Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 4, p. 797-826

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.

DOI : https://doi.org/10.1051/m2an:2005035
Classification:  65K10,  74G15,  74G70,  74N15
Keywords: multiscale methods, variational problems, continuum mechanics, discrete mechanics
@article{M2AN_2005__39_4_797_0,
     author = {Blanc, Xavier and Bris, Claude Le and Legoll, Fr\'ed\'eric},
     title = {Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {4},
     year = {2005},
     pages = {797-826},
     doi = {10.1051/m2an:2005035},
     zbl = {pre02213940},
     mrnumber = {2165680},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2005__39_4_797_0}
}
Blanc, Xavier; Bris, Claude Le; Legoll, Frédéric. Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 4, pp. 797-826. doi : 10.1051/m2an:2005035. http://www.numdam.org/item/M2AN_2005__39_4_797_0/

[1] G. Alberti and C. Mantegazza, A note on the theory of SBV functions. Bollettino U.M.I. Sez. B 7 (1997) 375-382. | Zbl 0877.49001

[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). | MR 1857292 | Zbl 0957.49001

[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

[4] X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics. Arch. Rational Mech. Anal. 164 (2002) 341-381. | Zbl 1028.74005

[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. | Zbl 0945.74006

[6] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer (1991). | Zbl 0804.65101

[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] P.G. Ciarlet, An O(h 2 ) method for a non-smooth boundary value problem. Aequationes Math. 2 (1968) 39-49. | Zbl 0159.11703

[9] P.G. Ciarlet, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. Studies in Mathematics and its Applications, North Holland (1988). | MR 936420 | Zbl 0648.73014

[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. | Zbl 0875.65086

[11] W. E and P. Ming, private communication.

[12] J. Knap and M. Ortiz, An analysis of the Quasicontinuum method. J. Mech. Phys. Solids 49 (2001) 1899-1923. | Zbl 1002.74008

[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. | Zbl 0875.73234

[14] F. Legoll, Méthodes moléculaires et multi-échelles pour la simulation numérique des matériaux

[15] J.E. Marsden and T.J.R. Hugues, Mathematical foundations of Elasticity. Dover (1994). | MR 1262126

[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] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer (1997). | MR 1299729 | Zbl 0803.65088

[18] E.B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529-1563.

[19] E.B. Tadmor and R. Phillips, Mixed atomistic and continuum models of deformation in solids. Langmuir 12 (1996) 4529-4534.

[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] 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] 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. | Zbl 0982.74071

[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. | Zbl 1068.74002

[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] 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] 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. | Zbl 1066.74008