Eight nodes or nine?

MacNeal, Richard H.; Harder, R. L.

doi:10.1002/nme.1620330510

Cited by 45 publications

(36 citation statements)

References 20 publications

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“…With the reduced order integration, the performance of QUAD9 is, however, worse than that of QUAD8. An excellent comparison of QUAD8 and QUAD9 is given by MacNeal and Harder [15].…”

Section: Introductionmentioning

confidence: 99%

“…MacNeal and Harder [15] proposed a method by which the shape functions of an improved 8-node quadrilateral plane element are obtained based on the shape functions of QUAD9 by 'constraining rigidly' the interior node. This results in an excellent element that can reproduce exactly a Cartesian quadratic displacement ÿeld when the element geometry is of bilinear shape.…”

Section: Introductionmentioning

confidence: 99%

“…Or, in other words, their element can reproduce exactly a Cartesian quadratic displacement ÿeld under angular distortion. For the cantilever beam and hemispherical shell test problems, however, the QUAD9 still shows a slightly better performance [15]. Although the technique is attractive and very e ective, explicit expressions for the modiÿed shape functions have not been presented by them.…”

Section: Introductionmentioning

confidence: 99%

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A novel unsymmetric 8‐node plane element immune to mesh distortion under a quadratic displacement field

Rajendran

Liew

2003

Numerical Meth Engineering

162

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SUMMARYAn 8-node quadrilateral plane ÿnite element is developed based on a novel unsymmetric formulation which is characterized by the use of two sets of shape functions, viz., the compatibility enforcing shape functions and completeness enforcing shape functions. The former are chosen to satisfy exactly the minimum inter-as well as intra-element displacement continuity requirements, while the latter are chosen to satisfy all the (linear and higher order) completeness requirements so as to reproduce exactly a quadratic displacement ÿeld. Numerical results from test problems reveal that the new element is indeed capable of reproducing exactly a complete quadratic displacement ÿeld under all types of admissible mesh distortions. In this respect, the proposed 8-node unsymmetric element emerges to be better than the existing symmetric QUAD8, QUAD8=9, QUAD9, QUAD12 and QUAD16 elements, and matches the performance of the quartic element, QUAD25. For test problems involving a cubic or higher order displacement ÿeld, the proposed element yields a solution accuracy that is comparable to or better than that of QUAD8, QUAD8=9 and QUAD9 elements. Furthermore, the element maintains a good accuracy even with the reduced 2 × 2 numerical integration.

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“…With the reduced order integration, the performance of QUAD9 is, however, worse than that of QUAD8. An excellent comparison of QUAD8 and QUAD9 is given by MacNeal and Harder [15].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A novel unsymmetric 8‐node plane element immune to mesh distortion under a quadratic displacement field

Rajendran

Liew

2003

Numerical Meth Engineering

162

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“…Thus, each original element is subdivided into four elements. Repeat this division again and again, the total number of the elements in the refined mesh after each action will be 4 A standard front method solver is used for elements Q8 and HSF-Q8, while a modified unsymmetric front method solver is used for elements Q8, HSF-Q8, US-QUAD8, and US-ATFQ8. The program…”

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confidence: 99%

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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“…The benefits and drawbacks of both elements have been discussed thoroughly in the literature, e.g. [48,49,101]. The 9 noded Lagrange element has the advantage of being less sensitive to the position of the midside nodes, and it passes the out-of-plane bending patch test when the sides are curved.…”

Section: Parameterization Of Quadratic Shear and Thickness Deformablementioning

confidence: 99%

Rectangular shell elements based on the absolute nodal coordinate formulation

Hyldahl¹

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This thesis concerns the development of tools that are useful for designing machine systems and components in an virtual environment using flexible multibody dynamics. Initially, flexible multibody dynamics is briefly reviewed to explain available formalisms and possible applications of this method. Subsequently, attention is turned towards the analysis of structures that undergo large deformations and rotations using shell finite elements based on the absolute nodal coordinate formulation, ANCF. This topic is the focus for the remainder of the thesis.The thesis is divided into three main parts. The first part acts as a general introduction to ANCF based shell finite elements and sums up available elements found in the research literature. In the end, a new ANCF element developed during this PhD project is presented which gives enhanced modeling capabilities of problems including e.g. moving boundary conditions. The second part concerns the performance and behavior of a certain class of ANCF shell elements that are developed for analysis of thin shell structures. This includes discussions on differences concerning their kinematic descriptions and disclosure of certain issues regarding their performance. Those being sensitivity to irregular mesh, pour representation of curved structures and load dependent convergence when analyzing curved structures.The final part concerns the development of a new versatile ANCF shell element. This element is distinguished by being able to describe both thin and thick curved structures. This part contains a thorough derivation of its kinematics and stiffness description, as well as numerical examples to demonstrate its performance. However, this part of the study is not yet complete.Finally, the findings of this PhD project are summed up in a conclusion and possibilities for further studies and perspectives are outlined.v ResuméDenne afhandling omhandler udviklingen af vaerktøjer, som er nyttige til at designe maskiner og maskindele i et virtuelt miljø vha. fleksibel flerlegemedynamik. Indledningsvist gives der en kort gennemgang af den fleksible flerlegeme-dynamik, for at give et overblik over metoden og dens mulige anvendelser. Dernaest flyttes opmaerksomheden over på analyse af strukturer som udsaettes for store deformationer og rotationer vha. endelige skalelementer (eng: shell finite elements) baseret på absolutte knude koordinater (eng: absolute nodal coordinate formulation, ANCF). Dette emne vil vaere fokus i resten af afhandlingen.Afhandlingen er inddelt i 3 dele. Den første del giver en general introduktion til ANCF skal-elementer, og opsummerer de forskellige elementklasser som kan findes i litteraturen. Mod slutningen introduceres der et nyt ANCF skal-element, som er udviklet under dette ph.d.-projekt. Dette element muliggør forbedret modellering af problemer med mobile randbetingelser.Anden del omhandler effektiviteten og egenskaberne hos en saerlig klasse af ANCF skal-elementer, som er udviklet specielt til analyse af tynde skalstrukturer. Heri diskuteres der...

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Eight nodes or nine?

Cited by 45 publications

References 20 publications

A novel unsymmetric 8‐node plane element immune to mesh distortion under a quadratic displacement field

A novel unsymmetric 8‐node plane element immune to mesh distortion under a quadratic displacement field

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

Rectangular shell elements based on the absolute nodal coordinate formulation

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