Global Formulation of Conservative Time Integration by the Increment of the Geometric Stiffness

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 internal forces by combining the mean value of stresses and virtual strains at the element level [

1

]. 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. The geometric stiffness matrix is usually available in assembled form in existing programs, and thus a global form is attained that avoids the need for modifying the classic element implementation.

The algorithm is derived by use of a state space formulation. In the state space format the extra term containing the increment of the geometric stiffness matrix is located in the same position as the viscous damping matrix, indicating that the effect of the extra incremental geometric stiffness term in the nonlinear algorithm is equivalent to a variable damping term, depending on the change of the state of stress over a time increment. This explains the observation made in the literature, that the simple mean value algorithm needs artificial damping to be stable, when used in kinematically nonlinear problems. After the derivation of the algorithm in the state space format, the velocity can be condensed out, leaving the resulting algorithm in ‘single step - single solve’ format, where the system of equations is of the same size as the equations of the corresponding quasi-static problem. The implementation and performance of the algorithm are illustrated by numerical examples. The basic properties including a discussion of damping were obtained for non-linear truss elements in [

2

].