Bi-discontinuous time step integration algorithms. II: Second-order equations. (English) Zbl 1083.74541

Fung, T. C.

Comput. Methods Appl. Mech. Eng. 192, No. 3-4, 351-374 (2003).

Summary: Bi-discontinuous time step integration algorithms for linear second-order equations are investigated. The second-order equations are manipulated directly. Both the initial and final displacement and velocity are only weakly enforced at the beginning and at the end of the time interval. As a result, there may be discontinuous jumps for both the displacement and velocity at the two ends of the time interval. If the displacement is assumed to be a polynomial of degree \(n+1\) within the time interval, there are \(n+4\) unknowns in the formulation. It is found that the order of accuracy would be at least \(n+1\) for displacement and \(n\) for velocity in general. It is shown that the relevant algorithmic parameters for the second-order equations can be related to the algorithmic parameters presented in Part 1 of this paper [see the foregoing entry] for the first-order equations. Unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation equivalent to the generalized Padé approximations with an order of accuracy \(2n+3\) can then be constructed systematically. The weighting parameters and the corresponding weighting functions are given explicitly. It is also shown that the accuracy of the particular solutions is compatible with the accuracy of the homogeneous solutions if the proposed weighting functions are employed.

MSC:

74H15	Numerical approximation of solutions of dynamical problems in solid mechanics
74S99	Numerical and other methods in solid mechanics
70-08	Computational methods for problems pertaining to mechanics of particles and systems

Keywords:

generalized Padé approximations; weighting functions

References:

[1]	Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs, NJ
[2]	Wood, W.L., Practical time-stepping schemes, (1990), Clarendon Press Oxford · Zbl 0694.65043
[3]	Zienkiewicz, O.C.; Taylor, R.L., The finite element method, (1991), McGraw-Hill New York · Zbl 0991.74002
[4]	Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations I, (1987), Springer Berlin · Zbl 0638.65058
[5]	Hairer, E.; Wanner, G., Solving ordinary differential equations II, (1991), Springer Berlin · Zbl 0729.65051
[6]	Fried, I., Finite-element analysis of time-dependent phenomena, Aiaa j, 7, 1170-1173, (1969) · Zbl 0179.55001
[7]	Oden, J.T., A general theory of finite element. 2. application, Int. J. numer. methods engrg, 1, 247-259, (1969) · Zbl 0263.73048
[8]	Argyris, J.H.; Scharpf, D.W., Finite elements in time and space, J. roy. aeronaut. soc, 73, 1041-1044, (1969)
[9]	Bailey, C.D., Application of hamilton’s law of varying action, Aiaa j, 13, 1154-1157, (1975) · Zbl 0323.70020
[10]	Bailey, C.D., The method of Ritz applied to the equation of Hamilton, Comput. methods appl. mech. engrg, 1, 235-247, (1976) · Zbl 0322.70015
[11]	Riff, R.; Baruch, M., Time finite element discretization of hamilton’s law of varying action, Aiaa j, 22, 1310-1318, (1984) · Zbl 0562.73060
[12]	Riff, R.; Baruch, M., Stability of time finite elements, Aiaa j, 22, 1171-1173, (1984) · Zbl 0568.73081
[13]	Kujawski, J.; Gallagher, R.H., A generalized least-squares family of algorithms for transient dynamic analysis, Earthquake engrg. struct. dyn, 18, 539-550, (1989)
[14]	Gellert, M., A new algorithm for integration of dynamic systems, Comput. struct, 9, 401-408, (1978) · Zbl 0405.65047
[15]	Fung, T.C., Unconditionally stable higher-order accurate Hermitian time finite elements, Int. J. numer. methods engrg, 39, 3475-3495, (1996) · Zbl 0884.73065
[16]	Simkins, T.E., Unconstrained variational statements for initial and boundary value problems, Aiaa j, 16, 559-563, (1978) · Zbl 0377.73006
[17]	Tiersten, H.F., Natural boundary and initial conditions from a modification of hamilton’s principle, J. math. phys, 9, 1445-1450, (1968) · Zbl 0159.16501
[18]	Cannarozzi, M.; Mancuso, M., Formulation and analysis of variational methods for time integration of linear elastodynamics, Comput. methods appl. mech. engrg, 127, 241-257, (1995) · Zbl 0862.73078
[19]	Hulbert, G.M., Time finite element methods for structural dynamics, Int. J. numer. methods engrg, 33, 307-331, (1992) · Zbl 0760.73064
[20]	Borri, M.; Bottasso, C., A general framework for interpreting time finite element formulations, Comput. mech, 13, 133-142, (1993) · Zbl 0789.70003
[21]	Borri, M.; Ghiringhelli, G.L.; Lanz, M.; Mantegazza, P.; Merlini, T., Dynamic response of mechanical systems by a weak Hamilton formulation, Comput. struct, 20, 495-508, (1985) · Zbl 0574.73091
[22]	Peters, D.A.; Izadpanah, A.P., hp-version finite elements for the space-time domain, Comput. mech, 3, 73-88, (1988) · Zbl 0627.73081
[23]	Fung, T.C., Third order newmark methods with controllable numerical dissipation, Commun. numer. methods engrg, 13, 307-315, (1997) · Zbl 0880.73075
[24]	Fung, T.C., Complex-time-step newmark methods with controllable numerical dissipation, Int. J. numer. methods engrg, 41, 64-93, (1998) · Zbl 0916.73080
[25]	Fung, T.C., Weighting parameters for unconditionally stable higher-order accurate time step integration algorithms–part 2. second order equations, Int. J. numer. methods engrg, 45, 971-1006, (1999) · Zbl 0943.74078
[26]	Clough, R.W.; Penzien, J., Dynamics of structures, (1993), McGraw-Hill