On the role of enrichment and statical admissibility of recovered fields in a-posteriori error estimation for enriched finite element methods

TL;DR

This paper evaluates how the statical admissibility and ability to represent singular solutions of recovered fields influence the accuracy of error estimators in enriched finite element methods, highlighting the importance of extended recovery techniques.

Contribution

It demonstrates that extended recovery procedures and statical admissibility are crucial for accurate error estimation in enriched finite element methods.

Findings

Extended recovery techniques improve error estimator effectiveness.

Statically admissible recovered solutions yield significant accuracy improvements.

Enrichment enhances the description of singular solutions in error estimation.

Abstract

Purpose: This paper aims at assessing the effect of (1) the statical admissibility of the recovered solution; (2) the ability of the recovered solution to represent the singular solution; on the accuracy, local and global effectivity of recovery-based error estimators for enriched finite element methods (e.g. the extended finite element method, XFEM). Design/methodology/approach: We study the performance of two recovery techniques. The first is a recently developed superconvergent patch recovery procedure with equilibration and enrichment (SPR-CX). The second is known as the extended moving least squares recovery (XMLS), which enriches the recovered solutions but does not enforce equilibrium constraints. Both are extended recovery techniques as the polynomial basis used in the recovery process is enriched with singular terms for a better description of the singular nature of the solution. Findings: Numerical results comparing the convergence and the effectivity index of both techniques with those obtained without the enrichment enhancement clearly show the need for the use of extended recovery techniques in Zienkiewicz-Zhu type error estimators for this class of problems. The results also reveal significant improvements in the effectivities yielded by statically admissible recovered solutions. Originality/value: This work shows that both extended recovery procedures and statical admissibility are key to an accurate assessment of the quality of enriched finite element approximations.