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Automatically inf-sup compliant diamond-mixed finite elements for Kirchhoff plates

Perotti, L. E. and Bompadre, A. and Ortiz, M. (2013) Automatically inf-sup compliant diamond-mixed finite elements for Kirchhoff plates. International Journal for Numerical Methods in Engineering, 96 (7). pp. 405-424. ISSN 0029-5981. doi:10.1002/nme.4555. https://resolver.caltech.edu/CaltechAUTHORS:20131108-093716091

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Abstract

We develop a mixed finite-element approximation scheme for Kirchhoff plate theory based on the reformulation of Kirchhoff plate theory of Ortiz and Morris [1]. In this reformulation the moment-equilibrium problem for the rotations is in direct analogy to the problem of incompressible two-dimensional elasticity. This analogy in turn opens the way for the application of diamond approximation schemes (Hauret et al. [2]) to Kirchhoff plate theory. We show that a special class of meshes derived from an arbitrary triangulation of the domain, the diamond meshes, results in the automatic satisfaction of the corresponding inf − sup condition for Kirchhoff plate theory. The attendant optimal convergence properties of the diamond approximation scheme are demonstrated by means of the several standard benchmark tests. We also provide a reinterpretation of the diamond approximation scheme for Kirchhoff plate theory within the framework of discrete mechanics. In this interpretation, the discrete moment-equilibrium problem is formally identical to the classical continuous problem, and the two differ only in the choice of differential structures. It also follows that the satisfaction of the inf − sup condition is a property of the cohomology of a certain discrete transverse differential complex. This close connection between the classical inf − sup condition and cohomology evinces the important role that the topology of the discretization plays in determining convergence in mixed problems.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1002/nme.4555DOIArticle
http://onlinelibrary.wiley.com/doi/10.1002/nme.4555/abstractPublisherArticle
ORCID:
AuthorORCID
Ortiz, M.0000-0001-5877-4824
Additional Information:© 2013 John Wiley & Sons, Ltd. Received 13 February 2013; Revised 2 July 2013; Accepted 15 July 2013. The support of the DOD MURI program on Mechanics and Mechanisms of Impulse Loading, Damage and Failure of Marine Structures and Materials, ONR Grant No. N00014-06-1-0730, program manager Dr. Y. D. S. Rajapakse, is gratefully acknowledged. MO also gratefully acknowledges the support of the US National Science Foundation through the Partnership for International Research and Education (PIRE) on Science at the Triple Point Between Mathematics, Mechanics and Materials Science, Award Number 0967140. AB and MO gratefully acknowledge the support of the Department of Energy National Nuclear Security Administration under Award Number DE-FC52-08NA28613 through Caltech’s ASC/PSAAP Center for the Predictive Modeling and Simulation of High Energy Density Dynamic Response of Materials.
Funders:
Funding AgencyGrant Number
Office of Naval Research (ONR)N00014-06-1-0730
NSF Partnership for International Research and Education (PIRE)0967140
Department of Energy (DOE) National Nuclear Security AdministrationDE-FC52-08NA28613
Subject Keywords:structures; finite element methods; plates; inf − sup condition; discrete mechanics; diamond mesh
Issue or Number:7
DOI:10.1002/nme.4555
Record Number:CaltechAUTHORS:20131108-093716091
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20131108-093716091
Official Citation:Perotti, L.E., Bompadre, A. and Ortiz, M. (2013), Automatically inf − sup compliant diamond-mixed finite elements for Kirchhoff plates. Int. J. Numer. Meth. Engng., 96: 405–424. doi: 10.1002/nme.4555
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:42336
Collection:CaltechAUTHORS
Deposited By: Jason Perez
Deposited On:08 Nov 2013 21:31
Last Modified:10 Nov 2021 16:21

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