Proceedings of ACMD06 A00688 OPTIMAL PRECONDITIONERS FOR THE SOLUTION OF CONSTRAINED MECHANICAL SYSTEMS IN INDEX-3 FORM Carlo L. Bottasso Daniel Dopico Dipartimento di Ingegneria Aerospaziale Escuela Polit´ ecnica Superior Politecnico di Milano University of La Coru˜ na Milano, 20156, ITALY Mendiz´abal s/n, 15403 Ferrol, SPAIN Email: carlo.bottasso@polimi.it Email: ddopico@udc.es Lorenzo Trainelli Dipartimento di Ingegneria Aerospaziale Politecnico di Milano Milano, 20156, ITALY Email: lorenzo.trainelli@polimi.it ABSTRACT Index-3 differential algebraic equations, such as those governing multibody system dynamics, often pose severe numerical difficulties for small time step sizes. These difficulties can be traced back to the effects of finite precision arithmetics. In this work a solution to this problem is found as a simple pre- conditioning for the governing equations that elim- inates the amplification of errors and the ill condi- tioning altogether. We develop a theoretical analy- sis, in particular for the case of the Newmark family of schemes, and show numerical experiments that confirm the predicted behavior. 1 INTRODUCTION Finite precision arithmetics and non-null conver- gence tolerances are at the root of well known nu- merical difficulties in the solution of high index dif- ferential algebraic equations (DAEs). Errors and perturbations pollute the numerical solution, re- sulting in disastrous effects for small values of the time step size. In fact, state variables and Lagrange multipliers are affected by increasing errors as the time step size decreases. Similarly the system Ja- cobian matrix becomes severely ill conditioned. Typically, proposed remedies in the multibody dynamics literature rely on the reduction of the in- dex from 3 to 2 or even 1. When this is done with- out appending additional constraints and multipli- ers, the well known drift of constraint violations is experienced. On the other hand, other approaches such as the GGL method (Gear et al. (1985)) or the more recent Embedded Projection Method (Borri et al. (2006)), avoid the drift effect by introduc- ing additional constraints and multipliers, yielding higher computational costs. Here we propose a different approach consisting in a preconditioning of the standard index-3 gov- erning equations. A proper scaling of equations and unknowns is found that completely eliminates the pollution problem, achieving perfect indepen- dence on the time step size, as observed in the case of ordinary differential equations. The recipe for the preconditioning is determined on the basis of a theoretical analysis of the pertur- bation problem in which we model the effects of finite precision arithmetics. We consider the case of the Newmark fam- ily of integration schemes, as representative of a larger class of commonly used time integrators (e.g. modified-α (HHT), generalized-α, etc.). Bot- tasso et al. (2006) reports in greater detail the proposed formulation, while Bottasso, Bauchau (2005a, 2006) consider the case of BDF methods. Finally, we show some numerical results obtained for a representative multibody problem that con- firm the predicted analysis. Copyright c 2006 by JSME