UNDERSTANDING THE DISCRETE ELEMENT METHOD SIMULATION OF NON-SPHERICAL PARTICLES FOR GRANULARAND MULTI-BODY SYSTEMS -- Copright -- Contents -- About the Authors -- Preface -- Acknowledgements -- List of Abbreviations -- 1 Mechanics -- 1.1 Degrees of freedom -- 1.1.1 Particle mechanics and constraints -- 1.1.2 From point particles to rigid bodies -- 1.1.3 More context and terminology -- 1.2 Dynamics of rectilinear degrees of freedom -- 1.3 Dynamics of angular degrees of freedom -- 1.3.1 Rotation in two dimensions -- 1.3.2 Moment of inertia -- 1.3.3 From two to three dimensions -- 1.3.4 Rotation matrix in three dimensions -- 1.3.5 Three-dimensional moments of inertia -- 1.3.6 Space-fixed and body-fixed coordinate systems and equations of motion -- 1.3.7 Problems with Euler angles -- 1.3.8 Rotations represented using complex numbers -- 1.3.9 Quaternions -- 1.3.10 Derivation of quaternion dynamics -- 1.4 The phase space -- 1.4.1 Qualitative discussion of the time dependence of linear oscillations -- 1.4.2 Resonance -- 1.4.3 The flow in phase space -- 1.5 Nonlinearities -- 1.5.1 Harmonic balance -- 1.5.2 Resonance in nonlinear systems -- 1.5.3 Higher harmonics and frequency mixing -- 1.5.4 The van der Pol oscillator -- 1.6 From higher harmonics to chaos -- 1.6.1 The bifurcation cascade -- 1.6.2 The nonlinear frictional oscillator and Poincaré maps -- 1.6.3 The route to chaos -- 1.6.4 Boundary conditions and many-particle systems -- 1.7 Stability and conservationlaws -- 1.7.1 Stability in statics -- 1.7.2 Stability in dynamics -- 1.7.3 Stable axes of rotation around the principal axis -- 1.7.4 Noether's theorem and conservation laws -- 1.8 Further reading -- Exercises -- References -- 2 Numerical Integration of Ordinary Differential Equations -- 2.1 Fundamentals of numerical analysis -- 2.1.1 Floating point numbers -- 2.1.2 Big-O notation.

2.1.3 Relative and absolute error -- 2.1.4 Truncation error -- 2.1.5 Local and global error -- 2.1.6 Stability -- 2.1.7 Stable integrators for unstable problems -- 2.2 Numerical analysis for ordinary differential equations -- 2.2.1 Variable notation and transformation of the order of a differential equation -- 2.2.2 Differences in the simulation of atoms and molecules, as compared to macroscopic particles -- 2.2.3 Truncation error for solutions of ordinary differential equations -- 2.2.4 Fundamental approaches -- 2.2.5 Explicit Euler method -- 2.2.6 Implicit Euler method -- 2.3 Runge-Kutta methods -- 2.3.1 Adaptive step-size control -- 2.3.2 Dense output and event location -- 2.3.3 Partitioned Runge-Kutta methods -- 2.4 Symplectic methods -- 2.4.1 The classical Verlet method -- 2.4.2 Velocity-Verlet methods -- 2.4.3 Higher-order velocity-Verlet methods -- 2.4.4 Pseudo-symplectic methods -- 2.4.5 Order, accuracy and energy conservation -- 2.4.6 Backward error analysis -- 2.4.7 Case study: the harmonic oscillator with and without viscous damping -- 2.5 Stiff problems -- 2.5.1 Evaluating computational costs -- 2.5.2 Stiff solutions and error as noise -- 2.5.3 Order reduction -- 2.6 Backward difference formulae -- 2.6.1 Implicit integrators of the predictor-corrector formulae -- 2.6.2 The corrector step -- 2.6.3 Multiple corrector steps -- 2.6.4 Program flow -- 2.6.5 Variable time-step and variable order -- 2.7 Other methods -- 2.7.1 Why not to use self-written or novel integrators -- 2.7.2 Stochastic differential equations -- 2.7.3 Extrapolation and high-order methods -- 2.7.4 Multi-rate integrators -- 2.7.5 Zero-order algorithms -- 2.8 Differential algebraic equations -- 2.8.1 The pendulum in Cartesian coordinates -- 2.8.2 Initial conditions -- 2.8.3 Drift and stabilization -- 2.9 Selecting an integrator -- 2.9.1 Performance and stability.

2.9.2 Angular degrees of freedom -- 2.9.3 Force equilibrium -- 2.9.4 Exploring new fields -- 2.9.5 ODE solvers unsuitable for DEM simulations -- 2.10 Further reading -- Exercises -- References -- 3 Friction -- 3.1 Sliding Coulomb friction -- 3.1.1 A block on a slope -- 3.1.2 Static and dynamic friction coefficients -- 3.1.3 Apparent and actual contact area -- 3.1.4 Roughness and the friction coefficient -- 3.1.5 Adhesion and chemical bonding -- 3.2 Other contact geometries of Coulomb friction -- 3.2.1 Rolling friction -- 3.2.2 Pivoting friction -- 3.2.3 Sliding and rolling friction: the billiard problem -- 3.2.4 Sliding and rolling friction: cylinder on a slope -- 3.2.5 Pivoting and rolling friction -- 3.3 Exact implementation of friction -- 3.3.1 Establishing the difference between dynamic and static friction -- 3.3.2 Single-particle contact -- 3.3.3 Frictional linear chain -- 3.3.4 Higher dimensions -- 3.4 Modeling and regularizations -- 3.4.1 The Cundall-Strack model -- 3.4.2 Cundall-Strack friction in three dimensions -- 3.5 Unfortunate treatment of Coulomb friction in the literature -- 3.5.1 Insufficient models -- 3.5.2 Misunderstandings concerning surface roughness and friction -- 3.5.3 The Painlevé paradox -- 3.6 Further reading -- Exercises -- References -- 4 Phenomenology of Granular Materials -- 4.1 Phenomenology of grains -- 4.1.1 Interaction -- 4.1.2 Friction and dissipation -- 4.1.3 Length and time scales -- 4.1.4 Particle shape, and rolling and sliding -- 4.2 General phenomenology of granular agglomerates -- 4.2.1 Disorder -- 4.2.2 Heap formation -- 4.2.3 Tri-axial compression and shear band formation -- 4.2.4 Arching -- 4.2.5 Clogging -- 4.3 History effects in granular materials -- 4.3.1 Hysteresis -- 4.3.2 Reynolds dilatancy -- 4.3.3 Pressure distribution under heaps -- 4.4 Further reading -- References.

5 Condensed Matter and Solid State Physics -- 5.1 Structure and properties of matter -- 5.1.1 Crystal structures in two dimensions -- 5.1.2 Crystal structures in three dimensions -- 5.1.3 From the Wigner-Seitz cell to the Voronoi construction -- 5.1.4 Strength parameters of materials -- 5.1.5 Strength of granular assemblies -- 5.2 From wave numbers to the Fourier transform -- 5.2.1 Wave numbers and the reciprocal lattice -- 5.2.2 The Fourier transform in one dimension -- 5.2.3 Properties of the FFT -- 5.2.4 Other Fourier variables -- 5.2.5 The power spectrum -- 5.3 Waves and dispersion -- 5.3.1 Phase and group velocities -- 5.3.2 Phase and group velocities for particle systems -- 5.3.3 Numerical computation of the dispersion relation -- 5.3.4 Density of states -- 5.3.5 Dispersion relation for disordered systems -- 5.3.6 Solitons -- 5.4 Further reading -- Exercises -- References -- 6 Modeling and Simulation -- 6.1 Experiments, theory and simulation -- 6.2 Computability, observables and auxiliary quantities -- 6.3 Experiments, theories and the discrete element method -- 6.4 The discrete element method and other particle simulation methods -- 6.5 Other simulation methods for granular materials -- 6.5.1 Continuum mechanics -- 6.5.2 Lattice models -- 6.5.3 The Monte Carlo method -- References -- 7 The Discrete Element Method in Two Dimensions -- 7.1 The discrete element method with soft particles -- 7.1.1 The bouncing ball as a prototype for the DEM approach -- 7.1.2 Using two different stiffness constants to model damping -- 7.1.3 Simulation of round DEM particles in one dimension -- 7.1.4 Simulation of round particles in two dimensions -- 7.2 Modeling of polygonal particles -- 7.2.1 Initializing two-dimensional particles -- 7.2.2 Computation of the mass, center of mass and moment of inertia -- 7.2.3 Non-convex polygons -- 7.3 Interaction.

7.3.1 Shape-dependent elastic force law -- 7.3.2 Computation of the overlap geometry -- 7.3.3 Computation of other dynamic quantities -- 7.3.4 Damping -- 7.3.5 Cohesive forces -- 7.3.6 Penetrating particle overlaps -- 7.4 Initial and boundary conditions -- 7.4.1 Initializing convex polygons -- 7.4.2 General considerations -- 7.4.3 Initial positions -- 7.4.4 Boundary conditions -- 7.5 Neighborhood algorithms -- 7.5.1 Algorithms not recommended for elongated particles -- 7.5.2 'Sort and sweep' -- 7.6 Time integration -- 7.7 Program issues -- 7.7.1 Program restart -- 7.7.2 Program initialization -- 7.7.3 Program flow -- 7.7.4 Proposed stages for the development of programs -- 7.7.5 Modularization -- 7.8 Computing observables -- 7.8.1 Computing averages -- 7.8.2 Homogenization and spatial averages -- 7.8.3 Computing error bars -- 7.8.4 Autocorrelation functions -- 7.9 Further reading -- Exercises -- References -- 8 The Discrete Element Method in Three Dimensions -- 8.1 Generalization of the force law to three dimensions -- 8.1.1 The elastic force -- 8.1.2 Contact velocity and related forces -- 8.2 Initialization of particles and their properties -- 8.2.1 Basic concepts and data structures -- 8.2.2 Particle generation and geometry update -- 8.2.3 Decomposition of a polyhedron into tetrahedra -- 8.2.4 Volume, mass and center of mass -- 8.2.5 Moment of inertia -- 8.3 Overlap computation -- 8.3.1 Triangle intersection by using the point-direction form -- 8.3.2 Triangle intersection by using the point-normal form -- 8.3.3 Comparison of the two algorithms -- 8.3.4 Determination of inherited vertices -- 8.3.5 Determination of generated vertices -- 8.3.6 Determination of the faces of the overlap polyhedron -- 8.3.7 Determination of the contact area and normal -- 8.4 Optimization for vertex computation -- 8.4.1 Determination of neighboring features.

8.4.2 Neighboring features for vertex computation.