The fast multipole boundary element method for potential problems: A tutorial

Liu, Yijun; Nishimura, Naoshi

doi:10.1016/j.enganabound.2005.11.006

Cited by 182 publications

(107 citation statements)

References 17 publications

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“…We note that FMMs have been used previously for boundary integrals and BEM [45][46][47] , as well as combining with GMRES 8, 48 . However, there are few publicly available parallel codes using these techniques, in particular highly efficient and scalable ones.…”

Section: Resultsmentioning

confidence: 99%

An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles

Jiang

Zhao

et al. 2016

The Journal of Chemical Physics

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we solve a boundary value problem for a ferroelectric/ferromagnetic volume in free space. In the second, we solve an electrostatic problem involving polarizable dielectric bodies in an unbounded dielectric medium. The results from these test cases show that our proposed parallel approach, which is built on a kernel-independent FMM, can enable highly efficient and accurate simulations and allow for considerable flexibility in a broad range of applications.

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Section: Resultsmentioning

confidence: 99%

An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles

Jiang

Zhao

et al. 2016

The Journal of Chemical Physics

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“…(36) to give sense to point wise evaluation of the single and double layer operators defined in Eqs. (30) and (31).…”

Section: Isogeometric Collocation Bem For Stokes Flowsmentioning

confidence: 99%

“…In the literature, the first issue is usually tackled by acceleration techniques such as the fast multipole method [39] (see [31] for a survey of this technique applied to BEM), which allow one to solve the system without assembling the full matri ces, and at the same time reduce the cost of the matrix vector multiplication from Oðn 2 Þ to OðnÞ (this technique has already been applied to Laplace problems in two dimensional IGA BEM in [44]). The second issue is more delicate and its solution depends highly on the degree of the discrete functional spaces and of the geometry description, remaining one of the most difficult aspects of the implementation of BEM.…”

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confidence: 99%

Nonsingular isogeometric boundary element method for Stokes flows in 3D

Heltai

Arroyo

DeSimone

2014

Computer Methods in Applied Mechanics and Engineering

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While the finite element isogeometric analysis (FE IGA) has received most of the attention of the scientific community, only few results are available for boundary element methods (BEM) and boundary integral equations (BIE) in general. In 2D, there are some works on potential flows, such as [36,44], and on elastostatic analysis [40]. Three dimensional appli cations are of great interest in the maritime community, and there are some works on NURBS based panel methods to study, for example, marine propellers [25] or the wavemaking resistance problem [10]. Three dimensional IGA BEM linear elasticity has recently been studied in [20] while some work on shape optimization has been presented in [29]. Boundary element isogeometric analysis (IGA BEM) is very attractive for the solution of homogeneous elliptic PDEs, both in the interior and in the exterior cases, since it requires the solution of integral equations only on the boundary of the enclos ing (or enclosed) domain, which is typically the only information that is generated by standard CAGD tools. In contrast, FE IGA requires the extension of the computational domain inside (or outside) the enclosing CAGD surface, an outstand ing problem in IGA.While the dimensional reduction of the domain diminishes significantly both the number of degrees of freedom and the computational cost associated with the handling of the geometry, there are two main drawbacks which require special attention, and are common to all boundary integral discretizations: (i) the system matrices that are produced with a bound ary integral method are full, and (ii) the integrands contain ð1=rÞ singularities or worse (in three dimensions) around the source point, arising from the fundamental solutions in formulating the boundary integrals. Both issues have perhaps con tributed in keeping the IGA community away from boundary integral formulations.In the literature, the first issue is usually tackled by acceleration techniques such as the fast multipole method [39] (see [31] for a survey of this technique applied to BEM), which allow one to solve the system without assembling the full matri ces, and at the same time reduce the cost of the matrix vector multiplication from Oðn 2 Þ to OðnÞ (this technique has already been applied to Laplace problems in two dimensional IGA BEM in [44]). The second issue is more delicate and its solution depends highly on the degree of the discrete functional spaces and of the geometry description, remaining one of the most difficult aspects of the implementation of BEM. Different approaches are possible, (see, for example, [23]) but high order methods and curved geometries still remain challenging.Popular approaches to tackle singular integration involve a local change of variables for the quadrature rules, following the principle that the Jacobian of the transformation can be used to eliminate locally the kernel singularities. Examples of these methods are given by Lachat and Watson [28] and by Telles [45]. Both methods are elegant and produce accurate re sults, but t...

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“…The expansions used for 2D potential problem for FMM are briefly summarized as table 2. Further details about the analytical derivations should be attained in [5,7]. Table 2.…”

Section: Fast Multipole Boundary Elementmentioning

confidence: 99%

“…Thereafter this new technique was applied for solving elasticity [3] and fluids [4] problems in large-scale. Some years after, [5] announced the FMM as one of the top algorithms in scientific computing that were developed in the 20th century. In this publication the authors had developed a complete tutorial which presents the basic concept and the main procedures in the FMM for solving boundary integral equations for 2D potential problems.…”

Section: Introductionmentioning

confidence: 99%

Optimization Using Topological Derivative and Boundary Element Method With Fast Multipole

Braga¹,

Anflor²,

Albuquerque³

2014

Proceedings of 10th World Congress on Computational Mechanics

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Abstract. The objective of this work is to compare topologies resulting from direct BEM (Boundary Element Method) with a BEM accelerated by Fast Multipole Method (FMM). A formulation of fast multipole boundary element (FMBEM) is introduced in order to turn the optimization process more attractive in the point of view of the computational cost. The formulation of the fast multipole is briefly summarized. A topological-shape sensitivity approach is used to select the points showing the lowest sensitivities, where material is removed by opening a cavity. As the iterative process evolves, the original domain has holes progressively removed, until a given stop criteria is achieved. A benchmark is investigated by

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The fast multipole boundary element method for potential problems: A tutorial

Cited by 182 publications

References 17 publications

An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles

An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles

Nonsingular isogeometric boundary element method for Stokes flows in 3D

Optimization Using Topological Derivative and Boundary Element Method With Fast Multipole

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