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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TLDR
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
Overview of adaptive finite element analysis in computational geodynamics
TL;DR: A review of the current status and usage of spatially adaptive finite element analysis in the field of geodynamics can be found in this article, along with a discussion pertaining to the issues related to using adaptive analysis techniques and perspectives for future research.
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Topology optimization driven by anisotropic mesh adaptation: Towards a free-form design
TL;DR: The new adaptive algorithm SIMPATY for topology optimization to design lightweight and stiff structures exhibiting free-form features is proposed by properly combining the classical SIMP method with an anisotropic mesh adaptation strategy based on a recovery-based a posteriori error estimator.
Journal ArticleDOI
A posteriori error estimation and error control for finite element approximations of the time-dependent Navier-Stokes equations
Serge Prudhomme,J. T. Oden +1 more
TL;DR: The error estimators and the error control procedure are based on the residuals of the Navier–Stokes equations, which are shown to be comparable to error components in the velocity variable.
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Coupling finite element method with meshless finite difference method in thermomechanical problems
J. Jakowiec,Sławomir Milewski +1 more
TL;DR: This paper focuses on coupling two different computational approaches, namely finite element method (FEM) and meshless finite difference method (MFDM), in one domain and the consistent formulation of the mixed problem for the coupled FEMMFDM method is derived.
Journal ArticleDOI
A recovery-based error estimator for anisotropic mesh adaptation in cfd
TL;DR: A gradient recovery type a posteriori error estimator for finite element approximations on anisotropic meshes with promising results, since it is shown numerically to yield quasi-optimal triangulations with respect to the error-vs-number of elements behaviour.
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.
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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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