Résolution du contact frottant entre objets déformables en temps réel et avec retour haptique. (Friction contact resolution between deformable objects in interactive simulation with haptic feedback)

This thesis deals with the solving of multiple friction contact problems between deformable bodies in the specific area of interactive simulation with force feedback. The general context of this work is the training of medical-surgical gestures via an haptic device. In this type of context, it is often necessary to take into account the global deformations of the organs and the local contact forces due to interactions soft tissues/ soft tissues or to interactions tools/soft tissues. These interactions are the consequences of the friction contact force (dry or viscous). We use the Finite Element Method based on the mechanics theory of continuous media which is certainly the most rigorous method to model linear or non linear behaviours of soft tissues. We also consider that the objects are deformed following the quasi-static equilibrium hypothesis because the operational gestures are slow and the handled objects have low masses. In order to solve these quasi-static equilibrium equations whose unknowns are the forces of contact and the elastic displacements, we use the incremental method of NewtonRaphson. We present various formulations, mainly based on the augmented lagrangian method (pseudo-potentials, bi-potential) to model the contacts laws (Signorini and friction of Coulomb) between deformable objects within a general framework (dynamic, large deformations, explicit or implicit numerical resolution). In order to disconnect the computation of the contact forces from the computation of the elastic displacements, we use the flexibility method which gives the possibility to consider a contact solver independently of the displacement solver. The contact solver is based on «Gauss-Seidel» and «Uzawa» techniques and allows a robust control of the contact forces relative to a given numerical precision. A version of this solver has been realised in which it is not necessary to know the models used for the deformations. A real-time simulator with haptic device has been produced and allowed to test the concepts introduced in this thesis.