zbMATH — the first resource for mathematics

A family of single-step Houbolt time integration algorithms for structural dynamics. (English) Zbl 0849.73079
Summary: A new family of implicit, single-step time integration methods is presented for solving structural dynamics problems. The proposed method is unconditionally stable, second-order accurate and asymptotically annihilating. It is spectrally equivalent to Houbolt’s method but is cast in single-step form rather than multistep form; thus the new algorithm computationally is more convenient. An explicit predictor-corrector algorithm is presented based upon the new implicit scheme. The explicit algorithm is spectrally equivalent to the central difference method. The two new algorithms are merged into an implicit-explicit method, resulting in an improved algorithm for solving structural dynamics problems composed of ‘soft’ and ‘stiff’ domains. Numerical results are presented demonstrating the improved performance of the new implicit-explicit method compared to previously developed implicit-explicit schemes for structural dynamics.

MSC:
 74S20 Finite difference methods applied to problems in solid mechanics 74H45 Vibrations in dynamical problems in solid mechanics
Full Text:
References:
 [1] Houbolt, J. C.: A recurrence matrix solution for the dynamic response of elastic aircraft. J. aeronant. Sci. 17, 540-550 (1950) [2] Newmark, N. M.: A method of computation for structural dynamics. J. engrg. Mech. div. ASCE 85, 67-94 (1959) [3] Bathe, K. J.; Wilson, E. L.: Stability and accuracy analysis of direct time integration methods. Earthquake engrg. Struct. dynam. 1, 283-291 (1973) [4] Park, K. C.: Evaluating time integration methods for nonlinear dynamic analysis. Finite element analysis of transient nonlinear behavior, 35-58 (1975) [5] Hilber, H. M.; Hughes, T. J. R.; Taylor, R. L.: Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake engrg. Struct. dynam. 5, 283-292 (1977) [6] Wood, W. L.; Bossak, M.; Zienkiewicz, O. C.: An alpha modification of newmark’s method. Intern. J. Numer. methods engrg. 15, 1562-1566 (1981) · Zbl 0441.73106 [7] Bazzi, G.; Anderheggen, E.: The \varrho-family of algorithms for time-step integration with improved numerical dissipation. Earthquake engrg. Struct. dynam. 10, 537-550 (1982) [8] Hoff, C.; Pahl, P. J.: Practical performance of the $${\theta}1$$ method and comparison with other dissipative algorithms in structural dynamics. Comput. methods appl. Mech. engrg. 67, 87-110 (1988) · Zbl 0619.73091 [9] Hoff, C.; Pahl, P. J.: Development of an implicit method with numerical dissipation from a generalized single-step algorithm for structural dynamics. Comput. methods appl. Mech. engrg. 67, 367-385 (1988) · Zbl 0619.73002 [10] Chung, J.; Hulbert, G. M.: A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-$${\alpha}$$ method. ASME J. Appl. mech. 60, 371-375 (1993) · Zbl 0775.73337 [11] G.M. Hulbert and J. Chung, On the (non-)importance of the spurious root of time integration algorithms for structural dynamics, Comm. Appl. Numer. Methods, in press. · Zbl 0800.73519 [12] Hulbert, G. M.: Limitations on linear multistep methods for structural dynamics. Earthquake engrg. Struct. dynam. 20, 191-196 (1991) [13] Hughes, T. J. R.; Liu, W. K.: Implicit-explicit finite elements in transient analysis: stability theory. ASME J. Appl. mech. 45, 371-374 (1978) · Zbl 0392.73076 [14] Hughes, T. J. R.; Liu, W. K.: Implicit-explicit finite elements in transient analysis: implementation and numerical examples. ASME J. Appl. mech. 45, 375-378 (1978) · Zbl 0392.73077 [15] Miranda, I.; Ferencz, R. M.; Hughes, T. J. R.: An improved implicit-explicit time integration method for structural dynamics. Earthquake engrg. Struct. dynam. 18, 643-653 (1989) [16] Hulbert, G. M.; Hughes, T. J. R.: An error analysis of truncated starting conditions in step-by-step time integration: consequences for structural dynamics. Earthquake engrg. Struct. dynam. 15, 901-910 (1987) [17] Katona, M. G.; Zienkiewicz, O. C.: A unite set of single-step algorithms, part 3: the beta-m method, a geralization of the nnewmarket scheme. J. numer. Methods engrg. 21, 1345-1359 (1985) · Zbl 0584.65044 [18] Hilber, H. M.; Hughes, T. J. R.: Collocation, dissipation and ’overshoot’ for time integration schemes in structural dynamics. Earthquake engrg. Struct. dynam. 6, 99-118 (1978) [19] Hughes, T. J. R.: The finite element method: linear static and dynamic finite element analysis. (1987) · Zbl 0634.73056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.