A Modification of the Canonical Grid Series Expansion in Order to Increase the Efficiency of the SMCG Method

Abstract

In this letter, we present a Green's function approximation valid in the weak-interaction region that can be used with the sparse-matrix canonical grid (SMCG) method. It can be easily introduced into existing SMCG codes, allowing a reduction in size of the neighborhood region and, consequently, of the dynamic memory and the computation time requirements.Index Terms-Electromagnetic scattering by rough surfaces, Green function, iterative methods, least mean square methods, moment methods. I NTEREST has grown in the Monte Carlo simulations of random rough surface scattering as a result of the increasing computational power of computers. Several techniques have recently been developed for a fast computation of electromagnetic scattering from a deterministic terrain profile. In particular, the banded matrix Iterative approach/canonical grid (BMIA/CAG) method [1]-[4] and the sparse-matrix canonical grid (SMCG) [5]- [7], which is its extension to the analysis of the scattering from two-dimensional rough surfaces, seem to be two of the most efficient methods, since they provide low dynamic memory and computation time.These techniques, based on a modification of the classical method of moments (MoM), allow a fast evaluation of the reaction integral and, when an iterative solver is used, a fast matrix-vector multiplication. In both methods, the MoM solution technique is applied to the surface integral equation, and the matrix equation is . The key feature is to introduce a neighborhood distance separating two regions: a near-interaction region and a weak-interaction region. The impedance matrix is then divided into the sum of a strong-and a weak-matrix, i.e., , where represents the strong near-field interaction and the weak nonnear-field interaction. The near-interaction region requires that several reaction integrals be evaluated and stored in the strong-matrix , which is a sparse matrix. Concerning the weak-matrix, for each planar distance between the basis and weighting functions greater than , a Taylor series expansion of the free space Green's function is introduced. This allows us to operate a canonical grid expansion of the exact weak-matrix elements into a series of translationally invariant terms times powers of the height differences between the basis and weighting functions. This rewritten matrix is then applied in a conjugate gradient solution of the matrix equation with Manuscript Fig. 1. Accuracy of the Taylor series approximation (TSA) and of the least square minimization (LSM) technique when only three terms are considered.Range of minimization: 0:3 < d < 1 ; N = 50, N = 25. Fig. 2. Accuracy of the Taylor series approximation (TSA) and of the least square minimization (LSM) technique when a different number of series terms are considered. Range of minimization: 0:3 < d < 1; N = 50, N = 25.required matrix-vector multiplies performed rapidly due to the sparse nature of the strong-matrix and the use of the fast Fourier transform (FFT) and its inverse for the weak-matrix multiplies. Storage limitations for larg...

Analysis of electromagnetic scattering by nearly periodic structures: an LDRD report.

“…This method is a modification of the Adaptive Integral Method (AIM) [13], which is a technique based on the projection of subdomain basis functions onto a rectangular grid. Like the methods presented in [32]- [35], the GIFFT algorithm extends AIM by using basis-function projections onto a rectangular grid through Lagrange interpolating polynomials. The use of a rectangular grid results in a matrix-vector product that is convolutional in form and can thus be evaluated using FFTs.…”

“…The use of a rectangular grid results in a matrix-vector product that is convolutional in form and can thus be evaluated using FFTs. Although GIFFT differs from [32]- [35] in various respects, the primary differences between the AIM approach [13] and this method is the latter's use of interpolation to represent the Green's function (GF) and its specialization to periodic structures by taking into account the reusability properties of matrices that arise from interactions between identical cell elements. In GIFFT, the GF projections serve as an approximation of the GF across the periodic structure and the projections of the basis functions are then performed by taking an inner product of the basis function with the interpolating polynomials.…”

GIFFT: A Fast Solver for Modeling Sources in a Metamaterial Environment of Finite Size

“…In principle GIFFT requires the storage of only 84 GF samples per cell when we choose a 3 7 7 × × points-per-cell interpolation scheme, so that a total of only 4 2.4 10 × GF samples are stored. In practice, the memory requirement is slightly higher because of the zero padding to the nearest power of 2 needed to apply the FTT.…”

“…Like the methods presented in [3]- [4], the GIFFT algorithm is an extension of the AIM method in that it uses basis-function projections onto a rectangular grid through Lagrange interpolating polynomials. The use of a rectangular grid results in a matrix-vector product that is convolutional in form and can thus be evaluated using FFTs.…”