On error estimation and adaptive refinement for element free Galerkin method: Part I: stress recovery and a posteriori error estimation

In this study, an adaptive refinement procedure using the element free Galerkin method (EFGM) for the solution of 2D elastostatic problem is suggested. This refinement procedure is based on the well-known Zienkiewicz and Zhu (Z–Z) error estimator for the a posteriori error estimation and a simple point refinement scheme for new point mesh generation. The presentation of the work is divided into two parts. In Part I, concentration will be paid on the stress recovery and the a posteriori error estimation processes for the EFGM. A comprehensive study on the theories and natures of some commonly used recovery schemes for the EFGM is given. The essential features and relationships among the studied recovery schemes are highlighted. It will be demonstrated that all these recovery schemes are particular cases of a general recovery scheme called the weighed continuous moving-least-square over stationary least square (MLS-SLS) recovery scheme. A numerical experience has also been carried out to study the performances of a number of different MLS-SLS recovery procedures generated by applying different fitting parameters to the continuous MLS-SLS recovery scheme. In Part II of the study, attention will be focused on the mesh refinement and the development of the adaptive algorithms for the proposed automatic adaptive EFGM procedure.