Towards large time simulation reduction for large scale mechanical systems: A robust control approach A. Gonzales, J. Lauber, T.M. Guerra, F. Massa, T. Tison Université de Valenciennes et du Hainaut Cambrésis, LAMIH laboratory (UMR CNRS 8201), Le Mont Houy, 59313 Valenciennes Cedex 9, France (e-mail: {qqq;guerra}@univ-valenciennes.fr) Abstract: In the context of mechanical engineering, one of the most common contact problems consisting of balancing two elastic structures under static loads. This problem has been reported to be actually very high costly in terms of time simulation, due to the large number of degree of freedom obtained after applying finite element methods. This note provides a novel approach based on robust control theory to tackle this problem that, to the best author’s knowledge, has been not exploited in previous literature. The main novelty of our proposal is therefore to have put the contact problem in terms of automatic control. Besides, a promising time simulation reduction with respect to existing solutions to this problem may be achieved since, as seen in the paper, the complexity of controller is independent of the number of degree of freedom. Keywords: Mechanical contact problem, Large scale system, Robust control theory, Linear Matrix Inequality, Nonlinear systems 1. INTRODUCTION In mechanical engineering, one of the most common contact problem deals with the unilateral contact between two elastic structures under external loads. The motivation of this work arises from the problem of finding the value of the contact load such that both structures are well balanced, in other words, to cancel the value of gap penetration in the contact surface. To deal with this problem, many solutions have already been proposed by the research community in mechanics and applied mathematics. Some of them are implemented in industrial Finite Elements (FE) software. Classical methods, such as the popular penalty approximation (Arnold, 1982) and «mixed » or « trial-and-error » methods appear, suitable for many applications. But in this kind of method, the contact boundary conditions and eventual friction laws are not satisfied accurately and it is tricky for the users to choose appropriate penalty factors (eg, the regularized contact law). They may fail for stiff problems because of unpleasant numerical oscillations among contact statuses. With the Lagrange multiplier method (Carpenter et al. 1991, Hild and Renard et al. 2010), the contact conditions are imposed by introducing new unknowns, the Lagrange multipliers, and new equations in the problem: the contact/friction constraints. The number of Lagrange multipliers is essentially determined by the number of contacting entity pairs and can considerably increase the size of the system to solve. This translates into high computational solution costs. Although numerical implementations of these methods in industrial software use condensation or sub-structuring techniques to reduce the size of the system, numerical problem or high computational costs may still arise due to the reduction scheme itself. Following these pioneer and popular methods, numerous contributions have been made to overcome these drawbacks. The augmented Lagrangian method appeared to deal with constrained minimization problems. Since friction problems are not minimum problems, the formulation needs to be extended. Alart and Curnier 1991, Simo and Laursen 1992 and De Saxce and Feng 1991 have obtained some extensions in mutually independent works. The third method differs from the others in the fact that it is based on two ingredients both proposed by the authors, a theory called ISM (Implicit Standard Materials) and a bi-potential method, in which the two minimum principles or the two variational inequalities are replaced by only one. Nevertheless, one cannot say that these complicated tools can be used as black-boxes. A minimum of theoretical background allows the effective application of the available computational tools. The effective application of finite element contact solutions needs a high degree of experience since the general robustness and stability cannot be guaranteed. For this reason the development of more efficient, fast and stable finite element discretization methods for the contact problem is still a hot topic, especially due to the fact that engineering applications are becoming more and more complex. Furthermore, if uncertainties are considered, numerous simulations (Monte Carlo simulation, for example) are required (Goldena et al. 2010). It is desirable that they can be performed in a time compatible with the design process and, as possible, without human intervention. Although in the past decade, substantial progresses have been made in the analysis of contact problems using finite element procedures: reviews may be consulted for an extensive list of references (Zhong and Macherle 1992, Klarbring, 1993, Wriggers 1995, Raous 1995) it is well known that this kind of solutions for this problem are still time consuming because