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.

1

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

2

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

3

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

4

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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