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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 75
Edited by: B.H.V. Topping and Z. Bittnar
Paper 15

Error Estimation and p-Adaptivity Based on the Partition of Unity Method

T. Pannachet+, P. Díez* and H. Askes+

+Faculty of Civil Engineering and Geosciences, Delft University of Technology, The Netherlands
*Departament de Matemàtica Aplicada III, Universitat Politèecnica de Catalunya, Barcelona, Spain

Full Bibliographic Reference for this paper
T. Pannachet, P. Díez, H. Askes, "Error Estimation and p-Adaptivity Based on the Partition of Unity Method", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 15, 2002. doi:10.4203/ccp.75.15
Keywords: error assessment, remeshing, p-adaptivity, partition of unity, goal-oriented error estimation.

Error estimation and adaptivity are indispensable tools to make finite element analysis applicable to engineering practice. Upper and lower bound error estimators, based on sound mathematical principles, are needed to assess the quality of a numerical solution.

Furthermore, flexible and robust adaptive remeshing algorithms are needed to optimise finite element discretisations, so that prescribed accuracies can be obtained at minimal computational costs. Recently, a new methodology has been developed for the formulation of interpolation functions, namely the Partition of Unity Method (PUM) [4]. Instead of an elementoriented methodology, a node-based strategy is taken. This makes -adaptivity much more flexible, since inter-element compatibility of the interpolation functions is straightforward with the PUM concept [3]. Moreover, functions other than polynomials are easily added to the interpolation space, so that analytical information for certain problems (singular behaviour in crack-tip problems, harmonic functions in higher-order gradient models) can be used to enrich the shape functions efficiently.

In this contribution, the PUM is applied for -adaptive analysis, and is driven by a residual-type error estimation. An element-based estimate is combined to a patchbased estimate, following the work of Díez [1]. Using Dirichlet boundary conditions, a lower-bound error estimate is obtained within a simple implementational context. Thus, the complete adaptive scheme including error assessment and remeshing is performed by the combination of the PUM and the local -refinement.

A related issue is the concept of the so-called goal-oriented error [2], whereby not the total (global) error is the quantity of interest, but derived quantities such as the stress in a given point, or the crack-opening displacement. The extension of the error estimation scheme in goal-oriented error assessment is addressed. Numerical examples show the capability of the -adaptive technique with the -error assessment.

P. Díez, J.J. Egozcue and A. Huerta, "A posteriori error estimation for standard finite element analysis", Computer Methods in Applied Mechanics and Engineering, 163, 141-157, 1998. doi:10.1016/S0045-7825(98)00009-7
S. Prudhomme and J.T. Oden, "On goal-oriented error estimation for elliptic problems application to the control of pointwise errors", Computer Methods in Applied Mechanics and Engineering, 176, 313-331, 1999. doi:10.1016/S0045-7825(98)00343-0
R. L. Taylor, O.C. Zienkiewicz and E. Oñate, "A hierarchical finite element method based on the partition of unity", Computer Methods in Applied Mechanics and Engineering, 152, 73-84, 1998. doi:10.1016/S0045-7825(97)00182-5
I. Babuska and J.M. Melenk, "The partition of unity method", International Journal for Numerical Methods in Engineering, 40, 727-758, 1997. doi:10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.3.CO;2-E

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