Browse Prior Art Database

Multipole Computations with Periodic Boundary Conditions

IP.com Disclosure Number: IPCOM000106000D
Original Publication Date: 1993-Sep-01
Included in the Prior Art Database: 2005-Mar-20
Document File: 2 page(s) / 51K

Publishing Venue

IBM

Related People

Berman, CL: AUTHOR [+2]

Abstract

A technique is disclosed which permits efficient computation of certain lattice sums of spherical harmonics which are needed to apply multipole algorithms in the context of periodic boundary conditions. Possible applications include computations concerning crystal lattices or particle simulations with Dirichlet boundary conditions.

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Multipole Computations with Periodic Boundary Conditions

      A technique is disclosed which permits  efficient computation
of certain lattice sums of spherical harmonics which are needed to
apply multipole algorithms in the context of periodic boundary
conditions.  Possible applications include computations concerning
crystal lattices or particle simulations with Dirichlet boundary
conditions.

      The method makes use two linear transformations.  The first F2F
maps a Laurent expansion of the solution to Laplace's equation
(centered at point P) to one centered at the origin.  The second,
F2L, maps a Laurent expansion (centered at point P) to a Taylor
expansion centered at the origin.  The exact form of both of these
linear transformations is given in [*].  In addition, other
terminology (such as well-separated) which is defined in [*]  is
used.

      In order to apply multipole methods under periodic boundary
conditions it is necessary to determine the local field due to the
well-separated copies of the unit cube.  This summation gives rise to
another linear transformation, OMEGA-1, which is determined by the
lattice sums desired for computation.  The exact form of OMEGA is
obtained by applying F2L to cubes centered at all lattice points and
summing the result.

      The method is based on an alternative method of computing
OMEGA-1 using the observation that the infinite array of unit cubes
can also be considered as an infinite array of cubes...