An enhanced method with local energy minimization for the robust a posteriori construction of equilibrated stress fields in finite element analyses

TL;DR

This paper introduces an enhanced flux-equilibration method with local energy minimization for constructing admissible stress fields in finite element analysis, improving the accuracy of error bounds with minimal additional computational effort.

Contribution

The paper presents a novel version of the EESPT method using a weak prolongation condition and local energy minimization to produce sharper error estimators in finite element analysis.

Findings

Enhanced method yields more accurate stress fields.

Improved estimators are computationally efficient.

Numerical experiments confirm the method's effectiveness.

Abstract

In the context of global/goal-oriented error estimation applied to computational mechanics, the need to obtain reliable and guaranteed bounds on the discretization error has motivated the use of residual error estimators. These estimators require the construction of admissible stress fields verifying the equilibrium exactly. This article focuses on a recent method, based on a flux-equilibration procedure and called the element equilibration + star-patch technique (EESPT), that provides for such stress fields. The standard version relies on a strong prolongation condition in order to calculate equilibrated tractions along finite element boundaries. Here, we propose an enhanced version, which is based on a weak prolongation condition resulting in a local minimization of the complementary energy and leads to optimal tractions in selected regions. Geometric and error estimate criteria are introduced to select the relevant zones for optimizing the tractions. We demonstrate how this optimization procedure is important and relevant to produce sharper estimators at affordable computational cost, especially when the error estimate criterion is used. Two- and three-dimensional numerical experiments demonstrate the efficiency of the improved technique.