In the work presented, we fashion and test two simple and effective local a posteriori error estimators (or indicators), specially adapted to partition of unity enriched finite element methods such as the extended finite element method (XFEM) [1,2]. The first estimator is based on extended moving least squares (XMLS) smoothing, the second on an extended global recovery technique, that we coin XGR, both are of the Zienkiewicz and Zhu type [4].

The XMLS estimation, based on an idea of Tabbara et al. [3] is explained first. 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 computed from the raw nodal XFEM displacements. The XMLS construction is taylored 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. This basis is 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 always close to 1. and improves upon mesh refinement. It is also shown that for the linear elastic fracture mechanics problems treated, the proposed estimator outperforms the Zienkiewicz and Zhu estimator of [4], which is expected, since the latter provides only C⁰ continuity of the recovered strains. Parametric studies of the general performance of the estimator are also carried out.

The XGR method, extending the idea of Zienkiewicz and Zhu [4] to enriched finite elements is as follows. The enhanced strain field is constructed as a linear combination of the finite element shape functions used for the displacement approximation, enriched with functions derived from the gradients of the near-tip fields. The coefficients of this linear aproximation are obtained by a global minimization of the difference between the raw solution and this enhanced solution. In all examples treated, the XGR method yields a recovered strain field which is not as smooth as the XMLS, but is computationally cheaper. As in the XMLS, the estimator always yields an effectivity which is close to 1. and improves upon mesh refinement.

1

T. Belytschko and T. Black. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 45:601-620, 1999. doi:10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S

2

N. Moës, J. Dolbow, and T. Belytschko. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46:131-150, 1999. doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J

3

M. Tabbara, T. Blacker, and T. Belytschko. Finite element derivative recovery by moving least square interpolants. Computer Methods in Applied Mechanics and Engineering, 117:211-223, 1994. doi:10.1016/0045-7825(94)90084-1

4

O. C. Zienkiewicz and J. Z. Zhu. A simple error estimator and adaptive procedure for practical engineering analysis. International Journal for Numerical Methods in Engineering, 24:337-357, 1987. doi:10.1002/nme.1620240206

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £75 +P&P)