Heterogeneous multiscale methods: A review

Communications in Computational Physics

Volumn 2, Issue 3, 2007, Pages 367-450

  E, Weinan ^a   Engquist, Bjorn ^b   Li, Xiantao ^c   Ren, Weiqing ^d   Vanden Eijnden, Eric ^d  

^a PRINCETON UNIVERSITY   (United States)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^c PENNSYLVANIA STATE UNIVERSITY   (United States)

^d NEW YORK UNIVERSITY   (United States)

Author keywords

Constrained micro scale solver; Data estimation; Heterogeneous multi scale method; Multi physics models; Multi scale modeling

Indexed keywords

EID: 34548474934     PISSN: 18152406     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Review

References (189)

1. A. Abdulle, Fourth order Chebychev methods with recurrence relations, SIAM J. Sci. Comput., 23(6) (2002), 2041-2054.
2. A. Abdulle, On a-priori error analysis of fully discrete heterogeneous multiscale FEM, preprint, Department Mathematik, Universität Basel, 2004-08, 2004.
3. A. Abdulle and W. E, Finite difference heterogeneous multi-scale method for homogenization problems, J. Comput. Phys., 191 (2003), 18-39.
4. A. Abdulle and C. Schwab, Heterogeneous multiscale FEM for diffusion problem on rough surface, SIAM Multiscale Model. Simul., 3 (2005), 195-220.
5. F. F. Abraham, J. Q. Broughton, N. Bernstein and E. Kaxiras, Concurrent coupling of length scales: Methodology and application, Phys. Rev. B, 60(4) (1999), 2391-2402.
6. F. F. Abraham, J. Q. Broughton, N. Bernstein and E. Kaxiras, Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture, Europhys. Lett., 44(6) (1998), 783-787.
7. D. Adalsteinsson and J. A. Sethian, The fast construction of extension velocities in level set methods, J. Comput. Phys., 148 (1999), 2-22.
8. D. Adalsteinsson and J. A. Sethian, A fast level set method for propagating interfaces, J. Comput. Phys., 118(2) (1995), 269-277.
9. M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley-Interscience, New York, 2000.
10. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, New York, 1987.
11. V. I. Arnold, Mathematical Methods in Classical Mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, 1989.
12. I. Babuška, Solution of interface by homogenization, I, II, III, SIAM J. Math. Anal., 7 (1976), 603-645;8 (1977), 923-937.
13. I. Babuška, Homogenization and its applications, mathematical and computational problems, in: B. Hubbard (Ed.), Numerical Solutions of Partial Differential Equations-III, (SYNSPADE. 1975, Maryland, May 1975), Academic Press, New York, 1976, pp. 89-116.
14. I. Babuška, G. Caloz and J. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal., 31 (1994), 945-981.
15. I. Babuška, U. Banerjee and J. Osborn, Survey of meshless and generalized finite element methods, Acta Numerica, 12 (2003), 1-125.
16. I. Babuška, U. Banerjee and J. Osborn, Generalized finite element methods - main ideas, results and perspective, Int. J. Comput. Meth., 1 (2004), 67-103.
17. C. S. Becquart, P. C. Clapp and J. A. Rifkin, Molecular-dynamics simulations of tweed structure and the ω phase in Ni-Al, Phys. Rev. B, 48 (1993), 6-13.
18. A. Bensoussan, J. L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Pub. Co., Amsterdam, 1978.
19. R. B. Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theory, John Wiley, New York, 1987.
20. D. L. Boley, Krylov space methods on state-space control models, Circ. Syst. Signal Proc., 13(6) (1994), 733-758.
21. J. F. Bourgat, Numerical experiments of the homogenization method for operators with periodic coefficients, in: Lecture Notes in Math., vol. 707, Springer-Verlag, 1977, pp. 330-356.
22. J. F. Bourgat, P. Le Tallec, B. Perthame and Y. Qiu, Coupling Boltzmann and Euler equations without overlapping, in: Domain Decomposition Methods in Science and Engineering (Como 1992), American Mathematics Society, 1994, pp. 377-398.
23. J. F. Bourgat, P. Le Tallec and M. Tidriri, Coupling Boltzmann and Navier-Stokes equations by friction, J. Comput. Phys., 127(2) (1996), 227-245.
24. J. H. Bramble, J. E. Pasciak and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring methods, Math. Comput., 47 (1986), 103-134.
25. A. Brandt, Multiscale scientific computation: Review 2001, in: T. J. Barth, et.al (Eds.), Multiscale and Multiresolution Methods: Theory and Applications, Yosemite Educational Symposium Conf. Proc., 2000, Lecture Notes in Comp. Sci. and Engrg., vol. 20, Springer-Verlag, 2002, pp. 3-96.
26. A. Brandt, Multigrid methods in lattice field computations, Nucl. Phys. B (Proc. Suppl.), 26B (1992), 137-180.
27. A. Brandt, Multi-level adaptive solutions to boundary value problems, Math. Comput., 31(138) (1977), 333-390.
28. A. Brandt, Multiscale research in molecular dynamics, Gauss Center Report WI/GC-3, 1995.
29. W. Cai, M. de Koning, V. V. Bulatov and S. Yip, Minimizing boundary reflections in coupled-domain simulations, Phys. Rev. Lett., 85(15) (2000), 3213-3216.
30. R. Car and M. Parrinello, Unified approach for molecular dynamics and density-functional theory, Phys. Rev. Lett., 55(22) (1985), 2471-2474.
31. S. Chakravorty, C. M. Wayman, Thermoelastic martensitic transformation in beta' Ni-Al alloys, Metall. Mater. Trans. A, 7(4) (1976), 555-582.
32. S. Chen and G. D. Doolen, Lattice Boltzmann methods for fluid flows, Ann. Rev. Fluid Mech., 30 (1998), 329-364.
33. S. Chen, The heterogeneous multi-scale method based on the discontinuous Galerkin and finite volume schemes, thesis, Brown University, 2005.
34. S. Chen, W. E and C. W. Shu, The heterogeneous multi-scale method based on discontinuous Galerkin method for hyperbolic and parabolic problems, SIAM Multiscale Model. Simul., 3 (2005), 871-894.
35. L. T. Cheng and W. E, The heterogeneous multiscale method for interface dynamics, in: S. Y. Cheng, C.-W. Shu and T. Tang (Eds), Recent Advances in Scientific Computing and Partial Differential Equations, Contemporary Mathematics, Vol. 330, AMS, 2003, pp. 43-53.
36. N. Choly, G. Lu, W. E and E. Kaxiras, Multiscale simulations in simple metals: A density-functional based methodology, Phys. Rev. B, 71 (2005), 094101.
37. A. J. Chorin, Conditional expectations and renormalization, SIAM J. Multiscale Model. Simul., 1 (2003), 105-118.
38. A. J. Chorin, Vorticity and Turbulence, Applied Mathematical Sciences, vol. 103, Springer-Verlag, 1998.
39. A. J. Chorin, Computational Fluid Mechanics: Selected Papers, Academic Press, New York, 1989.
40. A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), 745-762.
41. A. J. Chorin, O. Hald and R. Kupferman, Optimal prediction with memory, Physica D, 166(3-4) (2002), 239-257.
42. A. J. Chorin, A. P. Kast and R. Kupferman, Optimal prediction of underresolved dynamics, Proc. Natl. Acad. Sci. USA, 95(8) (1998), 4094-4098.
43. J. Cormier, J. M. Rickman and T. J. Delph, Stress calculation in atomistic simulations of perfect and imperfect solids, J. Appl. Phys., 89 (2001), 99-104.
44. J. Z. Cui and L. Q. Cao, The two-scale analysis method for woven composite materials, in: Z. H. Zhong (Ed.), Engineering Computation and Computer Simulation I, Hunan University Press, 1995, pp. 203-212.
45. S. Curtarolo and G. Ceder, Dynamics of an inhomogeneously coarse grained multiscale system, Phys. Rev. Lett., 88 (2002), 255504.
46. W. A. Curtin and R. E. Miller, Atomistic/continuum coupling in computational material science, Model. Simul. Mater. Sci. Engrg., 11 (2003), R33-R68.
47. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM Publications, 1992.
48. C. Denniston and M. O. Robbins, Molecular and continuum boundary conditions for a miscible binary fluid, Phys. Rev. Lett., 87(17) (2001), 178302.
49. S. M. Deshpande, Kinetic Flux Splitting Schemes, Computational Fluid Dynamics Review, John Wiley & Sons, Chichester, 1995.
50. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford Science Publications, 1986.
51. L. J. Durlofsky, Numerical calculation of equivalent grid block permeability tensors for heterogeneous poros-media, Water. Resour. Res., 27 (1991), 699-708.
52. E. B. Dussan, On the spreading of liquids on solid surfaces: Static and dynamic contact lines, Annu. Rev. Fluid Mech., 11 (1979), 371-400.
53. W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Commun. Math. Sci., 1(3) (2003), 423-436.
54. W. E, Homogenization of linear and nonlinear transport equations, Commun. Pur. Appl. Math., 45 (1992), 301-326.
55. W. E and B. Engquist, The heterogeneous multi-scale method for homogenization problems, preprint.
56. W. E and B. Engquist, The heterogeneous multi-scale methods, Commun. Math. Sci., 1 (2003), 87-133.
57. W. E and B. Engquist, Multiscale modeling and computation, Notices of the AMS, 50(9) (2003), 1062-1070.
58. W. E, B. Engquist and Z. Huang, Heterogeneous multiscale method: a general methodology for multiscale modeling, Phys. Rev. B, 67(9) (2003), 092101.
59. W. E and Z. Huang, Matching conditions in atomistic-continuum modeling of materials, Phys. Rev. Lett., 87(13) (2001), 135501.
60. W. E and Z. Huang, A dynamic atomistic-continuum method for the simulation of crystalline materials, J. Comput. Phys., 182 (2002), 234-261.
61. W. E, and X. Li, Analysis of the heterogeneous multiscale method for gas dynamics, Method. Appl. Anal., 11(4) (2004), 557-572.
62. W. E, X. Li and E. Vanden-Eijnden, Some recent progress in multiscale modeling, in: S. Attinger and P. Koumoutsakos (Eds.), Multiscale Modeling and Simulation, Lecture Notes in Computational Science and Engineering, vol. 39, Springer-Verlag, 2004, pp. 3-22.
63. W. E, D. Liu and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Commun. Pur. Appl. Math., 58(11) (2005), 1544-1585.
64. W. E and P. B. Ming, Analysis of multiscale methods, J. Comp. Math., 22 (2004), 210-219.
65. W. E, P. B. Ming and P. W. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Am. Math. Soc., 18(1) (2005), 121-156.
66. W. E and X. Y. Yue, Heterogeneous multiscale method for locally self-similar problems, Commun. Math. Sci., 2 (2004), 137-144.
67. B. Engquist, Computation of oscillatory solutions to hyperbolic differential equations, in: Lecture Notes in Math., vol. 1270, Springer-Verlag, 1987, pp. 10-22.
68. B. Engquist and O. Runborg, Wavelet-based numerical homogenization with applications, in: T. J. Barth, et.al (Eds.), Multiscale and Multiresolution Methods: Theory and Applications, Yosemite Educational Symposium Conf. Proc., 2000, Lecture Notes in Comp. Sci. and Engrg., vol. 20, Springer-Verlag, 2002, pp. 97-148.
69. B. Engquist and Y. H. Tsai, Heterogeneous multiscale methods for stiff ordinary differential equations, Math. Comput., 74 (2005), 1707-1742.
70. F. Ercolessi and J. B. Adams, Interatomic potentials from first-principles calculations: The force-matching method, Europhys. Lett., 26 (1993), 583.
71. K. Ericksson, C. Johnson and A. Logg, Explicit time-stepping for stiff ODEs, SIAM J. Sci. Comput., 25(4) (2003), 1142-1157.
72. I. Fatkullin and E. Vanden-Eijnden, A computational strategy for multiscale systems with applications to Lorenz 96 model, J. Comput. Phys., 200(2) (2004), 605-638.
73. E. G. Flekkøy, G. Wagner and J. Feder, Hybrid model for combined particle and continuum dynamics, Europhys. Lett., 52(3) (2000), 271-276.
74. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 2nd ed., Springer-Verlag, 1998.
75. D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, 2nd ed., Academic Press, 2001.
76. A. L. Garcia, J. B. Bell, W. Y. Crutchfield and B. J. Alder, Adaptive mesh and algorithm refinement using direct simulation Monte Carlo, J. Comput. Phys., 154 (1999), 134-155.
77. C. W. Gear and I. G. Kevrekidis, Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM J. Sci. Comput., 24(4) (2003), 1091-1106.
78. C. W. Gear, J. Li and I. G. Kevrekidis, The gap-tooth scheme for particle simulations, Phys. Lett. A, 316 (2003), 190-195.
79. J. Glimm, O. McBryan, R. Menikoff and D. H. Sharp, Front tracking applied to Rayleigh-Taylor instability, SIAM J. Sci. Stat. Comput., 7 (1986), 230-251.
80. E. Godlewski and P. A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer-Verlag, New York, 1996.
81. G. H. Golub and C. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, Maryland, 1983.
82. L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), 325-348.
83. W. Guo, T. P. Schulze and W. E, Simulation of impurity diffusion in a strained nanowire using off-lattice KMC, Commun. Comput. Phys., 2 (2007), 164-176.
84. N. G. Hadjiconstantinou, Hybrid atomistic-continuum formulations and the moving contact-line problem, J. Comput. Phys., 154(2) (1999), 245-265.
85. N. G. Hadjiconstantinou and A. T. Patera, Heterogeneous atomistic-continuum representation for dense fluid systems, Int. J. Mod. Phys. C, 8(4) (1997), 967-976.
86. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer-Verlag, 1991.
87. J. Helmsen, E. Puckett, P. Colella and M. Dorr, Two new methods for simulating photolithography development in 3D, in: SPIE 1996 Symposium Proc., Intl. Symposium on Microlithography, Santa Clara, CA, vol. 2726, 1996, pp. 253-261.
88. T. Y. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), 169-189.
89. B. D. Hughes, Random Walks and Random Environments, Vol. 2, Random Environments, Oxford University Press, 1996.
90. J. H. Irving and J. G. Kirkwood, The statistical mechanical theory of transport processes IV, J. Chem. Phys., 18 (1950), 817-829.
91. H. Kadowaki and W. K. Liu, Bridging multiscale method for localization problems, Comput. Method. Appl. Mech. Engrg., 193 (2004), 3267-3302.
92. E. G. Karpov, G. J. Wagner and W. K. Liu, A Green's function approach to deriving non-reflecting boundary conditions in molecular dynamics simulations, Int. J. Numer. Meth. Engrg., 62(9) (2005), 1250-1262.
93. E. G. Karpov, H. Yu, H. S. Park, W. K. Liu and J. Wang, Multiscale boundary conditions in crystalline solids: Theory and application to nanoindentation, Phys. Rev. B, submitted.
94. M. Katsoulakis, A. J. Majda and D. G. Vlachos, Coarse-grained stochastic processes for lattice systems, Proc. Natl. Acad. Sci. USA, 100 (2003), 782-787.
95. I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci., 1(4) (2003), 715-762.
96. P. E. Kloeden and E. Platen, Numerical Solutions of Stochastic Differential Equations, Applications of Mathematics, vol. 23, Springer-Verlag, 1992.
97. J. Koplik and J. R. Banavar, Continuum deductions from molecular hydrodynamics, Annu. Rev. Fluid Mech., 27 (1995), 257-292.
98. T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Funct. Anal., 12 (1973), 55-67.
99. L. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Pergamon Press, 1980.
100. L. Landau and E. M. Lifshitz, Theory of Elasticity, Pergamon Press, 1980.
101. V. I. Lebedev, Explicit difference schemes for solving stiff problems with a complex or separable spectrum, Zh. Vychisl. Mat. Mat. Fiz., 40(12) (2000), 1801-1812 (in Russian); Comput. Math. Math. Phys., 40(12) (2000),1729-1740 (English Translation).
102. V. I. Lebedev and S. I. Finogenov, Explicit methods of second order for the solution of stiff systems of ordinary differential equations, Zh. Vychisl. Mat. Mat. Fiz., 16(4) (1976), 895-910.
103. A. W. Lees and S. F. Edwards, The computer study of transport processes under extreme conditions, J. Phys. C, 5(15) (1972), 1921-1929.
104. P. Le Tallec and J. P. Perlat, Coupling kinetic models with Navier-Stokes equations, CID Review, 2 (1998), 833-855.
105. R. LeVeque, Numerical Methods for Conservation Laws, Birkhauser, 1990.
106. J. Li, D. Liao and S. Yip, Nearly exact solution for coupled continuum/ MD fluid simulation, J. Comput. Aid. Mater. Des., 6(2-3) (1999), 95-102.
107. T. Li, E. Vanden-Eijnden, P. Zhang and W. E, Stochastic models of polymeric fluids at small Deborah number, J. Non-Newton. Fluid Mech., 121(2-3) (2004), 117-125.
108. X. Li and W. E, Multiscale modeling of the dynamics of solids at finite temperature, J. Mech. Phys. Solids, 53 (2005), 1650-1685.
109. X. Li and W. E, Variational boundary conditions for molecular dynamics simulation of solids at low temperature, Commun. Comput. Phys., 1(1) (2006), 135-175.
110. X. Li and W. E, Variational boundary conditions for molecular dynamics simulation of solids at finite temperature, preprint.
111. P. L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished.
112. W. K. Liu, Y. F. Zhang and M. R. Ramirez, Multiple scale finite element methods, Int. J. Numer. Meth. Engrg., 32 (1991), 969-990.
113. W. K. Liu, E. G. Karpov, S. Zhang and H. S. Park, An introduction to computational nano mechanics and materials, Comput. Method. Appl. Mech. Engrg., 193 (2004), 1529-1578.
114. G. Lu and E. Kaxiras, An overview of multiscale simulation of materials, preprint, 2003.
115. A. J. Majda, I. Timofeyev and E. Vanden-Eijnden, A mathematical framework for stochastic climate models, Commun. Pur. Appl. Math., 54(8) (2001), 891-974.
116. R. Miller and E. B. Tadmor, The quasicontinuum method: Overview, applications and current directions, preprint, 2004.
117. R. Miller, E. B. Tadmor, R. Phillips and M. Ortiz, Quasicontinuum simulation of fracture at the atomic scale, Model. Simul. Mater. Sci. Engrg., 6(5) (1998), 607-638.
118. G. W. Milton, The Theory of Composites, Cambridge University Press, 2002.
119. P. B. Ming and X. Y. Yue, Numerical methods for multiscale elliptic problems, preprint, 2003.
120. P. B. Ming and P. W. Zhang, Analysis of the heterogeneous multiscale method for parabolic homogenization problems, Math. Comput., submitted.
121. F. Murat and L. Tartar, H-Convergence, Cours Peccot, 1977. Reprinted in: A. Cherkaev and R. Kohn (Eds.), Topics in the Mathematical Modeling of Composite Materials, Birkhauser, Boston, 1997, pp. 21-43.
122. H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 408-463.
123. X. Nie, S. Chen, W. E and M. O. Robbins, A continuum and molecular dynamics hybrid method for micro- and nano-fluid flow, J. Fluid Mech., 500 (2004), 55-64.
124. X. Nie, S. Chen, W. E and M. O. Robbins, Hybrid continuum-atomistic simulation of singular corner flow, Phys. Fluids, 16 (2004), 3579.
125. S. T. O'Connell and P. A. Thompson, Molecular dynamics-continuum hybrid computations: A tool for studying complex fluid flows, Phys. Rev. E, 52(6) (1995), R5792-R5795.
126. J. T. Oden and K. S. Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. J. Error estimates and adaptive algorithms, J. Comput. Phys., 164 (2000), 22-47.
127. S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49.
128. G. C. Papanicolaou, Introduction to the asymptotic analysis of stochastic equations, in: R. C. Diprima. (Ed.), Modern Modeling of Continuum Phenomena (Ninth Summer Sem. Appl. Math., Troy, N.Y., 1975), Lectures in Appl. Math., Amer. Math. Soc., vol. 16, 1977, pp. 109-147.
129. G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in: J. Fritz, et. al (Eds.), Random Fields, North-Holland, Amsterdam, vol. 27, 1981, pp. 835-873.
130. H. S. Park, E. G. Karpov, W. K. Liu and P. A. Klein, The bridging scale for two-dimensional atomistic/continuum coupling, Philos. Mag., 85(1) (2005), 79-113.
131. R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, 1989.
132. B. Perthame, Boltzmann-type schemes for gas dynamics and entropy property, SIAM J. Numer. Anal., 27 (1991), 1405-1421.
133. B. Perthame, Second order Boltzmann-type schemes for compressible Euler equations in one and two space dimensions, SIAM J. Numer. Anal., 29 (1992), 1-19.
134. C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comput. Phys., 25(3) (1977), 220-252.
135. C. S. Peskin and D. M. McQueen, Computational biofluid dynamics, Contemp. Math., 141 (1993), 161-186.
136. R. Phillips, Crystals, Defects and Microstructures, Cambridge University Press, 2001.
137. N. V. Priezjev and S. M. Troian, Molecular origin and dynamic behavior of slip in sheared polymer films, Phys. Rev. Lett., 92 (2004), 018302.
138. D. I. Pullin, Direct simulation methods for compressible inviscid ideal gas flow, J. Comput. Phys., 34 (1980), 231-244.
139. D. Qian, G. J. Wagner and W. K. Liu, A multiscale projection method for the analysis of carbon nanotubes, Comput. Method. Appl. Mech. Engrg., 193 (2004), 1603-1632.
140. T. Z. Qian and X. P. Wang, Sinusoidal fluid-solid interactions: from molecular simulations to hydrodynamic modeling, preprint.
141. T. Z. Qian, X. P. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306.
142. T. Qian, X. P. Wang and P. Sheng, Molecular hydrodynamics of the moving contact line in two-phase immiscible flows, Commun. Comput. Phys., 1 (2006), 1-52.
143. A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications, 1999.
144. W. Ren, Multiscale modeling of polymeric fluids, in preparation.
145. W. Ren, Accessment of the errors in multiscale methods for complex fluids, in preparation.
146. W. Ren and W. E, Heterogeneous multiscale method for the modeling of complex fluids and microfluidics, J. Comput. Phys., 204(1) (2005), 1-26.
147. W. Ren, X. Zhou and W. E, Constant rate of strain molecular dynamics, in preparation.
148. M. O. Robbins and M. H. Mülser, Computer simulations of friction, lubrication and wear, in: B. Bhushan (Ed.), Modern Tribology Handbook, CRC Press, 2001.
149. R. E. Rudd and J. Q. Broughton, Atomistic simulation of MEMS resonators through the coupling of length scales, J. Model. Simul. Microsyst., 1 (1999), 29-38.
150. R. E. Rudd and J. Q. Broughton, Coarse-grained molecular dynamics and the atomic limit of finite elements, Phys. Rev. B, 58(10) (1998), R5893-R5896.
151. O. Runborg, C. Theodoropoulos and I. G. Kevrekidis, Effective bifurcation analysis: Time-stepper based approach, Nonlinearity, 15(2) (2002), 491-511.
152. M. Sahimi, Flow and Transport in Porous Media and Fractured Rock, John Wiley, 1995.
153. R. H. Sanders and K. H. Prendergast, On the origin of the 3-kiloparsec arm, Astrophys. J., 188 (1974), 489-500.
154. S. Succi, O. Filippova, G. Mith and E. Kaxiras, Applying the lattice Boltzmann equation to multiscale fluid problems, Comput. Sci. Engrg., 3 (2001), 26.
155. H. Schmidt and R. Klein, A generalized level-set/in-cell-reconstruction approach for accelerating turbulent premixed flames, Combust. Theor. Model., 7 (2003), 243-267.
156. T. Schulze, P. Smereka and W. E, Coupling kinetic Monte-Carlo and continuum models with application to epitaxial growth, J. Comput. Phys., 189 (2003), 197-211.
157. C. Schwab and A. M. Matache, Generalized FEM for homogenization problems, in: T. J. Barth, et.al (Eds.), Multiscale and Multiresolution Methods: Theory and Applications, Yosemite Educational Symposium Conf. Proc., 2000, Lecture Notes in Comp. Sci. and Engrg., vol. 20, Springer-Verlag, 2002, pp. 197-237.
158. C. Schwab and A. M. Matache, Two-scale FEM for homogenization problems, in: I. Babuška, et. al (Eds.), Mathematical Modelling and Numerical Simulation in Continuum Mechanics Conf. Proc., Yamaguchi, Japan, 2000, Lecture Notes in Comp. Sci. and Engrg., Springer-Verlag, 2002.
159. J. A. Sethian, Fast marching level set methods for three dimensional photolithography development, in: SPIE 1996 Symposium Proc., Intl. Symposium on Microlithography, Santa Clara, CA, vol. 2726, 1996, pp. 261-272.
160. R. Sharp. Y. H. Tsai and B. Engquist, Multiple time scale methods for the inverted pendulum problem, in: Proceedings of Conference on Multiscale Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering, Springer Verlag, vol. 44, 2005, pp. 233-244.
161. V. B. Shenoy, R. Miller, E. B. Tadmor, R. Phillips and M. Ortiz, Quasicontinuum models of interfacial structure and deformation, Phys. Rev. Lett., 80(4) (1998), 742-745.
162. V. B. Shenoy, R. Miller, E. B. Tadmor, D. Rodney, R. Phillips and M. Ortiz, An adaptive finite element approach to atomic-scale mechanics - the quasicontinuum method, J. Mech. Phys. Solids, 47(3) (1999), 611-642.
163. L. E. Shilkrot, W. A. Curtin and R. E. Miller, A coupled atomistic/ continuum model of defects in solids, J. Mech. Phys. Solids, 50 (2002), 2085-2106.
164. L. E. Shilkrot, R. E. Miller and W. A. Curtin, Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics, J. Mech. Phys. Solids, 52 (2004), 755-787.
165. G. S. Smith, E. B. Tadmor and E. Kaxiras, Multiscale simulation of loading and electrical resistance in silicon nanoindentation, Phys. Rev. Lett., 84(6) (2000), 1260-1263.
166. H. Spohn, Large Scale Dynamics of Interacting Particles, Springer-Verlag, 1991.
167. Y. Sun, thesis, Princeton University, 2006.
168. Y. Sun and B. Engquist, Heterogeneous multiscale methods for interface tracking of combustion fronts, SIAM Multiscale Model. Simul., 5 (2006), 532-563.
169. E. B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in crystals, Philos. Mag., A73 (1996), 1529-1563.
170. L. Tartar, Solutions oscillantes des équations de Carleman, Goulaouic-Meyer-Schwartz Seminar, 1980-1981, Exp. No. XII, 15 pp., cole Polytech., Palaiseau, 1981.
171. P. A. Thompson and M. O. Robbins, Simulations of contact-line motion: slip and the dynamic contact angle, Phys. Rev. Lett., 63(7) (1989), 766-769.
172. P. A. Thompson and S. M. Troian, A general boundary condition for liquid flow at solid surfaces, Nature, 389 (1997), 360-362.
173. A. K. Tornberg and B. Engquist, Interface tracking in two-phase flows, in: Multifield Problems, State of the Art, Springer-Verlag, 2000, pp. 58-66.
174. S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer-Verlag, 2002.
175. E. Vanden-Eijnden, Numerical techniques for multiscale dynamical systems with stochastic effects, Commun. Math. Sci., 1(2) (2003), 385-391.
176. A. F. Voter and S. P. Chen, Accurate interatomic potentials for Ni, Al and Ni3Al, Mater. Res. Soc. Proc., 82 (1987), 175-180.
177. A. F. Voter, F. Montalenti and T. C. Germann, Extending the time scale in atomistic simulation of materials, Annu. Rev. Mater. Res., 32 (2002), 321-346.
178. G. J. Wagner and W. K. Liu, Coupling of atomistic and continuum simulations using a bridging scale decomposition, J. Comput. Phys., 190 (2003), 249-274.
179. A. Warshel and M. Levitt, Theoretical studies of enzymic reactions, J. Mol. Biol., 103 (1976), 227-249.
180. T. Werder, J. H. Walther, J. Asikainen and P. Koumoutsakos, Continuum-particle hybrid methods for dense fluids, in: S. Attinger and P. Koumoutsakos (Eds.), Multiscale Modeling and Simulation, Lecture Notes in Computational Science and Engineering, Springer, vol. 39, 2004, pp. 35-68.
181. S. P. Xiao and T. Belytschko, A bridging domain method for coupling continua with molecular dynamics, Comput. Method. Appl. Mech. Engrg., 193 (2004), 1645-1669.
182. J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33(5) (1996), 1759-1777.
183. K. Xu and K. H. Prendergast, Numerical Navier-Stokes solutions from gas kinetic theory, J. Comput. Phys., 114 (1994), 9-17.
184. J. Z. Yang and X. Li, Comparative study of boundary conditions for molecular dynamics simulations of solids at low temperature, Phys. Rev. B, 73 (2006), 224111.
185. H. T. Yau, Asymptotic solutions to dynamics of many-body systems and classical continuum equations, in: Current Development in Mathematics, International Press, 2000.
186. X. Y. Yue, The heterogeneous multi-scale method for modeling functionally graded materials, in preparation.
187. X. Y. Yue and W. E, Numerical methods for multiscale transport equations and application to two phase porous media flow, J. Comput. Phys., to appear.
188. X. Y. Yue and W. E, The local microscale problem in the multiscale modelling of strongly heterogeneous media: Effect of boundary conditions and cell size, J. Comput. Phys., to appear.
189. V. V. Yurinskii, Averaging of symmetric diffusion in random media, Sibirsk. Mat. Zh., 23(2) (1982), 176-188; Siberian Math. J., 27 (1986), 603-613 (English Translation).