Recovery by Equilibrium in Patches (Rep)

Boroomand, B.; Zienkiewicz, O. C.

doi:10.1002/(sici)1097-0207(19970115)40:1<137::aid-nme57>3.0.co;2-5

Cited by 155 publications

(55 citation statements)

References 25 publications

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“…Consider, in Figure , a plate with a hole subjected to a unidirectional tensile stress σ 0 =1 at an infinite distance from a cylindrical hole with the radius a =1. This plane strain elasticity problem has been extensively used in the literature as a benchmark test for stress recovery procedures,() and its exact solution is given by the following:

bold-italicu false(bold-italicx false) = \frac{false(1 + ν false) a σ_{0}}{2 E} ([], \begin{matrix} center \\ array 2 (1 - ν) (\frac{r}{a} + \frac{2 a}{r}) \cos θ + (\frac{a}{r} - \frac{a^{3}}{r^{3}}) \cos 3 θ \\ array - 2 (1 - 2 ν) \frac{a}{r} \sin θ - 2 ν \frac{r}{a} \sin θ + (\frac{a}{r} - \frac{a^{3}}{r^{3}}) \sin 3 θ \\ array 0 \end{matrix}),

boldσ false(bold-italicx false) = ([], \begin{matrix} center \\ array σ_{11} \\ array σ_{22} \\ array σ_{33} \\ array σ_{12} \\ array σ_{23} \\ array σ_{13} \end{matrix}) = σ_{0} ([], \begin{matrix} center \\ array 1 - \frac{a^{2}}{r^{2}} (\frac{3}{2} \cos 2 θ + \cos 4 θ) + \frac{3 a^{4}}{2 r^{2}} \cos 4 θ \\ array - \frac{a^{2}}{r^{2}} (\frac{1}{2} \cos 2 θ - \cos 4 θ) - \frac{3 a^{4}}{2 r^{2}} \cos 4 θ \\ array ν (1 - 2 \frac{a^{2}}{r^{2...}}) \end{matrix})

…”

Section: Examplesmentioning

confidence: 99%

An improved stress recovery technique for low‐order 3D finite elements

Sharma

Zhang

Langelaar

et al. 2018

Numerical Meth Engineering

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SummaryIn this paper, we propose a stress recovery procedure for low-order finite elements in 3D. For each finite element, the recovered stress field is obtained by satisfying equilibrium in an average sense and by projecting the directly calculated stress field onto a conveniently chosen space. Compared with existing recovery techniques, the current procedure gives more accurate stress fields, is simpler to implement, and can be applied to different types of elements without further modification. We demonstrate, through a set of examples in linear elasticity, that the recovered stresses converge at a higher rate than that of directly calculated stresses and that, in some cases, the rate of convergence is the same as that of the displacement field.

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bold-italicu false(bold-italicx false) = \frac{false(1 + ν false) a σ_{0}}{2 E} ([], \begin{matrix} center \\ array 2 (1 - ν) (\frac{r}{a} + \frac{2 a}{r}) \cos θ + (\frac{a}{r} - \frac{a^{3}}{r^{3}}) \cos 3 θ \\ array - 2 (1 - 2 ν) \frac{a}{r} \sin θ - 2 ν \frac{r}{a} \sin θ + (\frac{a}{r} - \frac{a^{3}}{r^{3}}) \sin 3 θ \\ array 0 \end{matrix}),

boldσ false(bold-italicx false) = ([], \begin{matrix} center \\ array σ_{11} \\ array σ_{22} \\ array σ_{33} \\ array σ_{12} \\ array σ_{23} \\ array σ_{13} \end{matrix}) = σ_{0} ([], \begin{matrix} center \\ array 1 - \frac{a^{2}}{r^{2}} (\frac{3}{2} \cos 2 θ + \cos 4 θ) + \frac{3 a^{4}}{2 r^{2}} \cos 4 θ \\ array - \frac{a^{2}}{r^{2}} (\frac{1}{2} \cos 2 θ - \cos 4 θ) - \frac{3 a^{4}}{2 r^{2}} \cos 4 θ \\ array ν (1 - 2 \frac{a^{2}}{r^{2...}}) \end{matrix})

…”

Section: Examplesmentioning

confidence: 99%

An improved stress recovery technique for low‐order 3D finite elements

Sharma

Zhang

Langelaar

et al. 2018

Numerical Meth Engineering

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show abstract

“…It is also found, for linear elements on irregular meshes, that SPR produces superconvergence of order O(h 1+α ) with α greater than zero [22]. Many variants of the patch recovery procedure have also been proposed [32][33][34][35][36][37][38][39]. However, the patch recovery procedure has been modified by various investigators [23][24][25][26][27][28][29][30][31].…”

Section: Recovery For Gradients and Stressesmentioning

confidence: 99%

Errors, Recovery Processes, and Error Estimates

Zienkiewicz¹,

Taylor²,

Zhu³

2013

The Finite Element Method: Its Basis and Fundamentals

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“…The idea behind the REP is to impose the fulfillment of equilibrium on the patch in a weak form while proceeding to a reinterpolation of stresses on a local polynomial base. Combinations of elementwise and boundary equilibrium conditions to be added to the standard REP formulation have been also proposed to enhance the accuracy of the obtained results . Other interesting examples of recovery procedures based on equilibrium considerations have been proposed, for instance, in related works and, more recently, in the work of Parés and Díez, where the devised technique is adopted to calculate error bounds for the advection‐diffusion‐reaction equation.…”

Section: Introductionmentioning

confidence: 99%

An equilibrium‐based stress recovery procedure for the VEM

Artioli

Miranda

Lovadina

et al. 2018

Numerical Meth Engineering

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Summary Within the framework of the displacement‐based virtual element method (VEM), namely, for plane elasticity, an important topic is the development of optimal techniques for the evaluation of the stress field. In fact, in the classical VEM formulation, the same projection operator used to approximate the strain field (and then evaluate the stiffness matrix) is employed to recover, via constitutive law, the stress field. Considering a first‐order formulation, strains are locally mapped onto constant functions, and stresses are piecewise constant. However, the virtual displacements might engender more complex strain fields for polygons, which are not triangles. This leads to an undesirable loss of information with respect to the underlying virtual stress field. The recovery by compatibility in patches, originally proposed for finite element schemes, is here extended to VEM, aiming at mitigating such an effect. Stresses are recovered by minimizing the complementary energy of patches of elements over an assumed set of equilibrated stress modes. The procedure is simple, efficient, and can be readily implemented in existing codes. Numerical tests confirm the good performance of the proposed technique in terms of accuracy and indicate an increase of convergence rate with respect to the classical approach in many cases.

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Recovery by Equilibrium in Patches (Rep)

Cited by 155 publications

References 25 publications

An improved stress recovery technique for low‐order 3D finite elements

An improved stress recovery technique for low‐order 3D finite elements

Errors, Recovery Processes, and Error Estimates

An equilibrium‐based stress recovery procedure for the VEM

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