TY - JOUR
T1 - Derivative recovery and a posteriori error estimate for extended finite elements
AU - Bordas, Stéphane
AU - Duflot, Marc
PY - 2007/7/15
Y1 - 2007/7/15
N2 - 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.
AB - 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.
KW - A posteriori error estimation
KW - Derivative recovery
KW - Enrichment
KW - Extended finite element method
KW - Extended moving least squares
KW - Fracture mechanics
KW - Partition of unity
KW - Three-dimensional cracks
UR - http://www.scopus.com/inward/record.url?scp=34250351160&partnerID=8YFLogxK
U2 - 10.1016/j.cma.2007.03.011
DO - 10.1016/j.cma.2007.03.011
M3 - Article
AN - SCOPUS:34250351160
VL - 196
SP - 3381
EP - 3399
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
SN - 0045-7825
IS - 35-36
ER -