Accelerated Cartesian Expansions for the Rapid Solution of Periodic Multiscale Problems

Abstract

We present an algorithm for the fast and efficient solution of integral equations that arise in the analysis of scattering from periodic arrays of PEC objects, such as multiband frequency selective surfaces (FSS) or metamaterial structures. Our approach relies upon the method of Accelerated Cartesian Expansions (ACE) to rapidly evaluate the requisite potential integrals. ACE is analogous to FMM in that it can be used to accelerate the matrix vector product used in the solution of systems discretized using MoM. Here, ACE provides linear scaling in both CPU time and memory. Details regarding the implementation of this method within the context of periodic systems are provided, as well as results that establish error convergence and scalability. We also demonstrate the applicability of this algorithm by studying several exemplary electrically dense systems.Index Terms-Fast methods, integral equations, periodic structures, frequency selective surfaces.

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

“…The results presented demonstrate error convergence as well as considerable acceleration. Further work is being submitted elsewhere to demonstrate the applicability of our method to the analysis of electromagnetic wave propagation [31]. We are presently working on adapting these techniques to solid-state electronic structure calculations involving defects.…”

“…We do not seek to tie this work to the solution of any particular problem (i.e. integral equation solvers, N -body dynamics, etc), although we note that we have adapted our method to the solution of integral equations which arise in the analysis of electromagnetic wave propagation, and have submitted it to a more appropriate forum [31]. The principal contributions of this paper are as follows:…”

Domains and defects in nuclear pasta

“…The exception being the largest of our runs, with 3 276 800 nucleons, which still had six large domains and many defects by the time we stopped evolving it due to its very high computational cost. The high cost of MD computations stemming from long range Coulomb repulsion between protons can be decreased with the implementation of robust fast multipole method algorithm for Yukawa-type potentials [105][106][107]. Excluding significant advances in computer performance, this may be the only way to simulate nuclear pasta systems beyond a few million nucleons that need to be evolved for tens of millions of times steps in order to reach equilibrium.…”

Differential operator acting on accelerated Cartesian expansion algorithm

“…By carefully handling the differential operator, an implementation which has higher numerical precision and better performance is achieved. The study is mainly based on 3D problem with homogeneous background, but the theorem can be extended to the ACE expansion for periodic Green's function [13], static Green's function [11] and other potential topics.…”