A posteriori error estimation techniques in practical finite element analysis

Grätsch, Thomas; Bathe, Klaus‐Jürgen

doi:10.1016/j.compstruc.2004.08.011

Cited by 220 publications

(165 citation statements)

References 71 publications

(106 reference statements)

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“…However, the areas of high stress gradients cannot be guaranteed to coincide with the areas of interest and this may lead to pollution error. Hence, the goal-oriented error estimation method could be improved by using a balance between the local error and the global error so that the mesh is refined to assure a high level of accuracy of the quantity of interest and at the same time to not underestimate the effect of the global error [14].…”

Section: (B) Numerical Resultsmentioning

confidence: 99%

A dual weighted residual method applied to complex periodic gratings

Lord

Mulholland²

2013

Proc. R. Soc. A.

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An extension of the dual weighted residual (DWR) method to the analysis of electromagnetic waves in a periodic diffraction grating is presented. Using the α,0-quasi-periodic transformation, an upper bound for the a posteriori error estimate is derived. This is then used to solve adaptively the associated Helmholtz problem. The goal is to achieve an acceptable accuracy in the computed diffraction efficiency while keeping the computational mesh relatively coarse. Numerical results are presented to illustrate the advantage of using DWR over the global a posteriori error estimate approach. The application of the method in biomimetic, to address the complex diffraction geometry of the Morpho butterfly wing is also discussed.

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Section: (B) Numerical Resultsmentioning

confidence: 99%

A dual weighted residual method applied to complex periodic gratings

Lord

Mulholland²

2013

Proc. R. Soc. A.

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“…Of course, we considered in this paper only a priori error estimates and corresponding convergence behaviors of shell elements, the a posteriori error estimation in shell analyses is another large and very important field, where much progress is still needed [27].…”

Section: Discussionmentioning

confidence: 99%

Measuring the convergence behavior of shell analysis schemes

Bathe

Lee

2011

Computers & Structures

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a b s t r a c tWhile shells have been analyzed abundantly for many years in engineering and the sciences, improved finite element and related analysis methods are still much desired and researched. More general and effective finite element procedures are needed for complex shell structures, including for the analysis of composite shells and the optimization of shells. In this paper we discuss how finite element methods, and other analysis techniques, should be tested in order to identify their reliability and effectiveness. We summarize some important theoretical results, present appropriate test problems and convergence measures, and we illustrate our discussion through some novel numerical results. An important conclusion is that the testing has to be performed very carefully in order to obtain relevant results, and we show how this is accomplished in detail.

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“…Our analysis approach is signficantly different from that of Mommer and Stevenson [18], combining the recent contraction frameworks of Cascon, Kreuzer, Nochetto and Siebert [7], of Nochetto, Siebert and Veeser [19], and of Holst, Tsogtgerel and Zhu [16]. We also give some numerical results comparing our goal-oriented method both to the one presented in [18] and the dual weighted residual (DWR) method as in [2,4,9,13,14,10], among others. Unlike the existing literature on the DWR method, we prove strong convergence of our goal-oriented method.…”

Section: Introductionmentioning

confidence: 99%

“…We also show a number of the adaptive meshes below, demonstrating that in many cases, the compared methods produce similar performance from qualitatively different adaptive refinements. A discussion of the DWR method may be found in [2,4,9,13,14,10] for example. In our DWR implementation, the finite element space for the primal problem V T k employs linear Lagrange elements as do HP and MS for both the primal and dual spaces.…”

Section: Numericsmentioning

confidence: 99%

Convergence of goal‐oriented adaptive finite element methods for nonsymmetric problems

Holst

Pollock

2015

Numerical Methods Partial

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ABSTRACT. In this article we develop convergence theory for a class of goal-oriented adaptive finite element algorithms for second order nonsymmetric linear elliptic equations. In particular, we establish contraction results for a method of this type for Dirichlet problems involving the elliptic operator Lu = ∇·(A∇u)−b·∇u−cu, with A Lipschitz, almost-everywhere symmetric positive definite, with b divergence-free, and with c ≥ 0. We first describe the problem class and review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We then describe a goal-oriented variation of standard AFEM (GOAFEM). Following the recent work of Mommer and Stevenson for symmetric problems, we establish contraction of GOAFEM and convergence in the sense of the goal function. Our analysis approach is signficantly different from that of Mommer and Stevenson, combining the recent contraction frameworks developed by Cascon, Kreuzer, Nochetto and Siebert; by Nochetto, Siebert and Veeser; and by Holst, Tsogtgerel and Zhu. We include numerical results demonstrating performance of our method with standard goal-oriented strategies on a convection problem .

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A posteriori error estimation techniques in practical finite element analysis

Cited by 220 publications

References 71 publications

A dual weighted residual method applied to complex periodic gratings

A dual weighted residual method applied to complex periodic gratings

Measuring the convergence behavior of shell analysis schemes

Convergence of goal‐oriented adaptive finite element methods for nonsymmetric problems

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