A multisymplectic integrator for elastodynamic frictionless impact problems

We present a structure preserving numerical algorithm for the collision of elastic bodies. Our integrator is derived from a discrete version of the field-theoretic (multisymplectic) variational description of nonsmooth Lagrangian continuum mechanics, combined with generalized Lagrange multipliers to handle inequality constraints. We test the resulting explicit integrator for the longitudinal impact of two elastic linear bar models, and for the collision of a nonlinear geometrically exact beam model with a rigid plane. Numerical simulations for various physical parameters are presented to illustrate the behavior and performance of our approach. (C) 2016 Elsevier B.V. All rights reserved.

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Alejandro Marcos Aragon, François Marie Alain Demoures, Mathieu Desbrun, François Gay-Balmaz, Tudor Ratiu
Elsevier, 2017
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https://infoscience.epfl.ch/record/226870?ln=fr

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Les mathématiques discrètes, parfois appelées mathématiques finies, sont l'étude des structures mathématiques fondamentalement discrètes, par opposition aux structures continues. Contrairement aux n

Multisymplectic Variational Integrators For Nonsmooth Lagrangian Continuum Mechanics

François Marie Alain Demoures, François Gay-Balmaz, Tudor Ratiu

This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In the presence of symmetry, a discrete version of the Noether theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

Cambridge Univ Press2016

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Geometric, variational discretization of continuum theories

Mathieu Desbrun

This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincare systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes. (C) 2011 Elsevier B.V. All rights reserved.

2011

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The existence of a critical length scale in regularised friction

Guillaume Anciaux, David Simon Kammer, Jean-François Molinari, Vladislav Yastrebov

We study a regularisation of Coulomb's friction law on the propagation of local slip at an interface between a deformable and a rigid solid. This regularisation, which was proposed based on experimental observations, smooths the effect of a sudden jump in the contact pressure over a characteristic length scale. We apply it in numerical simulations in order to analyse its influence on the behaviour of local slip. We first show that mesh convergence in dynamic simulations is achieved without any numerical damping in the bulk and draw a convergence map with respect to the characteristic length of the friction regularisation. By varying this length scale on the example of a given slip event, we observe that there is a critical length below which the friction regularisation does not affect anymore the propagation of the interface rupture. A spectral analysis of the regularisation on a periodic variation of Coulomb's friction is conducted to confirm the existence of this critical length. The results indicate that if the characteristic length of the friction regularisation is smaller than the critical length, a slip event behaves as if it was governed by Coulomb's law. We therefore propose that there is a domain of influence of the friction regularisation depending on its characteristic length and on the frequency content of the local slip event. A byproduct of the analysis is related to the existence of a physical length scale characterising a given frictional interface. We establish that the experimental determination of this interface property may be achieved by experimentally monitoring slip pulses whose frequency content is rich enough. (C) 2013 Elsevier Ltd. All rights reserved.

Elsevier2014

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