The Constitutive Relation Error Method: A General Verification Tool

Chamoin, Ludovic

doi:10.1007/978-3-319-20553-3_4

Cited by 9 publications

(10 citation statements)

References 54 publications

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“…being the traditional energy norms. Further information about posteriori error estimation for contact problems can be found in [62,63,64] Appendix B Quadratic second-order cone programming…”

Section: Discussionmentioning

confidence: 99%

Dual finite-element analysis using second-order cone programming for structures including contact

Boustani

Bleyer

Arquier³

et al. 2020

Engineering Structures

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Computation of elastic structures in contact is performed by means of a dual analysis combining displacement-based and equilibrium-based finite elements. Contact conditions are formulated in the framework of second-order cone programmaing (SOCP) and an efficient interior point method (IPM) algorithm is presented for the resolution of the associated optimization problems. The dual approach allows the user to assess the quality of convergence and to efficiently calculate a discretization error estimator which includes a contact error term. An efficient remeshing scheme, based on the local contributions of the elements to the global error, can then be used to efficiently improve the solution accuracy. The whole process is illustrated on some examples and applied to a typical steel assembly. Its efficiency, in particular concerning the IPM solver, is demonstrated in comparison with the industrial finite element code Abaqus.

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Section: Discussionmentioning

confidence: 99%

Dual finite-element analysis using second-order cone programming for structures including contact

Boustani

Bleyer

Arquier³

et al. 2020

Engineering Structures

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“…The idea behind this enrichment is to improve the ellipticity properties of the cost function by adding a term which relaxes unreliable parts of the model. Introduced in the 1980s for the purpose of FE verification (and extensively developed in [26][27][28][29][30][31][32] among many other references), the CRE concept was later adapted for model validation [33][34][35][36][37] in structural dynamics to define a modified CRE (mCRE) residual to be minimized with respect to the set of updated parameters. The mCRE functional reads:…”

Section: The Modified Constitutive Relation Error (Mcre)mentioning

confidence: 99%

Robust energy-based model updating framework for random processes in dynamics: Application to shaking-table experiments

Diaz

Charbonnel

Chamoin

2022

Computers & Structures

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“…The CRE concept, explained in full detail in the works of Ladevèze and Pelle and Ladevèze and Chamoin, has similitude with various methods in the literature, such as equilibrated residual or flux‐free approaches. They all share the idea of constructing a fully equilibrated flux field, which is actually the only way to recover guaranteed and computable (no unknown constant) estimates on the discretization error.…”

Section: Error Estimation and Adaptive Processmentioning

confidence: 99%

“…It is then possible to obtain a strict computable bounding on the error

{scriptE}_{Q} false(boldp false)

. It reads as follows:

false| {scriptE}_{Q} false(boldp false) - Q_{corr} false(boldp false) false| \leq \frac{1}{2} E_{CRE} ((), u_{m}^{h} false(\cdot, boldp false), {\overset{boldq}{true}}_{boldp}) . E_{CRE} ((), ũ_{m}^{h} false(\cdot, boldp false), {\overset{\overset{boldq}{true}}{true}}_{boldp}),

where Q corr ( p ) is a fully computable correction term. Consequently, a computable bounding on the exact value Q ( u p ) of the quantity of interest can be defined, under the form

Q^{-} false(boldp false) \leq Q false(u_{boldp} false) \leq Q^{+} false(boldp false)

with

\begin{matrix} right Q^{-} (p) & left = Q (u_{m}^{h} (\cdot, p)) + Q_{corr} (p) - \frac{1}{2} E_{CRE} (u_{m}^{h} (\cdot, p), {\hat{q}}_{p}) . E_{CRE} (ũ_{m}^{h} (\cdot, p),) \end{matrix}

…”

Section: Error Estimation and Adaptive Processmentioning

confidence: 99%

“…In order to control the quality of the approximation and as an additional scientific contribution in the paper, we introduce verification tools based on the Constitutive Relation Error (CRE) concept. This concept was widely used in FEA over the last 40 years . It provides for a general framework to obtain guaranteed, accurate, and fully computable bounds on the global error (in the energy norm) or on specific quantities of interest (goal‐oriented error estimation) using adjoint‐based techniques .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Certified real‐time shape optimization using isogeometric analysis, PGD model reduction, and a posteriori error estimation

Chamoin

Thai

2019

Numerical Meth Engineering

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Summary In this paper, we address the effective and accurate solution of problems with parameterized geometry. Considering the attractive framework of isogeometric analysis, which enables a natural and flexible link between computer‐aided design and simulation tools, the parameterization of the geometry is defined on the mapping from the isogeometric analysis parametric space to the physical space. From the subsequent multidimensional problem, model reduction based on the proper generalized decomposition technique with off‐line/online steps is introduced in order to describe the resulting manifold of parametric solutions with reduced CPU cost. Eventually, a posteriori estimation of various error sources inheriting from discretization and model reduction is performed in order to control the quality of the approximate solution, for any geometry, and feed a robust adaptive algorithm that optimizes the computational effort for prescribed accuracy. The overall approach thus constitutes an effective and reliable numerical tool for shape optimization analyses. Its performance is illustrated on several two‐ and three‐dimensional numerical experiments.

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The Constitutive Relation Error Method: A General Verification Tool

Cited by 9 publications

References 54 publications

Dual finite-element analysis using second-order cone programming for structures including contact

Dual finite-element analysis using second-order cone programming for structures including contact

Robust energy-based model updating framework for random processes in dynamics: Application to shaking-table experiments

Certified real‐time shape optimization using isogeometric analysis, PGD model reduction, and a posteriori error estimation

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