Positive Definiteness of the Blended Force-Based Quasicontinuum Method

The development of consistent and stable quasicontinuum models for multidimensional crystalline solids remains a challenge. For example, proving the stability of the force-based quasicontinuum (QCF) model [M. Dobson and M. Luskin, M2AN Math. Model. Numer. Anal., 42 (2008), pp. 113--139] remains an o...

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Bibliographic Details
Published in:Multiscale modeling & simulation 2012-01, Vol.10 (3), p.1023-1045
Main Authors: Li, Xingjie Helen, Luskin, Mitchell, Ortner, Christoph
Format: Article
Language:English
Subjects:
Applied mathematics
Approximation
Blended
Blending
Computer simulation
Continuums
Crystal structure
Equilibrium
Mathematical models
Methods
Stability
Two dimensional
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Summary:The development of consistent and stable quasicontinuum models for multidimensional crystalline solids remains a challenge. For example, proving the stability of the force-based quasicontinuum (QCF) model [M. Dobson and M. Luskin, M2AN Math. Model. Numer. Anal., 42 (2008), pp. 113--139] remains an open problem. In one and two dimensions, we show that by blending atomistic and Cauchy--Born continuum forces (instead of a sharp transition as in the QCF method) one obtains positive-definite blended force-based quasicontinuum (B-QCF) models. We establish sharp conditions on the required blending width. [PUBLICATION ABSTRACT]
ISSN:1540-3459
1540-3467
DOI:10.1137/110859270