The role of geometric stiffness in momentum and energy conserving time integration

A momentum and energy conserving time integration algorithm is developed for the motion of elastic bodies described in terms of the quadratic Green strain. Momentum conserving algorithms are formulated from an integral of the equations of motion and energy conservation has traditionally been obtained by evaluating the contribution from the internal forces by use of a combined mean value of stresses and virtual strains on the element level. It is here demonstrated that momentum and energy conservation can be obtained from the classic central difference formulation by including an extra global term in the form of the increment of the geometric stiffness matrix over the current time step, usually directly available in global form in existing finite element programmes. The theory is derived by the use of a state-space formulation, where this extra term is located in the same position as the viscous damping matrix, indicating that the effect of the extra incremental geometric stiffness term in the non-linear algorithm is equivalent to a variable damping term depending on the change of the state of stress over a time increment. In the actual numerical algorithm, the new value of the velocity vector is eliminated, leaving a non-linear equation for the displacement increment alone, followed by an explicit vector update of the velocity increment. Copyright q 2006 John Wiley & Sons, Ltd.