Symmetric Galerkin Boundary Element Method

Abstract

This review concerns a methodology for solving numerically, to 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, computer implementations. Areas and aspects which at present require further research are identified and comparative assessments are attempted with respect to traditional boundary integral-element methods.

Optical Properties of Metallic Nanoparticles

“…The method has remote deep roots in the history of mechanics 5 and its mathematical foundations include the theorems of Gauss, Green and Stokes -they allow the basic reduction from volume differential equations to boundary integral equations [100] as we have already seen in Chap. 3.…”

“…Examples of application cover the fields of elasticity, geomechanics, structural mechanics, electromagnetics, acoustics, hydraulics, biomechanics, and much more 5. A short overview about the historical development of the BEM can be found in[100] for example.…”

Application of Collocation Bem for Axisymmetric Transmission Problems in Electro- And Magnetostatics

“…Beginning in the mid-1970's, collocation solutions of integral equations for axisymmetric problems have been extensively considered in the literature, see e.g. [19,Chapter 6]. However, we did not find in the literature any theoretical result about convergence of collocation methods in the meridian plane.…”

“…Here ξ = (r, z), ξ 0 = (r 0 , z 0 ), n = (n r , n z ) = (−z , r )/|ψ (t)| the unit outward normal on Γ for the clock-wise change of the parameter t. An axisymmetric fundamental solution u * ax is defined in terms of the complete elliptic integral of the first kind K(m), see e.g. [19],…”