Petrov-Galerkin finite elements in time for rigid-body dynamics

The problem of rigid-body rotational dynamics is examined in the context of a novel Petrov-Galerkin mixed finite-element-in-time formulation. The present approach crucially differs from other formulations in the choice of the test functions that weight the constitutive equations in an integral average sense. Within this framework, two second-order accurate time-stepping algorithms are derived: the first attains a slightly greater accuracy than the second when compared with the analytical solution of a free-tumbling rigid body with an axis of material symmetry, but in general it does not preserve energy for force-free motion; the second is marginally less accurate, but it is associated with a discrete conservation law of the kinetic energy. Numerical simulations are presented that confirm the results of the analysis and illustrate the excellent performance of the proposed numerical approach.