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Computational Mechanics

, Volume 55, Issue 4, pp 617–642 | Cite as

A goal-oriented adaptive procedure for the quasi-continuum method with cluster approximation

  • Arash MemarnahavandiEmail author
  • Fredrik Larsson
  • Kenneth Runesson
Original Paper

Abstract

We present a strategy for adaptive error control for the quasi-continuum (QC) method applied to molecular statics problems. The QC-method is introduced in two steps: Firstly, introducing QC-interpolation while accounting for the exact summation of all the bond-energies, we compute goal-oriented error estimators in a straight-forward fashion based on the pertinent adjoint (dual) problem. Secondly, for large QC-elements the bond energy and its derivatives are typically computed using an appropriate discrete quadrature using cluster approximations, which introduces a model error. The combined error is estimated approximately based on the same dual problem in conjunction with a hierarchical strategy for approximating the residual. As a model problem, we carry out atomistic-to-continuum homogenization of a graphene monolayer, where the Carbon–Carbon energy bonds are modeled via the Tersoff–Brenner potential, which involves next-nearest neighbor couplings. In particular, we are interested in computing the representative response for an imperfect lattice. Within the goal-oriented framework it becomes natural to choose the macro-scale (continuum) stress as the “quantity of interest”. Two different formulations are adopted: The Basic formulation and the Global formulation. The presented numerical investigation shows the accuracy and robustness of the proposed error estimator and the pertinent adaptive algorithm.

Keywords

Goal-oriented adaptivity Quasi-continuum Cluster approximation Graphene 

Notes

Acknowledgments

The support from Swedish Research Council is gratefully acknowledged. We thank Prof. Ragnar Larsson for helpful discussions.

References

  1. 1.
    Arndt M, Luskin M (2008) Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel–Kontorova model. Comput Methods Appl Mech Eng 197:4298–4306CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Arroyo M, Belytschko T (2004) Finite element methods for the non-linear mechanics of crystalline sheets and nanotubes. Int J Numer Methods Eng 59:419–456CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Becker R, Rannacher R (2002) An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer 10:1–102MathSciNetGoogle Scholar
  4. 4.
    Brenner DW (1990) Empirical potential for hydrocarbons for use in simulating chemical vapor deposition of diamond films. Phys Rev B 42:9458–9471CrossRefGoogle Scholar
  5. 5.
    Broughton JQ, Abraham FF, Bernstein N, Kaxiras E (1999) Concurrent coupling of length scales: methodology and application. Phys Rev B 60:2391–2403CrossRefGoogle Scholar
  6. 6.
    Chamoin L, Prudhomme S, Dhia HB, Oden JT (2010) Ghost forces and spurious effects in atomic-to-continuum coupling methods by the arlequin approach. Int J Numer Methods Eng 83:1081–1113CrossRefzbMATHGoogle Scholar
  7. 7.
    Chung PW (2004) Computational method for atomistic homogenization of nanopatterned point defect structures. Int J Numer Methods Eng 60:833–859CrossRefzbMATHGoogle Scholar
  8. 8.
    Curtin WA, Miller RE (2003) Atomistic/continuum coupling in computational materials science. Model Simul Mater Sci 11:R33–R68CrossRefGoogle Scholar
  9. 9.
    Eidel B, Stukowski A (2009) A variational formulation of the quasicontinuum method based on energy sampling in clusters. J Mech Phys Solids 57:87–108CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Eriksson K, Estep D, Hansbo P, Johnson C (1995) Introduction to adaptive methods for differential equations. Acta Numer 4:105–158CrossRefMathSciNetGoogle Scholar
  11. 11.
    Fish J (2009) Multiscale methods: bridging the scales in science and engineering. Oxford University Press, OxfordCrossRefGoogle Scholar
  12. 12.
    Grantab R, Shenoy VB, Ruoff RS (2010) Anomalous strength characteristics of tilt grain boundaries in graphene. Science 330:946–948CrossRefGoogle Scholar
  13. 13.
    Hardikar K, Shenoy V, Phillips R (2001) Reconciliation of atomic-level and continuum notions concerning the interaction of dislocations and obstacles. J Mech Phys Solids 49:1951–1967CrossRefzbMATHGoogle Scholar
  14. 14.
    Knap J, Ortiz M (2001) An analysis of the the quasicontinuum method. J Mech Phys Solids 49:1899–1923CrossRefzbMATHGoogle Scholar
  15. 15.
    Kulkarni Y, Knap J, Ortiz M (2008) A variational approach to coarse graining of equilibrium and non-equilibrium atomistic description at finite temperature. J Mech Phys Solids 56:1417–1449CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Larsson F, Runesson K (2004) Modeling and discretization errors in hyperelasto-(visco-)plasticity with a view to hierarchical modeling. Comput Methods Appl Mech Eng 193:5283–5300CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Larsson F, Runesson K (2006) Adaptive computational meso-macro-scale modeling of elastic composites. Comput Methods Appl Mech Eng 195:324–338CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Larsson F, Runesson K (2008) Adaptive bridging of scales in continuum modeling based on error control. Int J Multiscale Com 6:371–392CrossRefGoogle Scholar
  19. 19.
    Larsson F, Hansbo P, Runesson K (2002) Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity. Int J Numer Methods Eng 55:879–894CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Larsson R, Samadikhah K (2011) Atomistic continuum modeling of graphene membranes. Comput Mater Sci 50:1744–1753CrossRefGoogle Scholar
  21. 21.
    Lee C, Wei X, Kysar JW, Hone J (2008) Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321:385–388CrossRefGoogle Scholar
  22. 22.
    Lee G, Cooper R, An S, Lee S, van der Zande A, Petrone N, Hammerberg A, Lee C, Crawford B, Oliver W, Kysar J, Hone J (2013) High-strength chemical-vapor deposited graphene and grain boundaries. Science 340:1073–1076CrossRefGoogle Scholar
  23. 23.
    Liu F, Ming P, Li J (2007) Ab initio calculation of ideal strength and phonon instability of graphene under tension. Phys Rev B 76:064,120CrossRefGoogle Scholar
  24. 24.
    Lu G, Kaxiras E (2004) An overview of multiscale simulations of materials. arXiv:cond-mat/0401073
  25. 25.
    Lu Q, Huang R (2009) Nonlinear mechanics of single-atomic-layer graphene sheets. Int J Appl Mech 1:443–467CrossRefGoogle Scholar
  26. 26.
    Lu Q, Gao W, Huang R (2011) Atomistic simulation and continuum modeling of graphene nanoribbons under uniaxial tension. Model Simul Mater Sci 19:054,006CrossRefGoogle Scholar
  27. 27.
    Luskin M, Ortner C (2009) An analysis of node-based cluster summation rules in the quasicontinuum method. SIAM J Numer Anal 47:3070–3086CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Luskin M, Ortner C, Koten BV (2013) Formulation and optimization of the energy-based blended quasicontinuum method. Comput Methods Appl Mech Eng 253:160–168CrossRefzbMATHGoogle Scholar
  29. 29.
    Marian J, Venturini G, Hansen BL, Knap J, Ortiz M, Campbell GH (2010) Finite-temperature extension of the quasicontinuum method using langevin dynamics: entropy losses and analysis of errors. Model Simul Mater Sci 18:015,003CrossRefGoogle Scholar
  30. 30.
    Miller R, Tadmor E (2002) The quasicontinuum method: overview, applications, and current directions. Comput Aided Des 9:203–239Google Scholar
  31. 31.
    Miller R, Tadmor EB, Phillips R, Ortiz M (1998) Quasicontinuum simulation of fracture at the atomic scale. Model Simul Mater Sci 6:607–638CrossRefGoogle Scholar
  32. 32.
    Miller RE, Tadmor EB (2009) A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model Simul Mater Sci 17:053,001CrossRefGoogle Scholar
  33. 33.
    Novoselov K, Geim A, Morozov S, Jiang D, Zhang Y, Dubonos S, Grigorieva I, Firsov A (2004) Electric field effect in atomically thin carbon films. Science 306:666–669CrossRefGoogle Scholar
  34. 34.
    Novoselov K, Jiang D, Schedin F, Booth T, Khotkevich V, Morozov S, Geim A (2005) Two-dimensional atomic crystals. Proc Natl Acad Sci USA 102:10,451–10,453CrossRefGoogle Scholar
  35. 35.
    Oden JT, Prudhomme S (2002) Estimation of modeling error in computational mechanics. J Comput Phys 182:496–515CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Oden JT, Prudhomme S, Romkes A, Bauman PT (2006) Multi-scale modeling of physical phenomena: adaptive control of models. SIAM J Sci Comput 28:2359–2389CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Prudhomme S, Baumann P, Oden J (2006) Error control for molecular statics problems. Int J Multiscale Comput 4:647–662CrossRefGoogle Scholar
  38. 38.
    Shan W, Nackenhorst U (2009) On the ghost forces in 3d quasicontinuum model. Proc Appl Math Mech 9:411–412CrossRefGoogle Scholar
  39. 39.
    Shan WZ, Nackenhorst U (2010) An adaptive FE-MD model coupling approach. Comput Mech 46:577–596CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Shenoy VB, Miller RE, Tadmor EB, Rodney D, Phillips R, Ortiz M (1999) An adaptive finite element approach to atomic-scale mechanics the quasicontinuum method. J Mech Phys Solids 47:611–642CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Shimokawa T, Mortensen JJ, Schiotz J, Jacobsen KW (2004) Matching conditions in the quasicontinuum method. Phys Rev B 69:214,104CrossRefGoogle Scholar
  42. 42.
    Smith GS, Tadmor EB, Bernstein N, Kaxiras E (2001) Multiscale simulations of silicon nanoindentation. Acta Mater 49:4089–4101CrossRefGoogle Scholar
  43. 43.
    Tadmor EB, Hai S (2003) A peierls criterion for the onset of deformation twinning at a crack tip. J Mech Phys Solids 51:765–793CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Tadmor EB, Miller RE (2011) Modeling materials: continuum, atomistic and multiscale techniques. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  45. 45.
    Tadmor EB, Ortiz M, Phillips R (1996) Quasicontinuum analysis of defects in solids. Philos Mag A 73:1529–1563CrossRefGoogle Scholar
  46. 46.
    Tadmor EB, Miller R, Phillips R, Ortiz M (1999) Nanoindentation and incipient plasticity. Mater Res 14:2233–2250CrossRefGoogle Scholar
  47. 47.
    Tadmor EB, Legoll F, Kim WK, Dupuy LM, Miller RE (2013) Finite-temperature quasi-continuum. Appl Mech Rev 65:010,803CrossRefGoogle Scholar
  48. 48.
    Wei X, Fragneaud B, Marianetti CA, Kysar JW (2009) Nonlinear elastic behavior of graphene: Ab initio calculations to continuum description. Phys Rev B 80:205,407CrossRefGoogle Scholar
  49. 49.
    Zhou J, Huang R (2008) Internal lattice relaxation of single-layer graphene under in-plane deformation. J Mech Phys Solids 56:1609–1623CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Arash Memarnahavandi
    • 1
    • 2
    Email author
  • Fredrik Larsson
    • 1
  • Kenneth Runesson
    • 1
  1. 1.Department of Applied MechanicsChalmers University of TechnologyGöteborgSweden
  2. 2.University of WaterlooWaterlooCanada

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