Library Subscription: Guest
Home Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Multiscale Computational Engineering

Impact factor: 0.768

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2011002388
pages 51-64

STRESS-BASED ATOMISTIC/CONTINUUM COUPLING: A NEW VARIANT OF THE QUASICONTINUUM APPROXIMATION

C. Makridakis
Department of Applied Mathematics, University of Crete, 71409 Heraklion-Crete, Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion-Crete
Christoph Ortner
University of Oxford
E. Suli
Mathematical Institute, 24-29 S. Giles', Oxford OX1 3LB, United Kingdom

ABSTRACT

The force-based quasicontinuum (QCF) approximation isthe principle that lies behind the most commonly usedatomistic/continuum hybrid models for crystalline solids. Recentanalyses have shown some potential pitfalls of the QCF method, particularly the lack of positive definiteness of the linearized QCF operator and the lack of uniform stability as the number ofatoms tends to infinity. We derive a weak variational representation of the QCF operator and identify the origin ofthese difficulties as the lack of an interface condition on thestresses. This leads us to propose an improved variant of the QCF method that can be understood as a coupling mechanism based onstresses rather than forces.

REFERENCES

  1. Blanc, X., Le Bris, C., and Lions, P.-L., From molecular models to continuum mechanics. DOI: 10.1007/s00205-002-0218-5

  2. Dobson, M. and Luskin, M., Analysis of a force-based quasicontinuum approximation. DOI: 10.1051/m2an:2007058

  3. Dobson, M., Luskin, M., and Ortner, C., Stability, instability, and error of the force-based quasicontinuum approximation.

  4. Dobson, M., Luskin, M., and Ortner, C., Sharp stability estimates for the force-based quasicontinuum method.

  5. Dobson, M., Luskin, M., and Ortner, C., Iterative methods for the force-based quasicontinuum approximation.

  6. Dobson, M., Luskin, M., and Ortner, C., Accuracy of quasicontinuum approximations near instabilities.

  7. Dobson, M., Ortner, C., and Shapeev, A., The spectrum of the force-based quasicontinuum operator for a homogeneous periodic chain.

  8. E, W., Lu, J., and Yang, J. Z., Uniform accuracy of the quasicontinuum method. DOI: 10.1103/PhysRevB.74.214115

  9. E, W. and Ming, P., Cauchy-Born rule and the stability of crystalline solids: static problems. DOI: 10.1007/s00205-006-0031-7

  10. Fish, J., Nuggehally, M. A., Shephard, M. S., Picu, C. R., Badia, S., Parks, M. L., and Gunzburger, M., Concurrent AtC coupling based on a blend of the continuum stress and the atomistic force. DOI: 10.1016/j.cma.2007.05.020

  11. Kohlhoff, S. and Schmauder, S., A new method for coupled elastic-atomistic modelling. DOI: 10.1007/978-1-4684-5703-2_42

  12. Makridakis, C., Ortner, C., and Sli, E., A priori error analysis of two force-based atomistic/continuum models of a periodic chain. DOI: 10.1007/s00211-011-0380-5

  13. Miller, R. E. and Tadmor, E. B., The quasicontinuum method: Overview, applications and current directions. DOI: 10.1023/A:1026098010127

  14. Ortiz, M., Phillips, R., and Tadmor, E. B., Quasicontinuum analysis of defects in solids. DOI: 10.1080/01418619608243000

  15. Ortner, C., A priori and a posteriori analysis of the quasi-nonlocal quasicontinuum method in 1d.

  16. Ortner, C. and Sli, E., Analysis of a quasicontinuum method in one dimension. DOI: 10.1051/m2an:2007057

  17. Shenoy, V. B., Miller, R., Tadmor, E. B., Rodney, D., Phillips, R., and Ortiz, M., An adaptive finite element approach to atomic-scale mechanicsthe quasicontinuum method. DOI: 10.1016/S0022-5096(98)00051-9

  18. Shimokawa, T., Mortensen, J. J., Schiotz, J., and Jacobsen, K. W., Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic region. DOI: 10.1103/PhysRevB.69.214104

  19. Tikhonov, A. N. and Samarskii, A. A., Homogeneous difference schemes. DOI: 10.1016/0041-5553(62)90005-8