A fast algorithm for particle simulations
Journal of Computational Physics
Journal of Computational Physics
Multipole translation theory for the three-dimensional Laplace and Helmholtz equations
SIAM Journal on Scientific Computing
A fast adaptive multipole algorithm in three dimensions
Journal of Computational Physics
A vortex particle method for two-dimensional compressible flow
Journal of Computational Physics
SIAM Journal on Scientific Computing
Fast multipole method for the biharmonic equation in three dimensions
Journal of Computational Physics
A scalar potential formulation and translation theory for the time-harmonic Maxwell equations
Journal of Computational Physics
A fast multipole method for the three-dimensional Stokes equations
Journal of Computational Physics
Fast multipole methods on graphics processors
Journal of Computational Physics
Scalable fast multipole methods on distributed heterogeneous architectures
Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis
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Vortex methods are used to efficiently simulate incompressible flows using Lagrangian techniques. Use of the FMM (Fast Multipole Method) allows considerable speed up of both velocity evaluation and vorticity evolution terms in these methods. Both equations require field evaluation of constrained (divergence free) vector valued quantities (velocity, vorticity) and cross terms from these. These are usually evaluated by performing several FMM accelerated sums of scalar harmonic functions. We present a formulation of vortex methods based on the Lamb-Helmholtz decomposition of the velocity in terms of two scalar potentials. In its original form, this decomposition is not invariant with respect to translation, violating a key requirement for the FMM. One of the key contributions of this paper is a theory for translation for this representation. The translation theory is developed by introducing ''conversion'' operators, which enable the representation to be restored in an arbitrary reference frame. Using this form, efficient vortex element computations can be made, which need evaluation of just two scalar harmonic FMM sums for evaluating the velocity and vorticity evolution terms. Details of the decomposition, translation and conversion formulae, and sample numerical results are presented.