COMPUTATIONALMECHANICS OFDISCONTINUA

WILEY SERIES IN COMPUTATIONAL MECHANICS

Series Advisors:

Rene de BorstPerumal NithiarasuTayfun E. TezduyarGenki YagawaTarek Zohdi

Introduction to Finite Element Analysis: Szabo and Babuska March 2011Formulation, Verification and Validation

COMPUTATIONALMECHANICS OFDISCONTINUA

Antonio A. MunjizaQueen Mary, University of London, UK

Earl E. KnightLos Alamos National Laboratory, USA

Esteban RougierLos Alamos National Laboratory, USA

A John Wiley & Sons, Ltd., Publication

This edition first published 2012© 2012 John Wiley & Sons, Ltd

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Library of Congress Cataloguing-in-Publication Data

Munjiza, Antonio A.Computational mechanics of discontinua / Antonio A. Munjiza,

Earl E. Knight and Esteban Rougier.p. cm. – (Wiley series in computational mechanics)

Includes bibliographical references and index.ISBN 978-0-470-97080-5 (hardback)1. Continuum mechanics. I. Knight, Earl E. II. Rougier, Esteban. III. Title.QA808.2.M87 2011531–dc23

2011020576

A catalogue record for this book is available from the British Library.

Print ISBN: 978-0-470-97080-5ePDF ISBN: 978-1-119-97118-4obook ISBN: 978-1-119-97116-0ePub ISBN: 978-1-119-97301-0Mobi ISBN: 978-1-119-97302-7

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Contents

Series Preface xi

Preface xiii

Acknowledgements xv

1 Introduction to Mechanics of Discontinua 11.1 The Concept of Discontinua 11.2 The Paradigm Shift 31.3 Some Problems of Mechanics of Discontinua 7

1.3.1 Packing 71.3.2 Fracture and Fragmentation 81.3.3 Demolition and Structures in Distress, Progressive Collapse 111.3.4 Nanotechnology 121.3.5 Block Caving 151.3.6 Mineral Processing 161.3.7 Discrete Populations in General 16References 18Further Reading 18

2 Methods of Mechanics of Discontinua 212.1 Introduction 212.2 Discrete Element Methods 21

2.2.1 Spherical Particles 222.2.2 Blocky Particles 232.2.3 Oblique and Super-Quadric Particles 232.2.4 Rigid Potential Field Particles 252.2.5 3D Real Shape Particles 252.2.6 Computer Games and Special Effects 26

2.3 The Combined Finite-Discrete Element Method 272.4 Molecular Dynamics 28

2.4.1 Common Potentials 292.5 Smooth Particle Hydrodynamics 31

viii Contents

2.6 Discrete Populations Approach 332.7 Algorithms and Solutions 35

References 36Further Reading 37

3 Disc to Edge Contact Interaction in 2D 393.1 Problem Description 393.2 Integration of Normal Contact Force 393.3 Tangential Force 443.4 Equivalent Nodal Forces 45

Further Reading 46

4 Triangle to Edge Contact Interaction in 2D 474.1 Problem Description 474.2 Integration of Normal Contact Force 474.3 Tangential Force 544.4 Equivalent Nodal Forces 55

Further Reading 56

5 Ball to Surface Contact Interaction in 3D 595.1 Problem Description 595.2 Integration of Normal Contact Force 595.3 Tangential Force 735.4 Equivalent Nodal Forces 74

Further Reading 75

6 Tetrahedron to Points Contact Interaction in 3D 776.1 Problem Description 776.2 Integration of Normal Contact Force 796.3 Tangential Force 846.4 Equivalent Nodal Forces 86

Further Reading 86

7 Tetrahedron to Triangle Contact Interaction in 3D 897.1 Problem Description 897.2 Integration of Normal Contact Force 897.3 Tangential Force 997.4 Equivalent Nodal Forces 101

Further Reading 102

8 Rock Joints 1038.1 Introduction 1038.2 Interaction between Mesh Entities in 2D 104

8.2.1 Interaction between a 2D Disk and a Straight Edge 1058.2.2 Numerical Integration of the Roller-Edge Interaction 111

Contents ix

8.3 Joint Dilation 1138.4 Shear Resistance of a 2D Rock Joint 1168.5 Numerical Examples 120

References 124Further Reading 124

9 MR Contact Detection Algorithm for Bodies of Similar Size 1259.1 The Challenge 1259.2 Constraints of MR Contact Detection Algorithm 1259.3 Space Decomposition 1279.4 Mapping of Spherical Bounding Boxes onto Cells 1279.5 Spatial Sorting 1299.6 Quick Sort Algorithm 1309.7 MR-Linear Sort Algorithm 1359.8 Implementation of the MR-Linear Sort Algorithm 1369.9 Quick Search Algorithm 1419.10 MR-Linear Search Algorithm 1439.11 CPU and RAM Performance 1459.12 CPU Performance and RAM Consumption 151

References 152Further Reading 152

10 MR Contact Detection Algorithm for Bodies of Different Sizes 15510.1 Introduction 15510.2 Description of the Multi-Step-MR Algorithm (MMR) 15510.3 Polydispersity 15610.4 CPU Performance 15710.5 RAM Requirements 15810.6 Robustness 15810.7 Applications 160

Further Reading 160

11 MR Contact Detection Algorithm for Complex Shapes in 2D 16311.1 Introduction 16311.2 Contactor Circle to Target Point MR Contact Detection Algorithm 163

11.2.1 Cell Size and Space Boundaries 16311.2.2 Rendering of 2D Target Points onto Cells 16611.2.3 Sorting of Target Cells 16711.2.4 Interrogation Tools for Sorted Target Cells 16711.2.5 Rendering of 2D Contactor Circles onto Cells 168

11.3 Contactor Circle to Target Edge MR Contact Detection Algorithm 17611.3.1 Rendering 2D Target Edges onto Cells 17611.3.2 Searching for Contacts 182

11.4 Contactor Triangle to Target Edge MR Contact Detection Algorithm 18411.4.1 Rendering 2D Triangles onto Cells 185

x Contents

11.5 Extension to Other Shapes 19211.6 Reporting of Contacting Couples 193

Further Reading 194

12 MR Contact Detection Algorithm for Complex Shapes in 3D 19712.1 Introduction 19712.2 Rendering Target Simplex Shapes 198

12.2.1 Rendering 3D Points onto Cells 19812.2.2 Rendering 3D Edges onto Cells 198

12.3 Sorting Target Cells 21012.4 Target Cells Interrogation Tools 21112.5 Searching for Contacts 212

12.5.1 Rendering Contactor Tetrahedron 21212.5.2 Rendering Contactor Triangular Facet 22612.5.3 Rendering Other Contactor Simplex Shapes 241Further Reading 241

13 Parallelization 24313.1 Introduction 24313.2 Domain Decomposition Approach 247

13.2.1 Communication Engine 25213.2.2 Broadcasting Engine 25413.2.3 Summing Engine 25413.2.4 Gathering Engine 25613.2.5 Distribution of Physical Objects across Processors 25713.2.6 Creating Proxies 25813.2.7 Relocating Originals 259

13.3 Graphics Processing Units (GPU) 26013.4 Structured Parallelization 262

Further Reading 263

Index 265

Series Preface

The series on Computational Mechanics is a conveniently identifiable set of books cover-ing interrelated subjects that have been receiving much attention in recent years and needto have a place in senior undergraduate and graduate school curricula, and in engineeringpractice. The subjects will cover applications and methods categories. They will rangefrom biomechanics to fluid-structure interactions to multiscale mechanics and from com-putational geometry to meshfree techniques to parallel and iterative computing methods.Application areas will be across the board in a wide range of industries, including civil,mechanical, aerospace, automotive, environmental and biomedical engineering. Practicingengineers, researchers and software developers at universities, industry and governmentlaboratories, and graduate students will find this book series to be an indispensible sourcefor new engineering approaches, interdisciplinary research, and a comprehensive learningexperience in computational mechanics.

Discrete element methods are used in a wide variety of applications – ranging fromfragmentation and mineral processing in engineering to the simulation of the dynamicsof galaxies in astrophysics. This book – written by leading experts in the field – providesa comprehensive overview of discrete element methods with an emphasis on algorithmicand implementation aspects. A unique feature is the in-depth treatment of accurate andfast methods for contact detection between particles, which is of pivotal importance for theefficiency of discrete element methods. Starting from basic concepts in discontinua, thebook further touches upon molecular dynamics simulations, smooth particle hydrodynam-ics and the combination of discrete element with finite element methods, and discussesparallel implementations of discrete element methods.

Preface

One of the more important breakthroughs of the modern scientific age was the develop-ment of differential calculus. The key to differential calculus is the concept of a pointwhich contains an instantaneous quantity such as point density or instantaneous velocity.Implicitly hidden is the assumption of smoothness of physical quantities, which translatesinto the assumption of a continuum. Based on this assumption, a whole range of scien-tific and engineering disciplines were developed, such as Fluid, Solid, and ContinuumMechanics. Common to all these is the existence of a set of governing partial differentialequations describing the physical problem as a continuum. With exponential advances incomputer hardware, fiber optics and related technologies, it has now become possible tosolve these governing equations using powerful computers and the associated numericalmethods of computational physics.

Modern science of the early decade of the 21st century is increasingly addressing prob-lems where the assumptions of smoothness and continuum are no longer true. The bestexample is Nano-Science and Nanotechnology where length scales are so small that thecontinuum assumption is simply not valid. Other examples include complex systems suchas biological systems, financial systems, crowds, hierarchical materials, mineral process-ing, powders, and so on. In these systems it is the presence of the interaction of a largenumber of individual atoms, molecules, particles, organisms, market players, individualpeople in the crowd or other individual building blocks of a complex system that producenew emergent properties and emergent phenomena such as a droplet of liquid, marketcrash, crowd stampede, and so on. A common feature of all of these is the departure fromthe continuum assumption towards an explicit adoption of the discontinuum. The newscientific discipline that has therefore emerged is called Mechanics of Discontinua.

While Continuum Mechanics smears out all the complex processes occurring at acertain length and time scale, Mechanics of Discontinua emphasizes these processes. Solv-ing equations of Continuum Mechanics produces numerical simulations which quantify“a priori” described physical quantities. In contrast, solving equations of Mechanics ofDiscontinua produces a virtual experiment that generates new qualities and properties,thus surprising the observer; for instance, from individual atoms a droplet of liquid or acrystal may appear; from individual market players a market crash may happen; from thebehaviour of individual people a stampede may occur.

Mechanics of Discontinua is a fundamental paradigm shift from the science that mea-sures “a priori” defined properties to the science that produces these as emergent propertiesand emergent phenomena.