Symmetric Galerkin Boundary Element Methods

Abstract

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.

Numerical methods for hydraulic fracture propagation: A review of recent trends

“…Among those, the Symmetric Galerkin BEM (SGBEM) is a conforming method to enforce variable continuity across element boundaries based on shape function like in the finite element method (see e.g. Bonnet et al (1998) for more details). The SGBEM formulation with dis-placement discontinuities as the main variables for the discretization of the fracture have been used to solve for the elastic deformation in some 3D hydraulic fracture simulators (Rungamornrat et al, 2005;Xu and Wong, 2013).…”

Three‐dimensional non‐planar crack growth by a coupled extended finite element and fast marching method

“…In the coupled method, the fast marching method maintains the location and motion of the crack front via signed distance functions, whereas the X-FEM is used to compute the local front velocity. In keeping with standard level set notation, we use in three dimensions are: finite element methods [12,13], boundary element-based techniques [14][15][16][17][18][19], and boundary integral equations [20,21]. Gao and Rice [22] and Lai et al [23] used perturbation analysis to study planar and non-planar cracks, whereas Lazarus and coworkers [24][25][26] conducted planar crack growth simulations.…”

Exploiting partial or complete geometrical symmetry in 3D symmetric Galerkin indirect BEM formulations

“…The operation [T n Aφ](x) gives rise to a hypersingular kernel involving a r −3 singularity. After a well-documented regularization process [3,7] involving two integrations by parts over ∂Ω, the resulting SGBIE formulation, on which the present development is based, reads:…”