A new bipenalty formulation for ensuring time step stability in time domain computational dynamics - Hetherington, Jack; Rodríguez-Ferran, Antonio; Askes, Harm (International journal for numerical methods in engineering, 2012)
SUMMARY It is well known that use of standard penalty methods can decrease the critical time step of time domain dynamic finite element analyses. The bipenalty method utilises both stiffness and mass penalties to impose constraints that have a minimal effect on the eigenfrequencies of the finite element system. One way of achieving this goal is to find a ratio of stiffness and mass penalty parameters—the critical penalty ratio (CPR)—that does not affect the maximum eigenfrequency (and therefore, for conditionally stable solution schemes, the critical time step) of a system. In this contribution, we develop a new method of calculating the CPR associated with a finite element formulation by examining the eigenvalue problem in detail. Advantages of the method compared with previous solutions include increased simplicity and generality and the ability to consider multiple constraints. The method is demonstrated by deriving CPRs for a few finite element formulations, which are then verified using simple numerical examples. The superiority of the bipenalty method over standard mass penalty methods is also demonstrated. Copyright © 2011 John Wiley & Sons, Ltd.