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Analysis of an Averaging Operator for Atomic-to-Continuum Coupling Methods by the Arlequin Approach

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Numerical Analysis of Multiscale Computations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 82))

Abstract

A new coupling term for blending particle and continuum models with the Arlequin framework is investigated in this work. The coupling term is based on an integral operator defined on the overlap region that matches the continuum and particle solutions in an average sense. The present exposition is essentially the continuation of a previous work (Bauman et al., On the application of the Arlequin method to the coupling of particle and continuum models, Computational Mechanics, 42, 511–530, 2008) in which coupling was performed in terms of an H 1-type norm. In that case, it was shown that the solution of the coupled problem was mesh-dependent or, said in another way, that the solution of the continuous coupled problem was not the intended solution. This new formulation is now consistent with the problem of interest and is virtually mesh-independent when considering a particle model consisting of a distribution of heterogeneous bonds. The mathematical properties of the formulation are studied for a one-dimensional model of harmonic springs, with varying stiffness parameters, coupled with a linear elastic bar, whose modulus is determined by classical homogenization. Numerical examples are presented for one-dimensional and two-dimensional model problems that illustrate the approximation properties of the new coupling term and the effect of mesh size.

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Acknowledgements

Support of this work by DOE under contract DE-FG02-05ER25701 is gratefully acknowledged. The second author, Robin Bouclier, would like also to thank ICES for hosting him for his internship during the Summer of 2009.

Author information

Authors and Affiliations

  1. Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, 1 University Station C0200, Austin, TX, 78712, USA

    Serge Prudhomme & J. Tinsley Oden

  2. Laboratoire de Mécanique et Technologie (LMT-Cachan), Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235, Cachan Cedex, France

    Robin Bouclier & Ludovic Chamoin

  3. Laboratoire de Mécanique des Sols, Structures et Matériaux (MSSMat), Ecole Centrale Paris, 1 Grande Voie des Vignes, 92290, Châtenay-Malabry, France

    Hachmi Ben Dhia

Authors
  1. Serge Prudhomme
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  2. Robin Bouclier
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  3. Ludovic Chamoin
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  4. Hachmi Ben Dhia
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  5. J. Tinsley Oden
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Corresponding author

Correspondence to Serge Prudhomme .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Texas, University Station C 1200 1, Austin, 78712-0257, Texas, USA

    Björn Engquist

  2. Dept. Numerical Analysis &, Computer Science (NADA), Royal Institute of Technology (KTH), Lindstedtsvägen 3, Stockholm, 100 44, Sweden

    Olof Runborg

  3. , Department of Mathematics, University of Texas at Austin, University Station C1200 1, Austin, 78712-0257, Texas, USA

    Yen-Hsi R. Tsai

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© 2012 Springer-Verlag Berlin Heidelberg

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Prudhomme, S., Bouclier, R., Chamoin, L., Dhia, H.B., Oden, J.T. (2012). Analysis of an Averaging Operator for Atomic-to-Continuum Coupling Methods by the Arlequin Approach. In: Engquist, B., Runborg, O., Tsai, YH. (eds) Numerical Analysis of Multiscale Computations. Lecture Notes in Computational Science and Engineering, vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21943-6_15

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