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
Modeling of crack propagation via an automatic adaptive mesh refinement based on modified superconvergent patch recovery technique
TL;DR: In this article, an automated adaptive remeshing procedure is presented for simulation of arbitrary shape crack growth in a 2D finite element mesh, where the Zienkiewicz-Zhu error estimator is employed in conjunction with a modified SPR technique based on the recovery of gradients using analytical crack-tip fields in order to obtain more accurate estimation of errors.
Journal ArticleDOI
Superconvergence of Mixed Finite Element Approximations over Quadrilaterals
TL;DR: In this paper, a superconvergence result was established for approximate solutions of second-order elliptic equations by mixed finite element methods over quadrilaterals over quadratic surfaces.
Journal ArticleDOI
Adaptivity and convergence in the Voronoi cell finite element model for analyzing heterogeneous materials
Suresh Moorthy,Somnath Ghosh +1 more
TL;DR: In this paper, an adaptive Voronoi cell finite element model is presented for analyzing micromechanical response of composites and porous materials, and two error measures, viz. traction reciprocity error and an error in the kinematic relation, are formulated as indicators of the quality of VCFEM solutions.
Journal ArticleDOI
Fluid–structure interactions of the mitral valve and left heart: Comprehensive strategies, past, present and future
Daniel R. Einstein,Facundo Del Pin,Xiangmin Jiao,Andrew P. Kuprat,James P. Carson,Karyn S. Kunzelman,Richard P. Cochran,Julius M. Guccione,Mark B. Ratcliffe +8 more
TL;DR: The work toward a comprehensive FSI model of the left heart is profile by reviewing the early work, presenting the current work and laying out the future work in four broad categories: data collection, geometry, FSI and validation.
Journal ArticleDOI
Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (II)
Zhimin Zhang,Jianzhong Zhu +1 more
TL;DR: It is proved that the recovered derivative by the least-squares fitting is superconvergent for the two-point boundary value problems and the a posteriori error estimator based on the recovery technique is shown to be asymptotically exact.
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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