1 Introduction ................................................. 1
1.1 Why Do We Need Numerical Simulations of Granular
Materials? .............................................. 2
1.2 Organization of the Book ................................ 8
1.3 Why Did We Write This Book As Is? ....................... 9
2 Molecular Dynamics .......................................... 13
2.1 Idea of Molecular Dynamics ............................. 13
2.1.1 Equations of Motion ............................. 13
2.1.2 Boundary Conditions ............................. 15
2.1.3 Initial Conditions .............................. 16
2.1.4 Models for Spherical Particles .................. 17
2.1.5 Gear's Integration Scheme ....................... 26
2.1.6 Sketch of the Molecular Dynamics Algorithm ...... 27
2.1.7 Vector Processing ............................... 29
2.2 Overview of the Presented Simulation Programs .......... 30
2.3 Molecular Dynamics Using Spherical Particles ........... 32
2.3.1 Basic Structure and Main Program ................ 32
2.3.2 Simple Algorithm for the Computation of
Forces .......................................... 36
2.3.3 Particle Class Sphere ........................... 38
2.3.4 Data Extraction ................................. 47
2.3.5 Examples ........................................ 48
2.3.6 Critical Discussion of the Particle Model ....... 52
2.4 Efficient Force Summation .............................. 53
2.4.1 Algorithmic Complexity of the Force Summation ... 53
2.4.2 Verlet Lists .................................... 54
2.4.3 Link Cell Algorithm ............................. 61
2.4.4 Lattice Algorithm ............................... 65
2.5 Quaternions for Three-Dimensional Simulations .......... 68
2.6 Composite Particles .................................... 75
2.6.1 Idea ............................................ 75
2.6.2 Geometrical Properties .......................... 76
2.6.3 Forces .......................................... 78
2.6.4 Implementation .................................. 79
2.6.5 Three-Dimensional Composite Particles ........... 84
2.6.6 Discussion ...................................... 85
2.7 Simulation of Sharp-Edged Particles .................... 86
2.7.1 Model ........................................... 86
2.7.2 Interaction of Colliding Triangles .............. 88
2.7.3 Contact Classification ......................... 100
2.7.4 Beam Forces .................................... 101
2.7.5 Examples ....................................... 104
2.7.6 Fragmentation of Sharp-Edged Particles ......... 108
2.8 Further Particle Models ............................... 108
2.9 Particle Fragmentation ................................ 110
2.9.1 Modeling of Fragmentation ...................... 110
2.9.2 Molecular Dynamics of Fragmenting Particles .... 111
2.9.3 Fragmentation Probability ...................... 113
2.9.4 Fragment Size Distribution ..................... 114
2.10 High Performance Computers ............................ 115
2.10.1 Vectorization .................................. 116
2.10.2 Parallelization ................................ 123
2.11 Vector Class .......................................... 129
Program Index .............................................. 133
3 Event-Driven Molecular Dynamics ............................ 135
3.1 Idea and Motivation ................................... 135
3.2 Collision of Particles ................................ 137
3.3 Uniqueness of the Collision Rule ...................... 140
3.4 Sketch of the Algorithm ............................... 141
3.5 Coefficients of Restitution ........................... 142
3.5.1 Coefficient of Normal Restitution en ........... 142
3.5.2 Tangential Coefficient of Restitution εt ....... 144
3.5.3 Relation between en and εt .................... 146
3.6 Simple Algorithm for Event-Driven Molecular
Dynamics .............................................. 147
3.6.1 Overview ....................................... 147
3.6.2 Force-Free Motion of Particles ................. 148
3.6.3 Pairwise Particle Collisions ................... 148
3.6.4 Wall Collisions ................................ 151
3.6.5 Initialization ................................. 153
3.6.6 Schedule of Collision Times .................... 153
3.6.7 Main Program ................................... 155
3.6.8 Output ......................................... 156
3.6.9 A Note on Numerical Errors ..................... 157
3.6.10 Critical Discussion of the Algorithm ........... 160
3.7 Improved Algorithm for Event-Driven Molecular
Dynamics .............................................. 160
3.7.1 Reduction of the Collision List ................ 160
3.7.2 Data Organization .............................. 162
3.7.3 Removal of Invalid Entries from the Event
Lists .......................................... 163
3.7.4 Collision-Free Motion of Particles ............. 163
3.7.5 Optimal Box Size ............................... 165
3.7.6 Scheduling Events .............................. 166
3.7.7 Update of Particle Positions ................... 167
3.7.8 Efficiency of the Algorithm .................... 167
3.8 Boundary Conditions ................................... 168
3.8.1 Reflecting Boundaries .......................... 169
3.8.2 Periodic Boundary Conditions ................... 170
3.8.3 Heated Walls ................................... 173
3.9 Inelastic Collapse .................................... 177
3.10 Granular Gases ........................................ 179
3.10.1 What are Granular Gases? ....................... 179
3.10.2 Cluster Instability ............................ 180
3.10.3 Some Open Problems ............................. 181
Program Index .............................................. 189
4 Direct Simulation Monte Carlo .............................. 191
4.1 Idea of Direct Simulation Monte Carlo ................. 191
4.2 Boltzmann Equation .................................... 193
4.3 Collision Frequency of a Uniform Hard Sphere Gas ...... 197
4.4 Integration of the Boltzmann Equation ................. 198
4.5 Implementation ........................................ 200
4.6 Application to a Force-Free Granular Gas .............. 206
5 Rigid-Body Dynamics ........................................ 211
5.1 Rigid Bodies .......................................... 211
5.2 Sketch of the Algorithm ............................... 215
5.3 Mathematical Description .............................. 215
5.3.1 Frictionless Particles ......................... 215
5.3.2 Particle Systems with Friction ................. 219
5.4 Dantzig's Algorithm for the Computation of the
Forces ................................................ 221
5.4.1 General Scheme ................................. 221
5.4.2 Application of the Algorithm to a Simple
Example ........................................ 224
5.5 Collisions ............................................ 232
5.6 Resolution of Static Indeterminacy .................... 236
5.7 Integration of the Equation of Motion ................. 238
5.8 Simple Examples ....................................... 239
5.9 Discussion of the Model ............................... 241
6 Cellular Automata .......................................... 243
6.1 Overview .............................................. 243
6.2 Heap Formation and Avalanches ......................... 243
6.3 Formation of Ripples .................................. 250
6.4 Lattice Gas Simulations ............................... 255
6.4.1 Lattice Gas Automaton .......................... 255
6.4.2 Granular Pipe Flow ............................. 257
6.4.3 Density Inhomogeneities in Granular Pipe
Flow ........................................... 268
7 Bottom-to-Top Reconstruction ............................... 271
7.1 Idea of the Method .................................... 271
7.2 Simulating a Heap ..................................... 273
7.3 Dynamic Simulations ................................... 287
7.4 Critical Analysis of the Model ........................ 290
8 Brownian Dynamics for the Simulation of Granular Flows ..... 293
8.1 Langevin Equation for Pipe Flow ....................... 293
8.2 Simulation of the Langevin Equation ................... 296
References .................................................... 303
Index ......................................................... 319
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