A partition-of-unity based ‘FE-Meshfree’ QUAD4 element with radial-polynomial basis functions for static analyses

Xu, Jianping; Rajendran, S.

doi:10.1016/j.cma.2011.08.005

Cited by 47 publications

(48 citation statements)

References 35 publications

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“…Various selections for weight functions are available in PUM, such as shape function of Quad4 [69] and smoothed weight function of Trig3-CNS [72]. The forms of nodal approximations are also various, including least-squares method [53] and orthogonalization leastsquares method (CO-LS) [58].…”

Section: Partition Of Unity Methodsmentioning

confidence: 99%

“…The present Quad4-RPIMcns is a development of Quad4-RPIM [69]. In order to get rid of the possible singularity of constructing nodal approximations , the radial-polynomial basis functions are explored in the least-square version of point interpolation [27].…”

Section: Formulation For Quad4-rpimcnsmentioning

confidence: 99%

“…By inheriting the advantages of Meshless method, FE-Meshless methods are able to achieve better accuracy and faster convergence rate, compared to four-node isoparametric quadrilateral element (Quad4) [53] and three-node triangle element (Trig3) [72]. However, FE-Meshless methods, which use pure polynomial basis functions to construct nodal approximations, also suffer from the singularity problem of constructing nodal approximation [27,69]. In order to avoid the singularity problem, radial basis functions introduced by [27] are used to construct nodal approximation in the context of hybrid FE-Meshless methods.…”

Section: Introductionmentioning

confidence: 97%

“…The present method can be regarded as the development of FEMeshless methods with radial basis functions [69] and Quad4-CNS [59]. In the present method, radial-polynomial basis functions [27] are used to construct the nodal approximations in the context of partition of unity, in order to avoid the singularity problem of constructing nodal approximation on arbitrary scattered nodes.…”

Section: Introductionmentioning

confidence: 99%

“…The concept of PUM is used to develop the hp clouds [48], generalized finite element method (GFEM) [56], particle-partition of unity method [55], numerical manifold method [9,10,71] and extended finite element method [6]. Furthermore, PUM provides a convenient approach to construct higher order global approximations without adding external nodes, thereby achieving higher accuracy and convergence rate [69].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A hybrid ‘FE-Meshless’ QUAD4 with continuous nodal stress using radial-polynomial basis functions

Yang

Zheng

2015

Engineering Analysis with Boundary Elements

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Section: Partition Of Unity Methodsmentioning

confidence: 99%

Section: Formulation For Quad4-rpimcnsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A hybrid ‘FE-Meshless’ QUAD4 with continuous nodal stress using radial-polynomial basis functions

Yang

Zheng

2015

Engineering Analysis with Boundary Elements

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A shape‐free 8‐node plane element unsymmetric analytical trial function method

Cen

Zhou

2012

Numerical Meth Engineering

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SUMMARYThe unsymmetric FEM is one of the effective techniques for developing finite element models immune to various mesh distortions. However, because of the inherent limitation of the metric shape functions, the resulting element models exhibit rotational frame dependence and interpolation failure under certain conditions. In this paper, by introducing the analytical trial function method used in the hybrid stress-function element method, an effort was made to naturally eliminate these defects and improve accuracy. The key point of the new strategy is that the monomial terms (the trial functions) in the assumed metric displacement fields are replaced by the fundamental analytical solutions of plane problems. Furthermore, some rational conditions are imposed on the trial functions so that the assumed displacement fields possess fourthorder completeness in Cartesian coordinates. The resulting element model, denoted by US-ATFQ8, can still work well when interpolation failure modes for original unsymmetric element occur, and provide the invariance for the coordinate rotation. Numerical results show that the exact solutions for constant strain/stress, pure bending and linear bending problems can be obtained by the new element US-ATFQ8 using arbitrary severely distorted meshes, and produce more accurate results for other more complicated problems.

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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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A partition-of-unity based ‘FE-Meshfree’ QUAD4 element with radial-polynomial basis functions for static analyses

Cited by 47 publications

References 35 publications

A hybrid ‘FE-Meshless’ QUAD4 with continuous nodal stress using radial-polynomial basis functions

A hybrid ‘FE-Meshless’ QUAD4 with continuous nodal stress using radial-polynomial basis functions

A shape‐free 8‐node plane element unsymmetric analytical trial function method

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

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