Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T11:51:17.374Z Has data issue: false hasContentIssue false
  • English
  • Français

A Constrained Cauchy-Born Elasticity Accelerated Multigrid Method for Nanoindentation

Published online by Cambridge University Press:  03 June 2015

Jingrun Chen ,
Pingbing Ming  and
Jerry Zhijian Yang
Jingrun Chen*
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, Beijing, 100190, P.R. China
Pingbing Ming*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, No. 55 East Road Zhong-Guan-Cun, Beijing, 100190, P.R. China
Jerry Zhijian Yang*
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

We introduce a new multigrid method to study the lattice statics model arising from nanoindentation. A constrained Cauchy-Born elasticity model is used as the coarse-grid operator. This method accelerates the relaxation process and considerably reduces the computational cost. In particular, it saves quite a bit when dislocations nucleate and move, as demonstrated by the simulation results.

Keywords

65B9965N3065Z0574G1574G6574S05Multigrid methodconstrained Cauchy-Born elasticitynanoindentationCauchy-Born rule
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 2 , February 2014 , pp. 470 - 486
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Mumps: a parallel sparse direct solver, http://graal.ens-lyon.fr/MUMPS/.Google Scholar
[2]Born, M. and Huang, K., Dynamical theory of crystal lattices, Oxford University Press, 1954.Google Scholar
[3]Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1977), 333–390.Google Scholar
[4]Brandt, A., Multigrid methods in lattice field computations, Nuclear Physics B (Proc. Suppl.) 26 (1992), 137–180.Google Scholar
[5]Chen, J. and Ming, P.B., An efficient multigrid method for molecular mechanics modeling in atomic solids, Commun. Comput. Phys. 10 (2011), 70–89.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.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]E, W. and Ming, P.B., Cauchy-Born rule and the stability of the crystalline solids: Static problems, Arch. Rational Mech. Anal. 183 (2007), 241–297.Google Scholar
[9]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
[10]Ercolessi, F. and Adams, J.B., Interatomic potentials from first-principles calculations: The force-matching method, Europhys. Lett. 26 (1994), 583.CrossRefGoogle Scholar
[11]Fischer-Cripps, A.C., Nanoindentation, Springer-Verlag, 2004.Google Scholar
[12]Goedecker, S., Lançon, F., and Deutsch, T., Linear scaling relaxation of the atomic positions in nanostructures, Phys. Rev. B 64 (2001), 161102.Google Scholar
[13]Gouldstone, A., Chollacoop, N., Dao, M., Li, J., Minor, A.M., and Shen, Y.-L., Indentation across size scales and disciplines: Recent developments in experimentation and modeling, Acta Mater. 55 (2007), 4015–4039.Google Scholar
[14]Knap, J. and Ortiz, M., Effect of indenter-radius size on Au(001) nanoindentation, Phys. Rev. Lett. 90 (2003), 226102.Google Scholar
[15]Lennard-Jones, J.E., On the determination of molecular fields. ii. from the equation of state of a gas, Proc. Roy. Soc. London Ser. A 106 (1924), 463–477.Google Scholar
[16]Li, J., Atomeye: an efficient atomistic configuration viewer, Model. Simul. Mater. Sci. Eng. 11 (2003), 173–177.Google Scholar
[17]Miller, R.E. and Tadmor, E.B., A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods, Model. Simul. Mater. Sci. Eng. 17 (2009), 053001.Google Scholar
[18]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 Materials 5 (20061), 697–702.Google Scholar
[19]Nocedal, J. and Wright, S., Numerical optimization, second ed., Springer-Verlag, 2006.Google Scholar
[20]Oden, J.T., Prudhomme, S., Romkes, A., and Bauman, P., Multiscale modeling of physical phenomena: Adaptive control of models, SIAM J. Sci. Comput. 28 (2006), 2359–2389.Google Scholar
[21]Poon, B., Rittel, D., and Ravichandran, G., An analysis of nanoindentation in linearly elastic solids, Solids, Int. J.Struct. 45 (2008), 6018–6033.Google Scholar
[22]Tadmor, E.B., Ortiz, M., and Phillips, R., Quasicontinuum analysis of defects in solids, Phil. Mag. A 73 (1996), 1529–1563.CrossRefGoogle Scholar
[23]Tadmor, Phillips R., Miller, E.B. and Ortiz, M., Nanoindentation and incipient plasticity, J. Mater. Res. 14 (1999), 2233–2250.Google Scholar
[24]Toselli, A. and Widlund, O., Domain decomposition methods – algorithms and theory, Springer Series in Computational Mathematics, 2004.Google Scholar
[25]Trottenberg, U., Oosterlee, C.W., and Schüller, A., Multigrid, Academic Press Inc., San Diego, CA, 2001, with contributions by Brandt, A. and Oswald, P. and Stüben, K.Google Scholar