Skip to main content
SpringerLink
Log in

Time history interfacial conditions in multiscale computations of lattice oscillations

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

Abstract

We make a systematic study on the time history interfacial conditions in multiscale computations for crystalline solids in one space dimension. The exact interfacial condition is derived rigorously. For a class of approximate time history kernel functions, the error is estimated. In particular, a cut-off of the exact kernel function is a special case. Moreover, an effective numerical algorithm is proposed for the convolution.

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

Similar content being viewed by others

References

  1. Abramowitz M, Stegun IA (1984) Pocketbook of mathematical functions. Abridged edition of “Handbook of mathematical functions”. Thun: Verlag Harri Deutsch

  2. Adelman S, Doll J (1974) Generalized Langevin equation approach for atom/solid-surface scattering: collinear atom/harmonic chain model. J Chem Phys 61: 4242–4245

    Article  Google Scholar 

  3. Born M, Huang K (1954) Dynamical theory of crystal lattices. Clarendon Press, Oxford

    MATH  Google Scholar 

  4. Cai W, Koning M, Bulatov V, Yip S (2000) Minimizing boundary reflections in coupled-domain simulations. Phys Rev Lett 85: 3213–3216

    Article  Google Scholar 

  5. Doetsch G (1971) Handbuch der Laplace-Transformation. Birkhäuser, Basel

    Google Scholar 

  6. E W, Huang Z (2002) A dynamic atomistic-continuum method for simulation of crystalline materials. J Comput Phys 182: 234–261

    Article  MATH  MathSciNet  Google Scholar 

  7. Karpov E, Park H, Liu W (2007) A phonon heat bath approach for the atomistic and multiscale simulation of solids. Int J Numer Methods Eng 70: 351–378

    Article  MathSciNet  Google Scholar 

  8. Karpov E, Wagner G, Liu W (2005) A Green’s function approach to deriving nonreflecting boundary conditions in molecular dynamics simulations. Int J Numer Methods Eng 62: 1250–1262

    Article  MATH  Google Scholar 

  9. Liu W, Karpov E, Park H (2005) Nano mechanics and materials: theory, multiscale methods and applications. Wiley, New York

    Google Scholar 

  10. Olver FW (1997) Asymptotics and special functions. AKP Classics, Wellesley

    MATH  Google Scholar 

  11. Park H, Karpov E, Klein P, Liu W (2005) Three-dimensional bridging scale analysis of dynamic fracture. J Comput Phys 207: 588–609

    Article  MATH  Google Scholar 

  12. Park H, Karpov E, Liu W, Klein P (2005) The bridging scale for two-dimensional atomistic/continuum coupling. Philos Magaz 85: 79–113

    Article  Google Scholar 

  13. Qian D, Gondhalekar R (2004) A virtual atom cluster approach to the mechanics of nanostructures. Int J Multiscale Comput Eng 2: 277–289

    Article  Google Scholar 

  14. Qian D, Wagner G, Liu W (2004) A multiscale projection method for the analysis of carbon nanotubes. Comput Methods Appl Mech Eng 193: 1603–1632

    Article  MATH  Google Scholar 

  15. Ren S (2005) Electronic states in crystals of finite size: quantum confinement of Bloch waves. Springer, New York

    Google Scholar 

  16. Tang S, Hou T, Liu W (2006) A mathematical framework of the bridging scale method. Int J Numer Methods Eng 65: 1688–1713

    Article  MATH  MathSciNet  Google Scholar 

  17. Tang S, Hou T, Liu W (2006) A pseudo-spectral multiscale method: interfacial conditions and coarse grid equations. J Comput Phys 213: 57–85

    Article  MATH  MathSciNet  Google Scholar 

  18. Wagner G, Liu W (2003) Coupling of atomistic and continuum simulations using a bridging scale decomposition. J Comput Phys 190: 249–274

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Konstanz, P.O. Box D187, 78457, Konstanz, Germany

    Michael Dreher

  2. LTCS, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing, 100871, China

    Shaoqiang Tang

  3. Center for Applied Physics and Technology, Peking University, Beijing, 100871, China

    Shaoqiang Tang

Authors
  1. Michael Dreher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shaoqiang Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shaoqiang Tang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dreher, M., Tang, S. Time history interfacial conditions in multiscale computations of lattice oscillations. Comput Mech 41, 683–698 (2008). https://doi.org/10.1007/s00466-007-0224-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-007-0224-4

Keywords

Search

Navigation