Existence of a Fixed Point of a Nonsmooth Function Arising in Numerical Mechanics

Florent Cadoux (1) , Jérôme Malick (1)

1 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems

A recent work (Acary et al. 2010) introduces a formulation as a nonsmooth fixed-point problem of a basic problem in numerical mechanics (namely the dynamical Coulomb friction problem in finite dimension with discretized time). Using this new formulation, the existence of a solution to the problem and its numerical resolution are then guaranteed under a strong assumption on the data of this problem. In this paper, we show that the fixed point problem admits solution under a natural, weaker assumption. This existence proof uses a perturbation argument combined with continuity properties of a set-valued mapping associated with the constraints of the problem.

Mots clés: Nonsmooth analysis; Numerical mechanics; Second-order cone programming; Sensitivity analysis; Fixed point

Domaines: Optimisation et contrôle [math.OC]

Florent Cadoux, Jérôme Malick. Existence of a Fixed Point of a Nonsmooth Function Arising in Numerical Mechanics. Set-Valued and Variational Analysis, 2010, 18 (3-4), pp.337-348. ⟨10.1007/s11228-010-0149-5⟩. ⟨hal-00651602⟩