A Low-order Nonconforming Finite Element for Reissner--Mindlin Plates

Lovadina, Carlo

doi:10.1137/040603474

Cited by 34 publications

(26 citation statements)

References 31 publications

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“…To deal with the discontinuity of V 2,h , we follow the idea in [11,16,18] and define for any vector-valued function ψ ∈ Π K∈J h H 1 (K) the jump across the edge e ∈ F as…”

Section: Reissner-mindlin Plate Modelmentioning

confidence: 99%

“…Various methods have been proposed to weaken or overcome the locking effect since the nineties of the last century, and most of them can be regarded as reduced integration methods. Recently, the discontinuous Galerkin method [1] has also been used to design finite element methods for the R-M plate problem (see, for instance, [2,11,16]). One common favorable feature of these discontinuous Galerkin R-M plate elements is that all the variables share the same nodes and consequently can also be extended to shell problems.…”

Section: Introductionmentioning

confidence: 99%

“…The second element differs from the first one only in the approximation of the rotation; which employs the modified NRQ 1 element for both components of the rotation, consequently all the variables in this method share the same nodes. Furthermore, the second finite element method enjoys the same promising features as the lower order triangular elements proposed in [2,11,16] and therefore is possible to be generalized to the shell problems.…”

Section: Introductionmentioning

confidence: 99%

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Two lower order nonconforming rectangular elements for the Reissner-Mindlin plate

Shi

2007

Math. Comp.

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Abstract. In this paper, we propose two lower order nonconforming rectangular elements for the Reissner-Mindlin plate. The first one uses the conforming bilinear element to approximate both components of the rotation, and the modified nonconforming rotated Q 1 element to approximate the displacement, whereas the second one uses the modified nonconforming rotated Q 1 element to approximate both the rotation and the displacement. Both elements employ a projection operator to overcome the shear force locking. We prove that both methods converge at optimal rates uniformly in the plate thickness t in both the H 1 -and L 2 -norms, and consequently they are locking free.

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“…To deal with the discontinuity of V 2,h , we follow the idea in [11,16,18] and define for any vector-valued function ψ ∈ Π K∈J h H 1 (K) the jump across the edge e ∈ F as…”

Section: Reissner-mindlin Plate Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two lower order nonconforming rectangular elements for the Reissner-Mindlin plate

Shi

2007

Math. Comp.

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“…See, for example, Arnold et al (2005), Arnold and Falk (1989), Brezzi and Marini (2003), Chinosi et al (2006), Lovadina (2005), and Ming and Shi (2001).…”

Section: Other Locking-free Elementsmentioning

confidence: 99%

Mixed Finite Element Methods

Auricchio

Veiga

Brezzi

et al. 2017

Encyclopedia of Computational Mechanics Second Edition

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Within the well‐known and highly effective finite element method for the computation of approximate solutions of complex boundary value problems, we focus on the often‐called mixed finite element methods , where in our terminology the word “mixed” indicates the fact that the problem discretization typically results in a linear algebraic system characterized by a null matrix on the main diagonal. Accordingly, the goals of the present article are (i) to sketch out that several physical problems share such an algebraic structure once a discretization is introduced; (ii) to present a simple, algebraic version of the abstract theory that rules most applications of mixed finite element methods; (iii) to give several examples of efficient mixed finite element methods; and (iv) finally, to give some hints on how to perform a stability and error analysis, focusing on a representative problem.

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“…A number of lockingfree individual finite elements and finite element families (e.g. [5,10,15,18,19,16,20,17]) have been obtained in this way.…”

Section: Introductionmentioning

confidence: 99%

Locking-free Reissner–Mindlin elements without reduced integration

Arnold

Brezzi

Falk

et al. 2007

Computer Methods in Applied Mechanics and Engineering

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In a recent paper of Arnold et al. [D.N. Arnold, F. Brezzi, L.D. Marini, A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate, J. Sci. Comput. 22 (2005) 25-45], the ideas of discontinuous Galerkin methods were used to obtain and analyze two new families of locking free finite element methods for the approximation of the Reissner-Mindlin plate problem. By following their basic approach, but making different choices of finite element spaces, we develop and analyze other families of locking free finite elements that eliminate the need for the introduction of a reduction operator, which has been a central feature of many locking-free methods. For k P 2, all the methods use piecewise polynomials of degree k to approximate the transverse displacement and (possibly subsets) of piecewise polynomials of degree k À 1 to approximate both the rotation and shear stress vectors. The approximation spaces for the rotation and the shear stress are always identical. The methods vary in the amount of interelement continuity required. In terms of smallest number of degrees of freedom, the simplest method approximates the transverse displacement with continuous, piecewise quadratics and both the rotation and shear stress with rotated linear Brezzi-Douglas-Marini elements.

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A Low-order Nonconforming Finite Element for Reissner--Mindlin Plates

Abstract: We report on recent results about some nonconforming finite elements for plates, introduced and analyzed in [6], [12] and [9]. All the elements are locking-free and exhibit optimal convergence rates.

Cited by 34 publications

References 31 publications

Two lower order nonconforming rectangular elements for the Reissner-Mindlin plate

Two lower order nonconforming rectangular elements for the Reissner-Mindlin plate

Mixed Finite Element Methods

Locking-free Reissner–Mindlin elements without reduced integration

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