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.

SUMMARYThis paper brings an attempt toward the systematic solution of the generalized non-linear, complexsymmetric eigenproblem, which are associated with the dynamic governing equations of a structure submitted to viscous damping, as laid out in the frame of an advanced mode superposition technique. The problem can be restated as (are complex-symmetric matrices given as power series of the complex eigenfrequencies , such that, if ( , /) is a solution eigenpair,The traditional Rayleigh quotient iteration and the more recent JacobiDavidson method are outlined for complex-symmetric linear problems and shown to be mathematically equivalent, both with asymptotically cubic convergence. The Jacobi-Davidson method is more robust and adequate for the solution of a set of eigenpairs. The non-linear eigenproblem subject of this paper can be dealt with in the exact frame of the linear analysis, thus also presenting cubic convergence. Two examples help us to visualize some of the basic concepts developed. Three more examples illustrate the applicability of the proposed algorithm to solve non-linear problems, in the general case of underdamping, but also for overdamping combined with multiple and close eigenvalues.

The most important features of a new numerical method are outlined. The mechanical, or variational consistency of the hybrid finite element method is extended to the conventional boundary element formulation, giving rise to naturally established symmetric force-displacement relations. The computational effort for the complete solution of a given problem, according to this method, is in some cases only a small fraction of the effort needed with traditional methods. This paper also outlines briefly the types of analyses which may be advantageously performed with this new method, many of which are already being implemented by the author and co-workers. Some numerical examples are provided.

The hybrid stress boundary element method (HSBEM) was introduced in 1987 on the basis of the Hellinger-Reissner potential, as a generalization of Pian's hybrid finite element method.This new two-field formulation makes use of fundamental solutions to interpolate the stress field in the domain of an elastic body, which ends up discretized as a superelement with arbitrary shape and arbitrary number of degrees of freedom located along the boundary. More recently, a variational counterpart -the hybrid displacement boundary element method (HDBEM) -was proposed, on the basis of three field functions, with equivalent advantages. The present paper discusses these methods as well as the traditional, collocation boundary element method (CBEM). The mechanical properties of the resulting matrix equations are investigated and a series of concepts in both HDBEM and CBEM that have not been properly considered by previous authors, particularly in which concerns body forces, are redefined. This is not a review paper, but rather a theoretical, comparative analysis of three methods, with many physical considerations, some innovations and a few academic illustrations.

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