EUROMECH Colloquium 524 - Program and Abstracts

Generalized-α time integration methods were originally designed for large scale problems in structural mechanics [1] but may be used as well for constrained systems of moderate dimension that are typical of multibody dynamics [2]. For more than a decade, the method has been used successfully in industrial simulation software that is based on a finite element approach in multibody dynamics [3]. In terms of numerical effort and stability, these Newmark type integrators are considered to be an interesting alternative to more classical ODE and DAE time integration methods in multibody numerics that are discussed, e.g., in [4]. For constrained systems, the Newmark like update formula qn+1 = qn + hvn + (0.5− β)han + βhan+1 , (1a) vn+1 = vn + (1− γ)han + γhan+1 (1b) for position coordinates q and velocity coordinates v and the generalized-α update scheme (1− αm)an+1 + αman = (1− αf )q̈n+1 + αf q̈n (1c) for the auxiliary vectors an are coupled to equilibrium equations Mq̈ = f −B>λ and constraints Φ = 0 at t = tn+1: M(qn+1)q̈n+1 = f(tn+1,qn+1,vn+1)−B(qn+1)λn+1 , (1d) 0 = Φ(qn+1) . (1e) In (1), the method parameters αf , αm, β, γ are chosen to satisfy the order condition γ = 1 2 + αf − αm, see [1], and the time step size h is for the moment considered to be constant for all time steps. Following the principles of classical mechanics, the nΦ constraints (1e) are coupled to the dynamical equations (1d) by constrained forces −B>(q)λ with B(q) := (∂Φ/∂q)(q) and Lagrange multipliers λ(t) ∈ RnΦ . All other forces and moments of the system are summarized in vector f = f(t,q,v). The mass matrix M(q) is assumed to be non-singular and symmetric, positive definite. With an appropriate scaling of the corrector equations [5], the fixed step size implementation of (1) works well for reasonable time step sizes h > 0. The convergence analysis for h→ 0 proves stability and second order convergence if the stability conditions αm < αf < 1/2 and β > 1/4 + (αf − αm)/2 are satisfied, see [6] and the extension to the generalized-α Lie group integrator of Brüls and Cardona [7] in [8]. Because of numerical damping, the generalized-α method (1) shows a favourable long-time behaviour. But in a short transient phase, the method may suffer from large errors that are damped out rapidly, see Fig. 1. ∗Email: [email protected]