Free-space Fast Multipole Method Research Articles

Adapting free-space fast multipole method for layered media Green's function: Algorithm and analysis

In this paper, we present a numerical algorithm for the accurate and efficient computation of the convolution of the frequency domain layered media Green's function with a given density function. Instead of compressing the convolution matrix directly as in the classical fast multipole method, fast direct solvers, and fast H-matrix algorithms, the new algorithm considers a translated form of the original matrix so that many existing building blocks from the highly optimized free-space fast multipole method can be easily adapted to the Sommerfeld integral representations of the layered media Green's function. An asymptotic analysis is performed on the Sommerfeld integrals for large orders to provide an estimate of the decay rate in the new "multipole" and "local" expansions. In order to avoid the highly oscillatory integrand in the original Sommerfeld integral representations when the source and target are close to each other, or when they are both close to the interface in the scattered field, mathematically equivalent alternative direction integral representations are introduced. The convergence of the multipole and local expansions and formulas and quadrature rules for the original and alternative direction integral representations are numerically validated.

A method to compute periodic sums

In a number of problems in computational physics, a finite sum of kernel functions centered at N particle locations located in a box in three dimensions must be extended by imposing periodic boundary conditions on box boundaries. Even though the finite sum can be efficiently computed via fast summation algorithms, such as the fast multipole method (FMM), the periodized extension is usually treated via a different algorithm, Ewald summation, accelerated via the fast Fourier transform (FFT). A different approach to compute this periodized sum just using a blackbox finite fast summation algorithm is presented in this paper. The method splits the periodized sum into two parts. The first, comprising the contribution of all points outside a large sphere enclosing the box, and some of its neighbors, is approximated inside the box by a collection of kernel functions (“sources”) placed on the surface of the sphere or using an expansion in terms of spectrally convergent local basis functions. The second part, comprising the part inside the sphere, and including the box and its immediate neighborhood, is treated via available summation algorithms. The coefficients of the sources are determined by least squares collocation of the periodicity condition of the total potential, imposed on a circumspherical surface for the box. While the method is presented in general, details are worked out for the case of evaluating electrostatic potentials and forces. Results show that when used with the FMM, the periodized sum can be computed to any specified accuracy, at an additional cost of the order of the free-space FMM. Several technical details and efficient algorithms for auxiliary computations are provided, as are numerical comparisons.

A Fast Multipole Method for Embedded Structure in a Stratified Medium

An efficient, static fast multipole method (FMM) based algorithm is presented in this paper for the evaluation of the parasitic capacitance of 3-D microstrip signal lines embedded in stratified dielectric media. The effect of dielectric interfaces on the capacitance matrix is included in the stage of FMM when outgoing multipole expansions are used to form local multipole expansions. The algorithm retains O(N) computational and memory complexity of the free-space FMM, where N is the number of conductor patches.

A Fast Multipole Method for Embedded Structure in a Stratified Medium - Abstract

An efficient, static fast multipole method (FMM) based algorithm is presented in this paper for the evaluation of the parasitic capacitance of 3-D microstrip signal lines embedded in stratified dielectric media. The effect of dielectric interfaces on the capacitance matrix is included in the stage of FMM when outgoing multipole expansions are used to form local multipole expansions. The algorithm retains O(N) computational and memory complexity of the free-space FMM, where N is the number of conductor patches.

A fast multipole-method-based calculation of the capacitance matrix for multiple conductors above stratified dielectric media

An efficient static fast-multipole-method (FMM)-based algorithm is presented in this paper for the evaluation of the parasitic capacitance of three-dimensional microstrip signal lines above stratified dielectric media. The effect of dielectric interfaces on the capacitance matrix is included in the stage of FMM when outgoing multipole expansions are used to form local multipole expansions by the use of interpolated image outgoing-to-local multipole translation functions. The increase in computation time and memory usage, compared to the free-space case, is, therefore, small. The algorithm retains O(N) computational and memory complexity of the free-space FMM, where N is the number of conductor patches.

A hierarchical fast-multipole method for stratified media

An efficient, static, fast-multipole method (FMM)-based algorithm is presented in this paper for the evaluation of the parasitic capacitance of 3-D microstrip signal lines above stratified dielectric media. A modified tree structure is used to perform the multilevel outgoing-to-local multipole translations. The algorithm, only marginally more expensive than the free-space FMM, retains its O(N) computational cost and memory use, where N is the number of conductor patches. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 27: 13–17, 2000.

Fast multipole method for scattering from 3-D PEC targets situated in a half-space environment

The fast multipole method (FMM) is extended to the problem of an arbitrary, three-dimensional perfect conductor situated above or below a lossy, dielectric half space. The interactions between basis and testing functions within an FMM cluster, and for nearby clusters, are handled via the rigorous dyadic Green's function, with the latter evaluated efficiently using the complex-image technique. Intercluster interactions are modeled as in the free-space FMM, with the dyadic Green's function approximated via real images and equivalent reflection coefficients; these approximations have proven to be highly accurate. Example results are presented for a large trihedral fiducial target, in free space and above a lossy, dispersive half space, with comparisons presented between the FMM and a rigorous method-of-moments (MoM) solution. ©1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 21: 399–405, 1999.