Fast and Efficient RCS Computation over Fast and Efficient RCS Computation over a Wide Frequency Band Using the a Wide Frequency Band Using the

Universal Characteristic Basis Functions Universal Characteristic Basis Functions (UCBFs)(UCBFs)

June 2007June 2007

Authors: Prof. Raj Mittra* Eugenio Lucente** Prof. Agostino Monorchio**

* PennState University (PA) USA** Pisa University (Pi) Italy

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Conventional MoM LimitationsConventional MoM Limitations

Long execution time Huge memory requirement Inefficient frequency analysis

Electrically large objects

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What is the Characteristic Basis Function What is the Characteristic Basis Function Method ( CBFM ) ?Method ( CBFM ) ?

The CBF method is an iteration-free, highly parallelizable MoM

approach based on macro-domain basis functions, namely

Characteristic Basis Function (CBFs), for solving large

multiscale electromagnetic scattering and radiation problems.

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How does CBFM work?How does CBFM work?

Step-1: Divide a complex structure into a number of smaller domains (blocks)

Geometry of a PEC plate divided into K blocks

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Compute CBFs of the ith block

ith block

Step-2a: Determine characteristic basis functions (CBFs) for each block: - solve “isolated” smaller blocks for a wide range of incident angles- each block is meshed by using RWG or other sub-domain basis- each block is analyzed via MoM technique. This results in a dense impedance matrix

- determination of CBFs can be a time and memory demanding task

Step-2b: Construct a new set of basis functions via the SVD approach.

Step 2

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Steps 3 - 4 - 5Steps 3 - 4 - 5 Step-3: Matrix reduction. Determine a reduced matrix, by using the

Galerkin method.

11 11 11 11 12 22 11 1

22 21 11 22 22 22 22 2

1 11 2 22

t t tM MM

t t tM MM

KM KM

t t tMM M MM M MM MM MM

J Z J J Z J J Z JJ Z J J Z J J Z J

Z

J Z J J Z J J Z J

Step-4: Solve the reduced linear system for the unknown weighting complex coefficients of CBFs

Step-5: Far field computation from the current distribution obtained in step-4.

Zred: reduced matrix, size KM by K M : unknown coefficient, b: new RHS J: current on the original geometry Jcbf: CBFs from each block

1

red

Mcbf

i ii

Z b

J J

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FeaturesFeaturesSize of the reduced matrix is much smaller than the original MoM matrix of

all structure

Reduced matrix is independent of angle of incidence

Reduced matrix equation can be solved efficiently for many incident

angles. It can be stored in a file and re-used whenever the structure is

analyzed for a new incident angle

For frequency sweep, CBFs must be generated anew for each frequency in the

band on interest. This leads to a huge time requirement. Reduction in the CPU

time is achieved by using universal CBFs rather than regular ones.

The CBF Method is highly parallelizable. Each block can be analyzed

independently. MPI-based parallel version has been developed

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Conventional Procedure for Generating CBFs on each block

Step-1: block is meshed by using a sub-domain scheme; typically triangular patch model.

Step-2: block is treated as an independent object illuminated by multiply incident Plane Waves (PWS)

Step-3: MoM technique is applied to the i-th block for obtaining the CBFs matrix equations

Step-4: Reducing number of initial CBFs via SVD by applying a thresholding procedure on Singular Values

Plane Wave Spectrum on Block

Individual Block

if /0

i Max ii

thresholdotherwise

i

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Observations on Conventional Procedure for Generating CBFs

Conventional CBFs Generation is time-consuming and memory-demanding task since it requires an LU decomposition for each block whose size can range from 1k to 14k unknowns

CBFs depend upon the frequency

CBFs must be generated anew for each frequency

Inefficient frequency sweep analysis

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In order to eliminate the frequency dependency , a new version of the CBFs is introduced, the so-called Universal CBFs (UCBFs)

UCBFs are generated only once, at the highest frequency, in the band of interest

They are used at lower frequencies, without going trough the time-consuming task of generating them anew

They can be used over 2 : 1 frequency band

Universal CBFs

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Universal Characteristic Basis Functions UCBFs

Physical Understanding of the UCBFs:The following figures show the behaviors of post-SVD CBFs for a 4 strip illuminated by a TE- and TM-polarized plane waves

Fig. 1. Magnitude of CBFs for a flat surface for TM polarization.

Fig. 2. Magnitude of CBFs for a flat surface for TE polarization

0 1 2 3 40

0.1

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The UCBFs have all the desired features of wavelets, through in

contrast to the wavelets, they are tailored to the geometry of the

object

The UCBFs, generated at the highest frequency, embody all the

spatial behaviors we would need to capture the corresponding

behaviors of the CBFs at lower frequencies, because they are less

oscillatory as the physics would suggest

Important Observations:

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Numerical Results

Scattering problem by a PEC cone: - Frequency range: 0.6 – 1.0 GHz - UCBFs are generated at 1.0 GHz - RCS is obtained at 0.6 GHz - The cone has been dived into 3 blocks ( 4500 unknowns ) - Total Number of Unknowns 12201

Block I

Block II Block III

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Fig. 3 – Comparison of RCS of a PEC cone for =0° at 0.6GHz. Continuous line:present approach; markers: conventional CBMoM solution.

Fig. 4 – Comparison of RCS of a PEC cone for=90° at 0.6GHz. Continuous line: present approach; markers: conventional CBMoM solution.

Scattering by a PEC Cone ( RCS )

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Scattering problem by a PEC sphere of 2 Radius: - Frequency range: 0.3 – 0.6 GHz - UCBFs are generated at 0.6 GHz - RCS is obtained at 0.3 GHz - The cone has been dived into 4 blocks

Fig. 5 –.Comparison of RCS of a PEC sphere with radius 2  for =0° at 0.3 GHz. Continuous line: present approach; markers: Mie solution.

Fig. 6 – Comparison of RCS of a PEC sphere with radius 2  for =90° at 0.3 GHz. Continuous line: present approach; markers: Mie solution.

Scattering problem by a PEC sphere

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Final Remarks

Fast and efficient frequency sweep

The UCBFs have all the desired features of wavelets

The UCBFs embody all the spatial behaviors at lower frequencies

Reduced computational effort