This review article concerns a methodology for solving numerically, for engineering purposes, boundary and initial-boundary value problems by a peculiar approach characterized by the following features: the continuous formulation is centered on integral equations based on the combined use of single-layer and double-layer sources, so that the integral operator turns out to be symmetric with respect to a suitable bilinear form. The discretization is performed either on a variational basis or by a Galerkin weighted residual procedure, the interpolation and weight functions being chosen so that the variables in the approximate formulation are generalized variables in Prager’s sense. As main consequences of the above provisions, symmetry is exhibited by matrices with a key role in the algebraized versions; some quadratic forms have a clear energy meaning; variational properties characterize the solutions and other results, invalid in traditional boundary element methods enrich the theory underlying the computational applications. The present survey outlines recent theoretical and computational developments of the title methodology with particular reference to linear elasticity, elastoplasticity, fracture mechanics, time-dependent problems, variational approaches, singular integrals, approximation issues, sensitivity analysis, coupling of boundary and finite elements, and computer implementations. Areas and aspects which at present require further research are identified, and comparative assessments are attempted with respect to traditional boundary integral-elements. This article includes 176 references.

https://hal.archives-ouvertes.fr/hal-00111259/document

Symmetric Galerkin Boundary Element Methods

A “symmetric” boundary element method based on a weighted residual Galerkin approach for elastoplastic analysis is revisited and its computer implementation for two-dimensional homogeneous problems is described.

A galerkin symmetric boundary‐element method in elasticity: Formulation and implementation

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.

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Recent Advances and Emerging Applications of the Boundary Element Method

The simple boundary element method for transient heat conduction in functionally graded materials

The direct boundary element method based on the Rayleigh-Green identity is employed for the static analysis of Kirchhoff plates. The starting point is a slightly modified version of Stern's equations. The focus is on the implementation of the method for linear elements and a Hermitian interpolation for the deflection w. The concept of element matrices is developed and the Cauchy principal values of the singular integrals are given in detail. The treatment of domain integrals, the handling of internal supports, the properties of the solution and the effect of singularities are discused. Numerical examples illustrate the various techniques. In the appendix the influence functions for the second and third derivatives of the deflection w are given.

The direct boundary element method in plate bending

This paper discusses the deficiencies of boundary element stiffness matrices, gives an account of the methods proposed to circumvent these defects and proposes a new procedure to obtain symmetric stiffness matrices.

B. Protopsaltis

Papers

Boundary elements and symmetry