Complex-variable, high-precision formulation of the consistent boundary element method for 2D potential and elasticity problems

Abstract

The collocation boundary element method was recently entirely revisited on the basis of a consistent derivation of Somigliana’s identity in terms of weighted residuals. Both conceptually and for the sake of code implementation, the correct traction force interpolation along generally curved boundaries, as for elasticity problems, leads to the enunciation of an inedited, actually long-sought, convergence theorem as well as to considerable numerical simplifications. Numerical evaluations for two-dimensional problems require exclusively Gauss–Legendre quadrature plus eventual correction terms obtained analytically regardless of the order or shape of the implemented boundary element interpolation. Arbitrarily high precision and accuracy is achievable for low-cost computation, as eventual mesh-subdivision refinements should take place only if the mechanical simulation demands — and not just for numerical evaluations. We now show that considerable simplification is obtained by switching the formulation from real to complex variable. Precision, round-off errors, and accuracy of a given numerical implementation may be kept – identifiably and separately – under control, as assessed for some potential and elasticity examples with extremely challenging topologies. In fact, source-field distances may be arbitrarily small — far smaller than deemed feasible in continuum mechanics, as we resort to nothing else than the problem’s mathematics.