electromagnetic scattering by surfaces of arbitrary shape
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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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