Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics

Xavier Blanc; Claude Le Bris; Frédéric Legoll

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

  • Volume: 39, Issue: 4, page 797-826
  • ISSN: 0764-583X

Abstract

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In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We provide a numerical analysis of the approach.

How to cite

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Blanc, Xavier, Bris, Claude Le, and Legoll, Frédéric. "Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.4 (2005): 797-826. <http://eudml.org/doc/245383>.

@article{Blanc2005,
abstract = {In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We provide a numerical analysis of the approach.},
author = {Blanc, Xavier, Bris, Claude Le, Legoll, Frédéric},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {multiscale methods; variational problems; continuum mechanics; discrete mechanics; Multiscale methods},
language = {eng},
number = {4},
pages = {797-826},
publisher = {EDP-Sciences},
title = {Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics},
url = {http://eudml.org/doc/245383},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Blanc, Xavier
AU - Bris, Claude Le
AU - Legoll, Frédéric
TI - Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 4
SP - 797
EP - 826
AB - In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We provide a numerical analysis of the approach.
LA - eng
KW - multiscale methods; variational problems; continuum mechanics; discrete mechanics; Multiscale methods
UR - http://eudml.org/doc/245383
ER -

References

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  1. [1] G. Alberti and C. Mantegazza, A note on the theory of SBV functions. Bollettino U.M.I. Sez. B 7 (1997) 375–382. Zbl0877.49001
  2. [2] L. Ambrosio, L. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York (2000). Zbl0957.49001MR1857292
  3. [3] X. Blanc, C. Le Bris and F. Legoll, work in preparation, and Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics, Preprint, Laboratoire Jacques-Louis Lions, Université Paris 6 (2004), available at http://www.ann.jussieu.fr/publications/2004/R04029.html Zbl1330.74066
  4. [4] X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics. Arch. Rational Mech. Anal. 164 (2002) 341–381. Zbl1028.74005
  5. [5] A. Braides, G. Dal Maso and A. Garroni, Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Rational Mech. Anal. 146 (1999) 23–58. Zbl0945.74006
  6. [6] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer (1991). Zbl0804.65101
  7. [7] J.Q. Broughton, F.F. Abraham, N. Bernstein and E. Kaxiras, Concurrent coupling of length scales: Methodology and application. Phys. Rev. B 60 (1999) 2391–2403. 
  8. [8] P.G. Ciarlet, An O ( h 2 ) method for a non-smooth boundary value problem. Aequationes Math. 2 (1968) 39–49. Zbl0159.11703
  9. [9] P.G. Ciarlet, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. Studies in Mathematics and its Applications, North Holland (1988). Zbl0648.73014MR936420
  10. [10] P.G. Ciarlet, Basic Error Estimates for Elliptic Problems, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet, J.-L. Lions, Eds., North-Holland (1991) 17–351. Zbl0875.65086
  11. [11] W. E and P. Ming, private communication. 
  12. [12] J. Knap and M. Ortiz, An analysis of the Quasicontinuum method. J. Mech. Phys. Solids 49 (2001) 1899–1923. Zbl1002.74008
  13. [13] P. Le Tallec, Numerical Methods for nonlinear three-dimensional elasticity, in Handbook of Numerical Analysis, Vol. III, P.G. Ciarlet, J.-L. Lions, Eds., North-Holland (1994) 465–622. Zbl0875.73234
  14. [14] F. Legoll, Méthodes moléculaires et multi-échelles pour la simulation numérique des matériaux(Molecular and multiscale methods for the numerical simulation of materials), Ph.D. Thesis, Université Pierre et Marie Curie (France), 2004, available at http://cermics.enpc.fr/ ∼ legoll/theseLegoll.pdf 
  15. [15] J.E. Marsden and T.J.R. Hugues, Mathematical foundations of Elasticity. Dover (1994). MR1262126
  16. [16] R. Miller, E.B. Tadmor, R. Phillips and M. Ortiz, Quasicontinuum simulation of fracture at the atomic scale. Model. Simul. Mater. Sci. Eng. 6 (1998) 607–638. 
  17. [17] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer (1997). Zbl0803.65088MR1299729
  18. [18] E.B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529–1563. 
  19. [19] E.B. Tadmor and R. Phillips, Mixed atomistic and continuum models of deformation in solids. Langmuir 12 (1996) 4529–4534. 
  20. [20] E.B. Tadmor, G.S. Smith, N. Bernstein and E. Kaxiras, Mixed finite element and atomistic formulation for complex crystals. Phys. Rev. B 59 (1999) 235–245. 
  21. [21] V.B. Shenoy, R. Miller, E.B. Tadmor, R. Phillips and M. Ortiz, Quasicontinuum models of interfacial structure and deformation. Phys. Rev. Lett. 80 (1998) 742–745. 
  22. [22] V.B. Shenoy, R. Miller, E.B. Tadmor, D. Rodney, R. Phillips and M. Ortiz, An adaptative finite element approach to atomic-scale mechanics – the Quasicontinuum method, J. Mech. Phys. Solids 47 (1999) 611–642. Zbl0982.74071
  23. [23] C. Truesdell and W. Noll, The nonlinear field theories of mechanics theory of elasticity. Handbuch der Physik, III/3, Springer Berlin (1965) 1–602. Zbl1068.74002
  24. [24] L. Truskinovsky, Fracture as a phase transformation, in Contemporary research in mechanics and mathematics of materials, Ericksen’s Symposium, R. Batra and M. Beatty, Eds., CIMNE, Barcelona (1996) 322–332. 
  25. [25] K.J. Van Vliet, J. Li, T. Zhu, S. Yip and S. Suresh, Quantifying the early stages of plasticity through nanoscale experiments and simulations. Phys. Rev. B 67 (2003) 104105. 
  26. [26] P. Zhang, P.A. Klein, Y. Huang, H. Gao and P.D. Wu, Numerical simulation of cohesive fracture by the virtual-internal-bond model. Comput. Model. Engrg. Sci. 3 (2002) 263–289. Zbl1066.74008

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