Abstract
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.
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References
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: 1803–1828
Belytschko, T.; Hughes, T. J. R. 1983; Computational Methods for Transient Analysis. North Holland, Amsterdam
Belytschko, T.; Lu, Y. Y. 1992: Stability analysis of elemental explicit-implicit partitions by Fourier methods, Comp. Meth. Appl. Mech. Engrg. 95: 87–96
Belytschko, T.; Mullen, R. 1977: Explicit integration of structural problems, In: Finite Elements in Nonlinear Mechanics, P. Bergen, et al. Eds Vol. 2, pp. 697–720
Belytschko, T.; Mullen, R. 1978: Stability of explicit-implicit mesh partitions in time integration, Int. J. Num. Meth. Engrg., 12: 1575–1586
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): 281–297
Belytschko, T.; Yen, H.-J.; Mullen, R. 1979: Mixed methods for time integration, Comp. Meth. Appl. Mech. Engrg. 17/18: 259–275
Donea, J.; Laval, H. 1988: Nodal partition of explicit finite element methods for unsteady diffusion problems, Comp. Meth. Appl. Mech. Engrg. 68: 189–204
Flanagan, D.; Belytschko, T. 1981: Simultaneous relaxation in structural dynamics, ASCE J. Engrg. Mech. Div. 107: 1039–1055
Hughes, T. J. R.; Liu, W. K. 1978: Implicit-explicit finite elements in transient analysis: stability theory, J. Appl. Mech. 45: 371–374
Irons, B. M. 1970: Applications of a theorem on eigenvalues to finite element problems (CR/132/70), University of Wales, Department of Civil Engineering, Swansea
Mizukami, A. 1986: Variable explicit finite element methods for unsteady heat conduction, Comp. Meth. Appl. Mech. Engrg. 59: 101–109
Neal, M. O.; Belytschko, T. 1989: Explicit-explicit subcycling with non-integer time step ratios for structural dynamics systems, Comp. Struct. 31 (6): 871–880
Park, K. C. 1980: Partitioned transient analysis procedure for coupled field problems: stability analysis, J. Appl. Mech. 47: 370–376
Smolinski, P. 1991: Stability of variable explicit time integration for unsteady diffusion problems, Comp. Meth. Appl. Mech. Engrg. 93: 247–252
Smolinski, P. 1992: Stability analysis of a multi-time step explicit integration method, Comp. Meth. Appl. Mech. Engrg. 95: 291–300
Smolinski, P.; Belytschko, T.; Neal, M. O. 1988: Multi-time step integration using nodal partitioning, Int. J. Numer. Meth. Engrg. 26: 349–359
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Communicated by S. N. Atluri, 5 March 1996
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Smolinski, P., Sleith, S. & Belytschko, T. Stability of an explicit multi-time step integration algorithm for linear structural dynamics equations. Computational Mechanics 18, 236–244 (1996). https://doi.org/10.1007/BF00369941
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DOI: https://doi.org/10.1007/BF00369941