Understanding the Discrete Element Method

Gives readers a more thorough understanding of DEM and equips researchers for independent work and an ability to judge methods related to simulation of polygonal particles

• Introduces DEM from the fundamental concepts (theoretical mechanics and solidstate physics), with 2D and 3D simulation methods for polygonal particles

• Provides the fundamentals of coding discrete element method (DEM) requiring little advance knowledge of granular matter or numerical simulation

• Highlights the numerical tricks and pitfalls that are usually only realized after years of experience, with relevant simple experiments as applications

• Presents a logical approach starting withthe mechanical and physical bases,followed by a description of the techniques and finally their applications

• Written by a key author presenting ideas on how to model the dynamics of angular particles using polygons and polyhedral

• Accompanying website includes MATLAB-Programs providing the simulation code for two-dimensional polygons

Recommended for researchers and graduate students who deal with particle models in areas such as fluid dynamics, multi-body engineering, finite-element methods, the geosciences, and multi-scale physics.

Autorentext
Hans-Georg Matuttis, The University of Electro-Communications, Japan Jian Chen, RIKEN Advanced Institute for Computational Science, Japan

Inhalt

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 Poincaré 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

Exercises 61

References 63

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 RungeKutta methods 79

2.3.1 Adaptive step-size control 79

2.3.2 Dense output and event location 81

2.3.3 Partitioned RungeKutta 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 predictorcorrector 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 i...