An enhanced-strain error estimator for Galerkin meshfree methods based on stabilized conforming nodal integration

Abstract

Gradient averaging-type a posteriori error estimators applied to the finite element method enjoy great popularity in the engineering community. This is mainly because they are easy in their construction and computer implementation and usually provide constant-free and sharp error estimates on the expense of losing the bounding property. However, it proves difficult to transfer this error estimation procedure to Galerkin meshfree methods. One reason is that the meshfree gradient approximation is generally rather accurate per se and difficult to improve. Moreover, in many Galerkin meshfree methods the shape functions are no interpolants, as required in the original idea of constructing the recovered and thus smooth gradient field. In this paper, a novel error estimation procedure is presented that is based on the enhanced assumed strain (EAS) method and applied to the reproducing kernel particle method (RKPM) as a representative of Galerkin meshfree methods. The error estimator is naturally tailored to Galerkin meshfree methods if stabilized conforming nodal integration (SCNI) is employed, which provides two gradient fields: a globally smooth one (the compatible strain) and a cellwise constant one (the enhanced assumed strain). It is shown that the difference (the enhanced strain) can be used to derive an error estimator that follows the notion of gradient averaging-type error estimators and resolves its issues when applied to Galerkin meshfree methods. In addition, the enhanced-strain error estimator is even easier to implement and computationally less expensive than gradient averaging-type error estimators. Numerical examples of engineering interest illustrate the performance of the enhanced-strain error estimator presented in this paper.