M. Borri, L. Trainelli, A. Croce

The Embedded Projection Method:

A General Index Reduction Procedure for Constrained System Dynamics

Computer Methods in Applied Mechanics and Engineering, submitted (2004).

Abstract

The dynamics of constrained systems with complex topology, as encountered, for example, in multibody system dynamics, control system dynamics, chemical system dynamics, typically requires the solution of differential-algebraic equations (DAEs). These equations impose severe robustness and efficiency requirements upon numerical integration tools, and the reliable solution of high-index DAEs is still an unsolved problem under many respects, as considerable numerical difficulties arise from the intimate coupling between the algebraic and the differential parts of the problem. The `Embedded Projection Method' (EPM) is designed to eliminate most of the troubles related to DAE solving, while preserving generality. By this approach, the algebraic and differential parts of the problem are completely uncoupled so that they can be solved separately. A modified state is introduced as a fully unconstrained variable, and an ordinary differential equation (ODE) is derived for it. This equation, complemented by additional variables defined through algebraic equations, can be solved by a suitable ODE integration algorithm, by-passing the need for a specialized DAE solver. This method is strictly equivalent to a consistent index reduction process from arbitrarily high values to one. The EPM numerical solution enjoys higher degrees of accuracy and stability with respect to conventional methodologies, resulting in enhanced overall algorithmic reliability and robustness, that balance the higher complexity of the procedure. The main features and properties of the EPM are illustrated with the aid of some representative applications featuring both holonomic and non-holonomic mechanical systems.