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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.
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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TL;DR: In this article , an adaptive clustering solution rewinding procedure and a dynamic adaptivity split factor strategy are suggested to further enhance the adaptive process, which can be used in other CROMs to extend them into ACROMs, opening new avenues to explore adaptivity in this context.
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TL;DR: In this paper, a metodologia for analizar the respuesta dinamica de estructuras sometidas a cargas moviles is presented, and un algoritmo de mallado adaptativo de elementos finitos que se mueve de forma solidaria a la carga movil actua sobre la estructura.
Abstract: Resumen En este articulo se presenta una metodologia para analizar la respuesta dinamica de estructuras sometidas a cargas moviles. Para ello se ha desarrollado un algoritmo de mallado adaptativo de elementos finitos que se mueve de forma solidaria a la carga movil que actua sobre la estructura. El algoritmo de mallado movil se ha validado con la solucion analitica de una carga movil que recorre una viga simplemente apoyada.
5 citations
TL;DR: In this paper, an interpolation mixte is presented based on elements of finis-particules, which resulte de la generalisation de plusieurs travaux dans ce domaine.
Abstract: Les methodes sans maillage emploient une interpolation associee a un ensemble de particules : aucune information concernant la connectivite ne doit etre fournie.
Un des atouts de ces methodes est que la discretisation
peut etre enrichie d'une facon tres simple, soit en augmentant le nombre de particules (analogue a la
strategie de raffinement h), soit en augmentant l'ordre de consistance (analogue
a la strategie de raffinement p). Neanmoins, le cout du calcul des fonctions
d'interpolation est tres eleve et ceci represente un inconvenient vis-a-vis
des elements finis. Cet article presente une interpolation mixte elements
finis-particules qui resulte de la generalisation de plusieurs travaux dans ce
domaine. La formulation de cette interpolation mixte est valable pour n'importe quel ordre de consistance. Dans ce contexte, on
enonce un estimateur d'erreur a priori dont la demonstration
se base dans les proprietes de l'interpolation mixte.
Ce resultat permet d'etudier la convergence de la
methode d'enrichissement et d'etablir les strategies de raffinement de
l'interpolation qui permettent d'atteindre une solution
avec une precision satisfaisante.
5 citations
TL;DR: This work proposes a 3D method of adaptivity with the latest version of the INRIA automatic mesh generator GAMHIC3D, based on a technique of error in constitutive relation and an efficient adaptive technique which automatically takes into account the steep gradient areas.
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TL;DR: In this article, an esquema adaptativo for problemas tridimensionales de conveccion-difusion discretizados mediante el Metodo de Elementos Finitos is presented, where remallado permite aumentar o disminuir drasticamente el tamano de los elementos in one single paso, de forma automatica.
Abstract: Resumen Se presenta un esquema adaptativo para problemas tridimensionales de conveccion-difusion discretizados mediante el Metodo de Elementos Finitos. El esquema adaptativo incluye una estrategia de remallado basada en la imposicion de un volumen maximo de los elementos a partir de una malla de referencia. El remallado permite aumentar o disminuir drasticamente el tamano de los elementos en un solo paso, de forma automatica. Se mantiene una calidad de la malla adecuada; y, como consecuencia, el numero de iteracciones necesarias para solucionar el sistema de ecuaciones lineales mediante algoritmos iterativos se mantiene constante. Se presentan 2 ejemplos de caracteristicas muy distintas que permiten analizar la propuesta para un amplio rango de situaciones. Uno es una extension tridimensional del problema de Smolarkiewicz y el otro es una version simplificada del problema de transporte de contaminacion emitida por un emisor puntual. Los resultados muestran la flexibilidad de la propuesta. Es posible encontrar un valor optimo de frecuencia de remallado, desde el punto de vista de coste computacional y precision de los resultados, para ambas tipologias extremas de problemas.
5 citations
References
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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
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
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.
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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
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