Conservation properties of a time FE method - part II: Time-stepping schemes for non-linear elastodynamics

In the present paper one-step implicit integration algorithms for non-linear elastodynamics are developed. The discretization process rests on Galerkin methods in space and time. In particular, the continuous Galerkin method applied to the Hamiltonian formulation of semidiscrete non-linear elastodynamics lies at the heart of the time-stepping schemes. Algorithmic conservation of energy and angular momentum are shown to be closely related to quadrature formulas that are required for the calculation of time integrals. We newly introduce the `assumed strain method in timeʹ which enables the design of energy{momentum conserving schemes and which can be interpreted as temporal counterpart of the well-established assumed strain method for nite elements in space. The numerical examples deal with quasi-rigid motion as well as large-strain motion