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This chapter deals with formulations based on boundary integral equations (BIEs) for elastic and plastic problems. After a brief review of the basic integral identities of solid mechanics and issues associated with the singular character of the fundamental solutions, the collocation and symmetric Galerkin BIE formulations and the associated boundary element methods (BEMs) are presented, in their conventional form where the complete matrix equation is set up using numerical integration and stored. This approach being inadequate for large problem sizes, fast solution techniques are then reviewed, with emphasis on the fast multipole method. Next, BEMs in both collocation and symmetric Galerkin form are described for fracture mechanics and small‐strain elastoplasticity. The treatment of the hypersingular integrals arising in the integral representation of tractions on the crack surface or of strains at potentially plastic interior points is discussed, along with other issues. Shape sensitivity analysis techniques are also presented, based on either the direct differentiation of the primary elastic BIE or an adjoint solution. Finally, the symmetric Galerkin BIE is used to define a symmetric formulation for BEM–FEM coupling.

Abstract. This article is devoted to some inverse problems arising in the context of linear elasticity, namely the identification of distributions of elastic moduli, model parameters, or buried objects such as cracks. These inverse problems are considered mainly for threedimensional elastic media under equilibrium or dynamical conditions, and also for thin elastic plates. The main goal is to overview some recent results, in an effort to bridge the gap between studies of a mathematical nature and problems defined from engineering practice. Accordingly, emphasis is given to formulations and solution techniques which are well suited to general-purpose numerical methods for solving elasticity problems on complex configurations, in particular the finite element method and the boundary element method. An underlying thread of the discussion is the fact that useful tools for the formulation, analysis and solution of inverse problems arising in linear elasticity, namely the reciprocity gap and the error in constitutive equation, stem from variational and virtual work principles, i.e. fundamental principles governing the mechanics of deformable solid continua. In addition, the virtual work principle is shown to be instrumental for establishing computationally efficient formulae for parameter or geometrical sensitivity, based on the adjoint solution method. Sensitivity formulae are presented for various situations, especially in connection with contact mechanics, cavity and crack shape perturbations, thus enriching the already extensive known repertoire of such results. Finally, the concept of topological derivative and its implementation for the identification of cavities or inclusions are expounded.

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.