Sentence examples for using the multipole expansion from inspiring English sources

Multiple scattering effects are described using the multipole expansion formalism and the addition theorem of cylindrical wave functions.

The time taken by the M2L kernel (using double precision without vectorisation) as function of expansion order p. The rotation accelerated version of Appendix A.6.2 is faster than the direct implementation of equation (3b) for p ≥ 5. Let us first consider the tree code, which also uses the multipole expansion but is algorithmically simpler than FMM.

This method arises from an observation that potential evaluation using the multipole to local expansion translation operator can be expressed as a series discrete convolutions of the multipole moments with their associated spherical harmonics functions.

A sextupole-like magnetic field could be excited when using the multipole magnet.

In that case, the two-dimensional borehole thermal resistance was determined using the multipole method, with recent improvements about boundary conditions (Claesson and Hellström 2011).

Finally, the dual tree walk is performed using the multipole-acceptance criterion θ < θ crit (5).

The multipole expansion method using vector spherical harmonics is utilized and appropriate beam-shape coefficients are derived in order to compute the radiation force cross-sections.

The multipole expansion is used to evaluate the integrals that are far away from the collocation point, whereas the direct approach is applied to evaluate the integrals on the remaining segments that are close to the collocation point.

Based on the multipole expansion of the kernel, the FMM can be used to accelerate the matrix-vector multiplication.

Based on the multipole expansion of kernel, we demonstrate that the approximate operator used in the collocation equation converges to the initial operator.

The key compression step in the H-FMM is based on a fact that the multipole expansion for the sources of the free-space Green's function can be used also to compress the far field of the sources of the layered-media or domain Green's function, and a similar result exists for the translation operators for the multipole and local expansions.