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A One-Dimensional Symmetric-Force-Based Blending Method for Atomistic-to-Continuum Coupling

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A³N²M: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 165))

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Abstract

Inspired by the blending method developed by [P. Seleson, S. Beneddine, and S. Prudhome, A Force-Based Coupling Scheme for Peridynamics and Classical Elasticity, (2013)] for the nonlocal-to-local coupling, we create a symmetric and consistent blended force-based atomistic-to-continuum (a/c) scheme for the atomistic chain in one-dimensional space. The conditions for the well-posedness of the underlying model are established by analyzing an optimal blending size and blending type to ensure the \(H^1\) semi-norm stability for the blended force-based operator. We present several numerical experiments to test and confirm the theoretical findings.

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Acknowledgements

Elaine Gorom-Alexander and Dr. X. Li are supported by NSF CAREER award: DMS-1847770 and the University of North Carolina at Charlotte Faculty Research Grant.

We would like to thank the helpful discussions from Dr. Pablo Seleson and Dr. Christoph Ortner. We also want to thank the opportunity provided by the 50th John H. Barrett Memorial Lectures organizing committee.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, USA

    Elaine Gorom-Alexander & Xingjie Helen Li

Authors
  1. Elaine Gorom-Alexander
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  2. Xingjie Helen Li
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Correspondence to Xingjie Helen Li .

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Editors and Affiliations

  1. Department of Mathematics, University of Tennessee, Knoxville, TN, USA

    Tadele Mengesha

  2. Department of Mathematics, University of Tennessee, Knoxville, TN, USA

    Abner J. Salgado

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Gorom-Alexander, E., Li, X.H. (2023). A One-Dimensional Symmetric-Force-Based Blending Method for Atomistic-to-Continuum Coupling. In: Mengesha, T., Salgado, A.J. (eds) A³N²M: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models. The IMA Volumes in Mathematics and its Applications, vol 165. Springer, Cham. https://doi.org/10.1007/978-3-031-34089-5_6

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