Recent Advances and Emerging Applications of the Boundary Element Method

Abstract

<jats:p>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.</jats:p>

A fast boundary element method for the solution of periodic many-inclusion problems via hierarchical matrix techniques

“…[12,16,25]. A related investigation on periodic problems for the Stokes equation can be found in [17].…”

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

“…The latter are known since a long time for many classical problems of physics such as electricity and magnetism, heat transfer, fluid flow, mechanics of deformable solids... As it exploits the conversion of partial differential equations supported on domains to integral equations supported on domain boundaries, the BEM is a mesh reduction method, subject to restrictive constitutive assumptions but yielding highly accurate solutions. The reader is referred to the recent review article [54] for an abundant bibliography on the general topic of BIE/BEM formulations which includes historical aspects and recent developments, see also [11,13].…”

Is the Burton–Miller formulation really free of fictitious eigenfrequencies?

“…In contrast to the finite element method (FEM), BEM has some advantages, such as the high accuracy, the reduction of dimensionality by one and the incomparable superiority in solving semiinfinite or infinite wave propagation problems [1,2]. However, some inherent shortcomings also exist [1,2]. One of them is the fictitious eigenfrequency problem, also called the non-uniqueness difficulty or the non-unique solution difficulty.…”