An Implicit Multistage Integration Method Including Projection for the Numerical Simulation of Constrained Multibody Systems

To be efficient, the simulation of multibody system dynamics requires fast and robust numerical algorithms for the time integration of the motion equations usually described by Differential Algebraic Equations (DAEs). Firstly, multistep schemes especially built up for second-order differential equations are developed. Some of them exhibit superior accuracy and stability properties than standard schemes for first-order equations. However, if unconditional stability is required, one must be satisfied with second-order accurate methods, like one-step schemes from the Newmark family.Multistage methods for which high accuracy is not contradictory with stringent stability requirements are then addressed. More precisely, a two-stage, third-order accurate Implicit Runge–Kutta (IRK) method which possesses the desirable properties of unconditional stability combined with high-frequency dissipation is proposed.Projection methods which correct the integrated estimates of positions, velocities and accelerations are suggested to keep the constraint equations satisfied during the numerical integration. The resulting time integration algorithm can be easily implemented in existing incremental/iterative codes. Numerical results indicate that this approach compares favourably with classical methods.