Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model.

*(English)*Zbl 1010.74003Summary: In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. We analyze the error between the approximate solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic-continuum approximation model. No numerical analysis has been done yet. We estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.

##### MSC:

74A15 | Thermodynamics in solid mechanics |

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

74S05 | Finite element methods applied to problems in solid mechanics |

82-08 | Computational methods (statistical mechanics) (MSC2010) |

##### Keywords:

lattice statics; material particle model; Lennard-Jones potential; global minimization; finite element method; error estimation; ill-posedness; quasi-continuum approximation; nearest neighbor interaction; full interaction; one-dimensional model; atomistic-continuum approximation
Full Text:
DOI

##### References:

[1] | J.L. Bassani, V. Vitek and E.S. Alber, Atomic-level elastic properties of interfaces and their relation to continua, Acta Metall. Mater., Vol. 40, 1992, S307-S320. |

[2] | L. Brillouin, Wave Propagation in Periodic Structures, International Series in Physics, McGraw-Hill Book Company, Inc., New York, 1946. · Zbl 0063.00607 |

[3] | F.C. Frank and J.H. van der Merwe, One-dimensional dislocations. I. Static theory, Proc. R. Soc. London A198, 1949, pp. 205-216. · Zbl 0033.04702 |

[4] | Isaak A. Kunin, Elastic media with microstructure. I, Springer Series in Solid-State Sciences, vol. 26, Springer-Verlag, Berlin-New York, 1982. One-dimensional models; Translated from the Russian. · Zbl 0527.73002 |

[5] | Mitchell Luskin, On the computation of crystalline microstructure, Acta numerica, 1996, Acta Numer., vol. 5, Cambridge Univ. Press, Cambridge, 1996, pp. 191 – 257. · Zbl 0867.65033 · doi:10.1017/S0962492900002658 · doi.org |

[6] | I.V. Markov, Crystal Growth for Beginners, World Scientific Publishing Co., 1995. |

[7] | E.B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids, Philosophical Magazine A, Vol. 73, No. 6, 1996, pp. 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.