A Continuation Problem for Computing Solutions of Discretised Evolution Problems with Application to Plane Quasi-Static Contact Problems with Friction

Tomas Ligursky (1) , Yves Renard (2, 3, 4)

1 DMAaAM PrF UP - Department of Mathematical Analysis and Applications of Mathematics
2 MMCS - Modélisation mathématique, calcul scientifique
3 ICJ - Institut Camille Jordan
4 LaMCoS - Laboratoire de Mécanique des Contacts et des Structures [Villeurbanne]

A continuation problem for finding successive solutions of discretised abstract first-order evolution problems is proposed and a general piecewise $C^{1}$ continuation problem is studied. A condition ensuring local existence and uniqueness of its solution curves is given. An analogy of the first-order system of smooth problems is derived and results of existence and uniqueness of its solutions are stated. Possibility of continuation of a solution curve along directions solving the first-order system is discussed. A technique for numerical continuation of the solution curves is developed. Furthermore, an application of the abstract continuation problem is presented for plane quasi-static contact problems with friction. Various formulations of the first-order system are derived for this case so that the analysis from the abstract frame can be developed and supplemented. Finally, the proposed numerical continuation is tested.

Mots clés: first-order system; Coulomb friction; continuation; piecewise-smooth system; predictor-corrector; contact problem

Domaines: Analyse numérique [math.NA]; Mécanique des solides [physics.class-ph]; Mécanique des solides [physics.class-ph]

Tomas Ligursky, Yves Renard. A Continuation Problem for Computing Solutions of Discretised Evolution Problems with Application to Plane Quasi-Static Contact Problems with Friction. Computer Methods in Applied Mechanics and Engineering, 2014, 280, pp.222-262. ⟨10.1016/j.cma.2014.07.011⟩. ⟨hal-00877716v2⟩