Atomistic to Continuum limits for computational materials science

 Access to full text

 Full (PDF)

 Access to full text

 Full (PDF)

1. R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal.36 (2004) 1–37.  
2. M. Anitescu, D. Negrut, P. Zapol and A. El-Azab, A note on the regularity of reduced models obtained by nonlocal quasi-continuum-like approach. Technical report ANL/MCS-P1303-1105, Argonne National Laboratory, Argonne, Illinois (2005). Available at http://www-unix.mcs.anl.gov/~anitescu/PUBLICATIONS/quasicont.pdf.  
3. N. Antonic, C.J. van Duijn, W. Jäger and A. Mikelic, Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives. Springer (2002).  
4. M. Arndt and M. Griebel, Derivation of higher order gradient continuum models from atomistic models for crystalline solids. SIAM J. Multiscale Model. Simul.4 (2005) 531–562.  
5. M. Arroyo and T. Belytshko, A finite deformation membrane based on inter-atomic potentials for the transverse mechanics of nanotubes. Mech. Mater.35 (2003) 175–622.  
6. N.W. Ashcroft and N.D. Mermin, Solid-State Physics. Saunders College Publishing (1976).  
7. A. Askar, Lattice dynamical foundations of continuum theories. World Scientific, Philadelphia (1985).  
8. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal.63 (1977) 337–403.  
9. J.M. Ball, Singularities and computation of miminizers for variational problems, in Foundations of Computational Mathematics, R. DeVore, A. Iserles and E. Suli Eds., Cambridge University Press London Mathematical Society Lecture Note Series284 (2001) 1–20.  
10. J.M. Ball, Some open problems in elasticity, in Geometry, Mechanics, and Dynamics. Springer (2002) 3–59.  
11. J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal.100 (1987) 13–52.  
12. J.M. Ball and R.D. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. Royal Soc. London A338 (1992) 389–450.  
13. J.M. Ball and F. Murat, W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984) 225–253.  
14. T.J. Barth, T. Chan and R. Haimes Eds., Multiscale and multiresolution methods, Lecture notes in computational science and engineering20. Springer (2002).  
15. P. Bénilan, H. Brezis and M. Crandall, A semilinear equation in $L^1\left({ℝ}^N\right)$. Ann. Sc. Norm. Sup. Pisa2 (1975) 523–555.  
16. A. Bensoussan, J.-L. Lions and G. Papnicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications5. North-Holland (1978).  
17. F. Bethuel, G. Huisken, S. Müller and K. Steffen, Variational models for microstructures and phase transition, in Calculus of Variations and Geometric Evolution Problems, Lecture Notes in Mathematics1713. Springer (1999) 85–210.  
18. K. Bhattacharya, Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect. Oxford Series on Materials Modelling, Oxford University Press (2003).  
19. K. Bhattacharya and G. Dolzmann, Relaxation of some multi-well problems. Proc. Royal Soc. Edinburgh A131 (2001) 279–320.  
20. X. Blanc, A mathematical insight into ab initio simulations of solid phase, in Mathematical Models and Methods for Ab Initio Quantum Chemistry, M. Defranceschi and C. Le Bris Eds., Lect. Notes Chem.74. Springer (2000) 133–158.  
21. X. Blanc, Geometry optimization for crystals in Thomas-Fermi type theories of solids. Comm. P.D.E.26 (2001) 651–696.  
22. X. Blanc, Unique solvability for system of nonlinear elliptic PDEs arising in solid state physics. SIAM J. Math. Anal.38 (2006) 1235–1248.  
23. X. Blanc and C. Le Bris, Optimisation de géométrie dans le cadre des théories de Thomas-Fermi pour les cristaux périodiques [Geometry optimization for Thomas-Fermi type theories of solids]. Note C.R. Acad. Sci. Sér. 1329 (1999) 551–556.  
24. X. Blanc and C. Le Bris, Thomas-Fermi type models for polymers and thin films. Adv. Diff. Equ.5 (2000) 977–1032.  
25. X. Blanc and C. Le Bris, Periodicity of the infinite-volume ground-state of a one-dimensional quantum model. Nonlinear Anal., T.M.A48 (2002) 791–803.  
26. X. Blanc and C. Le Bris, Définition d'énergies d'interfaces à partir de modèles atomiques. Note C.R. Acad. Sci. Sér. 1340 (2005) 535–540.  
27. X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: M2AN39 (2005) 797–826.  
28. X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics: the convex case. Acta Math. Appl. Sinica (to appear).  
29. X. Blanc, C. Le Bris and P.-L. Lions, Convergence de modèles moléculaires vers des modèles de mécanique des milieux continus [From molecular models to continuum mechanics]. Note C.R. Acad. Sci. Sér. 1332 (2001) 949–956.  
30. X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics. Arch. Rat. Mech. Anal.164 (2002) 341–381.  
31. X. Blanc, C. Le Bris and P.-L. Lions, A definition of the ground state energy for systems composed of infinitely many particles. Comm. P.D.E28 (2003) 439–475.  
32. X. Blanc, C. Le Bris and P.-L. Lions, Du discret au continu pour des modèles de réseaux aléatoires d'atomes [Discrete to continuum limit for some models of stochastic lattices of atoms]. Note C.R. Acad. Sci. Sér. 1.342 (2006) 627–633.  
33. X. Blanc, C. Le Bris and P.-L. Lions, On the energy of some microscopic stochastic lattices. Arch. Rat. Mech. Anal.184 (2007) 303–339.  
34. X. Blanc, C. Le Bris and P.-L. Lions (in preparation).  
35. A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications22. Oxford University Press, Oxford (2002).  
36. A. Braides, Non-local variational limits of discrete systems. Commun. Contemp. Math.2 (2000) 285–297.  
37. A. Braides and M.S. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids7 (2002) 41–66.  
38. A. Braides and M.S. Gelli, Limits of discrete systems with long-range interactions. J. Convex Anal.9 (2002) 363–399.  
39. A. Braides and M.S. Gelli, The passage from discrete to continuous variational problems: a nonlinear homogenization process. Preprint of the Scuola Normale Superiore di Pisa (2003). Available at  URIhttp://cvgmt.sns.it/cgi/get.cgi/papers/bragel03/
40. A. Braides, G. Dal Maso and A. Garroni, Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Rat. Mech. Anal.146 (1999) 23–58.  
41. A. Braides, M.S. Gelli and M. Sigalotti, The passage from nonconvex discrete systems to variational problems in Sobolev spaces: the one-dimensional case. Proc. Steklov Inst. Math.236 (2002) 395–414.  
42. L. Breimana, Probability, Classics in Applied Mathematics. SIAM, Philadelphia (1992).  
43. H. Brezis, Semilinear equations in ${ℝ}^N$ without condition at infinity. Appl. Math. Optim.12 (1984) 271–282.  
44. V.V. Bulatov and T. Diaz de la Rubia, Multiscale modelling of materials. MRS Bulletin26 (2001).  
45. D. Caillerie, A. Mourad and A. Raoult, Discrete homogenization in graphene sheet modeling, J. Elasticity84 (2006) 33–68.  
46. C. Carstensen, Numerical Analysis of Microstructure, in Theory and Numerics of Differential Equations, J.F. Blowey, J.P. Coleman and A.W. Craig Eds., Springer (2001) 59–126.  
47. C. Carstensen and T. Roubíček, Numerical approximation of young measuresin non-convex variational problems. Numer. Math.84 (2000) 395–415.  
48. I. Catto, C. Le Bris and P.-L. Lions, Limite thermodynamique pour des modèles de type Thomas-Fermi. Note C.R.A.S. Sér. 1322 (1996) 357–364.  
49. I. Catto, C. Le Bris and P.-L. Lions, Sur la limite thermodynamique pour des modèles de type Hartree et Hartree-Fock [On the thermodynamic limit for Hartree and Hartree-Fock type models]. Note C.R.A.S. Sér. 1327 (1998) 259–266.  
50. I. Catto, C. Le Bris and P.-L. Lions, Mathematical theory of thermodynamic limits: Thomas-Fermi type models. Oxford University Press (1998).  
51. I. Catto, C. Le Bris and P.-L. Lions, On the thermodynamic limit for Hartree-Fock type models. Ann. Inst. H. Poincaré, Anal. Non Linéaire18 (2001) 687–760.  
52. I. Catto, C. Le Bris and P.-L. Lions, On some periodic Hartree-type models for crystals. Ann. Inst. H. Poincaré, Anal. Non Linéaire19 (2002) 143–190.  
53. I. Catto, C. Le Bris and P.-L. Lions, From atoms to crytals: a mathematical journey. Bull. Amer. Math. Soc.42 (2005) 291–363.  
54. M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rat. Mech. Anal.103 (1988) 237–277.  
55. P.G. Ciarlet, Mathematical elasticity, Vol. 1. North Holland (1993).  
56. G. Csányi, T. Albaret, G. Moras, M.C. Payne and A. De Vita, Multiscale hybrid simulation methods for material systems J. Phys. Condens. Matt.17 (2005) R691.  
57. R. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag Berlin (1989).  
58. G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications8. Birkhäuser Boston, Inc., Boston, MA (1993).  
59. P. Deák, T. Frauenheim and M.R. Pederson, Eds., Computer simulation of materials at atomic level. Wiley (2000).  
60. B.N. Delaunay, N.P. Dolbilin, M.I. Shtogrin and R.V. Galiulin, A local criterion for regularity of a system of points. Sov. Math. Dokl.17 (1976) 319–322.  
61. G. Dolzmann, Variational Methods for Crystalline Microstructure – Analysis and Computation. Springer-Verlag (2003).  
62. W. E and B. Engquist, The Heterogeneous Multi-Scale Methods. Comm. Math. Sci.1 (2003) 87–132.  
63. W. E and Z. Huang, Matching conditions in atomistic-continuum modeling of materials. Phys. Rev. Lett.87 (2001) 135501.  
64. W. E and Z. Huang, A dynamic atomistic-continuum method for the simulation of crystalline materials. J. Comp. Phys.182 (2002) 234–261.  
65. W. E and P.B. Ming, Atomistic and continuum theory of solids, I. Preprint (2003).  
66. W. E and P.B. Ming, Analysis of multiscale methods. J. Comp. Math.22 (2004) 210–219.  
67. W. E and P.B. Ming, Cauchy-Born rule and stability of crystals: static problems. Arch. Rat. Mech. Anal.183 (2007) 241–297.  
68. M. Fago, R.L. Hayes, E.A. Carter and M. Ortiz, Density-functional-theory-based local quasicontinuum method: Prediction of dislocation nucleation. Phys. Rev. B70 (2004) 100102(R).  
69. I. Fonseca, Variational methods for elastic crystals. Arch. Rat. Mech. Anal.97 (1987) 187–220.  
70. I. Fonseca, The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl.67 (1988) 175–195.  
71. G. Friesecke and R.D. James, A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods. J. Mech. Phys. Solids48 (2000) 1519–1540.  
72. G. Friesecke, R.D. James and S. Müller, Rigorous derivation of nonlinear plate theory and geometric rigidity. C.R. Acad. Sci. Paris Sér. I334 (2002) 173–178.  
73. G. Friesecke and F. Theil, Validity and failure of the Cauchy-Born hypothesis in a Two-Dimensional Mass-Spring Lattice. J. Nonlinear Sci.12 (2002) 445–478.  
74. C.S. Gardner and C. Radin, The infinite-volume ground state of the Lennard-Jones potential. J. Stat. Phys.20 (1979) 719–724.  
75. G. Geymonat, F. Krasucki and S. Lenci, Analyse asymptotique du comportement d'un assemblage collé [Asymptotic analysis of the behaviour of a bonded joint]. C.R. Acad. Sci. Paris Sér. I322 (1996) 1107–1112.  
76. G. Geymonat, F. Krasucki and S. Lenci, Mathematical analysis of a bonded joint with a soft thin adhesive. Math. Mech. Solids4 (1999) 201–225.  
77. WJ. Hehre, L. Radom, P.V.R. Shleyer and J. Pople, Ab initio molecular orbital theory. Wiley (1986).  
78. O. Iosifescu, C. Licht and G. Michaille, Variational limit of a one dimensional discrete and statistically homogeneous system of material points. Asymptot. Anal.28 (2001) 309–329.  
79. O. Iosifescu, C. Licht and G. Michaille, Variational limit of a one-dimensional discrete and statistically homogeneous system of material points. C.R. Acad. Sci. Paris Sér. I Math.332 (2001) 575–580.  
80. F. John, Rotation and strain. Comm. Pure Appl. Math.14 (1961) 391–413.  
81. F. John, Bounds for deformations in terms of average strains, in Inequalities III, O. Shisha Ed. (1972) 129–144.  
82. D. Kinderlehrer, Remarks about equilibrium configurations of crystals, in Material instabilities in contiuum mechanics and related mathematical problems, J.M. Ball Ed., Oxford University Press (1998) 217–242.  
83. D. Kinderlehrer and P. Pedregal, Characterization of Young measures generated by gradients. Arch. Rat. Mech. Anal.115 (1991) 329–365.  
84. D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal.4 (1994) 59–90.  
85. O. Kirchner, L.P. Kubin and V. Pontikis Eds., Computer simulation in materials science, Kluwer (1996).  
86. H. Kitagawa, T. Aihara Jr. and Y. Kawazoe Eds., Mesoscopic dynamics of fracture, Advances in Materials Research. Springer (1998).  
87. C. Kittel, Introduction to Solid State Physics. 7th edn. Wiley (1996).  
88. J. Knap and M. Ortiz, An Analysis of the QuasiContinuum Method. J. Mech. Phys. Solids49 (2001) 1899.  
89. R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. I-II-III. Comm. Pure Appl. Math.39 (1986) 113–137, 139–182, 353–377.  
90. U. Krengel, Ergodic theorems, Studies in Mathematics6. de Gruyter (1985).  
91. J. Kristensen, On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré, Anal. Non Linéaire16 (1999) 1–13.  
92. C. Le Bris, Computational Chemistry, in Handbook of numerical analysis, Vol. X, P.G. Ciarlet Ed., North-Holland (2003).  
93. C. Le Bris, Computational chemistry from the perspective of numerical analysis, Acta Numer.14 (2005) 363–444.  
94. J. Li, K.J. Van Vliet, T. Zhu, S. Suresh and S. Yip, Atomistic mechanisms governing elastic limit and incipient plasticity in crystals. Nature418 (2002) 307.  
95. C. Licht, Comportement asymptotique d'une bande dissipative mince de faible rigidité [Asymptotic behaviour of a thin dissipative layer with low stiffness]. C.R. Acad. Sci. Paris Sér. I317 (1993) 429–433.  
96. C. Licht and G. Michaille, Une modélisation du comportement d'un joint collé élastique [A modelling of elastic adhesively bonding joints]. C.R. Acad. Sci. Paris Sér. I322 (1996) 295–300.  
97. E.H. Lieb, Thomas-Fermi and related theories of atoms and molecules. Rev. Modern Phys.53 (1981) 603–641 .  
98. E.H. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math.23 (1977) 22–116.  
99. P. Lin, A nonlinear wave equation of mixed type for fracture dynamics. Research report No. 777, Department of Mathematics, The National University of Singapore, August 2000. Available at  matlinp/WWW/linsiap.pdf  URIhttp://www.math.nus.edu.sg/
100. P. Lin, Theoretical and numerical analysis of the quasi-continuum approximation of a material particle model. Math. Comput.72 (2003) 657–675.  
101. P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material. Preprint 2005-80 of the Institute for mathematical sciences, National University of Singapore (2005). Available at  URIhttp://www.ims.nus.edu.sg/preprints/2005-80.pdf
102. P. Lin and C.W. Shu, Numerical solution of a virtual internal bond model for material fracture. Physica D167 (2002) 101–121.  
103. W.K. Liu, D. Qian and M.F. Horstemeyer, Special Issue on Multiple Scale Methods for Nanoscale Mechanics and Materials. Comp. Meth. Appl. Mech. Eng.193 (2004) 17–20.  
104. M. Luskin, On the computation of crystalline microstructure. Acta Numer.5 (1996) 191–258.  
105. M. Luskin, Computational modeling of microstructure, in Proceedings of the International Congress of Mathematicians, ICM, Beijing (2002) 707–716.  
106. R. Miller and E.B. Tadmor, The Quasicontinuum Method: Overview, applications and current directions. J. Computer-Aided Materials Design9 (2002) 203–239.  
107. R. Miller, E.B. Tadmor, R. Phillips and M. Ortiz, Quasicontinuum simulation of fracture at the atomic scale. Modelling Simul. Mater. Sci. Eng.6 (1998) 607.  
108. C.B. Morrey Jr., Quasi-convexity and the lower semi-continuity of multiple integrals. Pacific J. Math.2 (1952) 25–53.  
109. S. Müller, Variational models for microstructure and phase transitions, in Calculus of Variations and Geometric Evolution Problems. Lect. Notes Math.1713. Springer Verlag, Berlin (1999) 85–210.  
110. B.R.A. Nijboer and W.J. Ventevogel, On the configuration of systems of interacting particles with minimum potential energy per particle. Physica98A (1979) 274.  
111. B.R.A Nijboer and W.J. Ventevogel, On the configuration of systems of interacting particles with minimum potential energy per particle. Physica99A (1979) 569.  
112. C. Ortner, Continuum limit of a one-dimensional atomistic energy based on local minimization. Technical report 05/11, Oxford University Computing Laboratory (2005).  
113. S. Pagano and R. Paroni, A simple model for phase transitions: from the discrete to the continuum problem. Quart. Appl. Math.61 (2003) 89–109.  
114. P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser (1997).  
115. P. Pedregal, Variational Methods in Nonlinear Elasticity. SIAM (2000).  
116. C. Pisani Ed., Quantum mechanical ab initio calculation of the properties of crystalline materials, Lecture Notes in Chemistry67. Springer (1996).  
117. D. Raabe, Computational Material Science. Wiley (1998).  
118. C. Radin, Ground states for soft disks. J. Stat. Phys.26 (1981) 365.  
119. Y.G. Reshetnyak, Liouville's theory on conformal mappings under minimal regularity assumptions. Sibirskii Math.8 (1967) 69–85.  
120. M.O. Rieger and J. Zimmer, Young measure flow as a model for damage, SIAM J. Math. Anal. (2005) (to appear).  
121. R.E. Rudd and J.Q. Broughton, Concurrent coupling of length scales in solid state system, in [59] 251–291.  
122. B. Schmidt, On the passage form atomic to continuum theory for thin films. Preprint 82/2005 of the Max Planck Institute of Leipzig (2005). Available at  URIhttp://www.mis.mpg.de/preprints/2005/prepr2005_82.html
123. B. Schmidt, Qualitative properties of a continuum theory for thin films. Preprint 83/2005 of the Max Planck Institute of Leipzig (2005). Available at  URIhttp://www.mis.mpg.de/preprints/2005/prepr2005_83.html
124. B. Schmidt, A derivation of continuum nonlinear plate theory form atomistic models. Preprint 90/2005 of the Max Planck Institute of Leipzig (2005). Available at  URIhttp://www.mis.mpg.de/preprints/2005/prepr2005_90.html
125. V.B. Shenoy, R. Miller, E.B. Tadmor, R. Phillips and M. Ortiz, Quasicontinuum models of interfacial structure and deformation. Phys. Rev. Lett.80 (1998) 742.  
126. V.B. Shenoy, R. Miller, E.B. Tadmor, D. Rodney, R. Phillips and M. Ortiz, An adaptative finite element approach to atomic-scale mechanics – the QuasiContinuum Method. J. Mech. Phys. Solids47 (1999) 611.  
127. J.P. Solovej, Universality in the Thomas-Fermi-von Weizsäcker model of atoms and molecules. Comm. Math. Phys.129 (1990) 561–598.  
128. V. Šveràk, On regularity for Monge-Ampère equations. Preprint, Heriott-Watt University (1991).  
129. V. Šveràk, Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh A120 (1992) 185–189.  
130. V. Šveràk, On the problem of two wells, in Microstructure and phase transition, IMA Vol. Math. Appl.54. Springer, New York, (1993) 183–189.  
131. A. Szabo and N.S. Ostlund, Modern quantum chemistry: an introduction. Macmillan (1982).  
132. E.B. Tadmor and R. Phillips, Mixed atomistic and continuum models of deformation in solids. Langmuir12 (1996) 4529.  
133. E.B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids. Phil. Mag. A.73 (1996) 1529–1563.  
134. E.B. Tadmor, G.S. Smith, N. Bernstein and E. Kaxiras, Mixed finite element and atomistic formulation for complex crystals. Phys. Rev. B59 (1999) 235.  
135. F. Theil, A proof of crystallization in two dimensions. Comm. Math. Phys.262 (2006) 209–236.  
136. L. Truskinovsky, Fracture as a phase transformation, in Contemp. Res. in Mech. and Math. of Materials, Ericksen's symposium, R. Batra and M. Beatty Eds., CIMNE, Barcelone (1996) 322–332.  
137. W.J. Ventevogel, On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle. Physica92A (1978) 343.  
138. S. Yip, Synergistic materials science. Nature Mater.2 (2003) 3–5.  
139. L.C. Young, Lectures on the calculus of variations and optimal control theory. W.B. Saunders Co., Philadelphia-London-Toronto (1969).  
140. F. Zaittouni, F. Lebon and C. Licht, Étude théorique et numérique du comportement d'un assemblage de plaques [Theoretical study of the behaviour of bonded plates]. C.R. Mécanique330 (2002) 359–364.  

1. Matthew Dobson, Mitchell Luskin, Analysis of a force-based quasicontinuum approximation