Archive for Rational Mechanics and Analysis

, Volume 222, Issue 3, pp 1217–1268 | Cite as

Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations

  • V. Ehrlacher
  • C. OrtnerEmail author
  • A. V. Shapeev
Open Access


Numerical simulations of crystal defects are necessarily restricted to finite computational domains, supplying artificial boundary conditions that emulate the effect of embedding the defect in an effectively infinite crystalline environment. This work develops a rigorous framework within which the accuracy of different types of boundary conditions can be precisely assessed. We formulate the equilibration of crystal defects as variational problems in a discrete energy space and establish qualitatively sharp regularity estimates for minimisers. Using this foundation we then present rigorous error estimates for (i) a truncation method (Dirichlet boundary conditions), (ii) periodic boundary conditions, (iii) boundary conditions from linear elasticity, and (iv) boundary conditions from nonlinear elasticity. Numerical results confirm the sharpness of the analysis.


  1. 1.
    Alicandro R., De Luca L., Garroni A., Ponsiglione M.: Metastability and dynamics of discrete topological singularities in two dimensions: a \({\Gamma}\) -convergence approach. Arch. Ration. Mech. Anal. 214, 269–330 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ariza, M.P., Ortiz, M.: Discrete crystal elasticity and discrete dislocations in crystals. Arch. Ration. Mech. Anal. 178(2) 2005Google Scholar
  3. 3.
    Balluffi R.W.: Introduction to Elasticity Theory for Crystal Defects. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bernstein N., Kermode J.R., Csanyi G.: Hybrid atomistic simulation methods for materials systems. Rep. Prog. Phys. 72, 026501 (2009)ADSCrossRefGoogle Scholar
  5. 5.
    Blanc X., Bris C.L., Lions P.L.: Homogenization approach for the numerical simulation of periodic microstructures with defects: proof of concept. Milan J. Math. 80(2), 351–367 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blanc X., Le Bris C., Lions P.L.: From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164(4), 341–381 (2002). doi: 10.1007/s00205-002-0218-5 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bulatov V.V., Cai W.: Computer simulations of dislocations. Oxford Series on Materials Modelling, Vol. 3. Oxford University Press, Oxford (2006)Google Scholar
  8. 8.
    Cai W., Bulatov V.V., Chang J., Li J., Yip S.: Periodic image effects in dislocation modelling. Philos. Mag. 83, 539–567 (2003)ADSCrossRefGoogle Scholar
  9. 9.
    Cances E., Bris C.L.: Mathematical modelling of point defects in materials science. Math. Models Methods Appl. Sci. 23, 1795 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cances E., Deleurence A., Lewin M.: A new approach to the modeling of local defects in crystals: the reduced hartree-fock case. Commun. Math. Phys. 281, 129–177 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Catto I., Bris C.L., Lions P.L.: The Mathematical Theory of Thermodynamic Limits: Thomas–Fermi Type Models. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  12. 12.
    Chen, H., Ortner, C.: QM/MM Methods for Crystalline Defects. Part 2: Consistent Energy and Force-Mixing. arxiv:1509.06627
  13. 13.
    Dedner, A., Ortner, C., Wu, H.: Higher-order finite elements in atomistic/continuum coupling. (in preparation)Google Scholar
  14. 14.
    Weinan, E., Lu, J.,Yang, J.Z.: Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74(21): 214115 2006Google Scholar
  15. 15.
    Weinan, E., Ming, P.: Cauchy–Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183(2), 241–297 2007Google Scholar
  16. 16.
    Ehrlacher, V., Ortner, C., Shapeev, A.V.: Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations. arxiv:1306.5334v4
  17. 17.
    Hine, N., Frensch, K., Foulkes, W., Finnis, M.: Supercell size scaling of density functional theory formation energies of charged defects. Phys. Rev. B 2008, 1–13 2009. doi: 10.1103/PhysRevB.79.024112.
  18. 18.
    Hirth, J.P., Lothe, J.: Theory of Dislocations. Krieger, Malabar, 1982Google Scholar
  19. 19.
    Hudson T., Ortner C.: On the stability of Bravais lattices and their Cauchy–Born approximations. ESAIM:M2AN 46, 81–110 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hudson T., Ortner C.: Existence and stability of a screw dislocation under anti-plane deformation. Arch. Ration. Mech. Anal. 213(3), 887–929 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hudson T., Ortner C.: Analysis of stable screw dislocation configurations in an anti-plane lattice model. SIAM J. Math. Anal. 41, 291–320 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Li X.: Effcient boundary condition for molecular statics models of solids. Phys. Rev. B 80, 104112 (2009)ADSCrossRefGoogle Scholar
  23. 23.
    Li X., Luskin M., Ortner C., Shapeev A.: Theory-based benchmarking of the blended force-based quasicontinuum method. Comput. Methods Appl. Mech. Eng. 268, 763–781 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Li, X.H., Ortner, C., Shapeev, A., Koten, B.V.: Analysis of blended atomistic/continuum hybrid methods. Numer. Math. 2015. doi: 10.1007/s00211-015-0772-z
  25. 25.
    Luskin, M., Ortner, C.: Atomistic-to-continuum-coupling. Acta Numer. 2013Google Scholar
  26. 26.
    Luskin, M., Ortner, C., Van Koten, B.: Formulation and optimization of the energy-based blended quasicontinuum method. Comput. Methods Appl. Mech. Eng. 253 2013Google Scholar
  27. 27.
    Makov, G., Payne, M.C.: Periodic boundary conditions in ab initio calculations. Phys. Rev. B 51, 4014–4022 1995. doi: 10.1103/PhysRevB.51.4014
  28. 28.
    Makridakis C., Mitsoudis D., Rosakis P.: On atomistic-to-continuum couplings without ghost forces in three dimensions. Appl. Math. Res. Express 2014, 87–113 (2012)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Morrey C.B.: Multiple Integrals in the Calculus of Variations. Springer, New York (1966)zbMATHGoogle Scholar
  30. 30.
    Ortiz M., Phillips R., Tadmor E.B.: Quasicontinuum analysis of defects in solids. Philos. Mag. A 73(6), 1529–1563 (1996)ADSCrossRefGoogle Scholar
  31. 31.
    Ortner, C.: The role of the patch test in 2D atomistic-to-continuum coupling methods. ESAIM Math. Model. Numer. Anal. 46 2012Google Scholar
  32. 32.
    Ortner, C., Shapeev, A., Interpolants of lattice functions for the analysis of atomistic/continuum multiscale methods. arxiv:1204.3705 2012
  33. 33.
    Ortner, C., Shapeev, A.V.: Analysis of an energy-based atomistic/continuum coupling approximation of a vacancy in the 2D triangular lattice. Math. Comput. 82 2013Google Scholar
  34. 34.
    Ortner, C., Theil, F.: Justification of the Cauchy–Born approximation of elastodynamics. Arch. Ration. Mech. Anal. 207 2013Google Scholar
  35. 35.
    Ortner, C., Zhang, L.: Construction and sharp consistency estimates for atomistic/continuum coupling methods with general interfaces: a 2D model problem. SIAM J. Numer. Anal. 50 2012Google Scholar
  36. 36.
    Ortner, C., Zhang, L.: Atomistic/continuum blending with ghost force correction. SIAM J. Sci. Comput. 38 2016Google Scholar
  37. 37.
    Packwood, D., Kermode, J., Mones, L., Bernstein, N., Woolley, J., Gould, N.I.M., Ortner, C., Csanyi, G.: A universal preconditioner for simulating condensed phase materials (2016)Google Scholar
  38. 38.
    Shapeev A.V.: Consistent energy-based atomistic/continuum coupling for two-body potential: 1D and 2D case. Multiscale Model. Simul. 9(3), 905–932 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Shapeev, A.V.: Consistent energy-based atomistic/continuum coupling for two-body potentials in three dimensions. SIAM J. Sci. Comput. 34(3), B335–B360 (2012). doi: 10.1137/110844544.
  40. 40.
    Shenoy, V.B., Miller, R., Tadmor, E.B., Rodney, D., Phillips, R., Ortiz, M.: An adaptive finite element approach to atomic-scale mechanics—the quasicontinuum method. J. Mech. Phys. Solids 47(3), 611–642 (1999)Google Scholar
  41. 41.
    Shimokawa T., Mortensen J.J., Schiotz J., Jacobsen K.W.: 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 69(21), 214104 (2004)ADSCrossRefGoogle Scholar
  42. 42.
    Sinclair J.E.: Improved atomistic model of a bcc dislocation core. J. Appl. Phys. 42, 5231 (1971)CrossRefGoogle Scholar
  43. 43.
    Trinkle D.R.: Lattice green function for extended defect calculations: computation and error estimation with long-range forces. Phys. Rev. B 78, 014110 (2008)ADSCrossRefGoogle Scholar
  44. 44.
    Wallace D.: Thermodynamics of Crystals. Dover Publications, New York (1998)Google Scholar
  45. 45.
    Woodward C., Rao S.: Flexible ab initio boundary conditions: simulating isolated dislocations in bcc Mo and Ta. Phys. Rev. Lett. 88, 216402 (2002)ADSCrossRefGoogle Scholar
  46. 46.
    Xiao, S.P., Belytschko, T.: A bridging domain method for coupling continua with molecular dynamics. Comput. Methods Appl. Mech. Eng. 193(17–20), 1645–1669 (2004). doi: 10.1016/j.cma.2003.12.053.
  47. 47.
    Yip, S. (ed.): Handbook of Materials Modellin. Springer, New York, 2005Google Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.CERMICS, ENPCMarne la Valle Cedex 2France
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK
  3. 3.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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