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

Recent development in the multiscale method based on the bridging scale concept is presented with an emphasis on complex material systems. The bridging scale method (BSM) was originally proposed by Wagner and Liu (J. Comput. Phys. 2003; 190:249–274) as an effective way of treating the interface in coupled atomistic/continuum simulation. Since its publication, the BSM has become a very useful paradigm that has been applied to solve a host of problems in mechanical sciences of complex material systems. In this paper, we present a review on the recent developments. We first describe the application of BSM to the coupled atomistic/continuum simulation of dynamic fracture. The important extensions within the framework of space–time method and multiscale non-equilibrium molecular dynamics are then presented. We then focus on the multiresolution continuum theory that inherits the BSM concept in the analysis of heterogeneous material structures. Recent work of incorporating statistical factors into this model based on the concurrent nested homogenization of randomness of the material structures is highlighted. Finally, we present the use of the bridging scale concept in resolving the electron-mechanical coupling mechanism. The robustness of the BSM is demonstrated through many benchmark problems and application examples. Copyright © 2010 John Wiley & Sons, Ltd.

[1]  John E. Pask,et al.  Classical and enriched finite element formulations for Bloch‐periodic boundary conditions , 2009 .

[2]  Philippe H. Geubelle,et al.  The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials , 2002 .

[3]  Rong Tian,et al.  Complexity science of multiscale materials via stochastic computations , 2009 .

[4]  Wing Kam Liu,et al.  On criteria for dynamic adiabatic shear band propagation , 2007 .

[5]  Massimo Ruzzene,et al.  Bridging Scales Analysis of Wave Propagation in Heterogeneous Structures with Imperfections , 2008 .

[6]  Thomas Y. Hou,et al.  A pseudo-spectral multiscale method: Interfacial conditions and coarse grid equations , 2006, J. Comput. Phys..

[7]  W. E,et al.  Matching conditions in atomistic-continuum modeling of materials. , 2001, Physical review letters.

[8]  Harold S. Park,et al.  Three-dimensional bridging scale analysis of dynamic fracture , 2005 .

[9]  Shaofan Li,et al.  A MULTI-SCALE NONEQUILIBRIUM MOLECULAR DYNAMICS ALGORITHM AND ITS APPLICATIONS , 2009 .

[10]  M. A. Dokainish,et al.  Simulation of the (001) plane crack in α-iron employing a new boundary scheme , 1982 .

[11]  Robert E. Rudd,et al.  Concurrent Coupling of Length Scales in Solid State Systems , 2000 .

[12]  Thomas J. R. Hughes,et al.  Space-time finite element methods for second-order hyperbolic equations , 1990 .

[13]  T. Belytschko,et al.  A bridging domain method for coupling continua with molecular dynamics , 2004 .

[14]  Linda Hung,et al.  Accurate simulations of metals at the mesoscale: Explicit treatment of 1 million atoms with quantum mechanics , 2009 .

[15]  Harold S. Park,et al.  Nano Mechanics and Materials: Theory, Multiscale Methods and Applications , 2006 .

[16]  J. Q. Broughton,et al.  Concurrent coupling of length scales: Methodology and application , 1999 .

[17]  Hiroshi Kadowaki,et al.  Bridging multi-scale method for localization problems , 2004 .

[18]  Wing Kam Liu,et al.  Multiresolution analysis for material design , 2006 .

[19]  K. K. Choi,et al.  Reliability-based design optimization of problems with correlated input variables using a Gaussian Copula , 2009 .

[20]  Wing Kam Liu,et al.  Multiresolution continuum modeling of micro-void assisted dynamic adiabatic shear band propagation , 2010 .

[21]  Ted Belytschko,et al.  Discontinuous enrichment in finite elements with a partition of unity method , 2000 .

[22]  Wei Chen,et al.  Statistical volume element method for predicting microstructure–constitutive property relations , 2008 .

[23]  Dong Qian,et al.  Concurrent quantum/continuum coupling analysis of nanostructures , 2008 .

[24]  Dimos C. Charmpis,et al.  The need for linking micromechanics of materials with stochastic finite elements: A challenge for materials science , 2007 .

[25]  Wing Kam Liu,et al.  Predictive multiscale theory for design of heterogeneous materials , 2008 .

[26]  J. Pask,et al.  Finite element methods in ab initio electronic structure calculations , 2005 .

[27]  J. Linnett,et al.  Quantum mechanics , 1975, Nature.

[28]  Zhou,et al.  Dynamic crack processes via molecular dynamics. , 1996, Physical review letters.

[29]  Ted Belytschko,et al.  An atomistic-based finite deformation membrane for single layer crystalline films , 2002 .

[30]  Huajian Gao,et al.  Mechanism-based strain gradient plasticity—II. Analysis , 2000 .

[31]  W. Cai,et al.  Minimizing boundary reflections in coupled-domain simulations. , 2000, Physical review letters.

[32]  F. Bloch.  Über die Quantenmechanik der Elektronen in Kristallgittern , 1929 .

[33]  R. Landauer,et al.  Generalized many-channel conductance formula with application to small rings. , 1985, Physical review. B, Condensed matter.

[34]  R. Feynman.  Forces in Molecules , 1939 .

[35]  E Weinan,et al.  A dynamic atomistic-continuum method for the simulation of crystalline materials , 2001 .

[36]  Harold S. Park,et al.  An introduction to computational nanomechanics and materials , 2004 .

[37]  M. Ortiz,et al.  Vacancy clustering and prismatic dislocation loop formation in aluminum , 2007 .

[38]  H. Johnson,et al.  The influence of mechanical deformation on the electrical properties of single wall carbon nanotubes , 2004 .

[39]  H. Rafii-Tabar,et al.  Multiscale numerical modelling of cracK propagation in tvvo-dimensional metal plate , 1998 .

[40]  R. Hill.  Elastic properties of reinforced solids: some theoretical principles , 1963 .

[41]  Ted Belytschko,et al.  Conservation properties of the bridging domain method for coupled molecular/continuum dynamics , 2008 .

[42]  Wei Chen,et al.  Efficient Random Field Uncertainty Propagation in Design Using Multiscale Analysis , 2009 .

[43]  Holian,et al.  Fracture simulations using large-scale molecular dynamics. , 1995, Physical review. B, Condensed matter.

[44]  B. Schweizer,et al.  On Nonparametric Measures of Dependence for Random Variables , 1981 .

[45]  J. D. Doll,et al.  Generalized Langevin equation approach for atom/solid‐surface scattering: Collinear atom/harmonic chain model , 1974 .

[46]  Kaushik Bhattacharya,et al.  Non-periodic finite-element formulation of orbital-free density functional theory , 2006 .

[47]  J. W. Walter,et al.  The asymptotic structure of an adiabatic shear band in antiplane motion , 1996 .

[48]  Harold S. Park,et al.  An introduction and tutorial on multiple-scale analysis in solids , 2004 .

[49]  Harold S. Park,et al.  A phonon heat bath approach for the atomistic and multiscale simulation of solids , 2007 .

[50]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[51]  Thomas Y. Hou,et al.  A mathematical framework of the bridging scale method , 2006 .

[52]  T. Belytschko,et al.  Extended finite element method for three-dimensional crack modelling , 2000 .

[53]  Toshio Mura,et al.  Micromechanics of defects in solids , 1982 .

[54]  D. Pfeffermann,et al.  Small area estimation , 2011 .

[55]  Franck J. Vernerey,et al.  Multi-scale micromorphic theory for hierarchical materials , 2007 .

[56]  William T. Weeks,et al.  Numerical Inversion of Laplace Transforms Using Laguerre Functions , 1966, JACM.

[57]  Dong Qian,et al.  A Virtual Atom Cluster Approach to the Mechanics of Nanostructures , 2004 .

[58]  Hanqing Jiang,et al.  A Finite-Temperature Continuum Theory Based on Interatomic , 2005 .

[59]  Jacob Fish,et al.  Discrete-to-continuum bridging based on multigrid principles , 2004 .

[60]  Gregory J. Wagner,et al.  A multiscale projection method for the analysis of carbon nanotubes , 2004 .

[61]  Ted Belytschko,et al.  Arbitrary discontinuities in space–time finite elements by level sets and X‐FEM , 2004 .

[62]  E. Carter,et al.  Quantum simulation of materials at micron scales and beyond , 2008 .

[63]  R.K. Kalia,et al.  Multiscale simulation of nanosystems , 2001, Comput. Sci. Eng..

[64]  Cahal McVeigh,et al.  Multiresolution modeling of ductile reinforced brittle composites , 2009 .

[65]  M. Ortiz,et al.  Quasicontinuum analysis of defects in solids , 1996 .

[66]  R. Ghanem,et al.  Polynomial chaos decomposition for the simulation of non-gaussian nonstationary stochastic processes , 2002 .

[67]  Shaofan Li,et al.  On multiscale non‐equilibrium molecular dynamics simulations , 2010 .

[68]  Brad Lee Holian,et al.  Molecular dynamics investigation of dynamic crack stability , 1997 .

[69]  Ted Belytschko,et al.  Mechanics of defects in carbon nanotubes: Atomistic and multiscale simulations , 2005 .

[70]  Noam Bernstein,et al.  Mixed finite element and atomistic formulation for complex crystals , 1999 .

[71]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[72]  T. Belytschko,et al.  An atomistic-based finite deformation membrane for crystalline films one atom thick , 2001 .

[73]  H. Fischmeister,et al.  Crack propagation in b.c.c. crystals studied with a combined finite-element and atomistic model , 1991 .

[74]  Kenny S. Crump,et al.  Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation , 1976, J. ACM.

[75]  R. D. Mindlin.  Micro-structure in linear elasticity , 1964 .

[76]  Bin Liu,et al.  Electron Transport in Deformed Carbon Nanotubes , 2004 .

[77]  Abraham.  Dynamics of Brittle Fracture with Variable Elasticity. , 1996, Physical review letters.

[78]  Cahal McVeigh,et al.  Linking microstructure and properties through a predictive multiresolution continuum , 2008 .

[79]  Ted Belytschko,et al.  Modeling fracture in Mindlin–Reissner plates with the extended finite element method , 2000 .

[80]  J. H. Weiner,et al.  Statistical Mechanics of Elasticity , 1983 .

[81]  Hoover,et al.  Canonical dynamics: Equilibrium phase-space distributions. , 1985, Physical review. A, General physics.

[82]  Noam Bernstein,et al.  Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture , 1998 .

[83]  X. Frank Xu,et al.  A multiscale stochastic finite element method on elliptic problems involving uncertainties , 2007 .

[84]  Nils-Erik Wiberg,et al.  STRUCTURAL DYNAMIC ANALYSIS BY A TIME‐DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD , 1996 .

[85]  Robert E. Rudd,et al.  COARSE-GRAINED MOLECULAR DYNAMICS AND THE ATOMIC LIMIT OF FINITE ELEMENTS , 1998 .

[86]  Harold S. Park,et al.  The bridging scale for two-dimensional atomistic/continuum coupling , 2005 .

[87]  T. Belytschko,et al.  Atomistic simulations of nanotube fracture , 2002 .

[88]  Wing Kam Liu,et al.  Computational Nanomechanics of Materials , 2008 .

[89]  Huafeng Liu,et al.  Meshfree particle method , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[90]  M. Ortiz,et al.  An analysis of the quasicontinuum method , 2001, cond-mat/0103455.

[91]  Shaofan Li,et al.  A non-equilibrium multiscale simulation paradigm , 2008 .

[92]  S. Nosé.  A molecular dynamics method for simulations in the canonical ensemble , 1984 .

[93]  M. Cross,et al.  A multi-scale atomistic-continuum modelling of crack propagation in a two-dimensional macroscopic plate , 1998 .

[94]  J. D. Doll,et al.  Generalized Langevin equation approach for atom/solid-surface scattering: General formulation for classical scattering off harmonic solids , 1976 .

[95]  Shaofan Li,et al.  Nonequilibrium multiscale computational model. , 2007, The Journal of chemical physics.

[96]  Gregory J. Wagner,et al.  Coupling of atomistic and continuum simulations using a bridging scale decomposition , 2003 .

[97]  R. Ghanem,et al.  Probabilistic equivalence and stochastic model reduction in multiscale analysis , 2008 .

[98]  M. Ashby,et al.  Strain gradient plasticity: Theory and experiment , 1994 .

[99]  Li,et al.  Moving least-square reproducing kernel methods (I) Methodology and convergence , 1997 .

[100]  S. Nosé,et al.  An extension of the canonical ensemble molecular dynamics method , 1986 .

[101]  Su Hao,et al.  Computer implementation of damage models by finite element and meshfree methods , 2000 .

[102]  Huajian Gao,et al.  Mechanism-based strain gradient plasticity— I. Theory , 1999 .

[103]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[104]  Kaushik Bhattacharya,et al.  Quasi-continuum orbital-free density-functional theory (QC-OFDFT) , 2007 .

[105]  T. Beck.  Real-space mesh techniques in density-functional theory , 2000, cond-mat/0006239.

[106]  J. Tinsley Oden,et al.  Adaptive multiscale modeling of polymeric materials with Arlequin coupling and Goals algorithms , 2009 .

[107]  Shardool U. Chirputkar,et al.  Coupled Atomistic/Continuum Simulation based on Extended Space-Time Finite Element Method , 2008 .

[108]  Andrea Hawkins-Daarud,et al.  Toward a predictive model of tumor growth , 2011 .

[109]  Wing Kam Liu,et al.  Multi-length scale micromorphic process zone model , 2009 .

[110]  Eduard G. Karpov,et al.  A Green's function approach to deriving non‐reflecting boundary conditions in molecular dynamics simulations , 2005 .

[111]  Dong Qian,et al.  Mechanics of carbon nanotubes , 2002 .

[112]  David Farrell,et al.  Algorithms for bridging scale method parameters , 2007 .

[113]  Harold S. Park,et al.  Implementation aspects of the bridging scale method and application to intersonic crack propagation , 2007 .

[114]  Franck J. Vernerey,et al.  A micromorphic model for the multiple scale failure of heterogeneous materials , 2008 .

[115]  R. D. Mindlin.  MICROSTRUCTURE IN LINEAR ELASTICITY , 1999 .

[116]  I. Jasiuk,et al.  Fracture of random matrix-inclusion composites: scale effects and statistics , 1998 .

[117]  Wing Kam Liu,et al.  Multi-scale solid oxide fuel cell materials modeling , 2009 .

[118]  Franck J. Vernerey,et al.  An interactive micro-void shear localization mechanism in high strength steels , 2007 .

[119]  Wing Kam Liu,et al.  Reproducing kernel particle methods for structural dynamics , 1995 .

[120]  C. Kittel.  Introduction to solid state physics , 1954 .