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An Efficient Multigrid Method for Molecular Mechanics Modeling in Atomic Solids

Published online by Cambridge University Press:  20 August 2015

Jingrun Chen  and
Pingbing Ming
Jingrun Chen*
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, Beijing 100190, China
Pingbing Ming*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, Beijing 100190, China
*
Email:chenjr@lsec.cc.ac.cn
Corresponding author.Email:mpb@lsec.cc.ac.cn
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Abstract

We propose a multigrid method to solve the molecular mechanics model (molecular dynamics at zero temperature). The Cauchy-Born elasticity model is employed as the coarse grid operator and the elastically deformed state as the initial guess of the molecular mechanics model. The efficiency of the algorithm is demonstrated by three examples with homogeneous deformation, namely, one dimensional chain under tensile deformation and aluminum under tension and shear deformations. The method exhibits linear-scaling computational complexity, and is insensitive to parameters arising from iterative solvers. In addition, we study two examples with inhomogeneous deformation: vacancy and nanoindentation of aluminum. The results are still satisfactory while the linear-scaling property is lost for the latter example.

Keywords

65B9965N3065Z0574G1574G6574S05Multigrid methodlinear-scaling algorithmCauchy-Born rulenanoindentation
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 1 , July 2011 , pp. 70 - 89
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Annett, J. F., Efficiency of algorithms for Kohn-Sham density functional theory, Comput. ater. Sci., 4 (1995), 23–42.Google Scholar
[2]Blanc, X., Le Bris, C., and Lions, P.-L., From molecular models to continuum mechanics, Arch. Rational. Mech. Anal., 164 (2002), 341–381.CrossRefGoogle Scholar
[3]Born, M. and Huang, K., Dynamical Theory of Crystal Lattices, Oxford University Press, 1954.Google Scholar
[4]Broughton, J. Q., Abraham, F. F., Bernstein, N., and Kaxiras, E., Concurrent coupling of length scales: methodology and application, Phys. Rev. B., 60 (1999), 2391–2403.CrossRefGoogle Scholar
[5]Chipot, M. and Kinderlehrer, D., Equilibrium configurations of crystals, Arch. Rational. Mech. Anal., 103 (1988), 237–277.Google Scholar
[6]Daw, M. S. and Baskes, M. I., Embedded-atom-method: derivation and application to impurities, surfaces and other defects in metals, Phys. Rev. B., 29 (1984), 6443–6453.Google Scholar
[7]Deuflhard, P., Leinen, P., and Yserentant, H., Concepts of an adaptive hierarchical finite element code, IMPACT Comput. Sci. Engrg., 1 (1989), 3–35.Google Scholar
[8]Doye, J. P. K., Wales, D. J., and Berry, R. S., The effect of the range of the potential on the structures of clusters, J. Chem. Phys., 103 (1995), 4234–4249.Google Scholar
[9]Doye, J. P. K., Miller, M. A., and Wales, D. J., Evolution of the potential energy surface with size for Lennard-Jones clusters, J. Chem. Phys., 111 (1999), 8417–8428.Google Scholar
[10]W. E, and Engquist, B., The heterogeneous multiscale methods, Commun. ath. Sci., 1 (2003), 87–132.Google Scholar
[11]W. E, and Ming, P. B., Analysis of multiscale methods, J. Comput. Math., 22 (2004), 210–219.Google Scholar
[12]W. E, and Ming, P. B., Cauchy-Born rule and the stability of crystalline solids: static problems, Arch. ational. Mech. Anal., 183 (2007), 241–297.Google Scholar
[13]Eisenstat, S. C and Walker, H. F., Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17 (1996), 16–32.Google Scholar
[14]Engel, P., Geometric Crystallography: An Axiomatix Introduction to Crystallography, D. Reidel Publishing Company, Dordrecht, Holland, 1986.Google Scholar
[15]Ercolessi, F. and Adams, J. B., Interatomic potentials from first-principles calculations: the force-matching method, Europhys. Lett., 26 (1994), 583–588.Google Scholar
[16]Ericksen, J. L., The Cauchy and Born hypothesis for crystals, Phase Transformations and Material Instabilities in Solids, Gurtin, M. E. ed., Academic Press, 1984, 61–77.Google Scholar
[17]Fonseca, I., The lower quasiconvex envelop of the stored energy for an elastic crystal, J. Math. Pures. Appl., 67 (1988), 175–195.Google Scholar
[18]Friesecke, G. and Theil, F., Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice, J. Nonlinear. Sci., 12 (2002), 445–478.Google Scholar
[19]Goedecker, S., Linear scaling electronic structure methods, Rev. Mod. Phys., 71 (1999), 1085–1123.Google Scholar
[20] Hypre: a library of high performance preconditioners that features parallel multigrid methods for structured and unstructured grid problems: https://computation.lini.gov/casc/linear_solvers/sls_hypre.html.Google Scholar
[21]Keating, P. N., Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure, Phys. Rev., 145 (1966), 637–645.Google Scholar
[22]Knap, J. and Ortiz, M., Effect of indenter-radius size on Au (001) nanoindentation, Phys. Rev. Lett., 90 (2003), 226102.Google Scholar
[23]Lennard-Jones, J. E., On the determination of molecular fields, II, from the equation of state of a gas, Proc. oy. Soc. London. Ser. A., 106 (1924), 463–477.Google Scholar
[24]Li, J., Atom Eye: an efficient atomistic configuration viewer, Model. Simul. Mater. Sci. Eng., 11 (2003), 173–177.CrossRefGoogle Scholar
[25]Li, J., Van Vliet, K. J., Zhu, T., Yip, S., and Suresh, S., Atomistic mechanisms governing the elastic limit and incipient plasticity, Nature., 418 (2002), 307–310.Google Scholar
[26]Marchuk, G. I. and Shaidurov, V. V., Difference Methods and Their Extrapolations, Springer, New York Berlin Heidelberg, 1983.Google Scholar
[27]Miller, R. and Rodney, D., On the nonlocal nature of dislocation nucleation during nanoindentation, J. Mech. Phys. Solids., 56 (2008), 1203–1223.Google Scholar
[28]Ming, P. B., Chen, J., and Jerry Yang, Z., A constrained Cauchy-Born elasticity accelerated multigrid method for nanoindentation problems, preprint, 2010.Google Scholar
[29]Minor, A. M., Asif, S. A. S., Shan, Z. W., Stach, E. A., Cyrankowski, E., Wyrobek, T. J., and Warren, O. L., A new view of the onset of plasticity during the nanoindentation of aluminium, Nature. Mater., 5 (2006), 697–702.Google Scholar
[30]MUMPS: a parallel sparse direct solver: http://graal.ens-lyon.fr/MUMPS/.Google Scholar
[31]Nocedal, J. and Wright, S., Numerical Optimization, Springer-Verlag, 2nd ed., 2006.Google Scholar
[32]Sih, G. and Liebowitz, H., Mathematical theories of brittle fracture, in Fracture: An Advanced Treatise, Mathematical Fundamentals, Liebowitz, H. ed., Academic Press, New York, 1968, 67–190.Google Scholar
[33]Tadmor, E. B., Miller, R., and Phillips, R., Nanoindentation and incipient plasticity, J. Mater. Res., 14 (1999), 2233–2250.CrossRefGoogle Scholar
[34]Tadmor, E. B., Ortiz, M., and Phillips, R., Quasicontinuum analysis of defects in solids, Phil. Mag. A., 73 (1996), 1529–1563.CrossRefGoogle Scholar
[35]Tadmor, E. B., Smith, G. S., Bernstein, N., and Kaxiras, E., Mixed finite element and atomistic formulation for complex crystals, Phys. Rev. B., 59 (1999), 235–245.Google Scholar
[36]Ting, T. C. T., Anisotropic Elasticity: Theory and Applicatoins, Oxford University Press, 1996.Google Scholar
[37]Trottenberg, U., Oosterlee, C. W., and Schuüller, A., Multigrid, Academic Press Inc., San Diego, CA, 2001, with contributions by Brandt, A., Oswald, P., and Stuben, K..Google Scholar
[38]Truskinovsky, L., Fracture as a phase transition, in Comtemporary Resaerch in the Mechanics and Mathematics of Materials, Batra, R. C., and Beatty, M. F. eds., © CIMNE, Barcelona, 1996, 322–332.Google Scholar
[39]Yavari, A., Ortiz, M., and Bhattacharya, K., A theory of anharmonic lattice statics for analysis of defective crytsals, J. Elasticity., 86 (2007), 41–83.CrossRefGoogle Scholar