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Journal ArticleDOI

Adaptive finite element strategies based on error assessment

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.
Abstract: Two 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.

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Citations
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Journal ArticleDOI
TL;DR: In this article, a bibliographical review of the finite element methods applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view is presented.
Abstract: This paper gives a bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view. The bibliography at the end of the paper contains more than 1330 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1999–2002.

34 citations

Journal ArticleDOI
TL;DR: In this paper, an adaptive approach, with remeshing as essential ingredient, towards robust and efficient simulation techniques for fast transient, highly non-linear processes including contact is discussed.
Abstract: In this paper, an adaptive approach, with remeshing as essential ingredient, towards robust and efficient simulation techniques for fast transient, highly non-linear processes including contact is discussed. The necessity for remeshing stems from two sources: the capability to deal with large deformations that might even require topological changes of the mesh and the desire for an error driven distribution of computational resources. The overall computational approach is sketched, the adaptive remeshing strategy is presented and the crucial aspect, the choice of suitable error indicator(s), is discussed in more detail. Several numerical examples demonstrate the performance of the approach.

33 citations

Journal ArticleDOI
TL;DR: It is shown that the combined rh‐adaptive approach is superior to its components r‐ and h‐adaptivity, in that higher accuracies can be obtained compared to a purely r‐ adaptive approach, while the computational costs are lower than that of a purely h‐ Adaptive approach.
Abstract: An adaptive scheme is proposed in which the domain is split into two subdomains. One subdomain consists of regions where the discretization is refined with an h-adaptive approach, whereas in the other subdomain node relocation or r-adaptivity is used. Through this subdivision the advantageous properties of both remeshing strategies (accuracy and low computer costs, respectively) can be exploited in greater depth. The subdivision of the domain is based on the formulation of a desired element size, which renders the approach suitable for coupling with various error assessment tools. Two-dimensional linear examples where the analytical solution is known illustrate the approach. It is shown that the combined rh-adaptive approach is superior to its components r- and h-adaptivity, in that higher accuracies can be obtained compared to a purely r-adaptive approach, while the computational costs are lower than that of a purely h-adaptive approach. As such, a more flexible formulation of adaptive strategies is given, in which the relative importance of attaining a pre-set accuracy and speeding-up the computational process can be set by the user. Copyright © 2001 John Wiley & Sons, Ltd.

31 citations


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

  • ...As a consequence, these schemes oer topological exibility, but the expenses in terms of computer time associated to these adaptive techniques are relatively high [ 2 ]....

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  • ...In general, three methods of assessing the error exist, where we use the classication of Huerta and coworkers [ 2 ]:...

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  • ...If the approach is to be extended to other application elds, then error estimators and error indicators must be included [ 2 ]....

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  • ...Error estimators must have a solid mathematical basis [ 2 ; 14]....

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  • ...Since no elements are added to this subdomain, relative information is needed, that is, in which parts of the subdomain nodes should be concentrated, and from which parts of the subdomain nodes can be taken away [ 2 ]....

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Journal ArticleDOI
TL;DR: The proposed GSM for adaptive procedure has excellent stability that is crucial for adaptive analysis and is found very stable and can be easily applied to arbitrarily irregular triangular meshes for complex geometry.

29 citations


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

  • ...In advanced design of products of high precision, adaptive analysis is becoming an important tool in practical numerical computations [4]....

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Journal ArticleDOI
TL;DR: In this paper, the NURBS-enhanced finite element method (NEFEM) combined with a hybridisable discontinuous Galerkin (HDG) approach is presented for the first time.
Abstract: The NURBS-enhanced finite element method (NEFEM) combined with a hybridisable discontinuous Galerkin (HDG) approach is presented for the first time. The proposed technique completely eliminates the uncertainty induced by a polynomial approximation of curved boundaries that is common within an isoparametric approach and, compared to other DG methods, provides a significant reduction in number of degrees of freedom. In addition, by exploiting the ability of HDG to compute a postprocessed solution and by using a local a priori error estimate valid for elliptic problems, an inexpensive, reliable and computable error estimator is devised. The proposed methodology is used to solve Stokes flow problems using automatic degree adaptation. Particular attention is paid to the importance of an accurate boundary representation when changing the degree of approximation in curved elements. Several strategies are compared and the superiority and reliability of HDG-NEFEM with degree adaptation is illustrated.

27 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.

2,449 citations

Journal ArticleDOI
TL;DR: In this article, 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.
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.

1,211 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.

1,079 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.

815 citations

Journal ArticleDOI
Juan C. Simo1
TL;DR: In this article, the authors proposed a hyperelastic J2-flow theory for elastoplastic tangent moduli, which reduces to a trivial modification of the classical radial return algorithm which is amenable to exact linearization.

475 citations