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Smolinski, P.; Sleith, S.; Belytschko, T.

Stability of an explicit multi-time step integration algorithm for linear structural dynamics equations. (English) Zbl 0865.73075

Comput. Mech. 18, No. 3, 236-244 (1996).
Summary: A proof of stability is presented for an explicit multi-time step integration method for 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.

Cited in 29 Documents

MSC:

74S20 Finite difference methods applied to problems in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Keywords:

semidiscretization; eigenvalues; amplification matrices; unit circle; pseudo-energy
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\textit{P. Smolinski} et al., Comput. Mech. 18, No. 3, 236--244 (1996; Zbl 0865.73075)
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References:

[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: 1803-1828 · doi:10.1002/nme.1620350906
[2] Belytschko, T.; Hughes, T. J. R. 1983; Computational Methods for Transient Analysis. North Holland, Amsterdam · Zbl 0521.00025
[3] Belytschko, T.; Lu, Y. Y. 1992: Stability analysis of elemental explicit-implicit partitions by Fourier methods, Comp. Meth. Appl. Mech. Engrg. 95: 87-96 · Zbl 0757.65079 · doi:10.1016/0045-7825(92)90083-V
[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. 697-720
[5] Belytschko, T.; Mullen, R. 1978: Stability of explicit-implicit mesh partitions in time integration, Int. J. Num. Meth. Engrg., 12: 1575-1586 · Zbl 0398.65059 · doi:10.1002/nme.1620121008
[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): 281-297 · Zbl 0599.65060 · doi:10.1016/0045-7825(85)90126-4
[7] Belytschko, T.; Yen, H.-J.; Mullen, R. 1979: Mixed methods for time integration, Comp. Meth. Appl. Mech. Engrg. 17/18: 259-275 · Zbl 0403.73002 · doi:10.1016/0045-7825(79)90022-7
[8] Donea, J.; Laval, H. 1988: Nodal partition of explicit finite element methods for unsteady diffusion problems, Comp. Meth. Appl. Mech. Engrg. 68: 189-204 · Zbl 0626.76090 · doi:10.1016/0045-7825(88)90115-6
[9] Flanagan, D.; Belytschko, T. 1981: Simultaneous relaxation in structural dynamics, ASCE J. Engrg. Mech. Div. 107: 1039-1055
[10] Hughes, T. J. R.; Liu, W. K. 1978: Implicit-explicit finite elements in transient analysis: stability theory, J. Appl. Mech. 45: 371-374 · Zbl 0392.73076 · doi:10.1115/1.3424304
[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, Swansea · Zbl 0265.73064
[12] Mizukami, A. 1986: Variable explicit finite element methods for unsteady heat conduction, Comp. Meth. Appl. Mech. Engrg. 59: 101-109 · Zbl 0595.73130 · doi:10.1016/0045-7825(86)90026-5
[13] Neal, M. O.; Belytschko, T. 1989: Explicit-explicit subcycling with non-integer time step ratios for structural dynamics systems, Comp. Struct. 31 (6): 871-880 · Zbl 0705.73251 · doi:10.1016/0045-7949(89)90272-1
[14] Park, K. C. 1980: Partitioned transient analysis procedure for coupled field problems: stability analysis, J. Appl. Mech. 47: 370-376 · Zbl 0437.73072 · doi:10.1115/1.3153671
[15] Smolinski, P. 1991: Stability of variable explicit time integration for unsteady diffusion problems, Comp. Meth. Appl. Mech. Engrg. 93: 247-252 · Zbl 0825.76783 · doi:10.1016/0045-7825(91)90153-W
[16] Smolinski, P. 1992: Stability analysis of a multi-time step explicit integration method, Comp. Meth. Appl. Mech. Engrg. 95: 291-300 · Zbl 0755.73095 · doi:10.1016/0045-7825(92)90188-P
[17] Smolinski, P.; Belytschko, T.; Neal, M. O. 1988: Multi-time step integration using nodal partitioning, Int. J. Numer. Meth. Engrg. 26: 349-359 · Zbl 0629.73076 · doi:10.1002/nme.1620260205
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