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January  2019, 24(1): 351-361. doi: 10.3934/dcdsb.2018086

A locking free Reissner-Mindlin element with weak Galerkin rotations

Ruishu Wang 1, , Lin Mu 2,3,, and  Xiu Ye 2,3,

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

Received  October 2016 Revised  October 2017 Published  January 2019 Early access  March 2018

Fund Project: 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.

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.

Keywords: Weak Galerkin, finite element methods, the Reissner-Mindlin plate.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Ruishu Wang, Lin Mu, Xiu Ye. A locking free Reissner-Mindlin element with weak Galerkin rotations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 351-361. doi: 10.3934/dcdsb.2018086
References:
[1]

D. Arnold, F. Brezzi and D. Marini, A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate, J. Sci. Comput., 22 (2005), 25-45.  doi: 10.1007/s10915-004-4134-8.

[2]

D. Arnold, F. Brezzi, R. Falk and D. Marini, Locking-free Reissner-Mindlin elements without reduced integration, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3660-3671.  doi: 10.1016/j.cma.2006.10.023.

[3]

D. Arnold and R. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal., 26 (1989), 1276-1290.  doi: 10.1137/0726074.

[4]

D. N. Arnold and X. Liu, Interior estimates for a low order finite element method for the Reissner-Mindlin plate model, Adv. in Comp. Math., 7 (1997), 337-360.  doi: 10.1023/A:1018907205385.

[5]

S. Brenner, Korn's inequalities for piecewise $H^1$ vector fields, Math. Comput., 73 (2004), 1067-1087. 

[6]

F. Brezzi and M. Fortin, Numerical approximation of Mindlin-Reissner plates, Math. Comp., 47 (1986), 151-158.  doi: 10.1090/S0025-5718-1986-0842127-7.

[7]

F. Brezzi, K. Bathe and M. Fortin, Mixed interpolated elements for Reissner-Mindlin plates, Int. J. Numer. Methods Eng., 28 (1989), 1787-1801.  doi: 10.1002/nme.1620280806.

[8]

G. Brezzi, M. Fortin and R. Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Mathematical Models and Methods in Applied Sciences, 1 (1991), 125-151.  doi: 10.1142/S0218202591000083.

[9]

D. Chapelle and R. Stenberg, An optimal low-order locking-free finite element method for Reissner-Mindlin plates, Mathematical Models and Methods in Applied Sciences, 8 (1998), 407-430.  doi: 10.1142/S0218202598000172.

[10]

R. Duran and E. Liberman, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp., 58 (1992), 561-573.  doi: 10.1090/S0025-5718-1992-1106965-0.

[11]

R. Falk and T. Tu, Locking-free finite elements for the Reissner-Mindlin plate, Math. Comp., 69 (2000), 911-928. 

[12]

P. Hansbo, D. Heintz and M. Larson, A finite element method with discontinuous rotations for the Mindlin-Reissner plate model, Comput. Methods Appl. Mech. Engrg., 200 (2011), 638-648.  doi: 10.1016/j.cma.2010.09.009.

[13]

C. Lovadina and D. Marini, Nonconforming locking-free finite elements for Reissner-Mindlin plates, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3448-3460.  doi: 10.1016/j.cma.2005.06.025.

[14]

L. Mu, J. Wang and X. Ye, A weak Galerkin finite element method with polynomial reduction, Journal of Computational and Applied Mathematics, 285 (2015), 45-58.  doi: 10.1016/j.cam.2015.02.001.

[15]

L. Mu, J. Wang and X. Ye, Weak Galerkin finite element method for second-order elliptic problems on polytopal meshes, International Journal of Numerical Analysis and Modeling, 12 (2015), 31-53. 

[16]

R. Pierre, Convergence Properties and Numerical Approximation of the Solution of the Mindlin Plate Bending Problem, Math Comp., 51 (1988), 15-25.  doi: 10.1090/S0025-5718-1988-0942141-9.

[17]

J. Wang and X. Ye, A superconvergent finite element scheme for the reissner-mindlin plate by projection methods, International Journal of Numeerical Analysis and Modeling, 1 (2004), 99-110. 

[18]

J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comp. and Appl. Math, 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.

[19]

J. Wang and X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp., Math. Comp., 83 (2014), 2101-2126.  doi: 10.1090/S0025-5718-2014-02852-4.

[20]

X. Ye, Stabilized finite element approximations for the Reissner-Mindlin plate, Advances in Computational Mathematics, 13 (2000), 375-386.  doi: 10.1023/A:1016693613626.

[21]

X. Ye, A Rectangular Element for the Reissner-Mindlin Plate, Numer. Method for PDE, 16 (2000), 184-193.  doi: 10.1002/(SICI)1098-2426(200003)16:2<184::AID-NUM3>3.0.CO;2-B.

show all references

References:
[1]

D. Arnold, F. Brezzi and D. Marini, A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate, J. Sci. Comput., 22 (2005), 25-45.  doi: 10.1007/s10915-004-4134-8.

[2]

D. Arnold, F. Brezzi, R. Falk and D. Marini, Locking-free Reissner-Mindlin elements without reduced integration, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3660-3671.  doi: 10.1016/j.cma.2006.10.023.

[3]

D. Arnold and R. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal., 26 (1989), 1276-1290.  doi: 10.1137/0726074.

[4]

D. N. Arnold and X. Liu, Interior estimates for a low order finite element method for the Reissner-Mindlin plate model, Adv. in Comp. Math., 7 (1997), 337-360.  doi: 10.1023/A:1018907205385.

[5]

S. Brenner, Korn's inequalities for piecewise $H^1$ vector fields, Math. Comput., 73 (2004), 1067-1087. 

[6]

F. Brezzi and M. Fortin, Numerical approximation of Mindlin-Reissner plates, Math. Comp., 47 (1986), 151-158.  doi: 10.1090/S0025-5718-1986-0842127-7.

[7]

F. Brezzi, K. Bathe and M. Fortin, Mixed interpolated elements for Reissner-Mindlin plates, Int. J. Numer. Methods Eng., 28 (1989), 1787-1801.  doi: 10.1002/nme.1620280806.

[8]

G. Brezzi, M. Fortin and R. Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Mathematical Models and Methods in Applied Sciences, 1 (1991), 125-151.  doi: 10.1142/S0218202591000083.

[9]

D. Chapelle and R. Stenberg, An optimal low-order locking-free finite element method for Reissner-Mindlin plates, Mathematical Models and Methods in Applied Sciences, 8 (1998), 407-430.  doi: 10.1142/S0218202598000172.

[10]

R. Duran and E. Liberman, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp., 58 (1992), 561-573.  doi: 10.1090/S0025-5718-1992-1106965-0.

[11]

R. Falk and T. Tu, Locking-free finite elements for the Reissner-Mindlin plate, Math. Comp., 69 (2000), 911-928. 

[12]

P. Hansbo, D. Heintz and M. Larson, A finite element method with discontinuous rotations for the Mindlin-Reissner plate model, Comput. Methods Appl. Mech. Engrg., 200 (2011), 638-648.  doi: 10.1016/j.cma.2010.09.009.

[13]

C. Lovadina and D. Marini, Nonconforming locking-free finite elements for Reissner-Mindlin plates, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3448-3460.  doi: 10.1016/j.cma.2005.06.025.

[14]

L. Mu, J. Wang and X. Ye, A weak Galerkin finite element method with polynomial reduction, Journal of Computational and Applied Mathematics, 285 (2015), 45-58.  doi: 10.1016/j.cam.2015.02.001.

[15]

L. Mu, J. Wang and X. Ye, Weak Galerkin finite element method for second-order elliptic problems on polytopal meshes, International Journal of Numerical Analysis and Modeling, 12 (2015), 31-53. 

[16]

R. Pierre, Convergence Properties and Numerical Approximation of the Solution of the Mindlin Plate Bending Problem, Math Comp., 51 (1988), 15-25.  doi: 10.1090/S0025-5718-1988-0942141-9.

[17]

J. Wang and X. Ye, A superconvergent finite element scheme for the reissner-mindlin plate by projection methods, International Journal of Numeerical Analysis and Modeling, 1 (2004), 99-110. 

[18]

J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comp. and Appl. Math, 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.

[19]

J. Wang and X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp., Math. Comp., 83 (2014), 2101-2126.  doi: 10.1090/S0025-5718-2014-02852-4.

[20]

X. Ye, Stabilized finite element approximations for the Reissner-Mindlin plate, Advances in Computational Mathematics, 13 (2000), 375-386.  doi: 10.1023/A:1016693613626.

[21]

X. Ye, A Rectangular Element for the Reissner-Mindlin Plate, Numer. Method for PDE, 16 (2000), 184-193.  doi: 10.1002/(SICI)1098-2426(200003)16:2<184::AID-NUM3>3.0.CO;2-B.

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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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
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
1/81.7700e-22.012.2528e-10.943.0173e-42.164.7669e-42.02
1/164.4178e-32.001.1386e-10.987.3026e-52.051.1882e-42.00
1/321.1038e-32.005.7085e-21.001.8108e-52.012.9695e-52.00
1/642.7591e-42.002.8562e-21.004.5181e-62.007.4241e-62.00
$t=1e-3$
1/47.8909e-27.8920e-21.5185e-22.1876e-2
1/81.9680e-22.001.9692e-22.003.4940e-32.125.3275e-32.04
1/164.9063e-32.004.9182e-32.008.6948e-42.011.3334e-32.00
1/321.2254e-32.001.2373e-31.992.1650e-42.013.3216e-42.01
1/643.0652e-42.003.1821e-41.965.2591e-52.048.0763e-52.04
$t=1e-6$
1/47.8904e-27.8904e-21.5196e-22.1889e-2
1/81.9658e-22.011.9658e-22.013.5363e-32.105.3726e-32.03
1/164.8250e-32.034.8250e-32.031.0325e-31.781.5119e-31.83
1/321.2063e-32.001.2063e-32.002.5812e-42.003.7798e-42.00
1/643.0156e-42.003.0156e-42.006.4531e-52.009.4494e-52.00
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$h$ $\frac{\|{\boldsymbol{\rm{e}}}_0\|}{\|{\boldsymbol{\rm{Q}}}_0\mathit{\boldsymbol{\theta}}\|}$ 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
1/81.7700e-22.012.2528e-10.943.0173e-42.164.7669e-42.02
1/164.4178e-32.001.1386e-10.987.3026e-52.051.1882e-42.00
1/321.1038e-32.005.7085e-21.001.8108e-52.012.9695e-52.00
1/642.7591e-42.002.8562e-21.004.5181e-62.007.4241e-62.00
$t=1e-3$
1/47.8909e-27.8920e-21.5185e-22.1876e-2
1/81.9680e-22.001.9692e-22.003.4940e-32.125.3275e-32.04
1/164.9063e-32.004.9182e-32.008.6948e-42.011.3334e-32.00
1/321.2254e-32.001.2373e-31.992.1650e-42.013.3216e-42.01
1/643.0652e-42.003.1821e-41.965.2591e-52.048.0763e-52.04
$t=1e-6$
1/47.8904e-27.8904e-21.5196e-22.1889e-2
1/81.9658e-22.011.9658e-22.013.5363e-32.105.3726e-32.03
1/164.8250e-32.034.8250e-32.031.0325e-31.781.5119e-31.83
1/321.2063e-32.001.2063e-32.002.5812e-42.003.7798e-42.00
1/643.0156e-42.003.0156e-42.006.4531e-52.009.4494e-52.00
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
$t=1$
1/42.8817e-18.7886e-13.4967e-22.8229e-2
1/86.4481e-22.165.8311e-10.594.9549e-32.824.7686e-32.57
1/161.5738e-22.033.2520e-10.847.6558e-42.697.2651e-42.71
1/323.9112e-32.011.6784e-10.951.4638e-42.391.1930e-42.60
1/649.7633e-42.008.4614e-20.993.3163e-52.142.3538e-52.34
$t=1e-3$
1/41.9791e-11.9802e-12.8790e-12.8340e-01
1/85.5770e-21.835.5851e-21.831.0222e-11.491.0240e-011.47
1/161.5321e-21.861.5393e-21.862.8291e-21.852.8723e-021.83
1/323.8890e-31.983.9578e-31.967.2000e-31.977.3429e-031.97
1/649.1919e-42.089.8893e-42.001.7371e-32.051.7727e-032.05
$t=1e-6$
1/41.9793e-11.9793e-12.8800e-12.8350e-01
1/85.5887e-21.825.5887e-21.821.0241e-11.491.0258e-011.47
1/161.5435e-21.861.5435e-21.862.8442e-21.852.8873e-021.83
1/323.3377e-32.213.3377e-32.214.4595e-32.674.9441e-032.55
1/648.3442e-42.008.3442e-42.001.1149e-32.001.2360e-032.00
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$h$ $\frac{\|{\boldsymbol{\rm{e}}}_0\|}{\|{\boldsymbol{\rm{Q}}}_0\mathit{\boldsymbol{\theta}}\|}$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
$t=1$
1/42.8817e-18.7886e-13.4967e-22.8229e-2
1/86.4481e-22.165.8311e-10.594.9549e-32.824.7686e-32.57
1/161.5738e-22.033.2520e-10.847.6558e-42.697.2651e-42.71
1/323.9112e-32.011.6784e-10.951.4638e-42.391.1930e-42.60
1/649.7633e-42.008.4614e-20.993.3163e-52.142.3538e-52.34
$t=1e-3$
1/41.9791e-11.9802e-12.8790e-12.8340e-01
1/85.5770e-21.835.5851e-21.831.0222e-11.491.0240e-011.47
1/161.5321e-21.861.5393e-21.862.8291e-21.852.8723e-021.83
1/323.8890e-31.983.9578e-31.967.2000e-31.977.3429e-031.97
1/649.1919e-42.089.8893e-42.001.7371e-32.051.7727e-032.05
$t=1e-6$
1/41.9793e-11.9793e-12.8800e-12.8350e-01
1/85.5887e-21.825.5887e-21.821.0241e-11.491.0258e-011.47
1/161.5435e-21.861.5435e-21.862.8442e-21.852.8873e-021.83
1/323.3377e-32.213.3377e-32.214.4595e-32.674.9441e-032.55
1/648.3442e-42.008.3442e-42.001.1149e-32.001.2360e-032.00
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