Section: New Results
Numerical analysis of multibody mechanical systems with constraints
This scientific theme concerns the numerical analysis of mechanical systems with bilateral and unilateral constraints, with or without friction [1]. They form a particular class of dynamical systems whose simulation requires the development of specific methods for analysis and dedicated simulators [57].
Numerical solvers for frictional contact problems.
Participants : Vincent Acary, Maurice Brémond, Paul Armand.
In [34], we review several formulations of the discrete frictional contact problem that arises in space and time discretized mechanical systems with unilateral contact and threedimensional Coulomb’s friction. Most of these formulations are well–known concepts in the optimization community, or more generally, in the mathematical programming community. To cite a few, the discrete frictional contact problem can be formulated as variational inequalities, generalized or semi–smooth equations, second–order cone complementarity problems, or as optimization problems such as quadratic programming problems over secondorder cones. Thanks to these multiple formulations, various numerical methods emerge naturally for solving the problem. We review the main numerical techniques that are wellknown in the literature and we also propose new applications of methods such as the fixed point and extragradient methods with selfadaptive step rules for variational inequalities or the proximal point algorithm for generalized equations. All these numerical techniques are compared over a large set of test examples using performance profiles. One of the main conclusion is that there is no universal solver. Nevertheless, we are able to give some hints to choose a solver with respect to the main characteristics of the set of tests.
Recently, new developments have been carried out on two new applications of wellknown numerical methods in Optimization:

Interior point methods With the visit of Paul Armand, Université de Limoges, we cosupervise a M2 internship, Maksym Shpakovych on the application of interior point methods for quadratic problem with secondorder cone constraints. The results are encouraging and a publication in computational mechanics is in progress.

Alternating Direction Method of Multipliers. In collaboration with Yoshihiro Kanno, University of Tokyo, the use of the Alternating Direction Method of Multipliers (ADMM) has been adapted to the discrete frictional contact problems. With the help of some acceleration and restart techniques for firstorder optimization methods and a residual balancing technique for adapting the proximal penalty parameter, the method proved to be efficient and robust on our test bench examples. A publication is also in preparation on this subject.
Modeling and numerical methods for frictional contact problems with rolling resistance
Participants : Vincent Acary, Franck Bourrier.
In [19], the Coulomb friction model is enriched to take into account the resistance to rolling, also known as rolling friction. Introducing the rolling friction cone, an extended Coulomb's cone and its dual, a formulation of the Coulomb friction with rolling resistance as a cone complementarity problem is shown to be equivalent to the standard formulation of the Coulomb friction with rolling resistance. Based on this complementarity formulation, the maximum dissipation principle and the bipotential function are derived. Several iterative numerical methods based on projected fixed point iterations for variational inequalities and blocksplitting techniques are given. The efficiency of these method strongly relies on the computation of the projection onto the rolling friction cone. In this article, an original closedform formulae for the projection on the rolling friction cone is derived. The abilities of the model and the numerical methods are illustrated on the examples of a single sphere sliding and rolling on a plane, and of the evolution of spheres piles under gravity.
Finite element modeling of cable structures
Participants : Vincent Acary, Charlélie Bertrand.
Standard finite element discretization for cable structures suffer from several drawbacks. The first one is related to the mechanical assumption that the cable can not support compression. Standard formulations do not take into account this assumption. The second drawback comes from the high stiffness of the cable model when we deal with large lengths with high Young modulus such as cable ropeways installations. In this context, standard finite element applications cannot avoid compressive solutions and have huge difficulties to converge. In a forthcoming paper, we propose to a formulation based on a piecewise linear modeling of the cable constitutive behavior where the elasticity in compression is canceled. Furthermore, a dimensional analysis help us to formulate a problem that is wellbalanced and the conditioning of the problem is diminished. The finite element discretization of this problem yields a robust method where convergence is observed with the number of elements and the nonlinear solver based on nonsmooth Newton strategy is converging up to tight tolerances. The convergence with the number of element allows one to refine the mesh as much as we want that will be of utmost importance for applications with contact and friction. Indeed, a fine discretization with respect to the whole length of the cable will be possible in the contact zone.
Wellposedness of the contact problem
We continue in [3] the analysis of the socalled contact problem for Lagrangian systems with bilateral and unilateral constraints, with setvalued Coulomb's friction. The problem that is analysed this time concerns sticking contacts (in both the normal and the tangential directions), i.e., does there exist a solution (possibly unique) to the contact problem (that takes the form of a complementarity problem) when all contacts are sticking ? An algorithm is proposed that allows in principle to compute solutions. We rely strongly on results of existence and uniqueness of solutions to variational inequality of the second kind, obtained in the team some years ago. Let us note also the erratum/addendum of the monograph [45] in [17], which is regularly updated.