Journal ArticleDOI
The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique
O. C. Zienkiewicz,J. Z. Zhu +1 more
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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.read more
Citations
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Journal ArticleDOI
Goal-oriented a posteriori error estimates for transport problems
Dmitri Kuzmin,Sergey Korotov +1 more
TL;DR: A node-based approach to localization of global errors in the quantities of interest is pursued and a possible violation of Galerkin orthogonality is taken into account.
Journal ArticleDOI
Goal-Oriented Error Estimates Based on Different FE-Spaces for the Primal and the Dual Problem with Applications to Fracture Mechanics
TL;DR: In this paper, the authors derived goal-oriented a posteriori error estimators for the error obtained while approximately evaluating the nonlinear J-integral as a fracture criterion in linear elastic fracture mechanics using the finite element method.
Journal ArticleDOI
Coulomb frictional contact by explicit projection in the cone for finite displacement quasi-static problems
TL;DR: Six problems previously indicated as difficult to solve by the node-to-segment discretization and the operator split algorithm are here solved with the new projection algorithm.
Journal ArticleDOI
Superconvergent patch recovery—a key to quality assessed FE solutions
TL;DR: The Superconvergent Patch Recovery (SPR) method introduced by Zienkiewicz & Zhu (Int. Numer. Meth. Engng, 1992, 33, 1331, 1382) was the breakthrough of an accurate and cheap post-processing method to estimate errors in the FE solution as discussed by the authors.
Journal ArticleDOI
Positivity preservation and adaptive solution of two-equation models of turbulence
TL;DR: In this article, a simple change of dependent variables that guarantees positivity of turbulence variables in numerical simulation codes is proposed, which is valid for any numerical scheme be it a finite difference, a finite volume, or a finite element method.
References
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Journal ArticleDOI
A simple error estimator and adaptive procedure for practical engineerng analysis
O. C. Zienkiewicz,J. Z. Zhu +1 more
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.
Journal ArticleDOI
Local and global smoothing of discontinuous finite element functions using a least squares method
E. Hinton,John S. Campbell +1 more
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.
Journal ArticleDOI
Optimal stress locations in finite element models
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.
Journal ArticleDOI
The post-processing approach in the finite element method—part 1: Calculation of displacements, stresses and other higher derivatives of the displacements
Ivo Babuška,A. Miller +1 more
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.
Journal ArticleDOI
Higher order local accuracy by averaging in the finite element method
James H. Bramble,A. H. Schatz +1 more
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.
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