Computational Mechanics

, Volume 18, Issue 3, pp 236–244 | Cite as

Stability of an explicit multi-time step integration algorithm for linear structural dynamics equations

  • P. Smolinski
  • S. Sleith
  • T. Belytschko


A proof of stability is developed for an explicit multi-time step integration method of the second order differential equations which result from a semidiscretization of the equations of structural dynamics. The proof is applicable to an algorithm that partitions the mesh into subdomains according to nodal groups which are updated with different time steps. The stability of the algorithm is demonstrated by showing that the eigenvalues of the amplification matrices lie within the unit circle and that a pseudo-energy remains constant. Bounds on the stable time steps for the nodal partitions are developed in terms of element frequencies.


Differential Equation Information Theory Dynamic Equation Unit Circle Integration Method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Belytschko, T.; Gilbertsen, N. D. 1992: Implementation of Mixed Time Integration Techniques on a Vectorized Computer with Shared Memory. Int. J. Numer. Meth. Engrg. 35: 1803–1828Google Scholar
  2. Belytschko, T.; Hughes, T. J. R. 1983; Computational Methods for Transient Analysis. North Holland, AmsterdamGoogle Scholar
  3. Belytschko, T.; Lu, Y. Y. 1992: Stability analysis of elemental explicit-implicit partitions by Fourier methods, Comp. Meth. Appl. Mech. Engrg. 95: 87–96Google Scholar
  4. Belytschko, T.; Mullen, R. 1977: Explicit integration of structural problems, In: Finite Elements in Nonlinear Mechanics, P. Bergen, et al. Eds Vol. 2, pp. 697–720Google Scholar
  5. Belytschko, T.; Mullen, R. 1978: Stability of explicit-implicit mesh partitions in time integration, Int. J. Num. Meth. Engrg., 12: 1575–1586Google Scholar
  6. Belytschko, T.; Smolinski, P.; Liu, W. K. 1985: Stability of multi-time partitioned integrators for first-order systems, Comp. Meth. Appl. Mech. Engrg. 49 (3): 281–297Google Scholar
  7. Belytschko, T.; Yen, H.-J.; Mullen, R. 1979: Mixed methods for time integration, Comp. Meth. Appl. Mech. Engrg. 17/18: 259–275Google Scholar
  8. Donea, J.; Laval, H. 1988: Nodal partition of explicit finite element methods for unsteady diffusion problems, Comp. Meth. Appl. Mech. Engrg. 68: 189–204Google Scholar
  9. Flanagan, D.; Belytschko, T. 1981: Simultaneous relaxation in structural dynamics, ASCE J. Engrg. Mech. Div. 107: 1039–1055Google Scholar
  10. Hughes, T. J. R.; Liu, W. K. 1978: Implicit-explicit finite elements in transient analysis: stability theory, J. Appl. Mech. 45: 371–374Google Scholar
  11. Irons, B. M. 1970: Applications of a theorem on eigenvalues to finite element problems (CR/132/70), University of Wales, Department of Civil Engineering, SwanseaGoogle Scholar
  12. Mizukami, A. 1986: Variable explicit finite element methods for unsteady heat conduction, Comp. Meth. Appl. Mech. Engrg. 59: 101–109Google Scholar
  13. Neal, M. O.; Belytschko, T. 1989: Explicit-explicit subcycling with non-integer time step ratios for structural dynamics systems, Comp. Struct. 31 (6): 871–880Google Scholar
  14. Park, K. C. 1980: Partitioned transient analysis procedure for coupled field problems: stability analysis, J. Appl. Mech. 47: 370–376Google Scholar
  15. Smolinski, P. 1991: Stability of variable explicit time integration for unsteady diffusion problems, Comp. Meth. Appl. Mech. Engrg. 93: 247–252Google Scholar
  16. Smolinski, P. 1992: Stability analysis of a multi-time step explicit integration method, Comp. Meth. Appl. Mech. Engrg. 95: 291–300Google Scholar
  17. Smolinski, P.; Belytschko, T.; Neal, M. O. 1988: Multi-time step integration using nodal partitioning, Int. J. Numer. Meth. Engrg. 26: 349–359Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • P. Smolinski
    • 1
  • S. Sleith
    • 1
  • T. Belytschko
    • 2
    • 3
  1. 1.Department of Mechanical EngineeringUniversity of PittsburghPittsburghUSA
  2. 2.Department of CivilNorthwestern UniversityEvanstonUSA
  3. 3.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA

Personalised recommendations