Skip to main content

Analysis of Geometrically Consistent Schemes with Finite Range Interaction

  • Hongliang Li (a1) and Pingbing Ming (a2)

We analyze the geometrically consistent schemes proposed by E. Lu and Yang [6] for one-dimensional problem with finite range interaction. The existence of the reconstruction coefficients is proved, and optimal error estimate is derived under sharp stability condition. Numerical experiments are performed to confirm the theoretical results.

Corresponding author
*Corresponding author. Email addresses: L. Li), B. Ming)
Hide All
[1] Badia, S., Parks, M. L., Gunzburger, M., Bochev, P. B., and Lehoucq, R. B., On atomistic-tocontinuum coupling by blending, Multiscale Model. Simul., 7 (2008), 381406.
[2] Born, M. and Huang, K., Dynamical Theory of Crystal Lattices, Oxford University Press, 1954.
[3] Dobson, M., There is no pointwise consistent quasicontinuum energy, IMA J. Numer. Anal., 279 (2014), 2945.
[4] Dobson, M. and Luskin, M., An analysis of the effect of ghost force oscillation on quasicontinuum error, ESAIM:M2AN, 43 (2009), 591604.
[5] W. E, , Principles of Multiscale Modeling, Cambridge University Press, 2011.
[6] W. E, , Lu, J., and Yang, J. Z., Uniform accuracy of the quasicontinuum method, Phys. Rev. B, 74 (2006), 214115.
[7] W. E, and Ming, P. B., Cauchy-Born rule and the stability of crystalline solids: static problems, Arch. Ration. Mech. Anal., 183 (2007), 241297.
[8] Houchmandzadeh, B., Lajzerowicz, J., and Salje, E., Relaxations near surfaces and interfaces for first-, second- and third-neighbour interactions: theory and applications to polytypism, J. Phys. Condens. Matter, 4 (1992), 97799794.
[9] Hudson, T. and Ortner, C., On the stability of Bravais lattices and their Cauchy-Born approximations, ESAIM:M2AN, 46 (2012), 81110.
[10] Lennard-Jones, J. E., On the force between atoms and ions, Proc. R. Soc. London Ser. A, 109 (1925), 584597.
[11] Li, X. H. and Luskin, M., Lattice stability for atomistic chains modeled by local approximations of the embedded atom method, Comput. Mater. Sci., 66 (2013), 96103.
[12] Li, X. H. and Luskin, M., A generalized quasinonlocal atomistic-to-continuum coupling method with finite-range interaction, IMA J. Numer. Anal., 32 (2012), 373393.
[13] Losonczi, László, On some discrete quadratic inequalities, Int. Ser. Numer. Math., 80 (1987), 7385.
[14] Lu, J. and Ming, P. B., Convergence of a force-based hybrid method in three dimensions, Commun. Pure Appl. Math., 66 (2013), 83108.
[15] Lu, Jianfeng and Ming, Pingbing, Stability of a force-based hybridmethod in three dimension with sharp interface, SIAM J. Numer. Anal., 52 (2014), 20052026.
[16] Luskin, M. and Ortner, C., Atomistic-to-continuum coupling, Acta Numer., (2013), 397508.
[17] Ming, P. B. and Yang, J. Z., Analysis of a one-dimensional nonlocal quasicontinuum method, Multiscale Model. Simul., 7 (2009), 18381875.
[18] Morse, P. M., Diatomic molecules according to the wave mechanics, II, Viberational levels, Phys. Rev., 34 (1929), 5764.
[19] Olson, D., Bochev, P. B., Luskin, M., and Shapeev, A. V., An optimization-based atomistic-to-continuum coupling method, SIAM J. Numer. Anal., 52 (2014), 21832204.
[20] Ortner, C., A priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1D, Math. Comput., 80 (2011), 12651285.
[21] Ortner, C., Shapeev, A., and Zhang, L., (in-)stability and stabilization of QNL-type atomistic-to-continuum coupling methods, Multiscale Model. Simul., 12 (2014), 12581293.
[22] Ortner, C. and Theil, F., Justification of the Cauchy-Born approximation of elastodynamics, Arch. Rational Mech. Anal., 207 (2013), 10251073.
[23] Ortner, C. and Wang, H., A priori error estimates for energy-based quasicontinuum approximations of a periodic chain, Math. Models Methods Appl. Sci., 21 (2011), 24912521.
[24] Ortner, C. and Zhang, L., Construction and sharp consistency estimates for atomistic/continuum coupling methods with general interfaces: a two-dimensional model problems, SIAM J. Numer. Anal., 50 (2012), 29402965.
[25] Ortner, C. and Zhang, L., Energy-based atomistic-to-continuum coupling without ghost forces, Comput. Methods Appl. Mech. Eng., 279 (2014), 2945.
[26] Parks, M. L., Bochev, P. B., and Lehoucq, R. B., Connecting atomistic-to-continuum coupling and domain decompositiong, Multiscale Model. Simul., 7 (2008), 362380.
[27] Shapeev, A. V., Consistent energy-based atomistic/continuum coupling for two-body potentials in one and two dimension, Multiscale Model. Simul., 9 (2011), 905932.
[28] Shapeev, Alexander V., Consistent energy-based atomistic/continuum for two-body potential in three dimensions, SIAMJ. Sci. Comput., 34(3) (2012), B335B360.
[29] Shenoy, V. B., Miller, R., Tadmor, E. B., Rodney, D., Phillips, R., and Ortiz, M., An adaptive finite element approach to atomic-scale mechanics-the quasicontinuum method, J. Mech. Phys. Solids, 47 (1999), 611642.
[30] Shimokawa, T., Mortensen, J. J., Schiøtz, J., and Jacobsen, K. W., Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic region, Phys. Rev. B, 69 (2004), 214104.
[31] Tadmor, E. B., Ortiz, M., and Phillips, R., Quasicontinuum analysis of defects in solids, Philos. Mag. A, 73 (1996), 15291563.
[32] Van Koten, B. and Luskin, M., Analysis of energy-based bnended quasi-continuum approximations, SIAM J. Numer. Anal., 49 (2011), 21822209.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 20 *
Loading metrics...

Abstract views

Total abstract views: 104 *
Loading metrics...

* Views captured on Cambridge Core between 31st October 2017 - 20th July 2018. This data will be updated every 24 hours.