Adaptation and Performance of the Cartesian Coordinates Fast Multipole Method for Nanomagnetic Simulations

TL;DR

This paper presents a simplified Cartesian coordinate implementation of the Fast Multipole Method tailored for nanomagnetic simulations, optimizing performance and accuracy for systems with long-range dipolar interactions.

Contribution

It introduces a Cartesian polynomial expansion for FMM, derives multipole moments for dipoles, and compares its efficiency against traditional methods in nanomagnetic contexts.

Findings

Cartesian FMM outperforms spherical harmonics FMM and FFT in nanomagnetic simulations.

Optimal parameters for balancing accuracy and computational efficiency are identified.

A practical rule for dipole distribution in cells is proposed.

Abstract

An implementation of the fast multiple method (FMM) is performed for magnetic systems with long-ranged dipolar interactions. Expansion in spherical harmonics of the original FMM is replaced by expansion of polynomials in Cartesian coordinates, which is considerably simpler. Under open boundary conditions, an expression for multipole moments of point dipoles in a cell is derived. These make the program appropriate for nanomagnetic simulations, including magnetic nanoparticles and ferrofluids. The performance is optimized in terms of cell size and parameter set (expansion order and opening angle) and the trade off between computing time and accuracy is quantitatively studied. A rule of thumb is proposed to decide the appropriate average number of dipoles in the smallest cells, and an optimal choice of parameter set is suggested. Finally, the superiority of Cartesian coordinate FMM is demonstrated by comparison to spherical harmonics FMM and FFT.