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**Analysis of a force-based quasicontinuum approximation.**
*(English)*
Zbl 1140.74006

Summary: We analyze a force-based quasicontinuum approximation to a one-dimensional system of atoms that interact by a classical atomistic potential. This force-based quasicontinuum approximation can be derived as the modification of an energy-based quasicontinuum approximation by the addition of nonconservative forces to correct nonphysical “ghost” forces that occur in the atomistic to continuum interface during constant strain. The algorithmic simplicity and consistency with the purely atomistic model at constant strain has made the force-based quasicontinuum approximation popular for large-scale quasicontinuum computations. We prove that the force-based quasicontinuum equations have a unique solution when the magnitude of external forces satisfy explicit bounds. For Lennard-Jones next-nearest-neighbor interactions, we show that unique solutions exist for external forces that extend the system nearly to its tensile limit. We give an analysis of the convergence of the ghost force iteration method to solve the equilibrium equations for the force-based quasicontinuum approximation. We show that the ghost force iteration is a contraction and give an analysis for its convergence rate.

### MSC:

74A25 | Molecular, statistical, and kinetic theories in solid mechanics |

82B21 | Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics |

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\textit{M. Dobson} and \textit{M. Luskin}, ESAIM, Math. Model. Numer. Anal. 42, No. 1, 113--139 (2008; Zbl 1140.74006)

### References:

[1] | S. Antman, Nonlinear problems of elasticity, Applied Mathematical Sciences107. Springer, New York, second edition (2005). |

[2] | 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 |

[3] | X. Blanc, C. Le Bris and P.-L. Lions, Atomistic to continuum limits for computational materials science. ESAIM: M2AN41 (2007) 391-426. Zbl1144.82018 · Zbl 1144.82018 |

[4] | R.F. Brown, A Topological Introduction to Nonlinear Analysis. Birkhäuser (2004). · Zbl 1061.47001 |

[5] | W. E and P. Ming, Analysis of multiscale methods. J. Comput. Math.22 (2004) 210-219. · Zbl 1046.65108 |

[6] | 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, Singapore (2005) 18-32. |

[7] | W. E and P. Ming, Cauchy-born rule and the stabilitiy of crystalline solids: Static problems. Arch. Ration. Mech. Anal.183 (2007) 241-297. · Zbl 1106.74019 |

[8] | W. E, J. Lu and J. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B74 (2006) 214115. |

[9] | W. Fleming, Functions of Several Variables. Springer-Verlag (1977). · Zbl 0348.26002 |

[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. · Zbl 1220.74010 |

[13] | M. Marder, Condensed Matter Physics. John Wiley & Sons (2000). |

[14] | R. Miller and E. Tadmor, The quasicontinuum method: Overview, applications and current directions. J. Comput. Aided Mater. Des.9 (2002) 203-239. |

[15] | 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. |

[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. Technical report, Oxford Numerical Analysis Group (2006). |

[18] | C. Ortner and E. Süli, A priori analysis of the quasicontinuum method in one dimension. Technical report, Oxford Numerical Analysis Group (2006). |

[19] | S. Prudhomme, P.T. Bauman and J.T. Oden, Error control for molecular statics problems. Int. J. Multiscale Comput. Eng.4 (2006) 647-662. |

[20] | D. Rodney and R. Phillips, Structure and strength of dislocation junctions: An atomic level analysis. Phys. Rev. Lett.82 (1999) 1704-1707. |

[21] | D. Serre, Matrices: Theory and applications, Graduate Texts in Mathematics216. Springer-Verlag, New York (2002). Translated from the 2001 French original. |

[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] | E. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids. Phil. Mag. A73 (1996) 1529-1563. |

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