1 Introduction 1.1 Potential of Nanoscale Engineering 1.2 Motivation for Multiple Scale Modeling 1.3 Educational Approach 2 Classical Molecular Dynamics 2.1 Mechanics of a System of Particles 2.1.1Generalized Coordinates 2.1.2 Mechanical Forces and Potential Energy 2.1.3 Lagrange Equations of Motion 2.1.4 Integrals of Motion and Symmetric Fields 2.1.5 Newtonian Equations 2.1.6 Examples 2.2 Molecular Forces 2.2.1 External Fields 2.2.2 Pair-Wise Interaction 2.2.3 Multibody Interaction 2.2.4 Exercises 2.3 Molecular Dynamics Applications 3 Lattice Mechanics 3.1 Elements of Lattice Symmetries 3.1.1 Bravais Lattices 3.1.2 Basic Symmetry Principles 3.1.3 Crystallographic Directions and Planes 3.2 Equation of Motion of a Regular Lattice 3.2.1 Unit Cell and the Associate Substructure 3.2.2 Lattice Lagrangian and Equations of Motion 3.2.3 Examples 3.3 Transforms 3.3.1 Fourier Transform 3.3.2 Laplace Transform 3.3.3 Discrete Fourier Transform 3.4 Standing Waves in Lattices 3.4.1 NormalModes and Dispersion Branches 3.4.2 Examples 3.5 Green\\\'s Function Methods 3.5.1 Solution for a Unit Pulse 3.5.2 Free Lattice with Initial Perturbations 3.5.3 Solution for Arbitrary Dynamic Loads 3.5.4 General Inhomogeneous Solution 3.5.5 Boundary Value Problems and the Time History Kernel 3.5.6 Examples 3.6 Quasi-Static Approximation 3.6.1 Equilibrium State Equation 3.6.2 Quasi-Static Green\\\'s Function 3.6.3 Multiscale Boundary Conditions 4 Methods of Thermodynamics and Statistical Mechanics 4.1 Basic Results of the Thermodynamic Method 4.1.1State Equations 4.1.2 Energy Conservation Principle 4.1.3 Entropy and the Second Law of Thermodynamics 4.1.4 Nernst\\\'s Postulate 4.1.5 Thermodynamic Potentials 4.2 Statistics of Multiparticle Systems in Thermodynamic Equilibrium 4.2.1 Hamiltonian Formulation 4.2.2 Statistical Description of Multiparticle Systems 4.2.3 Microcanonical Ensemble 4.2.4 Canonical Ensemble 4.2.5 Maxwell-Boltzmann Distribution 4.2.6 Thermal Properties of Periodic Lattices 4.3 Numerical Heat Bath Techniques 4.3.1 Berendsen Thermostat 4.3.2 Nosé-Hoover Heat Bath 4.3.3 Phonon Method for Solid-Solid Interfaces 5 Introduction to Multiple Scale Modeling 5.1 MAAD 5.2 Coarse-Grained Molecular Dynamics 5.3 Quasi-Continuum Method 5.4 CADD 5.5 Bridging Domain 6 Introduction to Bridging Scale 6.1 Bridging Scale Fundamentals 6.1.1 Multiscale Equations of Motion 6.2 Removing Fine Scale Degrees of Freedom in Coarse Scale Region 6.2.1 Relationship of Lattice Mechanics to Finite Elements 6.2.2 Linearized MD Equation of Motion 6.2.3 Elimination of Fine Scale Degrees of Freedom 6.2.4 Commentary on Reduced Multiscale Formulation 6.2.5 Elimination of Fine Scale Degrees of Freedom:3D Generalization 6.2.6 Numerical Implementation of Impedance Force 6.2.7 Numerical Implementation of Coupling Force 6.3 Discussion on the Damping Kernel Technique 6.3.1 Programming Algorithm for Time History Kernel 6.4 Cauchy-Born Rule 6.5 Virtual Atom Cluster Method, 6.5.1 Motivations and General Formulation 6.5.2 General Idea of the VAC Model 6.5.3 Three-Way Concurrent Coupling with QM Method 6.5.4 Tight-Binding Method for Carbon Systems 6.5.5 Coupling with the VAC Model 6.6 Staggered Time Integration Algorithm 6.6.1 MD Update 6.6.2 FE Update 6.7 Summary of Bridging Scale Equations 6.8 Discussion on the Bridging Scale Method 7 Bridging Scale Numerical Examples 7.1 Comments on Time History Kernel 7.2 1 D Bridging Scale Numerical Examples 7.2.1 Lennard-Jones Numerical Examples 7.2.2 Comparison of VAC Method and Cauchy-Born Rule 7.2.3 Truncation of Time History Kernel 7.3 2D/3D Bridging Scale Numerical Examples 7.4 Two-Dimensional Wave Propagation 7.5 Dynamic Crack Propagation in Two Dimensions 7.6 Dynamic Crack Propagation in Three Dimensions 7.7 Virtual Atom Cluster Numerical Examples 7.7.1 Bending of Carbon Nanotubes 7.7.2 VAC Coupling with Tight Binding 8 Non-Nearest Neighbor MD Boundary Condition 8.1 Introduction 8.2 Theoretical Formulation in 3D 8.2.1 Force Boundary Condition: 1D Illustration 8.2.2 Displacement Boundary Condition: 1D Illustration 8.2.3 Comparison to Nearest Neighbors Formulation 8.2.4 Advantages of Displacement Formulation 8.3 Numerical Examples: 1D Wave Propagation 8.4 Time-History Kernels for FCC Gold 8.5 Conclusion for the Bridging Scale Method 8.5.1 Bridging Scale Perspectives 9 Multiscale Methods for Material Design 9.1 Multiresolution Continuum Analysis 9.1.1 Generalized Stress and Deformation Measures 9.1.2 Interaction between Scales 9.1.3 Multiscale Materials Modeling 9.2 Multiscale Constitutive Modeling of Steels 9.2.1 Methodology and Approach 9.2.2 First-Principles Calculation 9.2.3 Hierarchical Unit Cell and Constitutive Model 9.2.4 Laboratory Specimen Scale: Simulation and Results 9.3 Bio-Inspired Materials 9.3.1 Mechanisms of Self-Healing in Materials 9.3.2 Shape-Memory Composites 9.3.3 Multiscale Continuum Modeling of SMA Composites 9.3.4 Issues of Modeling and Simulation 9.4 Summary and Future Research Directions 10 Bio-Nano Interface 10.1 Introduction 10.2 Immersed Finite Element Method 10.2.1 Formulation 10.2.2 Computational Algorithm of IFEM 10.3 Vascular Flow and Blood Rheology 10.3.1 Heart Model 10.3.2 Flexible Valve-Viscous Fluid Interaction 10.3.3 Angioplasty Stent 10.3.4 Monocyte Deposition 10.3.5 Platelet Adhesion and Blood Clotting 10.3.6 RBC Aggregation and Interaction 10.4 Electrohydrodynamic Coupling 10.4.1 Maxwell Equations 10.4.2 Electro-manipulation 10.4.3 Rotation of CNTs Induced by Electroosmotic Flow 10.5 CNT/DNA Assembly Simulation 10.6 Cell Migration and Cell-Substrate Adhesion 10.7 Conclusions Appendix A Kernel Matrices for EAM Potential Bibliography Index
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