Skip to main content
SpringerLink
Log in

Semi-analytical solution for the generalized absorbing boundary condition in molecular dynamics simulations

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

We present a semi-analytical solution of a time-history kernel for the generalized absorbing boundary condition in molecular dynamics (MD) simulations. To facilitate the kernel derivation, the concept of virtual atoms in real space that can conform with an arbitrary boundary in an arbitrary lattice is adopted. The generalized Langevin equation is regularized using eigenvalue decomposition and, consequently, an analytical expression of an inverse Laplace transform is obtained. With construction of dynamical matrices in the virtual domain, a semi-analytical form of the time-history kernel functions for an arbitrary boundary in an arbitrary lattice can be found. The time-history kernel functions for different crystal lattices are derived to show the generality of the proposed method. Non-equilibrium MD simulations in a triangular lattice with and without the absorbing boundary condition are conducted to demonstrate the validity of the solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Moseler M, Nordiek J, Haberland H (1997) Reduction of the reflected pressure wave in the molecular-dynamics simulation of energetic particle-solid collisions. Phys Rev B 56(23):15439–15445. doi:10.1103/PhysRevB.56.15439

    Article  Google Scholar 

  2. Cai W, de Koning M, Bulatov VV, Yip S (2000) Minimizing boundary reflections in coupled-domain simulations. Phys Rev Lett 85(15):3213–3216. doi:10.1103/PhysRevLett.85.3213

    Article  Google Scholar 

  3. Jones RE, Kimmer CJ (2010) Efficient non-reflecting boundary condition constructed via optimization of damped layers. Phys Rev B 81(9):094301. doi:10.1103/PhysRevB.81.094301

    Article  Google Scholar 

  4. Tang S (2010) A two-way interfacial condition for lattice simulations. Adv Appl Math Mech 2(1):45–55. doi:10.4208/aamm.09-m0944

    MathSciNet  Google Scholar 

  5. Kantorovich L (2008) Generalized Langevin equation for solids. I. Rigorous derivation and main properties. Phys Rev B 78(9):094304. doi:10.1103/PhysRevB.78.094304

    Article  Google Scholar 

  6. Kantorovich L, Rompotis N (2008) Generalized Langevin equation for solids. II. Stochastic boundary conditions for nonequilibrium molecular dynamics simulations. Phys Rev B 78(9):094305. doi:10.1103/PhysRevB.78.094305

    Article  Google Scholar 

  7. Wang X, Tang S (2013) Matching boundary conditions for lattice dynamics. Int J Numer Methods Eng 93(12):1255–1285. doi:10.1002/nme.4426

    Article  MathSciNet  MATH  Google Scholar 

  8. Weinan E, Huang Z (2001) Matching conditions in atomistic-continuum modeling of materials. Phys Rev Lett 87(13):135501. doi:10.1103/PhysRevLett.87.135501

    Article  Google Scholar 

  9. Li X, Weinan E (2006) Variational boundary conditions for molecular dynamics simulations of solids at low temperature. Commun Comput Phys 1(1):135–175

    MATH  Google Scholar 

  10. Li X, Weinan E (2007) Variational boundary conditions for molecular dynamics simulations of crystalline solids at finite temperature: treatment of the thermal bath. Phys Rev B 76(10):104107-1–104107-22. doi:10.1103/PhysRevB.76.104107

  11. To AC, Li S (2005) Perfectly matched multiscale simulations. Phys Rev B 72(3):035414-1–035414-8. doi:10.1103/PhysRevB.72.035414

  12. Li S, Liu X, Agrawal A, To AC (2006) Perfectly matched multiscale simulations for discrete lattice systems: extension to multiple dimensions. Phys Rev B 74(4):045418-1–045418-14. doi:10.1103/PhysRevB.74.045418

  13. Guddati MN, Thirunavukkarasu S (2009) Phonon absorbing boundary conditions for molecular dynamics. J Comput Phys 228(21):8112–8134. doi:10.1016/j.jcp.2009.07.033

    Article  MathSciNet  MATH  Google Scholar 

  14. Adelman SA, Doll JD (1974) Generalized Langevin equation approach for atom/solid-surface scattering: Collinear atom/harmonic chain model. J Chem Phys 61(10):4242. doi:10.1063/1.1681723

    Article  Google Scholar 

  15. Doll JD, Myers LE, Adelman SA (1975) Generalized Langevin equation approach for atom/solid-surface scattering: Inelastic studies. J Chem Phys 63(11):4908. doi:10.1063/1.431234

    Article  Google Scholar 

  16. Adelman SA, Doll JD (1976) Generalized Langevin equation approach for atom/solid-surface scattering: General formulation for classical scattering off harmonic solids. J Chem Phys 64(6):2375. doi:10.1063/1.432526

    Article  Google Scholar 

  17. Wagner GJ, Liu WK (2003) Coupling of atomistic and continuum simulations using a bridging scale decomposition. J Comput Phys 190(1):249–274. doi:10.1016/s0021-9991(03)00273-0

    Article  MATH  Google Scholar 

  18. Karpov EG, Wagner GJ, Liu WK (2005) A Green’s function approach to deriving non-reflecting boundary conditions in molecular dynamics simulations. Int J Numer Methods Eng 62(9):1250–1262. doi:10.1002/nme.1234

    Article  MATH  Google Scholar 

  19. Wagner GJ, Karpov EG, Liu WK (2004) Molecular dynamics boundary conditions for regular crystal lattices. Comput Methods Appl Mech Eng 193(17–20):1579–1601. doi:10.1016/j.cma.2003.12.012

    Article  MathSciNet  MATH  Google Scholar 

  20. Park HS, Karpov EG, Liu WK, Klein PA (2005) The bridging scale for two-dimensional atomistic/continuum coupling. Philos Mag 85(1):79–113. doi:10.1080/14786430412331300163

    Article  Google Scholar 

  21. Karpov EG, Yu H, Park HS, Liu WK, Wang QJ, Qian D (2006) Multiscale boundary conditions in crystalline solids: Theory and application to nanoindentation. Int J Solids Struct 43(21):6359–6379. doi:10.1016/j.ijsolstr.2005.10.003

    Article  MATH  Google Scholar 

  22. Tang S, Hou TY, Liu WK (2006) A pseudo-spectral multiscale method: Interfacial conditions and coarse grid equations. J Comput Phys 213(1):57–85. doi:10.1016/j.jcp.2005.08.001

  23. Liu WK, Karpov EG, Park HS (2006) Nano mechanics and materials : theory, multiscale methods and applications. Wiley, Hoboken. ISBN: 9780470018514

  24. Medyanik SN, Karpov EG, Liu WK (2006) Domain reduction method for atomistic simulations. J Comput Phys 218(2):836–859. doi:10.1016/j.jcp.2006.03.008

    Article  MathSciNet  MATH  Google Scholar 

  25. Pang G, Tang S (2011) Time history kernel functions for square lattice. Comput Mech 48(6):699–711. doi:10.1007/s00466-011-0615-4

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was supported by the Ministry of Science and Technology, Taiwan, under grant no. 105-2221-E-002-025-MY3 and by National Taiwan University under grant no. 103R891805 (The NTU Excellence in Research Program). We are grateful to the National Center for High-Performance Computing and National Taiwan University for providing the computational resources.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, National Taiwan University, Taipei, 10617, Taiwan

    Chung-Shuo Lee, Yan-Yu Chen, Chi-Hua Yu, Yu-Chuan Hsu & Chuin-Shan Chen

Authors
  1. Chung-Shuo Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan-Yu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chi-Hua Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yu-Chuan Hsu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Chuin-Shan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chuin-Shan Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lee, CS., Chen, YY., Yu, CH. et al. Semi-analytical solution for the generalized absorbing boundary condition in molecular dynamics simulations. Comput Mech 60, 23–37 (2017). https://doi.org/10.1007/s00466-017-1389-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-017-1389-0

Keywords

Search

Navigation