Validation of recipes for the recovery of stresses and derivatives by a computer-based approach
01 Sep 1994-Mathematical and Computer Modelling (Elsevier Science Publishers B. V.)-Vol. 20, Iss: 6, pp 45-89
TL;DR: In this article, the authors present a methodology for checking the local quality of recipes for the recovery of stresses or derivatives from finite element solutions of linear elliptic problems, which can be used to obtain precise conclusions about the quality of a class of recipes, based on least-squares patch-recovery.
About: This article is published in Mathematical and Computer Modelling.The article was published on 1994-09-01 and is currently open access. It has received 16 citations till now. The article focuses on the topics: Partial differential equation & Finite element method.
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TL;DR: In this paper, the authors studied the pollution-error in the h-version of the finite element method and its effect on the quality of the local error indicators in the interior of the mesh.
Abstract: : We studied the pollution-error in the h-version of the finite element method and its effect on the quality of the local error indicators (resp. the quality of the derivatives recovered by local postprocessing) in the interior of the mesh. Here we show that it is possible to construct a-posteriori estimates of the pollution-error in a patch of elements by employing the local error indicators over the entire mesh. We also give an adaptive algorithm for the local control of the pollution-error in a patch of elements of interest.
128 citations
92 citations
TL;DR: In this article, the quality of the solution derivatives which are recovered from finite element solutions by local averaging schemes was investigated and it was shown that the recovered solution-derivatives have higher accuracy than the derivatives computed directly from the finite element solution.
80 citations
TL;DR: Mathematical proofs are presented for the derivative superconvergence obtained by a class of patch recovery techniques for both linear and bilinear nite elements in the approximation of second order elliptic problems.
Abstract: Mathematical proofs are presented for the derivative superconvergence obtained by a class of patch recovery techniques for both linear and bilinear nite elements in the approximation of second order elliptic problems.
75 citations
Cites methods from "Validation of recipes for the recov..."
...We refer to the recent work [1, 2, 3, 4] for a series of studies in a computer based approach on the superconvergence for finite element approximations....
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TL;DR: An error on the constitutive law (labeled the dissipation error) is used to measure the quality of finite element computations of plastic and viscoplastic structures whose behavior is described by internal variables.
62 citations
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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 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.
1,993 citations
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
1,097 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