A wideband fast multipole method for the Helmholtz equation in three dimensions

Cheng, Hui-Wen; Crutchfield, William Y.; Gimbutas, Zydrunas; Greengard, Leslie; Ethridge, J. Frank; Huang, Jingfang; Rokhlin, Vladimir; Yarvin, Norman; Zhao, Junsheng

doi:10.1016/j.jcp.2005.12.001

Cited by 246 publications

(243 citation statements)

References 27 publications

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“…The integrals in the layer potentials on each triangle were computed by "singularity subtraction": analytically for the static kernel 1/|x − y| and by high order Gaussian quadrature for the smoother remainder (e ik|x−y| − 1)/|x − y|. The calculation was accelerated by making use of the low frequency fast multipole method (FMM), described in [30]. FMM-accelerated integral equation methods permit the solution of electromagnetic scattering problems in O(N log N) time, where N denotes the number of degrees of freedom in the discretization of the scatterer.…”

Section: Calculation Of Electromagnetic Fieldsmentioning

confidence: 99%

Visualizing skin effects in conductors with MRI: 7Li MRI experiments and calculations

Ilott

Chandrashekar

Klöckner

et al. 2014

Journal of Magnetic Resonance

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While experiments on metals have been performed since the early days of NMR (and DNP), the use of bulk metal is normally avoided. Instead, often powders have been used in combination with low fields, so that skin depth effects could be neglected. Another complicating factor of acquiring NMR spectra or MRI images of bulk metal is the strong signal dependence on the orientation between the sample and the radio frequency (RF) coil, leading to non-intuitive image distortions and inaccurate quantification. Such factors are particularly important for NMR and MRI of batteries and other electrochemical devices. Here, we show results from a systematic study combining RF field calculations with experimental MRI of 7 Li metal to visualize skindepth effects directly and to analyze the RF field orientation effect on MRI of bulk metal. It is shown that a certain degree of selectivity can be achieved for particular faces of the metal, simply based on the orientation of the sample. By combining RF field calculations with bulk magnetic susceptibility calculations accurate NMR spectra can be obtained from first principles. Such analyses will become valuable in many applications involving battery systems, but also metals, in general.

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Section: Calculation Of Electromagnetic Fieldsmentioning

confidence: 99%

Visualizing skin effects in conductors with MRI: 7Li MRI experiments and calculations

Ilott

Chandrashekar

Klöckner

et al. 2014

Journal of Magnetic Resonance

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“…Trapezoidal quadrature rules are well suited to this type of integral, and are used here; they were also used in [7] for radial Fourier integrals involvingĜ given by (16a,b).…”

Section: Angular Integrationmentioning

confidence: 99%

“…We adopt instead the methodology proposed by Rokhlin and coauthors [4,8,36], which generates generalized Gaussian quadrature (GGQ) rules for specific types of integrals and is in particular useful for implementing some forms of the FMM. For instance, GGQ-generating algorithms have been developed in [36] and [8], respectively, for integrals involved in the low-frequency [19] and wideband [7] versions of the FMM for the 3D Helmholtz equation. The tabulated GGQ rules given…”

mentioning

confidence: 99%

A new Fast Multipole formulation for the elastodynamic half-space Greenʼs tensor

Chaillat

Bonnet

2014

Journal of Computational Physics

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Abstract. In this article, a version of the frequency-domain elastodynamic Fast Multipole-Boundary Element Method (FM-BEM) for semi-infinite media, based on the half-space Green's tensor (and hence avoiding any discretization of the planar traction-free surface), is presented. The half-space Green's tensor is often used (in non-multipole form until now) for computing elastic wave propagation in the context of soil-structure interaction, with applications to seismology or civil engineering. However, unlike the fullspace Green's tensor, the elastodynamic half-space Green's tensor cannot be expressed using derivatives of the Helmholtz fundamental solution. As a result, multipole expansions of that tensor cannot be obtained directly from known expansions, and are instead derived here by means of a partial Fourier transform with respect to the spatial coordinates parallel to the free surface. The obtained formulation critically requires an efficient quadrature for the Fourier integral, whose integrand is both singular and oscillatory. Under these conditions, classical Gaussian quadratures would perform poorly, fail or require a large number of points. Instead, a version custom-tailored for the present needs of a methodology proposed by Rokhlin and coauthors, which generates generalized Gaussian quadrature rules for specific types of integrals, has been implemented. The accuracy and efficiency of the proposed formulation is demonstrated through numerical experiments on single-layer elastodynamic potentials involving up to about N = 6×10 5 degrees of freedom. In particular, a complexity significantly lower than that of the non-multipole version is shown to be achieved. IntroductionThe main advantage of the boundary element method (BEM) is that only the domain boundaries (and possibly interfaces) are discretized, leading to a reduction of the number of degrees of freedom (DOFs) relative to domain-discretization methods. Moreover, elastodyamic field equations and radiation conditions are exactly satisfied by the formulation [26], which avoids cumulative effects of grid dispersion and makes the BEM well suited for modeling wave propagation in unbounded domains. However, the standard BEM [3] leads to fully-populated matrices, which results in high computational complexity (O(N 2 ) per iteration using an iterative solver such as GMRES, where N denotes the number of boundary degrees of freedom (DOFs) of the BE model) and severe problem size limitations induced by the O(N 2 ) size of the influence matrix.The advent of accelerated BE methodologies has dramatically improved the capabilities of BEMs for many areas of application, in a large part owing to the rapid development of the Fast Multipole Method (FMM) over the last 15-20 years [29]. Such approaches have resulted in considerable solution speedup, memory reduction and model size increase. The FMM inherently relies on iterative solvers (usually GMRES), so as to avoid actual computation and storage of the fully-populated influence matrix, and is known to require O(N log N )...

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“…The recent progress on the analysis and implementation of numerical-asymptotic hybridization of the boundary integral methods for the acoustic scattering problems was systemically given. In [56], the wideband fast multipole method for solving the Helmholtz equation in three dimensions was studied. In the high frequency regime, on invoking the fast multipole method, the computational workload is O(N log N ).…”

Section: Introductionmentioning

confidence: 99%

THE MODERN HIGH FREQUENCY METHODS FOR SOLVING ELECTROMAGNETIC SCATTERING PROBLEMS (Invited Paper)

Wu¹,

Chew²

2016

PIER

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Abstract-The high frequency scattering problems of electromagnetic fields scattered from electrically large scatterers are important and challenging. On the calculation of the reflected and diffracted wave fields, the high frequency methods could be classified into the current based method and the ray based method. In this paper, first, we give a review on the progress of the modern high frequency methods for solving the electromagnetic scattering problems. Next, due to the highly oscillatory property of the high frequency electromagnetic scattered fields, we propose the numerical steepest descent path method. Finally, we comprehensively address the high frequency wave physics, including the high frequency critical point contributions, the Keller's cone, the shadow and reflection boundaries and the creeping wave fields.

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A wideband fast multipole method for the Helmholtz equation in three dimensions

Cited by 246 publications

References 27 publications

Visualizing skin effects in conductors with MRI: 7Li MRI experiments and calculations

Visualizing skin effects in conductors with MRI: 7Li MRI experiments and calculations

A new Fast Multipole formulation for the elastodynamic half-space Greenʼs tensor

THE MODERN HIGH FREQUENCY METHODS FOR SOLVING ELECTROMAGNETIC SCATTERING PROBLEMS (Invited Paper)

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