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

arXiv:0810.0233v3 [physics.comp-ph] 21 Jul 2009 Adaptation and Performance of the Cartesian Coordinates Fast Multipole Method for Nanomagnetic Simulations Wen Zhang, Stephan Haas Department of Physics and Astronomy, University of Southern California Los Angeles, CA 90089, USA Abstract An implementation of the fast multiple method (FMM) is performed for mag- netic 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 opti- mized in terms of cell size and parameter set (expansion order and opening angle) and the trade offbetween computing time and accuracy is quantita- tively 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. Key words: FMM, cartesian, magnet PACS: 02.70.-c, 75.75.+a 1. Introduction Magnetic nanoparticles and fluids have attracted intensive investigation during the last two decades [1, 2, 3, 4, 5]. Micromagnetic simulation is an powerful tool to study such systems, where the main challenge is the dipolar Email address: zhangwen@usc.edu (Wen Zhang) Preprint submitted to Journal of Magnetism and Magnetic Materials November 3, 2018 interaction between magnetic moments, which can be dealt with either dis- cretely or in the continuum limit. Since the interaction is long ranged, the complexity of a brute-force calculation will be O(N2). One popular method to improve this performance is by Fast Fourier Transform (FFT), which re- duces the complexity to O(N ∗log(N)). However, FFT suffers from the fact that it requires a regular lattice arrangement of dipoles, and a large num- ber of padding areas have to be added when dealing with exotic geometries and open boundary conditions. Thus, the alternative, fast multipole method (FMM) has attracted increasingly more attention in the past several years [6, 7, 8, 9]. The FMM was first introduced by L. Greengard et al [10, 11], and has been used ever since to speed up large scale simulations involving long ranged interactions. It has the charming advantage of O(N) complexity. Further- more, there is no constraint on the distribution of particles and the bound- aries of the system. Hence it is extremely useful for magnetic fluid systems and nanomagnets with exotic geometries. Last but not least, it is a scal- able algorithm, i.e. it can be efficiently implemented in parallel [12]. With so many advantages though, the adoption of FMM in magnetic system is not widely applied, probably due to two reasons. One concern is that since there is a huge overhead to achieve O(N) complexity, the FMM becomes faster only for very large N. Actually this is true only when one considers a very high order expansion (say 10). In the context of micromagnetics, we will show that an expansion to the order of no more than 6 will be satisfy- ing, and with some optimization procedure FMM will be superior even in a system of 103 dipoles. The other reason is that the standard implementa- tion of the FMM algorithm is based on the well-known spherical harmonics expansion [13] of 1/r. For dipolar interaction which decays as 1/r2, how- ever, it is not straightforward to apply. Further, the additional complexity of calculating spherical harmonics acts as another barrier. To overcome these shortcomings, expansions in Cartesian coordinate were proposed[14, 15], but they were not applied successfully until recently[8]. Following these studies, here we present a very simple but useful formula (Eq. 3) to calculate the multipole moments in dipolar systems. In Section II and III, a brief description of the FMM algorithm and all the necessary equations is presented. In Section IV, performance issues will be discussed in detail. The number of dipoles contained in the smallest cell will be shown to be crucial in terms of performance and a rule of thumb on choosing the right number will be given (Eq. 17). Then, the optimal choice 2 of expansion order and opening angle to achieve certain error bounds will be discussed, followed by quantitative study of the trade offbetween computing time and accuracy. Finally, we compare the Cartesian coordinates FMM with spherical harmonics FMM and FFT. 2. Algorithm The crucial ideas of the FMM are: (i) chunk source points together into large cells whose field in remote cells is computed by a multipole expansion of the source points; (ii) use a single Taylor expansion to express the smooth field in a given cell contributed by the multiple expansions of all remote cells. As t

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