Computational Multiscale Modeling of Fluids and Solids (eBook)

Prof. Dr. Martin Oliver Steinhauser Fraunhofer Ernst-Mach-Institute for High-Speed Dynamics (EMI), Eckerstrasse 4 79104 Freiburg Germany

Preface to the 2nd Edition 6
Preface to the 1st Edition 8
Contents 14
List of Algorithms 19
List of Boxes 20
Part I Fundamentals 21
1 Introduction 22
1.1 Physics on Different Length- and Timescales 24
1.1.1 Electronic/Atomic Scale 25
1.1.2 Atomic/Microscopic Scale 26
1.1.3 Microscopic/Mesoscopic Scale 26
1.1.4 Mesoscopic/Macroscopic Scale 29
1.2 What are Fluids and Solids? 29
1.3 The Objective of Experimental and Theoretical Physics 32
1.4 Computer Simulations -- A Review 33
1.4.1 A Brief History of Computer Simulation 36
1.4.2 Computational Materials Science 44
1.5 Suggested Reading 46
2 Multiscale Computational Materials Science 48
2.1 Some Terminology 51
2.2 What Is Computational Material Science on Multiscales? 52
2.2.1 Experimental Investigations on Different Length Scales 53
2.3 What Is a Model? 56
2.3.1 The Scientific Method 57
2.4 Hierarchical Modeling Concepts Above the Atomic Scale 64
2.4.1 Example: Principle Model Hierarchies in Classical Mechanics 66
2.4.2 Structure-Property Paradigm 67
2.4.3 Physical and Mathematical Modeling 67
2.4.4 Numerical Modeling and Simulation 75
2.5 Unifications and Reductionism in Physical Theories 76
2.5.1 The Four Fundamental Interactions 78
2.5.2 The Standard Model 80
2.5.3 Symmetries, Fields, Particles and the Vacuum 82
2.5.4 Relativistic Wave Equations 89
2.5.5 Suggested Reading 96
2.6 Computer Science, Algorithms, Computability and Turing Machines 97
2.6.1 Recursion 100
2.6.2 Divide-and-Conquer 102
2.6.3 Local Search 105
2.6.4 Simulated Annealing and Stochastic Algorithms 107
2.6.5 Computability, Decidability and Turing Machines 108
2.6.6 Efficiency of Algorithms 118
2.6.7 Suggested Reading 125
3 Mathematical and Physical Prerequisites 128
3.1 Introduction 129
3.2 Sets and Set Operations 132
3.2.1 Cartesian Product, Product Set 136
3.2.2 Functions and Linear Spaces 137
3.3 Topological Spaces 145
3.3.1 Charts 152
3.3.2 Atlas 153
3.3.3 Manifolds 155
3.3.4 Tangent Vectors and Tangent Space 157
3.3.5 Covectors, Cotangent Space and One-Forms 160
3.3.6 Dual Spaces 165
3.3.7 Tensors and Tensor Spaces 167
3.3.8 Affine Connections and Covariant Derivative 172
3.4 Metric Spaces and Metric Connection 175
3.5 Riemannian Manifolds 178
3.5.1 Riemannian Curvature 179
3.6 The Problem of Inertia and Motion: Coordinate Systems in Physics 181
3.6.1 The Special and General Principle of Relativity 182
3.6.2 The Structure of Spacetime 186
3.7 Relativistic Field Equations 187
3.7.1 Relativistic Hydrodynamics 188
3.8 Suggested Reading 190
4 Fundamentals of Numerical Simulation 193
4.1 Basics of Ordinary and Partial Differential Equations in Physics 193
4.1.1 Elliptic Type 198
4.1.2 Parabolic Type 200
4.1.3 Hyperbolic Type 201
4.2 Numerical Solution of Differential Equations 203
4.2.1 Mesh-Based and Mesh-Free Methods 204
4.2.2 Finite Difference Methods 209
4.2.3 Finite Volume Method 212
4.2.4 Finite Element Methods 215
4.3 Elements of Software Design 217
4.3.1 Software Design 220
4.3.2 Writing a Routine 223
4.3.3 Code-Tuning Strategies 226
4.3.4 Suggested Reading 229
Part II Computational Methods on Multiscales 231
5 Computational Methods on Electronic/Atomistic Scale 233
5.1 Introduction 233
5.2 Ab-Initio Methods 235
5.3 Physical Foundations of Quantum Theory 238
5.3.1 A Short Historical Account of Quantum Theory 239
5.3.2 A Hamiltonian for a Condensed Matter System 242
5.3.3 The Born--Oppenheimer Approximation 242
5.4 Density Functional Theory 245
5.5 Car--Parinello Molecular Dynamics 246
5.5.1 Force Calculations: The Hellmann--Feynman Theorem 248
5.5.2 Calculating the Ground State 249
5.6 Solving Schrödinger's Equation for Many-Particle Systems: ƒ 251
5.6.1 The Hartree--Fock Approximation 252
5.7 What Holds a Solid Together? 262
5.7.1 Homonuclear Diatomic Molecules 263
5.8 Semi-empirical Methods 265
5.8.1 Tight-Binding Method 267
5.9 Bridging Scales: Quantum Mechanics (QM) - Molecular Mechanics (MM) 271
5.10 Concluding Remarks 272
6 Computational Methods on Atomistic/Microscopic Scale 274
6.1 Introduction 274
6.1.1 Thermodynamics and Statistical Ensembles 278
6.2 Fundamentals of Statistical Physics and Thermodynamics 279
6.2.1 Probabilities 280
6.2.2 Measurements and the Ergodic Hypotheses 281
6.2.3 Statistics in Phase Space and Statistical Ensembles 283
6.2.4 Virtual Ensembles 287
6.2.5 Entropy and Temperature 287
6.3 Classical Interatomic and Intermolecular Potentials 289
6.3.1 Charged Systems 289
6.3.2 Ewald Summation 291
6.3.3 The P3M Algorithm 292
6.3.4 Van der Waals Potential 293
6.3.5 Covalent Bonds 293
6.3.6 Embedded Atom Potentials 294
6.3.7 Pair Potentials 296
6.4 Classical Molecular Dynamics Simulations 298
6.4.1 Numerical Ingredients of MD Simulations 298
6.4.2 Integrating the Equations of Motion 304
6.4.3 Periodic Boundary Conditions 308
6.4.4 The Minimum Image Convention 309
6.4.5 Efficient Search Strategies for Interacting Particles 309
6.4.6 Making Measurements 312
6.5 Liquids, Soft Matter and Polymers 317
6.5.1 Scaling and Universality of Polymers 320
6.6 Monte Carlo Simulations 327
7 Computational Methods on Mesoscopic/Macroscopic Scale 330
7.1 Example: Meso- and Macroscale Shock-Wave Experiments with ceramics 334
7.2 Statistical Methods: Voronoi Tesselations and Power Diagrams for Modeling Microstructures of Ceramics 336
7.2.1 Reverse Monte Carlo Optimization 338
7.3 Dissipative Particle Dynamics 342
7.4 Ginzburg--Landau/Cahn--Hilliard Field Theoretic Mesoscale Simulation Method 344
7.5 Bridging Scales: Soft Particle Discrete Elements for Shock Wave Applications 345
7.6 Bridging Scales: Energetic Links between MD and FEM 355
7.6.1 Bridging Scales: Work-Hardening 357
7.7 Physical Theories for Macroscopic Phenomena: The Continuum Approach 358
7.7.1 The Description of Fluid Motion 359
7.8 Continuum Theory 360
7.8.1 The Continuum Hypothesis 361
7.9 Theory of Elasticity 363
7.9.1 Kinematic Equations 366
7.9.2 The Stress Tensor 371
7.9.3 Equations of Motion of the Theory of Elasticity 373
7.9.4 Constitutive Equations 373
7.10 Bridging Scale Application: Crack Propagation in a Brittle Specimen 376
8 Perspectives in Multiscale Materials Modeling 378
A Further Reading 381
B Mathematical Definitions 383
C Sample Code for the Main Routine of a MD Simulation 385
D A Sample Makefile 387
E Tables of Physical Constants 390
E.1 International System of Units (SI or mksA System) 390
E.2 Conversion Factors of Energy 390
References 398
Index 417