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

Abstract

23 pagesInternational audienceIn 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

Study of the strong prolongation equation for the construction of statically admissible stress fields: Implementation and optimization

“…Note that this lack of pertinence was known since a weak prolongation equation was proposed in (Florentin et al, 2003) to deal with very deformed elements and in (Pled et al, 2012) to improve the efficiency of the estimator in zones where the error is concentrated. On the other hand, the strong prolongation equation leads to lower computational costs for the error estimation.…”

“…Buildingσ is a more complex task, for which many techniques have been proposed, among others the element equilibration technique (Ladevèze and Leguillon, 1983;Ladevèze and Rougeot, 1997) (EET), the flux-free technique (Parés et al, 2006) and the star-patch element equilibration technique (Ladevèze et al, 2010;Pled et al, 2012). This paper revisits the EET technique, about which we give more details in Section 3.…”

A new equilibrated residual method improving accuracy and efficiency of flux-free error estimates

“…Therefore, for given a set of local weighted tractions {g i γ } γ⊂ω i verifying both (17) and (20), equations (21) and (22) A positive consequence of this fact is that, the optimization problem (26) turns to be a small constrained quadratic optimization problem. The remainder of the section is devoted to detail the computation of the local weighted tractions by exhaustively describing the constrained quadratic optimization problem to be solved.…”

“…A vast literature exists providing different approaches to compute the equilibrated tractions, see for instance [1,11,10,21,23,12]. The standard computation of the equilibrated tractions involves solving a local problem for each node x i of the mesh where the zeroth-order equilibration conditions (34) are imposed via the more restrictive first-order equilibration conditions…”

Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization

“…If we are interested in obtaining upper bounds, we should solve this problem with the stress-based FEM, however this is cumbersome and it is not used in practice. Some authors [94,95] use the standard displacement based FEM with a richer space, for instance p + 3 [95] where p is the interpolation degree of the FE solution u h .…”