Numerical integration in Natural Neighbour Galerkin methods

In this paper, issues regarding numerical integration of the discrete system of equations arising from natural neighbour (natural element) Galerkin methods are addressed. The sources of error in the traditional Delaunay triangle-based numerical integration are investigated. Two alternative numerical integration schemes are analysed. First, a ‘local’ approach in which nodal shape function supports are exactly decomposed into triangles and circle segments is shown not to give accurate enough results. Second, a stabilized nodal quadrature scheme is shown to render high levels of accuracy, while resulting specially appropriate in a Natural Neighbour Galerkin approximation method. The paper is completed with several examples showing the performance of the proposed techniques