An analysis of the effect of ghost force oscillation on quasicontinuum error. (English) Zbl 1165.81414

Summary: The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the rate \(O(h)\) in the discrete \(\ell^\infty\) and \(w^{1,1}\) norms where \(h\) is the interatomic spacing. We also give a proof that the error in the displacement gradient decays away from the interface to \(O(h)\) at distance \(O(h|\log h|)\) in the atomistic region and distance \(O(h)\) in the continuum region. Our work gives an explicit and simplified form for the decay of the effect of the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete \(\ell^\infty\) and \(w^{1,p}\) norms.


81V70 Many-body theory; quantum Hall effect
82B30 Statistical thermodynamics
65Z05 Applications to the sciences
70C20 Statics
Full Text: DOI arXiv EuDML


[1] M. Arndt and M. Luskin, Goal-oriented atomistic-continuum adaptivity for the quasicontinuum approximation. Int. J. Mult. Comp. Eng.5 (2007) 407-415.
[2] M. Arndt and M. Luskin, Error estimation and atomistic-continuum adaptivity for the quasicontinuum approximation of a Frenkel-Kontorova model. Multiscale Model. Simul.7 (2008) 147-170. Zbl1160.82313 · Zbl 1160.82313
[3] M. Arndt and M. Luskin, Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel-Kontorova model. Comp. Meth. App. Mech. Eng.197 (2008) 4298-4306. · Zbl 1194.74012
[4] S. Badia, M.L. Parks, P.B. Bochev, M. Gunzburger and R.B. Lehoucq, On atomistic-to-continuum (AtC) coupling by blending. Multiscale Model. Simul.7 (2008) 381-406. · Zbl 1160.65338
[5] X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: M2AN39 (2005) 797-826. · Zbl 1330.74066
[6] W. Curtin and R. Miller, Atomistic/continuum coupling in computational materials science. Model. Simul. Mater. Sc.11 (2003) R33-R68.
[7] M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum method. ESAIM: M2AN42 (2008) 113-139. · Zbl 1140.74006
[8] W. E and P. Ming. Analysis of the local quasicontinuum method, in Frontiers and Prospects of Contemporary Applied Mathematics, T. Li and P. Zhang Eds., Higher Education Press, World Scientific (2005) 18-32. · Zbl 1188.74019
[9] W. E., J. Lu and J. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B74 (2006) 214115.
[10] J. Knap and M. Ortiz, An analysis of the quasicontinuum method. J. Mech. Phys. Solids49 (2001) 1899-1923. · Zbl 1002.74008
[11] P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp.72 (2003) 657-675 (electronic). Zbl1010.74003 · Zbl 1010.74003
[12] P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material. SIAM J. Numer. Anal.45 (2007) 313-332. Zbl1220.74010 · Zbl 1220.74010
[13] R. Miller and E. Tadmor, The quasicontinuum method: Overview, applications and current directions. J. Comput. Aided Mater. Des.9 (2002) 203-239.
[14] R. Miller, L. Shilkrot and W. Curtin. A coupled atomistic and discrete dislocation plasticity simulation of nano-indentation into single crystal thin films. Acta Mater.52 (2003) 271-284.
[15] P. Ming and J.Z. Yang, Analysis of a one-dimensional nonlocal quasicontinuum method. Preprint. Zbl1177.74169 · Zbl 1177.74169
[16] J.T. Oden, S. Prudhomme, A. Romkes and P. Bauman, Multi-scale modeling of physical phenomena: Adaptive control of models. SIAM J. Sci. Comput.28 (2006) 2359-2389. · Zbl 1126.74006
[17] C. Ortner and E. Süli, A-posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension. Research Report NA-06/13, Oxford University Computing Laboratory (2006).
[18] C. Ortner and E. Süli, Analysis of a quasicontinuum method in one dimension. ESAIM: M2AN42 (2008) 57-91. · Zbl 1139.74004
[19] M.L. Parks, P.B. Bochev and R.B. Lehoucq, Connecting atomistic-to-continuum coupling and domain decomposition. Multiscale Model. Simul.7 (2008) 362-380. · Zbl 1160.65343
[20] S. Prudhomme, P.T. Bauman and J.T. Oden, Error control for molecular statics problems. Int. J. Mult. Comp. Eng.4 (2006) 647-662.
[21] D. Rodney and R. Phillips, Structure and strength of dislocation junctions: An atomic level analysis. Phys. Rev. Lett.82 (1999) 1704-1707.
[22] V. Shenoy, R. Miller, E. Tadmor, D. Rodney, R. Phillips and M. Ortiz, An adaptive finite element approach to atomic-scale mechanics - the quasicontinuum method. J. Mech. Phys. Solids47 (1999) 611-642. · Zbl 0982.74071
[23] T. Shimokawa, J. Mortensen, J. Schiotz and K. Jacobsen, Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic regions. Phys. Rev. B69 (2004) 214104.
[24] G. Strang and G. Fix, Analysis of the Finite Elements Method. Prentice Hall (1973). Zbl0356.65096 · Zbl 0356.65096
[25] E. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids. Phil. Mag. A73 (1996) 1529-1563.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.