Numerical aspects of the sweeping process

Abstract

The sweeping process, introduced some time ago by the author with motivation in plasticity theory, today remains an object of mathematical research. It is considered in this paper as the prototype of an evolution conditioned by inequality constraints. Since the governing differential requirements are only of order one with respect to time, this provides a simplified setting for analysing some numerical and theoretical features also present in unilateral dynamics. The latter is governed by differential inclusions of order two, for the numerical handling of which the existing literature proposes diverse strategies, briefly discussed. The paper is especially intended to offer an introduction to the numerical approach called 'contact dynamics'.

Conjugate gradient type algorithms for frictional multi-contact problems: applications to granular materials

“…Following the approach of Moreau [28], the main object of the computation is the velocity and a time stepping method is used as time integrator. The reader is referred to [9,30] for a discussion about the different integrators. For comparison with the previous case we consider the problem on a single time step; we have to predict the velocities distribution in a collection of rigid bodies at the end of this step.…”

“…A first study concerning the parallel treatment of the Gauss Seidel type method shows that the macroscopic features of the granular medium are preserved even if different deformation paths are followed when several processors are used [38]. More generally the Gauss Seidel type algorithms revealed to be efficient for granular problems [20,30], they converge even if the matrix is singular [12], but the optimization techniques are recommended in this case as long as the function stays lower bounded, which may be proved.…”

“…In a second section different whole processes are considered to take into account the variety of granular problems [6,23,34,35]. In these sections the CPG algorithm is essentially compared with the NLGS method generally associated with the NSCD approach [20,30]. All simulations are performed with the LMGC90 software [17] dedicated to multi-contact problems, very useful for granular materials.…”

“…A first study consisted in carrying out a parallel version of the non-linear Gauss Seidel algorithm associated with the Non-Smooth Contact Dynamics approach developed by Jean and Moreau [20,30]. This method is intrinsically sequential and we tried a simple multi-threading technique which consists in splitting the contact loop into several threads dedicated to different processors.…”

Models of non-smooth switches in electrical systems

“…This relation puts into duality the spaces of voltage measures and absolutely continuous virtual charge displacements. By using g adm i = w -i q from the mesh transformation (21) and by the axiomatic invariance of the virtual action under co-ordinate transformations, Equation (22) becomes…”