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Analysis of an energy-based atomistic/continuum approximation of a vacancy in the 2D triangular lattice


Authors: C. Ortner and A. V. Shapeev
Journal: Math. Comp. 82 (2013), 2191-2236
MSC (2010): Primary 65N12, 65N15, 70C20
DOI: https://doi.org/10.1090/S0025-5718-2013-02687-7
Published electronically: April 22, 2013
MathSciNet review: 3073196
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Abstract | References | Similar Articles | Additional Information

Abstract: We present an a priori error analysis of a practical energy based atomistic/continuum coupling method (A. V. Shapeev, Multiscale Model.
Simul., 9(3):905-932, 2011) in two dimensions, for finite-range pair-potential interactions, in the presence of vacancy defects.

We establish first-order consistency and stability of the method, from which we obtain a priori error estimates in the $ \textup {H}^1$-norm and the energy in terms of the mesh size and the ``smoothness'' of the atomistic solution in the continuum region. From these error estimates we obtain heuristics for an optimal choice of the atomistic region and the finite element mesh, as well as convergence rates in terms of the number of degrees of freedom. Our analytical predictions are supported by extensive numerical tests.


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Additional Information

C. Ortner
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: c.ortner@warwick.ac.uk

A. V. Shapeev
Affiliation: Section of Mathematics, Swiss Federal Institute of Technology (EPFL), Station 8, CH-1015, Lausanne, Switzerland
Address at time of publication: School of Mathematics, 206 Church St. SE, University of Minnesota, Minneapolis, Minnesota 55455
Email: ashapeev@umn.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02687-7
Keywords: Atomistic models, atomistic-to-continuum coupling, coarse graining
Received by editor(s): May 11, 2011
Received by editor(s) in revised form: January 13, 2012, and February 16, 2012
Published electronically: April 22, 2013
Additional Notes: This work was supported by the EPSRC Critical Mass Programme “New Frontiers in the Mathematics of Solids” (OxMoS), by the EPSRC grant “Analysis of atomistic-to-continuum coupling methods”, and by the ANMC Chair at EPFL (Prof. Assyr Abdulle)
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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