High-Order discontinuous boundary elements: Formulation, convergence and performance analysis on 3D elasticity

Resumo

The Boundary Element Method (BEM) is known for its ability to accurately approximate difficult engi-
neering problems in elasticity, such as stress concentration, contact problems, etc. Due to its inherent complexity,

element order has been historically kept as low as possible in the literature, being most of the time restricted to lin-
ear and quadratic elements, as they present an already adequate precision. Due to Green’s function singularities, a

large part of the BEM integration process is performed with high-order Gaussian quadratures, such that increasing
element order is, in many cases, more efficient than mesh size refinement. Moreover, it was recently highlighted
that the singularity subtraction method may impose the usage of high-order elements. This work’s objective is
to investigate the possible effects of employing a p-refinement on some selected benchmark problems. For that
purpose, we present a newly developed library and its formulation for computing arbitrary order shape functions
considering continuous or discontinuous, and Lagrangian or Serendipity type QUAD elements. This library (as
well as the BEM code) is written in GNU Octave, where the shape functions are automatically generated and tested
using a Computer Algebra System. This process avoids the manual implementation of these functions which in
general is cumbersome and prone to errors. The meshes for the benchmark cases are generated using GMSH,
which can generate arbitrary-order QUAD elements. Solution convergence is analyzed in terms of the L2 error
norm. The continuous versus discontinuous element subject is also briefly discussed.