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 ( ) operations for each matrix-vector multiply. The SMCG method reduces this complexity to ( log ) by defining a neighborhood distance, , by which particle interactions are decomposed into “strong” and “weak.” Strong interaction terms are calculated directly requiring ( ) 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 , 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 FFTs, respectively. Scattering results, computation times, and accuracy for large-scale problems with up to 2 gridpoints, 14 14 14 canonical grid size, fifth-order Taylor expansion, and 15 000 discrete scatterers are presented and compared against full solutions.