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Analysis of blended atomistic/continuum hybrid methods

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Abstract

We present a comprehensive error analysis of two prototypical atomistic-to-continuum coupling methods of blending type: the energy-based and the force-based quasicontinuum methods. Our results are valid in two and three dimensions, for finite range many-body interactions (e.g., EAM type), and in the presence of lattice defects (we consider point defects and dislocations). The two key ingredients in the analysis are (1) new force and energy consistency error estimates; and (2) a new technique for proving energy norm stability of a/c couplings that requires only the assumption that the exact atomistic solution is a stable equilibrium.

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Author notes

  1. Alexander V. Shapeev

    Present address: Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Building 3, Moscow, 143026, Russia

Authors and Affiliations

  1. The Division of Applied Mathematics at Brown University, 182 George St, Providence, RI, 02912, USA

    Xingjie Helen Li

  2. Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, UK

    Christoph Ortner

  3. School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN, 55455, USA

    Alexander V. Shapeev

  4. Department of Statistics, The University of Chicago, Eckhart Hall Room 108, 5734 S University Ave, Chicago, IL, 60637, USA

    Brian Van Koten

Authors
  1. Xingjie Helen Li
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  2. Christoph Ortner
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  3. Alexander V. Shapeev
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  4. Brian Van Koten
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Corresponding author

Correspondence to Alexander V. Shapeev.

Additional information

XHL was supported by an AMS-Simons Travel Grant. CO’s work was supported by EPSRC Grant EP/H003096, ERC Starting Grant 335120 and by the Leverhulme Trust through a Philip Leverhulme Prize. AVS was supported by the AFOSR Award FA9550–12–1–0187.

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Li, X.H., Ortner, C., Shapeev, A.V. et al. Analysis of blended atomistic/continuum hybrid methods. Numer. Math. 134, 275–326 (2016). https://doi.org/10.1007/s00211-015-0772-z

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