Analysis of the local quasicontinuum method

Frontiers And Prospects Of Contemporary Applied Mathematics

Volumn , Issue , 2006, Pages 18-32

  Weinan, E ^a,b   Ming, Pingbing ^c  

^a PRINCETON UNIVERSITY   (United States)

^b PEKING UNIVERSITY   (China)

^c INSTITUTE OF COMPUTATIONAL MATHEMATICS AND SCIENTIFIC ENGINEERING COMPUTING   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85115955160     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/9789812774194_0002     Document Type: Chapter

References (16)

1. R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press, second edition, 2003.
2. J. M. Ball and R. D. James, Proposed experimental tests of a theory of fine micro structure and the two-well problem, Phil. Trans. Roy. Soc. Lond. A., 338 (1992), 389-450.
3. X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical mul-tiscale method coupling atomistic and continuum mechanics, to appear in Math. Modelling and Numer. Anal.
4. M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, 1954.
5. W. E and P. B. Ming, Analysis of the multiscale methods, J. Comp. Math., 19 (2004), 209-220.
6. W. E and P. B. Ming, Cauchy-Born rule and the stability of crystals: static problem, 2005, preprint.
7. W. E, P. B. Ming and P. W. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc, 18 (2005), 121-156.
8. J. L. Ericksen, The Cauchy and Born hypotheses for crystals, In: Phases Transformations and Materials Instabilities in Solids, (M. Gurtin eds.), Academic Press, 1984, 61-77.
9. J. Knap and M. Ortiz, An analysis of the quasicontinuum method, J. Mech. Phys. Solids., 49 (2001), 1899-1923.
10. P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model, Math. Comp., 72 (2003), 657-675.
11. R. Miller and E. B. Tadmor, The quasicontinuum method: overview, applications and current directions, J. Computer-aided Material Design, 9 (2002), 203-239.
12. R. Rannacher and R. Scott, Some optimal error estimates for piece-wise linear finite element approximation, Math. Comp., 38 (1982), 437-445.
13. A. Schatz, An observation concerning Ritz-Garlerkin methods with infinite bilinear forms, Math. Comp., 28 (1974), 959-962.
14. V. B. Shenoy, R. Miller, E. B. Tadmor, R. Philips and M. Ortiz, An adaptive finite element approach to atomic scale mechanics-the quasicontinuum method, J. Mech. Phys. Solids., 47 (1999), 611-642.
15. E. B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids, Phil. Mag., A73 (1996), 1529-1563.
16. L. Truskinovsky, Fracture as a phase transition, In: Contemporary Research in the Mechanics and Mathematics of Materials (R. C. Batra and M. F. Beatty ed.), CIMNE, Barcelona, 1996, 322-332.