Backward Differentiation Formula and Newmark-Type Index-2 and Index-1 Integration Schemes for Constrained Mechanical Systems

Abstract

Various methods for solving systems of differential-algebraic equations (DAE systems) are known from literature. Here, an alternative approach is suggested, which is based on a collocated constraints approach (CCA). The basic idea of the method is to introduce intermediate time points. The approach is rather general and may basically be applied for solving arbitrary DAE systems. Here, the approach is discussed for constrained mechanical systems of index-3. Application of the presented formulations for nonmechanical higher index DAE systems is also possible. We discuss index-2 formulations with one intermediate time point and index-1 implementations with two intermediate time points. The presented technique is principally independent of the time discretization method and may be applied in connection with different time integration schemes. Here, implementations are investigated for backward differentiation formula (BDF) and Newmark-type integrator schemes. A direct application of the presented approach yields a system of discretized equations with larger dimensions. The increased dimension of the discretized system of equations may be considered as the main drawback of the presented technique. The main advantage is that the approach may be used in a very straightforward manner for solving rather arbitrary multiphysical DAE systems with arbitrary index. Hence, the method might, for instance, be attractive for general purpose DAE integrators, since the approach is not tailored for special DAE systems (e.g., constrained mechanical systems). Numerical examples will demonstrate the straightforward application of the approach.