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.
SUMMARYStatic discontinuities (i.e. distributions of forces along a line or a surface, implying a jump of tractions across it) and kinematic (displacement) discontinuities are considered simultaneously as sources acting on the unbounded elastic space R, along the boundary r of a homogeneous elastic body R embedded in R,. The auxiliary elastic state thus generated in the body is associated with the actual elastic state by a Betti reciprocity equation. Using suitable discretizations of actual and fictitious boundary variables, a symmetric Galerkin formulation of the direct boundary element method is generated.The foliowing topics are addressed: reciprocity relations among kernels with particular attention to the role of singularities; conditions to be satisfied by the boundary field modelling in order to achieve the symmetry of the coefficient matrix; variational properties of the solution.With reference to two-dimensional problems, a technique based on a complex-variable formalism is proposed to perform the double integrations involved in this apprgach. An implementation of this technique for elastic analysis is described assuming straight elements, with continuous linear displacements and piecewise-constant tractions; all the double integrations are carried out analytically.Comparisons, from the computational standpoint, with the traditional non-symmetric method based on collocation and single integration, demonstrate the effectiveness of the present approach.
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