NONLINEAR DYNAMICS OF 3-DIMENSIONAL RODS - EXACT ENERGY AND MOMENTUM CONSERVING ALGORITHMS

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 exactl y preserve fundamental constants of the motion such as the total linea r 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 al gorithmic stress resultants. Second, an algorithmic counterpart of the elastic constitutive equations is developed such that the law of cons ervation of total energy is exactly preserved. Conventional schemes ex hibiting no numerical dissipation, symplectic algorithms in particular , are shown to lead to unstable solutions when the high frequencies ar e not resolved. Compared to conventional schemes there is little, if a ny, additional computational cost involved in the proposed class of en ergy-momentum methods. The excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several num erical simulations.