Volume 38, Issue 1
A Robust Discretization of the Reissner-Mindlin Plate with Arbitrary Polynomial Degree

J. Comp. Math., 38 (2020), pp. 1-13.

Published online: 2020-02

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• Abstract

A numerical scheme for the Reissner–Mindlin plate model is proposed. The method is based on a discrete Helmholtz decomposition and can be viewed as a generalization of the nonconforming finite element scheme of Arnold and Falk [SIAM J. Numer. Anal., 26(6):1276-1290, 1989]. The two unknowns in the discrete formulation are the in-plane rotations and the gradient of the vertical displacement. The decomposition of the discrete shear variable leads to equivalence with the usual Stokes system with penalty term plus two Poisson equations and the proposed method is equivalent to a stabilized discretization of the Stokes system that generalizes the Mini element. The method is proved to satisfy a best-approximation result which is robust with respect to the thickness parameter $t$.

• Keywords

Reissner–Mindlin plate, Nonconforming finite element, Discrete Helmholtz decomposition, Robustness.

65N10, 65N15, 73K10, 73K25

dietmar.gallistl@uni-jena.de (Dietmar Gallistl)

mira.schedensack@math.uni-leipzig.de (Mira Schedensack)

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@Article{JCM-38-1, author = {Gallistl , Dietmar and Schedensack , Mira}, title = {A Robust Discretization of the Reissner-Mindlin Plate with Arbitrary Polynomial Degree}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {1}, pages = {1--13}, abstract = {

A numerical scheme for the Reissner–Mindlin plate model is proposed. The method is based on a discrete Helmholtz decomposition and can be viewed as a generalization of the nonconforming finite element scheme of Arnold and Falk [SIAM J. Numer. Anal., 26(6):1276-1290, 1989]. The two unknowns in the discrete formulation are the in-plane rotations and the gradient of the vertical displacement. The decomposition of the discrete shear variable leads to equivalence with the usual Stokes system with penalty term plus two Poisson equations and the proposed method is equivalent to a stabilized discretization of the Stokes system that generalizes the Mini element. The method is proved to satisfy a best-approximation result which is robust with respect to the thickness parameter $t$.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1902-m2018-0166}, url = {http://global-sci.org/intro/article_detail/jcm/13682.html} }
TY - JOUR T1 - A Robust Discretization of the Reissner-Mindlin Plate with Arbitrary Polynomial Degree AU - Gallistl , Dietmar AU - Schedensack , Mira JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 13 PY - 2020 DA - 2020/02 SN - 38 DO - http://doi.org/10.4208/jcm.1902-m2018-0166 UR - https://global-sci.org/intro/article_detail/jcm/13682.html KW - Reissner–Mindlin plate, Nonconforming finite element, Discrete Helmholtz decomposition, Robustness. AB -

A numerical scheme for the Reissner–Mindlin plate model is proposed. The method is based on a discrete Helmholtz decomposition and can be viewed as a generalization of the nonconforming finite element scheme of Arnold and Falk [SIAM J. Numer. Anal., 26(6):1276-1290, 1989]. The two unknowns in the discrete formulation are the in-plane rotations and the gradient of the vertical displacement. The decomposition of the discrete shear variable leads to equivalence with the usual Stokes system with penalty term plus two Poisson equations and the proposed method is equivalent to a stabilized discretization of the Stokes system that generalizes the Mini element. The method is proved to satisfy a best-approximation result which is robust with respect to the thickness parameter $t$.

Dietmar Gallistl & Mira Schedensack. (2020). A Robust Discretization of the Reissner-Mindlin Plate with Arbitrary Polynomial Degree. Journal of Computational Mathematics. 38 (1). 1-13. doi:10.4208/jcm.1902-m2018-0166
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