A multi-index variable time step method for the dynamic simulation of multibody systems

The paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. Various numerical problems that arise during the simulation process are described. The paper also describes ways to circumvent problems. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. It is also shown how the energy stored in the penalty system is suitable to measure the local integration error. Overall, the method is e¦cient and powerful; it is suitable for sti¤ and non-sti¤ systems, robust for all time step sizes, and it works for singular conÞgurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisÞed during the simulation process. The method is robust in the sense that it becomes more accurate as the time step size decreases. A section is devoted at the end of the paper to present numerical simulations that illustrate the performance of the proposed method. Copyright