A new non-intrusive technique for the construction of admissible stress fields in model verification

Ladevèze, Pierre; Chamoin, Ludovic; Florentin, Éric

doi:10.1016/j.cma.2009.11.007

Cited by 39 publications

(73 citation statements)

References 29 publications

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“…We will compare it with the Zienkiewicz and Zhu error estimator (14) based on the use of the SPR-C technique, which presents a high accuracy but does not have bounding properties. We have used two benchmark problems with analytical solution and a problem where only overkilled solution will be available in order to study the behavior of Q 1 (bilinear) and Q 2 (biquadratic) elements.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In the CRE technique the authors evaluate a stress field σ CRE which is statically admissible. Then, by using an expression similar to (14) they evaluate an upper bound of the error in the energy norm.…”

Section: Error Estimation a Spr-based Upper Bounding Techniquementioning

confidence: 99%

“…Then, with minimal loss in accuracy: (28) where C = C Ω . Now, we rewrite expression (15) composed by the ZZ error estimator (14) and the correction terms (16) as follows:…”

Section: Particular Case Using the Spr-c Recovery Techniquementioning

confidence: 99%

“…Some of these error estimators solve two global problems in parallel [10] whereas other post-process the FE solution [11,12,13]. Under this group we can also include the error estimators based on the Constitutive Relation Error (CRE) introduced by Ladevèze and Leguillon [11] and followed by several contributions for many applications, see for example [14,15,16,17]. This technique compares a kinematically admissible stress field with a statically admissible solution obtained by solving local problems, built using the strong prolongation condition.…”

Section: Introductionmentioning

confidence: 99%

“…Zienkiewicz and Zhu [18] introduced the so-called Zienkiewicz and Zhu (ZZ) error estimator (14) which consists of substituting the unknown field σ by the recovered stress field σ * , evaluated with a recovery procedure, in expression (13).…”

Section: Error Estimation a Spr-based Upper Bounding Techniquementioning

confidence: 99%

See 4 more Smart Citations

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

Nadal

Dı́ez

Ródenas

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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Significant research effort has been devoted to produce one-sided error estimates for Finite Element Analyses, in particular to provide upper bounds of the actual error. Typically, this has been achieved using residual-type estimates. One of the most popular and simpler (in terms of implementation) techniques used in commercial codes is the recovery-based error estimator. This technique produces accurate estimations of the exact error but is not designed to naturally produce upper bounds of the error in energy norm. Some attempts to remedy this situation provide bounds depending on unknown constants. Here, a new step towards obtaining error bounds from the recoverybased estimates is proposed. The idea is 1) to use a locally equilibrated recovery technique to obtain an accurate estimation of the exact error, 2) to add an explicit-type error bound of the lack of equilibrium of the recovered stresses in order to guarantee a bound of the actual error and 3) to efficiently and accurately evaluate the constants appearing in the bounding expressions, thus providing asymptotic bounds. The the numerical tests with h-adaptive refinement process show that the bounding property holds even for coarse meshes, providing upper bounds in practical applications.Error bounding, recovery techniques, explicit residual error estimator

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Error Estimation a Spr-based Upper Bounding Techniquementioning

confidence: 99%

“…Then, with minimal loss in accuracy: (28) where C = C Ω . Now, we rewrite expression (15) composed by the ZZ error estimator (14) and the correction terms (16) as follows:…”

Section: Particular Case Using the Spr-c Recovery Techniquementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Error Estimation a Spr-based Upper Bounding Techniquementioning

confidence: 99%

See 3 more Smart Citations

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

Nadal

Dı́ez

Ródenas

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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Calculation of strict error bounds for finite element approximations of non‐linear pointwise quantities of interest

Ladevèze

Chamoin

2010

Numerical Meth Engineering

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SUMMARYThis paper deals with the verification of simulations performed using the finite element method. More specifically, it addresses the calculation of strict bounds on the discretization errors affecting pointwise outputs of interest which may be non-linear with respect to the displacement field. The method is based on classical tools, such as the constitutive relation error and extraction techniques associated with the solution of an adjoint problem. However, it uses two specific and innovative techniques: the enrichment of the adjoint solution using a partition of unity method, which enables one to consider truly pointwise quantities of interest, and the decomposition of the non-linear quantities of interest by means of projection properties in order to take into account higher-order terms in establishing the bounds. Thus, no linearization is performed and the property that the local error bounds are guaranteed is preserved. The effectiveness of the approach and the quality of the bounds are illustrated with two-dimensional applications in the context of elastic fatigue problems.

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On the techniques for constructing admissible stress fields in model verification: Performances on engineering examples

Pled

Chamoin

Ladevèze

2011

Numerical Meth Engineering

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SUMMARYRobust global/goal-oriented error estimation is used nowadays to control the approximate finite element solutions obtained from simulation. In the context of Computational Mechanics, the construction of admissible stress fields (i.e. stress tensors which verify the equilibrium equations) is required to set up strict and guaranteed error bounds (using residual based error estimators) and plays an important role in the quality of the error estimates. This work focuses on the different procedures used in the calculation of admissible stress fields, which is a crucial and technically complicated point. The three main techniques that currently exist, called the element equilibration technique (EET), the star-patch equilibration technique (SPET), and the element equilibration + starpatch technique (EESPT), are investigated and compared with respect to three different criteria, namely the quality of associated error estimators, computational cost and easiness of practical implementation into commercial finite element codes.The numerical results which are presented focus on industrial problems; they highlight the main advantages and drawbacks of the different methods and show that the behavior of the three estimators, which have the same convergence rate as the exact global error, is consistent. Two-and three-dimensional experiments have been carried out in order to compare the performance and the computational cost of the three different approaches. The analysis of the results reveals that the SPET is more accurate than EET and EESPT methods, but the corresponding computational cost is higher. Overall, the numerical tests prove the interest of the hybrid method EESPT and show that it is a correct compromise between quality of the error estimate, practical implementation and computational cost. Furthermore, the influence of the cost function involved in the EET and the EESPT is studied in order to optimize the estimators.

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A new non-intrusive technique for the construction of admissible stress fields in model verification

Cited by 39 publications

References 29 publications

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

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

Calculation of strict error bounds for finite element approximations of non‐linear pointwise quantities of interest

On the techniques for constructing admissible stress fields in model verification: Performances on engineering examples

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