Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates

da Veiga, L.; Mora, D.; Rivera, G.

doi:10.1090/mcom/3331

Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates
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by L. Beirão da Veiga, D. Mora and G. Rivera;

Math. Comp. 88 (2019), 149-178

DOI: https://doi.org/10.1090/mcom/3331

Published electronically: April 5, 2018

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Abstract:

We present a virtual element method for the Reissner–Mindlin plate bending problem which uses shear strain and deflection as discrete variables without the need of any reduction operator. The proposed method is conforming in $[H^{1}(\Omega )]^2 \times H^2(\Omega )$ and has the advantages of using general polygonal meshes and yielding a direct approximation of the shear strains. The rotations are then obtained by a simple postprocess from the shear strain and deflection. We prove convergence estimates with involved constants that are uniform in the thickness $t$ of the plate. Finally, we report numerical experiments which allow us to assess the performance of the method.

References

B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013), no. 3, 376–391. MR 3073346, DOI 10.1016/j.camwa.2013.05.015
P. F. Antonietti, L. Beirão da Veiga, D. Mora, and M. Verani, A stream virtual element formulation of the Stokes problem on polygonal meshes, SIAM J. Numer. Anal. 52 (2014), no. 1, 386–404. MR 3164557, DOI 10.1137/13091141X
P. F. Antonietti, L. Beirão da Veiga, S. Scacchi, and M. Verani, A $C^1$ virtual element method for the Cahn-Hilliard equation with polygonal meshes, SIAM J. Numer. Anal. 54 (2016), no. 1, 34–56. MR 3439765, DOI 10.1137/15M1008117
P. F. Antonietti, P. Houston, X. Hu, M. Sarti, and M. Verani, Multigrid algorithms for $hp$-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes, Calcolo 54 (2017), no. 4, 1169–1198. MR 3735810, DOI 10.1007/s10092-017-0223-6
Douglas N. Arnold, Franco Brezzi, Richard S. Falk, and L. Donatella Marini, Locking-free Reissner-Mindlin elements without reduced integration, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3660–3671. MR 2339992, DOI 10.1016/j.cma.2006.10.023
Douglas N. Arnold and Richard S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal. 26 (1989), no. 6, 1276–1290. MR 1025088, DOI 10.1137/0726074
Blanca Ayuso de Dios, Konstantin Lipnikov, and Gianmarco Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 3, 879–904. MR 3507277, DOI 10.1051/m2an/2015090
L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199–214. MR 2997471, DOI 10.1142/S0218202512500492
L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo, The hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci. 24 (2014), no. 8, 1541–1573. MR 3200242, DOI 10.1142/S021820251440003X
L. Beirão Da Veiga, T. J. R. Hughes, J. Kiendl, C. Lovadina, J. Niiranen, A. Reali, and H. Speleers, A locking-free model for Reissner-Mindlin plates: analysis and isogeometric implementation via NURBS and triangular NURPS, Math. Models Methods Appl. Sci. 25 (2015), no. 8, 1519–1551. MR 3340708, DOI 10.1142/S0218202515500402
Lourenço Beirão da Veiga, Konstantin Lipnikov, and Gianmarco Manzini, The mimetic finite difference method for elliptic problems, MS&A. Modeling, Simulation and Applications, vol. 11, Springer, Cham, 2014. MR 3135418, DOI 10.1007/978-3-319-02663-3
L. Beirão da Veiga, C. Lovadina, and D. Mora, A virtual element method for elastic and inelastic problems on polytope meshes, Comput. Methods Appl. Mech. Engrg. 295 (2015), 327–346. MR 3388837, DOI 10.1016/j.cma.2015.07.013
Lourenço Beirão da Veiga, Carlo Lovadina, and Alessandro Russo, Stability analysis for the virtual element method, Math. Models Methods Appl. Sci. 27 (2017), no. 13, 2557–2594. MR 3714637, DOI 10.1142/S021820251750052X
Lourenco Beirão da Veiga, Carlo Lovadina, and Giuseppe Vacca, Divergence free virtual elements for the Stokes problem on polygonal meshes, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 2, 509–535. MR 3626409, DOI 10.1051/m2an/2016032
Lourenço Beirão da Veiga, David Mora, Gonzalo Rivera, and Rodolfo Rodríguez, A virtual element method for the acoustic vibration problem, Numer. Math. 136 (2017), no. 3, 725–763. MR 3660301, DOI 10.1007/s00211-016-0855-5
Matías Fernando Benedetto, Stefano Berrone, Andrea Borio, Sandra Pieraccini, and Stefano Scialò, A hybrid mortar virtual element method for discrete fracture network simulations, J. Comput. Phys. 306 (2016), 148–166. MR 3432346, DOI 10.1016/j.jcp.2015.11.034
Matías Fernando Benedetto, Stefano Berrone, Sandra Pieraccini, and Stefano Scialò, The virtual element method for discrete fracture network simulations, Comput. Methods Appl. Mech. Engrg. 280 (2014), 135–156. MR 3255535, DOI 10.1016/j.cma.2014.07.016
Silvia Bertoluzza, Substructuring preconditioners for the three fields domain decomposition method, Math. Comp. 73 (2004), no. 246, 659–689. MR 2031400, DOI 10.1090/S0025-5718-03-01550-3
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
F. Brezzi, Richard S. Falk, and L. Donatella Marini, Basic principles of mixed virtual element methods, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 4, 1227–1240. MR 3264352, DOI 10.1051/m2an/2013138
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Franco Brezzi, Michel Fortin, and Rolf Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models Methods Appl. Sci. 1 (1991), no. 2, 125–151. MR 1115287, DOI 10.1142/S0218202591000083
Franco Brezzi and L. Donatella Marini, Virtual element methods for plate bending problems, Comput. Methods Appl. Mech. Engrg. 253 (2013), 455–462. MR 3002804, DOI 10.1016/j.cma.2012.09.012
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 482275
Ernesto Cáceres and Gabriel N. Gatica, A mixed virtual element method for the pseudostress-velocity formulation of the Stokes problem, IMA J. Numer. Anal. 37 (2017), no. 1, 296–331. MR 3614887, DOI 10.1093/imanum/drw002
Andrea Cangiani, Emmanuil H. Georgoulis, and Paul Houston, $hp$-version discontinuous Galerkin methods on polygonal and polyhedral meshes, Math. Models Methods Appl. Sci. 24 (2014), no. 10, 2009–2041. MR 3211116, DOI 10.1142/S0218202514500146
C. Chinosi and C. Lovadina, Numerical analysis of some mixed finite element methods for Reissner-Mindlin plates, Comput. Mech., 16, (1995), no. 1, 36–44.
Claudia Chinosi and L. Donatella Marini, Virtual element method for fourth order problems: $L^2$-estimates, Comput. Math. Appl. 72 (2016), no. 8, 1959–1967. MR 3558291, DOI 10.1016/j.camwa.2016.02.001
Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
Philippe G. Ciarlet, Interpolation error estimates for the reduced Hsieh-Clough-Tocher triangle, Math. Comp. 32 (1978), no. 142, 335–344. MR 482249, DOI 10.1090/S0025-5718-1978-0482249-1
Daniele A. Di Pietro and Alexandre Ern, A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Engrg. 283 (2015), 1–21. MR 3283758, DOI 10.1016/j.cma.2014.09.009
Ricardo Durán and Elsa Liberman, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp. 58 (1992), no. 198, 561–573. MR 1106965, DOI 10.1090/S0025-5718-1992-1106965-0
R. Echter, B. Oesterle, and M. Bischoff, A hierarchic family of isogeometric shell finite elements, Comput. Methods Appl. Mech. Engrg. 254 (2013), 170–180. MR 3002769, DOI 10.1016/j.cma.2012.10.018
R. Falk, Finite elements for the Reissner-Mindlin plate, D. Boffi and L. Gastaldi, editors, Mixed finite elements, compatibility conditions, and applications, Springer, Berlin, 2008, 195–232.
Arun L. Gain, Cameron Talischi, and Glaucio H. Paulino, On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes, Comput. Methods Appl. Mech. Engrg. 282 (2014), 132–160. MR 3269894, DOI 10.1016/j.cma.2014.05.005
Quan Long, P. Burkhard Bornemann, and Fehmi Cirak, Shear-flexible subdivision shells, Internat. J. Numer. Methods Engrg. 90 (2012), no. 13, 1549–1577. MR 2931167, DOI 10.1002/nme.3368
C. Lovadina, A brief overview of plate finite element methods, Integral methods in science and engineering. Vol. 2, Birkhäuser Boston, Boston, MA, 2010, pp. 261–280. MR 2663168, DOI 10.1007/978-0-8176-4897-8_{2}5
Mikko Lyly, Jarkko Niiranen, and Rolf Stenberg, A refined error analysis of MITC plate elements, Math. Models Methods Appl. Sci. 16 (2006), no. 7, 967–977. MR 2250027, DOI 10.1142/S021820250600142X
David Mora, Gonzalo Rivera, and Rodolfo Rodríguez, A virtual element method for the Steklov eigenvalue problem, Math. Models Methods Appl. Sci. 25 (2015), no. 8, 1421–1445. MR 3340705, DOI 10.1142/S0218202515500372
Glaucio H. Paulino and Arun L. Gain, Bridging art and engineering using Escher-based virtual elements, Struct. Multidiscip. Optim. 51 (2015), no. 4, 867–883. MR 3353849, DOI 10.1007/s00158-014-1179-7
Ilaria Perugia, Paola Pietra, and Alessandro Russo, A plane wave virtual element method for the Helmholtz problem, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 3, 783–808. MR 3507273, DOI 10.1051/m2an/2015066
Sergej Rjasanow and Steffen Weißer, Higher order BEM-based FEM on polygonal meshes, SIAM J. Numer. Anal. 50 (2012), no. 5, 2357–2378. MR 3022222, DOI 10.1137/110849481
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. MR 290095
N. Sukumar and A. Tabarraei, Conforming polygonal finite elements, Internat. J. Numer. Methods Engrg. 61 (2004), no. 12, 2045–2066. MR 2101599, DOI 10.1002/nme.1141
C. Talischi, G. H. Paulino, A. Pereira and I. F. M. Menezes, Polygonal finite elements for topology optimization: A unifying paradigm, Internat. J. Numer. Methods Engrg., 82, (2010), no. 6, 671–698.
P. Wriggers, W. T. Rust, and B. D. Reddy, A virtual element method for contact, Comput. Mech. 58 (2016), no. 6, 1039–1050. MR 3572918, DOI 10.1007/s00466-016-1331-x
Jikun Zhao, Shaochun Chen, and Bei Zhang, The nonconforming virtual element method for plate bending problems, Math. Models Methods Appl. Sci. 26 (2016), no. 9, 1671–1687. MR 3529253, DOI 10.1142/S021820251650041X