Variational Multisymplectic Formulations of Nonsmooth Continuum Mechanics

  • R. C. Fetecau
  • J. E. Marsden
  • M. West

Abstract

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.

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References

  1. Abraham, R., J. E. Marsden, and T. S. Ratiu [ 1988 ], Manifolds, Tensor Analysis and Applications, second edition, Applied Mathematical Sciences, vol. 75, Springer-Verlag, New York.CrossRefGoogle Scholar
  2. Benjamin, T. B. [ 1984 ], Impulse, flow force and variational principles, IMA J. Appl. Math. 32, 3–68.MathSciNetMATHCrossRefGoogle Scholar
  3. Courant, R. and K. O. Friedrichs [ 1948 ], Supersonic Flow and Shock Waves, volume 1 of Pure and applied mathematics. Interscience Publishers, Inc., New York.Google Scholar
  4. Ebin, D. G. and J. E. Marsden [ 1970 ], Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92, 102–163.MathSciNetMATHCrossRefGoogle Scholar
  5. Fetecau, R., J. E. Marsden, M. Ortiz, and M. West [ 2002 ], Nonsmooth Lagrangian mechanics and variational collision algorithms, (preprint).Google Scholar
  6. Gotay, M., J. Isenberg, and J. E. Marsden [ 1997 ], Momentum Maps and the Hamiltonian Structure of Classical Relativistic Field Theories I, available at http: //www. cds. caltech. edu/“mrsden /.Google Scholar
  7. Graff, K. F. [ 1991 ], Wave Motion in Elastic Solids. Dover, New York, reprinted edition.Google Scholar
  8. Jaunzemis, W. [ 1967 ], Continuum Mechanics. The Macmillan Company, New York.MATHGoogle Scholar
  9. Kane, C., E. A. Repetto, M. Ortiz, and J. E. Marsden. [1999], Finite element analysis of nonsmooth contact, Computer Meth. in Appl. Mech. and Eng. 180,1–26.Google Scholar
  10. Kane, C., J. E. Marsden, M. Ortiz, and M. West [ 2000 ], Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, Int. J. Num. Math. Eng. 49, 1295–1325.MathSciNetMATHCrossRefGoogle Scholar
  11. Karamcheti, K. [ 1966 ], Principles of Ideal-Fluid Aerodynamics. Robert E. Krieger Publishing Company, Inc., Florida, corrected reprint edition, 1980.Google Scholar
  12. Kouranbaeva, S. and S. Shkoller [ 2000 ], A variational approach to second-order mulitysmplectic field theory, J. of Geom. and Phys. 35, 333–366.MathSciNetMATHCrossRefGoogle Scholar
  13. Lew, A., J. E. Marsden, M. Ortiz, and M. West [ 2002 ], Asynchronous variational integrators, Arch. Rat. Mech. Anal. (to appear).Google Scholar
  14. Lewis, D., J. E. Marsden, R. Montgomery, and T. S. Ratiu [ 1986 ], The Hamilto- nian structure for dynamic free boundary problems, Physica D 18, 391–404.Google Scholar
  15. Marsden, J. E. and T. J. R. Hughes [ 1983 ], Mathematical Foundations of Elasticity. Prentice Hall. Reprinted by Dover Publications, NY, 1994.MATHGoogle Scholar
  16. Marsden, J. E., G. W. Patrick, and S. Shkoller [ 1998 ], Multisymplectic geometry, variational integrators and nonlinear PDEs, Comm. Math. Phys. 199, 351–395.Google Scholar
  17. Marsden, J. E., S. Pekarsky, S. Shkoller, and M. West [ 2001 ], Variational methods, multisymplectic geometry and continuum mechanics, J. Geom. and Physics 38, 253–284.Google Scholar
  18. Marsden, J. E. and T. S. Ratiu [1999], Introduction to Mechanics and Symmetry, volume 17 of Texts in Applied Mathematics, vol. 17; 1994, Second Edition, 1999. Springer-Verlag.Google Scholar
  19. Marsden, J. E. and M. West [ 2001 ], Discrete mechanics and variational integrators, Acta Numerica 10, 357–514.Google Scholar
  20. Moreau, J.-J. [ 1982 ], Fluid dynamics and the calculus of horizontal variations, Internat. J. Engrg. Sci. 20, 389–411.MathSciNetMATHCrossRefGoogle Scholar
  21. Moreau, J. [ 1986 ], Une formulation du contact à frottement sec; application au calcul numérique, C. R. Acad. Sci. Paris Sér. II 302, 799–801.Google Scholar
  22. Moreau, J.-J. [ 1988 ], Free boundaries and nonsmooth solutions to some field equations: variational characterization through the transport method. In Boundary control and boundary variations (Nice, 1986), volume 100 of Lecture Notes in Comput. Sci., pages 235–264. Springer, New York.Google Scholar
  23. Palais, R. S. [ 1968 ], Foundations of Global Non-Linear Analysis. Ben-jamin/Cummings Publishing Co., Reading, MA.Google Scholar
  24. Pandolfi, A., C. Kane, J. E. Marsden, and M. Ortiz [ 2002 ], Time-discretized variational formulation of nonsmooth frictional contact, Int. J. Num. Methods in Engineering 53, 1801–1829.Google Scholar
  25. Truesdell, C. A. and R. Toupin [ 1960 ], The Classical Field Theories. In Flügge, S., editor, Encyclopedia of Physics, volume III/1. Springer-Verlag OHG, Berlin.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 2003

Authors and Affiliations

  • R. C. Fetecau
  • J. E. Marsden
  • M. West

There are no affiliations available

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