De Borst / Borst / Ramm Multiscale Methods in Computational Mechanics

Progress and Accomplishments
1. Auflage 2010
ISBN: 978-90-481-9809-2
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 55, 446 Seiten

Reihe: Lecture Notes in Applied and Computational Mechanics

This work gives a modern, up-to-date account of recent developments in computational multiscale mechanics. Both upscaling and concurrent computing methodologies will be addressed for a range of application areas in computational solid and fluid mechanics: Scale transitions in materials, turbulence in fluid-structure interaction problems, multiscale/multilevel optimization, multiscale poromechanics. A Dutch-German research group that consists of qualified and well-known researchers in the field has worked for six years on the topic of computational multiscale mechanics. This text provides a unique opportunity to consolidate and disseminate the knowledge gained in this project. The addition of chapters written by experts outside this working group provides a broad and multifaceted view of this rapidly evolving field.

Autoren/Hrsg.

Borst, René

Ramm, Ekkehard

De Borst, René

Inhaltsverzeichnis

1;Table of Contents;6
2;Preface;10
3;List of Authors;12
4;PART 1 Computational Fluid Dynamics;20
4.1;Residual-Based Variational Multiscale Theory of LES Turbulence Modeling;21
4.1.1;1 Variational Multiscale Formulation of the Incompressible Navier–Stokes Equations;21
4.1.1.1;1.1 Incompressible Navier–Stokes Equations;21
4.1.1.1.1;Global Space-Time Variational Formulation;22
4.1.1.1.2;Sliced Space-Time Variational Formulation;23
4.1.1.2;1.2 Scale Separation;24
4.1.1.3;1.3 Perturbation Series;27
4.1.2;2 Turbulent Channel Flow;30
4.1.3;3 Conclusions;32
4.1.4;Acknowledgements;36
4.1.5;References;36
4.2;A Posteriori Error Estimation for Computational Fluid Dynamics: The Variational Multiscale Approach;37
4.2.1;1 Introduction;37
4.2.2;2 The Variational Multiscale Approach to Error Estimation;38
4.2.2.1;2.1 The Abstract Problem;38
4.2.2.2;2.2 The Variational Multiscale Error Estimation Paradigm;39
4.2.3;3 The Smooth Paradigm for Error Estimation;40
4.2.3.1;3.1 Intrinsic Error Time Scales;41
4.2.3.1.1;Estimates in the L2 Norm;41
4.2.3.1.2;Example: One-Dimensional Advection-Diffusion;42
4.2.3.2;3.2 Error Upper Bounds;44
4.2.3.3;3.3 Relation to the Flow Time Scale Parameter;45
4.2.3.4;3.4 Extensions;45
4.2.4;4 Multidimensional Model;45
4.2.4.1;4.1 A Model for the Error Distribution;46
4.2.4.1.1;Element Interior Error;46
4.2.4.1.2;Element Boundary Error;46
4.2.4.2;4.2 Norms Based on the L8 Norm of the Residual;47
4.2.4.3;4.3 Summary of the Model;48
4.2.5;5 Multidimensional Error Scales for the Bilinear Quad;48
4.2.5.1;5.1 Hyperbolic Limit;48
4.2.5.2;5.2 Elliptic Limit;49
4.2.6;6 Numerical Example: L-shaped Domain Problem;50
4.2.7;7 Adaptivity;51
4.2.8;8 Conclusions;54
4.2.9;References;54
4.3;Advances in Variational Multiscale Methods for Turbulent Flows;57
4.3.1;1 Introduction;57
4.3.2;2 Residual-Based Variational Multiscale Method with Dynamic Subgrid Scales;59
4.3.3;3 The Algebraic Variational Multiscale-Multigrid Method;60
4.3.4;4 Using NURBS in Residual-Based Variational Multiscale Methods;62
4.3.5;5 Towards a Residual-Based Variational Multiscale Method for Turbulent Fluid-Structure Interaction;66
4.3.6;6 Conclusion;67
4.3.7;Acknowledgement;68
4.3.8;References;68
4.4;Variational Germano Approach for Multiscale Formulations;71
4.4.1;1 Introduction;71
4.4.2;2 General Discrete Germano Identity;72
4.4.2.1;2.1 Numerical Method as a Discrete Projector;72
4.4.2.2;2.2 Inverse Implication: Projector Implies Numerical Method;73
4.4.2.3;2.3 Multilevel Commutativity;74
4.4.2.4;2.4 Discrete Germano Identity;75
4.4.2.5;2.5 Partitioned Approach;76
4.4.3;3 Discrete Germano Approach for Stabilized Methods;76
4.4.3.1;3.1 Computation of Coarse Stabilization Parameter: Dissipation Method;77
4.4.3.1.1;Non-Homogenous Boundary Conditions;78
4.4.3.2;3.2 Computation of Coarse Stabilization Parameter: Least-Squares Method;78
4.4.3.2.1;Basis Independent Least-Squares Method;79
4.4.3.2.2;Basis Independent Least-Squares Method for a Spatial Varying Stability Parameter;80
4.4.3.3;3.3 Computation of Fine Stabilization Parameter;81
4.4.4;4 1D Convection-Diffusion;82
4.4.4.1;4.1 Stabilized Formulation;82
4.4.4.2;4.2 Stabilization Parameter Structure;82
4.4.4.3;4.3 Dissipation Method for Homogeneous Boundary Conditions;83
4.4.4.4;4.4 Dissipation Method for Non-Homogeneous Boundary Conditions;83
4.4.4.4.1;Reconstruction of the Lagrange Multipliers on Coarse Mesh;84
4.4.4.4.2;Injection of the Lagrange Multipliers from Fine Mesh;85
4.4.4.5;4.5 Naive Least-Squares Method;85
4.4.4.6;4.6 Basis-Independent Least-Squares Method;86
4.4.5;5 Numerical Results;86
4.4.5.1;5.1 Homogenous Boundary Conditions;86
4.4.5.2;5.2 Non-Homogenous Boundary Conditions;87
4.4.5.3;5.3 Basis Dependence of the Least-Squares Method;88
4.4.5.4;5.4 Computational Cost;89
4.4.6;6 Conclusion;90
4.4.7;Acknowledgments;90
4.4.8;References;91
4.5;Dissipative Structure and Long Term Behavior of a Finite Element Approximation of Incompressible Flows with Numerical Subgrid Scale Modeling;92
4.5.1;1 Introduction;92
4.5.2;2 Formulation;95
4.5.2.1;2.1 Continuous problem;95
4.5.2.2;2.2 Subgrid Scale Decomposition;96
4.5.2.3;2.3 Simplifying Assumptions;96
4.5.2.4;2.4 Final Formulation;97
4.5.3;3 Dissipative Structure and Backscatter;98
4.5.3.1;3.1 Local Kinetic Energy Balance Equations;98
4.5.3.2;3.2 Global Kinetic Energy Balance Equations;99
4.5.3.3;3.3 Backscatter;101
4.5.3.4;3.4 Flow over a Surface Mounted Obstacle;102
4.5.4;4 Long Term Stability;103
4.5.5;5 Long Term Simulations;105
4.5.5.1;5.1 Flow over a Plate;106
4.5.5.2;5.2 Flow around a Telescope;106
4.5.6;6 Conclusions;108
4.5.7;Acknowledgments;109
4.5.8;References;109
4.6;Large-Eddy Simulation of Multiscale Particle Dynamics at High Volume Concentration in Turbulent Channel Flow;111
4.6.1;1 Introduction;111
4.6.2;2 Mathematical Formulation;113
4.6.2.1;2.1 The Gas Phase;113
4.6.2.2;2.2 The Solids Phase;114
4.6.2.3;2.3 Subgrid-Modeling;118
4.6.2.4;2.4 The Numerical Method;119
4.6.3;3 Results;120
4.6.3.1;3.1 Turbulence Modulation;120
4.6.3.2;3.2 Effects of Collisions;121
4.6.3.3;3.3 Coherent Particle Structures;124
4.6.4;4 Concluding Remarks;125
4.6.5;Acknowledgments;126
4.6.6;References;127
5;PART 2 Materials with Microstructure;130
5.1;An Incremental Strategy for Modeling Laminate Microstructures in Finite Plasticity – Energy Reduction, Laminate Orientation and Cyclic Behavior;131
5.1.1;1 Introduction;131
5.1.2;2 Non-Convex Potentials and Relaxation;133
5.1.3;3 First-Order Laminate Microstructures;134
5.1.4;4 Incremental Numerical Scheme;139
5.1.5;5 Results;141
5.1.5.1;5.1 Evolution of the Internal Variables and Laminate Orientation;141
5.1.5.2;5.2 Energy Reduction;143
5.1.5.3;5.3 Cyclic Behavior;144
5.1.6;6 Discussion and Conclusions;146
5.1.7;References;146
5.2;The Micromorphic versus Phase Field Approach to Gradient Plasticity and Damage with Application to Cracking in Metal Single Crystals;149
5.2.1;1 Generalized Continua and Material Microstructure;149
5.2.2;2 Micromorphic Approach;150
5.2.2.1;2.1 Thermomechanics with Additional Degrees of Freedom;150
5.2.2.2;2.2 Non-Dissipative Contribution of Generalized Stresses and Micromorphic Model;152
5.2.2.2.1;Micromorphic Model;153
5.2.2.3;2.3 Viscous Generalized Stress and Phase Field Model;154
5.2.2.3.1;Phase Field Model;154
5.2.2.4;2.4 Elasto-Plastic Decomposition of Generalized Strains;155
5.2.3;3 Continuum Damage Model for Single Crystals and Its Regularization;156
5.2.3.1;3.1 Constitutive Equations;156
5.2.3.2;3.2 Microdamage Continuum;159
5.2.4;4 Finite Element Implementation;160
5.2.4.1;4.1 Variational Formulation and Discretization;160
5.2.4.2;4.2 Implicit Incremental Formulation;161
5.2.5;5 Numerical Examples;163
5.2.6;6 Conclusion;165
5.2.7;References;165
5.3;Homogenization and Multiscaling of Granular Media for Different Microscopic Constraints;168
5.3.1;1 Introduction;168
5.3.2;2 Quasi-Static Homogenization of Granular Aggregates;170
5.3.2.1;2.1 Deformation-Driven Homogenization of Microstructures;170
5.3.2.1.1;Definition of Particle Microstructures;170
5.3.2.1.2;Microscopic Boundary Conditions;171
5.3.2.1.3;Microscopic Equilibrium State;173
5.3.2.1.4;Macroscopic Boundary Conditions;176
5.3.2.2;2.2 Penalty-Type Implementation of Boundary Constraints;176
5.3.3;3 Microstructural Modeling of Granular Materials;178
5.3.3.1;3.1 Micromechanical Model for Interparticle Contact;178
5.3.3.2;3.2 Dynamic Relaxation of the Microstructural Response;179
5.3.4;4 Numerical Examples and Comparative Study;180
5.3.4.1;4.1 Specification of Basic Micromechanical Functions;181
5.3.4.2;4.2 Compression-Shear Mode for Cubic Microstructures;181
5.3.5;5 Multiple Scale Simulation of a Granular Medium;184
5.3.5.1;5.1 Two-Scale Simulations Based on DE-FE Coupling;184
5.3.5.2;5.2 Simulation of a Biaxial Compression Test of a Soil;185
5.3.5.2.1;Experimental Setup;185
5.3.5.2.2;Coupled FE-DE Two-Scale Model;185
5.3.5.2.3;Results and Discussion;186
5.3.6;6 Conclusion;188
5.3.7;Acknowledgement;188
5.3.8;References;188
5.4;Effective Hydraulic and Mechanical Properties of Heterogeneous Media with Interfaces;191
5.4.1;1 Introduction;191
5.4.2;2 Hydraulic Model for a Porous Matrix with Impermeable Inclusionary Phase;192
5.4.2.1;2.1 Mori–Tanaka Estimate;193
5.4.2.2;2.2 The Variational Approach;196
5.4.2.3;2.3 The Self-Consistent Approach;198
5.4.3;3 Mechanical Model for a Granular Cemented Rock;200
5.4.3.1;3.1 General Framework;200
5.4.3.2;3.2 The Self-Consistent Homogenization Scheme;202
5.4.4;4 Concluding Remarks;205
5.4.5;References;205
5.5;An Extended Finite Element Method for the Analysis of Submicron Heat Transfer Phenomena;207
5.5.1;1 Introduction;207
5.5.2;2 Level-Set Description of Material Layout;211
5.5.3;3 Gray Phonon Model;211
5.5.4;4 Discretization Methods;214
5.5.4.1;4.1 Discrete Ordinate Method;214
5.5.4.2;4.2 Extended Finite Element Method;215
5.5.4.3;4.3 Lagrange Multiplier Method;216
5.5.5;5 Numerical Examples;217
5.5.5.1;5.1 Verification Example;218
5.5.5.2;5.2 Analysis of Nano-Composites;218
5.5.5.3;5.3 Design Study;220
5.5.6;6 Conclusions;221
5.5.7;Acknowledgments;222
5.5.8;References;223
6;PART 3 Composites, Laminates, and Structures: Optimization;225
6.1;Multiscale Modeling and Simulation of Composite Materials and Structures;226
6.1.1;1 Introduction;226
6.1.2;2 Information-Passing Multiscale Approaches in Space;230
6.1.2.1;2.1 Direct Mathematical Homogenization for Nonlinear Problems;230
6.1.2.2;2.2 Eigendeformation-Based Reduced Order Homogenization;232
6.1.3;3 Concurrent Multiscale Methods in Space;234
6.1.3.1;3.1 Multiscale Enrichment Based on Partition of Unity (MEPU);235
6.1.3.2;3.2 Adaptive Model Selection;236
6.1.3.3;3.3 Numerical Example;236
6.1.4;4 Temporal Multiscale Model for Fatigue Life Prediction;237
6.1.5;References;240
6.2;Multiscale Modelling of the Failure Behaviour of Fibre-Reinforced Laminates;243
6.2.1;1 Introduction;243
6.2.2;2 Review of the Interface Damage Model;245
6.2.3;3 Mesoscale Simulations of a Centre-Cracked 2/1 GLARE Laminate;247
6.2.3.1;3.1 Geometry and Boundary Conditions;247
6.2.3.2;3.2 Results for a 2/1 Lay-up with Elastic Aluminium Layers;250
6.2.3.3;3.3 Results for a 2/1 Lay-up with Elasto-Plastic Aluminium Layers;253
6.2.4;4 Microscale Simulations of Single-Fibre Epoxy Systems;255
6.2.4.1;4.1 Fibre-Epoxy Interfacial Strength versus Epoxy Strength;256
6.2.5;5 Microscale Simulations of Multiple-Fibre Epoxy Systems;259
6.2.5.1;5.1 Influence of the Fibre Volume Fraction;259
6.2.6;6 Coupling between Microscale and Mesoscale Crack Modelling;260
6.2.6.1;6.1 Fibre-Epoxy Specimen Subjected to Uniaxial Tension;263
6.2.6.2;6.2 Influence of Sample Size;264
6.2.6.3;6.3 Influence of Imperfections;265
6.2.7;7 Concluding Remarks;266
6.2.8;Acknowledgements;268
6.2.9;References;268
6.3;Improved Multiscale Computational Strategies for Delamination;270
6.3.1;1 Introduction;270
6.3.2;2 Application of the Two-Scale Domain Decomposition Strategy to Delamination Analysis;272
6.3.2.1;2.1 The Substructured Delamination Problem;272
6.3.2.2;2.2 Two-Scale Iterative Resolution of the Substructured Problem;275
6.3.2.2.1;Introduction of the Macroscopic Scale;275
6.3.2.2.2;The Iterative Algorithm;275
6.3.2.3;2.3 First Example of a Delamination Analysis;278
6.3.3;3 Analysis of the Parameters of the Iterative Algorithm;280
6.3.4;4 The Three-Scale Domain Decomposition Strategy;281
6.3.4.1;4.1 Resolution of the Macroproblem through the Balancing Domain Decomposition Method;281
6.3.4.1.1;Partitioning of the Macroproblem;281
6.3.4.1.2;Resolution of the Super-Interface Problem;283
6.3.4.2;4.2 Results;283
6.3.5;5 Efficiency of the Strategy: Study of a Complex Test Case;284
6.3.6;6 Conclusion;286
6.3.7;References;286
6.4;Damage Propagation in Composites – Multiscale Modeling and Optimization;289
6.4.1;1 Introduction;289
6.4.2;2 Modeling of Discontinuities on Small Scale;291
6.4.2.1;2.1 Geometrical Description;291
6.4.2.2;2.2 Kinematic Description;292
6.4.2.3;2.3 Cohesive Law;293
6.4.2.4;2.4 Numerical Examples;294
6.4.2.4.1;Two-Phase Material under Tension [13];294
6.4.2.4.2;Pull-out of Fiber in Matrix;295
6.4.3;3 Multiscale Formulation [11, 12];295
6.4.3.1;3.1 Formulation Using Continuum Damage;296
6.4.3.1.1;Material Model;296
6.4.3.1.2;Concept;296
6.4.3.1.3;Numerical Examples [11];299
6.4.3.2;3.2 Discontinuum Model in Multiscale Formulation;302
6.4.3.2.1;Concept;302
6.4.3.2.2;Numerical Examples [13];303
6.4.4;4 Optimal Fiber Layout [18–20];304
6.4.4.1;4.1 Material Model;305
6.4.4.2;4.2 Optimization Concept;306
6.4.4.3;4.3 Multiphase Material Optimization;306
6.4.4.4;4.4 Shape Optimization of Fiber Geometry;307
6.4.4.5;4.5 Numerical Examples;307
6.4.4.5.1;Multiphase Material Optimization for a Beam;307
6.4.4.5.2;Multiphase and Shape Optimization for Deep Beam;308
6.4.5;5 Conclusions;309
6.4.6;Acknowledgement;310
6.4.7;References;310
6.5;Computational Multiscale Model for NATM Tunnels: Micromechanics-Supported Hybrid Analyses;313
6.5.1;1 Introduction;313
6.5.2;2 Continuum Micromechanics of Microheterogeneous Materials;315
6.5.2.1;2.1 Representative Volume Element (Separation of Scales);315
6.5.2.2;2.2 Homogenization of Elasticity;316
6.5.2.3;2.3 Homogenization of Strength;317
6.5.3;3 Micromechanics at the Cement Paste Level;318
6.5.3.1;3.1 Micromechanical Representation;318
6.5.3.2;3.2 Constitutive Behavior of Clinker, Water, Hydrates, and Air;319
6.5.3.3;3.3 Homogenized Elasticity of Cement Paste;320
6.5.3.4;3.4 Homogenized Strength of Cement Paste;321
6.5.4;4 Micromechanics at the Shotcrete Level;322
6.5.5;5 Experimental Validation of Micromechanics-Based Material Models;323
6.5.5.1;5.1 Mixture-Dependent Shotcrete Composition;323
6.5.5.2;5.2 Experimental Validation on Shotcrete Level;323
6.5.6;6 Micromechanics-Based Studies: Influence of Water-Cement and Aggregate-Cement Ratios on Evolutions of Elasticity and Strength of Shotcrete;324
6.5.7;7 Continuum Micromechanics-Based Safety Assessment of NATM Tunnel Shells;325
6.5.7.1;7.1 Water-Cement Ratio-Dependence of Structural Safety;328
6.5.7.2;7.2 Aggregate-Cement Ratio-Dependence of Structural Safety;328
6.5.8;8 Conclusions;330
6.5.9;Acknowledgements;331
6.5.10;References;331
6.6;Optimization of Corrugated Paperboard under Local and Global Buckling Constraints;337
6.6.1;1 Introduction;337
6.6.2;2 Problem Description and Methods Used;339
6.6.2.1;2.1 Unit Cell Approach;339
6.6.2.2;2.2 Formulation of the Optimization Problem;346
6.6.2.3;2.3 Homogenization Process Using Unit Cell Models;347
6.6.2.4;2.4 Unit Cell Approach for Local Buckling Analysis;347
6.6.2.5;2.5 Numerical Meso Structural Optimization Approach;348
6.6.3;3 Results of Optimization;349
6.6.4;4 Fold Formation in the Post-Buckling Regime;351
6.6.5;5 Conclusions;353
6.6.6;References;353
6.7;Framework for Multi-Level Optimization of Complex Systems;355
6.7.1;1 Introduction;355
6.7.2;2 A Unifying Multi-Level Notation;356
6.7.3;3 Decomposition;361
6.7.3.1;3.1 Physical Coupling;361
6.7.3.2;3.2 Problem Matrix;363
6.7.4;4 Coordination;370
6.7.5;5 Software Framework;372
6.7.6;6 Supersonic Business Jet Optimization;373
6.7.6.1;6.1 Multi-Level Optimization Problem;374
6.7.6.1.1;Hierarchic Decomposition;376
6.7.6.1.2;Non-Hierarchic Decomposition;378
6.7.6.1.3;Coordination;379
6.7.6.2;6.2 Numerical Results;380
6.7.6.2.1;Results;380
6.7.7;7 Conclusions;384
6.7.8;Acknowledgments;385
6.7.9;References;385
7;PART 4 Coupled Problems and Porous Media;386
7.1;Multiscale/Multiphysics Model for Concrete;387
7.1.1;1 Introduction;387
7.1.2;2 General Mathematical Model;388
7.1.3;3 Effective Stress Principle;391
7.1.4;4 Application of the Model to Concrete Structures at Elevated Temperature;393
7.1.4.1;4.1 Simulation of a Concrete Column under Fire with Fast Cooling;397
7.1.5;5 Application of the Model to Concrete Structures Subject to Leaching Process;402
7.1.5.1;5.1 Modelling Kinetics of Calcium Leaching Process;402
7.1.5.2;5.2 Numerical Simulation of the Non-Isothermal Leaching Process in a Concrete Wall;406
7.1.6;6 Conclusions;407
7.1.7;References;408
7.2;Swelling Phenomena in Electro-Chemically Active Hydrated Porous Media;411
7.2.1;1 Introduction;411
7.2.2;2 TPM Fundamentals;413
7.2.2.1;2.1 Immiscible Components and Volume Fractions;413
7.2.2.2;2.2 Miscible Components and Molar Concentrations;413
7.2.2.3;2.3 Constituent Balance Relations;414
7.2.3;3 Swelling Media as Biphasic, Four-Component Aggregates;415
7.2.3.1;3.1 Restrictions Obtained from the Entropy Inequality;417
7.2.3.2;3.2 The Fluid Components;419
7.2.3.3;3.3 Ion Diffusion and Fluid Flow;420
7.2.3.4;3.4 The Electrical Potential;422
7.2.3.5;3.5 The Solid Skeleton;422
7.2.4;4 Weak Forms and Basic Numerical Setting;423
7.2.5;5 Numerical Examples;424
7.2.5.1;5.1 Free Swelling Hydrogel;424
7.2.5.2;5.2 Electro-Active Polymers;426
7.2.5.3;5.3 Borehole Instability in Active Soil;427
7.2.5.4;5.4 Swelling of an Intervertebral Disc;427
7.2.6;6 Conclusion;429
7.2.7;References;429
7.3;Propagating Cracks in Saturated Ionized Porous Media;431
7.3.1;1 Introduction;431
7.3.2;2 Bulk Material;433
7.3.3;3 Discontinuity in the Solid Part;436
7.3.3.1;3.1 Cohesive Zone;436
7.3.3.2;3.2 Nonlocal Stress;437
7.3.4;4 Shearing Mode;438
7.3.4.1;4.1 Discontinuity in the Fluid Part;438
7.3.4.2;4.2 Numerical Example;439
7.3.5;5 Tensile Mode;441
7.3.5.1;5.1 Discontinuity in the Fluid Part;441
7.3.5.2;5.2 Numerical Example;443
7.3.6;6 Concluding Remarks;445
7.3.7;List of Symbols;445
7.3.8;Acknowledgement;446
7.3.9;References;446
8;Author Index;449
9;Subject Index;450