Abstract
In this work, an attempt is made at filling the apparent gap existing between the two major approaches evolved in the literature towards formulating space-time finite element methods. The first assumes Hamilton's Law as underlying concept, while the second performs a weighted residual approach on the ordinary differential equations emanating from the semidiscretization in the space dimension.
A general framework is proposed in the following pages, where the configuration space and the phase space forms of Hamilton's Law provide the general statements of the problem of motion. Within this framework, different families of integration algorithms are derived, according to different interpretations of the boundary terms. The bi-discontinuous form is obtained as the consequence of a consistent impulsive formulation of dynamics, while the discontinuous Galerkin form is obtained when the boundary terms at the end of the time interval are appropriately approximated.
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References
Argyris, J. H.; Scharpf, D. W. (1969): Finite elements in time and space. Aer. J. Royal Aer. Soc. 73, 1041–1044
Bailey, C. D. (1975): Application of Hamilton's Law of varying action. AIAA J. 13, 1154–1157
Bailey, C. D. (1976): Hamilton, Ritz, and elastodynamics. J. Appl. Mech. 43, 684–688
Baruch, M.; Riff, R. (1982): Hamilton's Principle, Hamilton's Law-6n correct formulations. AIAA J. 20, 687–691
Borri, M.; Ghiringhelli, G. L.; Lanz, M.; Mantegazza, P.; Merlini, T. (1985): Dynamic response of mechanical systems by a weak Hamiltonian formulation. Comp. and Struct. 20, 495–508
Borri, M.; Mello, F.; Atluri, S. N. (1990): Variational approaches for dynamics-numerical studies. In IUTAM symposium on dynamic problems of righd-elastic systems and structures. Moscow
Borri, M.; Mello, F.; Iura, M.; Atluri, N. (1991): Primal and mixed forms of Hamilton's Principle for constrained rigid and flexible dynamical systems: numerical studies. Comp. Mech. 7, 205–220
Borri, M.; Bottasso, C.; Mantegazza, P. (1992): Basic features of the time finite element approach for dynamics. Meccanica 27, 119–130
Fried, I. (1969): Finite element analysis of time dependent phenomena. AIAA J. 7, 1170–1173
Kujawski, J.; Desai, C. S. (1984): Generalized time finite element algorithm for non-linear dynamic problems. Eng. Comp. 1, 247–251
Hodges, D. H.; Hou, L. J. (1991): Shape functions for mixed p-version finite elements in the time domain. J. Sound and Vibration 2, 169–178
Hughers, T. J. R. (1983): Analysis of transient algorithms with particular reference to stability behavior. In: Belytschko, T.; Hughes, T. J. R. (eds.): Computational methods for transient analysis. Chapt. 2, 67–155, Elsevier Science Publishers
Hughes, T. J. R.; Hulbert, G. M. (1988): Space-time finite element methods for elastodynamics: formulations and error estimates. Comp. Meth. Appl. Mech. Eng. 63, 339–363
Hulbert, G. M. (1989): Space-time finite element methods for second-order hyperbolic equations. Ph.D. thesis, Stanford University
Hulbert, G. M. (1992): Time finite element methods for structural dynamics. Int. J. Num. Meth. Eng. 33, 307–331
Mello, F. (1989): Weak formulations in analytical dynamics, with applications to multi-rigid-body systems, using time finite elements. Ph.D. thesis, Georgia Institute of Technology
Peters, D. A.; Izadpanah, A. (1988): hp-version finite elements for the space time domain. Comp. Mech. 3, 73–88
Simkins, T. E. (1978): Unconstrained variational statements for initial and boundary value problems. AIAA J. 16, 559–563
Simkins, T. E. (1981): Finite elements for initial value problems in dynamics. AIAA J. 19, 1357–1362
Zienkiewicz, O. C. (1977): The finite element method. McGraw-Hill: London
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Communicated by T. A. Cruse, April 14, 1993
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Borri, M., Bottasso, C. A general framework for interpreting time finite element formulations. Computational Mechanics 13, 133–142 (1993). https://doi.org/10.1007/BF00370131
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DOI: https://doi.org/10.1007/BF00370131