AbstractsMathematics

In this thesis, we present a fast multipole algorithm (FMA) for solving a periodic scattering problem using method of moments (MOM). The difference between the standard integral equations (EFIE and MFIE) and the ones used for periodic geometry is the use of periodic Green's function (PGF) over free-space Green's function (FGF). When expressed as a spatial sum, PGF converges slowly making this representation unsuitable for numerical implementation. An alternative representation known as Ewald summation can be used to evaluate PGF with exponential convergence and high accuracy. Despite the use of Ewald summation to evaluate PGF quickly, the MOM solution for a periodic scattering problem is still too slow. The matrix fill time for a scatterer with only a few thousand unknowns requires several hours to complete. The same geometry treated as a single scatterer (as opposed to a unit cell in a periodic array) requires only seconds to compute. This means that even problems with a small number of unknowns can benefit from a fast method. The PGF can be factorized using usual methods of factorization. The convergence problems of PGF do not simply vanish in this factorization – instead they manifest in the form of a lattice constant. These constants are commonly found in low energy electron diffraction (LEED). The lattice constants can also be evaluated using Ewald summation and has been well documented by researchers. In this thesis, we present a fast algorithm for scattering by a 2-D lattice in 3-D. Using the factorization of PGF, we develop a multi-level periodic fast multipole algorithm (MLP-FMA) which performs on par with existing ML-FMA algorithms. The MLP-FMA is applied to some example geometry and compared to periodic MOM computed using Ewald summation. Advisors/Committee Members: Chew, Weng Cho (advisor).