The consistent boundary element method for potential and elasticity: Part II — machine-precision numerical evaluations for 2D problems

Abstract

The collocation boundary element method is outlined in a first companion paper for the general three-dimensional, static case of elasticity. This is done in a consistent way that sheds light on some essential conceptual and implementation-oriented aspects that have been up to now in part overseen and in part mistaken in the technical literature. We show in the present developments that actually only Gauss–Legendre quadratures are needed in a two-dimensional code implementation of potential and elasticity problems, with arbitrarily high – actually only machine-precision limited – computational accuracy of all results of interest independently of a problem’s geometry or topology. In fact, the higher the effect of a quasi-singularity the more accurate a result is achievable with a – for all practical means – restricted number of quadrature points by just having the mathematical aspects properly dealt with beforehand and general preliminary evaluations stored for use as black boxes. The developments are presented for generally curved boundary elements of any high order, with the mathematics related to improper, singular, real or complex quasi-singular integrals adequately outlined, which includes the accurate evaluation of results at internal points that may be arbitrarily close to the boundary. Numerical results are shown in a second companion paper and convergence features are assessed for problems with very challenging topology issues and even for subnanometer source-field distances.