The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique
30 May 1992-International Journal for Numerical Methods in Engineering (INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING)-Vol. 33, Iss: 7, pp 1331-1364
TL;DR: In this article, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes, which has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems.
Abstract: This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.
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01 Jan 2000TL;DR: In this paper, a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics is presented, focusing on methods for linear elliptic boundary value problems.
Abstract: This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.
2,607 citations
TL;DR: This review covers Verification, Validation, Confirmation and related subjects for computational fluid dynamics (CFD), including error taxonomies, error estimation and banding, convergence rates, surrogate estimators, nonlinear dynamics, and error estimation for grid adaptation vs Quantification of Uncertainty.
Abstract: This review covers Verification, Validation, Confirmation and related subjects for computational fluid dynamics (CFD), including error taxonomies, error estimation and banding, convergence rates, surrogate estimators, nonlinear dynamics, and error estimation for grid adaptation vs Quantification of Uncertainty.
1,654 citations
Cites background from "The superconvergent patch recovery ..."
...See Zhu & Zienkiewicz (1990), Zienkiewicz & Zhu (1987, 1992), and other FEM methods that have a similar flavor, such as those of Strouboulis & Oden (1990), Oden et al (1993), Babuska et al (1994), and Ewing et al (1990)....
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TL;DR: An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented in this article, which enables accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements.
Abstract: An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non-smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi-field problems. Copyright © 2010 John Wiley & Sons, Ltd.
1,228 citations
Cites methods from "The superconvergent patch recovery ..."
...This recovery technique has first been proposed for the classical FEM in [204, 205]....
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TL;DR: In this paper, the authors derived a theorem showing the dependence of the effectivity index for the Zienkiewicz-Zhu error estimator on the convergence rate of the recovered solution.
Abstract: In this second part of the paper, the issue of a posteriori error estimation is discussed. In particular, we derive a theorem showing the dependence of the effectivity index for the Zienkiewicz–Zhu error estimator on the convergence rate of the recovered solution. This shows that with superconvergent recovery the effectivity index tends asymptotically to unity. The superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the Zienkiewicz–Zhu error estimator to demonstrate accurate estimation of the exact error attainable. Numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient (stress) error estimation. Several examples of the performance of the error estimator in adaptive mesh refinement are also presented.
1,106 citations
01 Jan 2002
TL;DR: This document introduces Lua extensions for FEMM for scripting / batch run capabilities, and describes how to use these extensions in the context of a NoSQL database.
Abstract: Acknowledgements Thanks to the following people for their valuable contributions to FEMM: • Si Hang, for writing fast point location routines that greatly increase the speed at which FEMM evaluates line integrals; • Robin Cornelius, for adding Lua extensions for FEMM for scripting / batch run capabilities ; • Keith Gregory, for his detailed help and comments.
774 citations
Cites background from "The superconvergent patch recovery ..."
...The general approach to estimat ing the “true” value ofB at any node point is to fit a least-squares plane through the values of B at the Gauss points of all elements that surround a node of interest, and to take the value of the plane at the node point’s location as its smoothed value of B [13]....
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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, the concepts and potential advantages of local and global least squares smoothing of discontinuous finite element functions are introduced, and the relationship between local smoothing and the reduced integration technique is established.
Abstract: The concepts and potential advantages of local and global least squares smoothing of discontinuous finite element functions are introduced.
The relationship between local smoothing and the ‘reduced’ integration' technique is established.
Examples are presented to illustrate the application of the two smoothing techniques to the finite element stresses from several structural analysis problems.
The paper concludes with some practical recommendations for discontinuous finite element function smoothing.
613 citations
TL;DR: In this paper, the existence of optimal points for calculating accurate stresses within finite element models is discussed and a method for locating such points is proposed and applied to several popular finite elements.
Abstract: The existence of optimal points for calculating accurate stresses within finite element models is discussed. A method for locating such points is proposed and applied to several popular finite elements.
395 citations
TL;DR: In this article, a method for post-processing a finite element solution to obtain high accuracy approximations for displacements, stresses, stress intensity factors, etc. is presented.
Abstract: This is the first in a series of three papers in which we discuss a method for ‘post-processing’ a finite element solution to obtain high accuracy approximations for displacements, stresses, stress intensity factors, etc. Rather than take the values of these quantities ‘directly’ from the finite element solution, we evaluate certain weighted averages of the solution over the entire region. These yield approximations are of the same order of accuracy as the strain energy. We obtain error estimates, and also present some numerical examples to illustrate the practical effectiveness of the technique. In the third paper of this series we address the matters of adaptive mesh selection and a posteriori error estimation.
304 citations
TL;DR: In this paper, the authors describe the class of finite element subspaces and explain the main result on the accuracy of K h * u h, where K h is a fixed function, u h represents local averages, and * denotes convolution.
Abstract: This chapter describes the class of finite element subspaces and explains the main result on the accuracy of K h * u h ,where K h is a fixed function, u h represents local averages, and * denotes convolution. The function K h has the following properties: (1) K h has small support; (2) K h is independent of the specific choice of S h or the operator L; (3) K h * u h is easily computable from u h ; and (4) K h * u h approximates u to higher order than does u h . The chapter also discusses on notation, subspaces and the construction of K h .
263 citations