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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. J. C. SIMO. N. TARNOW AND M. DOBLARE class of mechanical systems incorporates finite rotations and large deformations without restrictions placed on the allowable flexibility, and furnishes the canonical model problem for nonlinear structural dynamics.Rod models of the type considered here are fairly classical. In fact, the local balance of momentum equations are essentially contained in the works of Euler, Clebch, Maxwell, Kirchhoff and others (see e.g. Reference 2). Extensions to incorporate effects such as transverse shear deformation and warping distortion have been addressed by a number of authors from different perspectives; see References 3-7 and references therein. The key point from a numerical analysis perspective concerns the specific choice of parametrization employed in the mathematical description of the kinematics of the rod. The parametrization adopted here, introduced in Reference 5, renders a form of the momentum equations which strongly resembles the classical Euler equations of rigid body dynamics and is well suited both for mathematical and numerical analysis (see References 8-10 and the work of Mielke.") The same parametrization is adopted in Reference 12; alternative approaches are discussed in References 13 and 14.The preceding non-linear rod model can be shown to define an infinite dimensional Hamiltonian system.' Energy-momentum schemes for finite dimensional Hamiltonian systems with a linear configuration space have been considered by a number of authors, see e.g. References 16-18 for the case of classical particle mechanics. However, considerable difficulties arise when the configuration manifold is non-linear, as in the case of rigid body mechanic^,'^*^^ or for the infinite dimensional case which is the situation of interest here...

SUMMARYA numerical time-integration scheme for the dynamics of non-linear elastic shells is presented that simultaneously and independent of the time-step size inherits exactly the conservation laws of total linear, total angular momentum as well as total energy. The proposed technique generalizes to non-linear shells recent work of the authors on non-linear elastodynamics and is ideally suited for long-term/large-scale simulations. The algorithm is second-order accurate and can be immediately extended with no modification to a fourth-order accurate scheme. The property of exact energy conservation induces a strong notion of non-linear numerical stability which manifests itself in actual simulations. The superior performance of the proposed scheme method relative to conventional time-integrators is demonstrated in numerical simulations exhibiting large strains coupled with a large overall rigid motion. These numerical experiments show that symplectic schemes often regarded as unconditionally stable, such as the mid-point rule, can exhibit a dramatic blow-up in finite time while the present method remains perfectly stable.

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