Advertisement

The Limitations and Successes of Concurrent Dynamic Multiscale Modeling Methods at the Mesoscale

  • Adrian DiazEmail author
  • David McDowell
  • Youping Chen
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 90)

Abstract

Dynamic concurrent multiscale modeling methods are reviewed and then analyzed based on their governing equations in terms of consistency in material descriptions between different scales, wave propagation across the numerical interfaces between the different descriptions, and advances in describing defects in the coarse-grained domain. The analysis finds that most methods suffer from the consequences of inconsistent materials descriptions between representations at different scales; a few methods such as Concurrent Atomistic Continuum (CAC), Coupled Atomistic Discrete Dislocation (CADD), and the coupled Extended Finite Element Method (XFEM) are capable of simulating moving defects in the coarse-scale domain to improve practicality and prediction. Application of multiscale simulation to coupled thermal and mechanical problems is showing promise. Mesoscale evolution of defects, largely beyond the reach of conventional atomistic methods, is still beyond the reach of many concurrent multiscale methods.

Notes

Acknowledgements

This paper is written in honor of Dr. Gerald Maugin. This material is based upon research supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award #DE-SC0006539.

References

  1. 1.
    Gracie, R., Belytschko, T.: Concurrently coupled atomistic and XFEM models for dislocations and cracks. Int. J. Numer. Methods Eng. 78(3), 354–378 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Hemminger, J., Crabtree, G., Sarrao J.: From quanta to the continuum: opportunities for mesoscale science. A Report from the Basic Energy Sciences Advisory Committee, Technical Report, pp. 1601–1606 (2012)Google Scholar
  3. 3.
    Ziolkowski, R.W.: Metamaterials: the early years in the USA. EPJ Appl. Metamaterials 1 (2014)CrossRefGoogle Scholar
  4. 4.
    Valentine, J., et al.: Three-dimensional optical metamaterial with a negative refractive index. Nature 455(7211), 376 (2008)CrossRefGoogle Scholar
  5. 5.
    Tsu, R.: Man-made superlattice and quantum wells: past and future. Waves Random Complex Media 24(3), 232–239 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Valentine, J., et al.: Three-dimensional optical metamaterial with a negative refractive index. Nature 455(7211), 376–379 (2008)CrossRefGoogle Scholar
  7. 7.
    Wegener, M.: Metamaterials beyond optics. Science 342(6161), 939–940 (2013)CrossRefGoogle Scholar
  8. 8.
    Gorishnyy, T., et al.: Hypersonic phononic crystals. Phys. Rev. Lett. 94(11), 115501 (2005)CrossRefGoogle Scholar
  9. 9.
    Hopkins, P.E., et al.: Reduction in the thermal conductivity of single crystalline silicon by phononic crystal patterning. Nano Lett. 11(1), 107–112 (2010)CrossRefGoogle Scholar
  10. 10.
    Maldovan, M.: Sound and heat revolutions in phononics. Nature 503(7475), 209–217 (2013)CrossRefGoogle Scholar
  11. 11.
    Zen, N., et al.: Engineering thermal conductance using a two-dimensional phononic crystal. Nat. Commun. 5 (2014)Google Scholar
  12. 12.
    Liu, Y., Zhang, X.: Metamaterials: a new frontier of science and technology. Chem. Soc. Rev. 40(5), 2494–2507 (2011)CrossRefGoogle Scholar
  13. 13.
    Tsu, R., Fiddy, M.A.: Waves in man-made materials: superlattice to metamaterials. Waves Random Complex Media 24(3), 250–263 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Shiari, B., Miller, R.E., Klug, D.D.: Multiscale modeling of solids at the nanoscale: dynamic approach. Can. J. Phys. 86(2), 391–400 (2008)CrossRefGoogle Scholar
  15. 15.
    Shilkrot, L., Miller, R., Curtin, W.: Coupled atomistic and discrete dislocation plasticity. Phys. Rev. Lett. 89(2), 025501 (2002)CrossRefGoogle Scholar
  16. 16.
    Shilkrot, L., Miller, R.E., Curtin, W.A.: Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics. J. Mech. Phys. Solids 52(4), 755–787 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Shiari, B., Miller, R.E.: Multiscale modeling of crack initiation and propagation at the nanoscale. J. Mech. Phys. Solids 88, 35–49 (2016)CrossRefGoogle Scholar
  18. 18.
    Miller, R.E., Tadmor, E.B.: A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model. Simul. Mater. Sci. Eng. 17(5), 053001 (2009)CrossRefGoogle Scholar
  19. 19.
    Pavia, F., Curtin, W.A.: Parallel algorithm for multiscale atomistic/continuum simulations using LAMMPS. Model. Simul. Mater. Sci. Eng. 23(5), 055002 (2015)CrossRefGoogle Scholar
  20. 20.
    Moseley, P., Oswald, J., Belytschko, T.: Adaptive atomistic-to-continuum modeling of propagating defects. Int. J. Numer. Method Eng. 92(10), 835–856 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Talebi, H., Silani, M., Rabczuk, T.: Concurrent multiscale modeling of three dimensional crack and dislocation propagation. Adv. Eng. Softw. 80, 82–92 (2015)CrossRefGoogle Scholar
  22. 22.
    Chen, Y., et al.: Assessment of atomistic coarse-graining methods. Int. J. Eng. Sci. 49(12), 1337–1349 (2011)zbMATHCrossRefGoogle Scholar
  23. 23.
    Dove, M.T.: Introduction to Lattice Dynamics. null. vol. null. (1993). nullGoogle Scholar
  24. 24.
    Kittel, C.: Introduction to Solid State Physics. null. vol. null. (1967). nullGoogle Scholar
  25. 25.
    Irving, J., Kirkwood, J.G.: The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18(6), 817–829 (1950)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kirkwood, J.G.: The statistical mechanical theory of transport processes I. General theory. J. Chem. Phys. 14(3), 180–201 (1946)CrossRefGoogle Scholar
  27. 27.
    Eringen, A.C.: Microcontinuum Field Theories: Foundations and Solids, vol. 487. Springer, New York (1999)zbMATHCrossRefGoogle Scholar
  28. 28.
    Chen, Y., Lee, J.D., Eskandarian, A.: Examining the physical foundation of continuum theories from the viewpoint of phonon dispersion relation. Int. J. Eng. Sci. 41(1), 61–83 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Eringen, A.C.: Mechanics of Micromorphic Continua. Springer (1968)Google Scholar
  30. 30.
    Chen, Y., Lee, J.D.: Connecting molecular dynamics to micromorphic theory. (I). Instantaneous and averaged mechanical variables. Phys. A 322, 359–376 (2003)zbMATHCrossRefGoogle Scholar
  31. 31.
    Chen, Y., Lee, J.D.: Connecting molecular dynamics to micromorphic theory. (II). Balance laws. Phys. A 322, 377–392 (2003)zbMATHCrossRefGoogle Scholar
  32. 32.
    Chen, Y., Lee, J., Eskandarian, A.: Atomistic counterpart of micromorphic theory. Acta Mech. 161(1–2), 81–102 (2003)zbMATHCrossRefGoogle Scholar
  33. 33.
    Chen, Y., Lee, J.D.: Determining material constants in micromorphic theory through phonon dispersion relations. Int. J. Eng. Sci. 41(8), 871–886 (2003)CrossRefGoogle Scholar
  34. 34.
    Maugin, G.A.: Some remarks on generalized continuum mechanics. Math. Mech. Solids 20(3), 280–291 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Maugin, G.A.: Generalized continuum mechanics: various paths. In: Continuum Mechanics Through the Twentieth Century, pp. 223–241. Springer (2013)Google Scholar
  36. 36.
    Maugin, G.A.: Generalized continuum mechanics: what do we mean by that? In: Mechanics of Generalized Continua, pp. 3–13 (2010)Google Scholar
  37. 37.
    Chen, Y.: Reformulation of microscopic balance equations for multiscale materials modeling. J. Chem. Phys. 130(13), 134706 (2009)CrossRefGoogle Scholar
  38. 38.
    Chen, Y.: Local stress and heat flux in atomistic systems involving three-body forces. J. Chem. Phys. 124(5), 054113 (2006)CrossRefGoogle Scholar
  39. 39.
    Chen, Y., Lee, J.: Atomistic formulation of a multiscale field theory for nano/micro solids. Philos. Mag. 85(33–35), 4095–4126 (2005)CrossRefGoogle Scholar
  40. 40.
    Deng, Q., Xiong, L., Chen, Y.: Coarse-graining atomistic dynamics of brittle fracture by finite element method. Int. J. Plast. 26(9), 1402–1414 (2010)zbMATHCrossRefGoogle Scholar
  41. 41.
    Deng, Q., Chen, Y.: A coarse-grained atomistic method for 3D dynamic fracture simulation. J. Multiscale Comput. Eng. 11(3), 227–237 (2013)CrossRefGoogle Scholar
  42. 42.
    Deng, Q.: Coarse-Graining Atomistic Dynamics of Fracture by Finite Element Method Formulation, Parallelization and Applications. Fla: University of Florida, Gainesville (2011)Google Scholar
  43. 43.
    Xiong, L., Chen, Y.: Coarse-grained simulations of single-crystal silicon. Model. Simul. Mater. Sci. Eng. 17, 035002 (2009)CrossRefGoogle Scholar
  44. 44.
    Xiong, L., et al.: Coarse-grained elastodynamics of fast moving dislocations. Acta Mater. 104, 143–155 (2016)CrossRefGoogle Scholar
  45. 45.
    Xu, S., et al.: An analysis of key characteristics of the Frank-Read source process in FCC metals. J. Mech. Phys. Solids 96, 460–476 (2016)CrossRefGoogle Scholar
  46. 46.
    Xiong, L., et al.: A concurrent scheme for passing dislocations from atomistic to continuum domains. Acta Mater. 60(3), 899–913 (2012)CrossRefGoogle Scholar
  47. 47.
    Xiong, L., et al.: Coarse-grained atomistic simulations of dislocations in Al, Ni and Cu crystals. Int. J. Plast. 38, 86–101 (2012)CrossRefGoogle Scholar
  48. 48.
    Xiong, L., McDowell, D.L., Chen, Y.: Nucleation and growth of dislocation loops in Cu, Al and Si by a concurrent atomistic-continuum method. Scr. Mater. 67(7), 633–636 (2012)CrossRefGoogle Scholar
  49. 49.
    Xiong, L., et al.: Coarse-grained atomistic simulation of dislocations. J. Mech. Phys. Solids 59(2), 160–177 (2011)zbMATHCrossRefGoogle Scholar
  50. 50.
    Yang, S., Chen, Y.: Concurrent atomistic-continuum simulation of polycrystalline strontium titanate (2014) (in preparation)Google Scholar
  51. 51.
    Xu, S., et al.: Comparing EAM Potentials to Model Slip Transfer of Sequential Mixed Character Dislocations Across Two Symmetric Tilt Grain Boundaries in Ni. JOM, 1–8 (2017)Google Scholar
  52. 52.
    Xu, S., et al.: Validation of the concurrent atomistic-continuum method on screw dislocation/stacking fault interactions. Crystals 7(5), 120 (2017)CrossRefGoogle Scholar
  53. 53.
    Xu, S., et al.: Sequential slip transfer of mixed-character dislocations across Σ3 coherent twin boundary in FCC metals: a concurrent atomistic-continuum study. npj Comput. Materials 2, 15016 (2016)Google Scholar
  54. 54.
    Xu, S., et al.: Edge dislocations bowing out from a row of collinear obstacles in Al. Scr. Mater. 123, 135–139 (2016)CrossRefGoogle Scholar
  55. 55.
    Xiong, L., et al.: Concurrent atomistic–continuum simulations of dislocation–void interactions in fcc crystals. Int. J. Plast. 65, 33–42 (2015)CrossRefGoogle Scholar
  56. 56.
    Xu, S., et al.: A quasistatic implementation of the concurrent atomistic-continuum method for FCC crystals. Int. J. Plast. 72, 91–126 (2015)CrossRefGoogle Scholar
  57. 57.
    Xiong, L., McDowell, D.L., Chen, Y.: Sub-THz Phonon drag on dislocations by coarse-grained atomistic simulations. Int. J. Plast. 55, 268–278 (2014)CrossRefGoogle Scholar
  58. 58.
    Chen, X., et al.: Effects of phonons on mobility of dislocations and dislocation arrays. Scr. Mater. 137, 22–26 (2017)CrossRefGoogle Scholar
  59. 59.
    Chen, X., et al.: Ballistic-diffusive phonon heat transport across grain boundaries. Acta Mater. 136(Supplement C), 355–365 (2017)CrossRefGoogle Scholar
  60. 60.
    Yang, S., Chen Y.: Concurrent Atomistic-Continuum Simulation of Defects in Polyatomic Ionic Materials, in Multiscale Materials Modeling for Nanomechanics, pp. 261–296. Springer International Publishing (2016)Google Scholar
  61. 61.
    Yang, S., Chen, Y.: Concurrent atomistic and continuum simulation of bi-crystal strontium titanate with tilt grain boundary. Proc. R. Soc. A Math. Phys. Eng. Sci. 471(2175) (2015)CrossRefGoogle Scholar
  62. 62.
    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620–630 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Venturini, G., et al.: Atomistic long-term simulation of heat and mass transport. J. Mech. Phys. Solids 73, 242–268 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Isihara, A.: The Gibbs-Bogoliubov inequality dagger. J. Phys. A: Gen. Phys. 1(5), 539 (1968)CrossRefGoogle Scholar
  65. 65.
    Ponga, M., Ortiz, M., Ariza, M.P.: Finite-temperature non-equilibrium quasi-continuum analysis of nanovoid growth in copper at low and high strain rates. Mech. Mater. 90, 253–267 (2015)CrossRefGoogle Scholar
  66. 66.
    Mauricio, P., et al.: Dynamic behavior of nano-voids in magnesium under hydrostatic tensile stress. Model. Simul. Mater. Sci. Eng. 24(6), 065003 (2016)CrossRefGoogle Scholar
  67. 67.
    Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithms 18(3), 209 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Ming, P., Yang, J.Z.: Analysis of a one-dimensional nonlocal quasi-continuum method. Multiscale Model. Simul. 7(4), 1838–1875 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Amelang, J.S., Venturini, G.N., Kochmann, D.M.: Summation rules for a fully nonlocal energy-based quasicontinuum method. J. Mech. Phys. Solids 82, 378–413 (2015)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Ortner, C., Zhang, L.: Atomistic/continuum blending with ghost force correction. SIAM J. Sci. Comput. 38(1), A346–A375 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Jeremy, A.T., Reese, E.J., Gregory, J.W.: Application of a field-based method to spatially varying thermal transport problems in molecular dynamics. Model. Simul. Mater. Sci. Eng. 18(8), 085007 (2010)CrossRefGoogle Scholar
  72. 72.
    Wagner, G.J., et al.: An atomistic-to-continuum coupling method for heat transfer in solids. Comput. Methods Appl. Mech. Eng. 197(41), 3351–3365 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Giessen, E.V.d., Needleman, A.: Discrete dislocation plasticity: a simple planar model. Model. Simul. Materials Sci. Eng. 3(5), 689 (1995)CrossRefGoogle Scholar
  74. 74.
    Jiang, L., Rogers, R.J.: Spurious wave reflections at an interface of different physical properties in finite-element wave solutions. Commun. Appl. Numer. Methods 7(8), 595–602 (1991)zbMATHCrossRefGoogle Scholar
  75. 75.
    Xu, M., Belytschko, T.: Conservation properties of the bridging domain method for coupled molecular/continuum dynamics. Int. J. Numer. Methods Eng. 76(3), 278–294 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Bažant, Z.P., Celep, Z.: Spurious reflection of elastic waves in nonuniform meshes of constant and linear strain unite elements. Comput. Struct. 15(4), 451–459 (1982)zbMATHCrossRefGoogle Scholar
  77. 77.
    Xiong, L., Chen, Y.: Multiscale modeling and simulation of single-crystal MgO through an atomistic field theory. Int. J. Solids Struct. 46(6), 1448–1455 (2009)zbMATHCrossRefGoogle Scholar
  78. 78.
    Yang, S., Zhang, N., Chen, Y.: Concurrent atomistic–continuum simulation of polycrystalline strontium titanate. Philos. Mag. 95(24), 2697–2716 (2015)CrossRefGoogle Scholar
  79. 79.
    Xu, S., et al.: Mesh refinement schemes for the concurrent atomistic-continuum method. Int. J. Solids Struct. 90, 144–152 (2016)CrossRefGoogle Scholar
  80. 80.
    Ariza, M.P., et al.: HotQC simulation of nanovoid growth under tension in copper. Int. J. Fract. 174(1), 75–85 (2012)CrossRefGoogle Scholar
  81. 81.
    Chernatynskiy, A., Phillpot, S.R.: Phonon-mediated thermal transport: confronting theory and microscopic simulation with experiment. Curr. Opin. Solid State Mater. Sci. 17(1), 1–9 (2013)CrossRefGoogle Scholar
  82. 82.
    Baz̆ant, Z.P.: Spurious reflection of elastic waves in nonuniform finite element grids. Comput. Methods Appl. Mech. Eng. 16(1), 91–100 (1978)CrossRefGoogle Scholar
  83. 83.
    Winsberg, E.: Models and theories at the nano-scale. Spontaneous Gener. J. Hist. Philos. Sci. 2(1), 139 (2009)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of FloridaGainesvilleUSA
  2. 2.Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA
  3. 3.School of Materials Science and EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations