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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A continuous shape sensitivity equation method for unsteady laminar flows
TL;DR: In this article, a general shape sensitivity equation method (SEM) is applied to the flow past a cylinder in ground proximity to predict the response of the unsteady flow response to changes in the ground to cylinder gap.
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A continuous Lagrangian sensitivity equation method for incompressible flow
TL;DR: It is proposed the use of pseudo-elasticity equations to provide a general framework to generate this mapping for unstructured meshes on complex geometries to produce significant improvements in terms of both accuracy and CPU time.
Proceedings Article
Local gradient and stress recovery for triangular elements in plane elasticity
TL;DR: In this article, a local gradient post-processing technique for linear, quadratic and cubic triangular elements is proposed for elasticity problems, based on the least-square residuals of the balance equation, the irrotationality condition and the superconvergence of the derivatives of the finite element solution at certain special points.
Journal ArticleDOI
Enhancing finite element approximation for eigenvalue problems by projection method
Huipo Liu,Ningning Yan +1 more
TL;DR: The superconvergence and the related recovery type a posteriori error estimators based on projection method for finite element approximation of the elliptic eigenvalue problems are established and can be employed to provide useful a posterioru error estimator in adaptive finite element computation under unstructured meshes.
Journal ArticleDOI
Stress projection, layerwise-equivalent, formulation for accurate predictions of transverse stresses in laminated plates and shells
TL;DR: A modified superconvergent patch recovery (MSPR) technique has been utilized to obtain accurate nodal in-plane stresses which are subsequently used with the thickness integration of the three-dimensional equilibrium equations to evaluate the transverse shear and normal stresses.
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.
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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.
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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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