A fast multipole method for Maxwell equations stable at all frequencies

Abstract

The solution of Helmholtz and Maxwell equations by integral formulations (kernel in exp(i kr)/r) leads to large dense linear systems. Using direct solvers requires large computational costs in O(N(3)). Using iterative solvers, the computational cost is reduced to large matrix-vector products. The fast multipole method provides a fast numerical way to compute convolution integrals. Its application to Maxwell and Helmholtz equations was initiated by Rokhlin, based on a multipole expansion of the interaction kernel. A second version, proposed by Chew, is based on a plane-wave expansion of the kernel. We propose a third approach, the stable-plane-wave expansion, which has a lower computational expense than the multipole expansion and does not have the accuracy and stability problems of the plane-wave expansion. The computational complexity is Nlog N as with the other methods.

A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain

“…A complete presentation of an elastodynamic FM-BEM formulation based on such transposition is the main purpose of this article. In particular, computational efficiency of fast elastodynamic BEMs in the mid-frequency regime is enhanced by using the so-called diagonal form for the Helmholtz Green's function [11,[43][44][45]; the upper limit stems from the fact that the size N becomes intractable at high frequencies, while the diagonal form breaks down at very low frequencies and must be replaced with other types of expansions [5,10,25]. Improving on [17], where the diagonal form was already adopted, the present work implements crucial features such as the adjustment of the truncation parameter in the multipole expansion to the subdivision level, known from recent studies for the Maxwell equations such as [8,25] to be necessary for achieving optimal computational efficiency.…”

Fast multipole method applied to 3-D frequency domain elastodynamics

“…This behavior is illustrated in Fig. 2 for several values of l. As a consequence, as discussed in [9,10], the FMM algorithm should avoid any evaluation of h (1) l (v) with l significantly larger than |v|.…”

“…The truncation parameters L P , L S depend upon the cell size through equation (9). Hence, in the multi-level framework, L P , L S are level-dependent: when applied at subdivision level ℓ, they are evaluated according to (9) with…”

“…Hence, in the multi-level framework, L P , L S are level-dependent: when applied at subdivision level ℓ, they are evaluated according to (9) with…”

“…The upper limit stems from the fact that the size N becomes intractable at high frequencies, while the diagonal form breaks down at very low frequencies and must be replaced with other types of expansions, based on e.g. the Wigner-3j symbols [8] or the so-called stable plane wave expansion [9]. The present work differs from [7] in details of the formulation that substantially impact the overall efficiency, and especially the fact that the truncation parameter in the multipole expansion should be adjusted to the subdivision level.…”