r-Factor Strategies for the Augmented Lagrangian Approach in Multi-Body Contact Mechanics

Multi-body system theory including unilateral constraints is now well established by means of setvalued force laws in connection with measure equations of motion. The crucial point consists in the numerical solution of such equations, especially when dealing with large systems as they often appear in practical problems of industrial relevance. Therefore the improvement of numerical algorithms is a focus of ongoing research.

In the meantime there are different approaches and algorithms in order to formulate and compute unilateral constrained mechanical systems. Besides (N)LCP-formulations the Augmented Lagrangian approach [

1

] becomes more and more popular in contact mechanics. Within this approach the equations of motion are augmented by projection equations representing the physical constraints. The overall set of non-smooth, nonlinear equations can be solved by a root-finding algorithm, e.g. a fixed-point iteration scheme. The projection equations depend on a non-negative auxiliar parameter

r

. Though this parameter

r

is arbitrary from the mathematical point of view, it plays a crucial role in view of the root- finding method. In particular, the problem of finding an optimal

r

-factor turns out to be a constrained optimization problem: on the one hand the parameter r may be bounded to ensure the convergence of the algorithm, on the other hand an appropriate choice improves the rate of convergence.

In the present paper two different

r

-factor strategies are presented considering a fixed-point iteration scheme in order to find the root of the Augmented Lagrangian. The first strategy proposes one global

r

-factor for all constraint equations. The second strategy considers local

r

-factors, i.e. a different parameter for each constraint. In both cases the conditions for convergence are given and an optimal choice of

r

is proposed. The paper discusses the treatment of planar and spatial contacts as well as systems that are statically indeterminate, where a unique solution for the constraint forces does not exist.

The presented

r

-factor strategies are applied to several non-smooth systems including a push belt CVT [

2

]. This large industrial problem is characterized by a hybrid multi-body model with a large number of unilateral and bilateral constraints.