Coupled model- and solution-adaptivity in the finite-element method

Stein, Erwin; Ohnimus, S.

doi:10.1016/s0045-7825(97)00082-0

Cited by 57 publications

(31 citation statements)

References 13 publications

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“…Recently, Houston, Senior and Süli (2003) presented Sobolev regularity in order to control hp-adaptivity. Simple indicators for hp-adaptivity were given by Stein and Ohnimus (1997) according to the following rule: If the a posteriori error estimator considered is of the size of the associated a priori error estimator (with an uncertainty factor two to three), then p is increased while h remains unchanged. If the a posteriori error estimator is considerably smaller, then h is reduced while p remains unchanged.…”

Section: Finite Element Methods For Elasticity With Error-controlled mentioning

confidence: 99%

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Finite Element Methods for Elasticity with Error‐Controlled Discretization and Model Adaptivity

Stein¹,

Rüter²

2004

Encyclopedia of Computational Mechanics

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The essential topics of the finite element method for linear and finite elastic deformations of solids are presented in this chapter from both the mechanical and mathematical point of view. As a starting point, the nonlinear, and linearized theory of elasticity are derived in a rigorous way, followed by the classical variational principles of elasticity, which are the basis for the finite element method in its various forms. More precisely, the discrete variational approach of the (one‐field) Dirichlet minimization principle of the total potential energy is presented, followed by concise representations of the (two‐field) Hellinger–Reissner stationary dual‐mixed principle and the (three‐field) Hu–Washizu stationary mixed principle, including the main features of the associated finite element methods. The main objective of this chapter is the systematic treatment of error estimation procedures and adaptivity for the linearized and finite elasticity problem covering both global and goal‐oriented a posteriori error estimators. The three basic classes, that is, residual‐, hierarchical‐, and averaging‐type error estimators are presented and applied to a fracture mechanics problem as an example. A further challenging topic is the combination of error‐controlled adaptive finite element solutions with hierarchical model and dimension adaptivity of the underlying mathematical model, especially for thin‐walled structures where model expansion is necessary in subdomains with boundary layers and other disturbances.

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Section: Finite Element Methods For Elasticity With Error-controlled mentioning

confidence: 99%

“…2. Regional model adaptivity in subdomains with boundary layers, especially of thin-walled plates and shells using global and regional error estimates see for example, Stein and Ohnimus (1997) and . 3.…”

Section: Introductionmentioning

confidence: 99%

Finite Element Methods for Elasticity with Error‐Controlled Discretization and Model Adaptivity

Stein¹,

Rüter²

2004

Encyclopedia of Computational Mechanics

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“…for a wide-spanned steel bridge, are often initiated by local buckling, combined with inelastic deformations or by crack propagations due to high cycle fatigue. To analyse these processes and scenarios by the finite element method, adaptive hierarchical modelling and analysis is necessary, see Reference [2], e.g. first using an enhanced beam theory, e.g.…”

Section: The Gap Between Computable Error Bounds and The Requirementsmentioning

confidence: 99%

Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends

Stein

Rüter

Ohnimus³

2004

Numerical Meth Engineering

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SUMMARYThe main purpose of this paper is two-fold: first, to present a critical review of available errorcontrolled adaptive finite element methods-with absolute global and goal-oriented error estimatesfor approximate solutions of a given mathematical model and the model error of the mathematical model considered and its dimensions, enhanced with our own recent results in fracture mechanics. The second related important matter is which physical/mathematical models-including various boundary layers and other disturbances-and which accuracy of the finite element solutions in which norm are prescribed and necessary for the safety and reliability of a certain engineering problem, e.g. in structural engineering, of course depending on materials, systems and the type as well as the magnitude of prescribed loadings (i.e. actions in a factorized concept) acting on a specific structure. It is hard to understand why the current codes for structural engineering design in Europe-especially the European Standards (Eurocodes) and the relevant German DIN-Codes (DIN: Deutsches Institut für Normung e. V.)-do not lay out any rules or bounds for the prescribed accuracy of the computation of material and structural resistance, i.e. the validity of mechanical modelling and the accuracy of associated numerical methods, usually today the finite element method. It can be observed that behind the existing codes there still lies the old thinking of, e.g., classical elastic beam theory of so-called second order for stability problems and the corresponding elastic potential for plates and shells in buckling. Modern flexible finite element modelling by deriving theories in variational form for thin-walled beams, plates and shells from rather general 3D theories, for both linear and non-linear problems, capturing inelastic deformations and layer effects by material and dimensional expansions are not addressed in the Eurocodes for structural engineering. The necessary consequence is that joint work of computational mechanics specialists and structural engineers, working in national and international code committees, is absolutely necessary, in order to avoid a lot of costly troubles and dangers of today by improving and updating the existing codes. THE GAP BETWEEN COMPUTABLE ERROR BOUNDS AND THE REQUIREMENTS OF TECHNICAL CODESThe goal of reliable and efficient engineering analysis is the proof of material and structural resistance against all possible loadings (actions) with sufficient safety at minimal cost, guaranteeing full usability within lifetime. Codes, standards and technical rules sufficiently support the engineering design of most technical components in all engineering fields. But requirements and instructions for adequate mechanical modelling and the prescribed accuracy of static and dynamic analysis are not included. The thinking behind these codes is obviously still influenced by classical beam and plate models for which 'exact' analytical or numerical solutions by series are possible under a couple of restricting conditions, bu...

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“…In [12] and [3] (see also [2]) the residual-type estimators were proposed and proved reliable and efficient under the assumptions that the right-hand side of the given equation is zero and the original domain is a plate with plane parallel faces. In [4] and [10] the implicit estimators based on the solution of local three-dimensional Neumann problems were developed for the hierarchical modelling of complex elastic plates. In [1] the estimator of Babuška and Schwab (see [2], [3]) was extended to take into account the discretization error stemming from the approximate solution of the reduced problem.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast to the above mentioned papers, which deal with the hierarchical modelling of the problems in thin domains, we only consider the so-called zero-order method of dimension reduction that is, however, very popular owing to its simplicity and purely two-dimensional formulation. At the same time, this method forms a basis for the hierarchical modelling of three-dimensional plates (see, e.g., [11], [3], [10]). We advocate the functional-type a posteriori error estimation approach (see [5], [6], [7], [8]) that essentially differs from the approaches taken in the aforementioned articles; however, surprisingly enough, it is possible to show that Babuška and Schwab's estimator for the zero-order reduced problem can be obtained as a particular case of our estimator when the right-hand side of the equation is zero and the original domain is a plate with plane parallel faces.…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Estimation of Dimension Reduction Errors

Repin

Sauter

Smolianski

2004

Numerical Mathematics and Advanced Applications

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A new a-posteriori error estimator is presented for the verification of the dimensionally reduced models stemming from the elliptic problems on thin domains. The original problem is considered in a general setting, without any specific assumptions on the domain geometry, coefficients and the right-hand sides. For the energy norm of the error of the zero-order dimension reduction method, the proposed estimator is shown to always provide a guaranteed upper bound. In the case when the original domain has constant thickness (but, possibly, non-plane upper and lower faces), the estimator demonstrates the optimal convergence rate as the thickness tends to zero. It is also flexible enough to successfully cope with the case of infinitely growing right-hand side of the equation when the domain thickness tends to zero. The numerical tests indicate the efficiency of the estimator and its ability to accurately represent the local error distribution needed for an adaptive improvement of the reduced model.2000 Mathematics Subject Classification: 35J20, 65N15, 65N30

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Coupled model- and solution-adaptivity in the finite-element method

Cited by 57 publications

References 13 publications

Finite Element Methods for Elasticity with Error‐Controlled Discretization and Model Adaptivity

Finite Element Methods for Elasticity with Error‐Controlled Discretization and Model Adaptivity

Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends

A Posteriori Estimation of Dimension Reduction Errors

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