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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: This paper presents a modification of the Giuliani method that generates non-distorted elements while preserving the element size and can be extended to three-dimensional cases.
Abstract: In the generation of quadrilateral unstructured meshes, special attention is focussed to the shape of the elements. This is because it is well known that the distortion of the elements and the accuracy of the analysis are closely related. However, in adaptive schemes it is also essential that the newly generated mesh meets the prescribed element sizes in order to obtain a solution with the desired precision. In 1982 Giuliani developed a robust rezoning algorithm based on geometrical criteria. It gives proven results in a smooth element size distribution, but elements do not verify the prescribed element size when sharp distributions appear. This paper presents a modification of the Giuliani method that generates non-distorted elements while preserving the element size. Similar to the original method, this modification can be extended to three-dimensional cases.

27 citations

Journal ArticleDOI
TL;DR: Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view.
Abstract: Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990‐2000

27 citations

Journal ArticleDOI
TL;DR: In this article, a moving mesh algorithm, integrated into a general finite element method, has been developed with regard to the contact between moving loads and cable structures, and the application of the method to the interaction between railway overhead lines and train pantograph has been carried out in order to demonstrate the flexibility and advantages of this proposal against traditional meshing techniques.

25 citations

Journal ArticleDOI
TL;DR: In this article, the h-adaptive finite element technique is employed to solve some complex geotechnical problems involving material nonlinearity and large deformations, and results are provided to illustrate the performance of the method.
Abstract: Adaptive finite element procedures automatically refine, coarsen, or relocate elements in a finite element mesh to obtain a solution with a specified accuracy. Although a significant amount of research has been devoted to adaptive finite element analysis, this method has not been widely applied to nonlinear geotechnical problems due to their complexity. In this paper, the h-adaptive finite element technique is employed to solve some complex geotechnical problems involving material nonlinearity and large deformations. The key components of h-adaptivity including robust mesh generation algorithms, error estimators and remapping procedures are discussed. This paper includes a brief literature review as well as formulation and implementation details of the h-adaptive technique. Finally, the method is used to solve some classical geotechnical problems and results are provided to illustrate the performance of the method.

24 citations


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

  • ...The h-adaptive FE strategy subdivides the integration region into successively smaller sub-regions [10], thus changing the density of the elements to yield a more accurate solution while keeping the element order constant....

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Journal ArticleDOI
TL;DR: A 3D method of adaptivity with the latest version of the INRIA automatic mesh generator GAMHIC3D is proposed, based on a technique of error in constitutive relation and an efficient adaptive technique which automatically takes into account the steep gradient areas.

23 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