Computational Mechanics

, Volume 43, Issue 2, pp 223–237 | Cite as

A numerical verification for an unconditionally stable FEM for elastodynamics

  • S. T. MillerEmail author
  • F. Costanzo
Original Paper


Numerical results for a time-discontinuous Galerkin space–time finite element formulation for second-order hyperbolic partial differential equations are presented. Discontinuities are allowed at finite, but not fixed, time increments. A method for h-adaptive refinement of the space–time mesh is proposed and demonstrated. Numerical results are presented for linear elastic problems in one space dimension. Numerical verification of unconditional stability, as proven in [7], is rendered. Comparison is made with analytic solutions when available. It is shown that the accuracy of the numerical solution can be increased without a major penalty on computational cost by using an adaptively refined mesh. Results are presented for a type of solid–solid dynamic phase transition problem where the trajectory of a moving surface of discontinuity is tracked.


Discontinuous Galerkin FEM Space–time finite element method Dynamic fracture 


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  1. 1.
    McOwen RC, Partial Differential Equations: Methods and Applications. Prentice-Hall, (2003)Google Scholar
  2. 2.
    Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  3. 3.
    Hughes TJR, Hulbert GM (1988) Space–time finite element methods for elastodynamics: formulations and error estimates. Comput Methods Appl Mech Eng 66(3): 339–363zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Hulbert GM, Hughes TJR (1990) Space–time finite element methods for second-order hyperbolic equations. Comput Methods Appl Mech Eng 84(3): 327–348zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Johnson C (1993) Discontinuous Galerkin finite element methods for second order hyperbolic problems. Comput Methods Appl Mech Eng 107: 117–129zbMATHCrossRefGoogle Scholar
  6. 6.
    Huang H, Costanzo F (2002) On the use of space–time finite elements in the solution of elasto-dynamic problems with strain-discontinuities. Comput Methods Appl Mech Eng 191(46): 1649–1679CrossRefMathSciNetGoogle Scholar
  7. 7.
    Costanzo F, Huang H (2005) Proof of unconditional stability for a single-field discontinuous galerkin finite element formulation for linear elasto-dynamics. Comput Methods Appl Mech Eng 194(18–20): 2059–2076zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Li XD, Wiberg N-E (1998) Implementation and adaptivity of a space–time finite element method for structural dynamics. Comput Methods Appl Mech Eng 156: 211–229zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Wiberg N-E, Li XD (1999) Adaptive finite element procedures for linear and non-linear dynamics. Int J Numer Methods Eng 46: 1781–1802zbMATHCrossRefGoogle Scholar
  10. 10.
    Thompson LL, He D (2005) Adaptive space–time finite element methods for the wave equation on unbounded domains. Comput Methods Appl Mech Eng 194: 1947–2000zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Idesman AV (2005) Solution of linear elastodynamics problems with space–time finite elements on structured and unstructured meshes. Comput Methods Appl Mech Eng 196: 1787–1815CrossRefGoogle Scholar
  12. 12.
    Abedi R, Petracovici B, Haber RB (2006) A space–time discontinuous Galerkin method for linearized elastodynamics with element-wise momentum balance. Comput Methods Appl Mech Eng 195: 3247–3473zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Courant R, Friedrichs KO, Lewy H (1928) Über die partiellen Differenzengleichungen der Mathematicschen Physik. Mathematische Annalen 100: 32–74zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lapidus L, Pinder GF (1982) Numerical solution of partial differential equations in science and engineering. Wiley, New YorkzbMATHGoogle Scholar
  15. 15.
    Gurtin ME (1981) An introduction to continuum mechanics. Academic Press, New YorkzbMATHGoogle Scholar
  16. 16.
    Bowen RM (1989) Introduction to continuum mechanics for engineers. Plenum Press, New YorkzbMATHGoogle Scholar
  17. 17.
    Abeyaratne R, Knowles JK (1990) On the driving traction acting on a surface of strain discontinuity in a continuum. J Mech Phys Solids 38(3): 345–360zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Gurtin ME (1995) The nature of configurational forces. Arch Ration Mech Anal 131: 67–100zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Bangerth W, Hartman R, Kanschat G (1998) deal.ii, Differential equations analysis library. Technical Reference.
  20. 20.
    Brenner S, Scott LR (1994) The mathematical theory of finite element methods. Springer, HeidelbergzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mechanical Science and EngineeringUniversity of Illinois Urbana-ChampaignUrbanaUSA
  2. 2.Department of Engineering Science and Mechanics, 212 Earth and Engineering Sciences BuildingThe Pennsylvania State UniversityUniversity ParkUSA

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