Higher Order Integration of Non-Smooth Dynamical Systems Using Parallel Computed Extrapolation Methods Based on Time-Stepping Schemes

摘要

Topic of this paper are integration methods for non-smooth multi-body systems (MBS). Non-smooth MBS are characterized by unilateral contacts, friction and frictional impacts leading to discrete jumps within the system's velocities e.g. due to closing rigid contacts. Therefore special numerical methods like time-stepping schemes are needed to integrate such systems. They are based on a constant time discretization of the system dynamics including the contact conditions. As a consequence jumps within the system velocities are allowed e.g. during impacts. ODE theories like error estimation and step size control are only applicable for the smooth part of the solution. In this paper step size selection and extrapolation methods are used to improve these integration schemes. Based on a step size control according to ODE theory for the smooth part some heuristic adjustments are made for the non-smooth part. For a better performance the time-stepping scheme is used as a base integration scheme for Richardson Extrapolation in order to increase the integration order from first to second or higher order during the smooth part. Examples with academic and industrial background like valve-train simulation are presented to show robustness and efficiency of the overall algorithm. As extrapolation methods themselves contain algorithmic parallelism they are ideal for the application of parallel computing techniques. All sub-step series can be calculated simultaneously by parallel computer architectures e.g. multicore CPUs to speed up numerical integration and to save computational time.