Jian Chen, RIKEN Advanced Institute for Computational Science, Japan

About the Authors xv

Preface xvii

Acknowledgements xix

List of Abbreviations xxi

1 Mechanics 1

1.1 Degrees of freedom 1

1.1.1 Particle mechanics and constraints 1

1.1.2 From point particles to rigid bodies 3

1.1.3 More context and terminology 4

1.2 Dynamics of rectilinear degrees of freedom 5

1.3 Dynamics of angular degrees of freedom 6

1.3.1 Rotation in two dimensions 6

1.3.2 Moment of inertia 7

1.3.3 From two to three dimensions 9

1.3.4 Rotation matrix in three dimensions 12

1.3.5 Three-dimensional moments of inertia 13

1.3.6 Space-fixed and body-fixed coordinate systems and equations of motion 16

1.3.7 Problems with Euler angles 19

1.3.8 Rotations represented using complex numbers 20

1.3.9 Quaternions 21

1.3.10 Derivation of quaternion dynamics 27

1.4 The phase space 29

1.4.1 Qualitative discussion of the time dependence of linear oscillations 31

1.4.2 Resonance 34

1.4.3 The flow in phase space 35

1.5 Nonlinearities 39

1.5.1 Harmonic balance 40

1.5.2 Resonance in nonlinear systems 42

1.5.3 Higher harmonics and frequency mixing 44

1.5.4 The van der Pol oscillator 45

1.6 From higher harmonics to chaos 47

1.6.1 The bifurcation cascade 47

1.6.2 The nonlinear frictional oscillator and Poincar´e maps 47

1.6.3 The route to chaos 51

1.6.4 Boundary conditions and many-particle systems 52

1.7 Stability and conservation laws 53

1.7.1 Stability in statics 54

1.7.2 Stability in dynamics 55

1.7.3 Stable axes of rotation around the principal axis 56

1.7.4 Noether's theorem and conservation laws 58

1.8 Further reading 61

2 Numerical Integration of Ordinary Differential Equations 65

2.1 Fundamentals of numerical analysis 65

2.1.1 Floating point numbers 65

2.1.2 Big-O notation 67

2.1.3 Relative and absolute error 69

2.1.4 Truncation error 69

2.1.5 Local and global error 71

2.1.6 Stability 74

2.1.7 Stable integrators for unstable problems 74

2.2 Numerical analysis for ordinary differential equations 75

2.2.1 Variable notation and transformation of the order of a differential equation 75

2.2.2 Differences in the simulation of atoms and molecules, as compared to macroscopic particles 76

2.2.3 Truncation error for solutions of ordinary differential equations 76

2.2.4 Fundamental approaches 77

2.2.5 Explicit Euler method 77

2.2.6 Implicit Euler method 78

2.3 Runge–Kutta methods 79

2.3.1 Adaptive step-size control 79

2.3.2 Dense output and event location 81

2.3.3 Partitioned Runge–Kutta methods 82

2.4 Symplectic methods 82

2.4.1 The classical Verlet method 82

2.4.2 Velocity-Verlet methods 83

2.4.3 Higher-order velocity-Verlet methods 85

2.4.4 Pseudo-symplectic methods 88

2.4.5 Order, accuracy and energy conservation 88

2.4.6 Backward error analysis 89

2.4.7 Case study: the harmonic oscillator with and without viscous damping 90

2.5 Stiff problems 92

2.5.1 Evaluating computational costs 93

2.5.2 Stiff solutions and error as noise 94

2.5.3 Order reduction 94

2.6 Backward difference formulae 94

2.6.1 Implicit integrators of the predictor–corrector formulae 94

2.6.2 The corrector step 96

2.6.3 Multiple corrector steps 97

2.6.4 Program flow 98

2.6.5 Variable time-step and variable order 98

2.7 Other methods 98

2.7.1 Why not to use self-written or novel integrators 98

2.7.2 Stochastic differential equations 100

2.7.3 Extrapolation and high-order methods 100

2.7.4 Multi-rate integrators 101

2.7.5 Zero-order algorithms 101

2.8 Differential algebraic equations 103

2.8.1 The pendulum in Cartesian coordinates 103

2.8.2 Initial conditions 106

2.8.3 Drift and stabilization 107

2.9 Selecting an integrator 109

2.9.1 Performance and stability 109

2.9.2 Angular degrees of freedom 109

2.9.3 Force equilibrium 109

2.9.4 Exploring new fields 110

2.9.5 ODE solvers unsuitable for DEM simulations 110

2.10 Further reading 111

3 Friction 129

3.1 Sliding Coulomb friction 129

3.1.1 A block on a slope 130

3.1.2 Static and dynamic friction coefficients 132

3.1.3 Apparent and actual contact area 134

3.1.4 Roughness and the friction coefficient 135

3.1.5 Adhesion and chemical bonding 136

3.2 Other contact geometries of Coulomb friction 136

3.2.1 Rolling friction 137

3.2.2 Pivoting friction 138

3.2.3 Sliding and rolling friction: the billiard problem 140

3.2.4 Sliding and rolling friction: cylinder on a slope 143

3.2.5 Pivoting and rolling friction 144

3.3 Exact implementation of friction 144

3.3.1 Establishing the difference between dynamic and static friction 145

3.3.2 Single-particle contact 148

3.3.3 Frictional linear chain 151

3.3.4 Higher dimensions 152

3.4 Modeling and regularizations 153

3.4.1 The Cundall–Strack model 153

3.4.2 Cundall-Strack friction in three dimensions 155

3.5 Unfortunate treatment of Coulomb friction in the literature 155

3.5.1 Insufficient models 156

3.5.2 Misunderstandings concerning surface roughness and friction 158

3.5.3 The Painlev´e paradox 158

3.6 Further reading 158

4 Phenomenology of Granular Materials 161

4.1 Phenomenology of grains 161

4.1.1 Interaction 161

4.1.2 Friction and dissipation 162

4.1.3 Length and time scales 162

4.1.4 Particle shape, and rolling and sliding 163

4.2 General phenomenology of granular agglomerates 164

4.2.1 Disorder 164

4.2.2 Heap formation 165

4.2.3 Tri-axial compression and shear band formation 166

4.2.4 Arching 168

4.2.5 Clogging 168

4.3 History effects in granular materials 168

4.3.1 Hysteresis 169

4.3.2 Reynolds dilatancy 170

4.3.3 Pressure distribution under heaps 171

4.4 Further reading 173

5 Condensed Matter and Solid State Physics 175

5.1 Structure and properties of matter 176

5.1.1 Crystal structures in two dimensions 176

5.1.2 Crystal structures in three dimensions 178

5.1.3 From the Wigner–Seitz cell to the Voronoi construction 180

5.1.4 Strength parameters of materials 182

5.1.5 Strength of granular assemblies 185

5.2 From wave numbers to the Fourier transform 186

5.2.1 Wave numbers and the reciprocal lattice 186

5.2.2 The Fourier transform in one dimension 188

5.2.3 Properties of the FFT 189

5.2.4 Other Fourier variables 193

5.2.5 The power spectrum 193

5.3 Waves and dispersion 194

5.3.1 Phase and group velocities 194

5.3.2 Phase and group velocities for particle systems 196

5.3.3 Numerical computation of the dispersion relation 199

5.3.4 Density of states 200

5.3.5 Dispersion relation for disordered systems 202

5.3.6 Solitons 204

5.4 Further reading 206

6 Modeling and Simulation 213

6.1 Experiments, theory and simulation 213

6.2 Computability, observables and auxiliary quantities 214

6.3 Experiments, theories and the discrete element method 215

6.4 The discrete element method and other particle simulation methods 217

6.5 Other simulation methods for granular materials 218

6.5.1 Continuum mechanics 218

6.5.2 Lattice models 219

6.5.3 The Monte Carlo method 220

7 The Discrete Element Method in Two Dimensions 223

7.1 The discrete element method with soft particles 223

7.1.1 The bouncing ball as a prototype for the DEM approach 224

7.1.2 Using two different stiffness constants to model damping 227

7.1.3 Simulation of round DEM particles in one dimension 228

7.1.4 Simulation of round particles in two dimensions 228

7.2 Modeling of polygonal particles 229

7.2.1 Initializing two-dimensional particles 229

7.2.2 Computation of the mass, center of mass and moment of inertia 231

7.2.3 Non-convex polygons 237

7.3 Interaction 237

7.3.1 Shape-dependent elastic force law 238

7.3.2 Computation of the overlap geometry 240

7.3.3 Computation of other dynamic quantities 244

7.3.4 Damping 246

7.3.5 Cohesive forces 248

7.3.6 Penetrating particle overlaps 249

7.4 Initial and boundary conditions 250

7.4.1 Initializing convex polygons 250

7.4.2 General considerations 252

7.4.3 Initial positions 253

7.4.4 Boundary conditions 255

7.5 Neighborhood algorithms 257

7.5.1 Algorithms not recommended for elongated particles 258

7.5.2 'Sort and sweep' 263

7.6 Time integration 271

7.7 Program issues 272

7.7.1 Program restart 272

7.7.2 Program initialization 274

7.7.3 Program flow 274

7.7.4 Proposed stages for the development of programs 276

7.7.5 Modularization 278

7.8 Computing observables 280

7.8.1 Computing averages 280

7.8.2 Homogenization and spatial averages 281

7.8.3 Computing error bars 282

7.8.4 Autocorrelation functions 284

7.9 Further reading 285

8 The Discrete Element Method in Three Dimensions 289

8.1 Generalization of the force law to three dimensions 289

8.1.1 The elastic force 290

8.1.2 Contact velocity and related forces 291

8.2 Initialization of particles and their properties 292

8.2.1 Basic concepts and data structures 292

8.2.2 Particle generation and geometry update 294

8.2.3 Decomposition of a polyhedron into tetrahedra 296

8.2.4 Volume, mass and center of mass 299

8.2.5 Moment of inertia 300

8.3 Overlap computation 301

8.3.1 Triangle intersection by using the point–direction form 301

8.3.2 Triangle intersection by using the point–normal form 305

8.3.3 Comparison of the two algorithms 309

8.3.4 Determination of inherited vertices 310

8.3.5 Determination of generated vertices 312

8.3.6 Determination of the faces of the overlap polyhedron 315

8.3.7 Determination of the contact area and normal 320

8.4 Optimization for vertex computation 322

8.4.1 Determination of neighboring features 323

8.4.2 Neighboring features for vertex computation 324

8.5 The neighborhood algorithm for polyhedra 325

8.5.1 'Sort and sweep' in three dimensions 325

8.5.2 Worst-case performance in three dimensions 326

8.5.3 Refinement of the contact list 327

8.6 Programming strategy for the polyhedral simulation 329

8.7 The effect of dimensionality and the choice of boundaries 332

8.7.1 Force networks and dimensionality 332

8.7.2 Quasi-two-dimensional geometries 332

8.7.3 Packings and sound propagation 333

8.8 Further reading 333

9 Alternative Modeling Approaches 335

9.1 Rigidly connected spheres 335

9.2 Elliptical shapes 336

9.2.1 Elliptical potentials 337

9.2.2 Overlap computation for ellipses 337

9.2.3 Newton–Raphson iteration 339

9.2.4 Ellipse intersection computed with generalized eigenvalues 340

9.2.5 Ellipsoids 344

9.2.6 Superquadrics 344

9.3 Composites of curves 345

9.3.1 Composites of arcs and cylinders 345

9.3.2 Spline curves 345

9.3.3 Level sets 347

9.4 Rigid particles 347

9.4.1 Collision dynamics ('event-driven method') 347

9.4.2 Contact mechanics 348

9.5 Discontinuous deformation analysis 349

9.6 Further reading 349

10 Running, Debugging and Optimizing Programs 353

10.1 Programming style 353

10.1.1 Literature 354

10.1.2 Choosing a programming language 355

10.1.3 Composite data types, strong typing and object orientation 356

10.1.4 Readability 356

10.1.5 Selecting variable names 357

10.1.6 Comments 359

10.1.7 Particle simulations versus solving ordinary differential equations 361

10.2 Hardware, memory and parallelism 362

10.2.1 Architecture and programming model 362

10.2.2 Memory hierarchy and cache 364

10.2.3 Multiprocessors, multi-core processors and shared memory 365

10.2.4 Peak performance and benchmarks 365

10.2.5 Amdahl's law, speed-up and efficiency 367

10.3 Program writing 369

10.3.1 Editors 370

10.3.2 Compilers 370

10.3.3 Makefiles 371

10.3.4 Writing and testing code 372

10.3.5 Debugging 377

10.4 Measuring load, time and profiles 378

10.4.1 The 'top' command 379

10.4.2 Xload 379

10.4.3 Performance monitor for multi-core processors 380

10.4.4 The 'time' command 380

10.4.5 The Unix profiler 383

10.4.6 Interactive profilers 383

10.5 Speeding up programs 383

10.5.1 Estimating the time consumption of operations 383

10.5.2 Compiler optimization options 384

10.5.3 Optimizations by hand 389

10.5.4 Avoiding unnecessary disk output 390

10.5.5 Look up or compute 390

10.5.6 Shared-memory parallelism and OpenMP 390

10.6 Further reading 391

11 Beyond the Scope of This Book 395

11.1 Non-convex particles 395

11.2 Contact dynamics and friction 395

11.3 Impact mechanics 396

11.4 Fragmentation and fracturing 396

11.5 Coupling codes for particles and elastic continua 396

11.6 Coupling of particles and fluid 398

11.6.1 Basic considerations for the fluid simulation 398

11.6.2 Verification of the fluid code 398

11.6.3 Macroscopic simulations 399

11.6.4 Microscopic simulations 399

11.6.5 Particle approach for both particles and fluid 400

11.6.6 Mesh-based modeling approaches 402

11.7 The finite element method for contact problems 402

11.8 Long-range interactions 403

A MATLAB R as Programming Language 407

A.1 Getting started with MATLAB R 407

A.2 Data types and names 408

A.3 Matrix functions and linear algebra 409

A.4 Syntax and control structures 413

A.5 Self-written functions 415

A.6 Function overwriting and overloading 416

A.7 Graphics 417

A.8 Solving ordinary differential equations 418

A.9 Pitfalls of using MATLAB R 420

A.10 Profiling and optimization 424

A.11 Free alternatives to MATLAB R 425

A.12 Further reading 425

Exercises 426

References 430

B Geometry and Computational Geometry 433

B.1 Trigonometric functions 433

B.2 Points, line segments and vectors 435

B.3 Products of vectors 436

B.3.1 Inner product (scalar product, dot product) 436

B.3.2 Orthogonality 437

B.3.3 Outer product 438

B.3.4 Vector product 438

B.3.5 Triple product 440

B.4 Projections and rejections 441

B.4.1 Projection of a vector onto another vector 441

B.4.2 Rejection of one vector with respect to another vector 442

B.5 Lines and planes 442

B.5.1 Lines and line segments 442

B.5.2 Planes 444

B.6 Oriented quantities: distance, area, volume etc. 446

B.7 Further reading 449

References 449

Index 451

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