Multiscale methods for mechanical science of complex materials: Bridging from quantum to stochastic multiresolution continuum

International Journal for Numerical Methods in Engineering

Volumn 83, Issue 8-9, 2010, Pages 1039-1080

  Liu, Wing Kam ^a   Qian, Dong ^b   Gonella, Stefano ^a   Li, Shaofan ^c   Chen, Wei ^a   Chirputkar, Shardool ^b  

^a Northwestern University   (United States)

^b University of Cincinnati   (United States)

^c Univ of California   (United States)

Author keywords

Bridging scale method; Coupled atomistic continuum simulation; Multiresolution continuum; Multiscale non equilibrium method; Quantum mechanical coupling; Virtual atomcluster

Indexed keywords

BRIDGING SCALE; COUPLED ATOMISTIC/CONTINUUM SIMULATION; MULTI-RESOLUTIONS; MULTISCALES; QUANTUM MECHANICAL; VIRTUAL ATOMCLUSTER;

BENCHMARKING; CONTINUUM MECHANICS; MATERIALS; MOLECULAR DYNAMICS;

DYNAMICS;

EID: 77956127506     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2915     Document Type: Review

References (121)

1. Schiff LI. Quantum Mechanics. McGraw-Hill: New York, 1968.
2. Liu WK, Karpov EG, Park HS. Nano Mechanics and Materials: Theory, Multiscale Methods and Applications. Wiley: New York, 2006.
3. Hao S, Liu WK, Chang CT. Computer implementation of damage models by finite element and meshfree methods. Computer Methods in Applied Mechanics and Engineering 2000; 187:401-440.
4. Liu WK, Jun S, Qian D. Computational nanomechanics of materials. Journal of Computational and Theoretical Nanoscience 2008; 5(5):970-996.
5. Zhou SJ, Lomdahl PS, Thomson R, Holian BL. Dynamic crack processes via molecular dynamics. Physical Review Letters 1996; 76(13):2318-2321.
6. Zhang S, Mielke SL, Khare R, Troya D, Ruoff RS, Schatz GC, Belytschko T. Mechanics of defects in carbon nanotubes: atomistic and multiscale simulations. Physical Review B 2005; 71(11):540301-540312.
7. Arroyo M, Belytschko T. An atomistic-based membrane for crystalline films one atom thick. Journal of Mechanics and Physics of Solids 2002; 50:1941-1977.
8. Belytschko T, Xiao SP, Schatz GC, Ruoff RS. Atomistic simulations of nanotube fracture. Physical Review B 2002; 65(23):543001-543008.
9. Mura T. Micromechanics of Defects in Solids. Kluwer: Dordrecht, 1991.
10. Broughton JQ, Abraham FF, Bernstein N, Kaxiras E. Concurrent coupling of length scales: methodology and application. Physical Review B 1999; 60(4):2391-2403.
11. Fish J, Wen J. Discrete-to-continuum bridging based on multigrid principles. Computer Methods in Applied Mechanics and Engineering 2004; 193:1693-1711.
12. Karpov EG, Wagner GJ, Liu WK. A Green's function approach to deriving non-reflecting boundary conditions in molecular dynamics simulations. International Journal for Numerical Methods in Engineering 2005; 62: 1250-1262.
13. Knap J, Ortiz M. An analysis of the quasicontinuum method. Journal of the Mechanics and Physics of Solids 2001; 49(9):1899-1923.
14. Park HS, Karpov EG, Klein PA, Liu WK. Three-dimensional bridging scale analysis of dynamic fracture. Journal of Computational Physics 2005; 207(2):588-609.
15. Park HS, Karpov EG, Liu WK, Klein PA. The bridging scale for two-dimensional atomistic/continuum coupling. Philosophical Magazine 2005; 85(1):79-113.
16. Park HS, Liu WK. An introduction and tutorial on multiple-scale analysis in solids. Computer Methods in Applied Mechanics and Engineering 2004; 193(17-20):1733-1772.
17. Bauman PT, Oden JT, Prudhomme S. Adaptive multiscale modeling of polymeric materials with Arlequin coupling and goals algorithms. Computer Methods in Applied Mechanics and Engineering 2009; 198(5-8): 799-818.
18. Hawkins A, Oden JT. Toward a predictive model of tumor growth. ICES Report, The university of Texas at Austin, 2008.
19. Xiao SP, Belytschko T. A bridging domain method for coupling continua with molecular dynamics. Computer Methods in Applied Mechanics and Engineering 2004; 193(17-20):1645-1669.
20. Xu M, Belytschko T. Conservation properties of the bridging domain method for coupled molecular/continuum dynamics. International Journal for Numerical Methods in Engineering 2008; 76(3):278-294.
21. Qian D, Wagner GJ, Liu WK. A multiscale projection method for the analysis of carbon nanotubes. Computer Methods in Applied Mechanics and Engineering 2004; 193(17-20):1603-1632.
22. Qian D, Liu WK, Zheng QJ. Concurrent quantum/continuum coupling analysis of nanostructures. Computer Methods in Applied Mechanics and Engineering 2008; 197(41-42):3291-3323.
23. Gao HJ, Huang Y, Nix WD, Hutchinson JW. Mechanism-based strain gradient theory-I. Theory. Journal of the Mechanics and Physics of Solids 1999; 47:1239-1263.
24. Huang Y, Gao HJ, Nix WD, Hutchinson JW. Mechanism-based strain gradient theory-II. Analysis. Journal of the Mechanics and Physics of Solids 2000; 48:99-128.
25. Fleck N, Muler GM, Ashby MF, Hutchinson JW. Strain gradient plasticity: theory and experiment. Acta Metallurgy and Materials 1994; 42:475-487.
26. Kadowaki H, Liu WK. Bridging multiscale method for localization problems. Computer Methods in Applied Mechanics and Engineering 2004; 193:3267-3302.
27. Mindlin RD. Microstructure in linear elasticity. Archive for Rational Mechanics and Analysis 1964; 16:15-78.
28. McVeigh C, Vernerey F, Liu WK, Brinson LC. Multiresolution analysis for material design. Computer Methods in Applied Mechanics and Engineering 2006; 195(37-40):5053-5076.
29. Yin X, Chen W, To AC, McVeigh C, Liu WK. Statistical volume element method for predicting microstructure-constitutive property relations. Computer Methods in Applied Mechanics and Engineering 2008; 197(43-44):3516-3529.
30. Yin X, Lee S, Chen W, Liu WK. Efficient random field uncertainty propagation in design using multiscale analysis. ASME Journal of Mechanical Design 2009; 131(2):021006.
31. Alzebdeh K, Al-Ostaz A, Jasiuk I, Ostoja-Starzewski M. Fracture of random matrix-inclusion composites: scale effects and statistics. International Journal of Solids and Structures 1998; 35(19):2537-2566.
32. Arnst M, Ghanem RG. Probabilistic equivalence and stochastic model reduction in multiscale analysis. Computer Methods in Applied Mechanics and Engineering 2008; 197(43-44):3584-3592.
33. Xu XF. A multiscale stochastic finite element method on elliptic problems involving uncertainties. Computer Methods in Applied Mechanics and Engineering 2007; 196(25-28):2723-2736.
34. Charmpris DC, Schueller GI, Pellissetti MF. The need to linking micromechanics of materials with stochastic finite elements: a challenge for materials science. Computational Materials Science 2007; 41(1):27-37.
35. Vernerey F, Liu WK, Moran B. Multi-scale micromorphic theory for hierarchical materials. Journal of the Mechanics and Physics of Solids 2007; 55(12):2603-2651.
36. Vernerey FJ, Liu WK, Moran B, Olson G. A micromorphic model for the multiple scale failure of heterogeneous materials. Journal of the Mechanics and Physics of Solids 2008; 56(4):1320-1347.
37. McVeigh C, Liu WK. Linking microstructure and properties through a predictive multiresolution continuum. Computer Methods in Applied Mechanics and Engineering 2008; 197(41-42):3268-3290.
38. McVeigh C, Liu WK. Multiresolution modeling of ductile reinforced brittle composites. Journal of the Mechanics and Physics of Solids 2009; 57(2):244-267.
39. Kohlhoff S, Gumbsch P, Fischmeister HF. Crack-propagation in Bcc crystals studied with a combined finiteelement and atomistic model. Philosophical Magazine A-Physics of Condensed Matter Structure Defects and Mechanical Properties 1991; 64(4):851-878.
40. Abraham FF. Dynamics of brittle fracture with variable elasticity. Physical Review Letters 1996; 77(5):869-872.
41. Mullins M, Dokainish MA. Simulation of the (001) plane crack in alpha-iron employing a new boundary scheme. Philosophical Magazine A 1982; 46(5):771-787.
42. Tadmor EB, Ortiz M, Phillips R. Quasicontinuum analysis of defects in solids. Philosophical Magazine A-Physics of Condensed Matter Structure Defects and Mechanical Properties 1996; 73(6):1529-1563.
43. Holian BL, Ravelo R. Fracture simulations using large-scale molecular-dynamics. Physical Review B 1995; 51(17):11275-11288.
44. Berendsen HJC, van Gunsteren WF. In Dynamics Simulation of Statistical Mechanical Systems, Ciccotti GPF, Hoover WG (eds), vol. 63. North-Holland: Amsterdam, 1986; 493.
45. Gumbsch P, Zhou SJ, Holian BL. Molecular dynamics investigation of dynamic crack stability. Physical Review B 1997; 55(6):3445-3455.
46. Adelman SA, Doll JD. Generalized Langevin equation approach for atom/solid-surface scattering-collinear atom/harmonic chain model. Journal of Chemical Physics 1974; 61(10):4242-4245.
47. Adelman SA, Doll JD. Generlized Langevin equation approach for atom/solid surface scattering-general formulation for classical scattering of harmonic solids. Journal of Chemical Physics 1976; 64(6):2375-2388.
48. Cai W, de Koning M, Bulatov VV, Yip S. Minimizing boundary reflections in coupled-domain simulations Physical Review Letters 2000; 85(15):3213-3216.
49. E W, Huang ZY. Matching conditions in atomistic-continuum modeling of materials. Physical Review Letters 2001; 97(13):5501-5504.
50. E W, Huang ZY. A dynamic atomistic-continuum method for the simulation of crystalline material. Journal of Computational Physics 2002; 182:234-261.
51. Wagner GJ, Liu WK. Coupling of atomistic and continuum simulations using a bridging scale decomposition. Journal of Computational Physics 2003; 190(1):249-274.
52. Liu WK, Jun S, Zhang YF. Reproducing Kernel particle methods. International Journal for Numerical Methods in Fluids 1995; 20(8-9):1081-1106.
53. Liu WK, Jun S, Li SF, Adee J, Belytschko T. Reproducing kernel particle methods for structural dynamics. International Journal for Numerical Methods in Engineering 1995; 38(10):1655-1679.
54. Liu WK, Li SF, Belytschko T. Moving least-square reproducing kernel methods. 1. Methodology and convergence. Computer Methods in Applied Mechanics and Engineering 1997; 143(1-2):113-154.
55. Gonella S, Ruzzene M. Bridging scales analysis of wave propagation in heterogeneous structures with imperfections. Wave Motion 2008; 45(4):481-497.
56. Tang SQ, Hou TY, Liu WK. A pseudo-spectral multiscale method: interfacial conditions and coarse grid equations. Journal of Computational Physics 2006; 213(1):57-85.
57. Tang SQ, Hou TY, Liu WK. A mathematical framework of the bridging scale method. International Journal for Numerical Methods in Engineering 2006; 65(10):1688-1713.
58. Karpov EG, Park HS, Liu WK. Phonon heat bath for the atomistic and multiscale simulation of solids. International Journal for Numerical Methods in Engineering 2007; 70(3):351-378.
59. Farrell DE, Karpov EG, Liu WK. Algorithms for bridging scale method parameters. Computational Mechanics 2007; 40(6):965-978.
60. Liu WK, Karpov EG, Zhang S, Park HS. An introduction to computational nanomechanics and materials. Computer Methods in Applied Mechanics and Engineering 2004; 193(17-20):1529-1578.
61. Weeks WT. Numerical inversion of Laplace transforms using Laguerre functions. Journal of the ACM 1966; 13(3):419-429.
62. Farrell DE, Park HS, Liu WK. Implementation aspects of the bridging scale method and application to intersonic crack propagation. International Journal for Numerical Methods in Engineering 2007; 71(5):583-605.
63. Crump KS. Numerical inversion of Laplace transforms using a Fourier-series approximation. Journal of the ACM 1976; 23(1):89-96.
64. Chirputkar SU, Qian D. Coupled atomistic/continuum simulation based on extended space-time finite element method. CMES-Computer Modeling in Engineering and Sciences 2008; 24(2-3):185-202.
65. Hulbert GM, Hughes TJR. Space-time finite-element methods for 2nd-order hyperbolic-equations. Computer Methods in Applied Mechanics and Engineering 1990; 84(3):327-348.
66. Li XD, Wiberg NE. Structural dynamic analysis by a time-discontinuous Galerkin finite element method. International Journal for Numerical Methods in Engineering 1996; 39(12):2131-2152.
67. Chessa J, Belytschko T. Arbitrary discontinuities in space-time finite elements by level sets and X-FEM. International Journal for Numerical Methods in Engineering 2004; 61(15):2595-2614.
68. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 1999; 45(5):601-620.
69. Dolbow J, Moes N, Belytschko T. Modeling fracture in Mindlin-Reissner plates with the extended finite element method. International Journal of Solids and Structures 2000; 37(48-50):7161-7183.
70. Dolbow J, Moes N, Belytschko T. Discontinuous enrichment in finite elements with a partition of unity method. Finite Elements in Analysis and Design 2000; 36(3-4):235-260.
71. Sukumar N, Moes N, Moran B, Belytschko T. Extended finite element method for three-dimensional crack modelling. International Journal for Numerical Methods in Engineering 2000; 48(11):1549-1570.
72. Liu XH, Li S. Non-equilibrium multiscale computation model. Journal of Chemical Physics 2007; 126:124105.
73. Li S, Sheng N, Liu XH. A non-equilibrium multiscale paradigm. Chemical Physics Letters 2007; 451:293-300.
74. Sheng N, Li S. A multiscale non-equilibrium molecular dynamics algorithm and its applications. International Journal of Applied Mechanics 2009; 1(3):1-16.
75. Li S, Sheng N. On multiscale non-equilibrium molecular dynamics simulations (in this Issue). International Journal for Numerical Methods in Engineering 2010; DOI: 10.1002/nme.2849.
76. Du Q, Faber V, Gunzburger M. Centroidal Voronoi tessellations: applications and algorithms. SIAM Review 1999; 41(4):637-676.
77. Nose S. An extension of the canonical ensemble molecular-dynamics method. Molecular Physics 1986; 57(1):187-191.
78. Hoover WG. Canonical dynamics-equilibrium phase-space distributions. Physical Review A 1985; 31(3): 1695-1697.
79. Nose S. A molecular-dynamics method for simulations in the canonical ensemble. Molecular Physics 1984; 52(2):255-268.
80. Weiner JH. Statistical Mechanics of Elasticity. Wiley: New York, 1983.
81. Jiang H, Huang Y, Hwang KC. A finite-temperature continuum theory based on interatomic potentials. Journal of Engineering Materials and Technology-Transactions of the ASME 2005; 127(4):408-416.
82. Kim JH, Liu WK, Lee C. Multi-scale solid oxide fuel cell materials modeling. Computational Mechanics 2009; 44(5):683-703.
83. Hill R. Elastic properties of reinforced solids-some theoretical principles. Journal of the Mechanics and Physics of Solids 1963; 11(5):357-372.
84. Vernerey F, Liu WK, Moran B, Olson G. Multi-length scale micromorphic process zone model. Computational Mechanics 2009; 44(3):433-445.
85. Medyanik SN, Liu WK, Li SF. On criteria for dynamic adiabatic shear band propagation. Journal of the Mechanics and Physics of Solids 2007; 55(7):1439-1461.
86. Wright TW, Walter JW. The asymptotic structure of an adiabatic shear band in antiplane motion. Journal of the Mechanics and Physics of Solids 1996; 44(1):77-97.
87. McVeigh C, Vernerey F, Liu WK, Moran B, Olson G. An interactive micro-void shear localization mechanism in high strength steels. Journal of the Mechanics and Physics of Solids 2007; 55(2):225-244.
88. Siad L, Liu WK, Benabbes A. Explicit numerical study of softening in porous ductile solids. Mechanics Research Communications 2009; 36(2):236-245.
89. Mcveigh C, Liu WK. Multiresolution continuum modeling of micro-void assisted dynamic adiabatic shear band propagation. Journal of the Mechanics and Physics of Solids 2010; 58(2):187-205.
90. Qian D, Gondhalekar RH. A virtual atom cluster approach to the mechanics of nanostructures. International Journal for Multiscale Computational Engineering 2004; 2(2):277-289.
91. Liu WK, McVeigh C. Predictive multiscale theory for design of heterogeneous materials. Computational Mechanics 2008; 42(2):147-170.
92. Liu WK, Larbi S, Tian R, Lee SH, Lee D, Yin X, Chen W, Chan S, Olson GB, Lindgeen LE, Horstemeyer MF, Chang YS, Choi JB, Kim YJ. Complexity science of multiscale materials via stochastic computations. International Journal for Numerical Methods in Engineering 2009; 80(6):932-978.
93. Sakamoto S, Ghanem R. Polynomial chaos decomposition for the simulation of non-Gaussian nonstationary stochastic processes. Journal for Engineering Mechanics 2002; 128(2):190-201.
94. Schweizer B, Wolff EF. On nonparametric measures of dependence for random variables. The Annals of Statistics 1981; 9(4):879-885.
95. Fisher NI. Copulas. Encyclopedia of Statistical Sciences. Wiley: New York, 1997.
96. Noh Y, Choi KK, Liu D. Reliability-based design optimization of problems with correlated input variables using a Gaussian Copula. Structural and Multidisciplinary Optimization 2009; 38(1):1-16.
97. Greene MS, Liu Y, Chen W, Liu WK. Stochastic multiscale analysis for complex material systems. Tenth US National Congress on Computational Mechanics, Columbus, OH, 2009.
98. Qian D, Wagner GJ, Liu WK, Yu MF, Ruoff RS. Mechanics of carbon nanotubes. Applied Mechanics Reviews 2002; 55(6):495-533.
99. Pask JE, Sterne PA. Finite element methods in ab initio electronic structure calculations. Modeling and Simulation in Materials Sciences and Engineering 2005; 13:R71-R96.
100. Beck TL. Real-space mesh techniques in density-functional theory. Reviews of Modern Physics 2000; 72(4): 1041-1080.
101. Sukumar N, Pask JE. Classical and enriched finite element formulations for Bloch-periodic boundary conditions. International Journal for Numerical Methods in Engineering 2009; 77(8):1121-1138.
102. Abraham FF, Broughton JQ, Bernstein N, Kaxiras E. Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture. Europhysics Letters 1998; 44(6):783-787.
103. Nakano A, Bachlechner ME, Kalia RK, Lidorikis E, Vashishta P, Voyiadjis GZ, Campbell TJ, Ogata S, Shimojo F. Multiscale simulation of nanosystems. Computing in Science and Engineering 2001; 3(4):56-66.
104. Rafii-Tabar H, Hua L, Cross M. Multiscale numerical modelling of crack propagation in two-dimensional metal plate. Materials Science and Technology 1998; 14(6):544-548.
105. Rafii-Tabar H, Hua L, Cross M. A multi-scale atomistic-continuum modelling of crack propagation in a two-dimensional macroscopic plate. Journal of Physics-Condensed Matter 1998; 10(11):2375-2387.
106. Rudd RE, Broughton JQ. Coarse-grained molecular dynamics and the atomic limit of finite elements. Physical Review B 1998; 58(10):R5893-R5896.
107. Rudd RE, Broughton JQ. Concurrent coupling of length scales in solid state systems. Physica Status Solidi B-Basic Research 2000; 217(1):251-291.
108. Gavini V, Bhattacharya K, Ortiz M. Quasi-continuum orbital-free density-functional theory: a route to multimillion atom non-periodic DFT calculation. Journal of the Mechanics and Physics of Solids 2007; 55(4):697-718.
109. Gavini V, Knap J, Bhattacharya K, Ortiz M. Non-periodic finite-element formulation of orbital-free density functional theory. Journal of the Mechanics and Physics of Solids 2007; 55(4):669-696.
110. Gavini V, Bhattacharya K, Ortiz M. Vacancy clustering and prismatic dislocation loop formation in aluminum. Physical Review B 2007; 76(18).
111. Hung L, Carter EA. Accurate simulations of metals at the mesoscale: explicit treatment of 1 million atoms with quantum mechanics. Chemical Physics Letters 2009; 475(4-6):163-170.
112. Peng Q, Zhang X, Hung L, Carter EA, Lu G. Quantum simulation of materials at micron scales and beyond. Physical Review B 2008; 78(5):054118.
113. Tadmor EB, Smith GS, Bernstein N, Kaxiras E. Mixed finite element and atomistic formulation for complex crystals. Physical Review B 1999; 59(1):235-245.
114. Zhang P, Huang Y, Geubelle PH, Klein P, Hwang KC. The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials. International Journal of Solids and Structures 2002; 39(13-14):3893-3906.
115. Liu B, Jiang H, Johnson HT, Huang Y. The influence of mechanical deformation on the electrical properties of single wall carbon nanotubes. Journal of the Mechanics and Physics of Solids 2004; 52(1):1-26.
116. Johnson HT, Liu B, Huang Y. Electron transport in deformed carbon nanotubes. Journal of Engineering Materials and Technology 2004; 126:222-229.
117. Bloch F. Uber die Quantenmechanik der Elektronen in Kristallgittern. Zeitschrift für Physik 1928; 52:555-600.
118. Kittel C. Introduction to Solid State Physics. Wiley: New York, London, Sydney, 1966.
119. Feynman RP. Forces in molecules. Physical Review 1939; 56:340-343.
120. Li SF, Liu WK. Meshfree Particle Methods. Springer: Berlin, 2004.
121. Buttiker M, Imry Y, Landauer R, Pinhas S. Generalized many-channel conductance formula with application to small rings. Physical Review B 1985; 31(10):6207-6215.