Error estimation based on Superconvergent Patch Recovery using statically admissible stress fields

Abstract

In this paper we investigate an approach for a posteriori error estimation based on recovery of an improved stress ÿeld. The qualitative properties of the recovered stress ÿeld necessary to obtain a conservative error estimator, i.e. an upper bound on the true error, are given. A speciÿc procedure for recovery of an improved stress ÿeld is then developed. The procedure can be classiÿed as Superconvergent Patch Recovery (SPR) enhanced with approximate satisfaction of the interior equilibrium and the natural boundary conditions. Herein the interior equilibrium is satisÿed a priori within each nodal patch. Compared to the original SPR-method, which usually underestimates the true error, the present approach gives a more conservative estimate. The performance of the developed error estimator is illustrated by investigating two plane strain problems with known closed-form solutions. ? 1998 John Wiley & Sons, Ltd.

A separated representation of an error indicator for the mesh refinement process under the proper generalized decomposition framework

“…As commented before the quality of the error indicator is slightly poor near the boundaries since the error in that zones is small. This problem, common in smoothing techniques, has been already reported by other authors [24,11,30], providing a variety of improved smoothing techniques which efficiently solves this lack of accuracy along the boundaries in the standard FE framework. Figure 8 shows the local effectivity index when considering the influence of all modes (1 in this case) and the corresponding weighting according expressions (31) and (32).…”

“…Generally, it is well known that the main drawback of recovery type techniques is the lack of accuracy along the boundaries of the domain. Further improvements to the SPR technique have been introduced [24][25][26][27] in order to prevent the lack of accuracy of the smoothed field along the boundaries of the domain and also to increase the accuracy into the domain.…”

A recovery-explicit error estimator in energy norm for linear elasticity

“…Wiberg and Abdulwahab [45,25] proposed to take into account the equilibrium of the recovered field by using a penalty method, Blacker and Belytschko [26] introduced the "Conjoint Polynomial Enhancement" to improve the recovered field along the boundaries. Other techniques looking for equilibrated recovered solutions for upper bounding purposes can be found in [46,47,48,49], but always presenting small lacks of equilibrium even at patch level, thus preventing the strict upper bound property. More recently, Ródenas and coworkers introduced the so-called SPR-C technique [28], where the "C" stands for "constraint", which was later applied in the XFEM framework by Ródenas et al [34] and finally adapted to geometry-mesh independent FE formulations [35], such as the cgFEM.…”

Estimación y acotación del error de discretización en el modelado de grietas mediante el método extendido de los elementos finitos

“…Kvamsdal y Okstad (1998) trabajaron en la idea de obtener un campo de tensiones estáticamente admisible usando una técnica de reconstrucción de la solución. Sin embargo, la técnica que presentaron no cumple rigurosamente las condiciones para obtener cotas superiores del error.…”