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Electromagnetic Scattering by Surfaces of Arbitrary Shape

By: Sadasiva M. Rao, Donald R. Wiltson & Allen W. Glisson

Presented By: Brian Cordill

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Overview

Method of Moment Summary Paper Presentation

Electric Field Integral Equation (EFIE) Triangle Subsectioning Basis Function Testing Procedure Matrix Element Calculation Efficiency Numeric Examples

Summary & Paper Evaluation

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Method of MomentsIntegral Equation

Sub-sectionalize the geometry

Choose Basis Functions

n nf fα=∑

Lf g=n nLf gα =∑

Choose Testing functions

, ,m n n mw Lf w gα⟨ ⟩ =⟨ ⟩∑Create matrix equation

[ ][ ] [ ]mn n ml gα =

Diagonal Terms

nnlOff-Diagonal

Termsmnl

Inverse Matrix[ ] [ ] [ ]1

n mn ml gα−

=Use to determine

parameters of interest

nα

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General Formulation Lf=g

Operator L operates on unknown quantity f yielding known quantity g

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Approximate f

Approximate f by an infinite summation of weighted basis functions fn

Can either use: Whole domain basis functions Subdivided basis functions

€

f = α n fn

n

∑

€

αnL fn( )n

∑ = g

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Testing Procedure

The testing procedure uses symmetric products to produce N equations and N unknowns from 1 equation and N unknowns.

Matrix I can now be solved by inverting lmn€

αn wm,L fn( )n

∑ = wm,g

€

l mn[ ] α n[ ] = gm[ ]

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Problem Statement

Is it possible to create a method to apply MoM to any arbitrary scattering surface?

Will need unified: Operator Basis Functions Testing Functions

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Problem Statement II

Sadasiva, Donald and Allen say yes!

Stipulate by using EFIE and triangular subsectioning they can develop an appropriate basis and testing function.

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Operator :Electric Field Integral Equation

€

E s = − jωA −∇Φ

€

A r r ( ) =

μ

4πJ

e− jkR

Rd ′ s

S

∫

€

Φ r

r ( ) =1

4πεσ

e− jkR

Rd ′ s

S

∫

€

∇s • J = − jωσ

€

n × E i + E s( ) = 0

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Basis Functions : Triangle Subsectioning Defined over two

subsections that share a common edge.

Current flows along p+ across ln and down p-

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Basis Function II Current has no

component normal to two of the boundaries.

Why is this important? Want to avoid

discontinuous current. Impresses line charge to

be present in the solution that is not present reality

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Basis Function III Component of current

normal to the nth edges constant and continuous across the edge.

Flux density normal to the edge is unit. No discontinuity of

current

€

An =l nNc

2

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Basis Function IV

Since:

€

±1

ρ n±

∂ ρ n± fn( )

∂ρ n±

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Test Procedure Reuse basis function

as testing function, i.e. Galerkin’s method

Take N symmetric products

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Test Procedure II

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Matrix Equation

€

ZI = V

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Efficient Numerical Evaluation of Matrix Elements

Each Zmn requires the evaluation of up to 8 integrals.

Tn+

Tn-

Tm-

Tm+

€

Amn+ =

μ

4πfn

e− jkRm+

Rm+

dsTn

+

∫ + fn

e− jkRm+

Rm+

dsTn

−

∫ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

Amn− =

μ

4πfn

e− jkRm−

Rm−

dsTn

+

∫ + fn

e− jkRm−

Rm−

dsTn

−

∫ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

Φmn+ =

1

4πjωε∇ s • fn

e− jkRm+

Rm+

dsTn

+

∫ + ∇ s • fn

e− jkRm+

Rm+

dsTn

−

∫ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

Φmn− =

1

4πjωε∇ s • fn

e− jkRm−

Rm−

dsTn

+

∫ + ∇ s • fn

e− jkRm−

Rm−

dsTn

−

∫ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

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Efficient Numerical Evaluation of Matrix Elements II Integrals are

repeated for adjacent elements.

Tn+

Tn-

Tm-

Tm+

To+

€

Amn+ =

μ

4πfn

e− jkRm+

Rm+

dsTo

+

∫ + fn

e− jkRm+

Rm+

dsTn

−

∫ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

Amn− =

μ

4πfn

e− jkRm−

Rm−

dsTo

+

∫ + fn

e− jkRm−

Rm−

dsTn

−

∫ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

Φmn+ =

1

4πjωε∇ s • fn

e− jkRm+

Rm+

dsTo

+

∫ + ∇ s • fn

e− jkRm+

Rm+

dsTn

−

∫ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

Φmn− =

1

4πjωε∇ s • fn

e− jkRm−

Rm−

dsTo

+

∫ + ∇ s • fn

e− jkRm−

Rm−

dsTn

−

∫ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

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Efficient Numerical Evaluation of Matrix Elements III Efficiency can be

gained by evaluating face-pair combinations rather then edge-pair combination

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Efficient Numerical Evaluation of Matrix Elements IV

A1, A2 and A3 are interrelated and we can define a normalized area coordinates:

Note that p1, p2, and p3 subdivide triangle q into 3 sub-triangles, areas A1, A2 and A3.

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Efficient Numerical Evaluation of Matrix Elements V Can now use the

normalized area coordinates to recast the integrations

where

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Numerical Results 4 Representative

Examples Square Plate Bent Plate Circular Disk Sphere

Plate and disk examples evaluate the EFIE approach when edges are present.

Circular disk presents an example with curved edges.

Sphere exemplifies both a closed surface and a doubly curved surface.

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Numerical Results : Square Plate Figure 5 : Current on a 0.15

plate

Figure 6 : Current on a 1.0 plate

Results match closely to other numerical techniques.

Note : The number of patches for each result is not representative of convergence rate.

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Numerical Results : Square Plate II Figure 7 : RCS of flat

plate

Not surprisingly give that the current matched well the so does the RCS.

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Numerical Results : Bent Plate Figure 8 : Current on

a plate bent 50º away from the incident wave.

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Numerical Results : Circular Disk Figure 9 : Current on

a Circular Disk

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Numerical Results : Sphere Figure 10 : Current

on a 0.2 Sphere

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Summary

Developed a method of applying MoM to an arbitrary scattering body. Utilized EFIE to handle both open and closed

bodies. Developed a basis function that insures

current continuity. Paid close attention to evaluating the

moment matrix efficiently.

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Paper Evaluation Very clear if one has a strong working

knowledge of MoM. Fell short in a few areas:

Did not address evaluating the diagonal terms of the moment matrix.

Did not address EFIE’s inability to handle closed bodies near resonance frequency.

Good stepping stone, but without addressing the above it falls shorts of its goal of evaluating on an arbitrary surface.

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