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 integrat ion of the equations of motion of constrained multibody systems in descript or form. The basis of the method is the augmented Lagrangian formulation wi th projections in index-3 and index-1. The method takes advantage of the be tter 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 fr om 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 problem s. The variable time stepping is accomplished through the use of an integral o f motion, which in the case of conservative systems becomes the total energ y. The error introduced by the numerical integrator in the integral of moti on during consecutive time steps provides a good measure of the local integ ration error, and permits a simple and reliable strategy for varying the ti me step. It is also shown how the energy stored in the penalty system is su itable to measure the local integration error. Overall, the method is effic ient and powerful; it is suitable for stiff and non-stiff systems, robust f or all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velo cities and accelerations are satisfied during the simulation process. The m ethod 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 simulatio ns that illustrate the performance of the proposed method. Copyright (C) 19 99 John Wiley & Sons, Ltd.