Virtual Element Methods for three-dimensional Hellinger-Reissner elastostatic problems

Carlo Lovadina; Michele Visinoni

doi:10.2478/caim-2022-0005

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Virtual Element Methods for three-dimensional Hellinger-Reissner elastostatic problems

Communications in Applied and Industrial Mathematics

Volume 13 (2022): Issue 1 (January 2022)

By: Carlo Lovadina and Michele Visinoni

Open Access

|Oct 2022

Abstract

This note aims at illustrating the application of the Virtual Element Method to elasticity problems in mixed form, following the Hellinger-Reissner variational principle. In order to highlight the potential and the flexibility of our approach, we focus on a three-dimensional low-order Virtual Element scheme, but similar considerations apply to two-dimensional and higher-order methods.

References

1. L. Beir˜ao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, Basic principles of Virtual Element Methods, Mathematical Models and Methods in Applied Sciences, vol. 23, pp. 119–214, 2013.10.1142/S0218202512500492
Search in Google Scholar Back to article
2. L. Beir˜ao da Veiga, K. Lipnikov, and G. Manzini, The mimetic finite difference method for elliptic problems, vol. 11 of Modeling, Simulation and Applications. Springer, 2014.10.1007/978-3-319-02663-3
Search in Google Scholar Back to article
3. D. Boffi, F. Brezzi, and M. Fortin, Mixed finite element methods and applications, vol. 44 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2013.10.1007/978-3-642-36519-5
Search in Google Scholar Back to article
4. E. Artioli, S. de Miranda, C. Lovadina, and L. Patruno, A stress/displacement virtual element method for plane elasticity problems, Comp. Methods Appl. Mech. Engrg., vol. 325, pp. 155–174, 2017.10.1016/j.cma.2017.06.036
Search in Google Scholar Back to article
5. E. Artioli, S. de Miranda, C. Lovadina, and L. Patruno, A family of virtual element methods for plane elasticity problems based on the hellinger-reissner principle, Comp. Methods Appl. Mech. Engrg., vol. 340, pp. 978–999, 2018.10.1016/j.cma.2018.06.020
Search in Google Scholar Back to article
6. M. Visinoni, A family of three-dimensional virtual elements for hellinger-reissner elasticity problems, in preparation.
Search in Google Scholar Back to article
7. B. F. De Veubeke and O. C. Zienkiewicz, Displacement and equilibrium models in the finite element method, International Journal for Numerical Methods in Engineering, vol. 52, no. 3, pp. 287–342, 2001.10.1002/nme.339
Search in Google Scholar Back to article
8. D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, ESAIM: Mathematical Modelling and Numerical Analysis, vol. 19, pp. 7–32, 1985.10.1051/m2an/1985190100071
Search in Google Scholar Back to article
9. F. Dassi, C. Lovadina, and M. Visinoni, A three-dimensional Hellinger-Reissner virtual element method for linear elasticity problems, Computer Methods in Applied Mechanics and Engineering, vol. 364, p. 112910, 2020.10.1016/j.cma.2020.112910
Search in Google Scholar Back to article
10. D. Braess, Finite elements. Theory, fast solvers, and applications in elasticity theory. Cambridge University Press, third ed., 2007.10.1017/CBO9780511618635
Search in Google Scholar Back to article
11. D. N. Arnold and R. Winther, Mixed finite elements for elasticity, Numerische Mathematik, vol. 92, pp. 401–419, 2002.10.1007/s002110100348
Search in Google Scholar Back to article
12. D. N. Arnold, G. Awanou, and R. Winther, Finite elements for symmetric tensors in three dimensions, Mathematics of Computation, vol. 77, pp. 1229–1251, 2008.10.1090/S0025-5718-08-02071-1
Search in Google Scholar Back to article
13. L. Beir˜ao da Veiga, F. Brezzi, L. D. Marini, and A. Russo, The hitchhiker’s guide to the virtual element method, Mathematical Models and Methods in Applied Sciences, vol. 24, no. 8, pp. 1541–1573, 2014.10.1142/S021820251440003X
Search in Google Scholar Back to article
14. F. Dassi, C. Lovadina, and M. Visinoni, Hybridization of the virtual element method for linear elasticity problems, Mathematical Models and Methods in Applied Sciences, vol. 31, no. 14, pp. 2979–3008, 2021.10.1142/S0218202521500676
Search in Google Scholar Back to article
15. B. Ayuso, K. Lipnikov, and G. Manzini, The nonconforming virtual element method, ESAIM: Mathematical Modelling and Numerical Analysis, vol. 50, no. 3, pp. 879–904, 2016.10.1051/m2an/2015090
Search in Google Scholar Back to article
16. S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelmanm, L. Dalcin, A. Dener, V. Eijkhout, W. D. Gropp, D. Karpeyev, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang, PETSc Web page. https://www.mcs.anl.gov/petsc, 2019.
Search in Google Scholar Back to article
17. L. Beir˜ao da Veiga, F. Brezzi, and L. D. Marini, Virtual Elements for linear elasticity problems, SIAM J. Numer. Anal., vol. 51, pp. 794–812, 2013.10.1137/120874746
Search in Google Scholar Back to article
18. L. Beir˜ao da Veiga, C. Lovadina, and G. Vacca, Divergence free virtual elements for the Stokes problem on polygonal meshes, ESAIM: Mathematical Modelling and Numerical Analysis, vol. 51, no. 2, pp. 509–535, 2017.10.1051/m2an/2016032
Search in Google Scholar Back to article
19. L. Beir˜ao da Veiga, D. Mora, and G. Vacca, The Stokes complex for virtual elements with application to Navier-Stokes flows, Journal of Scientific Computing, vol. 81, no. 2, pp. 990–1018, 2019.10.1007/s10915-019-01049-3
Search in Google Scholar Back to article
20. D. Arnold, Finite Element Exterior Calculus, vol. 93 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 2018.10.1137/1.9781611975543
Search in Google Scholar Back to article
21. D. Arnold, D. Boffi, and F. Bonizzoni, Finite element differential forms on curvilinear cubic meshes and their approximation properties, Numerische Mathematik, vol. 129, no. 1, pp. 1–20, 2015.10.1007/s00211-014-0631-3
Search in Google Scholar Back to article
22. F. Bonizzoni and G. Kanschat, H₁-conforming finite element cochain complexes and commuting quasi-interpolation operators on cartesian meshes, Calcolo, vol. 58, no. 18, 2021.10.1007/s10092-021-00409-6859168434803174
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/caim-2022-0005 | Journal eISSN: 2038-0909

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Language: English

Page range: 57 - 69

Submitted on: Jul 30, 2022

Accepted on: Sep 30, 2022

Published on: Oct 22, 2022

Published by: Italian Society for Applied and Industrial Mathemathics

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Virtual element methods,

elasticity problems,

hybridization procedure

Related subjects:

Mathematics,

Numerical and computational mathematics,

Applied mathematics

© 2022 Carlo Lovadina, Michele Visinoni, published by Italian Society for Applied and Industrial Mathemathics
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 13 (2022): Issue 1 (January 2022)