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

• Published in: International journal for numerical methods in engineering Vol. 96; no. 7; pp. 405 - 424
• Main Authors: Perotti, L.E., Bompadre, A., Ortiz, M.
• Format: Journal Article
• Language: English
• Published: Chichester Blackwell Publishing Ltd 16.11.2013; Wiley; Wiley Subscription Services, Inc
• Subjects: Analogies; Approximation; Convergence; diamond mesh; Diamonds; discrete mechanics; Discretization; Exact sciences and technology; Finite element method; finite element methods; Fundamental areas of phenomenology (including applications); inf − sup condition; Mathematical analysis; Physics; Plate theory; plates; Solid mechanics; Structural and continuum mechanics; Structural mechanics (beam, string...); structures; Theory and numerical methods; Finite element method; inf ― sup condition; discrete mechanics; plates; Discretization; Variational methods; finite element methods; Kirchhoff law; Analytical method; structures; diamond mesh; Plate
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.4555

SUMMARYWe 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. Copyright © 2013 John Wiley & Sons, Ltd.