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.
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References
Gracie, R., Belytschko, T.: Concurrently coupled atomistic and XFEM models for dislocations and cracks. Int. J. Numer. Methods Eng. 78(3), 354–378 (2009)
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)
Ziolkowski, R.W.: Metamaterials: the early years in the USA. EPJ Appl. Metamaterials 1 (2014)
Valentine, J., et al.: Three-dimensional optical metamaterial with a negative refractive index. Nature 455(7211), 376 (2008)
Tsu, R.: Man-made superlattice and quantum wells: past and future. Waves Random Complex Media 24(3), 232–239 (2014)
Valentine, J., et al.: Three-dimensional optical metamaterial with a negative refractive index. Nature 455(7211), 376–379 (2008)
Wegener, M.: Metamaterials beyond optics. Science 342(6161), 939–940 (2013)
Gorishnyy, T., et al.: Hypersonic phononic crystals. Phys. Rev. Lett. 94(11), 115501 (2005)
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)
Maldovan, M.: Sound and heat revolutions in phononics. Nature 503(7475), 209–217 (2013)
Zen, N., et al.: Engineering thermal conductance using a two-dimensional phononic crystal. Nat. Commun. 5 (2014)
Liu, Y., Zhang, X.: Metamaterials: a new frontier of science and technology. Chem. Soc. Rev. 40(5), 2494–2507 (2011)
Tsu, R., Fiddy, M.A.: Waves in man-made materials: superlattice to metamaterials. Waves Random Complex Media 24(3), 250–263 (2014)
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)
Shilkrot, L., Miller, R., Curtin, W.: Coupled atomistic and discrete dislocation plasticity. Phys. Rev. Lett. 89(2), 025501 (2002)
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)
Shiari, B., Miller, R.E.: Multiscale modeling of crack initiation and propagation at the nanoscale. J. Mech. Phys. Solids 88, 35–49 (2016)
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)
Pavia, F., Curtin, W.A.: Parallel algorithm for multiscale atomistic/continuum simulations using LAMMPS. Model. Simul. Mater. Sci. Eng. 23(5), 055002 (2015)
Moseley, P., Oswald, J., Belytschko, T.: Adaptive atomistic-to-continuum modeling of propagating defects. Int. J. Numer. Method Eng. 92(10), 835–856 (2012)
Talebi, H., Silani, M., Rabczuk, T.: Concurrent multiscale modeling of three dimensional crack and dislocation propagation. Adv. Eng. Softw. 80, 82–92 (2015)
Chen, Y., et al.: Assessment of atomistic coarse-graining methods. Int. J. Eng. Sci. 49(12), 1337–1349 (2011)
Dove, M.T.: Introduction to Lattice Dynamics. null. vol. null. (1993). null
Kittel, C.: Introduction to Solid State Physics. null. vol. null. (1967). null
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)
Kirkwood, J.G.: The statistical mechanical theory of transport processes I. General theory. J. Chem. Phys. 14(3), 180–201 (1946)
Eringen, A.C.: Microcontinuum Field Theories: Foundations and Solids, vol. 487. Springer, New York (1999)
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)
Eringen, A.C.: Mechanics of Micromorphic Continua. Springer (1968)
Chen, Y., Lee, J.D.: Connecting molecular dynamics to micromorphic theory. (I). Instantaneous and averaged mechanical variables. Phys. A 322, 359–376 (2003)
Chen, Y., Lee, J.D.: Connecting molecular dynamics to micromorphic theory. (II). Balance laws. Phys. A 322, 377–392 (2003)
Chen, Y., Lee, J., Eskandarian, A.: Atomistic counterpart of micromorphic theory. Acta Mech. 161(1–2), 81–102 (2003)
Chen, Y., Lee, J.D.: Determining material constants in micromorphic theory through phonon dispersion relations. Int. J. Eng. Sci. 41(8), 871–886 (2003)
Maugin, G.A.: Some remarks on generalized continuum mechanics. Math. Mech. Solids 20(3), 280–291 (2015)
Maugin, G.A.: Generalized continuum mechanics: various paths. In: Continuum Mechanics Through the Twentieth Century, pp. 223–241. Springer (2013)
Maugin, G.A.: Generalized continuum mechanics: what do we mean by that? In: Mechanics of Generalized Continua, pp. 3–13 (2010)
Chen, Y.: Reformulation of microscopic balance equations for multiscale materials modeling. J. Chem. Phys. 130(13), 134706 (2009)
Chen, Y.: Local stress and heat flux in atomistic systems involving three-body forces. J. Chem. Phys. 124(5), 054113 (2006)
Chen, Y., Lee, J.: Atomistic formulation of a multiscale field theory for nano/micro solids. Philos. Mag. 85(33–35), 4095–4126 (2005)
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)
Deng, Q., Chen, Y.: A coarse-grained atomistic method for 3D dynamic fracture simulation. J. Multiscale Comput. Eng. 11(3), 227–237 (2013)
Deng, Q.: Coarse-Graining Atomistic Dynamics of Fracture by Finite Element Method Formulation, Parallelization and Applications. Fla: University of Florida, Gainesville (2011)
Xiong, L., Chen, Y.: Coarse-grained simulations of single-crystal silicon. Model. Simul. Mater. Sci. Eng. 17, 035002 (2009)
Xiong, L., et al.: Coarse-grained elastodynamics of fast moving dislocations. Acta Mater. 104, 143–155 (2016)
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)
Xiong, L., et al.: A concurrent scheme for passing dislocations from atomistic to continuum domains. Acta Mater. 60(3), 899–913 (2012)
Xiong, L., et al.: Coarse-grained atomistic simulations of dislocations in Al, Ni and Cu crystals. Int. J. Plast. 38, 86–101 (2012)
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)
Xiong, L., et al.: Coarse-grained atomistic simulation of dislocations. J. Mech. Phys. Solids 59(2), 160–177 (2011)
Yang, S., Chen, Y.: Concurrent atomistic-continuum simulation of polycrystalline strontium titanate (2014) (in preparation)
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)
Xu, S., et al.: Validation of the concurrent atomistic-continuum method on screw dislocation/stacking fault interactions. Crystals 7(5), 120 (2017)
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)
Xu, S., et al.: Edge dislocations bowing out from a row of collinear obstacles in Al. Scr. Mater. 123, 135–139 (2016)
Xiong, L., et al.: Concurrent atomistic–continuum simulations of dislocation–void interactions in fcc crystals. Int. J. Plast. 65, 33–42 (2015)
Xu, S., et al.: A quasistatic implementation of the concurrent atomistic-continuum method for FCC crystals. Int. J. Plast. 72, 91–126 (2015)
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)
Chen, X., et al.: Effects of phonons on mobility of dislocations and dislocation arrays. Scr. Mater. 137, 22–26 (2017)
Chen, X., et al.: Ballistic-diffusive phonon heat transport across grain boundaries. Acta Mater. 136(Supplement C), 355–365 (2017)
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)
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)
Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620–630 (1957)
Venturini, G., et al.: Atomistic long-term simulation of heat and mass transport. J. Mech. Phys. Solids 73, 242–268 (2014)
Isihara, A.: The Gibbs-Bogoliubov inequality dagger. J. Phys. A: Gen. Phys. 1(5), 539 (1968)
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)
Mauricio, P., et al.: Dynamic behavior of nano-voids in magnesium under hydrostatic tensile stress. Model. Simul. Mater. Sci. Eng. 24(6), 065003 (2016)
Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithms 18(3), 209 (1998)
Ming, P., Yang, J.Z.: Analysis of a one-dimensional nonlocal quasi-continuum method. Multiscale Model. Simul. 7(4), 1838–1875 (2009)
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)
Ortner, C., Zhang, L.: Atomistic/continuum blending with ghost force correction. SIAM J. Sci. Comput. 38(1), A346–A375 (2016)
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)
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)
Giessen, E.V.d., Needleman, A.: Discrete dislocation plasticity: a simple planar model. Model. Simul. Materials Sci. Eng. 3(5), 689 (1995)
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)
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)
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)
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)
Yang, S., Zhang, N., Chen, Y.: Concurrent atomistic–continuum simulation of polycrystalline strontium titanate. Philos. Mag. 95(24), 2697–2716 (2015)
Xu, S., et al.: Mesh refinement schemes for the concurrent atomistic-continuum method. Int. J. Solids Struct. 90, 144–152 (2016)
Ariza, M.P., et al.: HotQC simulation of nanovoid growth under tension in copper. Int. J. Fract. 174(1), 75–85 (2012)
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)
Baz̆ant, Z.P.: Spurious reflection of elastic waves in nonuniform finite element grids. Comput. Methods Appl. Mech. Eng. 16(1), 91–100 (1978)
Winsberg, E.: Models and theories at the nano-scale. Spontaneous Gener. J. Hist. Philos. Sci. 2(1), 139 (2009)
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.
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Diaz, A., McDowell, D., Chen, Y. (2018). The Limitations and Successes of Concurrent Dynamic Multiscale Modeling Methods at the Mesoscale. In: Altenbach, H., Pouget, J., Rousseau, M., Collet, B., Michelitsch, T. (eds) Generalized Models and Non-classical Approaches in Complex Materials 2. Advanced Structured Materials, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-77504-3_3
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