An O(N) method for the rapid analysis of periodic problems using Accelerated Cartesian Expansions (ACE)

Abstract

The evaluation of long-range potentials in periodic, many-body systems arises as a necessary step in the numerical modeling of a multitude of interesting physical problems. Direct evaluation of these potentials requires O(N 2 ) operations and O(N 2 ) storage, where N is the number of interacting bodies. In this work, we present a method, which requires O(N ) operations and O(N ) storage, for the evaluation of periodic Helmholtz, Coulomb, and Yukawa potentials with periodicity in 1-, 2-, and 3-dimensions, using the method of Accelerated Cartesian Expansions (ACE). We present all aspects necessary to effect this acceleration within the framework of ACE including the necessary translation operators, and appropriately modifying the hierarchical computational algorithm. We also present several results that validate the efficacy of this method with respect to both error convergence and cost scaling, and derive error bounds for one exemplary potential.

Fast Periodic Interpolation Method for Periodic Unit Cell Problems

“…Hierarchical fast multipole method (FMM) type approaches with computational cost for static periodic problems in free space have been known for two decades [17], [18]. However, developing such approaches for electromagnetics is more challenging and only a few recent works have been reported [19]- [22]. Introducing fast hierarchical IE methods for general periodic unit cell problems would catalyze the development of efficient IE methods in electromagnetics and other fields of study in which -body problems on a periodic unit cell appear.…”

Applications of periodic Accelerated Cartesian Expansions to the analysis of electrically dense frequency selective structures

“…3) demonstrates the accuracy and speed of our method for a modestly sized problem. Here we utilize the Minkowski patch FSS taken originally from [4] and used to validate the un-accelerated solver in [2]. It is discretized with N = 2, 148 unknowns and swept across 27 frequencies from 0.4 GHz to 2.00 GHz.…”

“…The efficient computation of the translation operator plays a significant role in reducing precomputation costs, especially for periodic problems, as the translation operator is rendered as a set of infinite Ewald-like summations, the precise form of which can be found in [2]. To this end, as the translation operator only depends upon the distance between boxes, we can take advantage of the regularity of the octree structure by only computing unique translation operators for unique box-box separations.…”

“…error is independent of tree height), and its nearly kernel-independent framework. Whereas previous work by Baczewski and Shanker has demonstrated error convergence and linear scaling of the kernel itself [2], the original contribution of this work is a discussion of the ACE algorithm's integration into a fast EFIE solver. In the course of this discussion, we focus on changes that must be made to the tree construction algorithm for periodic domains (relative to previous, non-periodic implementations), as well as results which demonstrate the algorithms accuracy, and its utility in solving a number of exemplary periodic systems of varying size.…”

Rapid analysis of scattering from periodic dielectric structures using accelerated Cartesian expansions

“…The ACE algorithm has previously been applied to the efficient computation of potentials of the form R −ν [19], Lienard-Wiechert potentials [20], diffusion, lossy wave, and Klein-Gordon potentials [21], and periodic Helmholtz, Yukawa, and Coulomb potentials [22]. Like FMM, it is based upon a hierarchical decomposition of the computational domain mapped onto an octree data structure, wherein a distinction between near and farfield source-observer aggregates is made, and an addition theorem is used to effect the interaction of all bodies with linear scaling.…”