Practical Multiscaling

Preface xi

Acknowledgments xv

1 Introduction to Multiscale Methods 1

1.1 The Rationale for Multiscale Computations 1

1.2 The Hype and the Reality 2

1.3 Examples and Qualification of Multiscale Methods 3

1.4 Nomenclature and definitions 5

1.5 Notation 6

1.5.1 Index and matrix notation 6

1.5.2 M ultiple Spatial Scale Coordinates 8

1.5.3 Domains and boundaries 9

1.5.4 Spatial and Temporal Derivatives 9

1.5.5 Special symbols 10

References 11

2 Upscaling/Downscaling of Continua 13

2.1 Introduction 13

2.2 Homogenizaton of Linear Heterogeneous Media 16

2.2.1 Two-Scale Formulation 16

2.2.2 Two-Scale Formulation - Variational Form 23

2.2.3 Hill-Mandel Macrohomogeneity Condition and Hill-Reuss-Voigt Bounds 25

2.2.4 N umerical Implementation 27

2.2.5 B oundary Layers 38

2.2.6 Convergence Estimates 41

2.3 Upscaling Based on Enhanced Kinematics 47

2.3.1 M ultiscale Finite Element Method 48

2.3.2 Variational Multiscale Method 48

2.3.3 M ultiscale Enrichment Based on Partition of Unity 49

2.4 Homogenization of Nonlinear Heterogeneous Media 50

2.4.1 Asymptotic Expansion for Nonlinear Problems 50

2.4.2 Formulation of the Coarse-Scale Problem 54

2.4.3 Formulation of the Unit Cell Problem 58

2.4.4 Example Problems 61

2.5 Higher Order Homogenization 64

2.5.1 Introduction 64

2.5.2 Formulation 65

2.6 Multiple-Scale Homogenization 69

2.7 Going Beyond Upscaling - Homogenization-Based Multigrid 71

2.7.1 Relaxation 73

2.7.2 Coarse-grid Correction 77

2.7.3 Two-grid Convergence for a Model Problem in a Periodic Heterogeneous Medium 79

2.7.4 Upscaling-Based Prolongation and Restriction Operators 81

2.7.5 Homogenization-based Multigrid and Multigrid Acceleration 83

2.7.6 N onlinear Multigrid 84

2.7.7 M ultigrid for Indefinite Systems 86

Problems 87

References 91

3 Upscaling/Downscaling of Atomistic/Continuum Media 95

3.1 Introduction 95

3.2 Governing Equations 96

3.2.1 M olecular Dynamics Equation of Motion 96

3.2.2 M ultiple Spatial and Temporal Scales and Rescaling of the MD Equations 98

3.3 Generalized Mathematical Homogenization 100

3.3.1 M ultiple-Scale Asymptotic Analysis 100

3.3.2 The Dynamic Atomistic Unit Cell Problem 102

3.3.3 The Coarse-Scale Equations of Motion 103

3.3.4 Continuum Description of Equation of Motion 106

3.3.5 The Thermal Equation 107

3.3.6 Extension to Multi-Body Potentials 112

3.4 Finite Element Implementation and Numerical Verification 113

3.4.1 Weak Forms and Semidiscretization of Coarse-Scale Equations 113

3.4.2 The Fine-Scale (Atomistic) Problem 115

3.5 Statistical Ensemble 118

3.6 Verification 120

3.7 Going Beyond Upscaling 126

3.7.1 Spatial Multilevel Method Versus Space-Time Multilevel Method 127

3.7.2 The WR Scheme 129

3.7.3 Space-Time FAS 130

Problems 131

References 133

4 Reduced Order Homogenization 137

4.1 Introduction 137

4.2 Reduced Order Homogenization for Two-Scale Problems 139

4.2.1 Governing Equations 139

4.2.2 Residual-Free Fields and Model Reduction 141

4.2.3 Reduced Order System of Equations 148

4.2.4 One-Dimensional Model Problem 150

4.2.5 Computational Aspects 154

4.3 Lower Order Approximation of Eigenstrains 156

4.3.1 The Pitfalls of a Piecewise Constant One-Partition-Per-Phase Model 157

4.3.2 Impotent Eigenstrain 159

4.3.3 Hybrid Impotent-Incompatible Eigenstrain Mode Estimators 163

4.3.4 Chaboche Modification 164

4.3.5 Analytical Relations for Various Approximations of Eigenstrain Influence Functions 165

4.3.6 Eigenstrain Upwinding 172

4.3.7 Enhancing Constitutive Laws of Phases 175

4.3.8 Validation of the Hybrid Impotent-Incompatible Reduced Order Model with Eigenstrain Upwinding and Enhanced Constitutive Model of Phases 180

4.4 Extension to Nonlocal Heterogeneous Media 184

4.4.1 Staggered Nonlocal Model for Homogeneous Materials 186

4.4.2 Staggered Nonlocal Multiscale Model 188

4.4.3 Validation of the Nonlocal Model 189

4.4.4 Rescaling Constitutive Equations 193

4.5 Extension to Dispersive Heterogeneous Media 197

4.5.1 Dispersive Coarse-Scale Problem 199

4.5.2 The Quasi-Dynamic Unit Cell Problem 201

4.5.3 Linear Model Problem 204

4.5.4 N onlinear Model Problem 205

4.5.5 Implicit and Explicit Formulations 208

4.6 Extension to Multiple Spatial Scales 209

4.6.1 Residual-Free Governing Equations at Multiple Scales 210

4.6.2 M ultiple-Scale Reduced Order Model 211

4.7 Extension to Large Deformations 214

4.8 Extension to Multiple Temporal Scales with Application to Fatigue 219

4.8.1 Temporal Homogenization 220

4.8.2 M ultiple Temporal and Spatial Scales 224

4.8.3 Fatigue Constitutive Equation 225

4.8.4 Verfication of the Multiscale Fatigue Model 226

4.9 Extension to Multiphysics Problems 227

4.9.1 Reduced Order Coupled Vector-Scalar Field Model at Multiple Scales 228

4.9.2 Environmental Degradation of PMC 232

4.9.3 Validation of the Multiphysics Model 235

4.10 Multiscale Characterization 239

4.10.1 Formulation of the Inverse Problem 239

4.10.2 Characterization of Model Parameters in ROH 241

Problems 241

References 243

5 Scale-separation-free Upscaling/Downscaling of Continua 249

5.1 Introduction 249

5.2 Computational Continua (C2) 251

5.2.1 N onlocal Quadrature 251

5.2.2 Coarse-Scale Problem 254

5.2.3 Computational Unit Cell Problem 257

5.2.4 One-dimensional model problem 260

5.3 Reduced Order Computational Continua (RC2) 265

5.3.1 Residual-Free Computational Unit Cell Problem 266

5.3.2 The Coarse-Scale Weak Form 274

5.3.3 Coarse-Scale Consistent Tangent Stiffness Matrix 275

5.4 Nonlocal Quadrature in Multidimensions 278

5.4.1 Tetrahedral Elements 278

5.4.2 Triangular Elements 287

5.4.3 Quadrilateral and Hexahedral Elements 292

5.5 Model Verification 297

5.5.1 The Beam Problem 300

Problems 302

References 303

6 Multiscale Design Software 305

6.1 Introduction 305

6.2 Microanalysis with MDS-Lite 308

6.2.1 Familiarity with the GUI 309

6.2.2 Labeling Data Files 312

6.2.3 The First Walkthrough MDS-Micro Example 312

6.2.4 The Second Walkthrough MDS-Micro Example 318

6.2.5 Parametric Library of Unit Cell Models 331

6.3 Macroanalysis with MDS-Lite 340

6.3.1 First Walkthrough MDS-Macro Example 341

6.3.2 Second Walkthrough MDS-Macro Example 362

6.3.3 Third Walkthrough Example 373

6.3.4 Fourth Walkthrough Example 379

Problems 391

References 393

Index 395