Numerical accuracy of multipole expansion for 2-D MLFMA

Abstract.We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave e iŝ· v in terms of spherical harmonics {Y ,m (ŝ)} |m|≤ ≤∞ . We consider the truncated series where the summation is performed over the ( , m)'s satisfying |m| ≤ ≤ L. We prove that if v = | v| is large enough, the truncated series gives rise to an error lower than as soon aswhere W is the Lambert function and C , K, δ, γ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates for the error in the fast multipole method for scattering computation.Mathematics Subject Classification. 33C10, 33C55, 41A80.

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“…This numerical result agrees with our analysis. gives an error asymptotically less than when v is large (this agrees with the results in [18]). …”

Section: A Uniform Estimatesupporting

confidence: 89%

Error estimates in the fast multipole method for scattering problems Part 1: Truncation of the Jacobi-Anger series

Carayol

Collino

2004

ESAIM: M2AN

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“…A similar approach, based on fine-tuning the coefficient C of another empirical truncation formula with a O((kd) 1/3 ) term replacing the logarithmic term of (9), is reported in [18]. Using either approach, L is roughly proportional to the cell size.…”

Section: Numerical Quadrature Over the Unit Spherementioning

confidence: 99%

Fast multipole method applied to 3-D frequency domain elastodynamics

Sanz

Bonnet

Domínguez

2008

Engineering Analysis with Boundary Elements

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This article is concerned with the formulation and implementation of a fast multipoleaccelerated BEM for 3-D elastodynamics in the frequency domain, based on the so-called diagonal form for the expansion of the elastodynamic fundamental solution, a multi-level strategy. As usual with the FM-BEM, the linear system of BEM equations is solved by GMRES, and the matrix is never explicitly formed. The truncation parameter in the multipole expansion is adjusted to the level, a feature known from recent published studies for the Maxwell equations. A preconditioning strategy based on the concept of sparse approximate inverse (SPAI) is presented and implemented. The proposed formulation is assessed on numerical examples involving O(10 5 ) BEM unknowns, which show in particular that, as expected, the proposed FM-BEM is much faster than the traditional BEM, and that the GMRES iteration count is significantly reduced when the SPAI preconditioner is used.

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“…Therefore, the group size can not be too small in order to meet a specific error criteria. This fact has been defined as the low frequency break down problem for the zero-order Hankel function [25]. For traditional FMM, this problem is not so critical since only zero-order Hankel function is involved and the group size can be as small as half wavelength.…”

Section: Implementation Of the Fma-smmmentioning

confidence: 99%

Fast Multipole Accelerated Scattering Matrix Method for Multiple Scattering of a Large Number of Cylinders

Zhang¹,

Li²

2007

PIER

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Abstract-The lowering and raising operators of cylindrical harmonics are used to derive the general fast multipole expressions of arbitrary order Hankel functions. These expressions are then employed to transform the dense matrix in the scattering matrix method (SMM) into a combination of sparse matrices (aggregation, translation and disaggregation matrices). The novel method is referred to as fast multipole accelerated scattering matrix method (FMA-SMM). Theoretical study shows FMA-SMM has lower complexity O(N 1.5 ) instead of SMM's O(N 2 ), where N stands for total harmonics number used. An empirical formula is derived to relate the minimum group size in FMA-SMM to the highest order Hankel functions involved. The various implementation parameters are carefully investigated to guarantee the algorithm's accuracy and efficiency. The impact of the cylinders density on convergence rate of iterative solvers (BiCGStab(2) here), memory cost as well as CPU time is also investigated. Up to thousands of cylinders can be easily simulated and potential applications in photonic crystal devices are illustrated. 106 Zhang and Li

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Numerical accuracy of multipole expansion for 2-D MLFMA

Cited by 24 publications

References 21 publications

Error estimates in the fast multipole method for scattering problems Part 1: Truncation of the Jacobi-Anger series

Error estimates in the fast multipole method for scattering problems Part 1: Truncation of the Jacobi-Anger series

Fast multipole method applied to 3-D frequency domain elastodynamics

Fast Multipole Accelerated Scattering Matrix Method for Multiple Scattering of a Large Number of Cylinders

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