An analysis of the effect of ghost force oscillation on quasicontinuum error

Dobson, Matthew; Luskin, Mitchell

ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 3, pp. 591-604.

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($h|logh|$) 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.

Classification : 65Z05, 70C20
Mots clés : quasicontinuum, atomistic to continuum, ghost force

[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. | MR | Zbl

[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. | MR

[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. | MR | Zbl

[5] X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: M2AN 39 (2005) 797-826. | Numdam | MR

[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: M2AN 42 (2008) 113-139. | Numdam | MR | Zbl

[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. | MR

[9] W. E, J. Lu and J. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74 (2006) 214115.

[10] J. Knap and M. Ortiz, An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49 (2001) 1899-1923. | Zbl

[11] P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp. 72 (2003) 657-675 (electronic). | MR | Zbl

[12] P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material. SIAM J. Numer. Anal. 45 (2007) 313-332. | MR

[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. | Zbl

[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. | MR | Zbl

[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: M2AN 42 (2008) 57-91. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. Solids 47 (1999) 611-642. | MR | Zbl

[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. B 69 (2004) 214104.

[24] G. Strang and G. Fix, Analysis of the Finite Elements Method. Prentice Hall (1973). | MR | Zbl

[25] E. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids. Phil. Mag. A 73 (1996) 1529-1563.