Error estimation and atomistic-continuum adaptivity for the quasicontinuum approximation of a frenkel-kontorova model

Multiscale Modeling and Simulation

Volumn 7, Issue 1, 2008, Pages 147-170

  Arndt, Marcel ^a   Luskin, Mitchell ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

A posteriori; Atomistic continuum modeling; Defect; Dislocation; Error estimation; Frenkel.Kontorova model; Goal oriented; Quasicontinuum

Indexed keywords

A POSTERIORI; ATOMISTIC-CONTINUUM MODELING; DISLOCATION; ERROR ESTIMATION; FRENKEL.KONTOROVA MODEL; GOAL-ORIENTED; QUASICONTINUUM;

CONTINUUM MECHANICS; ERROR ANALYSIS;

MODAL ANALYSIS;

EID: 51749093823     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/070688559     Document Type: Article

References (25)

1. M. AINSWORTH and J. T. ODEN, A Posteriori Error Estimation in Finite Element Analysis, Wiley, New York, 2000.
2. M. ARNDT and M. GRIEBEL, Derivation of higher order gradient continuum models from atomistic models for crystalline solids, Multiscale Model. Simul., 4 (2005), pp. 531-562.
3. W. BANGERTH and R. RANNACHER, Adaptive Finite Element Methods for Differential Equations, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2003.
4. X. BLANC, C. LE BRIS, and F. LEGOLL, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics, Math. Model. Numer. Anal., 39 (2005), pp. 797-826.
5. P. M. CHAIKIN and T. C. LUBENSKY, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, UK, 2000.
6. M. S. DAW and M. I. BASKES, Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals, Phys. Rev. Lett., 50 (1983), pp. 1285-1288.
7. M. S. DAW and M. I. BASKES, Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals, Phys. Rev. B, 29 (1984), pp. 6443-6453.
8. M. DOBSON and M. LUSKIN, Analysis of a force-based quasicontinuum approximation, Math. Model. Numer. Anal., 42 (2008), pp. 113-139.
9. W. E, J. LU, and J. Z. YANG, Uniform accuracy of the quasicontinuum method, Phys. Rev. B, 74 (2006), 214115.
10. W. E and P. MING, Analysis of multiscale methods, J. Comput. Math., 22 (2004), pp. 210-219.
11. E. KAXIRAS, Atomic and Electronic Structure of Solids, Cambridge University Press, Cambridge, UK, 2003.
12. J. KNAP and M. ORTIZ, An analysis of the quasicontinuum method, J. Mech. Phys. Solids, 49 (2001), pp. 1899-1923.
13. P. LIN, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model, Math. Comp., 72 (2003), pp. 657-675.
14. P. LIN, Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects, SIAM J. Numer. Anal., 45 (2007), pp. 313-332.
15. M. MARDER, Condensed Matter Physics, John Wiley and Sons, New York, 2000.
16. J. T. ODEN and S. PRUDHOMME, Estimation of modeling error in computational mechanics, J. Comput. Phys., 182 (2002), pp. 496-515.
17. J. T. ODEN, S. PRUDHOMME, and P. BAUMAN, Error control for molecular statics problems, Int. J. Multiscale Comput. Eng., 4 (2006), pp. 647-662.
18. J. T. ODEN, S. PRUDHOMME, A. ROMKES, and P. T. BAUMAN, Multiscale modeling of physical phenomena: Adaptive control of models, SIAM J. Sci. Comput., 28 (2006), pp. 2359-2389.
19. J. T. ODEN and K. S. VEMAGANTI, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials: Part I: Error estimates and adaptive algorithms, J. Comput. Phys., 164 (2000), pp. 22-47.
20. C. ORTNER and E. SÜLI, A-posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension, Math. Model. Numer. Anal., to appear.
21. C. ORTNER and E. SÜLI, Analysis of a quasicontinuum method in one dimension, Math. Model. Numer. Anal., 42 (2008), pp. 57-91.
22. T. SHIMOKAWA, J. J. MORTENSEN, J. SCHIø TZ, and K. W. JACOBSEN, Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic region, Phys. Rev. B, 61 (2004), 214104.
23. E. B. TADMOR, R. MILLER, R. PHILLIPS, and M. ORTIZ, Nanoindentation and incipient plasticity, J. Mater. Res., 14 (1999), pp. 2233-2250.
24. E. B. TADMOR, M. ORTIZ, and R. PHILLIPS, Quasicontinuum analysis of defects in solids, Philos. Mag. A, 73 (1996), pp. 1529-1563.
25. E. B. TADMOR, R. PHILLIPS, and M. ORTIZ, Mixed atomistic and continuum models of deformation in solids, Langmuir, 12 (1996), pp. 4529-4534.