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Hans-Georg Matuttis | Jian Chen
Element Method Simulation of Non-Spherical Particles for Granular and Multi-body Systems
Hans-Georg Matuttis, The University of Electro-Communications, Japan Jian Chen, RIKEN Advanced Institute for Computational Science, Japan
Understanding the Discrete Element Method Simulation of Non-Spherical Particles for Granular and Multi-body Systems
The aim of this book is to advance the fi eld of granular and multi-body studies while giving readers a more thorough understanding of the discrete element method (DEM). By working through this volume, researchers will be better equipped for independent work and will develop an ability to judge methods related to the simulation of polygonal particles.
When materials are not handled as fl uids, they are dealt with mostly in granular form (e.g. cement, sand, grains, powders). Granular materials are characterized by abrupt transitions from loose to dense, from fl owing to static states, and vice versa. Many problems in natural disasters (earthquakes, landslides, etc.) are also of a “granular” nature. Continuum methods have been applied in these fi elds, but lack any intrinsic mechanism to account for the transitions, behavior that is inherently discontinuous. The “natural” approach is to use particle simulation methods, often called the “discrete element method”, where bodies in the physical system and the simulation match one to one. The fi eld of discrete element simulation has changed little since the early 1990s, when simulations predominantly used spherical particles. The aim of this book is to show the practicability and usefulness of non-spherical discrete element simulations. Phenomena from related fi elds (mechanics, solid state physics, etc.) are discussed, which as test cases are sometimes not applicable due to intriguing reasons. Understanding both the pitfalls and applications will help one to predict the outcome of simulations and use the predictions for the design of future experiments.
• Introduces the discrete element method (DEM) starting from the fundamental concepts (theoretical mechanics and solid state physics), with 2D and 3D simulation methods for polygonal and polyhedral particles
• Explains the basics of coding DEM, requiring little previous knowledge of granular matter or numerical simulation • Highlights numerical tricks and pitfalls that are usually only recognized after years of experience, using
relevant simple experiments to illustrate applications • Presents a logical approach starting with the mechanical and physical bases, followed by a description of the
techniques and their applications • Written by key authors presenting ideas on how to model the dynamics of angular particles using polygons
and polyhedrals • Accompanying website includes MATLAB® programs providing the simulation code for two-dimensional
convex polygons
This book is ideal for researchers and graduate students who deal with particle models in areas such as fl uid dynamics, multi-body engineering, fi nite element methods, virtual reality, the geosciences, and multi-scale physics. Computer scientists involved with solid modeling and game programming will also fi nd this book a useful reference for the design of physics engines.
Matuttis Chen
Hans-Georg Matuttis The University of Electro-Communications, Japan
Jian Chen RIKEN Advanced Institute for Computational Science, Japan
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ISBN: 978-1-118-56720-3
Set in 10/12pt Times by SPi Publisher Services, Pondicherry, India
1 2014
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
vi Contents
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 Poincare 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 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
Contents vii
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 Exercises 113 References 125
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
3.3.4 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 Painleve paradox 158
3.6 Further reading 158 Exercises 159 References 159
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 References 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
Contents ix
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 Exercises 206 References 210
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 References 221
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
x Contents
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 Exercises 286 References 286
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 References 333
Contents xi
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
References 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
xii Contents
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 Exercises 392 References 392
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
References 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
Contents xiii
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
About the Authors
Hans-Georg Matuttis did his Diploma and PhD on Quantum Monte Carlo methods at the University of Regensburg, Germany, and later worked on granular materials as assistant of professor Hans Herrmann at the university of Stuttgart. After three years research stay at The University of Tokyo, in 2002 H.-G. Matuttis became Associate Professor at the University of Electro-Communications (Tokyo, Japan).
Preface
While the discrete element method (DEM) for round particles has been around for decades (more than three if one starts with the algorithms of P. A. Cundall, more than five if one starts with the methods developed by B. Alder), the use of simulation techniques with non- spherical particles has spread only slowly, in spite of the much richer phenomenology and better geometric verisimilitude. Kepler’s dictum ‘ubi materia, ibi geometria’ (‘where you have to deal with matter, you have to deal with geometry’) is particularly true for the discrete element method.
Accordingly, we have focused more on concrete illustrations and motivations than on abstract derivations. This book basically consists of material that we wish we could have had in hand when we started to work with discrete element methods ourselves. We hope that the reader will find it useful.
Particle modeling needs a clear geometrical imagination, much more than modeling with continuum equations, where algebraic methods may also succeed. Geometrical arguments and approaches are given in detail, because we have found that they often don’t form part of the curricula in the fields to which this book may be relevant.
The reader is assumed to have reasonable familiarity with Newtonian mechanics, linear algebra, ordinary differential equations and at least one procedural programming language. We use MATLAB R© as the programming language in this book, as it has proved to be a flexi- ble tool for fast prototyping, even for complex algorithms; we have found that it improves readability and reduces development time compared to more ‘traditional’ programming languages.
As we intend the book to be accessible to graduates in the physical sciences, engineering and computer science, we have formulated many ideas more explicitly than if the book had been aimed at a single community alone. We have tried to make sure that the exposition is self-contained for the broadest readership we could envision. Nevertheless, depending on the reader’s background, some chapters will be more easily understandable and others more difficult.
It was the intention of the authors to make the book self-contained, i.e. all the important con- cepts are explained in the book without the need to use other reference material. In particular, ‘the internet’ contains some rather dubious resources for the learner.
xviii Preface
have derailed the schedules of many researchers for months and even years. We recommend starting with simpler algorithms and lower dimensions; only after gaining experience in these simpler settings should one move to more complex problems (such as polyhedra in three dimensions). We know of several projects which failed because the researchers began with full three-dimensional simulations right away, without first gaining experience with simpler algorithms in lower dimensions.
Acknowledgements
We are indebted to Wang Xiao Xing, Dominik Krengel, Shi Han Ng and Robin Tenhagen for reading the chapters and making valuable suggestions. Shi Han Ng is thanked in particular for programming various set-ups and algorithms, as well as for taking photos and recording movies.
Chen Wei Shen is thanked for ‘giving a hand’ in the pictures of the experiments. H.-G. M. would like to thank Professor Christian Lubich of the University of Tubingen for
introducing him to Filippov-type solutions for Coulomb friction problems. The authors gratefully acknowledge the pleasant cooperation with Clarissa Lim and James
List of Abbreviations
× three-dimensional vector product · scalar, matrix or quaternion product (depending on context) • placeholder for a variable in an operation ∝ proportional to 1 unit operator or identity matrix C field of complex numbers R field of real numbers | • | absolute value • length of a vector or norm of a matrix •∗ conjugate of a complex number or quaternion •−1 inverse of a matrix or quaternion or number arctan 2(y, x) two-argument arctangent (first argument is y, not x!) BDF backward difference formula (Gear predictor–corrector) BLAS basic linear algebra subroutines d differential increment in dx, dt, d/dx etc. DAE differential algebraic equation DDA discontinuous deformation analysis DEM discrete element method ED event-driven FEM finite element method i imaginary unit
√−1 g(· · · ) constraint function MD molecular dynamics MPS moving particle semi-implicit O order (Taylor order, Landau notation) ODE ordinary differential equation OpenMP open multi-processing
sgn(x)
= −1 x < 0
SPH smoothed particle hydrodynamics AT transpose Aji of matrix Aij .
T torque τ time-step
1 Mechanics
We start with an outline of classical mechanics, to provide a framework for the discrete ele- ment method (DEM). While most of the material in this chapter can be found scattered in various books on mechanics, no text seems to be available which covers concisely the con- cepts needed for DEM simulation. This chapter is intended as a crash course in theoretical mechanics, with an emphasis on issues relevant to computer implementation and testing. We give a list of secondary literature that the reader may refer to for further details.
1.1 Degrees of freedom
Before discussing the dynamics of a mechanical system, we need to understand the nature of the variables in the system. There are independent variables on the one hand, usually called ‘degrees of freedom’, and then there are dependent variables which depend on the degrees of freedom, via algebraic relations or derivatives.
1.1.1 Particle mechanics and constraints
The concept of a ‘mass point’ means that we neglect the size of the mass and are interested only in its trajectory. The position of a single mass point moving along the Cartesian x-axis is described by the value of x, which corresponds to a single degree of freedom. A point moving in the xy-plane has two degrees of freedom, r2D = (x, y), and a point moving in three- dimensional real space will have three degrees of freedom, r3D = (x, y, z). Although we can describe the motion of a point in three-dimensional space by four ‘space–time coordinates’ using the tuple (x, y, z, t), in classical mechanics t is not considered a degree of freedom but rather a parameter, i.e. an independent variable which cannot be influenced.
Two mass points moving independently along the x-axis represent two degrees of freedom, r1 and r2 (here and in the following, we assume equal masses). If we ‘glue’ these two particles together at distance d = r1 − r2 as in Figure 1.1, one degree of freedom gets lost, and we are
ndof = 2
ndof = 2 2 – 1 = 3
ndof = 3 · 2 – 3 = 3
ndof = 4 · 2 – 5 = 3
Figure 1.1 In two dimensions, the number of degrees of freedom ndof for 1, 2, 3 or 4 constrained particles with an increasing number of constraints introduced. Newly added constraints are in black; previous constraints are in gray.
left with only a single degree of freedom; in this case we can use either of r1, r2 or the average (r1+r2)/2 to determine the position of both particles uniquely. This means that one constraint between two position variables eliminates one degree of freedom.
In two dimensions, for two point particles at r1 = (x1, y1) and r2 = (x2, y2) we have four degrees of freedom, x1, y1, x2 and y2. If we again fix the distance between the particles at a constant distance d, so that
√ (x2 − x1)2 + (y2 − y1)2 = d, (1.1)
we can choose any three variables from {x1, y1, x2, y2} and the fourth will then be determined from (1.1) by elementary geometry. Alternatively, we can introduce new variables, such as the position of the center of mass, (x, y) = (r1 + r2)/2 for particles of the same mass, the displacement (x, y) = (x2 − x1, y2 − y1) between the particles, and the angle θ that the line segment between the two particles makes with the x-axis. In any case, we end up with three independent variables to describe the positions of the two particles fully. This means that a single constraint (1.1) reduces the number of degrees of freedom, i.e. the number of independent variables in the system, by 1.
In three-dimensional space, for two particles at positions (x1, y1, z1) and (x2, y2, z2) as shown in Figure 1.2, a constraint
√ (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 = d (1.2)
will again reduce the number of degrees of freedom by 1, so if we want to work with the center of mass
(x, y, z) = 1
2 {(x1, y1, z1) + (x2, y2, z2)},
ndof = 3 2 – 1 = 5 ndof = 3 · 4 – 6 = 6
ndof = 3 · 3 – 3 = 6 ndof = 3 · 5 – 9 = 6
Figure 1.2 In three dimensions, the number of degrees of freedom ndof for 1, 2, 3, 4 or 5 particles constrained so that the resulting cluster has no internal degrees of freedom. Newly added constraints are in black; previous constraints are in gray.
the center of mass and two angles, or with three Cartesian coordinates for one endpoint and two angles. In each case the number of degrees of freedom is the same, namely 5.
1.1.2 From point particles to rigid bodies
When we introduce one more point mass at (x3, y3, z3) to our set-up, we have 9 variables in total. If we connect this new point to both ends of our rod with the additional constraints
√ (x3 − x1)2 + (y3 − y1)2 + (z3 − z1)2 = d2, (1.3)
√ (x3 − x2)2 + (y3 − y2)2 + (z3 − z2)2 = d3, (1.4)
we get a triangle, as in the middle diagram of Figure 1.2. Again, we can give an alternative description of its position in space using the center of mass, and use three angles, φ, θ and ψ,
to describe the orientation. So the formula
(degrees of freedom) 6
= (variables) 9
− (constraints) 3
4 Understanding the Discrete Element Method
‘Mathematically’ one can define a point particle as an object having ‘zero extension’ and a rigid body as one having ‘zero deformation’. A more pragmatic definition of a point particle is an object whose extent is much smaller than the distances that it covers in the processes under investigation; after all, the Earth is pretty extended, but the point-mass approach to describing its trajectory around the sun works rather well. Likewise, a rigid body is an object for which the deformations are much smaller than the scales that are of interest in the processes being investigated.
1.1.3 More context and terminology
In principle, a ‘continuum’ has infinitely many degrees of freedom; but in order to solve continuum problems with a computer, we have to first discretize the continuum to obtain a finite number of degrees of freedom. We could, for instance, decompose the continuum into representative mass points and model the elasticity by springs between the mass points. The deformation of a spring can be computed from the positions of the bodies, so the springs will not be degrees of freedom, while the coordinates of the mass points will be degrees of freedom. With a finite element discretization, we decompose the elastic continuum into a space-filling partition of elements for which elastic stress relations hold, and the degrees of freedom are the nodes of the elements. Depending on the choice of boundary conditions, there may be as many nodes as there are elements, or more; therefore, from the nodes one can calculate the center of mass of the elements, but not vice versa. Describing the physics via the motion of particles, for example of centers of mass, is called the ‘Lagrangian representation’. This approach is natural for particulate systems, so we will adopt it in this book. Formulating the physics for a reference system in which, e.g., density amplitudes change is called the ‘Eulerian representation’; this representation is preferable for many continuum problems. In a Lagrangian representation, velocities of mechanical bodies are not degrees of freedom: they can be obtained as the time derivatives of the positions on which they depend. On the other hand, when we simulate a fluid volume where velocities are assigned to the nodes of a finite element or finite difference approximation in ‘Eulerian representation’, it is the velocities that are the degrees of freedom.
In the previous two subsections, we introduced constraints as algebraic relations between positions, but we remark here that constraints (whose associated functions are usually denoted by g in formulae) can also be imposed on velocities. For a pendulum of length l swing- ing around the origin as in Figure 1.3(a), the constraint g(x, y) stating that the bob (whose diameter we will neglect) stays at constant distance from the origin is
x2 + y2 = l2. (1.5)
(b)
z
Figure 1.3 (a) Pendulum as a constrained problem; (b) coupled pendulum–wheel–mass system, where transformation into polar coordinates does not simplify the calculation.
1.2 Dynamics of rectilinear degrees of freedom
Labeling the coordinates with different letters such as x1, y1, z1, . . . will soon become incon- venient, so let us rename them as follows: x1 = r
(1) 1 , y1 = r
(2) 1 , z1 = r
(3) 1 , x2 = r
r (2) 2 , z2 = r
(3) 2 , . . . , where the lower index represents the particle and the upper index in
parentheses represents the dimension. The corresponding velocities can then be obtained as time derivatives:
v (j) i = d
(j) i .
If all the velocities vanish, we say that the system is static; if the velocities (which may be non-zero) do not change, we say that the system is stationary. The accelerations are the time derivatives of the velocities, or the second derivatives of the positions with respect to time: a
(j) i = v
(j) i = r
(j) i . If the acceleration is constant, we also refer to it as ‘uniform’; in this
case the velocity changes at a constant rate. For a particle i with mass mi, Newton’s equation of motion1 expresses the relationship between the force F
(j) i applied to the particle and the
acceleration a (j) i in coordinate j as
F (j) i = mix
(1.6)
Numerical analysis prefers to deal with first-order equations, so often it is necessary to rewrite the second-order equation (1.6) as a first-order system by defining the velocity as an auxiliary variable:
F (j) i = miv
(j) i . (1.8)
6 Understanding the Discrete Element Method
Thus, instead of 3n second-order equations for n particles in three dimensions, we end up with 6n first-order equations. So, for a mechanical problem, one can choose whether to describe the system using first- or second-order differential equations. Consequently, physicists tend to call any equation with a first- or second-order time derivative on one side an ‘equation of motion’. For example, the quantum-mechanical wave equations are called ‘equations of motion of the probability’ due to their relation with probability densities [1], and the time-dependent heat equation is sometimes called the ‘equation of motion of heat’ [2].
1.3 Dynamics of angular degrees of freedom
1.3.1 Rotation in two dimensions
In two dimensions, we have three degrees of freedom: two for translation and one for rotation. Rotation of a vector r = (x, y)T by an angle φ in the xy-plane is represented by the rotation matrix for counterclockwise rotations,
Aφ = (
r ′ = Aφr = (
) . (1.10)
The inverse transformation of a rotation by φ is represented by the transpose of the original rotation matrix. That the inverse is equal to the transpose characterizes an orthogonal matrix, a matrix whose columns are orthogonal to each other (i.e. have scalar product zero). The determinant of an orthonormal matrix is 1, so the length of a vector r which is rotated using a matrix of the form (1.9) does not change, and if two different vectors r1 and r2 are rotated into
x
y
x
y
O
(a)
x
y
x
y
O
(b)
x
y
O
(c)
Hans-Georg Matuttis | Jian Chen
Element Method Simulation of Non-Spherical Particles for Granular and Multi-body Systems
Hans-Georg Matuttis, The University of Electro-Communications, Japan Jian Chen, RIKEN Advanced Institute for Computational Science, Japan
Understanding the Discrete Element Method Simulation of Non-Spherical Particles for Granular and Multi-body Systems
The aim of this book is to advance the fi eld of granular and multi-body studies while giving readers a more thorough understanding of the discrete element method (DEM). By working through this volume, researchers will be better equipped for independent work and will develop an ability to judge methods related to the simulation of polygonal particles.
When materials are not handled as fl uids, they are dealt with mostly in granular form (e.g. cement, sand, grains, powders). Granular materials are characterized by abrupt transitions from loose to dense, from fl owing to static states, and vice versa. Many problems in natural disasters (earthquakes, landslides, etc.) are also of a “granular” nature. Continuum methods have been applied in these fi elds, but lack any intrinsic mechanism to account for the transitions, behavior that is inherently discontinuous. The “natural” approach is to use particle simulation methods, often called the “discrete element method”, where bodies in the physical system and the simulation match one to one. The fi eld of discrete element simulation has changed little since the early 1990s, when simulations predominantly used spherical particles. The aim of this book is to show the practicability and usefulness of non-spherical discrete element simulations. Phenomena from related fi elds (mechanics, solid state physics, etc.) are discussed, which as test cases are sometimes not applicable due to intriguing reasons. Understanding both the pitfalls and applications will help one to predict the outcome of simulations and use the predictions for the design of future experiments.
• Introduces the discrete element method (DEM) starting from the fundamental concepts (theoretical mechanics and solid state physics), with 2D and 3D simulation methods for polygonal and polyhedral particles
• Explains the basics of coding DEM, requiring little previous knowledge of granular matter or numerical simulation • Highlights numerical tricks and pitfalls that are usually only recognized after years of experience, using
relevant simple experiments to illustrate applications • Presents a logical approach starting with the mechanical and physical bases, followed by a description of the
techniques and their applications • Written by key authors presenting ideas on how to model the dynamics of angular particles using polygons
and polyhedrals • Accompanying website includes MATLAB® programs providing the simulation code for two-dimensional
convex polygons
This book is ideal for researchers and graduate students who deal with particle models in areas such as fl uid dynamics, multi-body engineering, fi nite element methods, virtual reality, the geosciences, and multi-scale physics. Computer scientists involved with solid modeling and game programming will also fi nd this book a useful reference for the design of physics engines.
Matuttis Chen
Hans-Georg Matuttis The University of Electro-Communications, Japan
Jian Chen RIKEN Advanced Institute for Computational Science, Japan
This edition first published 2014 © 2014 John Wiley & Sons, Singapore Pte. Ltd.
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Library of Congress Cataloging-in-Publication data has been applied for.
ISBN: 978-1-118-56720-3
Set in 10/12pt Times by SPi Publisher Services, Pondicherry, India
1 2014
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
vi Contents
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 Poincare 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 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
Contents vii
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 Exercises 113 References 125
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
3.3.4 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 Painleve paradox 158
3.6 Further reading 158 Exercises 159 References 159
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 References 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
Contents ix
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 Exercises 206 References 210
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 References 221
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
x Contents
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 Exercises 286 References 286
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 References 333
Contents xi
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
References 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
xii Contents
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 Exercises 392 References 392
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
References 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
Contents xiii
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
About the Authors
Hans-Georg Matuttis did his Diploma and PhD on Quantum Monte Carlo methods at the University of Regensburg, Germany, and later worked on granular materials as assistant of professor Hans Herrmann at the university of Stuttgart. After three years research stay at The University of Tokyo, in 2002 H.-G. Matuttis became Associate Professor at the University of Electro-Communications (Tokyo, Japan).
Preface
While the discrete element method (DEM) for round particles has been around for decades (more than three if one starts with the algorithms of P. A. Cundall, more than five if one starts with the methods developed by B. Alder), the use of simulation techniques with non- spherical particles has spread only slowly, in spite of the much richer phenomenology and better geometric verisimilitude. Kepler’s dictum ‘ubi materia, ibi geometria’ (‘where you have to deal with matter, you have to deal with geometry’) is particularly true for the discrete element method.
Accordingly, we have focused more on concrete illustrations and motivations than on abstract derivations. This book basically consists of material that we wish we could have had in hand when we started to work with discrete element methods ourselves. We hope that the reader will find it useful.
Particle modeling needs a clear geometrical imagination, much more than modeling with continuum equations, where algebraic methods may also succeed. Geometrical arguments and approaches are given in detail, because we have found that they often don’t form part of the curricula in the fields to which this book may be relevant.
The reader is assumed to have reasonable familiarity with Newtonian mechanics, linear algebra, ordinary differential equations and at least one procedural programming language. We use MATLAB R© as the programming language in this book, as it has proved to be a flexi- ble tool for fast prototyping, even for complex algorithms; we have found that it improves readability and reduces development time compared to more ‘traditional’ programming languages.
As we intend the book to be accessible to graduates in the physical sciences, engineering and computer science, we have formulated many ideas more explicitly than if the book had been aimed at a single community alone. We have tried to make sure that the exposition is self-contained for the broadest readership we could envision. Nevertheless, depending on the reader’s background, some chapters will be more easily understandable and others more difficult.
It was the intention of the authors to make the book self-contained, i.e. all the important con- cepts are explained in the book without the need to use other reference material. In particular, ‘the internet’ contains some rather dubious resources for the learner.
xviii Preface
have derailed the schedules of many researchers for months and even years. We recommend starting with simpler algorithms and lower dimensions; only after gaining experience in these simpler settings should one move to more complex problems (such as polyhedra in three dimensions). We know of several projects which failed because the researchers began with full three-dimensional simulations right away, without first gaining experience with simpler algorithms in lower dimensions.
Acknowledgements
We are indebted to Wang Xiao Xing, Dominik Krengel, Shi Han Ng and Robin Tenhagen for reading the chapters and making valuable suggestions. Shi Han Ng is thanked in particular for programming various set-ups and algorithms, as well as for taking photos and recording movies.
Chen Wei Shen is thanked for ‘giving a hand’ in the pictures of the experiments. H.-G. M. would like to thank Professor Christian Lubich of the University of Tubingen for
introducing him to Filippov-type solutions for Coulomb friction problems. The authors gratefully acknowledge the pleasant cooperation with Clarissa Lim and James
List of Abbreviations
× three-dimensional vector product · scalar, matrix or quaternion product (depending on context) • placeholder for a variable in an operation ∝ proportional to 1 unit operator or identity matrix C field of complex numbers R field of real numbers | • | absolute value • length of a vector or norm of a matrix •∗ conjugate of a complex number or quaternion •−1 inverse of a matrix or quaternion or number arctan 2(y, x) two-argument arctangent (first argument is y, not x!) BDF backward difference formula (Gear predictor–corrector) BLAS basic linear algebra subroutines d differential increment in dx, dt, d/dx etc. DAE differential algebraic equation DDA discontinuous deformation analysis DEM discrete element method ED event-driven FEM finite element method i imaginary unit
√−1 g(· · · ) constraint function MD molecular dynamics MPS moving particle semi-implicit O order (Taylor order, Landau notation) ODE ordinary differential equation OpenMP open multi-processing
sgn(x)
= −1 x < 0
SPH smoothed particle hydrodynamics AT transpose Aji of matrix Aij .
T torque τ time-step
1 Mechanics
We start with an outline of classical mechanics, to provide a framework for the discrete ele- ment method (DEM). While most of the material in this chapter can be found scattered in various books on mechanics, no text seems to be available which covers concisely the con- cepts needed for DEM simulation. This chapter is intended as a crash course in theoretical mechanics, with an emphasis on issues relevant to computer implementation and testing. We give a list of secondary literature that the reader may refer to for further details.
1.1 Degrees of freedom
Before discussing the dynamics of a mechanical system, we need to understand the nature of the variables in the system. There are independent variables on the one hand, usually called ‘degrees of freedom’, and then there are dependent variables which depend on the degrees of freedom, via algebraic relations or derivatives.
1.1.1 Particle mechanics and constraints
The concept of a ‘mass point’ means that we neglect the size of the mass and are interested only in its trajectory. The position of a single mass point moving along the Cartesian x-axis is described by the value of x, which corresponds to a single degree of freedom. A point moving in the xy-plane has two degrees of freedom, r2D = (x, y), and a point moving in three- dimensional real space will have three degrees of freedom, r3D = (x, y, z). Although we can describe the motion of a point in three-dimensional space by four ‘space–time coordinates’ using the tuple (x, y, z, t), in classical mechanics t is not considered a degree of freedom but rather a parameter, i.e. an independent variable which cannot be influenced.
Two mass points moving independently along the x-axis represent two degrees of freedom, r1 and r2 (here and in the following, we assume equal masses). If we ‘glue’ these two particles together at distance d = r1 − r2 as in Figure 1.1, one degree of freedom gets lost, and we are
ndof = 2
ndof = 2 2 – 1 = 3
ndof = 3 · 2 – 3 = 3
ndof = 4 · 2 – 5 = 3
Figure 1.1 In two dimensions, the number of degrees of freedom ndof for 1, 2, 3 or 4 constrained particles with an increasing number of constraints introduced. Newly added constraints are in black; previous constraints are in gray.
left with only a single degree of freedom; in this case we can use either of r1, r2 or the average (r1+r2)/2 to determine the position of both particles uniquely. This means that one constraint between two position variables eliminates one degree of freedom.
In two dimensions, for two point particles at r1 = (x1, y1) and r2 = (x2, y2) we have four degrees of freedom, x1, y1, x2 and y2. If we again fix the distance between the particles at a constant distance d, so that
√ (x2 − x1)2 + (y2 − y1)2 = d, (1.1)
we can choose any three variables from {x1, y1, x2, y2} and the fourth will then be determined from (1.1) by elementary geometry. Alternatively, we can introduce new variables, such as the position of the center of mass, (x, y) = (r1 + r2)/2 for particles of the same mass, the displacement (x, y) = (x2 − x1, y2 − y1) between the particles, and the angle θ that the line segment between the two particles makes with the x-axis. In any case, we end up with three independent variables to describe the positions of the two particles fully. This means that a single constraint (1.1) reduces the number of degrees of freedom, i.e. the number of independent variables in the system, by 1.
In three-dimensional space, for two particles at positions (x1, y1, z1) and (x2, y2, z2) as shown in Figure 1.2, a constraint
√ (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 = d (1.2)
will again reduce the number of degrees of freedom by 1, so if we want to work with the center of mass
(x, y, z) = 1
2 {(x1, y1, z1) + (x2, y2, z2)},
ndof = 3 2 – 1 = 5 ndof = 3 · 4 – 6 = 6
ndof = 3 · 3 – 3 = 6 ndof = 3 · 5 – 9 = 6
Figure 1.2 In three dimensions, the number of degrees of freedom ndof for 1, 2, 3, 4 or 5 particles constrained so that the resulting cluster has no internal degrees of freedom. Newly added constraints are in black; previous constraints are in gray.
the center of mass and two angles, or with three Cartesian coordinates for one endpoint and two angles. In each case the number of degrees of freedom is the same, namely 5.
1.1.2 From point particles to rigid bodies
When we introduce one more point mass at (x3, y3, z3) to our set-up, we have 9 variables in total. If we connect this new point to both ends of our rod with the additional constraints
√ (x3 − x1)2 + (y3 − y1)2 + (z3 − z1)2 = d2, (1.3)
√ (x3 − x2)2 + (y3 − y2)2 + (z3 − z2)2 = d3, (1.4)
we get a triangle, as in the middle diagram of Figure 1.2. Again, we can give an alternative description of its position in space using the center of mass, and use three angles, φ, θ and ψ,
to describe the orientation. So the formula
(degrees of freedom) 6
= (variables) 9
− (constraints) 3
4 Understanding the Discrete Element Method
‘Mathematically’ one can define a point particle as an object having ‘zero extension’ and a rigid body as one having ‘zero deformation’. A more pragmatic definition of a point particle is an object whose extent is much smaller than the distances that it covers in the processes under investigation; after all, the Earth is pretty extended, but the point-mass approach to describing its trajectory around the sun works rather well. Likewise, a rigid body is an object for which the deformations are much smaller than the scales that are of interest in the processes being investigated.
1.1.3 More context and terminology
In principle, a ‘continuum’ has infinitely many degrees of freedom; but in order to solve continuum problems with a computer, we have to first discretize the continuum to obtain a finite number of degrees of freedom. We could, for instance, decompose the continuum into representative mass points and model the elasticity by springs between the mass points. The deformation of a spring can be computed from the positions of the bodies, so the springs will not be degrees of freedom, while the coordinates of the mass points will be degrees of freedom. With a finite element discretization, we decompose the elastic continuum into a space-filling partition of elements for which elastic stress relations hold, and the degrees of freedom are the nodes of the elements. Depending on the choice of boundary conditions, there may be as many nodes as there are elements, or more; therefore, from the nodes one can calculate the center of mass of the elements, but not vice versa. Describing the physics via the motion of particles, for example of centers of mass, is called the ‘Lagrangian representation’. This approach is natural for particulate systems, so we will adopt it in this book. Formulating the physics for a reference system in which, e.g., density amplitudes change is called the ‘Eulerian representation’; this representation is preferable for many continuum problems. In a Lagrangian representation, velocities of mechanical bodies are not degrees of freedom: they can be obtained as the time derivatives of the positions on which they depend. On the other hand, when we simulate a fluid volume where velocities are assigned to the nodes of a finite element or finite difference approximation in ‘Eulerian representation’, it is the velocities that are the degrees of freedom.
In the previous two subsections, we introduced constraints as algebraic relations between positions, but we remark here that constraints (whose associated functions are usually denoted by g in formulae) can also be imposed on velocities. For a pendulum of length l swing- ing around the origin as in Figure 1.3(a), the constraint g(x, y) stating that the bob (whose diameter we will neglect) stays at constant distance from the origin is
x2 + y2 = l2. (1.5)
(b)
z
Figure 1.3 (a) Pendulum as a constrained problem; (b) coupled pendulum–wheel–mass system, where transformation into polar coordinates does not simplify the calculation.
1.2 Dynamics of rectilinear degrees of freedom
Labeling the coordinates with different letters such as x1, y1, z1, . . . will soon become incon- venient, so let us rename them as follows: x1 = r
(1) 1 , y1 = r
(2) 1 , z1 = r
(3) 1 , x2 = r
r (2) 2 , z2 = r
(3) 2 , . . . , where the lower index represents the particle and the upper index in
parentheses represents the dimension. The corresponding velocities can then be obtained as time derivatives:
v (j) i = d
(j) i .
If all the velocities vanish, we say that the system is static; if the velocities (which may be non-zero) do not change, we say that the system is stationary. The accelerations are the time derivatives of the velocities, or the second derivatives of the positions with respect to time: a
(j) i = v
(j) i = r
(j) i . If the acceleration is constant, we also refer to it as ‘uniform’; in this
case the velocity changes at a constant rate. For a particle i with mass mi, Newton’s equation of motion1 expresses the relationship between the force F
(j) i applied to the particle and the
acceleration a (j) i in coordinate j as
F (j) i = mix
(1.6)
Numerical analysis prefers to deal with first-order equations, so often it is necessary to rewrite the second-order equation (1.6) as a first-order system by defining the velocity as an auxiliary variable:
F (j) i = miv
(j) i . (1.8)
6 Understanding the Discrete Element Method
Thus, instead of 3n second-order equations for n particles in three dimensions, we end up with 6n first-order equations. So, for a mechanical problem, one can choose whether to describe the system using first- or second-order differential equations. Consequently, physicists tend to call any equation with a first- or second-order time derivative on one side an ‘equation of motion’. For example, the quantum-mechanical wave equations are called ‘equations of motion of the probability’ due to their relation with probability densities [1], and the time-dependent heat equation is sometimes called the ‘equation of motion of heat’ [2].
1.3 Dynamics of angular degrees of freedom
1.3.1 Rotation in two dimensions
In two dimensions, we have three degrees of freedom: two for translation and one for rotation. Rotation of a vector r = (x, y)T by an angle φ in the xy-plane is represented by the rotation matrix for counterclockwise rotations,
Aφ = (
r ′ = Aφr = (
) . (1.10)
The inverse transformation of a rotation by φ is represented by the transpose of the original rotation matrix. That the inverse is equal to the transpose characterizes an orthogonal matrix, a matrix whose columns are orthogonal to each other (i.e. have scalar product zero). The determinant of an orthonormal matrix is 1, so the length of a vector r which is rotated using a matrix of the form (1.9) does not change, and if two different vectors r1 and r2 are rotated into
x
y
x
y
O
(a)
x
y
x
y
O
(b)
x
y
O
(c)