Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model
HTML articles powered by AMS MathViewer

by Ping Lin PDF
Math. Comp. 72 (2003), 657-675 Request permission

Abstract:

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. In this paper we shall analyze the approximate error between the 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. In the paper we shall 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.
References
  • 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.
  • A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8, DOI 10.1017/S0370164600012281
  • F.C. Frank and J.H. van der Merwe, One-dimensional dislocations. I. Static theory, Proc. R. Soc. London A198, 1949, pp. 205-216.
  • 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. MR 664203, DOI 10.1007/978-3-642-81748-9
  • Mitchell Luskin, On the computation of crystalline microstructure, Acta numerica, 1996, Acta Numer., vol. 5, Cambridge Univ. Press, Cambridge, 1996, pp. 191–257. MR 1624603, DOI 10.1017/S0962492900002658
  • I.V. Markov, Crystal Growth for Beginners, World Scientific Publishing Co., 1995.
  • 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.
Similar Articles
Additional Information
  • Ping Lin
  • Affiliation: Department of Mathematics, The National University of Singapore, 2 Science Drive 2, Singapore 117543
  • Email: matlinp@math.nus.edu.sg
  • Received by editor(s): June 9, 1998
  • Received by editor(s) in revised form: May 29, 2001
  • Published electronically: June 4, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Math. Comp. 72 (2003), 657-675
  • MSC (2000): Primary 65C20, 65K10, 65M15, 65M60, 74N15, 74G65
  • DOI: https://doi.org/10.1090/S0025-5718-02-01456-4
  • MathSciNet review: 1954960