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MULTISYMPLECTIC VARIATIONAL INTEGRATORS FOR NONSMOOTH LAGRANGIAN CONTINUUM MECHANICS

Part of: Numerical methods in calculus of variations and optimal control Numerical approximation and computational geometry Special kinds of problems Existence theories Equations of mathematical physics and other areas of application Elastic materials Symplectic geometry, contact geometry

Published online by Cambridge University Press:  08 July 2016

FRANÇOIS DEMOURES ,
FRANÇOIS GAY-BALMAZ  and
TUDOR S. RATIU
FRANÇOIS DEMOURES
Affiliation:
Department of Mathematics, Imperial College, London, UK; demoures@lmd.ens.fr LMD/IPSL, CNRS, École Normale Supérieure, PSL Research University, Université Paris-Saclay, Sorbonne Universités, Paris, France; francois.gay-balmaz@lmd.ens.fr
FRANÇOIS GAY-BALMAZ
Affiliation:
LMD/IPSL, CNRS, École Normale Supérieure, PSL Research University, Université Paris-Saclay, Sorbonne Universités, Paris, France; francois.gay-balmaz@lmd.ens.fr
TUDOR S. RATIU
Affiliation:
Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District, 200240 Shanghai, China Section de Mathématiques, École Polytechnique Fédérale de Lausanne, CH 1015 Lausanne, Switzerland; tudor.ratiu@epfl.ch

Abstract

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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.

MSC classification

Primary: 65D99: None of the above, but in this section 49M30: Other methods 74B20: Nonlinear elasticity 74M20: Impact
Secondary: 49J52: Nonsmooth analysis 53D20: Momentum maps; symplectic reduction 35Q74: PDEs in connection with mechanics of deformable solids
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e19
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

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