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Energy-momentum conserving integration schemes for molecular dynamics

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Abstract

We address the formulation and analysis of energy and momentum conserving time integration schemes in the context of particle dynamics, and in particular atomic systems. The article identifies three critical aspects of these models that demand a careful analysis when discretized: first, the treatment of periodic boundary conditions; second, the formulation of approximations of systems with three-body interaction forces; third, their extension to atomic systems with functional potentials. These issues, and in particular their interplay with Energy-Momentum integrators, are studied in detail. Novel expressions for these time integration schemes are proposed and numerical examples are given to illustrate their performance.

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  1. https://lammps.sandia.gov.

References

  1. Allen MP, Tildesley DJ (1989) Computer simulation of liquids. Clarendon Press

  2. Baskes M (1999) Many-body effects in fcc metals: a Lennard-Jones embedded-atom potential. Phys Rev Lett 83(13):2592

    Article  Google Scholar 

  3. Baskes MI (1992) Modified embedded-atom potentials for cubic materials and impurities. Phys Rev B 46(5):2727

    Article  Google Scholar 

  4. Betsch P, Uhlar S (2007) Energy-momentum conserving integration of multibody dynamics. Multibody Syst Dyn 17:243–249

    Article  MathSciNet  Google Scholar 

  5. Brooks BR, Bruccoleri RE, Olafson BD, States DJ, Sa Swaminathan, Karplus M (1983) Charmm: a program for macromolecular energy, minimization, and dynamics calculations. J Comput Chem 4(2):187–217

    Article  Google Scholar 

  6. Campello EMB, Pimenta PM, Wriggers P (2011) An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 2: shells. Comput Mech 48(2):195–211

  7. Celledoni E, Grimm V, McLachlan RI, McLaren DI, O’Neale D, Owren B, Quispel GRW (2012) Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method. J Comput Phys 231(20):6770–6789

  8. Daw MS (1989) Model of metallic cohesion: the embedded-atom method. Phys Rev B 39(11):7441

    Article  Google Scholar 

  9. Daw MS, Baskes MI (1984) Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals. Phys Rev B 29(12):6443

    Article  Google Scholar 

  10. Daw MS, Foiles SM, Baskes MI (1993) The embedded-atom method: a review of theory and applications. Mater Sci Rep 9(7–8):251–310

    Article  Google Scholar 

  11. García Orden JC, Goicolea JM (2000) Conserving properties in constrained dynamics of flexible multibody systems. Multibody Syst Dyn 4:225–244

    Article  MathSciNet  Google Scholar 

  12. Goldstein H, Poole C, Safko J (2002) Classical mechanics

  13. Gonzalez O (1996) Design and analysis of conserving integrators for nonlinear hamiltonian systems with symmetry. Ph.D. thesis, Stanford University Stanford, CA

  14. Gonzalez O (2000) Exact energy-momentum conserving algorithms for general models in nonlinear elasticity. Comput Methods Appl Mech Eng 190:1763–1783

    Article  MathSciNet  Google Scholar 

  15. Griebel M, Zumbusch G, Knapek S (2007) Numerical simulation in molecular dynamics. Numerics, algorithms, parallelization. Texts in Computational Science and Engineering. Springer, Berlin

  16. Haile J, Johnston I, Mallinckrodt AJ, McKay S (1993) Molecular dynamics simulation: elementary methods. Comput Phys 7(6):625–625

    Article  Google Scholar 

  17. Hairer E, Lubich C, Wanner G (2006) Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, vol 31. Springer, Berlin

  18. Kim S (2014) Issues on the choice of a proper time step in molecular dynamics. Phys Procedia 53:60–62

    Article  Google Scholar 

  19. Kreher DL, Stinson DR (1999) Combinatorial algorithms. Generation, enumeration, and search. CRC Press, Boca Raton

  20. Kuzkin VA (2015) On angular momentum balance for particle systems with periodic boundary conditions. ZAMM - Z Angew Math Mech 95(11):1290–1295

    Article  MathSciNet  Google Scholar 

  21. LaBudde RA, Greenspan D (1975) Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion. Numer Math 25(4):323–346

    Article  MathSciNet  Google Scholar 

  22. LaBudde RA, Greenspan D (1976) Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion. Numer Math 26(1):1–16

    Article  MathSciNet  Google Scholar 

  23. Laursen TA, Meng XN (2001) A new solution procedure for application of energy-conserving algorithms to general constitutive models in nonlinear elastodynamics. Comput Methods Appl Mech Eng 190:6300–6309

    Article  MathSciNet  Google Scholar 

  24. Leimkuhler B, Reich S (2004) Simulating hamiltonian dynamics, vol 14. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  25. Lennard-Jones JE (1924) On the determination of molecular fields. II. From the equation of state of gas. Proc R Soc A 106:463–477

    Google Scholar 

  26. McLachlan RI, Quispel GRW, Robidoux N (1999) Geometric integration using discrete gradients. Philos Trans R Soc Lond Ser A 357:1021–1045

    Article  MathSciNet  Google Scholar 

  27. Pimenta PM, Campello EMB, Wriggers P (2008) An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: Rods. Comput Mech 42(5):715–732

  28. Plimpton S (1993) Fast parallel algorithms for short-range molecular dynamics. Tech. rep., Sandia National Labs., Albuquerque, NM (United States)

  29. Rapaport DC (2004) The art of molecular dynamics simulation. Cambridge University Press, Cambridge

    Book  Google Scholar 

  30. Romero I (2012) An analysis of the stress formula for energy-momentum methods in nonlinear elastodynamics. Comput Mech 50(5):603–610

    Article  MathSciNet  Google Scholar 

  31. Romero I, Armero F (2002) An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum conserving scheme in dynamics. Int J Numer Methods Eng 54(12):1683–1716

    Article  MathSciNet  Google Scholar 

  32. Saluena C, Avalos JB (2014) Molecular dynamics algorithm enforcing energy conservation for microcanonical simulations. Phys Rev E 89(5):053314

    Article  Google Scholar 

  33. Simo J, Tarnow N (1992) The discrete energy-momentum method. conserving algorithms for nonlinear elastodynamics. ZAMM - Z Angew Math Mech 43(5):757–792

  34. Simo JC, Tarnow N (1994) A new energy and momentum conserving algorithm for the non-linear dynamics of shells. Int J Numer Methods Eng 37(15):2527–2549

    Article  MathSciNet  Google Scholar 

  35. Simo JC, Tarnow N, Doblaré M (1995) Non-linear dynamics of three-dimensional rods: exact energy and momentum conserving algorithms. Int J Numer Methods Eng 38(9):1431–1473

    Article  MathSciNet  Google Scholar 

  36. Simo JC, Tarnow N, Wong KK (1992) Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Comput Methods Appl Mech Eng 100(1):63–116

    Article  MathSciNet  Google Scholar 

  37. Skeel RD, Hardy DJ, Phillips JC (2007) Correcting mesh-based force calculations to conserve both energy and momentum in molecular dynamics simulations. J Comput Phys 225(1):1–5

    Article  MathSciNet  Google Scholar 

  38. Srinivasan S, Baskes M (2004) On the Lennard–Jones EAM potential. Proc R Soc Lond A Mat 460(2046):1649–1672

    Article  Google Scholar 

  39. Stillinger FH, Weber TA (1985) Computer simulation of local order in condensed phases of silicon. Phys Rev B 31(8):5262

    Article  Google Scholar 

  40. Tadmor EB, Miller RE (2011) Modeling materials: continuum, atomistic and multiscale techniques. Cambridge University Press, Cambridge

    Book  Google Scholar 

  41. Toxvaerd S (1983) Energy conservation in molecular dynamics. J Comput Phys 52(1):214–216

    Article  Google Scholar 

  42. Tuckerman M (2010) Statistical mechanics: theory and molecular simulation. OUP Oxford

  43. Zhong HG, Crisfield MA (1998) An energy-conserving co-rotational procedure for the dynamics of shell structures. Eng Comput 15(5):552–576

    Article  Google Scholar 

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Acknowledgements

Support for M.S. was provided by the Deutsche Forschungsgemeinschaft (DFG) under Grant BE 2285/13-1 and the Research Travel Grant of the Karlsruhe House of Young Scientists (KYHS). This support is gratefully acknowledged.

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Authors and Affiliations

  1. Institut für Mechanik, Karlsruher Institut für Technologie, Gebäude 10.30, Otto-Ammann-Platz 9, 76131, Karlsruhe, Germany

    Mark Schiebl

  2. IMDEA Materials Institute, Eric Kandel 2, Tecnogetafe, 28906, Madrid, Spain

    Ignacio Romero

  3. Universidad Politécnica de Madrid, José Gutiérrez Abascal, 2, 28006, Madrid, Spain

    Ignacio Romero

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  1. Mark Schiebl
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Correspondence to Ignacio Romero.

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Schiebl, M., Romero, I. Energy-momentum conserving integration schemes for molecular dynamics. Comput Mech 67, 915–935 (2021). https://doi.org/10.1007/s00466-020-01971-6

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  • DOI: https://doi.org/10.1007/s00466-020-01971-6

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