Fast Periodic Interpolation Method for Periodic Unit Cell Problems

Abstract

A fast periodic interpolation method (FPIM) is presented for rapidly computing fields in a unit cell of an infinitely periodic array. For low and moderate frequencies (for unit cells smaller than or on the order of the wavelength) the FPIM has the computational cost of ( ) and it requires only (1) periodic Green's function (PGF) evaluations, for sources and observers. For high-or mixed-frequencies the computational cost scales aswhere is the domain size within the unit cell and is the wavelength. FPIM is based on splitting the field into the near-field from the sources around the unit cell and the far-field from the remaining sources. The near-field component can be evaluated rapidly using any available fast method. The far-field component is computed by tabulating the PGF at sparse source and observer grids, using this table to calculate the field at the observation grid, and interpolating from the observation grid to the actual observers. The FPIM is kernel independent and allows using any method for evaluating the PGF, including simple Floquet expansions. The computational times can be comparable to those of conventional (non-periodic) -body electromagnetic problems. The presented method can be used to accelerate integral equations for periodic unit cell problems with many applications in microwave engineering and optics. Diego, where he currently holds the position of Associate Professor. His research interests include computational electromagnetics, micromagnetics, the analysis of microwave phenomena on structured surfaces, the analysis of optical phenomena in photonic nanostructures, and the analysis of magnetization dynamics in magnetic nanostructures.

Fast and Accurate MoM Analysis of Periodic Arrays of Multilayered Stacked Rectangular Patches With Application to the Design of Reflectarray Antennas

“…Note that a value of N c = 7 suffices to obtain the interpolated MPGF with four significant figures. This means that only 64 samples of the MPGF are required to generate this interpolation, which is substantially less than the number of samples required in [25] and [26] to obtain the interpolated periodic Green's functions with the same accuracy of four significant figures.…”

“…The CPU time required for this spatial domain computation of the MoM matrix entries is substantially reduced by introducing two improvements. The first improvement is that the MPGF with 2-D periodicity for the potentials are judiciously interpolated in the spatial domain [25], [26] in terms of 2-D Chebyshev polynomials after extracting the behavior of the MGPF around the source points. The second improvement in the spatial domain computation of the MoM entries has to do with the efficient computation of 4-D singular integrals.…”

Accelerated Cartesian Expansions for the Rapid Solution of Periodic Multiscale Problems

“…As is evident from (13), each component requires a number of operations, , which is approximately commensurate with the evaluation of an Ewald Sum. As the nearfield matrix elements are themselves Ewald Sums, the precomputation cost scales as (15) While the first contribution is clearly linear, it is not immediately clear as to how the latter dependent contribution will scale. As the translation operator only depends upon the separation between boxes, and the tree structures imposes a uniform superstructure on the computational domain, there is a significant redundancy that can be exploited in this stage of precomputation.…”

“…The subject remained largely unexplored for over a decade, until Otani and Nishimura [14] used a periodic FMM for the analysis of photonic crystals. Other types of hierarchical methods have been presented more recently, namely interpolatory methods such as those due to Li [15] and Chan [16]. Instead of exploiting addition theorems for the Green's Function, these methods hinge upon the projection of the Green's Function onto a hierarchy of increasingly sparse interpolatory grids.…”

An Efficient 1-D Periodic Boundary Integral Equation Technique to Analyze Radiation onto Straight and Meandering Microstrip Lines

“…An overview of techniques is provided in [12]. Additionally, a new interesting method based on fast periodic interpolations is reported in [13].…”