Adaptive Implicit Runge–Kutta Methods for Stabilized Index‐One Differential Algebraic Equations in Constrained Flexible Multibody Systems

ABSTRACT

This communication discusses the application of implicit Runge–Kutta methods to constrained flexible multibody systems. Since the resulting systems of differential algebraic equations (DAEs) are typically considered to be very stiff, the family of stiffly accurate ‐ and ‐stable Radau IIA collocation methods is investigated. When naively applied to an index three DAE an order reduction phenomenon appears for the velocities and constraint forces. A remedy to that is given by so‐called index reduction techniques that at the same time lead to constraint drift that has to be compensated. Therefore, stabilized formulations are used that reduce the index and at the same time preserve the original constraints such that no drift occurs. The well‐known stabilized index‐two formulation, proposed by Gear, Gupta, and Leimkuhler, still requires a modified solution technique due to the obtained order reduction in the constraint forces. In this investigation, it is shown that a stabilized index‐one formulation can be used without any modification of the resulting nonlinear system and the applied step‐size controller. The relation of this approach to the scaled error norms commonly used for higher‐index formulations is discussed, and it is extended to unstructured differential algebraic equations of index one, which are beyond the scope of integrators that require a constant mass matrix. Numerical examples with closed‐form solutions confirm that the classical order of convergence can be obtained when applied to constrained mechanical systems. Work–precision diagrams and implementation‐independent cost measures compare the implementation with the established solvers RADAU5 and IDA.