Computational Mechanics

, 45:157 | Cite as

Imposition of time-dependent boundary conditions in FEM formulations for elastodynamics: critical assessment of penalty-type methods

  • Elias A. Paraskevopoulos
  • Christos G. Panagiotopoulos
  • George D. ManolisEmail author
Original Paper


This paper is concerned with the variationally consistent incorporation of time dependent boundary conditions. The proposed methodology avoids ad hoc procedures and is applicable to both linear as well as nonlinear problems. An integral formulation of the dynamic problem serves as a basis for the imposition of the corresponding constraints, which are enforced via the consistent form of the penalty method, e.g. a form that complies with the norm and inner product of the functional space where the weak formulation is posed. Also, it is shown that well known and broadly implemented modelling techniques such as “large mass” and “large spring” methods, arise as limiting cases of the penalty formulation. Further extension of the proposed methodology is provided in the case where a generalized weak formulation of the dynamic problem with an independent velocity field is assumed. Finally, a few examples serve to illustrate the above concepts.


Time-dependent boundary conditions Elastodynamics Penalty method Large mass method Large spring method Transient dynamics Finite elements 


  1. 1.
    Arnold VI (1988) Mathematical methods of classical mechanics. Springer, New YorkGoogle Scholar
  2. 2.
    Stakgold I (1998) Green’s functions and boundary value problem. Wiley, New YorkGoogle Scholar
  3. 3.
    Mindlin RD, Goodman LE (1950) Beam vibrations with time-dependent boundary conditions. ASME J Appl Mech 17: 377–380zbMATHMathSciNetGoogle Scholar
  4. 4.
    Babuška I (1973) The finite element method with penalty. Math Comput 27: 221–228zbMATHCrossRefGoogle Scholar
  5. 5.
    Cook RD, Malkus DS, Plesha ME (1989) Concepts and applications of finite element analysis. Wiley, New YorkzbMATHGoogle Scholar
  6. 6.
    Leger P, Idet IM, Paultre P (1990) Multiple-support seismic analysis of large structures. Comput Struct 36: 1153–1158CrossRefGoogle Scholar
  7. 7.
    Babuška I (1973) The finite element method with Lagrangian multipliers. Numer Math 20: 179–192zbMATHCrossRefGoogle Scholar
  8. 8.
    Brezis H (1983) Analyse Functionelle, Thèorie et Applications. Masson, ParisGoogle Scholar
  9. 9.
    Rektorys K (1977) Variational methods in mathematics, science and engineering. D. Reidel Publishing Company, DordrechtGoogle Scholar
  10. 10.
    Ženišek A (1990) Nonlinear elliptic and evolution problems and their finite element. Approximations Academic Press, LondonzbMATHGoogle Scholar
  11. 11.
    Courant R, Hilbert D (1989) Methods of mathematical physics, vol 2. Wiley-VCH, BerlinGoogle Scholar
  12. 12.
    Hughes TJR, Hulbert GM (1988) Space-time finite element methods for elastodynamics: formulations and error estimates. Comput Methods Appl Mech Eng 66: 339–363zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Pitarresi JM, Manolis GD (1991) The temporal finite element method in structural dynamics. Comput Struct 41: 647–655zbMATHCrossRefGoogle Scholar
  14. 14.
    Idesman AV (2007) Solution of linear elastodynamic problems with space-time finite elements on structured and unstructured meshes. Comput Methods Appl Mech Eng 196: 1787–1815zbMATHCrossRefGoogle Scholar
  15. 15.
    Bathe KJ (1995) Finite element procedures. Prentice-Hall, Englewood CliffsGoogle Scholar
  16. 16.
    Brenan K, Campbell S, Petzold L (1989) Numerical solution of initial-value problems in differential-algebraic equations. North-Holland, New YorkzbMATHGoogle Scholar
  17. 17.
    Geradin M, Cardona A (2001) Flexible multibody dynamics: a finite element approach. Wiley, New YorkGoogle Scholar
  18. 18.
    Belytschko T, Liu WK, Moran B (2001) Nonlinear finite elements for continua and structures. Wiley, New YorkGoogle Scholar
  19. 19.
    Lanczos C (1974) The variational principles of mechanics, 4th edn. University of Toronto PressGoogle Scholar
  20. 20.
    Pars LA (1965) A treatise on analytical dynamics. Wiley, New YorkzbMATHGoogle Scholar
  21. 21.
    Hughes TJR (1976) Reduction scheme for some structural eigenvalue problems by a variational theorem. Int J Numer Methods Eng 10: 845–852zbMATHCrossRefGoogle Scholar
  22. 22.
    Paraskevopoulos EA, Talaslides DG (2004) A rational approach to mass matrix diagonalization in two dimensional elastodynamics. Int J Numer Methods Eng 61: 2639–2659zbMATHCrossRefGoogle Scholar
  23. 23.
    Paraskevopoulos EA, Panagiotopoulos CG, Talaslides DG (2008) Rational derivation of conserving time integration schemes: the moving mass case. In: Papadrakakis M, Charmpis DC, Lagaros ND, Tsompanakis Y (eds) Progress in computational dynamics and earthquake engineering. Taylor and Francis, LondonGoogle Scholar
  24. 24.
    Panagiotopoulos CG, Manolis GD (2009) Velocity-based reciprocal theorems in elastodynamics and BIEM implementation issues. Archives of Applied Mechanics, accepted, to appear in 2009Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Elias A. Paraskevopoulos
    • 1
  • Christos G. Panagiotopoulos
    • 1
  • George D. Manolis
    • 1
    Email author
  1. 1.Department of Civil EngineeringAristotle University of ThessalonikiThessalonikiGreece

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