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On a stress resultant geometrically exact shell model. Part VI: Conserving algorithms for non‐linear dynamics
Author(s) -
Simo J. C.,
Rifai M. S.,
Fox D. D.
Publication year - 1992
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620340108
Subject(s) - discretization , mathematics , finite element method , hamiltonian (control theory) , angular momentum , conservation law , conservation of energy , energy conservation , algorithm , mathematical analysis , mathematical optimization , classical mechanics , physics , ecology , biology , thermodynamics
In Parts I to V of the present work, the formulation and finite element implementation of a non‐linear stress resultant shell model are considered in detail. This paper is concerned with the extension of these results to incorporate completely general non‐linear dynamic response. Of special interest here is the dynamics of very flexible shells undergoing large overall motion which conserves the total linear and angular momentum and, for the Hamiltonian case, the total energy. A main goal of this paper is the design of non‐linear time‐stepping algorithms, and the construction of finite element interpolations, which preserve exactly these fundamental constants of motion. It is shown that only a very special class of algorithms, namely a formulation of the mid‐point rule in conservation form , exactly preserves the total linear and angular momentum. For the Hamiltonian case, a somewhat surprising result is proved: regardless of the degree of non‐linearity in the stored‐energy function, a generalized mid‐point rule algorithm always exists which exactly conserves energy The conservation properties of a time‐stepping algorithm need not, and in general will not, be preserved by the spatial discretization. Precise conditions which ensure preservation of these conservation properties are derived. A number of numerical simulations are presented which illustrate the exact conservation properties of the proposed methodology.