(1) Computer Science Department - University of Alcalá, SPAIN

1. A Numerically Efficient Technique for Orthogonalizing the Basis Functions Arising in the Solution of Electromagnetic Scattering Problems Using the CBFMCarlos Delgado(1), Felipe Cátedra(1) and Raj Mittra(2) (1) Computer Science Department - University of Alcalá, SPAIN (2) Electromagnetic Communication Laboratory - Penn State University, USA

2. Outline • Introduction • Characteristic Basis Function Method (CBFM): General features • Speeding up the CBF generation with the Physical Optics approach • CBF orthogonalization via multi-step SVD computations • Numerical Results • Conclusions

3. Introduction-I • When analysing electrically large problems, the high sampling rate required by the conventional Method of Moments (MoM), typically around l/10, usually leads to huge linear systems of equations described by dense matrices, which involves serious computational burden. • Some efficient numerical techniques improve the MoM performance by rigorously computing and storing only the near-field terms of the coupling matrix, highly improving the CPU-time via an efficient evaluation of the matrix-vector product operations in the iterative system solution process and also reducing the memory requirements. • Other family of approaches rely on the reduction of the total number of unknowns defining macro-basis functions over enlarged domains, in order to avoid the bottleneck imposed by the conventional MoM sampling rate.

4. Introduction-II • The Characteristic BasisFunction Method (CBFM) introduces a new system of equations related to a set of high-level basis functions defined over relatively large domains. • Each CBF can be expressed in terms of theconventional MoM subdomain functions (RWG, Rooftops, etc). • The CBFs do not rely upon predetermined fixed shapes. They are estimated for each problem. • The application of high-level macro-basis functions introduces the possibility of using direct solvers for the solutionof some problems where iterative solvers are the only possible choice if using the low-level discretization. • The CBFM is a kernel-independent approach.

5. The Characteristic Basis Function Method (CBFM): General Features-I • What are the motivations for reducing of the number of unknowns via the CBFM? • Reduction of the matrix size • Maintain a good condition number • This high-level basis functions are defined over large subdomains called blocks. It is necessary to separate the geometry into blocks during the preprocessing stage. • In each of these blocks, the current distribution is reprsented in terms of a superposition of several Characteristic Basis Functions (CBFs) whose weights are to be determined solving the system that describes the problem. • It is important to minimize the number of CBFs, since it is going to determine the size of the Reduced matrix that represents the coupling between the blocks

6. Off-Diagonal blocks: Mutual Coupling Diagonal blocks: Self-Interactions The Characteristic Basis Function Method (CBFM): General Features-II • Each CBF is defined in terms of weighted low-level subdomain functions, such as Rooftops, RWGs, etc. • The reduced matrix represents the coupling between the CBFs Reduced Matrix:

7. The Characteristic Basis Function Method (CBFM): General Features-III • The generation of CBFs is very flexible • Primary & secondary CBFs vs. PWS • Different approaches can be used (MoM, PO, ...) • To avoid the computation of secondary CBFs, the excitation can be derived from several plane waves with different incidence angles • This approach renders the reduced matrix independent of the excitation • Orthogonalization techniques are applied in order to reduce the number of basis functions, and to imporve the condition number • The same set of CBFs can be used for different blocks

8. The Characteristic Basis Function Method (CBFM): General Features-IV • If the MoM is applied to obtain the currents induced by an incident plane wave, each contour segment is extended a distance Δu and Δv, so that there is a physical overlap between thr extended blocks, in order to avoid the artificially introduced edge behaviour at the interface between blocks • After obtaining the induced currents, only the values contained within the original (non-extended) blocks are retained. • For large and smooth surfaces, it is possible to use the PO approach instead of the MoM to obtain these currents. This avoids matrix solutions while generating the CBFs, and saves considerable time.

9. Generation of CBFs Using Physical Optics-I • Application of MoM to obtain the currents induced by a set of plane waves entails solving the following matrix problem in a rigorous way. • When the object geometry is subdivided into a relatively large number of medium-sized blocks, and when the geometries of these blocks do not have fine features, it is considerably more efficient to resort to high-frequency techniques, such as the Physical Optics (PO) approach, for deriving the CBFs. This is because, in contrast to MoM, no matrix solution is needed in this approach.

10. Sum of all the low-level basis functions included in the block Low-level basis function “n” Incident field Sampling point over the “n-th” basis function Incidence direction of the plane wave Generation of CBFs Using Physical Optics-II • The current induced by the incident plane waves on the surface are calculated by applying the PO approach in the middle point of the domain where each low-level basis function is defined.

11. Multi-Step SVD Computation for Obtaining the CBFs-I • The ratio between the number of CBFs and low-level basis functions decreases as the block size is increased, which makes it desirable to develop techniques that calculate the CBFs efficiently over large surfaces.

12. The rows of [V]* are the new basis vectors We only retain vectors for which the associated singular value is above a certain threshold: Multi-Step SVD Computation for Obtaining the CBFs-II • The Singular Value Decomposition can be used to obtain an orthogonal set of vectors which replace the original induced currents. Sample points (low-level basis functions) … … … Currents induced by plane waves A = = [U][σ][V]* …

13. Multi-Step SVD Computation for Obtaining the CBFs-III • When electrically large blocks are considered, the Singular Value Decomposition can present considerable computational burden if the entire Plane Wave Spectrum is considered. Also, this can lead to: • Increase in CPU-time • Possible inaccuracies due to internal error accumulation, which forces us to increase the numerical precision for the computation • A better alternative is to perform the computation in multiple steps, limiting the number of “input basis vectors” or rows in the A matrix for each computation. The resulting basis vector from one step serve as the input vectors for the next level. Finally, the resulting set of basis vectors in the last step are retained as the CBFs for the purpose of Reduced Matrix generaion.

14. Multi-Step SVD Computation for Obtaining the CBFs-IV • The PWS can be represented as a set of points in the (θ,φ) plane containing all the possible incident plane waves within the visible range. • The plane waves are compartmentalized into several groups. • After performing the SVD for each group, the resulting basis vectors corresponding to adjacent regions are grouped one again and a new SVD operation is performed. • Finally, we retain the result of the highest-level SVD computation as the desired CBFs.

15. Numerical Example-I • Scattering problem considered: PEC sphere (R=1m) modeled by means of 8 NURBS patches. • Simulations run on a Pentium IV at 2.8 GHz PC with 1.5 Gbytes of RAM memory. • Bistatic RCS obtained at a frequency of 1.5 GHz for both polarizations. • Using a λ/10 discretization, 12324 low-level subdomains are needed for each block. In contrast to this, the CBFM retains only 806 CBFs per block.

16. Numerical Example-II • CBFM results have been compared with the analytical solution (MIE).

17. Numerical Example-III • CPU times for the CBF generation process have been compared when considering different levels.

18. Numerical Example-IV • The threshold value used for retaining the output vectors can be relaxed after the first-level SVD computation. The number of basis vectors retained after the SVD computation for level-L is given by the condition:

19. Conclusions • A methodology for enlarging the block sizes in the CBFM, without increasing the associated CPU-time needed to compute the CBFs, has been presented. • Larger blocks usually yield fewer CBFs, leading, in turn, to a smaller-size Reduced Matrix. • The PO approach circumvents the matrix solution for generating the CBFs, and is well-suited for smooth parts of the geometries of electrically large blocks. • The total number of CBFs obtained with the present approach is of the same order as that obtained by carrying out a single (no multi-level) SVD computation.