Bridging the scales in nano engineering and science
Source abstract
Abstract
This review article describes various multiscale approaches, development of which was spurred by the emergence of nanotechnology. The multiscale approaches are grouped into two main categories: information-passing and concurrent. In the concurrent multiscale methods both, the discrete and continuum scales are simultaneously resolved, whereas in the information-passing schemes, the discrete scale is modelled and its gross response is infused into the continuum scale. Most of the information-passing approaches provide sublinear computational complexity, (i.e., scales sublinearly with the cost of solving a fine scale problem), but the quantities of interest are limited to or defined only on the coarse scale. The issues of appropriate scale selection and uncertainty quantification are also reviewed.
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OpenAlex topics: Advanced Mathematical Modeling in Engineering · Composite Material Mechanics · Probabilistic and Robust Engineering Design
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References (100+)
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1. Adhikari N.P., X.H. Peng, et al. 2004. Multiscale modeling of the surfactant mediated synthesis and supramolecular assembly of cobalt nanodots. Phys. Rev. Lett. 93(18), 188301‐1–410.1103/PhysRevLett.93.188301 ↗
2. Aidun J., 2005. Private Communications. Sandia National Laboratory
3. Antoulas A., D. Sorensen, et al. (2004). Model reduction of large-scale dynamical systems. Comput. Sci. – ICCS 2004, Pt.␣3, Proceedings 3038: 740–74710.1007/978-3-540-24688-6_96 ↗
4. Aubry S., D.J. Bammann, et al. 2004. A Robust, Coupled Approach for Atomistic-Continuum Simulation. Albuquerque, SAND2004-477810.2172/919657 ↗
5. Babuska I., Caloz G., et al. (1994). Special finite-element methods for a class of 2nd-order elliptic problems with rough coefficients. Siam J. Num. Anal. 31(4): 945–98110.1137/0731051 ↗
6. Babuska I. and Osborn J.E. (1983). Generalized finite-element methods – their performance and their relation to mixed methods. Siam J. Num. Anal. 20(3): 510–53610.1137/0720034 ↗
7. Belytschko T. and Xiao S. (2003). Coupling methods for continuum model with molecular model. Int. J. Multiscale Comput. Eng. 1: 115–12610.1615/IntJMultCompEng.v1.i1.100 ↗
8. Ben Dhia H. and Rateau G. (2005). The Arlequin method as a flexible engineering design tool. Int. J. Int. J. Num. Meth. Eng. 62(11): 1442–146210.1002/nme.1229 ↗
9. Bottasso C.L. (2002). Multiscale temporal integration. Computer Meth. Appl. Mech. Eng. 191(25–26): 2815-283010.1016/S0045-7825(02)00219-0 ↗
10. Brenner D.W. (1990). Empirical potential for hydrocarbons for use in simulating the chemical vapor-deposition of diamond films. Phys. Rev. B 42(15): 9458–947110.1103/PhysRevB.42.9458 ↗
11. Broughton J.Q., Abraham F.F., et al. (1999). Concurrent coupling of length scales: methodology and application. Phys. Rev. B 60(4): 2391–240310.1103/PhysRevB.60.2391 ↗
12. Cai W., de Koning M., et al. (2000). Minimizing boundary reflections in coupled-domain simulations. Phys. Rev. Lett. 85(15): 3213–321610.1103/PhysRevLett.85.3213 ↗
13. Car R., Parrinello M.(1985). Unified approach for molecular-dynamics and density-functional theory. Phys. Rev. Lett. 55(22): 2471–247410.1103/PhysRevLett.55.2471 ↗
14. Chen W. & J. Fish, 2006. A generalized space–time mathematical homogenization theory for bridging atomistic and continuum scales. Int. J. Num. Meth. Eng. 67, 253–27110.1002/nme.1630 ↗
15. Chen W. & J. Fish 2006. ``A mathematical homogenization perspective of virial stress''. Int. J. Num. Meth. Engng. 6, 189–20710.1002/nme.1622 ↗
16. Chong K.P. (2004). Nanoscience and engineering in mechanics and materials. J. Phys. Chem. Solids 65(8–9): 1501–150610.1016/j.jpcs.2003.09.032 ↗
17. Chung P.W. (2004). Computational method for atomistic homogenization of nanopatterned point defect structures. Int. J. Num. Meth. Eng. 60(4): 833–85910.1002/nme.989 ↗
18. Cuitino A.M. (1997). A time-dependent deformation mechanism in metallic fcc crystals. Acta Mater. 45(6): 2509–252210.1016/S1359-6454(96)00328-X ↗
19. Curtin W.A. and Miller R.E. (2003). Atomistic/continuum coupling in computational materials science. Model. Simul. Mater. Sci. Eng. 11(3): R33–R6810.1088/0965-0393/11/3/201 ↗
20. Datta D.K., Picu C., et al. (2004). Composite grid atomistic continuum method, an adaptive approach to bridge continuum with atomistic analysis. Int. J. Multiscale Comput. Eng. 2(3): 401–41910.1615/IntJMultCompEng.v2.i3.40 ↗
21. E W., Engquist B., et al. (2003). Heterogeneous multiscale method: a general methodology for multiscale modeling. Phys. Rev. B 67(9): 1–4
22. E W.N. & Z.Y. Huang, 2001. Matching conditions in atomistic-continuum modeling of materials. Phys. Rev. Lett. 87(13), 13501-1–410.1103/PhysRevLett.87.135501 ↗
23. Fish J. and Belsky V. (1995). Multi-grid method for periodic heterogeneous media part 2: multiscale modeling and quality control in multidimensional case. Comput. Meth. Appl. Mech. Eng. 126: 17–3810.1016/0045-7825(95)00812-F ↗
24. Fish J. & V. Belsky, 1995. Multigrid method for periodic heterogeneous media part 1: convergence studies for one-dimensional case. Comput. Meth. Appl. Mech. Eng. 126: 1–1610.1016/0045-7825(95)00811-E ↗
25. Fish J. and Belsky V. (1997). Generalized aggregation multilevel solver. Int. J. Num. Meth. Eng. 40(23): 4341–436110.1002/(SICI)1097-0207(19971215)40:23<4341::AID-NME261>3.0.CO;2-C ↗
26. Fish J. and Chen W. (2001). Higher-order homogenization of initial/boundary-value problem. J. Eng. Mech.-Asce 127(12): 1223–123010.1061/(ASCE)0733-9399(2001)127:12(1223) ↗
27. Fish J. and Chen W. (2004). Discrete-to-continuum bridging based on multigrid principles. Comput. Meth. Appl. Mech. Eng. 193: 1693–171110.1016/j.cma.2003.12.022 ↗
28. Fish J., W. Chen & Y. Tang, 2005. ``Generalized mathematical homogenization of atomistic media at finite temperatures''. Int. J. Mult. Comput. Engng. 3(4), 393–41310.1615/IntJMultCompEng.v3.i4.10 ↗
29. Fish J., Chen W., et al. (2002). Non-local dispersive model for wave propagation in heterogeneous media: multi-dimensional case. Int. J. Num. Meth. Eng. 54(3): 34710.1002/nme.424 ↗
30. Fish J., Chen W., et al. (2002). Non-local dispersive model for wave propagation in heterogeneous media: one-dimensional case. Int. J. Num. Meth. Eng. 54(3): 331–34610.1002/nme.423 ↗
31. Fish J., W. Chen, et al. 2005. Generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions. Comp. Meth. Appl. Mech. Engng. (in print)
32. Fish J. and Markolefas S. (1992). The S-version of the finite-element method for multilayer laminates. Int. J. Num. Meth. Eng. 33(5): 1081–110510.1002/nme.1620330512 ↗
33. Fish J., Markolefas S., et al. (1994). “On Adaptive Multilevel Superposition of Finite-Element Meshes for Linear Elastostatics”. Appl. Num. Math. 14(1–3): 135–16410.1016/0168-9274(94)90023-X ↗
34. Fish J., Nayak P., et al. (1994). Microscale reduction error indicators and estimators for a periodic heterogeneous medium. Comput. Mech. 14(4): 323–33810.1007/BF00350003 ↗
35. Fish J. and Qu Y. (2000). Global-basis two-level method for indefinite systems. Part 1: convergence studies. Int. J. Num. Meth. Eng. 49(3): 439–46010.1002/1097-0207(20000930)49:3<439::AID-NME981>3.0.CO;2-A ↗
36. Fish J. and Schwob C. (2003). Towards constitutive model based on atomistics. J. Multiscale Comput. Eng. 1: 43–56
37. Fish J. and Yuan Z. (2005). Multiscale enrichment based on partition of unity. Int. J. Num. Meth. Eng. 62(10): 1341–135910.1002/nme.1230 ↗
38. Ghanem R. and Pellissetti M. (2002). “Adaptive data refinement in the spectral stochastic finite element method”. Commun. Num. Meth. Eng. 18(2): 141–15110.1002/cnm.476 ↗
39. Ghanem R. & J. Shi, 2002. A stochastic multi-scale characterization of heterogeneous materials. 4th Int. Conf. on Computational Stochastic Mechanics, Corfu, Greece
40. Ghoniem N.M., 2004. Computational Dislocation Dynamics: A Practioner’s Manual. UCLA Publication
41. Ghoniem N.M. and Cho K. (2002). The emerging role of multiscale modeling in nano- and micro-mechanics of materials. CMES-Comput. Model. Eng. Sci. 3(2): 147–173
42. Giladi E. and Keller H.B. (2002). Space-time domain decomposition for parabolic problems. Numer. Math. 93(2): 279–31310.1007/s002110100345 ↗
43. Gressick W., Wen J., et al. (2005). Order reduction for large scale finite element models: a system perspective. Int. J. Multiscale Comput. Eng. 3(3): 337–36210.1615/IntJMultCompEng.v3.i3.60 ↗
44. Grujicic M., Cao G., et al. (2003). Multiscale modelling of delamination and fracture of polycrystaline lamellar alloys. Int. J. Multiscale Comput. Eng. 1:1–2210.1615/IntJMultCompEng.v1.i1.20 ↗
45. Hardy R.J. (1982). Formulas for determining local properties in molecular-dynamics simulations – shock-waves. J. Chem. Phys. 76(1): 622–62810.1063/1.442714 ↗
46. Heine D.R., Grest G.S., et al. (2005). Diverse spreading behavior of binary polymer nanodroplets. Langmuir 21(17): 7959–796310.1021/la050139t ↗
47. Horton G. and Vandewalle S. (1995). A space–time multigrid method for parabolic partial-differential equations. Siam J. Sci. Comput. 16(4): 848–86410.1137/0916050 ↗
48. Hou T.Y. and Wu X.H. (1997). A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1): 169–18910.1006/jcph.1997.5682 ↗
49. Hughes T.J.R. (1995). Multiscale phenomena – greens-functions, the Dirichlet-to-Neumann Formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Meth. Appl. Mech. Eng. 127(1–4): 387–40110.1016/0045-7825(95)00844-9 ↗
50. Hurst J.L. and Wen J. (2005). Computation of Shear Viscosity: A Systems Approach. American Control Conference, Portland, OR
51. Irving J.H. and Kirkwood J.G. (1950). The statistical mechanical theory of transport processes.4. The equations of hydrodynamics. J. Chem. Phys. 18(6): 817–82910.1063/1.1747782 ↗
52. Johnson H.T., Bose R., et al. (203). Effects of externally applied stress on the properties of quantum dot nanostructutres. Int. J. Multiscale Comput. Eng. 1: 33–4210.1615/IntJMultCompEng.v1.i1.40 ↗
53. Kevrekidis I.G., Gear C.W., et al. (2004). Equation-free: the computer-aided analysis of comptex multiscale systems. AIChE J. 50(7): 1346–135510.1002/aic.10106 ↗
54. Kleiber M. & D. Tran, 1992. The Stochastic Finite Element Method: Basic Perturbation Technique and Computer Implementation. NY: Wiley
55. Knapek S. (1999). Matrix-dependent multigrid-homogenization for diffusion problems. SIAM J. Sci. Comput. 20: 515–53310.1137/S1064827596304848 ↗
56. Kulkarni A., Zhou M., et al. (2005). Orientation and size dependence of the elastic properties of zinc oxide nanobelts. Nanotechnology 16: 2749–275610.1088/0957-4484/16/12/001 ↗
57. Liang W. and Zhou M. (2004). Response of copper nanowires in dynamic tensile deformation. Proc. Inst. Mech. Eng. Part C – J. Mech. Eng. Sci. 218(6): 599–60610.1243/095440604774202231 ↗
58. Liang W. and Zhou M. (2005). Superelasticity of single crystalline Cu nanowires through reversible lattice reorientations. J. Eng. Mater. Tech. 127(4): 423–43310.1115/1.1928915 ↗
59. Liu W.K., Karpov E.G., et al. (2004). An introduction to computational nanomechanics and materials. Comput. Meth. Appl. Mech. Eng. 193(17–20): 1529–157810.1016/j.cma.2003.12.008 ↗
60. Lumley J.L. (1967). Similarity and turbulent energy spectrum. Phys. Fluids 10(4): 85510.1063/1.1762200 ↗
61. Marburger J.H., M.C. Roco, et al., 2004. National Nanotechnology Initiative. Supplement to the President’s FY 2004 Budget
62. McVeigh C., F. Vernerey, et al., 2006. Multiresolution Analysis for Material Design. Comp. Meth. Appl. Mech. Eng. (in print)
63. Melenk J.M. and Babuska I. (1996). The partition of unity finite element method: basic theory and applications. Comput. Meth. Appl. Mech. Eng. 139(1–4): 289–31410.1016/S0045-7825(96)01087-0 ↗
64. Miekkala U. and Nevanlinna O. (1987). Convergence of dynamic iteration methods for initial-value problems. Siam J. Sci. Stat. Computing 8(4): 459–48210.1137/0908046 ↗
65. Miller R.E. (2003). Direct coupling of atomistic and continuum mechanics in computational material science. Int. J. Multiscale Comput. Eng. 1: 57–7210.1615/IntJMultCompEng.v1.i1.60 ↗
66. Moes N., Dolbow J., et al. (1999). A finite element method for crack growth without remeshing. Int. J. Num. Meth. Eng. 46(1): 131–15010.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J ↗
67. Moriarty J.A., Vitek V., et al. (2002). Atomistic simulations of dislocations and defects. J. Comput.-Aided Mater. Design 9(2): 99–13210.1023/A:1026022602578 ↗
68. Moulton J.D., Dendy J.E., et al. (1998). The black box multigrid numerical homogenization algorithm. J. Comput. Phys. 142(1): 80–10810.1006/jcph.1998.5911 ↗
69. Muralidharan K., Deymier P.A., et al. (2003). A concurrent multiscale finite difference time domain/molecular dynamics method for bridging an elastic continuum to an atomic system. Model. Simul. Mater. Sci. Eng. 11(4): 487–50110.1088/0965-0393/11/4/306 ↗
70. Namba K., 2004. Revealing the mystery of the bacterial flagellum – a self-assembling nanomachine with fine switching capability. Japan Nanonet Bulletin (11th Issue)10.1240/sav_gbm_2004_h_000705 ↗
71. NSF, 2001. Nanoscale Science and Engineering. National Science Foundation. N. 03-043. Arlington, VA
72. Oden J.T., T. Belytschko, et al., 2006. Simulation-based engineering science. National Science Foundation
73. Oden J.T., A. Prudhomme, et al., 2005. Multi-scale modelling of physical phenomena: adaptive control of models. ICES Report 05-13, Austin
74. Oden J.T. and Vemaganti K.S. (2000). Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials I. Error estimates and adaptive algorithms. J. Comput. Phys. 164(1): 22–4710.1006/jcph.2000.6585 ↗
75. Park H.S., Karpov E.G., et al. (2005). Three-dimensional bridging scale analysis of dynamic fracture. J. Comput. Phys. 207(2): 588–60910.1016/j.jcp.2005.01.028 ↗
76. Park H.S., Karpov E.G., et al. (2005). The bridging scale for two-dimensional atomistic/continumn coupling. Philos. Mag. 85(1): 79–11310.1080/14786430412331300163 ↗
77. Park H.S. & J.A. Zimmerman, 2005. Modeling inelasticity and failure in gold nanowires. Phys. Rev. B 72(5), 54106-1–910.1103/PhysRevB.72.054106 ↗
78. Parr R.G. and Yang W. (1989). Density-Functional Theory of Atoms and Molecules. New York, Oxford University Press
79. Picu C. (2003). A Nonlocal Formulation of Rubber Elasticity. Int. J. Multiscale Comput. Eng. 1: 23–32
80. Qu Y. and Fish J. (2000). Global-basis two-level method for indefinite systems. Part 2: computational issues. Int. J. Numerical Meth. Eng. 49(3): 461–47810.1002/1097-0207(20000930)49:3<461::AID-NME982>3.0.CO;2-S ↗
81. Roco M.C. (2000). Interagency Working Group on Nano Science, Engineering and Technology. Nanotechnology Research Directions. Kluwer, Dordrecht :37–44
82. Rudd R.E. (2004). Coarse-Grained Molecular Dynamics for Computer Modeling of Nanomechanical Systems. Int. J. Multiscale Comput. Eng. 2(2): 203–22010.1615/IntJMultCompEng.v2.i2.30 ↗
83. Rudd R.E. and Broughton J.Q. (1998). Coarse-grained molecular dynamics and the atomic limit of finite elements. Phys. Rev. B 58(10): R5893–R589610.1103/PhysRevB.58.R5893 ↗
84. Sakamoto S. and Ghanem R. (2002). Simulation of multi-dimensional non-gaussian non-stationary random fields. Probabilistic Eng. Mech. 17(2): 167–17610.1016/S0266-8920(01)00037-6 ↗
85. Shilkrot L.E., R.E. Miller, et al., 2002. Coupled atomistic and discrete dislocation plasticity. Phys. Rev. Lett. 89(2), 025501-1–410.1103/PhysRevLett.89.025501 ↗
86. SIAM, Multiscale Modelling and Simulation: A SIAM Interdisciplinary Journal
87. Smolinski P., Belytschko T., et al. (1988). Multi-Time-Step Integration Using Nodal Partitioning. Int. J. Numerical Meth. Eng. 26(2): 349–35910.1002/nme.1620260205 ↗
88. Sobh N., Huang J., et al. (2000). A discontinuous Galerkin model for precipitate nucleation and growth in aluminium alloy quench processes. Int. J. Numerical Meth. Eng. 47(1–3): 749–76710.1002/(SICI)1097-0207(20000110/30)47:1/3<749::AID-NME792>3.0.CO;2-3 ↗
89. Strachan, A., A. C. T. van Duin, et al. (2003). Shock waves in high-energy materials: The initial chemical events in nitramine RDX. Phys. Rev. Lett. 91(9), 098301-1–410.1103/PhysRevLett.91.098301 ↗
90. Strouboulis T., Copps K., et al. (2001). The generalized finite element method. Computer Meth. Appl. Mech. Eng. 190(32–33): 4081–419310.1016/S0045-7825(01)00188-8 ↗
91. Tadmor E.B., Ortiz M., et al. (1996). Quasicontinuum analysis of defects in solids. Philos. Mag. – Phys. Condensed Matter Struct. Defects Mech. Prop. 73(6): 1529–1563
92. Tadmor E.B., Smith G.S., et al. (1999). Mixed finite element and atomistic formulation for complex crystals. Phys. Rev. B 59(1): 235–24510.1103/PhysRevB.59.235 ↗
93. van Duin A.C.T., Dasgupta S., et al. (2001). ReaxFF: a reactive force field for hydrocarbons. J. Phys. Chem. A 105(41): 9396–940910.1021/jp004368u ↗
94. Vemaganti K.S. and Oden J.T. (2001). Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials Part II: a computational environment for adaptive modeling of heterogeneous elastic solids. Comput. Meth. Appl. Mech. Eng. 190(46–47): 6089–612410.1016/S0045-7825(01)00217-1 ↗
95. Wagner G.J., Karpov E.G., et al. (2004). Molecular dynamics boundary conditions for regular crystal lattices. Comput. Meth. Appl. Mech. Eng. 193(17–20): 1579–160110.1016/j.cma.2003.12.012 ↗
96. Wagner G.J. and Liu W.K. (2003). Coupling of atomistic and continuum simulations using a bridging scale decomposition. J. Comput. Phys. 190(1): 249–27410.1016/S0021-9991(03)00273-0 ↗
97. Waisman H. & J. Fish, 2006. A space–time multilevel method for molecular dynamics simulations. Comp. Meth. Appl. Mech. Engng. 195(44–47), 6542–655910.1016/j.cma.2006.02.006 ↗
98. Waisman H., Fish J., et al. (2004). The generalized global basis (GGB) method. Int. J. Num. Meth. Eng. 61(8): 1243–126910.1002/nme.1107 ↗
99. Wang Z.L. (2004). Bright future for ZnO At the frontier of transparent oxides. Mater. Today 7: 26–3310.1016/S1369-7021(04)00286-X ↗
100. Wang Z.L., Kong X.Y., et al. (2004). Semiconductiong and piezoelectric oxide nanostructures induced by polar surfaces. Adv. Funct. Mater. 14: 943–95610.1002/adfm.200400180 ↗