A locking free Reissner-Mindlin element with weak Galerkin rotations

Ruishu Wang; Lin Mu; Xiu Ye

doi:10.3934/dcdsb.2018086

Article Contents

2019, Volume 24, Issue 1: 351-361. Doi: 10.3934/dcdsb.2018086

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A locking free Reissner-Mindlin element with weak Galerkin rotations

1.
Department of Mathematics, Jilin University, Changchun, China
2.
Computer Science and Mathematics Division Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA
3.
Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA

* Corresponding author: Lin Mu
* Corresponding author: Lin Mu

Received: October 2016

Revised: October 2017

Early access: March 2018

Published: January 2019

The research of second author was supported in part by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under award number ERKJE45; and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC., for the U.S. Department of Energy under Contract DE-AC05-00OR22725. This research of third author was supported in part by National Science Foundation Grant DMS-1620016.

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Abstract

A locking free finite element method is developed for the Reissner-Mindlin equations in their primary form. In this method, the transverse displacement is approximated by continuous piecewise polynomials of degree $k+1$ and the rotation is approximated by weak Galerkin elements of degree $k$ for $k≥1$ . A uniform convergence in thickness of the plate is established for this finite element approximation. The numerical examples demonstrate locking free of the method.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Table 1. Example 1. Error profile and convergence rate on triangular mesh.

h	$\frac{\\|{\boldsymbol{\rm{e}}}_0\\|}{\\|{\boldsymbol{\rm{Q}}}_0\mathit{\boldsymbol{\theta}}\\|}$	Rate	$\frac{\|\|\|{\boldsymbol{\rm{e}}}_h\|\|\|_{\theta}}{\|\|\|{\boldsymbol{\rm{Q}}}_h\mathit{\boldsymbol{\theta}}\|\|\|}$	Rate	$\frac{\\| \xi_h\\|}{\\|w_I\\|}$	Rate	$\frac{\|\|\| \xi_h\|\|\|_w}{\|\|\| w_I\|\|\|_w}$	Rate
t=1
1/4	2.2928e-1		8.2826e-1		1.1156e-1		9.4025e-2
1/8	7.2951e-2	1.65	6.1481e-1	0.43	1.4546e-2	2.94	1.7212e-2	2.45
1/16	1.8952e-2	1.94	3.6061e-1	0.77	1.8584e-3	2.97	2.4738e-3	2.80
1/32	4.7756e-3	1.99	1.8925e-1	0.93	2.4951e-4	2.90	3.3612e-4	2.88
1/64	1.1961e-3	2.00	9.5844e-2	0.98	3.8498e-5	2.70	4.8047e-5	2.81
t = 1e − 3
1/4	2.3633e-1		2.3639e-1		2.7175e-1		2.9153e-1
1/8	6.8597e-2	1.78	6.8675e-2	1.78	8.7506e-2	1.63	1.1448e-1	1.35
1/16	1.9470e-2	1.82	1.9542e-2	1.81	2.4404e-2	1.84	3.4550e-2	1.73
1/32	5.0660e-3	1.94	5.1359e-3	1.93	6.2578e-3	1.96	9.0799e-3	1.93
1/64	1.2169e-3	2.06	1.2876e-3	2.00	1.5154e-3	2.05	2.2184e-3	2.03
t = 1e − 6
1/4	2.3636e-1		2.3636e-1		2.7183e-1		2.9161e-1
1/8	6.8703e-2	1.78	6.8703e-2	1.78	8.7716e-2	1.63	1.1467e-1	1.35
1/16	1.9610e-2	1.81	1.9610e-2	1.81	2.4634e-2	1.83	3.4772e-2	1.72
1/32	5.0329e-3	1.96	5.0329e-3	1.96	4.4698e-3	2.46	7.7775e-3	2.16
1/64	1.2582e-3	2.00	1.2582e-3	2.00	1.1175e-3	2.00	1.9444e-3	2.00

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Table 2. Example 2. Error profile and convergence rate on triangular mesh.

$h$	$\frac{\\|{\boldsymbol{\rm{e}}}_0\\|}{\\|{\boldsymbol{\rm{Q}}}_0\mathit{\boldsymbol{\theta}}\\|}$	Rate	$\frac{\|\|\|{\boldsymbol{\rm{e}}}_h\|\|\|_{\theta}}{\|\|\|{\boldsymbol{\rm{Q}}}_h\mathit{\boldsymbol{\theta}}\|\|\|}$	Rate	$\frac{\\| \xi_h\\|}{\\|w_I\\|}$	Rate	$\frac{\|\|\| \xi_h\|\|\|_w}{\|\|\| w_I\|\|\|_w}$	Rate
$t=1$
1/4	7.1054e-2		4.3252e-1		1.3468e-3		1.9340e-3
1/8	1.7700e-2	2.01	2.2528e-1	0.94	3.0173e-4	2.16	4.7669e-4	2.02
1/16	4.4178e-3	2.00	1.1386e-1	0.98	7.3026e-5	2.05	1.1882e-4	2.00
1/32	1.1038e-3	2.00	5.7085e-2	1.00	1.8108e-5	2.01	2.9695e-5	2.00
1/64	2.7591e-4	2.00	2.8562e-2	1.00	4.5181e-6	2.00	7.4241e-6	2.00
$t=1e-3$
1/4	7.8909e-2		7.8920e-2		1.5185e-2		2.1876e-2
1/8	1.9680e-2	2.00	1.9692e-2	2.00	3.4940e-3	2.12	5.3275e-3	2.04
1/16	4.9063e-3	2.00	4.9182e-3	2.00	8.6948e-4	2.01	1.3334e-3	2.00
1/32	1.2254e-3	2.00	1.2373e-3	1.99	2.1650e-4	2.01	3.3216e-4	2.01
1/64	3.0652e-4	2.00	3.1821e-4	1.96	5.2591e-5	2.04	8.0763e-5	2.04
$t=1e-6$
1/4	7.8904e-2		7.8904e-2		1.5196e-2		2.1889e-2
1/8	1.9658e-2	2.01	1.9658e-2	2.01	3.5363e-3	2.10	5.3726e-3	2.03
1/16	4.8250e-3	2.03	4.8250e-3	2.03	1.0325e-3	1.78	1.5119e-3	1.83
1/32	1.2063e-3	2.00	1.2063e-3	2.00	2.5812e-4	2.00	3.7798e-4	2.00
1/64	3.0156e-4	2.00	3.0156e-4	2.00	6.4531e-5	2.00	9.4494e-5	2.00

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Table 3. Example 3. Error profile and convergence rate on triangular mesh.

$h$	$\frac{\\|{\boldsymbol{\rm{e}}}_0\\|}{\\|{\boldsymbol{\rm{Q}}}_0\mathit{\boldsymbol{\theta}}\\|}$	Rate	$\frac{\|\|\|{\boldsymbol{\rm{e}}}_h\|\|\|_{\theta}}{\|\|\|{\boldsymbol{\rm{Q}}}_h\mathit{\boldsymbol{\theta}}\|\|\|}$	Rate	$\frac{\\| \xi_h\\|}{\\|w_I\\|}$	Rate	$\frac{\|\|\| \xi_h\|\|\|_w}{\|\|\| w_I\|\|\|_w}$	Rate
$t=1$
1/4	2.8817e-1		8.7886e-1		3.4967e-2		2.8229e-2
1/8	6.4481e-2	2.16	5.8311e-1	0.59	4.9549e-3	2.82	4.7686e-3	2.57
1/16	1.5738e-2	2.03	3.2520e-1	0.84	7.6558e-4	2.69	7.2651e-4	2.71
1/32	3.9112e-3	2.01	1.6784e-1	0.95	1.4638e-4	2.39	1.1930e-4	2.60
1/64	9.7633e-4	2.00	8.4614e-2	0.99	3.3163e-5	2.14	2.3538e-5	2.34
$t=1e-3$
1/4	1.9791e-1		1.9802e-1		2.8790e-1		2.8340e-01
1/8	5.5770e-2	1.83	5.5851e-2	1.83	1.0222e-1	1.49	1.0240e-01	1.47
1/16	1.5321e-2	1.86	1.5393e-2	1.86	2.8291e-2	1.85	2.8723e-02	1.83
1/32	3.8890e-3	1.98	3.9578e-3	1.96	7.2000e-3	1.97	7.3429e-03	1.97
1/64	9.1919e-4	2.08	9.8893e-4	2.00	1.7371e-3	2.05	1.7727e-03	2.05
$t=1e-6$
1/4	1.9793e-1		1.9793e-1		2.8800e-1		2.8350e-01
1/8	5.5887e-2	1.82	5.5887e-2	1.82	1.0241e-1	1.49	1.0258e-01	1.47
1/16	1.5435e-2	1.86	1.5435e-2	1.86	2.8442e-2	1.85	2.8873e-02	1.83
1/32	3.3377e-3	2.21	3.3377e-3	2.21	4.4595e-3	2.67	4.9441e-03	2.55
1/64	8.3442e-4	2.00	8.3442e-4	2.00	1.1149e-3	2.00	1.2360e-03	2.00

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