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Journal ArticleDOI

Adaptive finite element strategies based on error assessment

10 Dec 1999-International Journal for Numerical Methods in Engineering (Wiley)-Vol. 46, Iss: 10, pp 1803-1818

TL;DR: Two main ingredients are needed for adaptive finite element computations: the error of a given solution must be assessed, and a new spatial discretization must be defined via h-, p- or r-adaptivity.

AbstractTwo main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or error indicators. After that, a new spatial discretization must be defined via h-, p- or r-adaptivity. In principle, any of the approaches for error assessment may be combined with any of the procedures for adapting the discretization. However, some combinations are clearly preferable. The advantages and limitations of the various alternatives are discussed. The most adequate strategies are illustrated by means of several applications in solid mechanics. Copyright © 1999 John Wiley & Sons, Ltd.

Topics: Mixed finite element method (63%), Discretization (61%), Finite element method (54%), Estimator (53%)

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Citations
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Reference EntryDOI
15 Nov 2004
Abstract: The aim of the present chapter is to provide an in-depth survey of arbitrary Lagrangian–Eulerian (ALE) methods, including both conceptual aspects of the mixed kinematical description and numerical implementation details. Applications are discussed in fluid dynamics, nonlinear solid mechanics and coupled problems describing fluid–structure interaction. The need for an adequate mesh-update strategy is underlined, and various automatic mesh-displacement prescription algorithms are reviewed. This includes mesh-regularization methods essentially based on geometrical concepts, as well as mesh-adaptation techniques aimed at optimizing the computational mesh according to some error indicator. Emphasis is then placed on particular issues related to the modeling of compressible and incompressible flow and nonlinear solid mechanics problems. This includes the treatment of convective terms in the conservation equations for mass, momentum, and energy, as well as a discussion of stress-update procedures for materials with history-dependent constitutive behavior. Keywords: ALE description; convective transport; finite elements; stabilization techniques; mesh regularization and adaptation; fluid dynamics; nonlinear solid mechanics; stress-update procedures; fluid–structure interaction

835 citations


Cites background or methods from "Adaptive finite element strategies ..."

  • ...Studies on the use of ALE as a mesh adaptation technique in solid mechanics are reported, among others, by Pijaudier-Cabot et al. (1995), Huerta et al. (1999), Askes and Sluys (2000), Askes and Rodŕıguez-Ferran (2001), Askes et al. (2001) and Rodŕıguez-Ferran et al. (2002)....

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  • ...The smoothed field ¤̃ can be obtained, for instance, by least-squares approximation, see Huétink et al. (1990). Another possibility is to retain the discontinuous field ¤ and devise appropriate algorithms that account for this fact....

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  • ...Similarly, the crack propagation problems discussed by Koh and Haber (1986) and Koh et al. (1988), where the crack path is known a priori, also allow the use of this kind of mesh update procedure....

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  • ...Similarly, the crack propagation problems discussed by Koh and Haber (1986) and Koh et al. (1988), where the crack path is known a priori, also allow the use of this kind of mesh update procedure. Other examples of prescribed mesh motion in nonlinear solid mechanics can be found in the works by Liu et al. (1986), Huétink et al....

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Journal ArticleDOI
Abstract: A method for characterisation of materials subjected to large strains beyond the levels when plastic instability occurs in standard tension tests is presented. Thin sheets of two types of hot-rolle ...

225 citations


Journal ArticleDOI
Abstract: The paper describes the application of the finite element method to predict the thermal, material and mechanical effects of welding; Computational Welding Mechanics (CWM). The most common numerical approach used in CWM is only described shortly. The paper focuses on different modelling aspects. The most important modelling issues are the models for heat input and material behaviour. Finally, some recent applications are reviewed and future developments are discussed.

224 citations


Journal ArticleDOI
Abstract: A mixed hierarchical approximation based on finite elements and meshless methods is presented. Two cases are considered. The first one couples regions where finite elements or meshless methods are used to interpolate: continuity and consistency is preserved. The second one enriches a finite element mesh with particles. Thus, there is no need to remesh in adaptive refinement processes. In both cases the same formulation is used, convergence is studied and examples are shown. Copyright © 2000 John Wiley & Sons, Ltd.

193 citations


Journal ArticleDOI
TL;DR: The basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem are reviewed and it is concluded that the actually practical error estimation techniques do not provide mathematically proven bounds on the error and need to be used with care.
Abstract: In this paper we review the basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem. We give the basic ideas to establish global error estimates for the energy norm as well as goal-oriented error estimates. While we show how these error estimation techniques are employed for our simple model problem, the emphasis of the paper is on assessing whether these procedures are ready for use in practical linear finite element analysis. We conclude that the actually practical error estimation techniques do not provide mathematically proven bounds on the error and need to be used with care. The more accurate estimation procedures also do not provide proven bounds that, in general, can be computed efficiently. We also briefly comment upon the state of error estimations in nonlinear and transient analyses and when mixed methods are used.

188 citations


References
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Journal ArticleDOI
TL;DR: A new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes.
Abstract: A new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes. The estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors. It can thus be combined with a full adaptive process of refinement or, more simply, provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials. When combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable. Estimates other than the energy norm have successfully been applied giving, for instance, a predetermined accuracy of stresses.

2,348 citations


Journal ArticleDOI
Abstract: Computable a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one-dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self-adjoint operator of the second order. Generalizations to more general one-dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a-posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.

1,178 citations


Journal ArticleDOI
TL;DR: An adaptive mesh procedure for improving the quality of steady state solutions of the Euler equations in two dimensions is described, implemented in conjunction with a finite element solution algorithm, using linear triangular elements, and an explicit time-stepping scheme.
Abstract: The present, quality-enhancing adaptive-mesh procedure for two-dimensional Euler equation steady state solutions is implemented by means of linear triangular elements and an explicit time-stepping scheme, in conjunction with a finite element solution algorithm. The meshes thus generated typically take the form of stretched elements in the vicinity of one-dimensional flow features; a considerable variation in element size may thereby emerge which allows the desired high-quality solutions to be obtained with commensurately high computational efficiency.

1,061 citations


Journal ArticleDOI
TL;DR: Three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations are presented and it is proved that as the mesh size decreases, under suitable assumptions, two of the error estimator approach upper bounds on the norm of the true error.
Abstract: We present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations. The estimators are based on solving local Neumann problems in each element. The estimators differ in how they enforce consistency of the Neumann problems. We prove that as the mesh size decreases, under suitable assumptions, two of the error estimators approach upper bounds on the norm of the true error, and all three error estimators are within multiplicative constants of the norm of the true error. We present numerical results in which one of the error estimators appears to converge to the norm of the true error.

791 citations


Journal ArticleDOI
Juan C. Simo1
Abstract: Computational aspects of the formulation of elastoplasticity proposed in Part I of this work, employing a multiplicative decomposition and based on the principle of maximum plastic dissipation, are examined in detail. In particular, the implementation of the proposed hyperelastic J2-flow theory is shown to reduce to a trivial modification of the classical radial return algorithm which is amenable to exact linearization. This results in a closed-form expression for the consistent elastoplastic tangent moduli. The algorithmic treatment of the general case is considered subsequently. It is emphasized that objective rates and incrementally objective algorithms play no role in the present approach. A number of numerical experiments are presented that illustrate the excellent performance of the proposed formulation.

450 citations