Multiscale Methods in Computational Mechanics (eBook)

Table of Contents 6
Preface 10
List of Authors 12
PART 1 Computational Fluid Dynamics 20
Residual-Based Variational Multiscale Theory of LES Turbulence Modeling 21
1 Variational Multiscale Formulation of the Incompressible Navier–Stokes Equations 21
1.1 Incompressible Navier–Stokes Equations 21
Global Space-Time Variational Formulation 22
Sliced Space-Time Variational Formulation 23
1.2 Scale Separation 24
1.3 Perturbation Series 27
2 Turbulent Channel Flow 30
3 Conclusions 32
Acknowledgements 36
References 36
A Posteriori Error Estimation for Computational Fluid Dynamics: The Variational Multiscale Approach 37
1 Introduction 37
2 The Variational Multiscale Approach to Error Estimation 38
2.1 The Abstract Problem 38
2.2 The Variational Multiscale Error Estimation Paradigm 39
3 The Smooth Paradigm for Error Estimation 40
3.1 Intrinsic Error Time Scales 41
Estimates in the L2 Norm 41
Example: One-Dimensional Advection-Diffusion 42
3.2 Error Upper Bounds 44
3.3 Relation to the Flow Time Scale Parameter 45
3.4 Extensions 45
4 Multidimensional Model 45
4.1 A Model for the Error Distribution 46
Element Interior Error 46
Element Boundary Error 46
4.2 Norms Based on the L8 Norm of the Residual 47
4.3 Summary of the Model 48
5 Multidimensional Error Scales for the Bilinear Quad 48
5.1 Hyperbolic Limit 48
5.2 Elliptic Limit 49
6 Numerical Example: L-shaped Domain Problem 50
7 Adaptivity 51
8 Conclusions 54
References 54
Advances in Variational Multiscale Methods for Turbulent Flows 57
1 Introduction 57
2 Residual-Based Variational Multiscale Method with Dynamic Subgrid Scales 59
3 The Algebraic Variational Multiscale-Multigrid Method 60
4 Using NURBS in Residual-Based Variational Multiscale Methods 62
5 Towards a Residual-Based Variational Multiscale Method for Turbulent Fluid-Structure Interaction 66
6 Conclusion 67
Acknowledgement 68
References 68
Variational Germano Approach for Multiscale Formulations 71
1 Introduction 71
2 General Discrete Germano Identity 72
2.1 Numerical Method as a Discrete Projector 72
2.2 Inverse Implication: Projector Implies Numerical Method 73
2.3 Multilevel Commutativity 74
2.4 Discrete Germano Identity 75
2.5 Partitioned Approach 76
3 Discrete Germano Approach for Stabilized Methods 76
3.1 Computation of Coarse Stabilization Parameter: Dissipation Method 77
Non-Homogenous Boundary Conditions 78
3.2 Computation of Coarse Stabilization Parameter: Least-Squares Method 78
Basis Independent Least-Squares Method 79
Basis Independent Least-Squares Method for a Spatial Varying Stability Parameter 80
3.3 Computation of Fine Stabilization Parameter 81
4 1D Convection-Diffusion 82
4.1 Stabilized Formulation 82
4.2 Stabilization Parameter Structure 82
4.3 Dissipation Method for Homogeneous Boundary Conditions 83
4.4 Dissipation Method for Non-Homogeneous Boundary Conditions 83
Reconstruction of the Lagrange Multipliers on Coarse Mesh 84
Injection of the Lagrange Multipliers from Fine Mesh 85
4.5 Naive Least-Squares Method 85
4.6 Basis-Independent Least-Squares Method 86
5 Numerical Results 86
5.1 Homogenous Boundary Conditions 86
5.2 Non-Homogenous Boundary Conditions 87
5.3 Basis Dependence of the Least-Squares Method 88
5.4 Computational Cost 89
6 Conclusion 90
Acknowledgments 90
References 91
Dissipative Structure and Long Term Behavior of a Finite Element Approximation of Incompressible Flows with Numerical Subgrid Scale Modeling 92
1 Introduction 92
2 Formulation 95
2.1 Continuous problem 95
2.2 Subgrid Scale Decomposition 96
2.3 Simplifying Assumptions 96
2.4 Final Formulation 97
3 Dissipative Structure and Backscatter 98
3.1 Local Kinetic Energy Balance Equations 98
3.2 Global Kinetic Energy Balance Equations 99
3.3 Backscatter 101
3.4 Flow over a Surface Mounted Obstacle 102
4 Long Term Stability 103
5 Long Term Simulations 105
5.1 Flow over a Plate 106
5.2 Flow around a Telescope 106
6 Conclusions 108
Acknowledgments 109
References 109
Large-Eddy Simulation of Multiscale Particle Dynamics at High Volume Concentration in Turbulent Channel Flow 111
1 Introduction 111
2 Mathematical Formulation 113
2.1 The Gas Phase 113
2.2 The Solids Phase 114
2.3 Subgrid-Modeling 118
2.4 The Numerical Method 119
3 Results 120
3.1 Turbulence Modulation 120
3.2 Effects of Collisions 121
3.3 Coherent Particle Structures 124
4 Concluding Remarks 125
Acknowledgments 126
References 127
PART 2 Materials with Microstructure 130
An Incremental Strategy for Modeling Laminate Microstructures in Finite Plasticity – Energy Reduction, Laminate Orientation and Cyclic Behavior 131
1 Introduction 131
2 Non-Convex Potentials and Relaxation 133
3 First-Order Laminate Microstructures 134
4 Incremental Numerical Scheme 139
5 Results 141
5.1 Evolution of the Internal Variables and Laminate Orientation 141
5.2 Energy Reduction 143
5.3 Cyclic Behavior 144
6 Discussion and Conclusions 146
References 146
The Micromorphic versus Phase Field Approach to Gradient Plasticity and Damage with Application to Cracking in Metal Single Crystals 149
1 Generalized Continua and Material Microstructure 149
2 Micromorphic Approach 150
2.1 Thermomechanics with Additional Degrees of Freedom 150
2.2 Non-Dissipative Contribution of Generalized Stresses and Micromorphic Model 152
Micromorphic Model 153
2.3 Viscous Generalized Stress and Phase Field Model 154
Phase Field Model 154
2.4 Elasto-Plastic Decomposition of Generalized Strains 155
3 Continuum Damage Model for Single Crystals and Its Regularization 156
3.1 Constitutive Equations 156
3.2 Microdamage Continuum 159
4 Finite Element Implementation 160
4.1 Variational Formulation and Discretization 160
4.2 Implicit Incremental Formulation 161
5 Numerical Examples 163
6 Conclusion 165
References 165
Homogenization and Multiscaling of Granular Media for Different Microscopic Constraints 168
1 Introduction 168
2 Quasi-Static Homogenization of Granular Aggregates 170
2.1 Deformation-Driven Homogenization of Microstructures 170
Definition of Particle Microstructures 170
Microscopic Boundary Conditions 171
Microscopic Equilibrium State 173
Macroscopic Boundary Conditions 176
2.2 Penalty-Type Implementation of Boundary Constraints 176
3 Microstructural Modeling of Granular Materials 178
3.1 Micromechanical Model for Interparticle Contact 178
3.2 Dynamic Relaxation of the Microstructural Response 179
4 Numerical Examples and Comparative Study 180
4.1 Specification of Basic Micromechanical Functions 181
4.2 Compression-Shear Mode for Cubic Microstructures 181
5 Multiple Scale Simulation of a Granular Medium 184
5.1 Two-Scale Simulations Based on DE-FE Coupling 184
5.2 Simulation of a Biaxial Compression Test of a Soil 185
Experimental Setup 185
Coupled FE-DE Two-Scale Model 185
Results and Discussion 186
6 Conclusion 188
Acknowledgement 188
References 188
Effective Hydraulic and Mechanical Properties of Heterogeneous Media with Interfaces 191
1 Introduction 191
2 Hydraulic Model for a Porous Matrix with Impermeable Inclusionary Phase 192
2.1 Mori–Tanaka Estimate 193
2.2 The Variational Approach 196
2.3 The Self-Consistent Approach 198
3 Mechanical Model for a Granular Cemented Rock 200
3.1 General Framework 200
3.2 The Self-Consistent Homogenization Scheme 202
4 Concluding Remarks 205
References 205
An Extended Finite Element Method for the Analysis of Submicron Heat Transfer Phenomena 207
1 Introduction 207
2 Level-Set Description of Material Layout 211
3 Gray Phonon Model 211
4 Discretization Methods 214
4.1 Discrete Ordinate Method 214
4.2 Extended Finite Element Method 215
4.3 Lagrange Multiplier Method 216
5 Numerical Examples 217
5.1 Verification Example 218
5.2 Analysis of Nano-Composites 218
5.3 Design Study 220
6 Conclusions 221
Acknowledgments 222
References 223
PART 3 Composites, Laminates, and Structures: Optimization 225
Multiscale Modeling and Simulation of Composite Materials and Structures 226
1 Introduction 226
2 Information-Passing Multiscale Approaches in Space 230
2.1 Direct Mathematical Homogenization for Nonlinear Problems 230
2.2 Eigendeformation-Based Reduced Order Homogenization 232
3 Concurrent Multiscale Methods in Space 234
3.1 Multiscale Enrichment Based on Partition of Unity (MEPU) 235
3.2 Adaptive Model Selection 236
3.3 Numerical Example 236
4 Temporal Multiscale Model for Fatigue Life Prediction 237
References 240
Multiscale Modelling of the Failure Behaviour of Fibre-Reinforced Laminates 243
1 Introduction 243
2 Review of the Interface Damage Model 245
3 Mesoscale Simulations of a Centre-Cracked 2/1 GLARE Laminate 247
3.1 Geometry and Boundary Conditions 247
3.2 Results for a 2/1 Lay-up with Elastic Aluminium Layers 250
3.3 Results for a 2/1 Lay-up with Elasto-Plastic Aluminium Layers 253
4 Microscale Simulations of Single-Fibre Epoxy Systems 255
4.1 Fibre-Epoxy Interfacial Strength versus Epoxy Strength 256
5 Microscale Simulations of Multiple-Fibre Epoxy Systems 259
5.1 Influence of the Fibre Volume Fraction 259
6 Coupling between Microscale and Mesoscale Crack Modelling 260
6.1 Fibre-Epoxy Specimen Subjected to Uniaxial Tension 263
6.2 Influence of Sample Size 264
6.3 Influence of Imperfections 265
7 Concluding Remarks 266
Acknowledgements 268
References 268
Improved Multiscale Computational Strategies for Delamination 270
1 Introduction 270
2 Application of the Two-Scale Domain Decomposition Strategy to Delamination Analysis 272
2.1 The Substructured Delamination Problem 272
2.2 Two-Scale Iterative Resolution of the Substructured Problem 275
Introduction of the Macroscopic Scale 275
The Iterative Algorithm 275
2.3 First Example of a Delamination Analysis 278
3 Analysis of the Parameters of the Iterative Algorithm 280
4 The Three-Scale Domain Decomposition Strategy 281
4.1 Resolution of the Macroproblem through the Balancing Domain Decomposition Method 281
Partitioning of the Macroproblem 281
Resolution of the Super-Interface Problem 283
4.2 Results 283
5 Efficiency of the Strategy: Study of a Complex Test Case 284
6 Conclusion 286
References 286
Damage Propagation in Composites – Multiscale Modeling and Optimization 289
1 Introduction 289
2 Modeling of Discontinuities on Small Scale 291
2.1 Geometrical Description 291
2.2 Kinematic Description 292
2.3 Cohesive Law 293
2.4 Numerical Examples 294
Two-Phase Material under Tension [13] 294
Pull-out of Fiber in Matrix 295
3 Multiscale Formulation [11, 12] 295
3.1 Formulation Using Continuum Damage 296
Material Model 296
Concept 296
Numerical Examples [11] 299
3.2 Discontinuum Model in Multiscale Formulation 302
Concept 302
Numerical Examples [13] 303
4 Optimal Fiber Layout [18–20] 304
4.1 Material Model 305
4.2 Optimization Concept 306
4.3 Multiphase Material Optimization 306
4.4 Shape Optimization of Fiber Geometry 307
4.5 Numerical Examples 307
Multiphase Material Optimization for a Beam 307
Multiphase and Shape Optimization for Deep Beam 308
5 Conclusions 309
Acknowledgement 310
References 310
Computational Multiscale Model for NATM Tunnels: Micromechanics-Supported Hybrid Analyses 313
1 Introduction 313
2 Continuum Micromechanics of Microheterogeneous Materials 315
2.1 Representative Volume Element (Separation of Scales) 315
2.2 Homogenization of Elasticity 316
2.3 Homogenization of Strength 317
3 Micromechanics at the Cement Paste Level 318
3.1 Micromechanical Representation 318
3.2 Constitutive Behavior of Clinker, Water, Hydrates, and Air 319
3.3 Homogenized Elasticity of Cement Paste 320
3.4 Homogenized Strength of Cement Paste 321
4 Micromechanics at the Shotcrete Level 322
5 Experimental Validation of Micromechanics-Based Material Models 323
5.1 Mixture-Dependent Shotcrete Composition 323
5.2 Experimental Validation on Shotcrete Level 323
6 Micromechanics-Based Studies: Influence of Water-Cement and Aggregate-Cement Ratios on Evolutions of Elasticity and Strength of Shotcrete 324
7 Continuum Micromechanics-Based Safety Assessment of NATM Tunnel Shells 325
7.1 Water-Cement Ratio-Dependence of Structural Safety 328
7.2 Aggregate-Cement Ratio-Dependence of Structural Safety 328
8 Conclusions 330
Acknowledgements 331
References 331
Optimization of Corrugated Paperboard under Local and Global Buckling Constraints 337
1 Introduction 337
2 Problem Description and Methods Used 339
2.1 Unit Cell Approach 339
2.2 Formulation of the Optimization Problem 346
2.3 Homogenization Process Using Unit Cell Models 347
2.4 Unit Cell Approach for Local Buckling Analysis 347
2.5 Numerical Meso Structural Optimization Approach 348
3 Results of Optimization 349
4 Fold Formation in the Post-Buckling Regime 351
5 Conclusions 353
References 353
Framework for Multi-Level Optimization of Complex Systems 355
1 Introduction 355
2 A Unifying Multi-Level Notation 356
3 Decomposition 361
3.1 Physical Coupling 361
3.2 Problem Matrix 363
4 Coordination 370
5 Software Framework 372
6 Supersonic Business Jet Optimization 373
6.1 Multi-Level Optimization Problem 374
Hierarchic Decomposition 376
Non-Hierarchic Decomposition 378
Coordination 379
6.2 Numerical Results 380
Results 380
7 Conclusions 384
Acknowledgments 385
References 385
PART 4 Coupled Problems and Porous Media 386
Multiscale/Multiphysics Model for Concrete 387
1 Introduction 387
2 General Mathematical Model 388
3 Effective Stress Principle 391
4 Application of the Model to Concrete Structures at Elevated Temperature 393
4.1 Simulation of a Concrete Column under Fire with Fast Cooling 397
5 Application of the Model to Concrete Structures Subject to Leaching Process 402
5.1 Modelling Kinetics of Calcium Leaching Process 402
5.2 Numerical Simulation of the Non-Isothermal Leaching Process in a Concrete Wall 406
6 Conclusions 407
References 408
Swelling Phenomena in Electro-Chemically Active Hydrated Porous Media 411
1 Introduction 411
2 TPM Fundamentals 413
2.1 Immiscible Components and Volume Fractions 413
2.2 Miscible Components and Molar Concentrations 413
2.3 Constituent Balance Relations 414
3 Swelling Media as Biphasic, Four-Component Aggregates 415
3.1 Restrictions Obtained from the Entropy Inequality 417
3.2 The Fluid Components 419
3.3 Ion Diffusion and Fluid Flow 420
3.4 The Electrical Potential 422
3.5 The Solid Skeleton 422
4 Weak Forms and Basic Numerical Setting 423
5 Numerical Examples 424
5.1 Free Swelling Hydrogel 424
5.2 Electro-Active Polymers 426
5.3 Borehole Instability in Active Soil 427
5.4 Swelling of an Intervertebral Disc 427
6 Conclusion 429
References 429
Propagating Cracks in Saturated Ionized Porous Media 431
1 Introduction 431
2 Bulk Material 433
3 Discontinuity in the Solid Part 436
3.1 Cohesive Zone 436
3.2 Nonlocal Stress 437
4 Shearing Mode 438
4.1 Discontinuity in the Fluid Part 438
4.2 Numerical Example 439
5 Tensile Mode 441
5.1 Discontinuity in the Fluid Part 441
5.2 Numerical Example 443
6 Concluding Remarks 445
List of Symbols 445
Acknowledgement 446
References 446
Author Index 449
Subject Index 450