Efficient Calculation of the Reduced Matrix in the Characteristic Basis Functions Method Jaime Laviada*(1), Marcos R. Pino(1), Fernando Las-Heras(1), and Raj Mittra(2) (1) Electrical Engineering Dep., University of Oviedo, Spain. (e-mail:
[email protected],
[email protected],
[email protected]) (2) Electromagnetic Communication Lab., 319 EEE, Pennsylvania State Univ., University Park, PA 16802, USA (e-mail:
[email protected])
Introduction Traditionally, basis functions in the Method of Moments (MoM) have been limited in the range of λ/10 to λ/6 in size. One of the reasons is that typical basis functions, such as RWG or rooftops, do not incorporate any phase variation and, hence, they are not able to model the behavior of the induced surface currents on large domains. Another important reason is that usual basis functions are defined over flat surfaces, and are unsuitable for modeling currents on large arbitrary shaped surfaces, except by using a subdomain approach. In contrast to the conventional MoM utilizing subdomain bases, the Characteristic Basis Function Method (CBFM) [1] defines a new set of “macro” basis functions, comprising of traditional low-level basis functions that depend on the geometry of the problem. Consequently, they are automatically adapted to the shape of the body and are also capable of modeling the phase variations in large domains. Once these characteristic basis functions (CBFs) have been generated by solving for the induced currents in subdomains into which the original problem geometry has been divided, the reaction integrals between the CBs that form the entries of the final reduced matrix system can be carried out by evaluating the matrix-vector products involving the sub-blocks of the conventional method of moment matrix. The reduced matrix, so-called because it is a compressed version of the conventional MoM matrix, is usually much smaller and, hence, is amenable to direct solvers, without resorting to iteration. In recent years, techniques based on the low-rank properties of the off-diagonal blocks have been applied successfully to model off-diagonal blocks of the regular MoM matrix [2-4]. These algorithms can also be applied to speed up the dot products involved in the calculation of the reduced matrix [4]. However, these algorithms are purely algebraic in nature, and they do not directly take into account of the physics of the problem. In addition, they can not deal with the case of contiguous blocks because of the poor, low-rank representation of these blocks. The objective of this paper is to present an efficient technique that not only takes advantage of the electromagnetic properties of the problem to generate the desired matrix- vector products efficiently, but is also capable of handling the case of adjacent blocks. This work has been supported by the “Ministerio de Educación y Ciencia” of Spain /FEDER under project TEC2005-03563, and by the “Cátedra Telefónica- Universidad de Oviedo”, “Fundación CTIC” under project FUO-EM-136-06 and “Plan de Ciencia, Tecnología e Innovación de Asturias” grant no. BP06-101.
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Efficient Calculation of the Reduced Matrix The calculation of the reduced matrix involves the calculation of the reaction integrals that take the form: G G J j (rG ) ⋅ EJGi (rG )dS (1)
∫
S G G where J j is the test current defined on the observation domain S, and E JGi is the electric G field due to the source current J i . The calculation of these terms involves the evaluation
of the electric field in the entire source domain. If the distance between the source and the observation domain is sufficiently far, the electric field will have a relatively smooth behavior. In the asymptotic region of the far field, the amplitude will decrease as the inverse of the distance and the phase will exhibit a linear variation. These facts suggest that if the field is sampled on a fine mesh in the observation domain, the sampled result will contain redundant information. We can take advantage of this fact by computing the field at only a sparse set of sample points and interpolating these fields to generate the desired field values at points on the original fine mesh via the use of interpolation in a computationally efficient manner. Obviously, the distance between the sample points should be sufficiently large in order for the method to be efficient; however, in this case an interpolation of the real and imaginary parts of the field will not work well, because it won’t be able to capture its oscillatory behavior. We may attempt to get around this problem by interpolating the amplitude and phase of the field instead, but this tactic still does not solve our problem quite satisfactorily yet. This is because the phase behavior contains ambiguity because of phase-wrapping, which must be resolved before performing the interpolation. Although there exist a number of global phase unwrap techniques in the literature (see [5]), we found that the best results are obtained when we use a simple local unwrap by looking for 2π gaps in the phase behavior. When the observation and source blocks are contiguous, the field radiated by the source will possess a singular behavior at the common edge. In addition, the phase will show rapid variations that would pose difficulties when we try to interpolate it. To overcome this problem, we carry out the coupling between the near regions of contiguous blocks by reverting to the blocks of the original MoM matrix.
Numerical results To illustrate the application of the procedure described above, we consider the problem of a moderately large PEC cube, whose dimensions 4λx4λ on the side. All the computations for which we present herein have been carried out on a desktop computer with an 2.4GHz Intel Core 2 Duo CPU with a 4GB RAM. (Note: only one core was used.). The cube faces are meshed with triangles whose edges are between λ/10 and λ/6 (Fig. 1a) and RWG basis functions are defined on these triangles. The total number of low-level basis functions is 36402. The CBFs were defined on 1λx1λ square blocks (Fig. 1b) by aggregating 380 lowlevel basis functions. To calculate the CBFs, the blocks were extended by 0.1λ and excited with 400 θ-polarized and φ-polarized plane waves. The threshold level for the SVD decomposition was chosen to be 10-3. After the SVD only 60 (average value) CBFs
were retained for each block. The total number of CBFs involved in the entire problem was 5767. The average regular MoM matrix relating two blocks is 380x380. For adjacent blocks, we define a 0.4λ strip for direct computation of the reaction integral, as opposed to interpolation. This procedure results in a 140x160 matrix, on average, which is approximately 6 times faster to fill. For the remaining cases a 7x7 sample grid is used on the surface of each non-contiguous block to carry out the evaluation of the tested field. Fig. 2a shows the bistatic RCS of the cube for a θ-cut in the φ=0º plane, together with the conventional MoM results, included here for the sake of completeness. The excitation is a G plane wave Ei = E0 exp(− jkz )xˆ . The time to fill the reduced matrix is 2500s using the conventional method, while it is only 420s with the speed-up approach. The time to solve the reduced equation system was 94s. We observe that the agreement between the conventional CBFM results and those obtained with the new speed-up is very good even in the null regions. Next, we turn to the monostatic RCS of the same cube for the θ-cut in the φ=0º plane (see Fig. 2b). A one-degree resolution was used that involved 91 θ-polarized plane waves. Times for the CBF generation and the reduced matrix calculation are the same as those for the bistatic example and the matrix solve-time for all the right hand sides is only ~1 second more than in the previous case. Once again we find that the agreement between the results is excellent.
Conclusions A new efficient technique for calculating the reaction between characteristics basis functions has been presented. The proposed method enables us to bypass the computation of the off-diagonal of the MoM matrix. For the case of contiguous blocks, an efficient hybrid approach has been developed that combines the conventional MoM matrix approach with the interpolation technique. Numerical results show that the technique is able to speed up the reduced matrix filling time without sacrificing the accuracy of the RCS results.
References: [1] V. Prakash and R. Mittra, “Characteristic basis function method: a new technique for efficient solution of method of moments matrix equations”, Micr. Opt. Techonol., vol. 36, pp. 95-100, Jan 2003. [2] K. Zhao, M. N. Vouvakis, and J. F. Lee, “The adative cross approximation algorithm for accelerated method of moments computations of EMC problems”, IEEE Trans. EMC, vol. 47, no. 4, pp. 763-773, Nov. 2005. [3] R. J. Burkholder and J.-F. Lee, “Fast dual-MGS block-factorization algorithm for dense MoM matrices”, IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1693-1699, Jul. 2004. [4] C. Delgado, R. Mittra, and F. Cátedra, “Analysis of Fast Numerical Techniques Applied to the Characteristic Basis Function Method”, IEEE AP-S International Symposium, pp. 4031-4034, Albuquerque, New Mexico, Jul. 2006. [5] D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods”, J. Opt. Soc. Am. A., vol. 11, no. 1, pp. 107-117, Jan. 1994.
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