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• W2573778312 abstract "The Fast Multipole Method (FMM) is in particular applied to many physical problems governed by equations of Helmholtz type which arise for e.g. linear acoustics, electromagnetic or elastic waves upon using Fourier transform in time or applying excitations under prescribed-frequency conditions. All cases involve integral equations whose (scalar or tensorial) kernels are defined, for three-dimensional formulations, in terms of the fundamental solution K(r) = exp(ikr)/4πr and its derivatives. The FMM crucially rests upon expansions of K(r) that allow a separation of variables between the “field” point y and the “source” point x (with r = y− x), which are available in several forms. These include the multipole expansion, based on the Gegenbauer addition theorem, and the plane-wave expansion of K(r) in diagonal form. For real values of the wavenumber k, which correspond to wave propagation problems in lossless media, FMMs based on either type of expansion have been extensively studied and implemented. These studies have in particular established that multipole expansions of K(r) are too costly except at low frequencies (because the expansion truncation threshold increases with k), while planewave expansions are well-suited to higher frequencies but break down in the low-wavenumber limit. It is also known that in the zero-wavenumber limit (i.e. static problems) an a-priori indication on the number N of multipoles to be employed in truncated expansions in order to obtain a prescribed accuracy in K(r), unaffected by the dimensions of the cells in the octree, is usually available. This nice feature (which ensures a O(ndof) complexity per iteration for multi-level FMMs applied to static problems) is no longer present in frequency-domain applications ([5, 1, 2]), where the implementation of the method is substantially complicated by the need to adapt the truncation order to the level of the octree (resulting in a O(ndof logndof) complexity per iteration). In contrast, only scattered efforts so far have been devoted to FMMs for Helmholtz-type problems involving complex wavenumbers k. Such formulations involve wavenumbers of the form k = (α+ iβ)θ (where θ is a parameter related to frequency and α,β are real constants). The paucity of available studies notwithstanding, complex-wavenumber FMMs arise for a number of different physical problems. First, such equations and kernels arise naturally upon considering wave propagation in lossy media (e.g. soils) in which mechanical or electromagnetic waves are damped. Such materials (e.g. viscoelastic materials) may be described, within frequency-domain approaches, in terms of linear constitutive relations involving complex moduli. This leads to complex-valued wavenumbers such that 0 < β < α, often with a small imaginary part, i.e. such that β/α 1, and represents the most direct generalization of Helmholtz-type equations with real wavenumbers. Another class of such problems correspond to parabolic problems involving elliptic partial differential operators in the space variables and first-order time derivatives, upon using Fourier transform in time or applying excitations at a prescribed frequency. They include heat conduction, transient Stokes flows, and eddy currents, and the associated Green’s functions or tensors involve K(r) with α = β = 1. Transient Stokes flows arise e.g. in connection with the analysis of dissipation in Micro-Systems. Also amenable to the general framework of Helmholtz-type equations (with purely imaginary frequency) is the computation of Casimir forces, which are attractive force arising between uncharged conductive surfaces in vacuum, a remarkable consequence of quantum electrodynamics In spite of its complicated nature, this phenomenon is now acquiring technological importance, as it has a major role in modern microand nano-technologies." @default.
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• W2573778312 date "2010-01-01" @default.
• W2573778312 modified "2023-10-18" @default.
• W2573778312 title "On the application of Fast Multipole Methods to Helmholtz problems with complex wavenumber" @default.
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