Research Article
On stabilized finite element methods for the Reissner–Mindlin plate model
Reijo Kouhia,
Corresponding Author
Reijo Kouhia
Laboratory of Structural Mechanics, Helsinki University of Technology, P.O. Box 2100, 02015 TKK Espoo, Finland
Laboratory of Structural Mechanics, Helsinki University of Technology, P.O. Box 2100, 02015 TKK Espoo, FinlandSearch for more papers by this authorReijo Kouhia,
Corresponding Author
Reijo Kouhia
Laboratory of Structural Mechanics, Helsinki University of Technology, P.O. Box 2100, 02015 TKK Espoo, Finland
Laboratory of Structural Mechanics, Helsinki University of Technology, P.O. Box 2100, 02015 TKK Espoo, FinlandSearch for more papers by this authorAbstract
Stabilized finite element formulation for the Reissner–Mindlin plate model is considered. Physical interpretation for the stabilization procedure for low-order elements is established. Explicit interpolation functions for linear and bilinear stabilized MITC elements are derived. Some numerical examples including buckling and frequency analyses are presented. Copyright © 2007 John Wiley & Sons, Ltd.
REFERENCES
- 1 Babuška I, Suri M. Locking effects in the finite element approximation of elasticity problems. Numerische Mathematik 1992; 62: 439–463.
- 2 Suri M, Babuška I, Schwab C. Locking effects in the finite element approximation of plate models. Mathematics of Computation 1995; 64: 461–482.
- 3 Arnold DN, Falk RS. A uniformly accurate finite element method for the Reissner–Mindlin plate. SIAM Journal on Numerical Analysis 1989; 26: 1276–1290.
- 4 Dill EH. A triangular finite element for thick and thin plates. Computers and Structures 1990; 35: 301–308.
- 5 Franca LP, Stenberg R. A modification of a low-order Reissner–Mindlin plate bending element. In The Mathematics of Finite Elements and Applications VII, MAFELAP 1990, JR Whiteman (ed.). Academic Press: New York, 1991; 436–452.
- 6 Brezzi F, Marini LD. A nonconforming element for the Reissner–Mindlin plate. Computers and Structures 2003; 81: 515–522.
- 7 Chinosi C, Lovadina C, Marini LD. Nonconforming locking-free finite elements for Reissner–Mindlin plates. Computer Methods in Applied Mechanics and Engineering 2006; 195: 3448–3460.
- 8 Auricchio F, Lovadina C. Analysis of kinematic linked interpolation methods for Reissner–Mindlin plate problems. Computer Methods in Applied Mechanics and Engineering 2001; 190: 2465–2482.
- 9 Batoz JL, Lardeur P. A discrete shear triangular nine d.o.f. element for the analysis of thick to very thin plates. International Journal for Numerical Methods in Engineering 1989; 28: 533–560.
- 10 Bletzinger K-U, Bischoff M, Ramm E. A unified approach for shear-locking-free triangular and rectangular shell finite elements. Computers and Structures 2000; 75: 321–334.
- 11 Bathe KJ, Dvorkin EN. A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. International Journal for Numerical Methods in Engineering 1985; 21: 367–383.
- 12 Dvorkin EN, Bathe K-J. A continuum mechanics based four-node shell element for general non-linear analysis. Engineering Computations 1984; 1: 77–88.
10.1108/eb023562 Google Scholar
- 13 Brezzi F, Fortin M, Stenberg R. Error analysis of mixed-interpolated elements for Reissner–Mindlin plates. Mathematical Models and Methods in Applied Sciences 1991; 1: 125–151.
10.1142/S0218202591000083 Google Scholar
- 14 Hughes TJR, Franca LP. Convergence of transverse shear stresses in the finite element analysis of plates. Communications in Applied Numerical Methods 1988; 4: 185–187.
- 15 Bergan PG, Wang X. Quadrilateral plate bending element with shear deformations. Computers and Structures 1984; 19: 25–34.
- 16 Stenberg R. A new finite element formulation for the plate bending. In Asymptotic Methods for Elastic Structures, PG Ciarlet, L Trabucho, JN Viaño (eds); International Conference held in Lisbon, Portugal, 4–8 October 1993. Walter de Gruyter & Co.: Berlin, 1995.
- 17 Lyly M, Stenberg R. Stabilized mitc plate bending elements. In Advances in Finite Element Techniques, M Papadrakakis, BHV Topping (eds). Civil Comp Press: Edinburgh, Scotland, 1994; 11–16.
- 18 Lyly M, Stenberg R. Stabilized Finite Elements for Reissner–Mindlin Plates. Institut für Mathematik und Geometrie, Universität Innsbruck, Forschungsbericht 4, 1999.
- 19 Lyly M. On the connection between some linear triangular Reissner–Mindlin plate bending elements. Numerische Mathematik 2000; 85: 77–107.
- 20 Tessler A, Hughes TJR. A three node Mindlin plate element with improved transverse shear. Computer Methods in Applied Mechanics and Engineering 1985; 50: 71–101.
- 21 Xu Z. A thick–thin triangular plate element. International Journal for Numerical Methods in Engineering 1992; 33: 963–973.
- 22 MacNeal RH. A simple quadrilateral shell element. Computers and Structures 1978; 8: 175–183.
- 23 Crisfield MA. A four-noded thin-plate bending element using shear constraints—a modified version of Lyons' element. Computer Methods in Applied Mechanics and Engineering 1983; 38: 93–120.
- 24 Crisfield MA. A quadratic Mindlin element using shear constraints. Computers and Structures 1984; 18: 833–852.
- 25 Kouhia R. On some low order plate bending elements (in Finnish). Rakenteiden Mekaniikka 1996; 29(3–4): 51–68.
- 26 Franca LP, Russo A. Unlocking with residual-free bubbles. Computer Methods in Applied Mechanics and Engineering 1997; 142: 361–364.
- 27 Freund J, Salonen E-M. Sensitizing according to Courant the Timoshenko beam finite element solution. International Journal for Numerical Methods in Engineering 2000; 47: 1621–1631.
- 28 Fried I, Yang SK. Triangular, nine-degrees-of-freedom C0 plate bending element with quadratic accuracy. Quarterly of Applied Mathematics 1973; 31: 303–312.
- 29 Pitkäranta J. Analysis of some low-order finite element schemes for Mindlin–Reissner and Kirchhoff plates. Numerische Mathematik 1988; 53: 237–254.
- 30 Lyly M, Stenberg R, Vihinen T. A stable bilinear element for the Reissner–Mindlin plate model. Computer Methods in Applied Mechanics and Engineering 1993; 110: 343–357.
- 31 Lyly M, Stenberg R. New three and four noded plate bending elements. Rakenteiden Mekaniikka 1994; 27(2): 3–29. Available at: http://math.tkk.fi/∼rstenber/Publications/RakMek.pdf.
- 32 Bogner FK, Fox RL, Schmit LA. The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulas. Conference of Matrix Methods in Structural Mechanics, Wright Patterson AFB, OH, 1965; 397–444.
- 33 Ajiz MA, Jennings A. A robust incomplete Cholesky conjugate gradient algorithm. International Journal for Numerical Methods in Engineering 1984; 20: 949–966.
- 34 Benzi M, Kouhia R, Tůma M. Stabilized and block approximate inverse preconditioners for problems in solid and structural mechanics. Computer Methods in Applied Mechanics and Engineering 2001; 190: 6533–6554.