Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics

Abstract

It is shown that widely used implicit schemes, in particular the classical Newmark family of algorithms and its variants, generally fail to conserve total angular momentum for nonlinear Hamiltonian systems including classical rigid body dynamics, nonlinear elastodynamics, nonlinear rods and nonlinear shells. For linear Hamiltonian systems, it is well known that only the Crank-Nicholson scheme exactly preserves the total energy of the system. This conservation property is typically lost in the nonlinear regime. A general class of implicit time-stepping algorithms is presented which preserves exactly the conservation laws present in a general Hamiltonian system with symmetry, in particular the total angular momentum and the total energy. Remarkably, the actual implementation of this class of algorithms can be effectively accomplished by means of a simple two-step solution scheme which results in essentially no added computational cost relative to standard implicit methods. A complete analysis of these algorithms and a related class of schemes referred to as symplectic integrators is given. The good performance of the proposed methodology is demonstrated by means of three numerical examples which constitute representative model problems of nonlinear elastodynamics, nonlinear rods and nonlinear shells.