A simple error indicator for meshfree methods based on natural neighbors

The main aim of this paper is the development of a refinement procedure able to operate in the context of the constrained natural element method (C-NEM). The C-NEM was proposed by the authors in a former work [Yvonnet J, Ryckelynck D, Lorong P, Chinesta F. A new extension of the natural element method for non-convex and discontinuous domains: the constrained natural element method (C-NEM). Int J Numer Methods Eng 2004;60:1451–1474] and its main meshless features, that allow to describe large domain changes as well as to handle fixed or moving discontinuities, were analyzed in [Yvonnet J, Chinesta F, Lorong P, Rynckelynck D. The constrained natural element method (C-NEM) for treating thermal models involving moving interfaces. Int J Thermal Sci 2005;44:559–569; Yvonnet J, Lorong P, Ryckelynck D, Chinesta F. Simulating dynamic thermo-elastoplasticity in large transformations with adaptive refinement in the natural element method: application to shear banding. Int J Forming Process 2005;8:347–363]. Sometimes, in order to improve the interpolation accuracy for describing boundary layers or an anisotropic behavior, new nodes must be added, removed or repositioned. The interpolation in the vast majority of meshless techniques is free of mesh quality requirement. Thus, introduction, elimination or repositioning of nodes is a trivial task, because no geometrical restrictions exist. In this way, nodes can be added without geometrical checks in the regions where the solution must be improved (identified by using an appropriate error indicator). For this purpose, in this paper an a posteriori error indicator will be proposed and tested in some linear elastostatic problems benchmarks involving different levels of difficulty (stress concentration, solution singularities, …) all of them with a known exact solution. The computational implementation of this error indicator is very simple, and when it is used in tandem with an efficient refinement procedure, which makes use of the meshless features of the C-NEM, provides an accurate adaptation procedure, specially appropriate in the C-NEM framework.