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

Static 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 Ω∞ along the boundary Γ of a homogeneous elastic body Ω embedded in Ω∞. 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 following 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 approach. 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.