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Exeter Thesis

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In this article, the recently proposed patch recovery scheme of Zienkiewicz and Zhu [4] is applied to the basic four-node isoparametric displacement membrane. The common configuration of a four element patch is considered. It is observed, for this element and configuration, that the proposed scheme can produce results that are dependent on the co-ordinate system in which the patch is defined. In attempting to overcome this problem the concept of a parent patch is proposed. Results from an error estimator based on this concept and applied to a set of problems similar to those laid down in [3] are presented.
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The existence of optimal points for calculating accurate stresses within finite element models is discussed. A method for locating such points is proposed and applied to several popular finite elements.
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A error estimate for the finite element solutions of elliptic boundary value problems is introduced. The error measure is defined by the energy-norm distance between the kinematically admissible stress field computed by the displacement finite element method, and a quasi-statically admissible stress field. The error estimate is defined using this distance normalized by the total strain energy of the deformed body. The element contribution to the above error estimate is used to define an error indicator and a local error measure which can be used in adaptive finite element method as a criteria for mesh refinement. This method of error estimate can be implemented in existing finite element programs in a straightforward manner.
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This chapter discusses the experience with the patch test for convergence of finite elements. The usual explanation for the astonishing popularity of finite elements is that the technique allows an engineer to specify any feasible problem geometry, and material properties. The program blocks are complex, but they are modular and readily checked. The storage and arithmetic requirements are nicely balanced for modern computers. Most of the innermost loops are such that the cost of the arithmetic can be halved by introducing, as options, only two or three very simple machine-language subroutines. The chapter reviews the origins of the patch test. For if the external nodes of any sub-assembly of a successful assembly of elements are given prescribed values corresponding to an arbitrary state of constant curvature, then the internal nodes must obediently take their correct values. An internal node is defined as one completely surrounded by elements. Conversely, if two overlapping patches can reproduce any given state of constant curvature, they should combine into a larger successful patch, provided that every external node lost is internal to one of the original patches.
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The reliabilities of several a posteriori error estimators, including those of Gago, Zienkiewicz-Zhu, and more recently those proposed by Beckers and Zhong are compared through a set of examples in plane elasticity. The examples range from those having analytic solutions to those having progressively stronger singularities. The examples generally use either 4 (or 8) node quadrilaterals for initial comparisons. The results of these examples indicate that, in certain cases, some of the error estimators are unreliable and do not appear to be asymptotically exact. Further studies are suggested to investigate the general validity of the initial conclusions.
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This paper shows how the Continuum Region Element (CRE) Method can be used exclusively for element sensitivity testing without the need for a finite element system. For the four-node quadrilateral membrane, the element's geometric and stress/strain relations can be expressed explicitly in terms of its shape parameters. The procedure is demonstrated for the isoparametric displacement membrane.
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A new definition of shape parameters for a quadrilateral is presented. It is shown that the shape parameters can be expressed in terms of simple polynomial coefficients with a clear physical meaning. It is also shown how the shape parameters can be evaluated from the Jacobian matrix. Element warpage and convexity are also covered.