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Non‐linear dynamics of three‐dimensional rods: Exact energy and momentum conserving algorithms
Author(s) -
Simo J. C.,
Tarnow N.,
Doblare M.
Publication year - 1995
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.1620380903
Subject(s) - angular momentum , mathematics , symplectic geometry , momentum (technical analysis) , hamiltonian (control theory) , dissipation , conservation law , numerical analysis , rod , energy–momentum relation , total angular momentum quantum number , classical mechanics , algorithm , mathematical analysis , physics , mathematical optimization , quantum mechanics , finance , economics , medicine , alternative medicine , pathology
The long‐term dynamic response of non‐linear geometrically exact rods under‐going finite extension, shear and bending, accompanied by large overall motions, is addressed in detail. The central objective is the design of unconditionally stable time‐stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum, the total angular momentum and, for the Hamiltonian case, the total energy. This objective is accomplished in two steps. First, a class of algorithms is introduced which conserves linear and angular momentum. This result holds independently of the definition of the algorithmic stress resultants. Second, an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved. Conventional schemes exhibiting no numerical dissipation, symplectic algorithms in particular, are shown to lead to unstable solutions when the high frequencies are not resolved. Compared to conventional schemes there is little, if any, additional computational cost involved in the proposed class of energy–momentum methods. The excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations.