Variational Multisymplectic Formulations of Nonsmooth Continuum Mechanics

This paper develops the foundations of the multisymplectic formulation of nonsmooth continuum mechanics. It may be regarded as a PDE generalization of previous techniques that developed a variational approach to collision problems. These methods have already proved of value in computational mechanics, particularly in the development of asynchronous integrators and efficient collision methods. The present formulation also includes solid—pfluid interactions and material interfaces and, in addition, lays the groundwork for a treatment of shocks.

[1]  Richard Courant,et al.  Supersonic Flow And Shock Waves , 1948 .

[2]  Krishnamurty Karamcheti,et al.  Principles of ideal-fluid aerodynamics , 1966 .

[3]  A. Spencer.  Continuum Mechanics , 1967, Nature.

[4]  Mario Bunge,et al.  Classical Field Theories , 1967 .

[5]  Richard S. Palais,et al.  Foundations of global non-linear analysis , 1968 .

[6]  J. Marsden,et al.  Groups of diffeomorphisms and the motion of an incompressible fluid , 1970 .

[7]  Fluid dynamics and the calculus of horizontal variations , 1982 .

[8]  R. Abraham,et al.  Manifolds, Tensor Analysis, and Applications , 1983 .

[9]  T. Benjamin.  Impulse, Flow Force and Variational Principles , 1984 .

[10]  J. Moreau.  Une formulation du contact à frottement sec; application au calcul numérique , 1986 .

[11]  Jerrold E. Marsden,et al.  The Hamiltonian structure for dynamic free boundary problems , 1986 .

[12]  R. Abraham,et al.  Manifolds, tensor analysis, and applications: 2nd edition , 1988 .

[13]  Free boundaries and non-smooth solutions to some field equations: Variational characterization through the transport method , 1988 .

[14]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[15]  J. Marsden,et al.  Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs , 1998, math/9807080.

[16]  Steve Shkoller,et al.  Second-order multisymplectic field theory: A variational approach to second-order multisymplectic , 1999 .

[17]  E. A. Repetto,et al.  Finite element analysis of nonsmooth contact , 1999 .

[18]  J. Marsden,et al.  Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems , 2000 .

[19]  Jerrold E. Marsden,et al.  Variational Methods, Multisymplectic Geometry and Continuum Mechanics , 2001 .

[20]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[21]  J. Marsden,et al.  Time‐discretized variational formulation of non‐smooth frictional contact , 2002 .

[22]  J. Marsden,et al.  Asynchronous Variational Integrators , 2003 .

[23]  Jerrold E. Marsden,et al.  Nonsmooth Lagrangian Mechanics and Variational Collision Integrators , 2003, SIAM J. Appl. Dyn. Syst..