An analysis of the quasicontinuum method. (English) Zbl 1002.74008

From the summary: We examine a version of quasicontinuum theory and analyze its accuracy and convergence characteristics. Specifically, we assess the effect of summation rules on accuracy; we determine the rate of convergence of the method in the presence of strong singularities, such as point loads; and we assess the effect of refinement tolerance, which controls the rate at which new nodes are inserted in the model, on the development of dislocation microstructures.


74A60 Micromechanical theories
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
Full Text: DOI arXiv


[2] Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0634.73056
[3] Joe, B., Three-dimensional triangulations from local transformations, SIAM J. Sci. Stat. Comp., 10, 718-741 (1989) · Zbl 0681.65087
[4] Joe, B., Construction of three-dimensional improved-quality triangulations using local transformations, SIAM J. Sci. Comput., 16, 6, 1292-1307 (1995) · Zbl 0851.65081
[5] Kelchner, C. L.; Plimpton, S. J.; Hamilton, J. C., Dislocation nucleation and defect structure during surface indentation, Phys. Rev., B 58, 11085-11088 (1998)
[6] Knuth, D. E., The Art of Computer Programming. (1997), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0191.17903
[7] Lennard-Jones, J. E.; Devonshire, A. F., Critical and cooperative phenomena. III. A theory of melting and the structure of liquids, Proc. Roy. Soc. Lond., A 169, 317-338 (1939) · Zbl 0020.32703
[8] Lennard-Jones, J. E.; Devonshire, A. F., Critical and cooperative phenomena. IV. A theory of disorder in solids and liquids and the process of melting, Proc. Roy. Soc. Lond., A 170, 464-484 (1939) · Zbl 0021.18401
[9] Miller, R.; Ortiz, M.; Phillips, R.; Shenoy, V.; Tadmor, E. B., Quasicontinuum models of fracture and plasticity, Eng. Fract. Mech., 61 (1998)
[10] Miller, R.; Tadmor, E. B.; Phillips, R.; Ortiz, M., Quasicontinuum simulation of fracture at the atomic scale, Model. Simul. Mater. Sci. Eng., 6, 607-638 (1998)
[11] Rodney, D.; Phillips, R., Structure and strength of dislocation junctions: an atomic level anlaysis, Phys. Rev. Lett., 82, 1704-1707 (1999)
[12] Shenoy, V. B.; Miller, R.; Tadmor, E. B.; Phillips, R.; Ortiz, M., Quasicontinuum models of interfacial structure and deformation, Phys. Rev. Lett., 80, 742-745 (1998)
[13] Shenoy, V. B.; Miller, R.; Tadmor, E. B.; Rodney, D.; Phillips, R.; Ortiz, M., Anadaptive finite element approach to atomic-scale mechanics—the quasicontinuum method, J. Mech. Phys. Solids, 47, 611-642 (1999) · Zbl 0982.74071
[14] Shenoy, V. B.; Phillips, R.; Tadmor, E. B., Nucleation of dislocations beneath a plane strain indenter, J. Mech. Phys. Solids, 48, 649-673 (2000) · Zbl 1007.74060
[15] Shephard, M. S.; Georges, M. K., Automatic Three-Dimensional Mesh Generation by the Finite Octree Technique, Int. J. Numer. Methods Eng., 32, 4, 709-749 (1991) · Zbl 0755.65116
[16] Smith, G. S.; Tadmor, E. B.; Kaxiras, E., Multiscale simulation of loading and electrical resistance in silicon nanoindentation, Phys. Rev. Lett., 84, 1260-1263 (2000)
[17] Tadmor, E. B.; Miller, R.; Phillips, R.; Ortiz, M., Nanoindentation and incipient plasticity, J. Mater. Res., 14, 2233-2250 (1999)
[18] Tadmor, E. B.; Ortiz, M.; Phillips, R., Quasicontinuum analysis of defects in solids, Philos. Mag., 73, 1529-1563 (1996)
[19] Tadmor, E. B.; Phillips, R.; Ortiz, M., Mixed atomistic and continuum models of deformation in solids, Langmuir, 12, 19, 4529-4534 (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.