Understanding the Discrete Element Method

About the Authors xv Preface xvii Acknowledgements xix List of Abbreviations xxi 1 Mechanics 1 1.1 Degrees of freedom 1 1.2 Dynamics of rectilinear degrees of freedom 5 1.3 Dynamics of angular degrees of freedom 6 1.4 The phase space 29 1.5 Nonlinearities 39 1.6 From higher harmonics to chaos 47 1.7 Stability and conservation laws 53 1.8 Further reading 61 2 Numerical Integration of Ordinary Differential Equations 65 2.1 Fundamentals of numerical analysis 65 2.2 Numerical analysis for ordinary differential equations 75 2.3 Runge-Kutta methods 79 2.4 Symplectic methods 82 2.5 Stiff problems 92 2.6 Backward difference formulae 94 2.7 Other methods 98 2.8 Differential algebraic equations 103 2.9 Selecting an integrator 109 2.10 Further reading 111 3 Friction 129 3.1 Sliding Coulomb friction 129 3.2 Other contact geometries of Coulomb friction 136 3.3 Exact implementation of friction 144 3.4 Modeling and regularizations 153 3.5 Unfortunate treatment of Coulomb friction in the literature 155 3.6 Further reading 158 4 Phenomenology of Granular Materials 161 4.1 Phenomenology of grains 161 4.2 General phenomenology of granular agglomerates 164 4.3 History effects in granular materials 168 4.4 Further reading 173 5 Condensed Matter and Solid State Physics 175 5.1 Structure and properties of matter 176 5.2 From wave numbers to the Fourier transform 186 5.3 Waves and dispersion 194 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 7 The Discrete Element Method in Two Dimensions 223 7.1 The discrete element method with soft particles 223 7.2 Modeling of polygonal particles 229 7.3 Interaction 237 7.4 Initial and boundary conditions 250 7.5 Neighborhood algorithms 257 7.6 Time integration 271 7.7 Program issues 272 7.8 Computing observables 280 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 8.2 Initialization of particles and their properties 292 8.3 Overlap computation 301 8.4 Optimization for vertex computation 322 8.5 The neighborhood algorithm for polyhedra 325 8.6 Programming strategy for the polyhedral simulation 329 8.7 The effect of dimensionality and the choice of boundaries 332 8.8 Further reading 333 9 Alternative Modeling Approaches 335 9.1 Rigidly connected spheres 335 9.2 Elliptical shapes 336 9.3 Composites of curves 345 9.4 Rigid particles 347 9.5 Discontinuous deformation analysis 349 9.6 Further reading 349 10 Running, Debugging and Optimizing Programs 353 10.1 Programming style 353 10.2 Hardware, memory and parallelism 362 10.3 Program writing 369 10.4 Measuring load, time and profiles 378 10.5 Speeding up programs 383 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 11.7 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.4 Projections and rejections 441 B.5 Lines and planes 442 B.6 Oriented quantities: distance, area, volume etc. 446 B.7 Further reading 449 References 449 Index 451