Derivative recovery and a posteriori error estimate for extended finite elements

By Stéphane P.A. Bordas, Marc Duflot. 2007

146 citations recorded by the source corpus.

This paper is the first attempt at error estimation for extended finite elements. The goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method (XFEM). In each element, the local estimator is the L2 norm of the difference between the raw XFEM strain field and an enhanced strain field computed by extended moving least squares (XMLS) derivative recovery obtained from the raw nodal XFEM displacements. The XMLS construction is tailored to the nature of the solution. The technique is applied to linear elastic fracture mechanics, in which near-tip asymptotic functions are added to the MLS basis. The XMLS shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity. The result is a very smooth enhanced strain solution including the singularity at the crack tip. Results are shown for two- and three-dimensional linear elastic fracture mechanics problems in mode I and mixed mode. The effectivity index of the estimator is close to 1 and improves upon mesh refinement for the studied near-tip problem. It is also shown that for the linear elastic fracture mechanics problems treated, the proposed estimator outperforms one of the superconvergent patch recovery technique of Zienkiewicz and Zhu, which is only C0. Parametric studies of the general performance of the estimator are also carried out.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/02cf292f4fcc5d414a1f5916d9c858b4ebe94d6a

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In Computational Mechanics

In Electrical and Electronic Engineering

In Mechanics of Materials

Citations in the corpus

A recovery‐type error estimator for the extended finite element method based on singular+smooth stress field splitting (2008, methodology): “[18] and Bordas and Duflot [19] have presented a recovery-based error estimator for XFEM.”
Computation of limit and shakedown loads using a node‐based smoothed finite element method (2012, methodology): “In addition to the above discussions, the authors believe that the method presented herein can be very promising to: (1) improve the load factor solution of limit and shakedown problems with strong discontinuities [81–83] in fracture structures by coupling the NS-FEM to the extended FEM (XFEM) [84–8”
On three-dimensional modelling of crack growth using partition of unity methods (2010, methodology): “Yet, on the negative side, this method is still not able to provide accurate stress intensity factors in 3D that are judged accurate enough for industrial applications [146–149], which has spurred recent interest in the development of a posteriori error indicators and estimators as well as adaptivit”
Accurate fracture modelling using meshless methods, the visibility criterion and level sets: Formulation and 2D modelling (2011, methodology): “the node splitting method proposed in [7] as well as other proposals [34, 46].”
A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies (2012, methodology): “A simple derivative recovery technique and its associated a posteriori error estimator have been proposed in [10, 11, 12, 38].”
Assessment of smoothed point interpolation methods for elastic mechanics (2010, background): “In addition, the extended finite element [20–26] has been proposed to solve the crack problem.”
Numerically determined enrichment functions for the extended finite element method and applications to bi‐material anisotropic fracture and polycrystals (2010, background): “A posteriori error estimators for X-FEM have already been proposed [18, 19, 23, 24] and provide the opportunity to develop adaptive refinement schemes.”
Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth (2017, methodology): “This has been used for 3D crack growth problems [29–31,27] as well as for industrial applications [32,21,33].”
Extended finite element method with global enrichment (2015, methodology): “In standard XFEM those higher order terms can be represented by the FE part of the approximation, in addition since the 13 Page 13 of 81 http://mc.manuscriptcentral.com/nme International Journal for Numerical Methods in Engineering Peer Review O nly tip enrichment functions are weighted by the FE sh”
A review of extended/generalized finite element methods for material modeling (2009, background): “Bordas and Duflot [18] have developed error estimators based on L2 projections.”