On the Optimal Scaling of Index Three Daes in Multibody Dynamics

Abstract

Errors and perturbations due to finite precision arithmetic pollute the numerical solution of high index differential algebraic equations. This pollution causes disastrous effects for small values of the time step size.

Various remedies have been so far offered in the multibody dynamics literature to this problem. All remedies point to the reduction of the index, which requires a rewriting of the equations of motion. This either increases the cost, since additional constraints and multipliers are introduced, or causes additional problems, like the drift of constraint violations.

In References [1], [2], we have shown that the pollution problem can be avoided for BDF schemes by a proper scaling of the index three problem. The procedure involves both a scaling of the equations and a scaling of the unknowns. This way, we were able to achieve perfect time step size independence in the sensitivity to perturbation, as in the case of well behaved ordinary differential equations.

In this paper we offer a new theoretical analysis of the perturbation problem, and we extend the scaling to the case of Newmark-type integration schemes. The predictions of the theory are confirmed by numerical experiments.