Efficient and accurate stress recovery procedure and a posteriori error estimator for the stable generalized/extended finite element method

Abstract

Summary This paper presents a new stress recovery technique for the generalized/extended finite element method (G/XFEM) and for the stable generalized FEM (SGFEM). The recovery procedure is based on a locally weighted L2 projection of raw stresses over element patches; the set of elements sharing a node. Such projection leads to a block‐diagonal system of equations for the recovered stresses. The recovery procedure can be used with GFEM and SGFEM approximations based on any choice of elements and enrichment functions. Here, the focus is on low‐order 2D approximations for linear elastic fracture problems. A procedure for computing recovered stresses at re‐entrant corners of any internal angle is also presented. The proposed stress recovery technique is used to define a Zienkiewicz‐Zhu (ZZ) a posteriori error estimator for the G/XFEM and the SGFEM. The accuracy, computational cost, and convergence rate of recovered stresses together with the quality of the ZZ estimator, including its effectivity index, are demonstrated in problems with smooth and singular solutions.

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

“…In addition, available extensions of the ZZ estimator to the G/XFEM have already shown that different basis functions can be used to generate the displacement and the recovered stress fields. For instance, Bordas and Duflot 34 and Bordas et al 35 used Westergaard solutions to generate a basis that approximates the near-tip recovered strain field, and Prange et al 31 and Lins et al 32,33 used the asymptotic near-tip stress field, that is, the stresses of a problem consisting of a crack inserted in an unbounded domain, to write the recovered stress field. It is noted that the methods presented in those works [31][32][33][34][35] are first-order.…”

“…Following the strategy proposed by Lins et al, 33 the right-hand side entries b d 𝛽j can first be reinterpreted as an L 2 inner product between σd (x) and  d 𝛽j (x), locally weighted by the PoU 𝜑 𝛽 (x), that is,…”

“…[18][19][20][21][22][23][24][25] The formulation of recovery-based error estimators has been recently adapted for other methods such as the nonconforming and discontinuous Galerkin methods, [26][27][28] the virtual element method, 29 and the Generalized/eXtended FEM. [30][31][32][33] This requires the development of stress or strain recovery approaches that are suitable for these methods. Within the G/XFEM context, several methodologies to build the recovered stress or strain field based on what is commonly used with the standard FEM were proposed and used for a posteriori error estimation.…”

“…This error estimator is computationally efficient and was obtained by modifying the conventional ZZ estimator by applying a locally weighted block-diagonal L 2 projection, originally conceived for the computation of lumped mass matrices. 38 In this article, extensions of the block-diagonal ZZ error estimator 33 to second-order G/XFEM and FEM approximations are presented. The focus is on LEFM problems but the proposed error estimator formulation is quite general and can be adapted to other types of problems such as those involving material interfaces.…”

An improved stress recovery technique for the unfitted finite element analysis of discontinuous gradient fields

“…later adapted this approach to improve the stress field within stable GFEM (SGFEM); because SGFEM modifies the X/GFEM enrichment functions to solve the issue of ill‐conditioned stiffness matrices, 46‐48 the enrichment functions used to interpolate the recovered stress field are also modified. Instead of using a global least squares projection, Lins et al 49 . proposed a more computationally efficient recovery technique, whereby a consistent block‐diagonal projection operator 50 is used to perform a locally weighted least squares projection of directly‐calculated stresses over patches of elements.…”