The grid-based fast multipole method – a massively parallel numerical scheme for calculating two-electron interaction energies

AbstractThe fast multipole method (FMM) is implemented in canonical ensemble particle simulations to compute non-bonded interactions efficiently with explicit error control. Multipole and local expansions have been derived to implement the FMM efficiently in Cartesian coordinates for soft-sphere (inverse power law), Lennard- Jones, Morse and Yukawa potential functions. Significant reductions in execution times have been achieved with respect to the direct method. For a given number, N, of particles the execution times of the direct method scale asO(N2). The FMM execution times scale asO(N) on sequential workstations and vector processors and asymptotically0(logN) on massively parallel computers. Connection Machine CM-2 and WAVETRACER-DTC parallel FMM implementations execute faster than the Cray-YMP vectorized FMM for ensemble sizes larger than 28k and 35k, respectively. For 256k particle ensembles the CM-2 parallel FMM is 12 times faster than the Cray-YMP vectorized direct method and 2.2 times faster than the vectorized FMM. For 256k particle ensembles the WAVETRACER-DTC parallel FMM is 33 times faster than the Cray-YMP vectorized direct method.

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A massively parallel adaptive fast-multipole method on heterogeneous architectures

Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis - SC '09 ◽

10.1145/1654059.1654118 ◽

2009 ◽

Cited By ~ 39

Author(s):

Ilya Lashuk ◽

George Biros ◽

Aparna Chandramowlishwaran ◽

Harper Langston ◽

Tuan-Anh Nguyen ◽

...

Keyword(s):

Fast Multipole Method ◽

Massively Parallel ◽

Fast Multipole ◽

Heterogeneous Architectures ◽

Multipole Method

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A Numerical Scheme for Generalized Peierls-Nabarro Model of Dislocations Based on the Fast Multipole Method and Iterative Grid Redistribution

Communications in Computational Physics ◽

10.4208/cicp.130114.270315a ◽

2015 ◽

Vol 18 (5) ◽

pp. 1282-1312 ◽

Cited By ~ 4

Author(s):

Aiyu Zhu ◽

Congming Jin ◽

Degang Zhao ◽

Yang Xiang ◽

Jingfang Huang

Keyword(s):

Boundary Conditions ◽

Long Range ◽

Numerical Scheme ◽

Fast Multipole Method ◽

Dislocation Core ◽

Elastic Interaction ◽

Epitaxial Thin Films ◽

Fast Multipole ◽

Multipole Method ◽

Grid Points

AbstractDislocations are line defects in crystalline materials. The Peierls-Nabarro models are hybrid models that incorporate atomic structure of dislocation core into continuum framework. In this paper, we present a numerical method for a generalized Peierls-Nabarro model for curved dislocations, based on the fast multipole method and the iterative grid redistribution. The fast multipole method enables the calculation of the long-range elastic interaction within operations that scale linearly with the total number of grid points. The iterative grid redistribution places more mesh nodes in the regions around the dislocations than in the rest of the domain, thus increases the accuracy and efficiency. This numerical scheme improves the available numerical methods in the literature in which the long-range elastic interactions are calculated directly from summations in the physical domains; and is more flexible to handle problems with general boundary conditions compared with the previous FFT based method which applies only under periodic boundary conditions. Numerical examples using this method on the core structures of dislocations in Al and Cu and in epitaxial thin films are presented.

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