Interaction and multiscale mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Multiscale simulation based on kriging based finite element method

  • Received : 2009.08.17
  • Accepted : 2009.11.15
  • Published : 2009.12.25
⟨ Previous Next ⟩

Abstract

A new seamless multiscale simulation was developed for coupling the continuum model with its molecular dynamics. Kriging-based Finite Element Method (K-FEM) is employed to model the continuum base of the entire domain, while the molecular dynamics (MD) is confined in a localized domain of interest. In the coupling zone, where the MD domain overlaps the continuum model, the overall Hamiltonian is postulated by contributions from the continuum and the molecular overlays, based on a quartic spline scaling parameter. The displacement compatibility in this coupling zone is then enforced by the Lagrange multiplier technique. A multiple-time-step velocity Verlet algorithm is adopted for its time integration. The validation of the present method is reported through numerical tests of one dimensional atomic lattice. The results reveal that at the continuum/MD interface, the commonly reported spurious waves in the literature are effectively eliminated in this study. In addition, the smoothness of the transition from MD to the continuum can be significantly improved by either increasing the size of the coupling zone or expanding the nodal domain of influence associated with K-FEM.

Keywords

References

  1. Abraham, F., Broughton, J., Bernstein, N. and Kaxiras, E. (1998), "Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture", EPL-Europhys. Lett., 44, 783-787. https://doi.org/10.1209/epl/i1998-00536-9
  2. Adelman, S.A. and Doll, J.D. (1976), "Generalized langevin theory for atom/solid-surface scattering: General formulation for classical scattering off harmonic solids", J. Chem. Phys., 64, 2375-2388. https://doi.org/10.1063/1.432526
  3. Arroyo, M. and Belytschko, T. (2002), "An atomistic-based finite deformation membrane for single layer crystalline films", J. Mech. Phys. Solids, 50, 1941-1977. https://doi.org/10.1016/S0022-5096(02)00002-9
  4. Arroyo, M. and Belytschko, T. (2003), "A finite deformation membrane based on inter-atomic potentials for the transverse mechanics of nanotubes", Mech. Mater., 35, 193-215. https://doi.org/10.1016/S0167-6636(02)00270-3
  5. Arroyo, M. and Belytschko, T. (2004), "Finite element methods for the non-linear mechanics of crystalline sheets and nanotubes", Int. J. Numer. Meth. Eng., 59, 419-456. https://doi.org/10.1002/nme.944
  6. Belytschko, T., Lu, Y.Y. and Gu, L. (1994), "Element-free Galerkin methods", Int. J. Numer. Meth. Eng., 37, 229-256. https://doi.org/10.1002/nme.1620370205
  7. Belytschko, T. and Xiao, S.P. (2003), "Coupling methods for continuum model with molecular model", Int. J. Multiscale Com., 1, 115-126. https://doi.org/10.1615/IntJMultCompEng.v1.i1.100
  8. Cressie, N. A. (1993), Statistics for Spatial data, Wiley, New York.
  9. Dai, K.Y., Liu, G.R., Lim, K.M. and Gu, Y.T. (2003), "Comparison between the radial point interpolation and the Kriging interpolation used in meshfree methods", Comput. Mech., 32, 60-70. https://doi.org/10.1007/s00466-003-0462-z
  10. Dolbow, J. and Belytschko, T. (1998), "An Introduction to Programming the Meshless Element Free Galerkin Method", Comp. Mech. Eng., 5, 207-241.
  11. Gu, L. (2003), "Moving Kriging interpolation and element-free Galerkin method", Int. J. Numer. Meth. Eng., 56, 1-11. https://doi.org/10.1002/nme.553
  12. Kanok-Nukulchai, W. and Plengkhom, K. (2004), "An extended FEM with element-free shape functions", Abstract presented at WCCM VI in conjunction with APCOM'04, Beijing, China.
  13. Kanok-Nukulchai, W., Plengkhom, K. and Tongsuk, P. (2004), "Element-free shape functions for superior performance of FEM", Abstract presented at IWACOM, Tokyo, Japan.
  14. Kanok-Nukulchai, W. and Wong, F.T. (2008). "A break-through enhancement of FEM using node-based Kriging interpolation", Bull. Int. Assoc. Comp. Mech., 23, 26-31.
  15. Liu, W.K., Karpov, E.G. and Park, H.S. (2006), Nano mechanics and materials: theory, multiscale methods and applications, John Wiley & Sons Ltd, England.
  16. Liu, W.K., Karpov, E.G., Zhang, S. and Park, H.S. (2004), "An introduction to computational nano mechanics and materials", Comput. Method. Appl. M., 193, 1529-1578. https://doi.org/10.1016/j.cma.2003.12.008
  17. Park, H.S. and Liu, W.K. (2004), "Introduction and tutorial on multiple scale analysis in solids", Comput. Method. Appl. M., 193, 1733-1772. https://doi.org/10.1016/j.cma.2003.12.054
  18. Park, H.S., Kapov, E.G., Liu, W.K. and Klein, P.A. (2005a), "The bridging scale for two-dimensional atomistic/continuum coupling", Philos. Mag., 85, 79-113. https://doi.org/10.1080/14786430412331300163
  19. Park, H.S., Karpov, E.G., Klein, P.A. and Liu, W.K. (2005b), "Three-dimensional bridging scale analysis of dynamic fracture", J. Comput. Phys., 207, 588- 609. https://doi.org/10.1016/j.jcp.2005.01.028
  20. Plengkhom, P. and Kanok-Nukulchai, W. (2005), "An enhancement of finite element method with moving Kriging shape functions", Int. J. Comput. Meth., 2, 451-475. https://doi.org/10.1142/S0219876205000594
  21. Qian, D. and Gondhalekar, R.H. (2004), "A virtual atom cluster approach to the mechanics of nanostructures", Int. J. Multiscale. Com., 2, 227-289.
  22. Rudd, R.E. and Broughton, J.Q. (1998), "Coarse-grained molecular dynamics and the atomic limit of finite element", Phys. Rev. B., 58, 5893-5896. https://doi.org/10.1103/PhysRevB.58.R5893
  23. Sommanawat, W. and Kanok-Nukulchai, W. (2008), "Multiscale simulation based on Moving Kriging method", Abstract presented at WCCM8 and ECCOMAS 2008, Venice, June - July.
  24. Tadmor, E., Ortiz, M. and Phillips, R. (1996), "Quasicontinuum analysis of defects in solids", Philos. Mag. A., 73, 1529-1563. https://doi.org/10.1080/01418619608243000
  25. Tang, S., Hou, T.Y. and Liu, W.K. (2006), "A mathematical framework of the bridging scale method", Int. J. Numer. Meth. Eng., 65, 1688-1713. https://doi.org/10.1002/nme.1514
  26. Tongsuk, P. and Kanok-Nukulchai, W. (2004), "Further investigation of element-free Galerkin method using moving Kriging interpolation", Int. J. Comput. Meth., 1, 345-365. https://doi.org/10.1142/S0219876204000162
  27. Xiao, S.P. and Belytschko, T. (2004), "A bridging domain method for coupling continua with molecular dynamics", Comput. Method. Appl. M., 193, 1645-1699. https://doi.org/10.1016/j.cma.2003.12.053
  28. Xu, M. and Belytschko, T. (2008), "Conservation properties of the bridging domain method for coupled molecular/continuum dynamics", Int. J. Numer. Meth. Eng., 76, 278-294. https://doi.org/10.1002/nme.2323
  29. Wagner, G.J. and Liu, W.K. (2003), "Coupling of atomistic and continuum simulations using a bridging scale decomposition", J. Comput. Phys., 190, 249-274. https://doi.org/10.1016/S0021-9991(03)00273-0
  30. Wong, F.T. and Kanok-Nukulchai, W. (2009), "On the convergence of the Kriging based finite element method", Int. J. Comput. Meth., 6, 93-118. https://doi.org/10.1142/S0219876209001784
  31. Zhang, S., Khare, R., Lu, Q. and Belytschko, T. (2007), "A bridging domain and strain computation method for coupled atomistic-continuum modeling of solids", Int. J. Numer. Meth. Eng., 70, 913-933. https://doi.org/10.1002/nme.1895