1. Fast and Efficient RCS Computation over a Wide Frequency Band Using the Universal Characteristic Basis Functions (UCBFs) June 2007 Authors: Prof. Raj Mittra* Eugenio Lucente** Prof. Agostino Monorchio** * PennState University (PA) USA ** Pisa University (Pi) Italy

2. 2/17 Conventional MoM Limitations  Long execution time  Huge memory requirement  Inefficient frequency analysis Electrically large objects

3. 3/17 What is the Characteristic Basis Function 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.

4. 4/17 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

5. 5/17 Compute CBFs of the i th block i th 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

6. 6/17 Steps 3 - 4 - 5 Step-3: Matrix reduction. Determine a reduced matrix, by using the Galerkin method. 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. Z red : reduced matrix, size KM by K M  : unknown coefficient, b: new RHS J: current on the original geometry J cbf : CBFs from each block

7. 7/17 Features  Size 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

8. 8/17 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 Observations

9. 9/17 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

10. 10/17 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

11. 11/17 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

12. 12/17  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:

13. 13/17 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 IIBlock III

14. 14/17 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 )

15. 15/17 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

16. 16/17 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