A recovery-type a posteriori error estimator for gradient elasticity

Abstract

An a posteriori error estimator is developed for Aifantis' gradient elasticity.C0-continuous interpolations suffice for the spatial discretisation.Singularities are avoided with gradient elasticity which simplifies error estimation.Effectivity index is very close to 1 which shows the effectiveness of the method. In this paper, an a posteriori error estimator of the recovery type is developed for the gradient elasticity theory of Aifantis. This version of gradient elasticity can be implemented in a staggered way, whereby solution of the classical equations of elasticity is followed by solving a reaction-diffusion equation that introduces the gradient enrichment and removes the singularities. With gradient elasticity, singularities in the stress field can be avoided, which simplifies error estimation. Thus, we develop an error estimator associated with the second step of the staggered algorithm. Stress-gradients are recovered based on the methodology of Zienkiewicz and Zhu, after which a suitable energy norm is discussed. The approach is illustrated with a number of examples that demonstrate its effectiveness.