Computational error estimates and adaptive processes for some nonlinear structural problems

Babuška, Ivo; Rheinboldt, Werner C.

doi:10.1016/0045-7825(82)90094-9

Cited by 52 publications

(14 citation statements)

References 29 publications

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“…Some results for nonlinear boundary value problems have been obtained, beginning with early work on a posteriori error estimation for nonlinear equations by Carey [23] and Babuška and Rheinboldt [24]. Their error indicators were based on local residuals and element nonlinear problems with Dirichlet boundary data, respectively.…”

Section: B R Carnes and G F Careymentioning

confidence: 98%

Estimating spatial and parameter error in parameterized nonlinear reaction–diffusion equations

Carnes

Carey

2006

Commun. Numer. Meth. Engng.

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SUMMARYA new approach is proposed for the a posteriori error estimation of both global spatial and parameter error in parameterized nonlinear reaction-diffusion problems. The technique is based on linear equations relating the linearized spatial and parameter error to the weak residual. Computable local element error indicators are derived for local contributions to the global spatial and parameter error, along with corresponding global error indicators. The effectiveness of the error indicators is demonstrated using model problems for the case of regular points and simple turning points. In addition, a new turning point predictor and adaptive algorithm for accurately computing turning points are introduced.

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Section: B R Carnes and G F Careymentioning

confidence: 98%

Estimating spatial and parameter error in parameterized nonlinear reaction–diffusion equations

Carnes

Carey

2006

Commun. Numer. Meth. Engng.

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“…For elliptic problems, the appropriate norm appears naturally as they energy norm and a fairly complete and general theory of error estimators is available [24,25). However, far less research has been directed towards hyperbolic problems and e detailed theory is still lacking.…”

Section: Mesh Enrichmentmentioning

confidence: 99%

High Speed Inviscid Compressible Flow by the Finite Element Method

Zienkiewicz

Loehner

Morgan

1985

The Mathematics of Finite Elements and Applications

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“…Indeed methods based on the use of SA-field, like the error in constitutive equation (Ladevèze, 1975;Ladevèze and Leguillon, 1983) or like the equilibrated residuals (Babuška and Rheinboldt, 1978;Babuška and Miller, 1987), provide efficient error estimators, for global or local quantities (Ladevèze and Moës, 1999;Prudhomme and Oden, 1999;Ohnimus et al, 2001;Becker and Rannacher, 2001), even in some nonlinear contexts (Babuška and Rheinboldt, 1982;Ladevèze et al, 1986;Ladevèze, 2001;Louf et al, 2003) and domain decomposition methods (Parret-Fréaud et al, 2010).…”

Section: Introductionmentioning

confidence: 99%

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

Rey¹,

Gosselet²,

Rey³

2014

Computer Methods in Applied Mechanics and Engineering

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This paper focuses on the construction of statically admissible stress fields (SA-fields) for a posteriori error estimation. In the context of verification, the recovery of such fields enables to provide strict upper bounds of the energy norm of the discretization error between the known finite element solution and the unavailable exact solution. The reconstruction is a difficult and decisive step insofar as the effectiveness of the estimator strongly depends on the quality of the SA-fields. This paper examines the strong prolongation hypothesis, which is the starting point of the Element Equilibration Technique (EET). We manage to characterize the full space of SA-fields satisfying the prolongation equation so that optimizing inside this space is possible. The computation exploits topological properties of the mesh so that implementation is easy and costs remain controlled. In this paper, we describe the new technique in details and compare it to the classical EET and to the flux-free technique for different 2D mechanical problems. The method is explained on first degree triangular elements, but we show how extensions to different elements and to 3D are straightforward.

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Computational error estimates and adaptive processes for some nonlinear structural problems

Cited by 52 publications

References 29 publications

Estimating spatial and parameter error in parameterized nonlinear reaction–diffusion equations

Estimating spatial and parameter error in parameterized nonlinear reaction–diffusion equations

High Speed Inviscid Compressible Flow by the Finite Element Method

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

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