Penalty methods for time domain computational dynamics based on positive and negative inertia

Positive and negative inertial penalties are used to impose constraints in time domain computational dynamics. Mathematical proofs are given to show that, asymptotically, positive and negative inertial penalties render the same solution. It is also proven that the constraint approximations calculated using positive and negative inertial penalties bound the constrained solution. Based on these observations, algorithms can be developed that improve the accuracy of restraint imposition in time domain computational dynamics. In particular, a scheme based on alternating signs and a scheme based on linear interpolation are discussed, both of which are effective in reducing the error due to the use of penalties. Finally, it is demonstrated that inertia penalties tend to increase the critical time step in conditionally stable time integration schemes, which is an important advantage over the conventional stiffness-type penalties.