Sparse matrix/canonical grid method applied to 3-D dense medium simulations

The sparse matrix/canonical grid (SMCG) Method, which has been shown to be an efficient method for calculating the scattering from one-dimensional and two-dimensional random rough surfaces, is extended to three-dimensional (3-D) dense media scattering. In particular, we study the scattering properties of media containing randomly positioned and oriented dielectric spheroids. Mutual interactions between scatterers are formulated using a Method of Moments solution of the volume integral equation. Iterative solvers for the resulting system matrix normally require (O) over dot (N-2) operations for each matrix-vector multiply. The SMCG method reduces this complexity to O(N log N) by defining a neighborhood distance, r(d), by which particle interactions are decomposed into "strong" and "weak." Strong interaction terms are calculated directly requiring O(N) operations for each iteration. Weak interaction terms are approximated by a multivariate Taylor series. expansion of the 3-D background dyadic Green's function between any given pair of particles. Greater accuracy may be achieved by increasing r(d), using a higher order Taylor expansion, and/or increasing mesh density at the cost of more interaction terms, more fast Fourier transforms (FFTs), and longer FF%TS, respectively. Scattering, results, computation times, and accuracy for large-scale problems with r(d) up to 2 gridpoints, 14 x 14 x 14 canonical grid size, fifth-order Taylor expansion, and 15 000 discrete scatterers are presented and compared against full solutions.