The present article is an overview of some mathematical results, which provide elements of rigorous basis for some multiscale computations in materials science. The emphasis is laid upon atomistic to continuum limits for crystalline materials. Various mathematical approaches are addressed. The setting is stationary. The relation to existing techniques used in the engineering literature is investigated.
Mots clés : problems of mechanics, variational problems, discrete to continuum limit, multiscale models, homogenization theory, $\Gamma $-limit, quasiconvexity, gradient flows, quasicontinuum method, adaptivity
@article{M2AN_2007__41_2_391_0,
author = {Blanc, Xavier and Bris, Claude Le and Lions, Pierre-Louis},
title = {Atomistic to continuum limits for computational materials science},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {391--426},
publisher = {EDP-Sciences},
volume = {41},
number = {2},
year = {2007},
doi = {10.1051/m2an:2007018},
mrnumber = {2339634},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007018/}
}
TY - JOUR AU - Blanc, Xavier AU - Bris, Claude Le AU - Lions, Pierre-Louis TI - Atomistic to continuum limits for computational materials science JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 391 EP - 426 VL - 41 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007018/ DO - 10.1051/m2an:2007018 LA - en ID - M2AN_2007__41_2_391_0 ER -
%0 Journal Article %A Blanc, Xavier %A Bris, Claude Le %A Lions, Pierre-Louis %T Atomistic to continuum limits for computational materials science %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 391-426 %V 41 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007018/ %R 10.1051/m2an:2007018 %G en %F M2AN_2007__41_2_391_0
Blanc, Xavier; Bris, Claude Le; Lions, Pierre-Louis. Atomistic to continuum limits for computational materials science. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 2, pp. 391-426. doi : 10.1051/m2an:2007018. http://www.numdam.org/articles/10.1051/m2an:2007018/
[1] and, A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36 (2004) 1-37. | MR | Zbl
[2] ,, and, 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. | MR | Zbl
[3] ,, and, Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives. Springer (2002). | MR | Zbl
[4] and, Derivation of higher order gradient continuum models from atomistic models for crystalline solids. SIAM J. Multiscale Model. Simul. 4 (2005) 531-562. | MR | Zbl
[5] and, A finite deformation membrane based on inter-atomic potentials for the transverse mechanics of nanotubes. Mech. Mater. 35 (2003) 175-622.
[6] and, Solid-State Physics. Saunders College Publishing (1976).
[7] , Lattice dynamical foundations of continuum theories. World Scientific, Philadelphia (1985). | MR
[8] , Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 63 (1977) 337-403. | MR | Zbl
[9] , 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 Series 284 (2001) 1-20. | MR | Zbl
[10] , Some open problems in elasticity, in Geometry, Mechanics, and Dynamics. Springer (2002) 3-59. | Zbl
[11] and, Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13-52. | Zbl
[12] and, Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. Royal Soc. London A 338 (1992) 389-450. | Zbl
[13] and, -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | Zbl
[14] , and Eds., Multiscale and multiresolution methods, Lecture notes in computational science and engineering 20. Springer (2002). | MR | Zbl
[15] , and, A semilinear equation in . Ann. Sc. Norm. Sup. Pisa 2 (1975) 523-555. | Numdam | Zbl
[16] , and, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications 5. North-Holland (1978). | MR | Zbl
[17] ,, and, Variational models for microstructures and phase transition, in Calculus of Variations and Geometric Evolution Problems, Lecture Notes in Mathematics 1713. Springer (1999) 85-210. | Zbl
[18] , 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). | MR | Zbl
[19] and, Relaxation of some multi-well problems. Proc. Royal Soc. Edinburgh A 131 (2001) 279-320. | Zbl
[20] , 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. | Zbl
[21] , Geometry optimization for crystals in Thomas-Fermi type theories of solids. Comm. P.D.E. 26 (2001) 651-696. | Zbl
[22] , Unique solvability for system of nonlinear elliptic PDEs arising in solid state physics. SIAM J. Math. Anal. 38 (2006) 1235-1248.
[23] and, 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. 1 329 (1999) 551-556. | Zbl
[24] and, Thomas-Fermi type models for polymers and thin films. Adv. Diff. Equ. 5 (2000) 977-1032.
[25] and, Periodicity of the infinite-volume ground-state of a one-dimensional quantum model. Nonlinear Anal., T.M.A 48 (2002) 791-803. | Zbl
[26] and, Définition d'énergies d'interfaces à partir de modèles atomiques. Note C.R. Acad. Sci. Sér. 1 340 (2005) 535-540. | Zbl
[27] , and, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: M2AN 39 (2005) 797-826. | Numdam
[28] , and, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics: the convex case. Acta Math. Appl. Sinica (to appear). | MR
[29] , and, Convergence de modèles moléculaires vers des modèles de mécanique des milieux continus [From molecular models to continuum mechanics]. Note C.R. Acad. Sci. Sér. 1 332 (2001) 949-956. | Zbl
[30] , and, From molecular models to continuum mechanics. Arch. Rat. Mech. Anal. 164 (2002) 341-381. | Zbl
[31] , and, A definition of the ground state energy for systems composed of infinitely many particles. Comm. P.D.E 28 (2003) 439-475. | Zbl
[32] , and, 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. | Zbl
[33] , and, On the energy of some microscopic stochastic lattices. Arch. Rat. Mech. Anal. 184 (2007) 303-339.
[34] , and (in preparation).
[35] , -convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, Oxford (2002). | MR
[36] , Non-local variational limits of discrete systems. Commun. Contemp. Math. 2 (2000) 285-297. | Zbl
[37] and, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7 (2002) 41-66. | Zbl
[38] and, Limits of discrete systems with long-range interactions. J. Convex Anal. 9 (2002) 363-399. | Zbl
[39] and, The passage from discrete to continuous variational problems: a nonlinear homogenization process. Preprint of the Scuola Normale Superiore di Pisa (2003). Available at http://cvgmt.sns.it/cgi/get.cgi/papers/bragel03/ | MR
[40] , and, Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Rat. Mech. Anal. 146 (1999) 23-58. | Zbl
[41] , and, The passage from nonconvex discrete systems to variational problems in Sobolev spaces: the one-dimensional case. Proc. Steklov Inst. Math. 236 (2002) 395-414. | Zbl
[42] , Probability, Classics in Applied Mathematics. SIAM, Philadelphia (1992). | MR | Zbl
[43] , Semilinear equations in without condition at infinity. Appl. Math. Optim. 12 (1984) 271-282. | Zbl
[44] and, Multiscale modelling of materials. MRS Bulletin 26 (2001).
[45] , and, Discrete homogenization in graphene sheet modeling, J. Elasticity 84 (2006) 33-68. | Zbl
[46] , 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. | Zbl
[47] and, Numerical approximation of young measuresin non-convex variational problems. Numer. Math. 84 (2000) 395-415. | Zbl
[48] , and, Limite thermodynamique pour des modèles de type Thomas-Fermi. Note C.R.A.S. Sér. 1 322 (1996) 357-364. | Zbl
[49] , and, 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. 1 327 (1998) 259-266. | Zbl
[50] , and, Mathematical theory of thermodynamic limits: Thomas-Fermi type models. Oxford University Press (1998). | MR | Zbl
[51] , and, On the thermodynamic limit for Hartree-Fock type models. Ann. Inst. H. Poincaré, Anal. Non Linéaire 18 (2001) 687-760. | Numdam | Zbl
[52] , and, On some periodic Hartree-type models for crystals. Ann. Inst. H. Poincaré, Anal. Non Linéaire 19 (2002) 143-190. | Numdam | Zbl
[53] , and, From atoms to crytals: a mathematical journey. Bull. Amer. Math. Soc. 42 (2005) 291-363. | Zbl
[54] and, Equilibrium configurations of crystals. Arch. Rat. Mech. Anal. 103 (1988) 237-277. | Zbl
[55] , Mathematical elasticity, Vol. 1. North Holland (1993). | Zbl
[56] ,,, and, Multiscale hybrid simulation methods for material systems J. Phys. Condens. Matt. 17 (2005) R691.
[57] , Direct methods in the calculus of variations. Springer-Verlag Berlin (1989). | MR | Zbl
[58] , An introduction to -convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser Boston, Inc., Boston, MA (1993). | MR | Zbl
[59] , and, Eds., Computer simulation of materials at atomic level. Wiley (2000).
[60] ,, and, A local criterion for regularity of a system of points. Sov. Math. Dokl. 17 (1976) 319-322. | Zbl
[61] , Variational Methods for Crystalline Microstructure - Analysis and Computation. Springer-Verlag (2003). | Zbl
[62] W. E and B. Engquist, The Heterogeneous Multi-Scale Methods. Comm. Math. Sci. 1 (2003) 87-132. | Zbl
[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. | Zbl
[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. | Zbl
[67] W. E and P.B. Ming, Cauchy-Born rule and stability of crystals: static problems. Arch. Rat. Mech. Anal. 183 (2007) 241-297. | Zbl
[68] ,, and, Density-functional-theory-based local quasicontinuum method: Prediction of dislocation nucleation. Phys. Rev. B 70 (2004) 100102(R).
[69] , Variational methods for elastic crystals. Arch. Rat. Mech. Anal. 97 (1987) 187-220. | Zbl
[70] , The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67 (1988) 175-195. | Zbl
[71] and, A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods. J. Mech. Phys. Solids 48 (2000) 1519-1540. | Zbl
[72] , and, Rigorous derivation of nonlinear plate theory and geometric rigidity. C.R. Acad. Sci. Paris Sér. I 334 (2002) 173-178. | Zbl
[73] and, Validity and failure of the Cauchy-Born hypothesis in a Two-Dimensional Mass-Spring Lattice. J. Nonlinear Sci. 12 (2002) 445-478. | Zbl
[74] and, The infinite-volume ground state of the Lennard-Jones potential. J. Stat. Phys. 20 (1979) 719-724.
[75] , and, Analyse asymptotique du comportement d'un assemblage collé [Asymptotic analysis of the behaviour of a bonded joint]. C.R. Acad. Sci. Paris Sér. I 322 (1996) 1107-1112. | Zbl
[76] , and, Mathematical analysis of a bonded joint with a soft thin adhesive. Math. Mech. Solids 4 (1999) 201-225. | Zbl
[77] WJ. Hehre, L. Radom, P.V.R. Shleyer and J. Pople, Ab initio molecular orbital theory. Wiley (1986).
[78] , and, Variational limit of a one dimensional discrete and statistically homogeneous system of material points. Asymptot. Anal. 28 (2001) 309-329. | Zbl
[79] , and, 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. | Zbl
[80] , Rotation and strain. Comm. Pure Appl. Math. 14 (1961) 391-413. | Zbl
[81] , Bounds for deformations in terms of average strains, in Inequalities III, O. Shisha Ed. (1972) 129-144. | Zbl
[82] , 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] and, Characterization of Young measures generated by gradients. Arch. Rat. Mech. Anal. 115 (1991) 329-365. | Zbl
[84] and, Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59-90. | Zbl
[85] , and Eds., Computer simulation in materials science, Kluwer (1996).
[86] , and Eds., Mesoscopic dynamics of fracture, Advances in Materials Research. Springer (1998).
[87] , Introduction to Solid State Physics. 7th edn. Wiley (1996). | Zbl
[88] and, An Analysis of the QuasiContinuum Method. J. Mech. Phys. Solids 49 (2001) 1899. | Zbl
[89] and, Optimal design and relaxation of variational problems. I-II-III. Comm. Pure Appl. Math. 39 (1986) 113-137, 139-182, 353-377. | Zbl
[90] , Ergodic theorems, Studies in Mathematics 6. de Gruyter (1985). | MR | Zbl
[91] , On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré, Anal. Non Linéaire 16 (1999) 1-13. | Numdam | Zbl
[92] , Computational Chemistry, in Handbook of numerical analysis, Vol. X, P.G. Ciarlet Ed., North-Holland (2003). | MR | Zbl
[93] , Computational chemistry from the perspective of numerical analysis, Acta Numer. 14 (2005) 363-444. | Zbl
[94] ,,, and, Atomistic mechanisms governing elastic limit and incipient plasticity in crystals. Nature 418 (2002) 307.
[95] , 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. I 317 (1993) 429-433. | Zbl
[96] and, Une modélisation du comportement d'un joint collé élastique [A modelling of elastic adhesively bonding joints]. C.R. Acad. Sci. Paris Sér. I 322 (1996) 295-300. | Zbl
[97] , Thomas-Fermi and related theories of atoms and molecules. Rev. Modern Phys. 53 (1981) 603-641 . | Zbl
[98] and, The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math. 23 (1977) 22-116. | Zbl
[99] , 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 http://www.math.nus.edu.sg/ matlinp/WWW/linsiap.pdf
[100] , Theoretical and numerical analysis of the quasi-continuum approximation of a material particle model. Math. Comput. 72 (2003) 657-675. | Zbl
[101] , 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 http://www.ims.nus.edu.sg/preprints/2005-80.pdf
[102] and, Numerical solution of a virtual internal bond model for material fracture. Physica D 167 (2002) 101-121. | Zbl
[103] , and, Special Issue on Multiple Scale Methods for Nanoscale Mechanics and Materials. Comp. Meth. Appl. Mech. Eng. 193 (2004) 17-20. | Zbl
[104] , On the computation of crystalline microstructure. Acta Numer. 5 (1996) 191-258. | Zbl
[105] , Computational modeling of microstructure, in Proceedings of the International Congress of Mathematicians, ICM, Beijing (2002) 707-716. | Zbl
[106] and, The Quasicontinuum Method: Overview, applications and current directions. J. Computer-Aided Materials Design 9 (2002) 203-239.
[107] ,, and, Quasicontinuum simulation of fracture at the atomic scale. Modelling Simul. Mater. Sci. Eng. 6 (1998) 607.
[108] , Quasi-convexity and the lower semi-continuity of multiple integrals. Pacific J. Math. 2 (1952) 25-53. | Zbl
[109] , 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. | Zbl
[110] and, On the configuration of systems of interacting particles with minimum potential energy per particle. Physica 98A (1979) 274. | MR
[111] and, On the configuration of systems of interacting particles with minimum potential energy per particle. Physica 99A (1979) 569. | MR
[112] , Continuum limit of a one-dimensional atomistic energy based on local minimization. Technical report 05/11, Oxford University Computing Laboratory (2005).
[113] and, A simple model for phase transitions: from the discrete to the continuum problem. Quart. Appl. Math. 61 (2003) 89-109. | Zbl
[114] , Parametrized Measures and Variational Principles. Birkhäuser (1997). | MR | Zbl
[115] , Variational Methods in Nonlinear Elasticity. SIAM (2000). | MR | Zbl
[116] ., Quantum mechanical ab initio calculation of the properties of crystalline materials, Lecture Notes in Chemistry 67. Springer (1996).
[117] , Computational Material Science. Wiley (1998).
[118] , Ground states for soft disks. J. Stat. Phys. 26 (1981) 365. | MR
[119] , Liouville's theory on conformal mappings under minimal regularity assumptions. Sibirskii Math. 8 (1967) 69-85. | Zbl
[120] and, Young measure flow as a model for damage, SIAM J. Math. Anal. (2005) (to appear). | MR | Zbl
[121] and, Concurrent coupling of length scales in solid state system, in [59] 251-291.
[122] , On the passage form atomic to continuum theory for thin films. Preprint 82/2005 of the Max Planck Institute of Leipzig (2005). Available at http://www.mis.mpg.de/preprints/2005/prepr2005_82.html | MR | Zbl
[123] , Qualitative properties of a continuum theory for thin films. Preprint 83/2005 of the Max Planck Institute of Leipzig (2005). Available at http://www.mis.mpg.de/preprints/2005/prepr2005_83.html
[124] , A derivation of continuum nonlinear plate theory form atomistic models. Preprint 90/2005 of the Max Planck Institute of Leipzig (2005). Available at http://www.mis.mpg.de/preprints/2005/prepr2005_90.html | MR | Zbl
[125] ,,, and, Quasicontinuum models of interfacial structure and deformation. Phys. Rev. Lett. 80 (1998) 742.
[126] ,,,, and, An adaptative finite element approach to atomic-scale mechanics - the QuasiContinuum Method. J. Mech. Phys. Solids 47 (1999) 611. | Zbl
[127] , Universality in the Thomas-Fermi-von Weizsäcker model of atoms and molecules. Comm. Math. Phys. 129 (1990) 561-598. | Zbl
[128] , On regularity for Monge-Ampère equations. Preprint, Heriott-Watt University (1991).
[129] , Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh A 120 (1992) 185-189. | Zbl
[130] , On the problem of two wells, in Microstructure and phase transition, IMA Vol. Math. Appl. 54. Springer, New York, (1993) 183-189. | Zbl
[131] and, Modern quantum chemistry: an introduction. Macmillan (1982).
[132] and, Mixed atomistic and continuum models of deformation in solids. Langmuir 12 (1996) 4529.
[133] , and, Quasicontinuum analysis of defects in solids. Phil. Mag. A. 73 (1996) 1529-1563.
[134] ,, and, Mixed finite element and atomistic formulation for complex crystals. Phys. Rev. B 59 (1999) 235.
[135] , A proof of crystallization in two dimensions. Comm. Math. Phys. 262 (2006) 209-236. | Zbl
[136] , 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] , On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle. Physica 92A (1978) 343.
[138] , Synergistic materials science. Nature Mater. 2 (2003) 3-5.
[139] , Lectures on the calculus of variations and optimal control theory. W.B. Saunders Co., Philadelphia-London-Toronto (1969). | Zbl
[140] , and, Étude théorique et numérique du comportement d'un assemblage de plaques [Theoretical study of the behaviour of bonded plates]. C.R. Mécanique 330 (2002) 359-364. | Zbl
Cité par Sources :
