New bounding techniques for goal‐oriented error estimation applied to linear problems

Abstract

The paper deals with the accuracy of guaranteed error bounds on outputs of interest computed from approximate methods such as the finite element method. A considerable improvement is introduced for linear problems, thanks to new bounding techniques based on Saint-Venant's principle. The main breakthrough of these optimized bounding techniques is the use of properties of homothetic domains that enables to cleverly derive guaranteed and accurate bounding of contributions to the global error estimate over a local region of the domain. Performances of these techniques are illustrated through several numerical experiments.h can be recovered from the data and the FE stress field h alone. Starting from an admissible solution . O u h , O h / provided by one of the existing techniques, one can measure the global residual on constitutive relation (3), called the CRE and hereafter

The Constitutive Relation Error Method: A General Verification Tool

“…We now derive upper bounds on the two terms in the right-hand side of (27). We only give the results here, but more details can be found in [20,29].…”

“…In [27], a new bounding technique using the Saint-Venant principle and homothetic domains is derived; this can give sharper bounds. The idea is to decompose the domain Ω in two disjoint zones: (i) zone ω λ , parameterized with scalar value λ, surrounding the zone where the quantity of interest is defined; (ii) complementary zone Ω/ω λ .…”

“…Second, we introduce the advances performed recently. They mainly concern key technical points [23][24][25][26][27] and also various engineering problems [21,22,[28][29][30][31][32][33]. Complex material models even in dynamics are considered and Constitutive Relation Errors are developed in [4].…”

Synergies between the constitutive relation error concept and PGD model reduction for simplified V&V procedures

“…The CRE concept was first introduced as a robust a posteriori error estimator in FE computations [5], enabling to compute both strict and effective discretization error bounds for linear and more generally convex structural mechanics problems, and to lead mesh adaptivity processes. It was primarily used for linear thermal and elasticity problems [6,7] before being extended to nonlinear time dependent problems [8,9] and to goal-oriented error estimation [10][11][12]. The use of CRE for model verification, for which a general overview can be found in [2], requires in particular the computation of admissible dual fields which are fully equilibrated.…”

Strict upper and lower bounds of quantities for beams on elastic foundation by dual analysis

“…Then, by utilizing the CRE estimation, strict bounds of quantities of interest are acquired. So far, CRE-based goal-oriented error estimation featuring strict bounds of quantities has been extensively exploited in analysis of various problems (Panetier et al, 2009;Chamoin and Ladevèze, 2008;Waeytens et al, 2012;Chamoin et al, 2012;Ladevèze et al, 2012Ladevèze et al, , 2013Charbonnel et al, 2013).…”