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Numerische Mathematik

, Volume 134, Issue 2, pp 275–326 | Cite as

Analysis of blended atomistic/continuum hybrid methods

  • Xingjie Helen Li
  • Christoph Ortner
  • Alexander V. Shapeev
  • Brian Van Koten
Article

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.

Keywords

Atomistic models Coarse graining Atomistic-to-continuum coupling Quasicontinuum method Blending 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Xingjie Helen Li
    • 1
  • Christoph Ortner
    • 2
  • Alexander V. Shapeev
    • 3
    • 4
  • Brian Van Koten
    • 5
  1. 1.The Division of Applied Mathematics at Brown UniversityProvidenceUSA
  2. 2.Mathematics Institute, Zeeman BuildingUniversity of WarwickCoventryUK
  3. 3.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  4. 4.Skolkovo Institute of Science and Technology, Skolkovo Innovation CenterMoscowRussia
  5. 5.Department of StatisticsThe University of ChicagoChicagoUSA

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