A stabilization matrix for the bilinear mindlin plate element

Belytschko, Ted; Tsay, Chen-Shyh; Liu, W.K.

doi:10.1016/0045-7825(81)90048-7

Cited by 121 publications

(31 citation statements)

References 8 publications

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“…As the shell thickness decreases, the convergence of the finite element solution rapidly deteriorates [8]. Researchers developed methods to overcome locking phenomena in flat triangular shell elements, such as the Mixed Interpolation of Tensorial Components method used in the MITC3 element [8], the line integration method used in TRIA3 element [42], the mixed or hybrid formulation [43][44][45], incompatible displacement methods [46][47], stabilization methods [29,[48][49][50][51][52][53][54], assumed strain methods [19,28,55], etc.…”

Section: Introductionmentioning

confidence: 99%

A 3-Node Co-Rotational Triangular Elasto-Plastic Shell Element Using Vectorial Rotational Variables

Li¹,

Izzuddin²,

Vu‐Quoc³

et al. 2017

Volume 13 Number 3

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A 3-node co-rotational triangular elasto-plastic shell element is developed. The local coordinate system of the element employs a zero-'macro spin' framework at the macro element level, thus reducing the material spin over the element domain, and resulting in an invariance of the element tangent stiffness matrix to the order of the node numbering. The two smallest components of each nodal orientation vector are defined as rotational variables, achieving the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing co-rotational finite-element formulations, both element tangent stiffness matrices in the local and global coordinate systems are symmetric owing to the commutativity of the nodal variables in calculating the second derivatives of strain energy with respect to the local nodal variables and, through chain differentiation with respect to the global nodal variables. For elasto-plastic analysis, the Maxwell-Huber-Hencky-von Mises yield criterion is employed together with the backward-Euler return-mapping method for the evaluation of the elasto-plastic stress state, where a consistent tangent modulus matrix is used. Assumed membrane strains and assumed shear strains---calculated respectively from the edge-member membrane strains and the edge-member transverse shear strains---are employed to overcome locking problems, and the residual bending flexibility is added to the transverse shear flexibility to improve further the accuracy of the element. The reliability and convergence of the proposed 3-node triangular shell element formulation are verified through two elastic plate patch tests as well as three elastic and three elasto-plastic plate/shell problems undergoing large displacements and large rotations.

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Section: Introductionmentioning

confidence: 99%

A 3-Node Co-Rotational Triangular Elasto-Plastic Shell Element Using Vectorial Rotational Variables

Li¹,

Izzuddin²,

Vu‐Quoc³

et al. 2017

Volume 13 Number 3

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“…In order to avoid shear locking, many effective modifications and numerical techniques have been proposed and tested, such as the reduced integration and selective reduced integration schemes proposed by Zienkiewicz et al [17] and Hughes et al [18]; the substitute shear strain method proposed by Hinton et al [19]; the stabilization procedure proposed by Belytschko et al [20,21]; free formulation method proposed by Bergan et al [22], etc. In 1982, Macneal [23] introduced a new concept called 'assumed natural strain: ANS method'.…”

Section: Introductionmentioning

confidence: 99%

Application of the quadrilateral area co‐ordinate method: a new element for Mindlin–Reissner plate

Cen

Long

Yao

et al. 2005

Numerical Meth Engineering

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SUMMARYThe quadrilateral area co-ordinate method is used to formulate a new quadrilateral element for Mindlin-Reissner plate bending problem. Firstly, an independent shear field is assumed based on the locking-free Timoshenko's beam formulae; secondly, a fourth-order deflection field is assumed by introducing some generalized conforming conditions; thirdly, the rotation field is determined by the strain-displacement relations. Furthermore, a hybrid post-processing procedure is suggested to improve the stress/internal force solutions. Following this procedure, a new 4-node, 12-dof quadrilateral element, named AC-MQ4, is successfully constructed. Since all formulations are expressed by the area coordinates, element AC-MQ4 presents some different, but beneficial characters when compared with other usual models. Numerical examples show the new element is free of shear locking, insensitive to mesh distortion, and possesses excellent accuracy in the analysis of both thick and thin plates. It has also been demonstrated that the area co-ordinate method, the generalized conforming condition method, and the hybrid post-processing procedure are efficient tools for developing simple, effective and reliable finite element models.

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“…However this technique can lead to instability known as the hour-glass phenominon. To overcome the instability problem or to suppress the zero-energy hour-glass modes, the addition of a stabilization matrix need to be introduced [4][5][6][7]15,21,[25][26][27]33]. This stabilization matrix should preserve the conditions of invariance and consistency which ensure that the element passes the patch test.…”

Section: Introductionmentioning

confidence: 99%

A novel FEM by scaling the gradient of strains with factor α (αFEM)

2008

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This paper presents a novel finite element method of quadrilateral elements by scaling the gradient of strains and Jacobian matrices with a scaling factor α (αFEM). We first prove that the solution of the αFEM is continuous for α ∈ [0, 1] and bounded from both below and above, and hence is convergent. A general procedure of the αFEM has been proposed to obtain the exact or best possible solution for a given problem, in which an exact-α approach is devised for overestimation problems and a zero-α approach is suggested for underestimation problems. Using the proposed αFEM approaches, much more stable and accurate solutions can be obtained compared to that of standard FEM. The theoretical analyses and intensive numerical studies also demonstrate that the αFEM effectively overcomes the following wellknown drawbacks of the standard FEM: (1) Overestimation of stiffness matrix when the full Gauss integration is used; (2) Instability problem known as hour-glass locking (presence of hour-glass modes or spurious zero-energy modes) when the reduced integration is used; (3) Volumetric locking in nearly incompressible problems when the bulk modulus becomes infinite.

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A stabilization matrix for the bilinear mindlin plate element

Cited by 121 publications

References 8 publications

A 3-Node Co-Rotational Triangular Elasto-Plastic Shell Element Using Vectorial Rotational Variables

A 3-Node Co-Rotational Triangular Elasto-Plastic Shell Element Using Vectorial Rotational Variables

Application of the quadrilateral area co‐ordinate method: a new element for Mindlin–Reissner plate

A novel FEM by scaling the gradient of strains with factor α (αFEM)

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