The consistent boundary element method for potential and elasticity: Part I — Formulation and convergence theorem

Abstract

The collocation boundary element method is outlined for the general three-dimensional, static case of elasticity on the basis of a weighted-residuals statement that leads to the Somigliana’s identity, as already proposed in the literature. However, this is done in a consistent way that sheds light on some conceptual and implementation-oriented aspects that should be – but are missing – in the textbooks on the subject. Arbitrary rigid-body displacements, as for elasticity, are naturally taken into account, and traction force parameters are always in balance independently of problem scale and mesh discretization. For generally curved boundaries, the correct definition of traction force interpolation functions enables the enunciation of a general convergence theorem, the introduction of patch and cut-out tests and, not least, a considerable simplification of the numerical implementations. A rather recreative example of a simple truss element is brought in an Appendix. It enables the demystification of apparently convoluted concepts that might be thought only pertaining to highly elaborated mechanics and mathematics for singularities related to infinity and to ungraspable linear algebra. Simple code schemes based on the outlined concepts are proposed in a companion paper for 2D problems of potential and elasticity, which rely exclusively on Gauss–Legendre quadrature and lead to arbitrarily high computational accuracy of all results of interest independently of a problem’s geometry or topology. Numerical results are shown in another companion paper and convergence features are assessed for 2D potential and elasticity problems with very challenging topology issues and even for subnanometer source-field distances. Summing up, whether a problem we are attempting to simulate consists of smooth fields or not, we are able to reproduce it — maybe with numerical approximations due to a coarse mesh discretization but never introducing unduly singularities.