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Review Articles

Recent Advances and Emerging Applications of the Boundary Element Method

[+] Author and Article Information
Y. J. Liu

Mechanical Engineering, University of Cincinnati,
598 Rhodes Hall, 2600 Clifton Avenue,
Cincinnati, OH 45221-0072
e-mail: yijun.liu@uc.edu

S. Mukherjee

Sibley School of Mechanical and
Aerospace Engineering, Cornell University,
Ithaca, NY 14853
e-mail: sm85@cornell.edu

N. Nishimura

Department of Applied Analysis and
Complex Dynamical Systems,
Graduate School of Informatics,
Kyoto University,
Kyoto 606-8501, Japan
e-mail: nchml@i.kyoto-u.ac.jp

M. Schanz

Institute of Applied Mechanics,
Graz University of Technology,
Technikerstr. 4, A-8010 Graz, Austria
e-mail: m.schanz@tugraz.at

W. Ye

Department of Mechanical Engineering,
Hong Kong University of Science and
Technology, Clearwater Bay, Kowloon,
Hong Kong, China
e-mail: mewye@ust.hk

A. Sutradhar

Department of Surgery and
Department of Mechanical Engineering,
The Ohio State University,
915 Olentangy River Road, Suite 2100,
Columbus, OH 43212
e-mail: Alok.Sutradhar@osumc.edu

E. Pan

Department of Civil Engineering,
The University of Akron, ASEC Room 534,
Akron, OH 44325-3905
e-mail: pan2@uakron.edu

N. A. Dumont

Pontifical Catholic University of Rio de Janeiro,
22451-900 Brazil
e-mail: dumont@puc-rio.br

A. Frangi

Department of Structural Engineering,
Politecnico of Milano, P.za L. da Vinci 32,
20133 Milano, Italy
e-mail: attilio.frangi@polimi.it

A. Saez

Department of Continuum Mechanics and
Structural Analysis, University of Seville,
Camino de los Descubrimientos s/n,
Seville, E-41092, Spain
e-mail: andres@us.es

1Corresponding author.

Manuscript received May 25, 2011; final manuscript received December 12, 2011; published online March 30, 2012. Transmitted by Editor: J. N. Reddy.

Appl. Mech. Rev 64(3), 030802 (Mar 30, 2012) (38 pages) doi:10.1115/1.4005491 History: Received May 25, 2011; Revised December 12, 2011

Sponsored by the U.S. National Science Foundation, a workshop on the boundary element method (BEM) was held on the campus of the University of Akron during September 1–3, 2010 (NSF, 2010, “Workshop on the Emerging Applications and Future Directions of the Boundary Element Method,” University of Akron, Ohio, September 1–3). This paper was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Green's functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the fast multipole method, the pre-corrected fast Fourier transformation method, and the adaptive cross approximation method are presented. Emerging applications of the BEM in solving microelectromechanical systems, composites, functionally graded materials, fracture mechanics, acoustic, elastic and electromagnetic waves, time-domain problems, and coupled methods are reviewed. Finally, future directions of the BEM as envisioned by the authors for the next five to ten years are discussed. This paper is intended for students, researchers, and engineers who are new in BEM research and wish to have an overview of the field. Technical details of the BEM and related approaches discussed in the review can be found in the Reference section with more than 400 papers cited in this review.

Copyright © 2011 by ASME
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References

Figures

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Fig. 1

A 3D domain V with boundary S

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Fig. 2

Illustration of the Galerkin weight functions for 2D BEM

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Fig. 3

Domain of dependence and range of influence. (a) The nodes 1, 2 and 3 lie within the DOD of the evaluation point E. The ROIs of nodes 1, 2, 3, 4 and 5 are shown as gray regions. In the standard BNM, the ROI of a node near an edge, e.g., node 4, is truncated at the edges of a panel. In the EBNM, the ROI can reach over to neighboring panels and contain edges and/or corners - see, e.g., node 5 (b) Gaussian weight function defined on the ROI of a node (from Ref. [149]).

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Fig. 4

ONE-step multilevel BNM cell refinement for the “sinsinh cube” problem: (a) initial configuration with 96 surface cells; (b) final adapted configuration (obtained in one step) with 1764 surface cells (from Ref. [131])

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Fig. 5

A hierarchical cell structure covering all the boundary elements

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Fig. 6

A hierarchical quad-tree structure for the 2D boundary element mesh

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Fig. 7

(a) 2D view of a parallelepiped superimposed onto a discretized sphere. The black dots indicate grid points and the cubes are represented by the solid lines; (b) Illustration of the four steps in the pFFT scheme; the gray region denotes the near-field region of the yellow element.

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Fig. 8

H-matrix with 3279 entries and rank numbers of ACA

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Fig. 9

Two parallel conducting beams

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Fig. 10

SEM pictures of (a) a laterally oscillating beam resonator, and (b) a biaxial accelerometer

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Fig. 11

Matrix domain V0 and n inclusions

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Fig. 12

2D models of fiber composites: (a) circular-shaped fibers, (b) star-shaped fibers (Ref. [44])

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Fig. 13

A 3D RVE with 5832 fibers and with the total DOFs = 10,532,592 [249]

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Fig. 14

Analysis of an FGM rotor part using the BEM [263]

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Fig. 15

Arrays of 12 × 12 × 12 penny-shaped cracks in an elastic domain solved by using the fast multipole BEM (total number of DOFs = 1,285,632) [297]

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Fig. 16

A BEM model of the Skipjack submarine impinged upon by an incident wave

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Fig. 17

Sound field from five wind turbines evaluated using the ACA and fast multipole BEM

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Fig. 18

BEM model of 1152 spherical cavities in an infinite elastic solid [343]. The spatial DOF is 1,105,290 and the number of time steps is 200. The CPU time is 10 h 47 min and the memory requirement is 152.8 GB.

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Fig. 19

Skin of worm: (a) model; (b) energy transmittance

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Fig. 20

Sound pressure Level (dB) at the boundary of the amphitheatre

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