Unilateral Contact and Dry Friction in Finite Freedom Dynamics

In a recent paper [ACLM11], we prove an abstract existence result for the Coulomb friction problem in discrete time. This problem must be solved at each time step when performing a simulation of the dynamics of a mechanical system involving unilateral contact and Coulomb friction (expressed here at the level of velocities). In this paper, we only recall this result and the gist of its proof and then give an overview of its range of applicability to show the power of our existence criterion. By considering several mechanical systems (Painlevé's example, granular material on a plan or in a drum) and several particular cases (cases with no moving external objects, cases without friction), we demonstrate the broad range of use-cases to which the criterion can be applied by pure abstract reasoning, without any computations. We also show counter-examples where the criterion does not apply. We then turn to more complicated situations where the existence result cannot be used trivially, and discuss the computational methods that are available to check the criterion in practice using optimization software. It turns out that in suffices to solve a linear program (LP) when the problem is bi-dimensional, and a second order cone program (SOCP) when the problem is tri-dimensional. Key-words:Contact Mechanics, Coulomb friction, Painlevé example, Brouwer's theorem, second-order cone programming (SOCP).

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“…This condition can be equivalently written at the velocity level [Mor88]. By denoting the Jacobian of the constraints by…”

Section: Time-discretized Dynamics Of Rigid and Flexible Bodiesmentioning

confidence: 99%

Applications of an Existence Result for the Coulomb Friction Problem

Acary

Cadoux

2013

Recent Advances in Contact Mechanics

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“…© i (q)¸0, see [15]. This model can also be interpreted as the Karush-Kuhn-Tucker first order conditions of the following equivalent maximum dissipation principle [16][17]:…”

Section: The Formulation Of the Equations Of Motionmentioning

confidence: 99%

GPU-Based Parallel Computing for the Simulation of Complex Multibody Systems with Unilateral and Bilateral Constraints: An Overview

Tasora

Negrut²,

Anitescu

2010

Computational Methods in Applied Sciences

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This work reports on advances in large-scale multibody dynamics simulation facilitated by the use of the Graphics Processing Unit (GPU). A description of the GPU execution model along with its memory spaces is provided to illustrate its potential parallel scientific computing. The equations of motion associated with the dynamics of large system of rigid bodies are introduced and a solution method is presented. The solution method is designed to map well on the parallel hardware, which is demonstrated by an order of magnitude reductions in simulation time for large systems that concern the dynamics of granular material. One of the salient attributes of the solution method is its linear scaling with the dimension of the problem. This is due to efficient algorithms that handle in linear time both the collision detection and the solution of the nonlinear complementarity problem associated with the proposed approach. The current implementation supports the simulation of systems with more than one million bodies on commodity desktops. Efforts are under way to extend this number to hundreds of millions of bodies on small affordable clusters.

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“…For a more in-depth presentation of these concepts and equations which may have subtle implications, the interested reader should definitely refer to [9] or [2,3].…”

Section: Some Geometry For Systems With Non-permanent Contactsmentioning

confidence: 99%

“…But we have seen in section 3.1 that discontinuities of the velocities may occur when the coordinates of such systems are constrained to stay inside closed sets. These classical differential equations must therefore be turned into measure differential equations [9] …”

Section: Nonsmooth Lagrangian Dynamical Systemsmentioning

confidence: 99%