A four‐node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation

Bathe, Klaus‐Jürgen; Dvorkin, Eduardo N.

doi:10.1002/nme.1620210213

Cited by 922 publications

(483 citation statements)

References 9 publications

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“…Then, standard properties of separation of isolated parts of the spectrum (see for instance [14]) yield the following result: Mixed finite elements for the load problem (2.4) have been introduced and analyzed in several papers (see for example [6,8,10,15,16]). The method that we will use here can be seen as the lowest degree member of the so-called MITC elements, which are based on relaxing the second equation of (2.3).…”

Section: Approximation Of the Eigenvalue Problemmentioning

confidence: 99%

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Approximation of the vibration modes of a plate by Reissner-Mindlin equations

et al. 1999

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Abstract. This paper deals with the approximation of the vibration modes of a plate modelled by the Reissner-Mindlin equations. It is well known that, in order to avoid locking, some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods. In particular, one of the most widely used procedures is the mixed interpolation tensorial components, based on the family of elements called MITC. We use the lowest order method of this family.Applying a general approximation theory for spectral problems, we obtain optimal order error estimates for the eigenvectors and the eigenvalues. Under mild assumptions, these estimates are valid with constants independent of the plate thickness. The optimal double order for the eigenvalues is derived from a corresponding L 2 -estimate for a load problem which is proven here. This optimal order L 2 -estimate is of interest in itself.Finally, we present several numerical examples showing the very good behavior of the numerical procedure even in some cases not covered by our theory.

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Section: Approximation Of the Eigenvalue Problemmentioning

confidence: 99%

“…Among these methods, the so-called MITC ones, introduced by Bathe and Dvorkin in [6], are very likely the most used in practice. Their application to load problems has also drawn much attention from the mathematical point of view ( [5,8,10,15,16]).…”

Section: Introductionmentioning

confidence: 99%

Approximation of the vibration modes of a plate by Reissner-Mindlin equations

et al. 1999

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“…Numerically, the standard FSDT-based finite elements are too stiff and lead to shear locking. To treat this phenomenon, some improved techniques such as reduced integration [26], mixed interpolation of tensorial components (MITC) [27,28], discrete shear gap (DSG) [17] elements, etc. were adopted.…”

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confidence: 99%

Isogeometric analysis for nonlinear thermomechanical stability of functionally graded plates

Tran

Phung‐Van

Lee

et al. 2016

Composite Structures

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“…Thereafter, the calculated components are evaluated at the Gauss points using bilinear or linear interpolation functions (depending on the number of sampling points) from their values at the sampling points. This method was originally applied for four shell elements by Dvorkin and Bathe [4] ant it has proven to be efficient, very robust and to give very good results for four node plate bending theory based on Mindlin/Reissner plate theory in [30] (applied on the transverse shear strain components E 13 , E 23 ). This method was later extended to overcome the curvature thickness locking by modifying the interpolation of the transverse normal strain component E 33 for four-node shell elements, as proposed by Betsch and Stein [29] and also Bischoff and Ramm [31].…”

Section: Introductionmentioning

confidence: 99%

On the comparison of two solid-shell formulations based on in-plane reduced and full integration schemes in linear and non-linear applications

Bettaieb

Sena

Sousa

et al. 2015

Finite Elements in Analysis and Design

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In the present paper, a detailed description of the formulation of the new SSH3D solid-shell element is presented. This formulation is compared with the previously proposed RESS solid-shell element [1,2]. Both elements were recently implemented within the LAGAMINE in-house research finite element code. These solid-shell elements possess eight nodes with only displacement nodal degrees of freedom (DOF). In order to overcome various locking pathologies, the SSH3D formulation employs the well known Enhanced Assumed Strain (EAS) concept originally introduced by Simo and Rifai [3] and based on the Hu-Veubeke-Washizu variational principle combined with the Assumed Natural Strain (ANS) technique based on the work of Dvorkin and Bathe [4]. For the RESS solid-shell element, on the other hand, only the EAS technique is used with a Reduced Integration (RI) Scheme. A particular characteristic of these elements is their special integration schemes, with an arbitrary number of integration points along the thickness direction, dedicated to analyze problems involving non-linear through-thickness distribution (i.e. metal forming applications) without requiring many element layers. The formulation of the SSH3D element is also particular, with regard to the solid-shell elements proposed in the literature, in the sense that it is characterized by an in-plane full integration and a large variety in terms of (i) enhancing parameters, (ii) the ANS version choice and (iii) the number of integration points through the thickness direction. The choice for these three parameters should be adapted to each problem so as to obtain accurate results and to keep the calculation time low. Numerous numerical examples are performed to investigate the performance of these elements. These examples illustrate the reliability and the efficiency of the proposed formulations in various cases including linear and non-linear problems. SSH3D element is more robust thanks to the various options proposed and its full in-plane integration scheme, while RESS element in more efficient from a computational point of view.

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A four‐node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation

Cited by 922 publications

References 9 publications

Approximation of the vibration modes of a plate by Reissner-Mindlin equations

Approximation of the vibration modes of a plate by Reissner-Mindlin equations

Isogeometric analysis for nonlinear thermomechanical stability of functionally graded plates

On the comparison of two solid-shell formulations based on in-plane reduced and full integration schemes in linear and non-linear applications

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