Advertisement

Stability of Some Finite Element Methods for Finite Elasticity Problems

  • F. Auricchio
  • L. Beirão da Veiga
  • C. Lovadina
  • A. Reali
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 509)

Abstract

We consider the finite elasticity problem for incompressible materials, proposing a simple bidimensional problem for which we provide indications on the solution stability. Furthermore, we study the stability of the discrete solution, obtained by means of some well-known finite elements, and we present several numerical experiments in order to evaluate and compare the performance of the different discrete schemes under investigation on the considered model problem.

Keywords

Mass Transfer Finite Element Method Numerical Experiment Model Problem Solution Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. F. Armero. On the locking and stability of finite elements in finite deformation plane strain problems. Computers & Structures, 75:261–290, 2000.CrossRefGoogle Scholar
  2. D. N. Arnold, F. Brezzi, and M. Fortin. A stable finite element for the stokes equation. Calcolo, 21:337–344, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  3. F. Auricchio, L. Beirão da Veiga, C. Lovadina, and A. Reali. An analysis of some mixed-enhanced finite element for plane linear elasticity. Computer Methods in Applied Mechanics and Engineering, 194:2947–2968, 2005a.zbMATHCrossRefMathSciNetGoogle Scholar
  4. F. Auricchio, L. Beirão da Veiga, C. Lovadina, and A. Reali. A stability study of some mixed finite elements for large deformation elasticity problems. Computer Methods in Applied Mechanics and Engineering, 194:1075–1092, 2005b.zbMATHCrossRefMathSciNetGoogle Scholar
  5. K. J. Bathe. Finite Element Procedures. Prentice Hall, Englewood Cliffs, NJ, 1996.Google Scholar
  6. J. Bonet and R. D. Wood. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, 1997.Google Scholar
  7. D. Braess. Enhanced assumed strain elements and locking in membrane problems. Computer Methods in Applied Mechanics and Engineering, 165:155–174, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  8. F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, 1991.zbMATHGoogle Scholar
  9. P. G. Ciarlet. The Finite Element Methods for Elliptic Problems. North-Holland, Amsterdam, 1978.Google Scholar
  10. S. Glaser and F. Armero. On the formulation of enhanced strain finite elements in finite deformations. Engineering Computations, 14:759–791, 1997.zbMATHCrossRefGoogle Scholar
  11. T. J. R. Hughes. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications, Mineola, New York, 2000.Google Scholar
  12. O. Klaas, A. M. Maniatty, and M. S. Shephard. A stabilized mixed petrovgalerkin finite element method for finite elasticity. formulation for linear displacement and pressure interpolation. Computer Methods in Applied Mechanics and Engineering, 180:65–79, 1999.zbMATHCrossRefGoogle Scholar
  13. P. Le Tallec. Existence and approximation results for nonlinear mixed problems: application to incompressible finite elasticity. Numerische Mathematik, 38:365–382, 1982.zbMATHCrossRefGoogle Scholar
  14. P. Le Tallec. Numerical methods for nonlinear three-dimensional elasticity. In P. G. Ciarlet and J. L. Lions, editors, Handbook of Numerical Analysis, volume III, pages 465–622. Elsevier Science, North Holland, 1994.Google Scholar
  15. C. Lovadina. Analysis of strain-pressure finite element methods for the stokes problem. Numerical Methods for PDE’s, 13:717–730, 1997.zbMATHMathSciNetGoogle Scholar
  16. C. Lovadina and F. Auricchio. On the enhanced strain technique for elasticity problems. Computers & Structures, 81:777–787, 2003.CrossRefMathSciNetGoogle Scholar
  17. A. M. Maniatty, Y. Liu, O. Klaas, and M. s. Shephard. Higher order stabilized finite element method for hyperelastic finite deformation. Computer Methods in Applied Mechanics and Engineering, 191:1491–1503, 2002.zbMATHCrossRefGoogle Scholar
  18. J. C. Nagtegaal and D. D. Fox. Using assumed enhanced strain elements for large compressive deformation. International Journal of Solids and Structures, 33:3151–3159, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  19. D. Pantuso and K. J. Bathe. A four-node quadrilateral mixed-interpolated elements for solids and fluids. Mathematical Models & Methods in Applied Sciences, 5:1113–1128, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  20. D. Pantuso and K. J. Bathe. On the stability of mixed finite elements in large strain analysis of incompressible solids. Finite Elements in Analysis and Design, 28:83–104, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  21. B. D. Reddy and J. C. Simo. Stability and convergence of a class of enhanced strain methods. SIAM Journal on Numerical Analysis, 32:705–1728, 1995.CrossRefMathSciNetGoogle Scholar
  22. S. Reese, M. Küssner, and B. D. Reddy. A new stabilization technique for finite elements in non-linear elasticity. International Journal for Numerical Methods in Engineering, 44:1617–1652, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  23. S. Reese and P. Wriggers. A stabilization technique to avoid hourglassing in finite elasticity. International Journal for Numerical Methods in Engineering, 48:79–109, 2000.zbMATHCrossRefGoogle Scholar
  24. J. C. Simo and F. Armero. Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 33:1413–1449, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  25. J. C. Simo and M. S. Rifai. A class of mixed assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 29:1595–1638, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  26. R. L. Taylor. FEAP: A Finite Element Analysis Program, programmer manual. http://www.ce.berkeley.edu/~rlt/feap/, 2001.Google Scholar
  27. P. Wriggers and S. Reese. A note on enhanced strain methods for large deformations. Computer Methods in Applied Mechanics and Engineering, 135:201–209, 1996.zbMATHCrossRefGoogle Scholar

Copyright information

© CISM, Udine 2009

Authors and Affiliations

  • F. Auricchio
    • 1
    • 4
    • 5
  • L. Beirão da Veiga
    • 3
    • 4
  • C. Lovadina
    • 2
    • 4
  • A. Reali
    • 1
    • 5
  1. 1.Dipartimento di Meccanica StrutturaleUniversità di PaviaItaly
  2. 2.Dipartimento di MatematicaUniversità di PaviaItaly
  3. 3.Dipartimento di MatematicaUniversità di MilanoItaly
  4. 4.IMATI - CNRPaviaItaly
  5. 5.ROSE SchoolUniversità di PaviaItaly

Personalised recommendations