A Posteriori Error Estimates and h-Adaptive Algorithms for Accuracy Control of a Second-Order Generalized/eXtended FEM for 3-D Linear Elastic Fracture Mechanics Problems

Abstract

The Generalized/eXtended Finite Element Method (G/XFEM) augments standard FEM approximation spaces with functions tailored to represent well specific behaviors of a problem, such as those introduced by cracks in linear elastic fracture mechanics (LEFM). The method allows mesh generation to be made independently of crack surfaces and achieves optimal first- and second-order convergence rates, while keeping the condition number of its global matrices under control. Regarding second-order G/XFEM formulations, it has been shown that when only singular enrichment functions able to represent the $\sqrt{r}$ singularity are adopted, the convergence rate is still bounded by the second-highest crack singularity. In this case, mesh refinement around crack fronts is a strategy to recover optimal convergence. In this work, this is addressed by h-adaptive mesh refinement algorithms. To this end, first, a Zienkiewicz and Zhu block-diagonal (ZZ-BD) error estimator is proposed for three-dimensional (3-D) LEFM problems. For that, the most challenging part is the definition of good recovery enrichment functions that are able to represent the crack singularity. These functions are proposed in this work based on the derivatives of enrichment functions commonly adopted in the G/XFEM context. It is shown that the use of these new singular recovery enrichment functions in the recovery process of the ZZ-BD error estimator leads to estimated discretization errors that are very close to the exact discretization errors. Also, the performance of the error estimator becomes much better than if one adopts recovery enrichment functions commonly used in 2-D. Finally, with a good error estimator able to also quantify the distribution of discretization errors over the domain and along crack fronts, adaptive algorithms are developed. Herein, h-adaptive techniques are proposed to recover optimal second-order convergence for G/XFEM and enhance its usability in such a way that final discretizations meeting a user's pre-specified tolerance on the discretization error are delivered on the fly by the adaptive procedure. 3-D LEFM numerical experiments with increasing levels of complexity are used to assess the ZZ-BD effectivity as well as to show that the proposed h-adaptive algorithms can, at a reasonable computational cost, deliver accurate discretizations at optimal convergence rates.