Reversed stability conditions in transient finite element analysis

Numerical methods which introduce artificially unstable modes are discussed. In structural and elastodynamics these results from optimal mass lumping with higher-order elements. In fluid mechanics an additional source of these modes can be a penalty function with alternating signs. These modes yield unstable modal equations; however, they do not necessarily imply unstable ttransient integration in the presence of algorithmic damping. Stable integration can be achieved by satisfying a stability condition in which the roles of space-step and time-step are reversed. Elastodynamics, the Navier-Stokes equations, and non-Newtonian fluids provide numerical examples.