Force‐stepping integrators in Lagrangian mechanics

We present a class of integration schemes for Lagrangian mechanics, referred to as energy-stepping integrators, that are momentum and energy conserving, symplectic and convergent. In order to achieve these properties we replace the original potential energy by a piecewise constant, or terraced approximation at steps of uniform height. By taking steps of diminishing height, an approximating sequence of energies is generated. The trajectories of the resulting approximating Lagrangians can be characterized explicitly and consist of intervals of piecewise rectilinear motion. We show that the energy-stepping trajectories are symplectic, exactly conserve all the momentum maps of the original system and, subject to a transversality condition, converge to trajectories of the original system when the energy step is decreased to zero. These properties, the excellent long-term behavior of energy-stepping and its automatic time-step selection property, are born out by selected examples of application, including the dynamics of a frozen Argon cluster, the spinning of an elastic cube and the collision of two elastic spheres.

[1]  Sergey Korotov,et al.  On Nonobtuse Simplicial Partitions , 2009, SIAM Rev..

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

[3]  J. Marsden,et al.  Symplectic-energy-momentum preserving variational integrators , 1999 .

[4]  C. Scovel,et al.  A survey of open problems in symplectic integration , 1993 .

[5]  Jerrold E. Marsden,et al.  Hamiltonian Reduction by Stages , 2007 .

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

[7]  J. Izaguirre.  Longer Time Steps for Molecular Dynamics , 1999 .

[8]  L. Chua,et al.  A global representation of multidimensional piecewise-linear functions with linear partitions , 1978 .

[9]  Peter Wriggers,et al.  Computational Contact Mechanics , 2002 .

[10]  K. Bathe.  Finite Element Procedures , 1995 .

[11]  J. Candy,et al.  Symplectic integrators for long-term integrations in celestial mechanics , 1991 .

[12]  Kenneth R. Meyer,et al.  Introduction to Hamiltonian Dynamical Systems and the N-Body Problem , 1991 .

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

[14]  Donald Greenspan,et al.  Discrete mechanics—A general treatment , 1974 .

[15]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[16]  J. Pesonen.  Vibration–rotation kinetic energy operators: A geometric algebra approach , 2001 .

[17]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[18]  Leon O. Chua,et al.  Efficient Computer Algorithms for Piecewise-Linear Analysis of Resistive Nonlinear Networks , 1971 .

[19]  H. Saunders.  Book Reviews : NUMERICAL METHODS IN FINITE ELEMENT ANALYSIS K.-J. Bathe and E.L. Wilson Prentice-Hall, Inc, Englewood Cliffs, NJ , 1978 .

[20]  Lemma Poincar\'e for L_infty,loc - forms , 2007, 0712.1682.

[21]  Eduard Zehnder,et al.  Symplectic Invariants and Hamiltonian Dynamics , 1994 .

[22]  Michael Ortiz,et al.  On the Γ-Convergence of Discrete Dynamics and Variational Integrators , 2004, J. Nonlinear Sci..

[23]  Ernest S. Kuh,et al.  Solving nonlinear resistive networks using piecewise-linear analysis and simplicial subdivision , 1977 .

[24]  S. Reich.  Backward Error Analysis for Numerical Integrators , 1999 .

[25]  Mari Paz Calvo,et al.  The Development of Variable-Step Symplectic Integrators, with Application to the Two-Body Problem , 1993, SIAM J. Sci. Comput..

[26]  Jerrold E. Marsden,et al.  An Overview of Variational Integrators , 2004 .

[27]  J. Katzenelson.  An algorithm for solving nonlinear resistor networks , 1965 .

[28]  W. Goldsmith,et al.  Impact: the theory and physical behaviour of colliding solids. , 1960 .

[29]  Ted Belytschko,et al.  On the Unconditional Stability of an Implicit Algorithm for Nonlinear Structural Dynamics , 1975 .

[30]  J. Marsden,et al.  Variational time integrators , 2004 .

[31]  J. Marsden,et al.  Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators , 1988 .

[32]  M. Ortiz,et al.  Energy-stepping integrators in Lagrangian mechanics , 2009 .

[33]  E. S. Kuh,et al.  Piecewise-Linear Theory of Nonlinear Networks , 1972 .

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

[35]  Littlejohn,et al.  Internal or shape coordinates in the n-body problem. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[36]  H. Whitney.  Geometric Integration Theory , 1957 .

[37]  Harold W. Kuhn,et al.  Some Combinatorial Lemmas in Topology , 1960, IBM J. Res. Dev..

[38]  B. Berne,et al.  A Multiple-Time-Step Molecular Dynamics Algorithm for Macromolecules , 1994 .

[39]  J. Marsden,et al.  Gyration-radius dynamics in structural transitions of atomic clusters. , 2007, The Journal of chemical physics.

[40]  R. Littlejohn,et al.  Gauge fields in the separation of rotations and internal motions in the n-body problem , 1997 .

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

[42]  Francis Edward Su,et al.  Lower Bounds for Simplicial Covers and Triangulations of Cubes , 2005, Discret. Comput. Geom..

[43]  P. Julián,et al.  High-level canonical piecewise linear representation using a simplicial partition , 1999 .

[44]  R. Taylor,et al.  Lagrange constraints for transient finite element surface contact , 1991 .

[45]  William W. Hager,et al.  Updating the Inverse of a Matrix , 1989, SIAM Rev..

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

[47]  Robert D. Skeel,et al.  Dangers of multiple time step methods , 1993 .

[48]  M. Chien.  Searching for multiple solutions of nonlinear systems , 1979 .

[49]  R. Unbehauen,et al.  A generalization of canonical piecewise-linear functions , 1994 .

[50]  Ernst Hairer,et al.  The life-span of backward error analysis for numerical integrators , 1997 .