A Uniformly Accurate Finite Element Method for the Reissner–Mindlin Plate

Arnold, Douglas N.; Falk, Richard S.

doi:10.1137/0726074

Cited by 252 publications

(241 citation statements)

References 13 publications

Supporting

Mentioning

235

Contrasting

Unclassified

Order By: Relevance

“…This, combined with known regularity results ( [2,7]), allow us to prove analogous estimates for the approximation of the eigenfunctions in vibration problems. Let us remark that this would not be the case for higher order methods, since the eigenfunctions are in general not regular enough for us to attain similar results.…”

Section: Introductionmentioning

confidence: 98%

“…In what follows we will make use of the known a priori estimates for the solutions of problems (2.4) and (2.5) (see for instance [2]):…”

Section: Proofsmentioning

confidence: 99%

“…In order to avoid this drawback, reduced integration or mixed methods are usually applied. To perform their mathematical analysis, a family of problems, one for each thickness t > 0, is considered and approximation results valid uniformly on t are sought (see for instance [2,7,10]). …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

et al. 1999

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 98%

“…In what follows we will make use of the known a priori estimates for the solutions of problems (2.4) and (2.5) (see for instance [2]):…”

Section: Proofsmentioning

confidence: 99%

See 1 more Smart Citation

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

et al. 1999

View full text Add to dashboard Cite

show abstract

“…conditions on the whole boundary for both β and ω, and impose the uniform loading g = 1 in Ω. We have solved this problem using the present approach and the non-conforming method from [2], which is known to be a locking-free finite element method. We have fixed a mesh of the same type as those from Figure 1, containing 12 divisions in each direction, and have solved the problem using the present methods (33) and (74), and the method from [2], on the same mesh.…”

Section: Assessing the Lack Of Locking: A Uniformily Loaded Platementioning

confidence: 99%

“…With the aim of avoiding this locking (i.e., in order to 1 design finite element methods whose error estimates are independent of the thickness of the plate) a considerable amount of work has been produced over the last couple of decades. Most of these works have been carried out using a mixed finite element approach, see, e.g., [2], [12], [19], [1], and [14] for an overview, and the references therein. All these references deal with a second order equation for the rotations, while the remaining variables were determined by first-order partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Stabilised finite element methods for a bending moment formulation of the Reissner-Mindlin plate model

Barrenechea

Barrios

Wachtel

2014

Calcolo

View full text Add to dashboard Cite

This version is available at https://strathprints.strath.ac.uk/50948/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.ukThe Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output.Stabilised finite element methods for a bending moment formulation of the Reissner-Mindlin plate model * Gabriel R. Barrenechea † , Tomás P. Barrios ‡ , and Andreas Wachtel § , AbstractThis work presents new stabilised finite element methods for a bending moments formulation of the Reissner-Mindlin plate model. The introduction of the bending moment as an extra unknown leads to a new weak formulation, where the symmetry of this variable is imposed strongly in the space. This weak problem is proved to be well-posed, and stabilised Galerkin schemes for its discretisation are presented and analysed. The finite element methods are such that the bending moment tensor is sought in a finite element space constituted of piecewise linear continuos and symmetric tensors. Optimal error estimates are proved, and these findings are illustrated by representative numerical experiments.Mathematics Subject Classifications (1991):

show abstract

Spline approximation of thin shell dynamics

Rosario

Smith

1997

Int. J. Numer. Meth. Engng.

View full text Add to dashboard Cite

SUMMARYA spline-based method for approximating thin shell dynamics is presented here. While the method is developed in the context of the Donnell-Mushtari thin shell equations, it can be easily extended to the Byrne-Flü gge-Lur'ye equations or other models for shells of revolution as warranted by applications. The primary requirements for the method include accuracy, flexibility and efficiency in smart material applications. To accomplish this, the method was designed to be flexible with regard to boundary conditions, material non-homogeneities due to sensors and actuators, and inputs from smart material actuators such as piezoceramic patches. The accuracy of the method was also of primary concern, both to guarantee full resolution of structural dynamics and to facilitate the development of PDE-based controllers which ultimately require real-time implementation. Several numerical examples provide initial evidence demonstrating the efficacy of the method.1997 by John Wiley & Sons, Ltd.

show abstract

A Uniformly Accurate Finite Element Method for the Reissner–Mindlin Plate

Abstract: This paper is dedicated to Jim Douglas, Jr., on the occasion of his 60th birthday.

Cited by 252 publications

References 13 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

Stabilised finite element methods for a bending moment formulation of the Reissner-Mindlin plate model

Spline approximation of thin shell dynamics

Contact Info

Product

Resources

About