Understanding the Discrete Element Method

Simulation of Non-Spherical Particles for Granular and Multi-body Systems

Hans-Georg Matuttis, Jian Chen

• Publisher: Wiley
• Year: 2014
• Print ISBN: 9781118567203
• eBook ISBN: 9781118567289
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Mechanics

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.

1
Mechanics

We start with an outline of classical mechanics, to provide a framework for the discrete element 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 concepts 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, r_2D = (x, y), and a point moving in three-dimensional real space will have three degrees of freedom, r_3D = (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 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.

Figure 1.1 In two dimensions, the number of degrees of freedom n_dof 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.

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 dista...

Table of contents

• Cover
• Dedication
• Title page
• Copyright page
• About the Authors
• Preface
• Acknowledgements
• List of Abbreviations
• 1 Mechanics
• 2 Numerical Integration of Ordinary Differential Equations
• 3 Friction
• 4 Phenomenology of Granular Materials
• 5 Condensed Matter and Solid State Physics
• 6 Modeling and Simulation
• 7 The Discrete Element Method in Two Dimensions
• 8 The Discrete Element Method in Three Dimensions
• 9 Alternative Modeling Approaches
• 10 Running, Debugging and Optimizing Programs
• 11 Beyond the Scope of This Book
• A MATLAB® as Programming Language
• B Geometry and Computational Geometry
• Index
• End User License Agreement