Recovery strategies, a posteriori error estimation, and local error indication for second‐order G/XFEM and FEM

Abstract

AbstractThis article presents a computationally efficient and straightforward to implement a posteriori error estimator for second‐order G/XFEM and FEM approximations. The formulation is based on the recently proposed block‐diagonal Zienkiewicz–Zhu (ZZ‐BD) a posteriori error estimator. The focus is on linear elastic fracture mechanics (LEFM) problems but the proposed error estimator formulation is general and can be adapted to other types of problems such as those involving material interfaces. The proposed ZZ‐BD error estimator is based on a new strategy to recover stress fields from second‐order G/XFEM and FEM approximations. This recovery procedure involves locally weighted projections of raw stresses on an approximation space for discontinuous and singular stress fields. The basis functions for these stress approximations are defined using a low‐order partition of unity together with polynomial, discontinuous, and singular recovery enrichment functions. These singular enrichments are provided by the gradient of G/XFEM enrichments for LEFM problems and are, thus, available in most G/XFEM implementations. They also lead to a more computationally efficient stress recovery procedure than the one in the original ZZ‐BD error estimator. The adopted G/XFEM are optimally convergent with the error in the energy norm of . Numerical experiments show that the error of the recovered stress field converges at the same rate as the discretization error. They also show that the effectivity index of the error estimator is close to the unity and that it provides good local error indicators for mesh adaptivity algorithms. Second‐order FEM for elasticity problems is a special case of the G/XFEM adopted here. This is exploited to define a block‐diagonal ZZ error estimator for the FEM as a special case of the proposed estimator for the G/XFEM. This brings the computational efficiency, simplicity, and accuracy of the ZZ‐BD error estimator, originally developed for the G/XFEM, to second‐order standard FEM.