On convergence of nonconforming convex quadrilateral finite elements AGQ6

Flajs, Rado; Cen, Song; Saje, Miran

doi:10.1016/j.cma.2010.02.006

Cited by 10 publications

(15 citation statements)

References 21 publications

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“…However, as the prediction of MacNeal's theorem, the two elements cannot strictly pass the C 0 patch test (only pass the weak form of the C 0 patch test ). So, their convergence raised some queries and discussions . Chen et al .…”

Section: Introductionmentioning

confidence: 99%

“…However, as the prediction of MacNeal's theorem, the two elements cannot strictly pass the C 0 patch test (only pass the weak form of the C 0 patch test [31]). So, their convergence raised some queries and discussions [41][42][43]. Chen et al [44] imposed some constraints on the elements to make them pass the C 0 patch test, but the trapezoidal locking in bending comes back.…”

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confidence: 99%

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An unsymmetric 4-node, 8-DOF plane membrane element perfectly breaking through MacNeal's theorem

Cen

Zhou

et al. 2015

Int. J. Numer. Meth. Engng

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Summary Among numerous finite element techniques, few models can perfectly (without any numerical problems) break through MacNeal's theorem: any 4‐node, 8‐DOF membrane element will either lock in in‐plane bending or fail to pass a C0 patch test when the element's shape is an isosceles trapezoid. In this paper, a 4‐node plane quadrilateral membrane element is developed following the unsymmetric formulation concept, which means two different sets of interpolation functions for displacement fields are simultaneously used. The first set employs the shape functions of the traditional 4‐node bilinear isoparametric element, while the second set adopts a novel composite coordinate interpolation scheme with analytical trail function method, in which the Cartesian coordinates (x,y) and the second form of quadrilateral area coordinates (QACM‐II) (S,T) are applied together. The resulting element US‐ATFQ4 exhibits amazing performance in rigorous numerical tests. It is insensitive to various serious mesh distortions, free of trapezoidal locking, and can satisfy both the classical first‐order patch test and the second‐order patch test for pure bending. Furthermore, because of usage of the second form of quadrilateral area coordinates (QACM‐II), the new element provides the invariance for the coordinate rotation. It seems that the behaviors of the present model are beyond the well‐known contradiction defined by MacNeal's theorem. Copyright © 2015 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

An unsymmetric 4-node, 8-DOF plane membrane element perfectly breaking through MacNeal's theorem

Cen

Zhou

et al. 2015

Int. J. Numer. Meth. Engng

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“…In this section, a series of classical benchmark problems are considered. The results obtained by the present new element US-Q4θ are compared with other 4-node quadrilateral elements available in literatures 12,20,[27][28][29]32,53,[61][62][63][64][65][66] to assess the capabilities of US-Q4θ.…”

Section: Numerical Validationmentioning

confidence: 99%

“…However, they cannot strictly pass the C 0 patch test . Thus, some queries and discussions concerned with their convergences were raised …”

Section: Introductionmentioning

confidence: 99%

“…27 Thus, some queries and discussions concerned with their convergences were raised. 28 Shi et al 29 developed Q4-like membrane elements at the base of the quasi-conforming element technique, 30,31 whose pioneer works can be traced back to Tang et al and Chen et al 32,33 The stiffness matrices of these elements in Shi et al 29 can be explicitly computed in Cartesian coordinates. Thus, they are very computationally efficient.…”

Section: Introductionmentioning

confidence: 99%

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4‐node unsymmetric quadrilateral membrane element with drilling DOFs insensitive to severe mesh‐distortion

Shang

Ouyang

2017

Numerical Meth Engineering

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Summary The unsymmetric finite element method is a promising technique to produce distortion‐immune finite elements. In this work, a simple but robust 4‐node 12‐DOF unsymmetric quadrilateral membrane element is formulated. The test function of this new element is determined by a concise isoparametric‐based displacement field that is enriched by the Allman‐type drilling degrees of freedom. Meanwhile, a rational stress field, instead of the displacement one in the original unsymmetric formulation, is directly adopted to be the element's trial function. This stress field is obtained based on the analytical solutions of the plane stress/strain problem and the quasi‐conforming technique. Thus, it can a priori satisfy related governing equations. Numerical tests show that the presented new unsymmetric element, named as US‐Q4θ, exhibits excellent capabilities in predicting results of both displacement and stress, in most cases, superior to other existing 4‐node element models. In particular, it can still work very well in severely distorted meshes even when the element shape deteriorates into concave quadrangle or degenerated triangle.

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An unsymmetric 8‐node hexahedral element with high distortion tolerance

Zhou

Cen

Huang

et al. 2016

Numerical Meth Engineering

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SUMMARYAmong all 3D 8-node hexahedral solid elements in current finite element library, the 'best' one can produce good results for bending problems using coarse regular meshes. However, once the mesh is distorted, the accuracy will drop dramatically. And how to solve this problem is still a challenge that remains outstanding. This paper develops an 8-node, 24-DOF (three conventional DOFs per node) hexahedral element based on the virtual work principle, in which two different sets of displacement fields are employed simultaneously to formulate an unsymmetric element stiffness matrix. The first set simply utilizes the formulations of the traditional 8-node tri-linear isoparametric element, while the second set mainly employs the analytical trial functions in terms of 3D oblique coordinates (R, S, T). The resulting element, denoted by US-ATFH8, contains no adjustable factor, and can be used for both isotropic and anisotropic cases. Numerical examples show it can strictly pass both the first-order (constant stress/strain) patch test and the second-order patch test for pure bending, remove the volume locking, and provide the

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On convergence of nonconforming convex quadrilateral finite elements AGQ6

Cited by 10 publications

References 21 publications

An unsymmetric 4-node, 8-DOF plane membrane element perfectly breaking through MacNeal's theorem

An unsymmetric 4-node, 8-DOF plane membrane element perfectly breaking through MacNeal's theorem

4‐node unsymmetric quadrilateral membrane element with drilling DOFs insensitive to severe mesh‐distortion

An unsymmetric 8‐node hexahedral element with high distortion tolerance

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