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Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. (English) Zbl 1010.74003
Summary: 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)
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[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.
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