w dI

q

jω ds

= − (2.54)

Application of (2.54) directly to (2.51) would lead to a charge distribution made of a series of point charges. This is a not suitable approximation for numerical purpose, so a finite difference is used for calculating the charge:

( )

1 1 2 1 1 1 Nwire w w w n n n n n n q p s j s s β β ω − − = − − = − −

∑

(2.55) Where 0 wire 0 w w N

β

=

β

= , and

( )

1 1 2 otherwise 1 0 n n n s r s p s + − ≤ ≤  =   (2.56) n

s

represents the segment endpoint in the local frame. The moment of the

p

n basis function is:

( )

(

) (

)

1 2 1 2 1 2 1 2 1 1 2 2 ' ' ' ' ' ' n n n n n n n n n n n s n n s T T s s n n s s n r dS p s dS p s dS p s dS r r r r φ + + − − + − − + ∪ ≅ = + = − + −

∫∫

∫

(2.57)

3) For the wire-surface junction, we will use a local coordinate system ς ψ ζ . We assume , ,

that the coordinates of junction vertex are ζ =1, ξ =η=0. Referring to Figure 2.7, we define a vector basis function associated with the th

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________________________________________________________________________________

( )

i n f r p s s s  − ∈  =    where :

and

α

n is the angle between the two edges of

shown in Figure 2.7. The surface divergence is:

(

i n

f r

∇ ⋅ =

The moment of the wire-surface junction basis function is:

( )

' ' ' i n n n i i i n n n T T T r dS f r dS f r ds r r r r φ + − ∪ = + = − + −

∫∫

∫

where r , ₀ 1 2 r , rcm are explained Figure 2. ________________________________________________________________________________

(

)

( )

1 2 2 i n 2 n 0 1 1 1 r T on the wire n n p s s s ς ρ α ρ   ₋ ₋    − ∈     ɵ

( )

1 2 0 0 otherwise 1 0 s s s p s =  ≤ ≤ 

is the angle between the two edges of Tni in common with the junction

The surface divergence is:

( )

(

)

( )

2 2 1 2 0 1 1 2 1 on the wire i n n n r T f r p s s s s

ς

α

ρ

 ₋ ∈   ∇ ⋅ =    ₋ 

surface junction basis function is:

( )

(

)

₍

₎

1 2 1 2 0 0 0 ' ' ' i n n n s c i i i m n n n s T T T r dS = f r dS + f r ds = r −r + r −r

∫∫

∫

are explained below.

.8 Local system of wire-surface junction

________________________________________________________________________________

(2.58)

(2.59)

the junction node, as

(2.60)

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________________________________________________________________________________

________________________________________________________________________________ 47

Now, we can write the total surface current and the total axial current on the wire, respectively, as:

( )

1 1 0 1 1 r on Surface s on Wire Surf Junct Wire Junct N N S _S _J _i n n n n n n N N W _W _J n n n n n J r r I f r I s p s p s I α φ β = = = = = + = +

∑

(2.62)

Where the superscript

J

stands for wire-surface junction, the superscript

S

indicate the surface, hence W indicate the wire. N_Surf,

N

Wire, and

N

Junct represent the number of unknowns

on the surface, the wire and the wire-surface junction (triangular patch that shared the junction node).

As seen in equation (2.30), the application of the MoM to the integral equation can be summarized in the form:

[

Zmn

][ ] [ ]

In = Vm (2.63)

In this case, the evaluation of

V

m elements require the integration of the electric field over each

triangle of the surface and on every wire segment. However, as explained in the previous paragraphs, it is possible to avoid testing integrations by approximating the electric field in the centroids of each triangle and in the midpoints of each wire segments, multiplying for the moment of the appropriate basis function

m

φ

. Some examples of

V

m element testing procedure

are given below:

1) Incident Plane Wave:

( )

(

)(

)

(

)(

)

( )(

)

( )(

)

1 2 1 2 1 2 1 2 0 1 2 1 2 0 on surface patch 2 2 on wire segment on junction c m c inc c inc c S m m m m m inc inc W m m m m m m m inc c c inc J m m m V l E r E r V E r r r E r r r V E r r r E r r r ρ + ρ − + − − − + +   =  +    = − + − = − + − (2.64) 2) Delta gap:

The delta gap can be placed in a wire or in a wire-surface junction. In both cases, we need just to modify the element of V that belongs to the unknown where the delta gap is placed. Therefore, the linear system has the following structure:

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________________________________________________________________________________ SS SW SJ S S WS WW WJ W W JS JS JJ J J Z Z S I V Z Z Z I V Z Z Z I V _ _ _ _ _ _ _ _ _ _       _ _{ }   _{ } _ _ _₌_ _       _ _ _ _ _ _ _ _ _ _       Eq (2.1) XY Z  

  is a NX ×NY sub-matrix obtained by testing X with Y , where X and Y can represent

observation point on: surface (

S

), wire (

W

) or junction (

J

). The first subscript indicate the testing point, while the second one the source point.

Elements on the diagonal of the impedance matrix can be evaluated as following:

2 2 c c SS SS m SS m SS SS mn m mn mn m mn mn Z j l

ω

A

ρ

A

ρ

l + − + − − +     =  + + _Φ − Φ _     (2.65)

(

1 2

)

(

1 2

)

1 2, 1 2, 1 2, 1 2, WW WW WW WW WW m m m m mn m n m n m n m n Z = jωA − r −r − +A + r + −r + Φ + − Φ − (2.66)

(

0

)

1 2,

(

1 2 0

)

1 2, c JJ JJ JJ JJ JJ m mn mn n n mn Z = j

ω

_A r −r +A r −r _+ Φ − Φ (2.67) where: ( ) ; ( ) S c c SS SS S n m m mn mn n A ±=A r ± Φ ±= Φ r± (2.68) 1 2 1 2 1 2, ( ) ; 1 2, ( ) W WW WW W n m m m n m n n A ± = A r ± Φ ± = Φ r ± (2.69) 1 2 1 2 1 2, 1 2, ( ) ; ( ) ; ( ) ; ( ) J c J c JJ JJ JJ J JJ J n m n m mn n mn n n n A =A r A =A r Φ = Φ r Φ = Φ r (2.70) and r , 0 c m

r , r1 2 are shown in Figure 2.8.

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Chapter 2: Method of Moments ________________________________________________________________________________ ________________________________________________________________________________ 49 2 2 c c SW SW m SW m SW SW mn m mn mn m mn mn Z j l

ω

A

ρ

A

ρ

l + − + − − +     =  + + _Φ − Φ _     (2.71) 2 2 c c SJ SJ m SJ m SJ SJ mn m mn mn m mn mn Z j l

ω

A

ρ

A

ρ

l + − + − − +     =  + + _Φ − Φ _     (2.72)

(

1 2

)

(

1 2

)

1 2, 1 2, 1 2, 1 2, WS WS WS WS WS m m m m mn m n m n m n m n Z = jωA − r −r − +A + r + −r + Φ + − Φ − (2.73)

(

1 2

)

(

1 2

)

1 2, 1 2, 1 2, 1 2, WJ WJ WJ WJ WJ m m m m mn m n m n m n m n Z = jωA − r −r − +A + r + −r + Φ + − Φ − (2.74)

(

0

)

1 2,

(

1 2 0

)

1 2, c JS JS JS JS JS m mn mn n n mn Z = j

ω

_A r −r +A r −r _+ Φ − Φ (2.75)

(

0

)

1 2,

(

1 2 0

)

1 2, c JW JW JW JW JW m mn mn n n mn Z = j

ω

_A r −r +A r −r _+ Φ − Φ (2.76) where ( ) ; ( ) W c c SW SW W n m m mn mn n A ± =A r ± Φ ± = Φ r ± (2.77) ( ) ; ( ) J c c SJ SJ J n m m mn mn n A ±=A r ± Φ ±= Φ r ± (2.78) 1 2 1 2 1 2, ( ) ; 1 2, ( ) S WS WS S n m m m n m n n A _± = A r _± Φ _± = Φ r _± (2.79) 1 2 1 2 1 2, ( ) ; 1 2, ( ) J WJ WJ J n m m m n m n n A ± =A r ± Φ ± = Φ r ± (2.80) 1 2 1 2 1 2, 1 2, ( ) ; ( ) ; ( ) ; ( ) S c S c JS JS JS S JS S n m n m mn n mn n n n A =A r A = A r Φ = Φ r Φ = Φ r (2.81)

( )

1 2

( )

1 2 1 2, 1 2, ( ) ; ; ( ) ; W c W c JW JW JW W JW W n m n m mn n mn n n n A =A r A =A r Φ = Φ r Φ = Φ r (2.82)

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________________________________________________________________________________

( )

1 2 1 2 1 0 0 0 ' ' 4 1 ' ' 4 ' ' ' ' 4 1 ' 4 B B n n n n n n jkR S n n S jkR S n s n S s s _jkR _jkR W n n n s s s _jkR W n n s e A r r dS R e r r dS j R e e A r p s ds p s ds R R e r p s j R µ φ π φ π ωε µ π π ωε + − − − − − − − = Φ = − ∇ ⋅     = +     Φ = −

∫∫

∫

( )

1 0 ' ' ' ' ' 4 1 ' ' 4 n n i n i n s _jkR n s jkR J i n n T jkR J i n s n T e ds p s ds R e A r f r dS R e r f r dS j R µ π π ωε + − − −   +       = Φ = − ∇ ⋅

∫

∫∫

(2.83)

2.9 Numerical Simulations

To prove the correctness of the method implementation, it will shown the monostatic or bistatic radar cross section of various metal objects, compared with commercial software, when possible.

2.9.1 PEC boat

The first example is a simply PEC boat. Its dimensions are: 10 meters along x axis, 3.15 meters along y and 3.11 meters along z. Below, the monostatic RCS for a frequency range of 5-30 MHz is shown. The greatest dimension changes from 0.167

λ

(in air) at 5MHz to 0.311

λ

to 30MHz. At these frequencies this object is very small. The mesh is made at

λ

/10

at the highest frequency, that is at 0.1 meter. In Figure 2.9 the mesh of the geometry of the boat is depicted, hence in Figure 2.10 we show the monostatic RCS for different elevations and azimuths. The results are compared with FEKO and we can note that the superposition is really good.

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________________________________________________________________________________

________________________________________________________________________________ 51

Figure 2.9 Mesh of the simply boat model

a)

b)

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________________________________________________________________________________

2.9.2 PEC vessel

Now, the bistatic RCS of a PEC vessel model will be presented. The maximum dimensions are: 46.35 meters along x axis, 7.9 meters along y and 10 meters along z. In Figure 2.11 the geometry of the vessel is displayed, while in Figure 2.11 b) its mesh at 10 MHz.

a)

b)

Figure 2.11 Model of vessel: a) geometry; b) triangular mesh

Below, in Figure 2.12 the bistatic RCS in decibel for square meter (dBsm) as

θ

change, from 0° to 90° , for φ =90° is depicted. The frequencies are 5MHz a) and 10 MHz b). Also in this case the results are compared with FEKO and the results are satisfying.

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Chapter 2: Method of Moments ________________________________________________________________________________ ________________________________________________________________________________ 53 a) b)

Figure 2.12 Bistatic RCS of PEC vessel for φ=90° at: a) 5 MHz; b) 10 MHz

2.9.3 Monopole on PEC boat

On the boat of the paragraph 2.9.3 a monopole of 1.5 meters long is inserted, as shown in Figure 2.13. We have chosen a diameter of 0.02 meters for the wire that simulates the antenna.

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________________________________________________________________________________

Figure 2.13 Monopole antenna on a simply PEC boat

The operating frequency is 100 MHz and the excitation is 1 volt. The comparison of the gain, in decibel, obtained with our code (CBMoM) and with FEKO is shown in Figure 2.14.

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________________________________________________________________________________

________________________________________________________________________________ 55

b)

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