Machine-precision implementation of higher-order, generally curved boundary elements for three-dimensional potential problems

Resumo

The author recently revisited the collocation boundary element method for its consistency. Arbitrary rigid-body displacements, as for elasticity of a finite domain, 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 for precision and accuracy assessments, and, most importantly, a considerable simplification of the numerical implementations. The formulation became extremely simple to code for two-dimensional elasticity problems in terms of a complex variable, with precision only limited by the machine and less liable to round-off errors than its real-variable counterpart: precision is only conditioned by the Gauss-Legendre quadrature of the regular parts of the integrands. We now present the three-dimensional counterpart of such general implementation for potential problems, which is far less intuitive but is based on the same concepts of a complex quasi-singularity. A more general mathematical framework is required, which resorts to the hypercomplex – but surprisingly simple – description of the problem’s geometry, as we have been able to develop. We show that the old concepts of geometric near singularities – with many badly and highly constrained approximating solutions – should be abandoned in the face of the proposed general, precise, and accurate methodology. We present the basics of this new formulation, discuss a geometry-preserving alternative (besides the isoparametric one), and close with some applications to topologically challenging problems.