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Numerische Mathematik

, Volume 140, Issue 3, pp 703–754 | Cite as

Force-based atomistic/continuum blending for multilattices

  • Derek Olson
  • Xingjie Li
  • Christoph Ortner
  • Brian Van Koten
Article
  • 30 Downloads

Abstract

We formulate the blended force-based quasicontinuum method for multilattices and develop rigorous error estimates in terms of the approximation parameters: choice of atomistic region, blending region, and continuum finite element mesh. Balancing the approximation parameters yields a convergent atomistic/continuum multiscale method for multilattices with point defects, including a rigorous convergence rate in terms of the computational cost. The analysis is illustrated with numerical results for a Stone–Wales defect in graphene.

Mathematics Subject Classification

65N12 74S30 

References

  1. 1.
    Abdulle, A., Lin, P., Shapeev, A.: Numerical methods for multilattices. Multiscale Model. Simul. 10(3), 696–726 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abdulle, A., Lin, P., Shapeev, A.: A priori and a posteriori \({W}^{1,\infty }\) error analysis of a qc method for complex lattices. SIAM J. Numer. Anal. 51(4), 2357–2379 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abraham, F., Broughton, J., Bernstein, N., Kaxiras, E.: Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture. Europhys. Lett. 44(6), 783 (1998)CrossRefGoogle Scholar
  4. 4.
    Arroyo, M., Belytschko, T.: Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule. Phys. Rev. B 69, 115415 (2004)CrossRefGoogle Scholar
  5. 5.
    Badia, S., Bochev, P., Gunzburger, M., Lehoucq, R., Parks, M.: Blending methods for coupling atomistic and continuum models. In: Fish, J. (ed.) Bridging the Scales in Science and Engineering, pp. 165–186. Oxford University Press, Oxford (2009)zbMATHGoogle Scholar
  6. 6.
    Badia, S., Bochev, P., Lehoucq, R., Parks, M., Fish, J., Nuggehally, M., Gunzburger, M.: A force-based blending model for atomistic-to-continuum coupling. Int. J. Multiscale Comput. Eng. 5(5), 387–406 (2007)CrossRefGoogle Scholar
  7. 7.
    Badia, S., Parks, M., Bochev, P., Gunzburger, M., Lehoucq, R.: On atomistic-to-continuum coupling by blending. Multiscale Model. Simul. 7(1), 381–406 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bauman, P., Ben Dhia, H., Elkhodja, N., Oden, J., Prudhomme, S., Prudhomme, S.: On the application of the Arlequin method to the coupling of particle and continuum models. Comput. Mech. 42, 511–530 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Born, M., Huang, K.: Dynamical Theory of Crystal Lattices, 1st edn. Clarendon Press, Oxford (1954)zbMATHGoogle Scholar
  10. 10.
    Brenner, S., Scott, R.: The Mathematical Theory of Finite Element Methods, vol. 15. Springer, New York (2008)zbMATHGoogle Scholar
  11. 11.
    Cauchy, A.L.: De la pression ou la tension dans un systeme de points materiels. In: Exercices de Mathematiques (1828)Google Scholar
  12. 12.
    Datta, D., Picu, C., Shephard, M.: Composite grid atomistic continuum method: an adaptive approach to bridge continuum with atomistic analysis. Int. J. Multiscale Comput. Eng. 2(3), 401 (2004)CrossRefGoogle Scholar
  13. 13.
    Dobson, M., Elliott, R., Luskin, M., Tadmor, E.: A multilattice quasicontinuum for phase transforming materials: Cascading Cauchy Born kinematics. J. Comput. Aided Mater. Des. 14(1), 219–237 (2007)CrossRefGoogle Scholar
  14. 14.
    Dobson, M., Luskin, M.: Analysis of a force-based quasicontinuum approximation. Esaim Math. Model. Numer. Anal. 42, 113–139, 0 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Dobson, M., Luskin, M., Ortner, C.: Sharp stability estimates for the force-based quasicontinuum approximation of homogeneous tensile deformation. Multiscale Model. Simul. 8(3), 782–802 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lu, W .E.J., Yang, J.Z.: Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74, 214115 (2006)CrossRefGoogle Scholar
  17. 17.
    Ehrlacher, V., Ortner, C., Shapeev, A.V.: Analysis of boundary conditions for crystal defect atomistic simulations. Arch. Ration. Mech. Anal. 222(3), 1217–1268 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Eidel, B., Stukowski, A.: A variational formulation of the quasicontinuum method based on energy sampling in clusters. J. Mech. Phys. Solids 57(1), 87–108 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ericksen, J.: On the Cauchy-Born rule. Math. Mech. Solids 13(3–4), 199–220 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hubbard, J., Hubbard, B.: Vector calculus, linear algebra, and differential forms: a unified approach. Matrix Editions, fourth edition (2009)Google 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)CrossRefGoogle Scholar
  22. 22.
    Van Koten, B., Ortner, C.: Symmetries of 2-lattices and second order accuracy of the Cauchy-Born model. SIAM Multiscale Model. Simul. 11, 615–634 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Li, X., Luskin, M., Ortner, C.: Positive definiteness of the blended force-based quasicontinuum method. Multiscale Model. Simul. 10(3), 1023–1045 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Li, X., Luskin, M., Ortner, C., Shapeev, A.V.: Theory-based benchmarking of the blended force-based quasicontinuum method. Comput. Methods Appl. Mech. Eng. 268, 763–781 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Li, X., Ortner, C., Shapeev, A.V., Van Koten, B.: Analysis of blended atomistic/continuum hybrid methods. Numer. Math. 134(2), 275–326 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lu, J., Ming, P.: Convergence of a force-based hybrid method in three dimensions. Commun. Pure Appl. Math. 66(1), 83–108 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lu, J., Ming, P.: Stability of a force-based hybrid method with planar sharp interface. SIAM J. Numer. Anal. 52, 2005–2026 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Luskin, M., Ortner, C.: Atomistic-to-continuum coupling. Acta Numerica 22, 397–508, 4 (2013)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Luskin, M., Ortner, C., Van Koten, B.: Formulation and optimization of the energy-based blended quasicontinuum method. Comput. Methods Appl. Mech. Eng. 253, 160–168 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Makridakis, C., Mitsoudis, D., Rosakis, P.: On atomistic-to-continuum couplings without ghost forces in three dimensions. Appl. Math. Res. Express 2014(1), 87–113 (2014)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Miller, R., Tadmor, E.: A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model. Simul. Mater. Sci. Eng. 17(5), 053001 (2009)CrossRefGoogle Scholar
  32. 32.
    Olson, D.: Formulation and Analysis of an Optimization-Based Atomistic-to-Continuum Coupling Algorithm. PhD thesis, University of Minnesota (2015)Google Scholar
  33. 33.
    Olson, D., Bochev, P., Luskin, M., Shapeev, A.: Development of an optimization-based atomistic-to-continuum coupling method. In: Lirkov, I., Margenov, S., Waniewski, J. (eds.) Large-Scale Scientific Computing, volume 8353 of Lecture Notes in Computer Science, pp. 33–44. Springer, Berlin (2014)Google Scholar
  34. 34.
    Olson, D., Li, X., Ortner, C., Van Koten, B.: Force-Based Atomistic/Continuum Blending for multilattices. ArXiv e-prints (2016) 909Google Scholar
  35. 35.
    Olson, D., Ortner, C.: Regularity and locality of point defects in multilattices. Appl. Math. Res. Express 2017, 297–337 (2017)MathSciNetGoogle Scholar
  36. 36.
    Olson, D., Shapeev, A., Bochev, P., Luskin, M.: Analysis of an optimization-based atomistic-to-continuum coupling method for point defects. ESAIM: M2AN 50(1), 1–41 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Ortiz, M., Phillips, R., Tadmor, E.: Quasicontinuum analysis of defects in solids. Philos. Mag. A 73(6), 1529–1563 (1996)CrossRefGoogle Scholar
  38. 38.
    Ortner, C.: A posteriori existence in numerical computations. SIAM J. Numer. Anal. 47(4), 2550–2577 (2009)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Ortner, C., Shapeev, A.: Interpolants of Lattice Functions for the Analysis of Atomistic/Continuum Multiscale Methods. ArXiv e-prints (2012). 1204.3705Google Scholar
  40. 40.
    Ortner, C., Süli, E.: A note on linear elliptic systems on \(\mathbb{R}^d\). ArXiv e-prints (2012) 1202.3970Google Scholar
  41. 41.
    Ortner, C., Theil, F.: Justification of the Cauchy-Born approximation of elastodynamics. Arch. Ration. Mech. Anal. 207, 1025–1073 (2013)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Ortner, C., Zhang, L.: Atomistic/continuum blending with ghost force correction. ArXiv e-prints, 1407.0053, (2014)Google Scholar
  43. 43.
    Scott, R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Seleson, P., Beneddine, S., Prudhomme, S.: A force-based coupling scheme for peridynamics and classical elasticity. Comput. Mater. Sci. 66, 34–49 (2013)CrossRefGoogle Scholar
  45. 45.
    Seleson, P., Gunzburger, M.: Bridging methods for atomistic-to-continuum coupling and their implementation. Commun. Comput. Phys. 7, 831–876 (2010)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Seleson, P., Ha, Y.D., Beneddine, S.: Concurrent coupling of bond-based peridynamics and the navier equation of classical elasticity by blending. Int. J. Multiscale Comput. Eng. 13, 91–113 (2015)CrossRefGoogle Scholar
  47. 47.
    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)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Shapeev, A.V.: Consistent energy-based atomistic/continuum coupling for two-body potentials in one and two dimensions. SIAM J. Multiscale Model. Simul. 9, 905–932 (2011)MathSciNetCrossRefGoogle Scholar
  49. 49.
    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)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Shilkrot, L., Miller, R., Curtin, W.: Coupled atomistic and discrete dislocation plasticity. Phys. Rev. Lett. 89(2), 025501 (2002)CrossRefGoogle Scholar
  51. 51.
    Shimokawa, T., Mortensen, J.J., Schiøtz, 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, 214104 (2004)CrossRefGoogle Scholar
  52. 52.
    Stillinger, F., Weber, T.: Computer simulation of local order in condensed phases of silicon. Phys. Rev. B 31(8), 5262 (1985)CrossRefGoogle Scholar
  53. 53.
    Tadmor, E., Miller, R.: Modeling Materials Continuum, Atomistic and Multiscale Techniques, 1st edn. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  54. 54.
    Xiao, S.P., Belytschko, T.: A bridging domain method for coupling continua with molecular dynamics. Comput. Methods Appl. Mech. Eng. 193(17), 1645–1669 (2004)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Yang, Jerry Z, Weinan, E.: Generalized Cauchy–Born rules for elastic deformation of sheets, plates, and rods: derivation of continuum models from atomistic models. Phys. Rev. B 74(18), 184110 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Derek Olson
    • 1
  • Xingjie Li
    • 2
  • Christoph Ortner
    • 3
  • Brian Van Koten
    • 4
  1. 1.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA
  2. 2.Department of Mathematics and StatisticsUniversity of North Carolina-CharlotteCharlotteUSA
  3. 3.Mathematics InstituteUniversity of WarwickCoventryUK
  4. 4.Department of StatisticsUniversity of ChicagoChicagoUSA

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