Particle-Based Methods: Fundamentals and Applications

Particle-Based Methods Computational Methods in Applied Sciences Volume 25 Series Editor E. Oñate International Cent...

Author: Eugenio Oñate | Roger Owen

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Particle-Based Methods

Computational Methods in Applied Sciences Volume 25

Series Editor E. Oñate International Center for Numerical Methods in Engineering (CIMNE) Technical University of Catalonia (UPC) Edificio C-1, Campus Norte UPC Gran Capitán, s/n 08034 Barcelona, Spain [email protected] www.cimne.com

For other titles published in this series, go to www.springer.com/series/6899

Eugenio Oñate • Roger Owen Editors

Particle-Based Methods Fundamentals and Applications

Editors Eugenio Oñate International Center for Numerical Methods in Engineering Technical University of Catalonia (UPC) Gran Capitan 08034 Barcelona Spain [email protected]

Roger Owen Civil and Computational Engineering Centre School of Engineering Swansea University Swansea SA2 8PP, Wales, UK [email protected]

ISSN 1871-3033 ISBN 978-94-007-0734-4 e-ISBN 978-94-007-0735-1 DOI 10.1007/978-94-007-0735-1 Springer Dordrecht Heidelberg London New York

Library of Congress Control Number: 2011922136 © Springer Science+Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: SPI Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Advances in the Particle Finite Element Method (PFEM) for Solving Coupled Problems in Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Oñate, S.R. Idelsohn, M.A. Celigueta, R. Rossi J. Marti, J.M. Carbonell, P. Ryzhakov and B. Suárez 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Basis of the Particle Finite Element Method . . . . . . . . . . . . . . . 2.1 Basic Steps of the PFEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 FIC/FEM Formulation for a Lagrangian Incompressible Thermal Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Discretization of the Equations . . . . . . . . . . . . . . . . . . . . . . 4 Overview of the Coupled FSI Algoritm . . . . . . . . . . . . . . . . . . . . . . . 5 Generation of a New Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Identification of Boundary Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Treatment of Contact Conditions in the PFEM . . . . . . . . . . . . . . . . . 7.1 Contact between the Fluid and a Fixed Boundary . . . . . . . 7.2 Contact between Solid-Solid Interfaces . . . . . . . . . . . . . . . 8 Modeling of Bed Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Modelling and Simulation of Excavation and Wear of Rock Cutting Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Rigid Objects Falling into Water . . . . . . . . . . . . . . . . . . . . . 10.2 Impact of Water Streams on Rigid Structures . . . . . . . . . . 10.3 Dragging of Objects by Water Streams . . . . . . . . . . . . . . . . 10.4 Impact of Sea Waves on Piers and Breakwaters . . . . . . . . . 10.5 Soil Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Melting, Spread and Burning of Polymer Objects in Fire . 10.7 Simulation of Excavation Process and Wear of Rock Cutting Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 3 4 7 7 9 10 13 15 17 17 18 18 23 25 25 27 27 28 29 34 40

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11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Advances in Computational Modelling of Multi-Physics in Particle-Fluid Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y.T. Feng, K. Han and D.R.J. Owen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Particle-Particle Interactions – Discrete Element Approach . . . . . . 2.1 Representation of Discrete Objects . . . . . . . . . . . . . . . . . . . 2.2 Contact Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Contact Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Governing Equations and Time Stepping . . . . . . . . . . . . . . 3 Fluid-Particle Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Lattice Boltzmann Method . . . . . . . . . . . . . . . . . . . . . . 3.2 Incorporating Turbulence Model in the Lattice Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hydrodynamic Forces for Fluid-Particle Interactions . . . . 3.4 Fluid and Particle Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 4 Thermal-Particle Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Convective Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Conductive Heat Transfer in Particles . . . . . . . . . . . . . . . . . 5 Fluid-Magnetic-Particle Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Magnetic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Fluid, Magnetic and Particle Coupling . . . . . . . . . . . . . . . . 6 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Example 1: Simulation of Particle Transport in a Vacuum Dredging System . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Example 2: Simulation of Heat Transfer in a Packed Bed 6.3 Example 3: Simulation of a Magnetorheological Fluid . . . 7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 53 54 54 55 56 57 57 59 60 61 63 63 64 69 69 73 73 73 77 77 86 87

Large Scale Simulation of Industrial, Engineering and Geophysical Flows Using Particle Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Paul W. Cleary, Mahesh Prakash, Matt D. Sinnott, Murray Rudman and Raj Das 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2 Industrial Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.1 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.2 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.3 Comminution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.4 Material Forming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.5 Bubbly and Reacting Multiphase Flows . . . . . . . . . . . . . . . 98 3 Fluid-Structure and Engineering Flows . . . . . . . . . . . . . . . . . . . . . . . 100 3.1 Rogue Wave Impact on an Moored Oil Platform . . . . . . . . 100 3.2 Ship Slamming and Green Water–Ship Interaction . . . . . . 101

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3.3 Spillway Flow and Dam Discharge . . . . . . . . . . . . . . . . . . . 101 3.4 Dam Wall Collapse under Earthquake Loading . . . . . . . . . 102 3.5 Fracture of a Structural Column during Projectile Impact 104 3.6 Excavation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4 Geo-Hazard and Extreme Geophysical Flows . . . . . . . . . . . . . . . . . . 105 4.1 Landslide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2 Flooding from Dam Wall Collapse . . . . . . . . . . . . . . . . . . . 107 4.3 Tsunami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Parallel Computation Particle Methods for Multi-Phase Fluid Flow with Application Oil Reservoir Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 113 John R. Williams, David Holmes and Peter Tilke 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 1.1 Oil Reservoir Characterization . . . . . . . . . . . . . . . . . . . . . . . 113 1.2 Multi-Core Parallel Computing . . . . . . . . . . . . . . . . . . . . . . 114 2 Computational Physics Using Particle Methods . . . . . . . . . . . . . . . . 116 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 2.2 SPH for Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 2.3 Testing and Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 2.4 Characterization of Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3 Application of SPH to Pore Scale Physics . . . . . . . . . . . . . . . . . . . . . 122 4 Parallel Computation on Multi-Core . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.1 Parallel Algorithms and the Ghost Region Issue . . . . . . . . 124 4.2 Spatial Hashing in Particle Methods . . . . . . . . . . . . . . . . . . 129 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 The Particle Finite Element Method for Multi-Fluid Flows . . . . . . . . . . . . 135 S.R. Idelsohn, M. Mier-Torrecilla, J. Marti and E. Oñate 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 2 Particle Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3 Multi-Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3.2 Discontinuities at the Interface . . . . . . . . . . . . . . . . . . . . . . 141 3.3 Interface Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.4 PFEM for Multi-Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . 144 3.5 Combustion Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.1 Bubble Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.2 Negatively Buoyant Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.3 Candle Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

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On Material Modeling by Polygonal Discrete Elements . . . . . . . . . . . . . . . . 159 B. Schneider, G.A. D’Addetta and E. Ramm 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 2 Basic Discrete Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3 Models for Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 3.1 Normal Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 3.2 Tangential Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 3.3 Contact with Background Plate . . . . . . . . . . . . . . . . . . . . . . 163 4 Models for Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.1 Brittle Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.2 Beam with Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.3 Softening Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.2 Representative Volume Element (RVE) . . . . . . . . . . . . . . . 169 5.3 Averaging Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.4 Extension to Higher Order Continua . . . . . . . . . . . . . . . . . . 172 5.5 Size of RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.1 Samples without Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.2 Samples with Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Discrete Numerical Analysis of Failure Modes in Granular Materials . . . . 187 Luc Sibille, Florent Prunier, François Nicot and Félix Darve 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 2.1 Kinetic Energy and Second-Order Work . . . . . . . . . . . . . . 189 2.2 DEM Investigation for Proportional Strain Loading Paths 192 3 Cones of Unstable Loading Directions, Bifurcation Domain . . . . . . 194 3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 3.2 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 3.3 Case of the Proportional Strain Loading Path . . . . . . . . . . 201 4 From Limit States to Failure Occurrence . . . . . . . . . . . . . . . . . . . . . . 202 4.1 Mixed Loading Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4.2 Stable and Unstable Loading Directions . . . . . . . . . . . . . . . 206 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Homogenization of Granular Material Modeled by a 3D DEM . . . . . . . . . 211 C. Wellmann and P. Wriggers 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 2 Discrete Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 3 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 4 Periodic Triaxial Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

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4.1 Periodic RVEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 4.3 Random Sample Generation . . . . . . . . . . . . . . . . . . . . . . . . . 220 5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 5.1 Packing Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5.2 RVE Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 5.3 Particle Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Some Consideration on Derivative Approximation of Particle Methods . . 233 Hitoshi Matsubara, Shigeo Iraha, Genki Yagawa and Doosam Song 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 2 Formulation and Some Remarks on Particle Methods . . . . . . . . . . . 234 3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 3.1 Energy Norm in Elasticity Field . . . . . . . . . . . . . . . . . . . . . 238 3.2 Strain Distributions in Complicated Displacement Field . 241 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Discrete Element Modelling of Rock Cutting . . . . . . . . . . . . . . . . . . . . . . . . 247 Jerzy Rojek, Eugenio Oñate, Carlos Labra and Hubert Kargl 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 2 Numerical Model of Rock Cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 3 Discrete Element Method Formulation . . . . . . . . . . . . . . . . . . . . . . . 250 4 Determination of Rock Model Parameters . . . . . . . . . . . . . . . . . . . . . 254 4.1 Dimensionless Micro-Macro Relationships . . . . . . . . . . . . 254 5 Simulation of Rock Cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 5.1 Simulation of Rock Cutting with a Single Roadheader Pick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 5.2 Simulation of the Linear Cutting Test . . . . . . . . . . . . . . . . . 264 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Preface

This volume contains the extended version of a selection of papers presented at the First International Conference on Particle-Based Methods (PARTICLES 2009), held in Barcelona, Spain on November 25–27, 2009. PARTICLES 2009 was a forum for practitioners in the computational mechanics field to discuss recent advances and identify future research directions for particlebased methods. The different chapters in the book have been selected with the aims of providing both the fundamental basis and the applicability of state of the art and new particlebased computational methods for solving a variety of problems in engineering and applied sciences. The content of the different chapters includes state of the art developments and applications of standard and innovative particle-based techniques such as the discrete element method (DEM), the smooth particle hydrodynamic method (SPH), the particle finite element method (PFEM), the material point method, and atomistic and quantum mechanics-based methods, among others. The coupling of these methods with standard numerical procedures, such as the finite element method and also with meshless techniques offers new possibilities to solve complex problems in engineering and sciences with an accurate representation of the physical phenomena at nano, micro and macro scales. The applications of the particle-based methods compiled in the book cover geomechanical and mining problems, industrial forming processes, fluid-structure interaction problems accounting for free surface flow effects in civil and marine engineering (water streams acting on constructions, wave loads in harbours and marine structures, ship hydrodynamics, etc.), multi-fracturing processes in impact situations, nano-micro-macroscopic effects in material science and bio-medical engineering, molecular dynamics, quantum mechanics problems, melting of polymers in fire situations and many others. This book includes contributions submitted directly by the authors. The editors cannot accept responsibility for any inaccuracies, comments and opinions contained in the text. The editors would like to thank all authors for submitting their contributions. Eugenio Oñate and Roger Owen

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Advances in the Particle Finite Element Method (PFEM) for Solving Coupled Problems in Engineering E. Oñate, S.R. Idelsohn∗, M.A. Celigueta, R. Rossi J. Marti, J.M. Carbonell, P. Ryzhakov and B. Suárez

Abstract We present some developments in the formulation of the Particle Finite Element Method (PFEM) for analysis of complex coupled problems on fluid and solid mechanics in engineering accounting for fluid-structure interaction and coupled thermal effects, material degradation and surface wear. The PFEM uses an updated Lagrangian description to model the motion of nodes (particles) in both the fluid and the structure domains. Nodes are viewed as material points which can freely move and even separate from the main analysis domain representing, for instance, the effect of water drops. A mesh connects the nodes defining the discretized domain where the governing equations are solved, as in the standard FEM. The necessary stabilization for dealing with the incompressibility of the fluid is introduced via the finite calculus (FIC) method. An incremental iterative scheme for the solution of the non linear transient coupled fluid-structure problem is described. The procedure for modelling frictional contact conditions at fluid-solid and solidsolid interfaces via mesh generation are described. A simple algorithm to treat soil erosion in fluid beds is presented. An straight forward extension of the PFEM to model excavation processes and wear of rock cutting tools is described. Examples of application of the PFEM to solve a wide number of coupled problems in engineering such as the effect of large waves on breakwaters and bridges, the large motions of floating and submerged bodies, bed erosion in open channel flows, the wear of rock cutting tools during excavation and tunneling and the melting, dripping and burning of polymers in fire situations are presented.

E. Oñate · S.R. Idelsohn · M.A. Celigueta · R. Rossi · J. Marti · J.M. Carbonell · P. Ryzhakov · B. Suárez International Center for Numerical Methods in Engineering (CIMNE), Technical University of Catalonia (UPC), Campus Norte UPC, 08034 Barcelona, Spain; e-mail: [email protected], www.cimne.com/eo, www.cimne.com/pfem ∗

ICREA Research Professor at CIMNE.

E. Oñate and R. Owen (eds.), Particle-Based Methods, Computational Methods in Applied Sciences 25, DOI 10.1007/978-94-007-0735-1_1, © Springer Science+Business Media B.V. 2011

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1 Introduction The analysis of problems involving the interaction of fluids and structures accounting for large motions of the fluid free surface and the existence of fully or partially submerged bodies which interact among themselves is of big relevance in many areas of engineering. Examples are common in ship hydrodynamics, off-shore and harbour structures, spill-ways in dams, free surface channel flows, environmental flows, liquid containers, stirring reactors, mould filling processes, etc. Typical difficulties of fluid-multibody interaction analysis in free surface flows using the FEM with both the Eulerian and ALE formulation include the treatment of the convective terms and the incompressibility constraint in the fluid equations, the modelling and tracking of the free surface in the fluid, the transfer of information between the fluid and the moving solid domains via the contact interfaces, the modeling of wave splashing, the possibility to deal with large motions of the bodies within the fluid domain, the efficient updating of the finite element meshes for both the structure and the fluid, etc. For a comprehensive list of references in FEM for fluid flow problems see [9, 49] and the references there included. A survey of recent works in fluid-structure interaction (FSI) analysis can be found in [26, 35, 47, 49]. Most of the above problems disappear if a Lagrangian description is used to formulate the governing equations of both the solid and the fluid domains. In the Lagrangian formulation the motion of the individual particles are followed and, consequently, nodes in a finite element mesh can be viewed as moving material points (hereforth called “particles”). Hence, the motion of the mesh discretizing the total domain (including both the fluid and solid parts) is followed during the transient solution. The authors have successfully developed in the past years a particular class of Lagrangian formulation for problems involving complex interactions between fluids and solids. The so called particle finite element method (PFEM, www.cimne.com/pfem), treats the nodes in the fluid and solid domains as particles which can freely move and even separate from the main fluid (or solid) domain representing, for instance, the effect of water drops. A mesh connects the nodes discretizing the domain where the governing equations are solved using a stabilized FEM. The FEM solution in the (incompressible) fluid domain implies solving the momentum and incompressibility equations. This is not a simple problem as the incompressibility condition limits the choice of the FE approximations for the velocity and pressure to overcome the well known div-stability condition [9,49]. In our work we use a stabilized mixed FEM based on the Finite Calculus (FIC) approach which allows for a linear approximation for the velocity and pressure variables. An advantage of the Lagrangian formulation is that the convective terms disappear from the fluid equations. The difficulty is however transferred to the problem of adequately (and efficiently) moving the mesh nodes. We use a mesh regeneration procedure blending elements of different shapes using an extended Delaunay tesselation with special shape functions [13, 15]. The theory and applications of the PFEM are reported in [2, 8, 13, 14, 16, 17, 34–36, 38, 40, 44–46].

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The PFEM has been recently extended to model the frictional interaction between water and solids, as well as between deformable solids accounting for surface wear situations. Successful applications of the PFEM in this field include the modeling of bed erosion in free surface flows [40], the simulation of excavation and tunneling problems and the study of wear in rock cutting tools [5, 6]. Yet another successful application of the PFEM is the study of how objects melt, drip and burn in presence of fire. The solution of this complex FSI problem requires solving the equations of a coupled thermal-flow in a multifluid environment including an appropriate combustion model and taking into account the large deformations and eventual loss of mass in the burning object [24, 41, 46]. The aim of this paper is to describe recent advances of the PFEM for (a) the the interaction between a collection of bodies which are fixed, floating and/or submerged in a fluid, (b) the soil erosion in open channel flows, (c) the wear of rock cutting tools and their performance during excavation and tunneling processes and (d) the melting, dripping and burning of polymer objects in fire situations. All these problems are of great relevance in many areas of engineering. It is shown that the PFEM provides a general analysis methodology for treat such complex problems in a simple and efficient manner. The layout of the paper is the following. In the next section the key ideas of the PFEM are outlined. Next the basic equations for an incompressible thermal flow using a Lagrangian description and the FIC formulation are presented. Then an algorithm for the transient solution is briefly described. The treatment of the coupled FSI problem and the methods for mesh generation and for identification of the free surface nodes are outlined. The procedure for treating at mesh generation level the contact conditions at fluid-wall interfaces and the frictional contact interaction between moving solids is explained. A methodology for modeling bed erosion due to fluid forces is described. The extension of this erosion technique to model excavation in soil/rock and wear of rock cutting tools with the PFEM is presented. The potencial of the PFEM is shown in its application to FSI problems involving large flow motions, surface waves, moving bodies in water and bed erosion. Other examples shown include the application of PFEM to excavation and tunneling problems and to the melting, dripping and burning of polymers in fire situations.

2 The Basis of the Particle Finite Element Method Let us consider a domain containing both fluid and solid subdomains. The moving fluid particles interact with the solid boundaries thereby inducing the deformation of the solid which in turn affects the flow motion and, therefore, the problem is fully coupled. In the PFEM both the fluid and the solid domains are modelled using an updated Lagrangian formulation. That is, all variables in the fluid and solid domains are assumed to be known in the current configuration at time t. The new set of variables in both domains are sought for in the next or updated configuration at time t + t

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Fig. 1 Updated lagrangian description for a continuum containing a fluid and a solid domain

(Figure 1). The finite element method (FEM) is used to solve the continuum equations in both domains. Hence a mesh discretizing these domains must be generated in order to solve the governing equations for both the fluid and solid problems in the standard FEM fashion. Recall that the nodes discretizing the fluid and solid domains are treated as material particles which motion is tracked during the transient solution. This is useful to model the separation of fluid particles from the main fluid domain in a splashing wave, or soil particles in a bed erosion problem, and to follow their subsequent motion as individual particles with a known density, an initial acceleration and velocity and subject to gravity forces. The mass of a given domain is obtained by integrating the density at the different material points over the domain. The quality of the numerical solution depends on the discretization chosen as in the standard FEM. Adaptive mesh refinement techniques can be used to improve the solution in zones where large motions of the fluid or the structure occur.

2.1 Basic Steps of the PFEM For clarity purposes we will define the collection or cloud of nodes (C) pertaining to the fluid and solid domains, the volume (V ) defining the analysis domain for the fluid and the solid and the mesh (M) discretizing both domains. A typical solution with the PFEM involves the following steps:

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Fig. 2 Sequence of steps to update a “cloud” of nodes representing a domain containing a fluid and a solid part from time n (t = tn ) to time n + 2 (t = tn + 2t)

1. The starting point at each time step is the cloud of points in the fluid and solid domains. For instance n C denotes the cloud at time t = tn (Figure 2). 2. Identify the boundaries for both the fluid and solid domains defining the analysis domain n V in the fluid and the solid. This is an essential step as some boundaries (such as the free surface in fluids) may be severely distorted during the solution, including separation and re-entering of nodes. The Alpha Shape method [10] is used for the boundary definition (Section 5). 3. Discretize the fluid and solid domains with a finite element mesh n M. In our work we use an innovative mesh generation scheme based on the extended Delaunay tesselation (Section 4) [13, 14, 16]. 4. Solve the coupled Lagrangian equations of motion for the fluid and the solid domains. Compute the state variables in both domains at the next (updated) configuration for t + t: velocities, pressure, viscous stresses and temperature in the fluid and displacements, stresses, strains and temperature in the solid. 5. Move the mesh nodes to a new position n+1 C where n + 1 denotes the time tn + t, in terms of the time increment size. This step is typically a consequence of the solution process of step 4.

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Fig. 3 Sequence of steps to update in time a “cloud” of nodes representing a polymer object under thermal loads represented by a prescribed boundary heat flux q. Crossed circles denote prescribed nodes at the boundary. The figure also explains the application of the PFEM to modelling a rock cutting problem

6. Go back to step 1 and repeat the solution process for the next time step to obtain n+2 C. The process is shown in Figure 2. Figure 3 shows another conceptual example of application of the PFEM to modelling the melting and dripping of a polymer object under a heat source q acting at a boundary. Figure 3 can be also used to explain the application of the PFEM to rock cutting problems. In those cases q represents the forces of the rock cutting tool acting on a

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Fig. 4 Breakage of a water column. (a) Discretization of the fluid domain and the solid walls. Boundary nodes are marked with circles. (b) and (c) Mesh in the fluid domain at two different times

rock mass represented by the cloud of points. The figure shows the detachment of the rock mass during the cutting process. Figure 4 shows a typical example of a PFEM solution of a free surface flow problem in 2D. The images correspond to the analysis of the problem of breakage of a water column [16, 36]. Figure 4a shows the initial grid of four-noded rectangles discretizing the fluid domain and the solid walls. Figures 4b and 4c show the deformed mesh at two later times.

3 FIC/FEM Formulation for a Lagrangian Incompressible Thermal Fluid 3.1 Governing Equations The key equations to be solved in the incompressible thermal flow problem, written in the Lagrangian frame of reference, are the following:

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Momentum ρ

∂σij ∂vi + bi = ∂t ∂xj

in

(1)

Mass balance ∂vi =0 ∂xi

in

(2)

Heat transport ∂ ∂T = ρc ∂t ∂xi

∂T k ∂xi

+Q

in

(3)

In the above equations vi is the velocity along the ith global (cartesian) axis, T is the temperature, ρ, c and k are the density (assumed constant), the specific heat and the conductivity of the material, respectively, bi and Q are the body forces and the heat source per unit mass, respectively and σij are the (Cauchy) stresses related to the velocities by the standard constitutive equation (for incompressible Newtonian material) σij = sij − pδij (4a) ∂vj 1 1 ∂vi (4b) sij = 2µ ε˙ ij − δij ε˙ ii , ε˙ ij = + 3 2 ∂xj ∂xi In Eqs. (4), sij is the deviatoric stresses, p is the pressure (assumed to be positive in compression), ε˙ ij is the rate of deformation, µ is the viscosity and δij is the Kronecker delta. In the following we will assume the viscosity µ to be a known function of temperature, i.e. µ = µ(T ). Indexes in Eqs. (1)–(4) range from i, j = 1, nd , where nd is the number of space dimensions of the problem (i.e. nd = 2 for two-dimensional problems). The standard sum notation for repeated indices is assumed, unless otherwise specified. Equations (1)–(4) are completed with the standard boundary conditions of prescribed velocities and surface tractions in the mechanical problem and prescribed temperature and prescribed normal heat flux in the thermal problem [2, 9]. We note that Eqs. (1)–(5) are the standard ones for modeling the deformation of viscoplastic materials using the so called “flow approach” [49–51]. In our work the dependence of the viscosity with the strain typical of viscoplastic flows has been simplified to the Newtonian form of Eq. (4b).

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3.2 Discretization of the Equations A key problem in the numerical solution of Eqs. (1)–(4) is the satisfaction of the incompressibility condition (Eq. (2)). A number of procedures to solve his problem exist in the finite element literature [9, 49]. In our approach we use a stabilized formulation based in the so-called finite calculus procedure [27–29, 36, 38, 40]. The essence of this method is the solution of a modified mass balance equation which is written as ∂ ∂p ∂vi +τ + πi = 0 i = 1, nd (5) ∂xi ∂xi ∂xi where τ is a stabilization parameter given by [10] τ=

8µ 2ρ|v| + 2 h 3h

−1 (6)

In the above, h is a characteristic length of each finite element (such as [A(e) ]1/2 for 2D elements) and |v| is the modulus of the velocity vector. In Eq. (5) πi are auxiliary pressure-gradient projection variables that are assumed to have a continuous distribution in the mesh. The difference between the discontinuous pressure gradient field within each element and the continuous distribution for the πi provides the necessary stabilization to solve the problem with the FEM. The set of governing equations for the velocities, the pressure and the πi variables is completed by adding the following constraint equation [36, 40] ∂p τ wi + πi dV = 0, i = 1, nd not sum in i (7) ∂xi V where wi are arbitrary weighting functions. The rest of the integral equations are obtained by applying the standard Galerkin technique to the governing equations (1), (2), (3), (5) and (7) and the corresponding boundary conditions [36, 40]. We interpolate next in the standard finite element fashion the set of problem variables. For 3D problems these are the three velocities vi , the pressure p, the temperature T and the three pressure gradient projections πi . In our work we use equal order linear interpolation for all variables over meshes of 3-noded triangles (in 2D) and 4-noded tetrahedra (in 3D) [36, 40, 53]. The resulting set of discretized equations has the following form:

Momentum Mv˙¯ + K(µ)¯v − Gp¯ = f

(8)

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Mass balance GT v¯ + Lp¯ + Qπ¯ = 0

(9)

ˆ π¯ + QT p¯ = 0 M

(10)

¯ =q CT˙¯ + HT

(11)

Pressure gradient projection

Heat transport

˙¯ = ∂/∂t (·). ¯ denotes nodal variables and (·) ¯ The different In Eqs. (8)–(11) (·) matrices and vectors are given in the Appendix. The solution in time of Eqs. (8)–(11) can be performed using any time integration scheme typical of the updated Lagrangian finite element method. A basic algorithm following the conceptual process described in Section 2.1 is presented in Box I. t +t (¯a)j +1 denotes the values of the nodal variables a¯ at time t + t and the j + 1 iterations. We note the coupling of the flow and thermal equations via the dependence of the viscosity µ with the temperature.

4 Overview of the Coupled FSI Algoritm Figure 5 shows a typical domain V with external boundaries V and t where the velocity and the surface tractions are prescribed, respectively. The domain V is formed by fluid (VF ) and solid (VS ) subdomains (i.e. V = VF ∪ VS ). Both subdomains interact at a common boundary F S where the surface tractions and the kinematic variables (displacements, velocities and acelerations) are the same for both subdomains. Note that both set of variables (the surface tractions and the kinematic variables) are equivalent in the equilibrium configuration. Let us define t S and t F the set of variables defining the kinematics and the stressstrain fields at the solid and fluid domains at time t, respectively, i.e. t t

S := [t xs , t us , t vs , t as , t εs , t σ s , t Ts ]T

F := [ xF , uF , vF , aF , ε˙ F , σ F , TF ] t

t

t

t

t

t

t

(12) T

(13)

where x is the nodal coordinate vector, u, v and a are the vector of displacements, velocities and accelerations, respectively, ε, ε˙ and σ are the strain vector, the strainrate (or rate of deformation) vectors and the Cauchy stress vector, respectively, T is the temperature and subscripts F and S denote the variables in the fluid and solid

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Box I. Flow chart of basic PFEM algorithm for the fluid domain

domains, respectively. In the discretized problem, a bar over these variables denotes nodal values. The coupled fluid-structure interaction (FSI) problem of Figure 4 is solved in this work using the following strongly coupled staggered scheme: 1. We assume that the variables in the solid and fluid domains at time t (t S and t F ) are known. 2. Solve for the variables at the solid domain at time t +t (t +t S) under prescribed surface tractions at the fluid-solid boundary F S . The boundary conditions at the part of the external boundary intersecting the domain are the standard ones in solid mechanics. The variables at the solid domain t +t S are found via the integration of the equations of dynamic motion in the solid written as [52]

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Fig. 5 Split of the analysis domain V into fluid and solid subdomains. Equality of surface tractions and kinematic variables at the common interface

Ms a¯ s + gs − fs = 0

(14)

where a¯ s is the vector of nodal accelerations and Ms , gs and fs are the mass matrix, the internal node force vector and the external nodal force vector in the solid domain. Indeed, the solid model can include any type of material and geometrical non-linearity using standard non-linear solid mechanics procedures [52]. The time integration of Eq. (14) is performed using a standard Newmark method. Solve for the variables at the fluid domain at time t +t (t +t F ) under prescribed surface tractions at the external boundary t and prescribed velocities at the external and internal boundaries V and F S , respectively. An incremental iterative scheme is implemented within each time step to account for non linear geometrical and material effects. Iterate between 1 and 2 until convergence. The above FSI solution algorithm is shown schematically in Box II.

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LOOP OVER TIME STEPS t = 1,...ntime Initial values:

t

S ,

t

F

LOOP OVER STAGGERED SOLUTION j = 1,...nstag Solve for solid variables (prescribed tractions at

t +∆t

ΓSF )

LOOP OVER ITERATIONS i = 1,...niter Solve for t +∆t S ij Integrate Eq.(14) using a Newmark scheme Check convergence → Yes: solve for fluid variables NO: Next iteration i ← i + 1 Solve for fluid variables (prescribed velocities at

t +∆t

Γ FS )

LOOP OVER ITERATIONS i = 1,...niter Solve for t +∆t Fji using the scheme of Section 4 Check convergence → Yes: go to C Next iteration i ← i + 1 C Check convergence of surface tractions at Yes: Next time step Next staggered solution j ← j + 1, i ← i + 1 Next time step

t +∆t

S ←t +∆t S ij ,

t +∆t

t +∆t

Γ FS

F ←t +∆t Fji

Box II. Staggered solution scheme for the FSI problem (Figure 5). S: variables in the solid domain. F : variables in the fluid domain

5 Generation of a New Mesh One of the key points for the success of the PFEM is the fast regeneration of a mesh at every time step on the basis of the position of the nodes in the space domain. Any fast meshing algorithm can be used for this purpose. In our work the mesh is generated at each time step using the extended Delaunay tesselation (EDT) [13, 15, 16]. The EDT allows one to generate non standard meshes combining elements of arbitrary polyhedrical shapes (triangles, quadrilaterals and other polygons in 2D and tetrahedra, hexahedra and arbitrary polyhedra in 3D) in a computing time of order n, where n is the total number of nodes in the mesh (Figure 6). The C ◦ continuous shape functions of the elements can be simply obtained using the so called meshless finite element interpolation (MFEM). In our work the simpler linear C ◦ interpolation has been chosen [13, 15, 16]. Figure 7 shows the evolution of the CPU time required for generating the mesh, for solving the system of equations and for assembling such a system in terms of

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Fig. 6 Generation of non standard meshes combining different polygons (in 2D) and polyhedra (in 3D) using the extended Delaunay technique.

Fig. 7 3D flow problem solved with the PFEM. CPU time for meshing, assembling and solving the system of equations at each time step in terms of the number of nodes

the number of nodes. the numbers correspond to the solution of a 3D flow in an open channel with the PFEM [40]. The figure shows the CPU time in seconds for each time step of the algorithm of Section 3.2. The CPU time required for meshing grows linearly with the number of nodes, as expected. Note also that the CPU time for solving the equations exceeds that required for meshing as the number of nodes increases. This situation has been found in all the problems solved with the PFEM. As a general rule, for large 3D problems (over 500000 nodes) meshing consumes around 20% of the total CPU time for each time step, while the solution of the

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Fig. 8 Identification of individual particles (or a group of particles) starting from a given collection of nodes

equations and the assembly of the system consume approximately 65 and 15% of the CPU time for each time step, respectively. These figures prove that the generation of the mesh has an acceptable cost in the PFEM.

6 Identification of Boundary Surfaces One of the main tasks in the PFEM is the correct definition of the boundary domain. Boundary nodes are sometimes explicitly identified. In other cases, the total set of nodes is the only information available and the algorithm must recognize the boundary nodes. In our work we use an extended Delaunay partition for recognizing boundary nodes. Considering that the nodes follow a variable h(x) distribution, where h(x) is typically the minimum distance between two nodes, the following criterion has been used. All nodes on an empty sphere with a radius greater than αh, are considered as boundary nodes. In practice α is a parameter close to, but greater than one. Values of α ranging between 1.3 and 1.5 have been found to be optimal in all examples analyzed. This criterion is coincident with the Alpha Shape concept [10]. Figure 8 shows an example of the boundary recognition using the Alpha Shape technique. Once a decision has been made concerning which nodes are on the boundaries, the boundary surface is defined by all the polyhedral surfaces (or polygons in 2D) having all their nodes on the boundary and belonging to just one polyhedron. The method described also allows one to identify isolated fluid particles outside the main fluid domain. These particles are treated as part of the external boundary where the pressure is fixed to the atmospheric value. We recall that each particle is a material point characterized by the density of the solid or fluid domain to which it belongs. The mass which is lost when a boundary element is eliminated due to departure of a node (a particle) from the main analysis domain is again regained when the “flying” node falls down and a new boundary element is created by the Alpha Shape algorithm (Figures 2 and 8).

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Fig. 9 Automatic treatment of contact conditions at the fluid-wall interface

The boundary recognition method above described is also useful for detecting contact conditions between the fluid domain and a fixed boundary, as well as between different solids interacting with each other. The contact detection procedure is detailed in the next section. We note that the main difference between the PFEM and the classical FEM is just the remeshing technique and the identification of the domain boundary at each

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Fig. 10 Contact conditions at a solid-solid interface

time step. The rest of the steps in the computation are coincident with those of the classical FEM.

7 Treatment of Contact Conditions in the PFEM 7.1 Contact between the Fluid and a Fixed Boundary The motion of the solid is governed by the action of the fluid flow forces induced by the pressure and the viscous stresses acting at the common boundary F S , as mentioned above. The condition of prescribed velocities at the fixed boundaries in the PFEM are applied in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. Contact between the fluid particles and the fixed boundaries is accounted for by the incompressibility condition which naturally prevents the fluid nodes to penetrate into the solid boundaries (Figure 9). This simple way to treat the fluid-wall contact at mesh generation level is a distinct and attractive feature of the PFEM.

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7.2 Contact between Solid-Solid Interfaces The contact between two solid interfaces is simply treated by introducing a layer of contact elements between the two interacting solid interfaces. This layer is automatically created during the mesh generation step by prescribing a minimum distance (hc ) between two solid boundaries. If the distance exceeds the minimum value (hc ) then the generated elements are treated as fluid elements. Otherwise the elements are treated as contact elements where a relationship between the tangential and normal forces and the corresponding displacement is introduced so as to model elastic and frictional contact effects in the normal and tangential directions, respectively (Figure 10). This algorithm has proven to be very effective and it allows to identifying and modeling complex frictional contact conditions between two or more interacting bodies moving in water in an extremely simple manner. Of course the accuracy of this contact model depends on the critical distance mentioned above. This contact algorithm can also be used effectively to model frictional contact conditions between rigid or elastic solids in standard structural mechanics applications. Figures 11–14 show examples of application of the contact algorithm to the bumping of a ball falling in a container, the failure of an arch formed by a collection of stone blocks under a seismic loading and the motion of five tetrapods as they fall and slip over an inclined plane, respectively. The images in Figures 11 and 14 show explicitely the layer of contact elements which controls the accuracy of the contact algorithm.

8 Modeling of Bed Erosion Prediction of bed erosion and sediment transport in open channel flows are important tasks in many areas of river and environmental engineering. Bed erosion can lead to instabilities of the river basin slopes. It can also undermine the foundation of bridge piles thereby favouring structural failure. Modeling of bed erosion is also relevant for predicting the evolution of surface material dragged in earth dams in overspill situations. Bed erosion is one of the main causes of environmental damage in floods. Bed erosion models are traditionally based on a relationship between the rate of erosion and the shear stress level [22, 48]. The effect of water velocity on soil erosion was studied in [42]. In a recent work we have proposed an extension of the PFEM to model bed erosion [39]. The erosion model is based on the frictional work at the bed surface originated by the shear stresses in the fluid. The resulting erosion model resembles Archard law typically used for modeling abrasive wear in surfaces under frictional contact conditions [1, 32, 43]. The algorithm for modeling the erosion of soil/rock particles at the fluid bed is the following:

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Fig. 11 Bumping of a ball within a container. The layer of contact elements is shown

1. Compute at every point of the bed surface the resultant tangential stress τˆ induced by the fluid motion. In 3D problems τˆ = (τs2 + τt )2 where τs and τt are the tangential stresses in the plane defined by the normal direction n at the bed node. The value of τˆ for 2D problems can be estimated as follows: τt = µγt with

(15a)

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Fig. 12 Failure of an arch formed by stone blocks under seismic loading

γt =

vk 1 ∂vt = t 2 ∂n 2hk

(15b)

where vtk is the modulus of the tangential velocity at the node k and hk is a prescribed distance along the normal of the bed node k. Typically hk is of the order of magnitude of the smallest fluid element adjacent to node k (Figure 15). 2. Compute the frictional work originated by the tangential stresses at the bed surface as t t k 2 µ vt τt γt dt = dt (16) Wf = hk ◦ ◦ 4

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Fig. 13 Motion of five tetrapods on an inclined plane

Eq. (17) is integrated in time using a simple scheme as n

Wf = n−1 Wf + τt γt t

(17)

3. The onset of erosion at a bed point occurs when n Wf exceeds a critical threshold value Wc defined empirically according to the specific properties of the bed material. 4. If n Wf > Wc at a bed node, then the node is detached from the bed region and it is allowed to move with the fluid flow, i.e. it becomes a fluid node. As a consequence, the mass of the patch of bed elements surrounding the bed node

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Fig. 14 Detail of five tetrapods on an inclined plane. The layer of elements modeling the frictional contact conditions is shown

Fig. 15 Modeling of bed erosion by dragging of bed material

vanishes in the bed domain and it is transferred to the new fluid node. This mass is subsequently transported with the fluid. Conservation of mass of the bed particles within the fluid is guaranteed by changing the density of the new fluid node so that the mass of the suspended sediment traveling with the fluid equals the mass

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originally assigned to the bed node. Recall that the mass assigned to a node is computed by multiplying the node density by the tributary domain of the node. 5. Sediment deposition can be modeled by an inverse process to that described in the previous step. Hence, a suspended node adjacent to the bed surface with a velocity below a threshold value is assigned to the bed surface. This automatically leads to the generation of new bed elements adjacent to the boundary of the bed region. The original mass of the bed region is recovered by adjusting the density of the newly generated bed elements. Figure 15 shows a schematic view of the bed erosion algorithm proposed.

9 Modelling and Simulation of Excavation and Wear of Rock Cutting Tools The PFEM has been successfully applied for modelling excavation processes in civil and mining engineering. The method can also accurately predict the wear of the rock cutting tools during the excavation. The process to model surface erosion and tool wear during excavation follows the lines explained for modelling soil erosion in river beds (Section 8). Material is removed from the excavation front or the tool surface when the work of the frictional forces at the rock/soil-tool interface exceeds a prescribed value. A new boundary is defined with the volume that remains in the analysis domain using the alpha-shape approach as it is typical in the PFEM (Section 6). The surface properties control the wear occurring during the frictional contact. Mass loss in a cutting tool and the amount of excavated material that is extracted by the machine is modeled via a wear rate function. When a steady state position in the wear mechanism is reached, wear rate is described by a linear Archard-type equation [1, 5, 43] as fn s (18) Vw = K H where Vw is the volume loss of the material along the contact surface due to wear, s (m) is the sliding distance, fn is the normal force vector to the contact surface and H is the hardness of the material. Constant K is a non-dimensional wear coefficient which depends on the relative contribution of the body under abrasion, adhesion and wear processes [5, 43]. In the PFEM each node on the contact surface has a mesh of elements associated to it. The volume of material wear is compared with the volume associated to each contact node. When both volumes coincide, the node is released and all the elements associated to it are eliminated. The incremental equation for updating the volume loss due to wear at a node is as follows: Vwt +t = Vwt + K

fn (vt · t) H

(19)

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Fig. 16 Removing material and boundary update in an excavation process

where all variables are nodal variables, vt is the relative tangent velocity between the contact surfaces and t is the time step. When the volume of worn material associated to a node and the volume of material are the same, the node is released. Elements that contain the released node are eliminated in the next time step. Some particles are also eliminated and hence the global volume of the problem changes. The historical value of the variables in these particles is lost as these particles do not contribute to the system anymore. A scheme of the geometry updating process is shown in Figure 16. The remeshing process allows the boundary recognition and the update of the analysis domain due to excavation. The geometry of the domain is changed at each time step as excavation moves forward. The flowchart for solving an excavation problem with the PFEM using an updated Lagrangian approach and an implicit integration scheme is the following: 1. Read initial geometrical, mechanical and kinematic conditions from a reference mesh. 2. Transfer the elemental variables to the particles (i.e. the nodes). 3. For each time step and each Newton iteration: • Compute internal forces and contact forces at nodes r := M¯as + gs + fc − fs where r is the residual force vector [52], fc is the contact force vector and the rest of the terms are defined in Eq. (14). • Compute displacement increments and update displacement values δ u¯ = A−1 r −→ t +t u¯ i+1 = t +t u¯ i + δ u¯ where A is the Jacobian matrix. Typically A=

1 M + KT + K c βt 2

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where β is a parameter of the Newmark scheme [52], KT is the tangent stiffness matrix of the solid mechanics problem accounting for material and nonlinear geometrical effects and Kc is the tangent stiffness matrix emanating from the contact forces [5, 6, 52]. 4. Compute internal variables, strains and stresses at integration points within each element. 5. Check convergence of Newton iterations. 6. Once the iterative solutions has converged • Update particle positions: t +t x = t x + t +t u¯ • Compute velocities (t +t v¯ ) and accelerations at particles (t +t a¯ ). • Transfer strains and stresses from elements to particles: t +t

σ p = t σ p + σ p

t +t

εp = t εp + εp

where (·)p denotes values at each particle. Note that the strain and stress history is stored at the particles. • Update constitutive law parameters. 7. Check damage and erosion (wear) on particles. Remove eroded particles from the excavation front and worn particles from cutting tools. 8. Boundary recognition via the alpha shape method. Create new mesh. Update problem dimensions if the number of particles has changed. 9. Identify interface elements for contact. 10. Initiate solution for next time step. A detailed description of above algorithm, together with many applications, can be found in [5, 6].

10 Examples 10.1 Rigid Objects Falling into Water The analysis of the motion of submerged or floating objects in water is of great interest in many areas of harbour and coastal engineering and naval architecture among others. Figure 17 shows the penetration and evolution of a cube and a cylinder of rigid shape in a container with water. The colours denote the different sizes of the elements at several times. In order to increase the accuracy of the FSI problem smaller size elements have been generated in the vicinity of the moving bodies during their motion (Figure 18).

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Fig. 17 2D simulation of the penetration and evolution of a cube and a cylinder in a water container. The colours denote the different sizes of the elements at several times

Fig. 18 Detail of element sizes during the motion of a rigid cylinder within a water container

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Fig. 19 Evolution of a water column within a prismatic container including a vertical cylinder

10.2 Impact of Water Streams on Rigid Structures Figure 19 shows an example of a wave breaking within a prismatic container including a vertical cylinder. Figure 20 shows the impact of a wave on a vertical column sustained by four pillars. The objective of this example was to model the impact of a water stream on a bridge pier accounting for the foundation effects.

10.3 Dragging of Objects by Water Streams Figure 21 shows the effect of a wave impacting on a rigid cube representing a vehicle. This situation is typical in flooding and Tsunami situations. Note the layer of contact elements modeling the frictional contact conditions between the cube and the bottom surface.

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Fig. 20 Impact of a wave on a prismatic column on a slab sustained by four pillars

10.4 Impact of Sea Waves on Piers and Breakwaters Figure 22 shows the 3D simulation of the interaction of a wave with a vertical pier formed by a collection of reinforced concrete cylinders. Figure 23 shows the simulation of the falling of two tetrapods in a water container. Figure 24 shows the motion of a collection of ten tetrapods placed in the slope of a breakwaters under an incident wave. Figure 25 shows a detail of the complex three-dimensional interactions between water particles and tetrapods and between the tetrapods themselves. Figures 26 and 27 show the analysis of the effect of breaking waves on two different sites of a breakwater containing reinforced concrete blocks (each one of 4 × 4 m). The figures correspond to the study of Langosteira harbour in A Coruña, Spain using PFEM. Figure 28 displays the effect of an overtopping wave on a truck circulating by the perimetral road of the harbour adjacent to the breakwater.

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Fig. 21 Dragging of a cubic object by a water stream

10.5 Soil Erosion Figure 29 shows a very illustrative example of the potential of the PFEM to model soil erosion in free surface flows. The example represents the erosion of an earth dam under a water stream running over the dam top. A schematic geometry of the dam has been chosen to simplify the computations. Sediment deposition is not considered in the solution. The images show the progressive erosion of the dam until the whole dam is dragged out by the fluid flow [39]. Figure 30 shows the capacity of the PFEM to modelling soil erosion, sediment transport and material deposition in a river bed. The soil particles are first detached from the bed surface under the action of the jet stream. Then they are transported by the flow and eventually fall down due to gravity forces and are deposited on the bed surface at a downstream point. Figure 31 shows the progressive erosion of the unprotected part of a break water slope in the Langosteira harbour in A Coruña, Spain. Note that the upper shoulder zone not protected by the concrete blocks is progressively eroded under the action of the sea waves. Figure 32 displays the progressive erosion and dragging of soil particles in a river bed adjacent to the foot of bridge pile due to a water stream (water is not shown in

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Fig. 22 Interaction of a wave with a vertical pier formed by reinforced concrete cylinders

Fig. 23 Motion of two tetrapods falling in a water container

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Fig. 24 Motion of ten tetrapods on a slope under an incident wave

Fig. 25 Detail of the motion of ten tetrapods on a slope under an incident wave. The figure shows the complex interactions between the water particles and the tetrapods

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Fig. 26 Effect of breaking waves on a breakwater slope containing reinforced concrete blocks. Detail of the mesh of 4-noded tetrahedra near the slope at two different times

Fig. 27 Study of breaking waves on the edge of a breakwater structure formed by reinforced concrete blocks

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Fig. 28 Effect of an overtopping wave on a truck passing by the perimetral road of a harbour adjacent to the breakwater

Fig. 29 Erosion of a 3D earth dam due to an overspill stream

the figure). Note the disclosure of the bridge foundation due to the removal of the adjacent soil due to erosion.

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Fig. 30 Erosion, transport and deposition of particles at a river bed due to a jet stream

10.6 Melting, Spread and Burning of Polymer Objects in Fire We show an application of the PFEM for simulating an experiment performed at the National Institute for Stanford and Technology (NIST) in which a slab of polymeric material is mounted vertically and exposed to uniform radiant heating on one face. It is assumed that the polymer melt flow is governed by the equations of an incompressible fluid with a temperature dependent viscosity. A quasi-rigid behaviour of the polymer object at room temperature is reproduced by using a very high value of the viscosity parameter. As temperature increases in the thermoplastic object due to heat exposure, the viscosity decreases in several orders of magnitude as a function of temperature and this induces the melt and flow of the particles in the heated zone. Polymer melt is captured by a pan below the sample. A rectangular polymeric sample of dimensions 10 cm high by 10 cm wide by 2.5 cm thick is mounted upright and exposed to uniform heating on one face from a radiant cone heater placed on its side (Figure 33). The sample is insulated on its lateral and rear faces. The melt flows down the heated face of the sample and drips onto a surface below. Measurements include the mass of polymer remaining in the sample, and the mass of polymer falling onto the catch surface [4]. Figure 33 shows all three curves of viscosity vs. temperature for the polypropylene type PP702N, a low viscosity commercial injection molding resin formulation. The relationship used in the model, as shown by the black line, connects the curve for the undegraded polymer to points A and B extrapolated from the viscosity curve for each melt sample to the temperature at which the sample was formed. The result

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Fig. 31 Erosion of unprotected part of a breakwater slope due to sea waves

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Fig. 32 Progressive erosion and dragging of soil particles in a river bed adjacent to the foot of a bridge pile due to a water stream. Water is not shown

Fig. 33 Polymer melt experiment. Viscosity vs. temperature for PP702N polypropylene in its initial undegraded form and after exposure to 30 and 40 kW/m2 heat fluxes. The black curve follows the extrapolation of viscosity to high temperatures

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Fig. 34 Polymer melt experiment. Evolution of the melt flow into the catch pan at t = 400, 550, 700 and 1000 s

is an empirical viscosity-temperature curve that implicitly accounts for molecular weight changes. The finite element mesh has 3098 nodes and 5832 triangular elements. No nodes are added during the course of the run. The addition of a catch pan to capture the dripping polymer melt tests the ability of the PFEM model to recover mass when a particle or set of particles reaches the catch surface. Heat flux is only applied to free surfaces above the midpoint between the catch pan and the base of the sample. However, every free surface is subject to radiative and convective heat losses. To keep the melt fluid, the catch pan is set to a temperature of 600 K. Figure 34 shows four snapshots of the melt flow into the catch pan. To test the ability of the PFEM to solve this type of problem in three dimensions, a 3D problem for flow from a heated sample was run. The same boundary conditions are used as in the 2D problem illustrated in Figure 33, but the initial dimensions of the sample are reduced to 10 × 2.5 × 2.5 cm. The initial size of the model is 22475 nodes and 97600 four-noded tetrahedra. The shape of the surface and temperature field at different times after heating begins are shown in Figure 35.

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Fig. 35 Simulation of a 3D polymer melt problem with the PFEM. Melt flow from a heated prismatic sample at different times

Although the resolution for this problem is not fine enough to achieve high accuracy, the qualitative agreement of the 3D model with 2D flow and the ability to carry out this problem in a reasonable amount of time suggest that the PFEM can be used to model melt flow and spread of complex 3D polymer geometry. Figure 36 shows results for the analysis of the melt flow of a triangular thermoplastic object into a catch pan. The material properties for the polymer are the same as for the previous example. The PFEM succeeds to predicting in a very realistic manner the progressive melting and slip of the polymer particles along the vertical wall separating the triangular object and the catch pan. The analysis follows until the whole object has fully melt and its mass is transferred to the catch pan. We note that the total mass was preserved with an accuracy of 0.5% in all these studies. Gasification, in-depth absorption or radiation were not taken into account in these analysis. More examples of application of the PFEM to the melting and dripping of polymers are reported in [41]. The PFEM has been recently extended for modelling the combined melting and burning of polymer objects under fire [21]. In [46] the surrounding air was induced

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Fig. 36 Melt flow of a heated triangular object into a catch pan

in the simulation and in [24] the equations governing the coupled thermal-flow problem were coupled to a combustion model governing the burning of combustible and the heat interchanges between the object and the air during combustion. Figure 37 shows a 2D application of the PFEM to the burning of a prismatic polymer object simulating a chair. The sequence of images shows the change of shape of the object as it burns, melts and drips on the floor surface and the intensity of the flame at different times.

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Fig. 37 Simulation of the burning, melting and dripping of a chair modelled as a 2D prismatic polymer object

10.7 Simulation of Excavation Process and Wear of Rock Cutting Tools 10.7.1 Disc Cutting of a Ground Section The first example is an elastic cutting disc in 2D acting against a solid wall. The disc has an imposed rotation in order to generate friction when contacting with the solid wall. The material is modelled with a simple damage law. The problem is solved first for the case of a soft wall material. Figure 38 shows that contact is detected when the disc comes near the wall. An interface mesh of contact elements is generated and it anticipates the contact area. The contacting forces are transmitted thought the contact elements to each domain. This interaction damages the solid wall until it crashes. Contact forces are computed at the axis of the disc in order to yield force and momentum reactions. The mesh is coarse so as to show better the process and the contact interface mesh. In a fine mesh contact elements are quite small and are difficult to visualize. It can be seen how as contact forces erode the wall, the excavated particles are taken away from the model. This generates a hollow in the surface while at the same time the material experiences large deformations. Figures 39 and 40 show a similar examples of excavation of a soft soil mass with rotating discs. Figure 41 displays the action of a rotating disc on a stiff wall. Note the change in the pattern of the excavation front and the progressive wear of the disc surface.

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Fig. 38 Simulation of a disc excavating a soft wall with the PFEM

10.7.2 Roadheader Penetrating in the Ground The next example is the simulation of a roadheader digging a portion of ground. This is an illustrative example of the capability of the PFEM for modeling ground excavation and wear of the cutting tools at the same time. The results are shown in Figure 42. A rotation and a displacement have been imposed to the roadheader. Note that contact elements only appear in the contact zone. The cone that models the roadheader loses material at the tip due to wear. Ground geometry suffers big changes during the simulation. Remeshing and detection of the boundary via the alpha-shape technique are crucial for capturing the fast and drastic changes of the domain boundary.

10.7.3 Simulation of an Excavation with a TBM Figures 43–45 show a simulation of a tunneling process with a TMB (Tunnel Boring Machine) acting on a 3D soil/rock domain. This example evidences the capability of

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Fig. 39 Example of application of the PFEM to the excavation of a soft soil mass with a rotating disc

Fig. 40 Simulation of the excavation of a soft soil mass with a rotating gear disc with the PFEM. Contour of the modulus of the acceleration vector in the soil at two instances

the PFEM to model complex excavation settings. The discretization of the TMB and the soil/rock region is displayed in Figure 43. Figure 44 shows an overview of the simulation as the tunneling process advance and the stress contour lines and Figure 45 shows the wear of the rock cutting discs in the TBM induced by the excavation forces. Far away from the rotating axis the displacement is bigger for the same rotation velocity and it generates larger friction forces at the edges of the tunneling head. The previous examples illustrate the good capabilities of the PFEM for modelling ground excavation processes.

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Fig. 41 Simulation of the excavation of a stiff rock wall with the PFEM. Note the change of the rotating disc edge due to wear

Fig. 42 Simulation of an excavation with a roadheader using the PFEM. Note the geometry change in the roadheader tip due the wear

11 Conclusions The particle finite element method (PFEM) is a powerful computational technique for solving coupled problems in engineering, involving fluid-structure interaction, large motion of fluid or solid particles, surface waves, water splashing, separation

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Fig. 43 Simulation of a tunneling process with a TBM using the PFEM. Discretization of soil mass and TBM geometry with 4-noded tetrahedra

of water drops, frictional contact situations, bed erosion, coupled thermal flows, melting, dripping and burning of objects, etc. The success of the PFEM lies in the accurate and efficient solution of the equations of fluid and of solid mechanics using an updated Lagrangian formulation and a stabilized finite element method, allowing the use of low order elements with equal order interpolation for all the variables. Other essential solution ingredients are the identification of the domain boundaries via the Alpha Shape technique and the efficient regeneration of the finite element mesh at each time step, and the algorithm to treat frictional contact conditions at fluid-solid and solid-solid interfaces via mesh generation. The examples presented have shown the potential of the PFEM for solving a wide class of coupled problems in engineering. Examples of validation of the PFEM results with data from experimental tests are reported in [23].

Appendix The matrices and vectors in Eqs. (8)–(11) for a 4-noded tetrahedron are: T ρNi Nj dV , Kij = BTi DBj dV Mij = Ve

Gij =

Ve

Ve

BTi mNj dV , fi = Lij =

Q = [Q1 , Q2 , Q3 ]

Ve

Ve

NTi bdV +

∇ T Ni τ ∇Nj dV ,

∇=

,

[Qk ]ij =

τ Ve

e

ˆ ij = NTi td , M

∂ ∂ ∂ , , ∂x1 ∂x2 ∂x3

∂Ni Nj dV , ∂xk

T

Ve

τ NTi Nj dV

m = [1, 1, 1, 0, 0, 0]T

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Fig. 44 Simulation of a tunneling operation with a TBM using the PFEM

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ B = [B1 , B2 , B3 , B4 ]; Bi = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

∂Ni ∂x 0

⎤ 0 ∂Ni ∂y

0 0 ∂Ni ∂z 0

0 ∂Ni ∂y

0 ∂Ni ∂x

∂Ni ∂z

0

∂Ni ∂x

0

∂Ni ∂z

∂Ni ∂y

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

D=µ

2I3 0 0 I3

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Fig. 45 Wear of the rock cutting discs in a TBM during the simulation of a tunneling operation using the PFEM. Circles denote worn cutting discs

N = [N1 , N2 , N3 , N4 ], Ni = Ni I3 , I3 : 3 × 3 unit matrix Cij = ρcNi Nj dV , Hij = ∇T Ni [k]∇Nj dV ⎡

Ve

Ve

⎤

k1 0 0 [k] = ⎣ 0 k2 0 ⎦ , 0 0 k3

qi =

Ve

Ni QdV −

qe

Ni qn d

In the above equations indexes i, j run from 1 to the number of element nodes (4 for a tetrahedron), qn is the heat flow prescribed at the external boundary q , t is the surface traction vector t = [tx , ty , tz ]T and V e and e are the element volume and the element boundary, respectively.

Acknowledgements Thanks are given to Mrs. M. de Mier for many useful suggestions. This research was partially supported by project SEDUREC of the Consolider Programme of the Ministerio de Educación y Ciencia (MEC) of Spain, project XPRES of the National I+D Programme of MEC (Spain) and projects REALTIME and SAFECON of the European Research Council (ERC). Thanks are also given to the Spanish construction company Dragados for financial support for the study of harbour engineering and tunneling problems.

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45. Ryzhakov, P.B., Rossi, R., Idelsohn, S., Oñate, E., A monolithic Lagrangian approach for fluid-structure interaction problems. Journal of Computational Mechanics 46(6):883–899, 2010. 46. Ryzhakov, P.B., Rossi, R., Oñate, E., An algorithm for polymer melting simulation. In: Conference Proceedings METNUM-2009, Barcelona, Spain, 29 June–2 July, 2009. 47. Tezduyar, T.E., Finite element method for fluid dynamics with moving boundaries and interface. In: Encyclopedia of Computational Mechanics, E. Stein, R. de Borst, T.J.R. Hughes (eds.), vol. 3, pp. 545–578, John Wiley, 2004. 48. Wan, C.F., Fell, R., Investigation of erosion of soils in embankment dams. Journal of Geotechnical and Geoenvironmental Engineering 130:373–380, 2004. 49. Zienkiewicz, O.C., Taylor, R.L., Nithiarasu, P., The Finite Element Method for Fluid Dynamics, Elsevier, 2006. 50. Zienkiewicz, O.C., Jain, P.C., Oñate, E., Flow of solids during forming and extrusion: Some aspects of numerical solutions. International Journal of Solids and Structures 14:15–38, 1978. 51. Zienkiewicz, O.C., Oñate, E., Heinrich, J.C., A general formulation for the coupled thermal flow of metals using finite elements. International Journal for Numerical Methods in Engineering 17:1497–1514, 1981. 52. Zienkiewicz, O.C., Taylor, R.L., The Finite Element Method for Solid and Structural Mechanics, Elsevier, 2005. 53. Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z., The Finite Element Method. Its Basis and Fundamentals, Elsevier, 2005.

Advances in Computational Modelling of Multi-Physics in Particle-Fluid Systems Y.T. Feng, K. Han and D.R.J. Owen

Abstract The current work presents the recent advances in computational modelling strategies for effective simulations of multi physics involving fluid, thermal and magnetic interactions in particle systems. The numerical procedures presented comprise the Discrete Element Method for simulating particle dynamics; the Lattice Boltzmann Method for modelling the mass and velocity field of the fluid flow; the Discrete Thermal Element Method and the Thermal Lattice Boltzmann Method for solving the temperature field. The coupling of the fields is realised through hydrodynamic and magnetic interaction force terms. Selected numerical examples are provided to illustrate the applicability of the proposed approach.

1 Introduction In recent years the modelling of coupled field problems, in which two or more physical fields contribute to the system response, has become a focus of major research activity. Among them, the quantitative study of fluid, thermal and magnetic interactions in particulate systems encountered in many engineering applications is of fundamental importance. For instance, the mineral recovery operation in the mining industry employs a suction process to extract rock fragments from the ocean or river bed. The computational modelling of this particle transport problem requires a fluid-particle interaction simulation. The motion of the particles is driven collectively by the gravity and the hydrodynamic forces exerted by the fluid, and may also be altered by the interaction between the particles. On the other hand, the fluid flow pattern can be greatly affected by the presence of the particles, and is often of a turbulent nature. In the nuclear industry, the process of a pebble bed nuclear reactor essentially involves the forced flow of gas through uranium enriched spheres that Y.T. Feng · K. Han · D.R.J. Owen Civil and Computational Engineering Centre, School of Engineering, Swansea University, Swansea SA2 8PP, UK; e-mail: [email protected]

E. Oñate and R. Owen (eds.), Particle-Based Methods, Computational Methods in Applied Sciences 25, DOI 10.1007/978-94-007-0735-1_2, © Springer Science+Business Media B.V. 2011

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are cyclically fed through a concentric column in order to extract thermal energy. In this situation, the introduction of additional field, thermal (heat transfer between the moving particles in the form of conduction, convection, and radiation, as well as transfer of heat to the gas stream), poses even more computational challenges. Another example is the modelling of magnetorheological fluids (MR fluids). An MR fluid is a type of smart fluid, which consists of micron-sized magnetizable particles dispersed in a non-magnetic carrier fluid. In the absence of a magnetic field, the rheological behaviour of an MR fluid is basically that of the carrier fluid, except that the suspended magnetizable particles makes the fluid ‘thicker’. When subjected to an external magnetic field, the particles become magnetized and acquire a dipole moment. Due to magnetic dipolar interactions, the particles line up and form chainlike structures in the direction of the applied field. This change in the suspension microstructure significantly alters the rheological properties of the fluid. To model the particle chain formation and the rheological properties of the MR fluid under an applied magnetic field, the magnetic, hydrodynamic and contact interactions should be fully resolved. The fundamental physical phenomena involved in these systems are generally not well understood and often described in an empirical fashion, mainly due to the intricate complexity of the hydrodynamic, thermodynamic and magnetic interactions exhibited and the non-existence of high-fidelity modelling capability. The Discrete Element Method [5], among other discontinuous methodologies, has become a promising numerical tool capable of simulating problems of a discrete or discontinuous nature. In the framework of the Discrete Element Method, a discrete system is considered as an assembly of individual discrete objects which are treated as rigid and represented by discrete elements as simple geometric entities. The dynamic response of discrete elements depends on the interaction forces which can be short-ranged, such as mechanical contact, and/or medium-ranged, such as attraction forces in liquid bridges, and obey Newton’s second law of motion. By tracking the motion of individual discrete elements and handling their interactions, the dynamic behaviour of a discrete system can be simulated. Conventional computational fluid dynamic methods have limited success in simulating particulate flows with a high number of particles due to the need to generate new, geometrically adapted grids, which is a very time-consuming task especially in three-dimensional situations [10]. In contrast, the Lattice Boltzmann Method [2, 36] overcomes the limitations of the conventional numerical methods by using a fixed, non-adaptive (Eulerian) grid system to represent the flow field. In particular, it can efficiently model fluid flows in complex geometries, as is the case of particulate flow under consideration. A rich publication in recent years (see, for instance, [1,4,10,13,15,16,21,28,29,33] and the references therein) has proved the effectiveness of the method. If an additional field, thermal, is introduced to a particulate system, the Thermal Lattice Boltzmann Method [25] may be employed to model heat transfer between particles and between particles and the surrounding fluid. Our numerical tests show, however, that the Thermal Lattice Boltzmann Method is not efficient for simulating heat conduction in particles. For this reason, a novel numerical scheme, termed

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the Discrete Thermal Element Method [14] is put forward. In this approach, each particle is treated as an individual element with the number of (temperature) unknowns equal to the number of particles that it is in contact with. The element thermal conductivity matrix can be very effectively evaluated and is entirely dependent on the contact characteristics. This new element shares the same form and properties with its conventional thermal finite element counterpart. In particular, the entire solution procedure can follow exactly the same steps as those involved in the finite element analysis. Unlike finite elements or other numerical techniques, no discretisation errors are involved in the Discrete Thermal Element Method. The numerical validation against the finite element solution indicates that the solution accuracy of this scheme is reasonable and highly efficient in particular. The interaction problems considered is often of a dynamic and transient nature. Although the Discrete Thermal Element Method is capable of modelling the steadystate heat conduction in large particulate systems efficiently, it is not trivial to be extended to transient situations. Meanwhile, its formulation is not compatible with that of the Discrete Element Method which accounts for particle-particle interactions. Therefore the Discrete Thermal Element Method needs to be modified to realise thermal-particle coupling. The pipe-network model is such a modification [15], in which each particle is replaced by a thermal pipe-network connecting the particle’s centre with each contact zone associated with the particle. For numerical modelling of magnetorheological fluids, in addition to the above numerical techniques, the magnetic forces formed between magnetized particles under an externally applied magnetic field need to be properly accounted for. This turns out not to be an easy issue since the mutual The objective of this work is to present our recent developments [13–16, 23, 24] on all essential computational procedures for the effective modelling of the above mentioned multi-physics problems involving fluid, thermal and/or magnetic interactions in particulate systems. In what follows, the basic formulations of the Discrete Element Method, the Lattice Boltzmann Method, the Discrete Thermal Element Method, the magnetic interactions and the coupling techniques, will be outlined. Selected numerical examples are provided to illustrate the applicability of the proposed approach.

2 Particle-Particle Interactions – Discrete Element Approach Interactions between the moving particles are modelled by the Discrete Element Method [5], in which each discrete object is treated as a geometrically simplified entity that interacts with other discrete objects through boundary contact. At each time step, objects in contact are identified with a contact detection algorithm; and the contact forces are evaluated based on appropriate interaction laws. The motion of each discrete object is governed by Newton’s second law of motion. A set of governing equations is built up and integrated with respect to time, to update each

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object’s position, velocity and acceleration. The main building blocks of the discrete element procedure are described as follows.

2.1 Representation of Discrete Objects In the Discrete Element Method, discrete objects are treated as rigid and represented either by regular geometric shapes, such as disks, spheres and superquadrics, or by irregular geometric shapes, such as polygons, polyhedrons, clustering or clumping of regular shapes to form compound shapes. Circular and spherical elements are the most used discrerete elements due to their geometric simplicity, smooth and continuous boundary. Contact resolution for this type of element is therefore trivial and computationally efficient. However, idealising materials such as grains and concrete aggregates as perfect disks (or spheres) is not always realistic and may not produce correct dynamic behaviour. One of the reasons is that circular and spherical elements cannot provide resistance to rolling motion. This has led to the introduction of more sophisticated elements to represent the discrete system more realistically. Contrary to the circular and spherical elements where only the radius can be modified, polygonal elements (polygons or polyhedrons) offer increased flexibility in terms of shape variation. Since the boundary of this type of element is not smooth, some complex situations such as corner/corner contact, often arise in the contact resolution. Higher order discrete elements can be used, such as superquadrics and hyperquadrics as proposed in [37], which may represent many simple geometric entities (for instance, disk, sphere, ellipse and ellipsoid) within the framework. However, this mathematical elegancy may be offset by the expensive computation involved in the contact resolution. Preparation of an initial packing configuration of particles is a very important issue both practically and numerically. There is only limited work reported. See [8, 9] for a very effective packing of disks/ploygons, and for [19] for spheres.

2.2 Contact Detection In the discrete element simulation of problems involving a large number of discrete objects, as much as 60–70% of the computational time could be spent in detecting and tracking the contact between discrete objects. Due to a large diversity of object shapes, many efficient contact processing algorithms often adopt a two-phase solution strategy. The first phase, termed contact detection or global search, identifies the discrete objects which are considered as potential contactors of a given object. The second phase, termed contact resolution or local search, resolves the details of the contact pairs based on their actual geometric shapes.

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Some search algorithms used in general computing technology and computer graphics have been adopted for this purpose. Algorithms such as bucket sorting, heap sorting, quick sorting, binary tree and quadrant tree data structure all originated from general computing algorithms. However, applications of these algorithms in discrete element codes need modifications to meet the needs of particular discrete element body representations and the kinematic resolution. For the detection of potential contact between a large number of discrete elements, a spatial search algorithm based on space-cell subdivision and incorporating a tree data storage structure possesses significant computational advantages. For instance, the augmented spatial digital tree [6] is a spatial binary tree based contact detection algorithm. It uses the lower corner vertex to represent a rectangle in a binary spatial tree, with the upper corner vertex serving as the augmented information. The algorithm is insensitive to the size distributions of the discrete objects. Numerical experiments in [6] indicate that this search algorithm can reduce the CPU requirement of a contact detection from an originally demanding level down to a more acceptable proportion of the computing time. Another type of the contact detection algorithms is the so-called cell based search [31, 32]. The main procedures in these algorithms involve: (1) dividing the domain that the discrete objects occupy into regular grid cells; (2) mapping each discrete object to one of the grid cells; and (3) for each discrete object in a cell, checking for possible contacts with other objects in the same cell and in the neighbouring cells. Provided the number of cell columns and rows is significantly less than the number of discrete objects, it can be proved that the memory requirement for the dynamic cell search algorithm is O(N). Also for a fixed cell size the computational time Top may be expressed as Top = O(N + ) where represents the costs associated with the maintenance of various lists used in the algorithm. Numerical tests conducted in [22] show that the dynamic cell search algorithm is even more efficient than the tree based search algorithms for large scale problems.

2.3 Contact Resolution The identified pairs with potential contact are then kinematically resolved based on their actual shapes. The contact forces are evaluated according to certain constitutive relationship or appropriate physically based interaction laws. In general, the interaction laws describe the relationship between the overlap and the corresponding repulsive force of a contact pair. For rigid discrete elements, the interaction laws may be developed on the basis of the physical phenomena involved. The Hertz normal contact model that governs elastic contact of two spheres (assumed rigid in discrete element modelling) in the normal direction is such an example, in which the normal contact force, Fn , and the contact overlap, δ, has the following relation

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√ 4E ∗ (R∗ ) 3/2 Fn = δ 3 where

(1)

1 − ν12 1 − ν22 1 = + E∗ E1 E2 1 1 1 = + R∗ R1 R2

with R1 and R2 being the radii; E1 , E2 , and ν1 , ν2 are the elastic properties (Young’s modulus and Poisson’s ratio) of the two spheres. For irregularly shaped particles, such as polygons and polyhedra, the contact interaction models can be an serious issue where the normal direction may not be uniquely defined. Energy based contact models, proposed in [10] for polygons and [11] for polyhedra, provide an elegant solution to the problem. An application to superquadrics is proposed in [20]. For ‘wet’ particles the interaction laws may include the effects of a liquid bridge. In other cases, adhesion may be considered. Energy dissipation due to plastic deformation, heat loss and material damping etc during contact is taken into account by adding a viscous damping term in the governing equation. Friction is one of the fundamental physical phenomena involved in particulate systems. Although the search for a quantitative understanding of the features of friction has been in progress for several centuries, a universally accepted friction model has not yet been achieved. One difficulty is associated with the nature of the friction force near zero relative velocity, where a strong nonlinear behaviour is exhibited. The classic Coulomb friction law is usually employed in engineering applications for its simplicity. The discontinuous nature of the friction force in this model, however, imposes some numerical difficulties when the relative sliding velocity reverses its direction and/or during the transition from sliding (sticking) to sticking (sliding). The difficulties are usually circumvented by artificially introducing a ‘transition zoneŠ which smears the discontinuity in the numerical computation. Nevertheless, the suitability of any friction model should be carefully examined and the associated numerical issues fully investigated in order to correctly capture the physical phenomena involved. Proper considering rolling friction is another challenging issue and many numerical issues remain outstanding [7]. A comprehensive study of the contact interaction laws can be found in [17, 18].

2.4 Governing Equations and Time Stepping The motion of the discrete objects is governed by Newton’s second law of motion as Mu¨ + Cd u˙ = Fc (2) J θ¨ = Tc

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where M and Cd are respectively the mass and damping matrices of the system, u, u˙ and u¨ are respectively the displacement, velocity and acceleration vectors, J is the moment of inertia, θ¨ the angular acceleration, Fc and Tc denote the contact force and torque, respectively. The configuration of the entire discrete system is evolved by employing an explicit time integration scheme. With this scheme, no global stiffness matrix needs to be formed and inverted, which makes the operations at each time step far less computationally intensive. However, any explicit time integration scheme is only conditionally stable. For a linear system the critical time step can be evaluated as tcr =

2

(3)

ωmax

where ωmax is the maximum eigenvalue of the system. However, the above result may not be valid since a contact system is generally nonlinear, as is demonstrated in [12]. To ensure a stable and reasonably accurate solution, the critical time step chosen should be much smaller than the value given in Eq. (3).

3 Fluid-Particle Interactions The interaction between fluid and particles is solved by a coupled technique: using the Lattice Boltzmann Method to simulate the fluid field, and the Discrete Element Method to model particle dynamics. The hydrodynamic interactions between fluid and particles are realised through an immersed boundary condition. The solution procedures are outlined as follows.

3.1 The Lattice Boltzmann Method In the Lattice Boltzmann Method, the problem domain is divided into regular lattice nodes. The fluid is modelled as a group of fluid particles that are allowed to move between lattice nodes or stay at rest. During each discrete time step of the simulation, fluid particles move to the nearest lattice node along their directions of motion, where they ‘collide’ with other fluid particles that arrive at the same node. By tracking the evolution of fluid particle distributions, the macroscopic variables, such as velocity and pressure, of the fluid field can be conveniently calculated from its first two moments. The lattice Boltzmann equation with a single relaxation time for the collision operator is expressed as fi (x + ei t, t + t) − fi (x, t) = −

1 eq fi (x, t) − fi (x, t) τ

(4)

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where fi is the density distribution function of the fluid particles with discrete veloeq city ei along the i-th direction; fi is the equilibrium distribution function; and τ is the relaxation time which controls the rate of approach to equilibrium. The left-hand side of Eq. (4) denotes a streaming process for fluid particles while the right-hand side models collisions through relaxation. In the widely used D2Q9 model [33], the fluid particles at each node move to their eight immediate neighbouring nodes with discrete velocities ei (i = 1, . . . , 8). eq The equilibrium distribution functions fi depend only on the fluid density, ρ, and velocity, v, which are defined in D2Q9 model as ⎧ 3 ⎪ eq ⎪ ⎨ f0 = ρ 1 − 2 v · v 2c (5) 3 9 3 ⎪ eq 2 ⎪ (i = 1, . . . , 8) ⎩ fi = wi ρ 1 + 2 ei · v + 4 (ei · v) − 2 v · v c 2c 2c in which c = x/t is the lattice speed with x and t being the lattice spacing and time step, respectively; wi is the weighting factor with w0 = 4/9, w1−4 = 1/9, w5−8 = 1/36. The macroscopic fluid variables, density ρ and velocity v, can be recovered from the distribution functions as ρ=

8

fi ,

ρv =

i=0

8

fi ei

(6)

i=1

The fluid pressure field p is determined by the following equation of state: p = cs2 ρ

(7)

where cs is termed the fluid speed of sound and is related to the lattice speed c by √ (8) cs = c/ 3 The kinematic viscosity, ν, of the fluid is implicitly determined by the model parameters, x, t and τ as 1 1 x 2 1 1 ν= τ− = τ− c x (9) 3 2 t 3 2 which indicates that the selection of these three parameters should be correlated to achieve a correct fluid viscosity. It can be proved that the lattice Boltzmann equation (4) recovers the incompressible Navier–Stokes equations to the second order in both space and time [2], which is the theoretical foundation for the success of the Lattice Boltzmann Method for modelling general fluid flow problems. However, since it is obtained by the linearised expansion of the original kinetic theory based Boltzmann equation, Eq. (4) is only valid for small velocities, or small ‘computational’ Mach number defined by

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Ma =

vmax c

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(10)

where vmax is the maximum simulated velocity in the flow. Generally smaller Mach number implies more accurate solution. It is therefore required that Ma 1 (11) i.e., the lattice speed c should be sufficiently larger than the maximum fluid velocity to ensure a reasonably accurate solution.

3.2 Incorporating Turbulence Model in the Lattice Boltzmann Equation As many fluid-particle interaction problems are turbulent in nature, a turbulence model should be incorporated into the lattice Boltzmann equation (4). The Large Eddy Simulation, amongst other turbulence models, solves large scale turbulent eddies directly but the smaller scale eddies using a sub-grid model. The separation of these scales is achieved through the filtering of the Navier–Stokes equations, from which the solutions to the resolved scales are directly obtained. Unresolved scales can be modelled by, for instance, the Smagorinsky sub-grid model [34] that assumes that the Reynolds stress tensor is dependent only on the local strain rate. Yu et al. [38] proposed to incorporate the Large Eddy Simulation in the lattice Boltzmann equation by including the eddy viscosity as 1 ˜ eq f˜i (x + ei t, t + t) = f˜i (x, t) − fi (x, t) − f˜i (x, t) τ∗

(12)

where f˜i and f˜i denote the distribution function and the equilibrium distribution function at the resolved scale, respectively. The effect of the unresolved scale motion is modelled through an effective collision relaxation time scale τt . Thus in Eq. (12) the total relaxation time equals eq

τ∗ = τ + τt where τ and τt are respectively the relaxation times corresponding to the true fluid viscosity ν and the turbulence viscosity ν∗ defined by a sub-grid turbulence model. Accordingly, ν∗ is given by 1 1 2 1 1 2 ν∗ = ν + νt = τ∗ − c t = τ + τt − c t 3 2 3 2 νt =

1 2 τt c t 3

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With the Smagorinsky model, the turbulence viscosity νt is explicitly calculated from the filtered strain rate tensor S˜ij = (∂j u˜ i + ∂i u˜ j )/2 and a filter length scale (which is equal to the lattice spacing x) as νt = (Sc x)2 Sˆ

(13)

where Sc is the Smagorinsky constant; and Sˆ the characteristic value of the filtered strain rate tensor S˜ S˜ij S˜ij Sˆ = i,j

An attractive feature of the model is that S˜ can be obtained directly from the second˜ of the non-equilibrium distribution function order moments, Q, S˜ =

Q˜ 2ρSc τ∗

(14)

in which Q˜ can be simply computed by the filtered density functions at the lattice nodes 8 eq Q˜ ij = eki ekj (f˜k − f˜k ) (15) k=1

where eki is the k-th component of the lattice velocity ei . Consequently Sˆ =

Qˆ 2ρSc τ∗

(16)

˜ with Qˆ the filtered mean momentum flux computed from Q ˆ = 2 Q Q˜ Q˜ ij ij

(17)

i,j

3.3 Hydrodynamic Forces for Fluid-Particle Interactions The modelling of the interaction between fluid and particles requires a physically correct ‘no-slip’ velocity condition imposed on their interface. In other words, the fluid adjacent to the particle surface should have identical velocity as that of the particle surface. Ladd [28] proposes a modification to the bounce-back rule so that the movement of a solid particle can be accommodated. This approach provides a relationship of the exchange of momentum between the fluid and the solid boundary nodes. It also assumes that the fluid fills the entire volume of the solid particle, or in other words, the particle is modelled as a ‘shell’ filled with fluid. As a result, both solid and fluid

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nodes on either side of the boundary surface are treated in an identical fashion. It has been observed, however, that the computed hydrodynamic forces may suffer from severe fluctuations when the particle moves across the grid with a large velocity. This is mainly caused by the stepwise representation of the solid particle boundary and the constant changing boundary configurations. To circumvent the fluctuation of the computed hydrodynamic forces with the modified bounce-back rule, Noble and Torczynski [30] proposed an immersed moving boundary method. In this approach, a control volume is introduced for each lattice node that is a x × x square around the node, as illustrated by the shadow area in Figure 1a. Meanwhile, a local fluid to solid ratio γ is defined, which is the volume fraction of the nodal cell covered by the particle as shown in Figure 1b. The lattice Boltzmann equation for those lattice nodes (fully or partially) covered by a particle is modified to enforce the ‘no-slip’ velocity condition as fi (x + ei t, t + t) = fi (x, t) −

1 eq (1 − β) fi (x, t) − fi + βfim τ

(18)

where β is a weighting function depending on the local fluid/solid ratio γ ; and fim is an additional term that accounts for the bounce back of the non-equilibrium part of the distribution function, computed by the following expressions: ⎧ ⎨ β = γ (τ −0.5) (1−γ )+(τ −0.5) (19) ⎩ f m = f (x, t) − f (x, t) + f eq (ρ, v ) − f eq (ρ, v) −i i b i i −i where −i denotes the opposite direction of i. The total hydrodynamic forces and torque exerted on a particle over n particlecovered nodes are summed up as m fi ei Ff = c x βn (20) n

Tf = c x

i

(x − xc ) × βn

n

fim ei

(21)

i

where xc is the coordinate of the particle center. With this approach, the computed hydrodynamic forces are sufficiently smooth, which is also confirmed in our previous numerical tests [13, 21].

3.4 Fluid and Particle Coupling Fluid and particle coupling at each time step is realised by first computing the fluid solution, and then updating the particle positions through the integration of the equations of motion given by

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(a) Control area of a node

(b) Nodal solid area fraction

Fig. 1 Immersed boundary scheme of Noble and Torczynski

ma + cd v = Fc + Ff + mg J θ¨ = Tc + Tf

(22)

where m and J are respectively the mass and the moment of inertia of the particle; θ¨ the angular acceleration; g the gravitational acceleration if considered; Ff and Tf are respectively the hydrodynamic force and torque; Fc and Tc denote the contact force and torque from other particles and/or boundary walls; cd is a damping coefficient and the term cd v represents a viscous force that accounts for the effect of all possible dissipation forces in the system. The static buoyancy force of the fluid is taken into account by reducing the gravitational acceleration to (1 − ρ/ρs ) g, where ρs is the density of a particle. This dynamic equation governing the evolution of the system can be solved by the central difference scheme. Some important computational issues regarding the solution are briefly discussed as follows: 1. Subcycling time integration. There are two time steps used in the coupled procedure, t for the fluid flow and tD for the particles. Since tD is generally smaller than t, it has to be reduced to ts so that the ratio between t and ts is an integer ns : t ts = (ns = t/tD + 1) (23) ns where · denotes an integer round-off operator. This basically gives rise to a socalled subcycling time integration for the discrete element part; in one step of the fluid computation, ns sub-steps of integration are performed for Eq. (22) using the time step ts ; whilst the hydrodynamic forces Ff and Tf are kept unchanged during the subcycling. 2. The dynamic equation in the lattice coordinate system. Since the lattice Boltzmann equation is implemented in the lattice coordinate system in this work, the dynamic equation (22) should be implemented in the same way. It can be de-

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rived that in the lattice coordinate system Eq. (22) takes the form of m¯ ¯ a + c¯d v¯ = F¯ c + F¯ f + m¯ ¯g ⎧ ¯ = m/ρs x 2 ⎨m a¯ = at/c; ⎩ c¯d = cxcd ;

where

(24)

v¯ = v/c g¯ = gt/c F¯ t = Ft /(ρ0 c2 x)

4 Thermal-Particle Interactions 4.1 Convective Heat Transfer If an additional field, thermal, exists, the Thermal Lattice Boltzmann Method is adopted to account for heat exchange between particles and between particles and the surrounding fluid. In the double-population model [25] , in addition to the evolution equation for fluid flow (Eq. (4)), an internal energy distribution function is also introduced to solve thermodynamics, as described by the following evolution equation: g¯i (x + ei t, t + t) − g¯i (x, t) = − where

τg 1 eq g¯i (x, t) − gi (x, t) − fi Zi τg + 0.5 τg + 0.5 (25)

0.5 eq f¯i = fi + (fi − fi ) τf g¯i = gi +

0.5 t eq (gi − gi ) + fi Zi τg 2

(26) (27)

in which gi is the internal energy distribution function with discrete velocity ei along eq the i-th direction; gi is the corresponding equilibrium distribution function; τg is the internal energy relaxation time which controls the rate of change to equilibrium. The term Zi = (ei − v) · [∂v/∂t + (ei · ∇)v] represents the effect of viscous heating and can be expressed as Zi =

(ei − v) · [v(x + ei t, t + t) − v(x, t)] t

For gas flow, the lattice speed c can be defined as c = 3RTm where R is the gas constant and Tm the average temperature.

(28)

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The internal energy equilibrium distribution functions gi are defined in the D2Q9 model as ⎧ 3(v · v) eq ⎪ ⎪ = w ρ − g 0 ⎪ 0 ⎪ 2c2 ⎪ ⎨ 3 3(e 9(ei · v)2 3(v · v) i · v) eq (i = 1, 2, 3, 4) (29) gi = wi ρ + + − 2 4 2 ⎪ 2 2c 2c 2c ⎪ ⎪ 2 ⎪ ⎪ ⎩ geq = wi ρ 3 + 6(ei · v) + 9(ei · v) − 3(v · v) (i = 5, 6, 7, 8) i 2 4 2 c 2c 2c in which wi are the weighting factors with the same values as defined in Section 3.1; and ρ denotes the internal energy. The internal energy per unit mass and heat flux q can be calculated from the zeroth and first order moments of the distribution functions as τg t t ei g¯i − ρv − fi Zi ; q = ei fi Zi ρ = g¯i − 2 2 τg + 0.5 (30) To evaluate the convective heat exchange between a solid particle and the surrounding fluid, the following approach is proposed in this work. Assume that a solid particle is mapped onto the lattice by a set of lattice nodes. The nodes on the boundary of the solid region are termed boundary nodes. If i is a link (or direction) between a boundary node and a fluid node, the convective heat exchange between the solid particle and the surrounding fluid can be evaluated as [g−i (x, t) − gi (x, t+ )] (31) q= i

where gi (x, t+ ) denotes the post collision distribution at the boundary node x, and −i is the opposite direction of i. Our numerical tests show that the Thermal Lattice Boltzmann Method can model natural or forced convection in particulate systems well, but is not efficient to simulate heat conduction between particles, particularly for systems comprising a large number of particles. For this reason, a novel numerical approach, termed the Discrete Thermal Element Method [14], is proposed, which is outlined in the following.

4.2 Conductive Heat Transfer in Particles Consider a circular particle of radius R in a particle assembly that is in contact with n neighboring particles, as shown in Figure 2a, in which heat is conducted only through the n contact zones on the boundary of the particle, and the rest of the particle boundary is fully insulated. A polar coordinate system (r, θ ) is established with the origin set at the centre of the particle. Each contact zone (assumed to be an arc) can be described by the position angle θ and the contact angle α in Figure 2b. In general situations the position angles are well spaced along the boundary and the

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qj

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qi Qj q j (θ )

q1

αj

αj

θj

θi +α i

qi (θ )

∂T =0 ∂n

Qi = R

∫

q(θ ) dθ

θi −α i

(resultant flux)

αi αi

θi ∂T =0 ∂n

qn

Qm

Qn

(b) Thermal particle representation

(a) A circular particle in an assembly

Fig. 2 Heat conduction in a simple particle system

contact angles αi are small. The position and contact angles of the n contact zones constitute the local element (contact) configuration of the particle. Furthermore, if the heat flux along the i-th contact zone is described by a (local) continuous function qi (θ ), then the heat flux on the boundary of the particle can be represented as qi (θ − θi ) θi − αi ≤ θ ≤ θi + αi (i = 1, . . . , n) q(θ ) = (32) 0 otherwise The heat flux equilibrium in the particle requires

2π

q(θ )dθ = 0

(33)

0

The temperature distribution T (r, θ ) within the particle domain = {(r, θ ) : 0 ≤ r ≤ R; 0 ≤ θ ≤ 2π} is governed by the Laplace equation as:

κT = 0 ∂T κ = q(θ ) ∂n

in on ∂

(34)

where κ is the thermal conductivity; ∂ denotes the boundary (circumference) of the particle; and ∂T /∂n is the temperature gradient along the normal direction to the boundary. Then the temperature at any point (r, θ ) ∈ can be expressed as T (r, θ ) = −

R 2πκ

0

2π

r 2 r q(φ) ln 1−2 cos(θ −φ)+ dφ+To , R R

where To is the temperature at the centre, i.e. To = T (0, 0).

(r, θ ) ∈ (35)

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The solutions (35) are in integral form which provide an explicit formulation to evaluate the temperature distribution over the particle when the input heat flux along the boundary is given. The temperature distribution along the i-th contact arc is given by Tci (θ )

n θ − φ − θj R αj =− qj (φ) ln sin dφ+To πκ 2 −αj

(θi −αi ≤ θ ≤ θi +αi )

j =1

(36) Define Ti and Qi respectively as the average temperature and the resultant flux on the i-th arc and further assume that qi (θ ) is constant. Then Ti can be obtained as Ti =

n j =1

−

Qj 4πκαi αj

or Ti =

n

αi −αi

αj −αj

θij + θ − φ ln sin dφdθ + To 2

hij Qj + To

(i = 1, . . . , n)

(37)

(38)

j =1

where hij = hj i

1 =− 4πκαi αj

αi −αi

αj −αj

θij + θ − φ ln sin dφdθ > 0 2

(39)

With the introduction of the particle (element) temperature vector Te = {T1 , . . . , Tn }T , the heat flux vector Qe = {Q1 , . . . , Qn }T , the particle (element) thermal resistance matrix He = {hij }n×n , and e = {1, . . . , 1}T , Eq. (38) can be expressed in matrix form as (40) Te − eTo = He Qe This is the heat conduction equation of the particle in terms of thermal resistance: the temperatures at the n contact zones, relative to the average temperature T0 , can be obtained when the fluxes Qe are known. The inverse form of Eq. (40) reads Ke (Te − eTo ) = Qe

(Ke = H−1 e )

(41)

In both Eqs. (40) and (41), the average temperature To can be treated as a unknown internal variable which can be obtained by a linear combination of the discrete boundary temperature Te as To = gTe Te /κe

(ge = Ke e,

κe = eT Ke e)

(42)

Eliminating To from Eq. (41) based on relation (42), we have Ke Te = Qe where

(43)

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Ke = Ke − ge gTe /κe is the heat conductivity matrix of the particle. Equation (43) is the heat conduction equation in discrete form for the particle, which is termed the discrete thermal element. It has an identical form as a thermal finite element. Thus the subsequent procedure to model heat conduction in the particle system can follow the same procedure as those of the conventional finite element analysis. This discrete thermal element approach provides a simple and accurate heat conduction model for a circular particle in which the temperature field within the particle is fully resolved, which is a distinct advantage over the existing isothermal models. In the discrete thermal element, the temperature distribution in a particle is a linear superposition of the contributions from all the heat fluxes at the thermal contact zones. Specifically, the temperature at the i-th zone, Ti , depends not only on the flux Qi of the zone, but also on other fluxes Qj . This coupling effect is accounted for by the off-diagonal terms, hij , in the thermal resistance matrix He . The numerical evaluation conducted in [14] shows that a typical value of hij is about 10 times smaller than that of the diagonal terms hii , which implies that the coupling effect between different zones is fairly weak. This observation promotes the development of a simplified version of the discrete thermal element formulation, termed the pipe-network model, in [15]. In the pipe-network model, the off-diagonal terms in the thermal resistance matrix He is neglected such that ¯ e = diag{hii } H Then the original equations (40) are fully decoupled: Ti − To = hii Qi

(i = 1, . . . , n)

(44)

The resulting decoupled thermal equations can be conceptually represented by a simple star-shaped ‘pipe’ network model, as shown in Figure 3. For an individual pipe i, the corresponding thermal resistance Ri and conductivity ki are given by Ri = hii ;

ki = 1/Ri = 1/ hii

(45)

and Eq. (44) can be rewritten as ki (Ti − To ) = Qi

(i = 1, . . . , n)

(46)

In this model, To plays a central role. If no external heat source is applied, the net ! flux at the centre must equal zero due to the heat flux equilibrium requirement Qi = 0. Then Eq. (42) can be further simplified as To =

n i=1

(ki Ti )

n " i=1

ki

(47)

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Qi Tj Rj kj

ki

Rm

Rn km

Qm

Ti

Ri To

kn

Tm

Tn

Qn

Fig. 3 Pipe-network model

With the pipe-network model, the transient analysis can be readily performed. The governing equation for the transient heat conduct analysis of a solid is expressed as ρcp T˙ + κT = 0 (48) where ρ and cp are the density and the specific heat capacity of the solid, respectively; T˙ = ∂T /∂t with t being the time. Within the pipe-network framework, the corresponding discrete version of the transient equation (48) for the i-th particle can be expressed as Ci T˙io +

n

Qij = 0

(49)

j =1

where Qij are the internal heat fluxes associated with the particle defined by Qij = kij (Tjo − Tio )

(50)

and Ci is the total heat capacity of the particle, given by Ci = πρcp Ri2 The global system of equations can be assembled as CT˙ o (t) + Kg To (t) = Q(t)

(51)

where the global heat capacity matrix C = diag{Ci } is a diagonal matrix, Kg is the global stiffness matrix, and To = {T1o , . . . , Tmo }T is the average temperature vector of the particles. The system can be solved either explicitly or implicitly. The formulation of the pipe-network model is compatible with that of the Discrete Element Method, which makes the thermal and mechanical coupling possible.

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5 Fluid-Magnetic-Particle Interactions The fluid-magnetic-particle interaction phenomenon exists in MR fluids. Numerical simulations of MR fluids require an accurate and computationally efficient approach to fully account for magnetic, hydrodynamic and contact interactions. Firstly, the scheme to be employed should be able to effectively model contact phenomena between the magnetizable particles during the evolution of the magnetic microstructure. The Discrete Element Method described in Section 2 is suitable for this purpose. Secondly, the interaction between the magnetic particles and the carrier fluid can be effectively modelled by the Lattice Boltzmann Method outlined earlier. Special attention in this section will be given to the calculation of magnetic interactions. The full version of the modelling methodology is reported in [23] for 2D cases and [24] for 3D cases.

5.1 Magnetic Forces The magnetic interaction in an MR fluid can be treated as a magnetostatic problem, which is described by Laplace’s equation subject to appropriate boundary conditions. The magnetic forces are resolved by formulating the Maxwell stress tensor from the resultant field. The solution procedure is outlined below, based on the work of [26]. Let H and B denote the magnetic field intensity and flux density, respectively. For a linear isotropic medium with the magnetic permeability µ, H and B are related by the constitutive equation B = µH (52) Assume that the external magnetic field H0 is applied along the z direction with a magnitude H0 , i.e. H0 = H0 z, where z denotes the unit vector of the z-axis. If µp and µf represent, respectively, the magnetic permeability of the particles and fluid, then the relative susceptibility, χ, and effective susceptibility, χe , of the particles are defined as µp 3(χ − 1) χ= χe = µf χ +2 5.1.1 Fixed Dipole Model When an external magnetic field is applied, each particle in an MR fluid is magnetized and acquires a magnetic dipole moment m which, when ignoring the presence of the other particles, is m=

4πR3 3(χ − 1) H0 = Cp H0 ; 3 χ +2

m = |m| = Cp H0

(53)

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Fig. 4 Magnetic forces on dipole moment m2 from dipole moment m1

where Cp = Vp χe ; Vp = 4πR 3 /3 is the volume of the particle. Consider one particle with dipole moment m1 = m. The magnetic field produced by this dipole at any point (with a relative position vector r to the dipole) in space can be expressed as [35] H1 (m1 , r) =

1 3(m1 · rˆ )ˆr − m1 4π r3

(54)

where r = |r| and rˆ = r/r is the unit vector of r. The corresponding flux density B is calculated as (55) B1 (r) = µH1 (r) If a second particle of magnetic moment m2 = m is placed in the magnetic field of m1 as illustrated in Figure 4, the magnetic force, Fm , acting on the second dipole due to the first one can be determined by Fm (r) = ∇(m2 · B1 (r))

(56)

with r = x2 − x1 . This force can be expressed more conveniently in a spherical coordinate system (r, θ, ϕ) with θ and ϕ being the zenith and azimuth angles, respectively. Particularly, the component of the force in the azimuth angle ϕ is zero, and the radial and transversal components, Fn and Fτ , can be computed as Fn (r, θ ) = −

3µ m1 m2 3µ m1 m2 1 [3 cos2 θ − 1] = − [3 cos 2θ + 1] 4π r 4 4π r 4 2

and Fτ (r, θ ) = −

3µ m1 m2 sin 2θ 4π r 4

(57)

(58)

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Depending on the angle θ , the normal component Fn can be attractive (when θ < θc ) or repulsive (when θ > θc ), where the critical angle θc = 54.47◦. Equations (57) and (58) define the magnetic interaction between any two magnetized particles, which is basically the classic magnetic dipole model, or the fixed dipole model. Owing to its simplicity, this model has been commonly used in modelling MR-fluids, especially when a large number of particles are involved. The pairwise nature of the model also makes it suitable for use within the discrete element modelling framework. This fixed dipole model is accurate if the separation distance (gap) of two magnetized particles is larger than their diameter 2R [26], which suggests a cut-off distance to be used in the later magnetic interaction computation, rc = 4R

(59)

However, a large error will be introduced when the separation distance between two particles is less than their radius, r < R. This error arises mainly from the strong interaction between the two magnetized fields of the particles, and will reach maximum when the particles touch each other. The numerical investigation performed by [26] shows that for χ = 5, the fixed dipole model underestimates the maximum attraction force by around 35%, while overestimates the maximum repulsive force by 50% or more. The error will become more pronounced for larger susceptibility values.

5.1.2 Mutual Dipole Model The fixed dipole model discussed above assumes no interactions between the particles’ magnetized fields. In fact, the presence of other magnetized particles will increase the magnetization of a particle, thereby enhancing its dipole strength and its interactions with other particles. If the mutual magnetization between the particles are taken into account, the accuracy of the fixed dipole model may be improved. More specifically, each particle is still viewed as a point dipole but is subjected to an additional secondary magnetization from the other particles’ magnetized fields. Note that the magnetization due to the external field is termed the primary magnetization and the magnetization by other particles’ magnetized fields is termed the secondary magnetization. The mutually magnetized moment of particle i, mi , can be evaluated as mi = Cp [H0 + H(xi )] (i = 1, . . . , N)

(60)

where N is the total number of particles in the system, and H(xi ) is the total secondary magnetic field generated by all the other magnetized particles at the centre of particle i,

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H(xi ) =

N j =1,j =i

Hj (mj , rij ) =

N j =1,j =i

1 3ˆrij (mj · rˆ ij ) − mj 4π rij3

(61)

with rij = xi − xj ; rij = |rij |; rˆ ij = rij /rij . Equations (60) and (61) define a 3N × 3N linear system of equations with mi unknown variables. After all the magnetic moments are solved, the magnetic forces between the particles can be determined by the fixed dipole model using these total magnetization moments. This is the idea behind the so-called mutual dipole model [26]. Nevertheless, the computational cost associated with the solution of the linear system of equation (61) for systems involving a large number of particles can be substantial, and in particular, the solution needs to be performed at every time step of the simulation. In the present work, the classic Gauss–Seidel algorithm is employed to iteratively solve the equations. Let mki be the approximate values at the k-th iteration, and m0i be a given initial values. Then at the k + 1-th iteration mi is computed as ⎡ ⎤ i−1 N = Cp ⎣H0 + Hj (mk+1 Hj (mkj , rij )⎦ ; k = 0, 1, 2, . . . mk+1 i j , rij ) + j =1

j =i+1

(62) where mi at the previous step serves as the initial value for the current step. As the time step is usually very small, it is a very good initial value and thus the convergence of the iterative scheme is rapid. The numerical tests conducted have shown that the above scheme is very effective, and a solution accuracy of 10−5 can be generally achieved in no more than three iterations. Our numerical investigations show that using this mutual dipole model for two particles in contact, the upper limit of the maximum increased magnetic moment is 33.33% for a perfectly magnetized material (χ = ∞), which gives a 77.78% increase of the attraction force; while the upper limit of the maximum decreased magnetic moment is 11.11% which results in a 20.99% decrease of the repulsive force. The effect is even more significant for a longer chain of particles. However, the exact maximum force between two particles in contact is larger than that predicted by the mutual dipole model. Particularly, it is infinite when χ = ∞. Further improvement to the mutual dipole model has been undertaken by [26]. After the total magnetized moments are obtained, the force between any two particles is calculated by using the two-body exact solution, a special case of the general solution to multiple particle problems [3]. Although some improvement is achieved, the exact solution is still not obtained since the two-body solution is not exact in general multiple particle cases. More importantly, from a computational point of view, this version of the mutual dipole model loses its original simplicity as a result of the substantial computational cost involved in the incorporation of the two-body exact solution.

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In view of the difficulties discussed above, a better approach for improving the accuracy but retaining the computational simplicity of the fixed or mutual dipole model, as proposed by [27], is to use some empirical formulae to describe the magnetic interaction when the particles are close to each other. However, the procedure involves substantial pre-computations for different susceptibility values and different relative positions between two particles.

5.2 Fluid, Magnetic and Particle Coupling The fluid field is governed by the lattice Boltzmann equation (4) and the particle dynamics is accounted for by the Discrete Element Method. The coupling between the magnetized particles and the carrier fluid is realised through the hydrodynamic and magnetic interactions as Mu¨ + Cd u˙ = Fm + Ff + Fc

(63)

which is solved by the central difference algorithm.

6 Numerical Illustration To assess the applicability of the proposed approach, some numerical experiments will be performed in this section.

6.1 Example 1: Simulation of Particle Transport in a Vacuum Dredging System The combined Lattice Boltzmann and the Discrete Element procedure described in Section 3 is employed to model a vacuum dredging system for mineral recovery. This recovery operation uses a suction process to extract rock fragments. The system consists of a rigid pipe connected to a slurry transport system, which is typically powered by a gravel pump. The gravel is transported to the pipe entrance via hydraulic entrainment. The front view of the problem is illustrated in Figure 5, where the suction pipe has an internal diameter of 101 mm and a tube thickness of 16 mm. The gravel is initially confined to a cylindrical region called the gravel bed of 300 mm in diameter and 70 mm in depth. The gravel particles are made of quartz and are assumed to be spherical with diameter in the range of 6–12 mm. A total of 5086 particles are randomly packed using the packing algorithm developed in [19] with an initial porosity of approximately 50%. Full gravity (g = 9.81 m/s2 ) is applied. The

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Fig. 5 The front view of the problem geometry

fluid inside the suction pipe is water and is expected to be fully turbulent, thereby the Large Eddy Simulation based Smagorinsky turbulence model is adopted with the Smagorinsky constant Sc = 0.1. The following parameters are chosen: particle density ρs = 2650 kg/m3 , normal contact stiffness kn = 5 × 108 N/m, contact damping ratio ξ = 0.5 and time step factor λ = 0.1, which gives a time step of tD = 1.16 × 10−5 for the Discrete Element simulation of the particles. The fluid domain is divided into regular lattice with lattice spacing x = 2.5 mm. The fluid properties are those of water at room temperature, i.e. density ρ = 1000 kg/m3 and kinematic viscosity ν = 10−6 m2 /s. A complete simulation is achieved with τ = 0.50002. This gives a time step t = 4.17 × 10−5 s and thus the corresponding lattice speed c = 60 m/s. The boundary conditions are set as follows. Except for the bottom of the gravel bed which is a solid stationary wall, the others are flow boundaries. A constant pressure boundary condition with ρin = ρ is imposed at the inlet walls, and a smaller pressure with ρout = 0.975ρ is applied to the outlet of the pipe. The flow is therefore driven by the pressure difference between the inlet and outlet. The relevant laboratory test has also been performed. During the test, video footage is captured using a high speed digital camera. Image processing is used to provide an indication of the gravel velocity history during the test. The final excavation profile and gravel volume removed during the test are also recorded. Figure 6 shows the images of the gravel motion at the start, during and towards the end of the experiment and simulation, respectively. Of greater importance are the flow velocity at the pipe outlet, the total weight of the gravel particles removed and the excavation profile. The calculated values are compared with those observed from experimentation. The predicted average velocity on the exit plane of the suction pipe is approximately 0.99 m/s, which agrees

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(a) (b)

(c) Gravel motion during the test (d)

(e) (f) Fig. 6 Gravel motion at three stages of test and simulation: at the start of the test (a) and simulation (b); during the test (c) and simulation (d); towards the end of the test (e) and simulation (f)

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Depth [mm]

0 10 20 30 40

100

50

0 50 Radial Position from the Centre [mm]

100

(a) Excavation profile of the test

(b) Excavation profile of the simulation Fig. 7 Excavation profiles of the experiment and simulation

well with the measured value (1.05 m/s). A volume of 678341 mm3 of gravel is removed in the test, which weighs approximately 1.09 kg (assuming a bulk density of 1600 kg/m3 ), while 1110 particles, weighting 1.22 kg in total, is excavated in the simulation. The final excavation profiles, measured respectively from experimentation and simulation, at the gravel bed are illustrated in Figure 7. Note that the excavation profile for the simulation is obtained by a radial mapping of all the particles onto the cross section and then rotating about the central axis to create an axisymmetrical profile to facilitate the comparison with the experiment. The simulated maximum fluid velocity is vmax = 1.36 m/s at the pipe outlet (with the characteristic length L = 0.101 m). Thus the maximum Mach number and Reynolds number are therefore estimated as Ma =

vmax = 0.0226 c

vmax L = 137360 ν The Mach number indicates that the results obtained are reasonably accurate. It can be seen that the overall correspondence between numerical results and experimental measurements is good. Re =

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6.2 Example 2: Simulation of Heat Transfer in a Packed Bed The problem considered is a randomly packed particle bed of dimensions 0.5 × 1.0 m. The initial temperature of the 512 particles is set in the range of [20, 100]◦C. A hot gas (100◦ C) flow is introduced from the bottom of the bed throughout the simulation, while the other boundaries are fully insulated. Full gravity is applied. To simulate the velocity and temperature fields of this moving particle system with heat transfer, the coupled procedures proposed in this work can be adopted: the gas-particle interaction is modelled by the coupled Lattice Boltzmann and the Discrete Element Methods; while the gas-thermal and particle-thermal interactions are simulated jointly by the Thermal Lattice Boltzmann Method and the Discrete Thermal Element Method. The physical properties are chosen as: for particles, radius R = 1−2 mm, density ρs = 2500 kg/m3 , heat capacity cs = 150 J/kgK, thermal conductivity ks = 35 W/mK; whereas for the gas, density ρf = 1.0 kg/m3 , kinematic viscosity ν = 10−5 m2 /s, heat capacity cf = 1005 J/kgK and thermal conductivity kf = 0.024 W/mK. The fluid domain is divided into a 250 × 500 square lattice with lattice spacing x = 2 mm. The initial packing of the particles is generated using the packing algorithm proposed in [8]. Figures 8(a)–(d) and 9(a)–(d) show snapshot images of the velocity and temperature field evolution. It can be seen that the initially motionless particles start to move upwards when the hydrodynamic forces counteract the gravitational forces acting on the particles. when the velocity of the particles is low, most of the particles are in contacts and the mechanism of conductive heat transfer is significant. As the the particles moves away from each other, the convective transfer of the particles with the surrounding gas becomes dominant. Meanwhile, the particles close to the hot gas inlet (the bottom bed) get heated first and then move upwards, while the particles with lower temperatures move downwards to pick up heat. The circulation patterns of the particles and gas can be clearly seen from the pictures.

6.3 Example 3: Simulation of a Magnetorheological Fluid A two-stage numerical experiment will be performed for an MR fluid. The simulation involves, at the first stage, the microstructure evolution of the MR fluid with four different particle concentration fractions under the action of an applied magnetic field; and at the second stage, the application of the particle chains established at the first stage as the initial configuration to investigate the rheological properties of the MR fluid under different shear loading conditions. A representative volume element (RVE) of the MR fluid system to be investigated is chosen to be a rectangular cuboid domain, 0.2 × 0.05 × 0.05 mm, parallel to the axes of a Cartesian coordinate system. As shearing loadings will be applied in the x-direction at the second stage, this dimension is chosen to be the largest for achieving a higher accuracy. The RVE is filled with a Newtonian fluid of dynamic

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(a) t = 0 s

(b) t = 3 s

(c) t = 5 s

(d) t = 10 s

(e) t = 13 s

(f) t = 15 s

Fig. 8 Velocity contours at six time instants

viscosity η = 0.1 Pa·s and density ρf = 1000 kg/m3 in which spherical magnetizable particles are dispersed. The particles are randomly distributed with an identical radius R = 2 µm and density ρp = 7ρf . The Young modulus of the particles is set to E = 10 GPa, and the Poisson ratio is 0.3. The permeability of the fluid is that of a free space, i.e. µf = 4π × 10−7 N/A2 , whereas µp = 2000µf is chosen for the

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(a) t = 0 s

(b) t = 3 s

(c) t = 5 s

(d) t = 10 s

(e) t = 13 s

(f) t = 15 s

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Fig. 9 Temperature contours at six time instants

particles, corresponding to χ = 2000. An external uniform magnetic field is applied in the z-axis direction with a magnitude of H0 = 1.33 × 104 A/m.

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(a) t = 0 ms

(b) t = 12 ms

(c) t = 35 ms

(d) t = 46 ms

Fig. 10 Particle dynamics evolution: 5% volume fraction

(a) t = 0 ms

(b) t = 8 ms

(c) t = 19 ms

(d) t = 33 ms

Fig. 11 Particle dynamics evolution: 10% volume fraction

6.3.1 Particle Chain Formation The particle chain formation under the action of the external magnetic field is simulated for four different samples of the fluid with 5, 10, 20 and 30% particle volume fractions, which correspond to 746, 1492, 2984 and 4476 particles respectively. The evolution of the particle dynamics is solved in the context of the discrete element method in which the magnetic forces are described by the mutual dipole model; the hydrodynamic forces are approximated with the Stokes formula since the fluid field is not resolved at this stage; and the contact forces are evaluated by the Hertzian model. Periodic boundary conditions are imposed on the RVE for the particles in all the directions, i.e. if a particle moves out of the RVE from one end, it re-enters from

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(a) t = 0 ms

(b) t = 4 ms

(c) t = 12 ms

(d) t = 16 ms

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Fig. 12 Particle dynamics evolution: 20% volume fraction

(a) t = 0 ms

(b) t = 3 ms

(c) t = 7 ms

(d) t = 10 ms

Fig. 13 Particle dynamics evolution: 30% volume fraction

the other end. The time step size used in the central difference time integration is 1.36 × 10−8 s. The simulation is terminated when the system reaches a steady state. The dynamic evolution of the entire system is monitored by the history plot of the total kinetic energy of the particles. A suitably small value of the total kinetic energy indicates that a steady state is reached. With this information, the response time of the MR fluid system can also be identified. Figures 10–13 depict the microstructure evolution of the particles at four time instants for the four different particle concentrations. It can be seen that with the application of the magnetic field, the particles become magnetized and acquire a magnetic dipole moment. Due to dipolar interactions, the particles aggregate to form short fragmented chains. As time progresses, these short chains merge together to

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(a) 5% volume fraction

(b) 10% volume fraction

(c) 20% volume fraction

(d) 30% volume fraction

Fig. 14 Top views of final formed particle chains for four volume fractions

form longer chains that align in the direction of the applied magnetic field. Theoretically, the final chain structure corresponds to a (possibly local) minimum energy state. The particle concentration in the MR fluid has significant effect on the formed chain configurations. For a lower volume fraction, greater mobility of the particles results in straighter chains of single particle width aligned with the applied magnetic field; while for a higher volume fraction, less mobility of the particles make it difficult to form straight chains. Some of the chains tangle with other chains to form multiple particle width strands or clusters. This is illustrated in Figure 14, where the top views of the final particle chains are displayed. Figure 15 is the time history plot (truncated after 25 ms) of the averaged kinetic energy per particle for the four particle volume fractions. The kinetic energy rapidly reaches the peak value within a few milliseconds, indicating an active particle motion at the initial particle aggregation. The subsequent decrease of kinetic energy corresponds to a further growth of particle chains until the final stable configurations are achieved. The local spikes, notably for the lower volume fractions (5 and 10%), represent merging of shorter chains. There are far fewer spikes in the higher volume fractions (20 and 30%) which indicates a continuous formation/merging of the (shorter) chains. The simulations have establish that the times for the systems to approach a steady state, i.e. the response time, are approximately inversely proportional to the particle volume fractions, which are about 46, 33, 16 and 10 milliseconds respectively for 5, 10, 20 and 30% volume fractions. Clearly the steady state is reached faster for a higher volume concentration of the particles, as expected. Additional simulations have also been performed for two different intensities of the applied magnetic field, 0.5H0 and 2H0 . Except that a stronger (weaker) magnetic field results in a shorter (longer) response time, the final chain configurations are not much different, implying that the particle volume fraction plays a dominant role in the particle dynamic simulation. In particular, the mutual dipole model, though inaccurate when the particles are very close, may be sufficient if only the microstructure of the particle chains is of interest.

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Fig. 15 History plot of the kinetic energy per particle for four different particle volume fractions

6.3.2 Rheological Properties under Shear Loading As shown in the previous subsection, with the application of an external magnetic field, columnar particle chains are formed which are perpendicular to the direction of the fluid flow in the MR fluid. As a result, the fluid motion is largely restricted. This change in the suspension microstructure greatly alters the rheological properties of the fluid. To examine the MR effect, the following numerical tests are performed to establish the relationship between the applied shear loading and the resulting shear stress or viscosity under different magnetic field strengths. The steady-state particle chains established at the first stage simulation are applied as the initial configuration of the MR fluid system. In the following simulations, the MR fluid with 10% particle volume fraction investigated in the previous subsection is chosen. The combined LBM and DEM procedure is employed to fully resolve the fluid field and particle-fluid interaction. The fluid domain is divided into a 400 × 100 × 100 cubic lattice with lattice spacing h = 0.5 µm. The relaxation time is chosen to be τ = 0.75 to match the viscosity of the fluid, which leads to a time step for the fluid of 2.08 × 10−10 s and a lattice speed c = 2400 m/s. The same time step is also used for the particles. The maximum computational Mach number encountered for all the simulations is Ma = 0.006, which is much smaller than 1.0, therefore the numerical results are reasonably accurate. A constant horizontal velocity v0 in the positive x-direction is applied to the top surface of the RVE, and the equivalent shear rate is γ˙ = v0 /W with W = 0.05mm the height of the domain. By changing the value of v0 , different shear rates can be applied. Three types of fluid boundary conditions are applied: ‘no-slip’ for the bottom surface; slip for the front and back surfaces; and periodic for the left and the right surfaces. Special treatment is made to the shared edges and vertices of

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(a) t = 3 ms

(b) t = 5 ms

(c) t = 7 ms

(d) t = 11 ms

Fig. 16 Four snapshots in a shear mode simulation: H0 = 1.33 × 104 A/m, γ˙ = 240 s−1

the surfaces with different boundary conditions. A pure shear flow case (without particles) is tested to ensure a linear velocity distribution along the z direction and an identical fluid field for all the vertical planes parallel to the x–z plane. Due to the shear loading applied, the boundary conditions for the particles are slightly modified. The particles are restrained between the top, bottom, front and back surfaces, which is achieved by implementing mechanical contact conditions between the particles and the boundaries. The periodic condition is applied in the x-direction, i.e. the particles are allowed to move out of the RVE from the right surface but re-enter the domain from the left surface. For the magnetic interaction computation, however, the same full periodic conditions as those in the previous particle dynamic simulations are imposed. During the course of the simulation, the total horizontal shear force, Fs , acting on the top surface is recorded. The final converged value, when divided by the total area, A = 0.0025 mm2 , of the top surface, gives the apparent stress σ = Fs /A. The apparent viscosity is then calculated as σ/γ˙ . Seven different shear rates, γ˙ = 24, 60, 120, 180, 240, 360, 480 s−1 and three different magnetic intensities H = 0.5H0 , H0 , 2H0 , which combine into 21 different cases, are simulated. Figure 16 depicts the total velocity contour of the MR fluid system (the particles and fluid at two cross-sections) at four time instants for the shear rate γ˙ = 240 s−1 and the external magnetic field H0 . These snapshots show a typical shear behaviour of an MR fluid. Under the shear operation, the particles close to the moving top surface break from the chains first (Figure 16a), then the (long) particle chains soon get deformed (Figure 16b), detach from the bottom surface (Figure 16c), and finally break into shorter chains that tend to re-group to form one layer of clusters (Figure 16d). These correspond to a sharp decrease in the shear force at the initial stage and then achieve a steady-state afterwards, as shown in Figure 17.

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Fig. 17 Shear force history in a shear mode simulation: H0 = 1.33 × 104 A/m, γ˙ = 240 s−1

Figure 18 depicts the shear stress and viscosity as a function of the applied shear rates for three different magnetic strengths. It can be seen that the MR fluid behaves like a Bingham fluid. Figure 18(b) indicates the shear thinning behaviour of the MR fluid, whereby the viscosity upon yielding decreases with the increased shear rate. This phenomenon can be explained by the fact that with increase of the shear rate, the microstructure formed is destroyed rapidly by the increased shear stresses; longer particle chains are broken into shorter chains, which improves the fluidity of the fluid and leads to a decrease in fluid viscosity. Figure 18 also shows that both apparent viscosity and shear stress increase with increase of the magnetic field strength, as expected, but in a nonlinear fashion, implying that numerical modelling may be an ideal tool to predict the rheological behaviour of MR fluids under a wide range of operational conditions. The magnetic interaction forces between the suspended particles increase with increase of the magnetic field strength, which causes larger resistance to the fluid flow and therefore the MR fluid gains larger viscosity and shear stress. Thus, unlike the particle chain formation, the accuracy of the magnetic interaction models has a major effect on the simulated rheological properties of an MR fluid. The dynamic yield stress is an important property of MR fluids. It is theoretically defined as the limiting value of the shear stress when the shear rate tends to zero. As observed in [27], it is computationally intensive to undertake the simulations at small shear rates since a large number of time increments have to be performed, especially in three-dimensional situations. The yielding behaviour of the MR fluid is not addressed in this work.

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7 Concluding Remarks The present work has established a computational framework for the effective coupling of multi field interactions in particulate systems, in which the motion of the

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particles is simulated by the Discrete Element Method; the mass and velocity field of the fluid flow is modelled by the Lattice Boltzmann Method; the temperature field of the heat transfer is solved jointly by the Discrete Thermal Element Method and the Thermal Lattice Boltzmann Method. The coupling of the hydrodynamic, thermodynamic and magnetic interactions are realised through the force terms. The applicability of the proposed approach has been illustrated via selected numerical examples.

References 1. Aidun, C.K., Lu, Y., Ding, E.G., Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation. Journal of Fluid Mechanics, 373:287–311, 1998. 2. Chen, S., Doolen, G., Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics, 30:329–364, 1998. 3. Clercx, H., Bossis, G., Many-body electrostatic interactions in electrorheological fluids. Physical Review E, 48(4):2721–2738, 1993. 4. Cook, B.K., Noble, D.R., Williams, J.R., A direct simulation method for particle-fluid systems. International Journal for Engineering Computations, 21(2-4):151–168, 2004. 5. Cundall, P.A., Strack, O.D.L., A discrete numerical model for granular assemblies. Géotechnique, 29:47–65, 1979. 6. Feng, Y.T., Owen, D.R.J., An augmented spatial digital tree algorithm for contact detection in computational mechanics. International Journal for Numerical Methods in Engineering 55:556–574, 2002. 7. Feng, Y.T., Han, K., Owen, D.R.J., Some computational issues on numerical simulation of particulate systems. In Proceedings of the Fifth World Congress on Computational Mechanics, H.A. Mang, F.G. Rammerstorfer, J. Eberhardsteiner (eds.), 2002. 8. Feng, Y.T., Han, K., Owen, D.R.J., Filling domains with disks: An advancing front approach. International Journal for Numerical Methods in Engineering, 56:699–713, 2003. 9. Feng, Y.T., Han, K., Owen, D.R.J., An advancing front packing of polygons, ellipses and spheres. In Proceedings of the Third International Conference on Discrete Element Methods, B.K. Cook and R.P. Jensen (eds.), pp. 93–98, 2002. 10. Feng, Y.T., Owen, D.R.J., A 2D polygon/polygon contact model: Algorithmic aspects. International Journal for Engineering Computations, 21:265–277, 2004. 11. Feng, Y.T., Han, K., Owen, D.R.J., An energy based polyhedron-to-polyhedron contact model. In Proceeding of 3rd MIT Conference of Computational Fluid and Solid Mechanics, MIT, USA, 14–17 June, 2005. 12. Feng, Y.T., On the central difference algorithm in discrete element modeling of impact. International Journal for Numerical Methods in Engineering, 64(14):1959–1980, 2005. 13. Feng, Y.T., Han, K., Owen, D.R.J., Coupled lattice Boltzmann method and discrete element modelling of particle transport in turbulent fluid flows: Computational issues. International Journal for Numerical Methods in Engineering, 72(9):1111–1134, 2007. 14. Feng, Y.T., Han, K., Li, C.F., Owen, D.R.J., Discrete thermal element modelling of heat conduction in particle systems: Basic Formulations. Journal of Computational Physics, 227:5072–5089, 2008. 15. Feng, Y.T., Han, K., Owen, D.R.J., Discrete thermal element modelling of heat conduction in particle systems: Pipe-network model and transient analysis. Powder Technology, 193(3):248– 256, 2009 16. Feng, Y.T., Han, K., Owen, D.R.J., Combined three-dimensional Lattice Boltzmann Method and Discrete Element Method for modelling fluid-particle interactions with experimental validation. International Journal for Numerical Methods in Engineering, 81(2):229–245, 2010.

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17. Han, K., Peric, D., Crook, A.J.L., Owen, D.R.J., Combined finite/discrete element simulation of shot peening process. Part I: Studies on 2D interaction laws. International Journal for Engineering Computations, 17(5):593–619, 2000. 18. Han, K., Peric, D., Owen, D.R.J., Yu, J., Combined finite/discrete element simulation of shot peening process. Part II: 3D interaction laws. International Journal for Engineering Computations, 17(6/7):683–702, 2000. 19. Han, K., Feng, Y.T., Owen, D.R.J., Sphere packing with a geometric based compression algorithm. Powder Technology, 155(1):33–41, 2005. 20. Han, K., Feng, Y.T., Owen, D.R.J., Polygon-based contact resolution for superquadrics. International Journal for Numerical Methods in Engineering, 66:485–501, 2006. 21. Han, K., Feng, Y.T., Owen, D.R.J., Numerical simulations of irregular particle transport in turbulent flows using coupled LBM-DEM. Computer Modeling in Engineering & Science, 18(2):87–100, 2007. 22. Han, K., Feng, Y.T., Owen, D.R.J., Performance comparisons of tree based and cell based contact detection algorithms. International Journal for Engineering Computations, 24(2):165– 181, 2007. 23. Han, K., Feng, Y.T., Owen, D.R.J., Modelling of Magnetorheological Fluids with Combined Lattice Boltzmann and Discrete Element Approach. Communications in Computional Physics, 7(5):1095–1117, 2010. 24. Han, K., Feng, Y.T., Owen, D.R.J., Three dimensional modelling and simulation of magnetorheological fluids. International Journal for Engineering Computations, Published Online: June 28, 2010. 25. He, X., Chen, S., Doolen, G.R., A novel thermal model for the lattice Boltzmann method in imcompressible limit. Journal of Computational Physics, 146:282–300, 1998. 26. Keaveny, E.E., Maxey, M.R., Modeling the magnetic interactions between paramagnetic beads in magnetorheological fluids. Journal of Computational Physics, 227:9554–9571, 2008. 27. Klingenberg, D., van Swol, F., Zukoski, C., The small shear rate response of electrorheological suspensions. II. Extension beyond the point-dipole limit. Journal of Chemical Physics, 94(9):6170–6178, 1991. 28. Ladd, A., Numerical simulations of fluid particulate suspensions via a discretized Boltzmann equation (Parts I & II). Journal of Fluid Mechanics, 271:285–339, 1994. 29. Ladd, A., Verberg, R., Lattice-Boltzmann simulations of particle-fluid suspensions. Journal of Statistical Physics, 104(5/6):1191–1251, 2001. 30. Noble, D., Torczynski, J., A lattice Boltzmann method for partially saturated cells. International Journal of Modern Physics C, 9:1189–1201, 1998. 31. Munjiza, A., Andrews, K.R.F., NBS contact detection algorithm for bodies of similar size. International Journal for Numerical Methods in Engineering, 43:131–149, 1998. 32. Perkins, E., Williams, J.R., A fast contact detection algorithm insensitive to object sizes. International Journal for Engineering Computations, 12:185–201, 1995. 33. Qian, Y., d’Humieres, D., Lallemand, P., Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17:479–484, 1992. 34. Smagorinsky, J., General circulation model of the atmosphere. Weather Rev., 99–164, 1963. 35. Stratton, J.A., Electromagnetic Theory, First Edition. McGraw-Hill Book Company, 1941. 36. Chen, H., Kandasamy, S., Orszag, S., Shock, R., Succi, S., Yakhot, V., Extended Boltzmann kinetic equation for turbulent flows. Science, 301:633–636, 2003 37. Williams, J.R., O’Connor, R., A linear complexity intersection algorithm for discrete element simulation of arbitrary geometries. International Journal for Engineering Computations, 12:185–201, 1995. 38. Yu, H., Girimaji, S., Luo, L., DNS and LES of decaying isotropic turbulence with and without frame rotation using lattice Boltzmann method. Journal of Computational Physics, 209:599– 616, 2005.

Large Scale Simulation of Industrial, Engineering and Geophysical Flows Using Particle Methods Paul W. Cleary, Mahesh Prakash, Matt D. Sinnott, Murray Rudman and Raj Das

Abstract Particle based computational methods, such as DEM and SPH, are shown to be widely applicable as tools to understand complex large scale particulate and fluid flows in industrial processing, civil, marine and coastal engineering and geohazards.

1 Introduction DEM has been developed and used over the past 30 years for modelling flows of particulate solids in many applications, starting with small systems in simple geometries in two dimensions [1–4]. It is now possible to model systems of tens of millions of particles on desktop computers [5, 6] enabling many complex particulate flows to be explored in depth. It is the most effective method for any flow controlled by collision of particulates. SPH is a powerful particle method that is suitable for solving complex multiphysics flow and deformation problems. It is particularly well suited to splashing free surface flows, interaction with dynamic moving bodies and discrete particles. It is also very well suited to situations where flow or material history is important. The method was first developed for incompressible flows by Monaghan [7]. Many examples of SPH applications are given in [8]. The inherent flexibility of these two Lagrangian techniques allows them to be easily and effectively applied to a wide range of different modelling problems with the only adjustments required being modifications to the detailed physics. In this chapter we describe a sub-set of the applications where such methods have been applied and highlight the diversity of physics and modelling scenarios that can be readily handled by DEM and SPH. Paul W. Cleary · Mahesh Prakash · Matt D. Sinnott · Murray Rudman · Raj Das CSIRO Mathematics, Informatics and Statistics, Private Bag 33, Clayton South, Victoria, 3168, Australia; e-mail: [email protected]

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2 Industrial Flows Industrial applications can be broadly classified according to their key physical processes, such as: • • • • • • • • • • • •

Mixing (fluid, particulate and fluid-particulate) Separation Comminution Agglomeration Storage/unloading/transport Material forming (casting, forging, extrusion, forming, etc.) Sampling Excavation Fracture and material failure Bio-medical and biomechanical Fluid-particulate flow Fluid-bubble flow

In this section we will explore the use of particle based modelling in predicting the behaviour of these flow based processes.

2.1 Mixing Mixing of granular materials occurs in applications as broad as minerals processing, chemical manufacture and pharmceutical preparation A variety of techniques have been developed to mix granular materials, often based on empiricism and intuition. This diversity is suggestive of a fundamental lack of rigorous understanding of granular mixing. DEM is an ideal technique for developoing such understanding. A peg mixer is one example of a device that uses an agitator to mix a bed of particles. It is commonly used to reduce residence times and thus equipment dimensions compared to tumbling blenders and bins. In Figure 1 we study a laboratoryscale version. This peg mixer has a 300 mm diameter, 600 mm long cylindrical shell. Agitation is achieved using an axially mounted peg-agitator consisting of 18 pins attached to a 50 mm shaft and arranged at equal intervals along a helical path. Each pin is 15 mm in diameter and 125 mm in length with ends that move just inside the surrounding shell. The mixer rotates anti-clockwise at 60 rpm for a period of 30 seconds. Feed material enters the mixer through a 60 mm inclined port at the left end. The feed rate is chosen to match the discharge rate at the other end so as to ensure that the bed mass remains constant at 22 kg. The particles are spherical with density 2000 kg/m3 and have a size range of 2.–8.8 mm. This is a collision dominated particulate system and consequently, DEM is the ideal technique to predict the flow behaviour. The particles have a coefficient of restitution of 0.3 and a friction coefficient of 0.75. A spring stiffness of 5000 N/m is used.

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Fig. 1 Mixing in a continuously fed peg mixer predicted using DEM

Figure 1 shows the progress of feed material (red) through the original bed of particles (blue). The feed material falls onto the rotating shaft and is distributed over the bed near the feed opening. Particles are moved around by the axially migrating recirculation zones set up by the peg movement. Direct interaction with the pegs gives the particles high velocities leading to strong local mixing, particularly close to the surface of the bed where the impacts with the pegs can move individual particles a significant distance. In Figure 1 the feed material dominates the first quarter of the mixer with some particles having already passed the mid-point. The flow behaviour is quite different to the plug flow observed in a conventional drum mixer and the interface between the blue and red particles is much less sharp in the mixer here. The effect of design choices for the agitator on axial transport and agglomerate break-up can be assessed using DEM simulation. For more details on how to quantify mixing and for assessment of particulate mixing in many types of mixers using DEM [9]. Figure 2 shows the progress of the mixing and submergence of buoyant particulates into a fluid in a 1 m diameter cylindrical tank. Here the modelling is performed using SPH with the particulates modelled as clusters of SPH particles having a pellet-like shape.The impeller is downward pumping and its motion sets up a swirl in the tank with a downward axial flow at the impeller shaft that leads to a bulk recirculation within the tank. At 1.0 s, this downdraft begins to pull down the highly buoyant pellets. By 1.5 s, the pellets have started to cluster near the centre of the tank. By 2 s (Figure 2a) the pellets are starting to be dragged down into the fluid with the leading pellets reaching half way down the shaft. The recirculating flow pattern is fully established around 2.5 s and a significant proportion of the pellets have been drawn down towards the impeller. By 3.0 s, (Figure 2b) pellets are flung outwards by the impeller and are re-circulated within the fluid. Very good agreement is obtained with experiment for the distribution of solids, the critical speed for

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Fig. 2 Mixing and submergence of buoyant particulates in a tank of water driven by a central impeller rotating at 200 rpm, after (a) 2 s and (b) 3 s

submergence and the rate of submergence. This demonstrates that SPH is a viable method for predicting mixing of particulate-fluid systems. For the detailed comparison to experiment [10].

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Fig. 3 Separation by a double deck banana screen with 5 g peak acceleration. The particles are coloured by size with red being coarse, green are the intermediate sizes and blue are the fines

2.2 Separation Screens are often used for separation of particulates into different size fractions. They consist of one or more decks that are fitted with screen panels with arrays of square or rectangular holes. The screen is vibrated at high frequency to generate peak accelerations of 3–10 g which separates particles flowing over the screen according to size. Material passing through the first and second panels of the top deck leads to the formation of a dense bed on the bottom deck. The blue (fine) material falls fairly quickly through the top deck bed so the visible particles are increasingly red to yellow in colour reflecting the ongoing removal of the fine material. The bed depth on the bottom deck increases along the screen as material falls from above and builds up. The colour of the lower deck bed changes from blue to green/yellow over the third and fourth panels again reflecting the removal of fines that fall through to the lower collection chute as an underflow stream. The contribution to the separation efficiency of each panel of each deck can be understood allowing the screen design and/or operating conditions to be optimized. For detailed analysis of the product size distributions, evaluation of the contribution of each panel to the separation of each deck and wear and power consumption [11, 12].

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Fig. 4 Particle distribution in the cone crusher with particles coloured by size. The material is appreciably finer (blue) after passing the choke point of maximum constriction. The concave is the sectioned outer object and the mantle is the nutating conical object over which particles flow

2.3 Comminution Comminution is the process of reducing the size of particles by breakage. It starts with large particles which are typically broken by one or more stages of crushing. The intermediate size particles produced are then fed to grinding mills that grind the particles down to mm or micron size. DEM simulations of a crusher and a mill are presented. Figure 4 shows a DEM model of a cone crusher (around 0.6 m wide and 0.4 m high). The cone crusher is choke fed from above with medium size material (10–

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Fig. 5 Ball flow pattern in the second chamber of a cement mill. The particles are coloured by velocity with red being high (5 m/s) and blue being slow (< 1 m/s)

40 mm). The mantle (the moving conical section in the middle) is inclined at 1o and rotates with a nutating motion at 600 rpm. The concave (the outer object) is stationary. At any circumferential location the mantle oscillates away from and toward the concave causing particles to fall into the crushing zone and be compressed and fractured. The DEM model uses dynamic breakage of the particles (with a compression breakage rule) as described in [13]. This permits the coarse parent particles to break and the daughter particles to move lower in the crusher and be re-broken before exiting the crusher at the bottom. This allows prediction of key machine outputs, including the power draw which was 9.5 kW and the throughput of 11.5 tonnes/hr. Grinding of clinker for cement production is often performed in a two chamber ball mill. In the first shorter chamber, raw clinker feed is ground with the product being transferred to the second longer chamber. Here smaller balls are used to grind the product material even finer. Figure 5 shows such a second chamber of a cement ball mill. It has an inner diameter of 3.85 m, is 8.4 m long and rotates at 16.13 rpm (75% critical speed). It has a classifying liner with a symmetric wave profile and 120 wave peaks around the circumference of the mill. The feed end of each lifter has a vertical step up to its highest point. The height then decreases along the lifter. There are 17 sets of these lifters along the axis of the mill. The ball size ranges from 15 to 50 mm and the fill level is 30% by volume, leading to a ball charge of 136.4 tonnes consisting of 3.2 million balls. Figure 5 shows the charge motion in the mill with particles being dragged around by the mill shell to a shoulder position where they become mobile and flow down

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as a cascading stream to the toe position. They are then re-captured by the liner and begin to circulate again. The free surface has the characteristic S shape. The particles move slowly near the shoulder and toe. As they flow down the surface they accelerate reaching peak speeds of 5 m/s in the steep central section. The particles near the mill shell are transported around with the mill rotation at speeds of close to 3 m/s. There is a narrow band of dark blue connecting the shoulder and toe. This material is very slow moving. Between the slow moving semi-circular blue band and the mill shell is a region of strong shear. Similarly, there is high shear between the slow moving layer and the high speed cascading flow above. Fine clinker particles are systematically trapped and crushed between the balls as they pass each other in these separate sliding layers.

2.4 Material Forming SPH has been demonstrated to be a very effective method for predicting complex fluid flow in the High Pressure Die Casting (HPDC) process [14, 15]. Here we show the casting of an automatic differential cover. This component has a very complex three dimensional shape. The base plate is about 250 mm × 250 mm square in area. A thin-walled dome-like structure rises from the base plate and has an average section thickness of about 6.5 mm. Two cylindrical bosses blend into the dome and several bolt plates are raised from the surface to allow structural attachment to the car. Liquid aluminium is injected into the die cavity through the curved gates attached to one side of the base plate. The gates are are fed from a runner system attached to the shot sleeve. The liquid metal in the shot sleeve is pressurised by a plunger that forces the metal out into the runner system, through the gate and into the part. In this simulation an SPH particle size of 0.75 mm is used. When the cavity is completely filled, the total number of particles is about 900,000. The liquid aluminium viscosity used is 0.01 Pa s and the density is 2700 kg/m3 . Figure 6 shows the filling pattern at 40 ms. Fluid initially enters the die at 10 ms, forming two broad jets at diverging 45◦ angles, partially fragmenting to spray across the cavity. In regions of die curvature, the wall drag causes the fluid to slow and the following fluid catches up leading to the formation of moderately coherent streams with fragmented boundaries. By 40 ms (Figure 6), almost the entire half of the die on the far side of the gate is filled and the detailed topographic structures are becoming clear. There is a strong back flow along the sides and along the base plate towards the gates. The dominant void regions are now just the areas on either side of the incoming streams from the gate, with some residual voids present on the top, behind the horizontal boss, and on the sides of the vertical boss. All the exit vents that are attached to the base plate are now blocked and all the remaining air in the die (represented here as the void regions) is trapped, leading to porosity formation. By 50 ms, there has been significant back filling as fluid flows back from the far side of the die to fill upcavities in the leading half of the die. Filling is complete after 60 ms with the last areas to fill being adjacent to the gates.

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Fig. 6 Filling of a differential cover with the molten aluminium. The fluid is coloured by speed with blue being slow and red being fast

SPH is also well suited to simulating mechanical forming processes such as forging and extrusion due to its ability to model complex free surface behaviour, its ability to tolerate high levels of deformation and its history tracking capability. Here we show an SPH prediction of cold extrusion of aluminium alloy A6061 through a simple orifice. The initial billet dimension is 50 × 50 mm and the ratio of the extruded product width to the billet width is 1:2. A punch speed of 25 m/min is used for the simulation. An SPH particle separation of 1 mm is used giving a total of around 2,700 particles. Figure 7 shows the progress of the billet being extruded. On the left material is coloured in vertical strata according to the initial material position so we can track internal deformation. On the right, the particles are coloured by their plastic strain. Once the billet corners contact the converging walls of the die, the metal quickly becomes elastically loaded and begins to undergo plastic deformation. By 20 ms, the leading edge of the billet has reached the end of the convergent section and mild plastic strains of up to 50% are found in these regions. By 60 ms, the leading edge emerges from the die. High strains of around 200% are observed in the regions just adjacent to the die walls. The distribution of plastic strain is fairly uniform along the length of the extruded rod but has significant variation across the width. The strong predisposition of the metal in the middle of the billet to flow preferentially along the centreline of the die is easily observed due to the frictional resistance of the walls. For more details about the application of SPH to solids forming processes [16].

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Fig. 7 Cold extrusion of aluminium using an elastoplastic SPH model. The particles are coloured (left) in four bands based on their initial position to show the deformation pattern and (right) by plastic strain with red being 1.8 and dark blue corresponding to 0.0

2.5 Bubbly and Reacting Multiphase Flows Figure 8a shows a model of a bubbly flow, where bubbles are sparged into the fluid at the bottom. The fluid is modelled using SPH to enable prediction of the free surface and the oscillating gas source. The bubbles are represented as spherical discrete elements and can be created from either gas sources or a nucleation model. Nucleation sites are typically surface defects that dissolved gas diffuses to, creating bubbles via a phase change. Expanding bubbles evnetually separate from the surface, generating plumes of bubbles whose motions are coupled to the fluid flow [17] for details.

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Fig. 8 (a) Bubbly flow with discrete bubbles coupled to an SPH flow, and (b) reacting particulates immersed in a hot liquid with fluid coloured by its volume fraction of reaction products

SPH can also be used to solve for thermal evolution [18] and for natural convective motion [19]. Combining these with the ability to model floating particles (see Figure 2) and the ability to predict the transport of the product gas through the liquid (including coupling of gas bouyancy into the fluid motion [8]) allows reacting multiphase flow to be modelled. Figure 8b shows the motion of bouyant reacting pellets in a liquid bath. The colour represents the volume fraction of product gas. It shows the generation of gas from the pellets and its transport through the liquid bath. The buoyant pellets float rapidly and cluster near the surface because they are positively buoyant. Note that the gas motion around and above each pellet creates a buoyant plume that tends to entrain fluid which in turn pushes the pellet upwards. So the natural buoyancy of the pellets is enhanced by the generation of buoyant gas plumes from the reacting pellets.

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Fig. 9 Impact of a rogue wave on a 60 m high floating oil platform, after (a) 7.2 s and (b) 10.4 s

3 Fluid-Structure and Engineering Flows Particle methods also provide powerful capabilities for modelling fluids and solids behaviour in civil, marine, ship and coastal engineering, construction and structural failure applications.

3.1 Rogue Wave Impact on an Moored Oil Platform Figure 9 shows the impact of a 30 m rogue wave on a floating oil platform that is moored to the ocean floor using a Taut Spread Mooring (TSM) system. In Figure 9a, the wave can be seen approaching from behind. It is about 40 m away and just beginning to break with the top of the wave travelling in excess of 25 m/s. Wave impact starts at about 6 s, with contact occurring across the entire leading surface of

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the platform. The top of the wave is just below the top deck. The extreme pressure of the water pushes the platform to the right (termed surge) and tilts it sharply clockwise (termed pitch). Figure 9b shows the platform at 10.4 s when the rogue wave has just passed the back of the platform. The surge at 15 m and the pitch of 8o are now substantial. Water splashing over the super-structure inundates much of the top of the platform. Over the next few seconds the platform starts to straighten with the pitch halving. The maximum surge has passed and the platform starts to move back to the left. The rogue wave has passed then beyond the platform, but the equally dangerous following trough is now directly under the platform. For more details of the recovery process and the comparative performance to other rigging systems and the effect of large conventional waves [20].

3.2 Ship Slamming and Green Water–Ship Interaction The ability to easily couple free surface fluid motion to the dynamic response of objects subject to fluid forces is an attractive feature of the SPH technique. One important application is in the area of marine hydrodynamics where a range of different physical phenomena need to be captured. Two of these are known as “slamming” and “green water on deck” (see Figure 10). Slamming occurs when the combination of swell position and pitch of the vessel causes the bow (or stern) to rise completely above the sea surface. As the pitch and swell change, the bow (or stern) can slam down onto the sea surface, giving rise to high pressures and structural loads that can damage the vessel structure. Green water on deck occurs in similar conditions to slamming and is caused when the pitch of the vessel causes the next wave to wash over its bow. This water is not in the form of spray or foam, instead being a large coherent volume of water, hence the term “green water”. The volume and speed of the water mean that significant danger to crew arises and damage can be done to the deck, the superstructure and infrastructure such as lifeboats. Green water can also take a long time to drain from the deck, temporarily increasing the weight and behaviour of the vessel and decreasing manoeuvrability.

3.3 Spillway Flow and Dam Discharge Spillways are structures used for the controlled release of water from a dam or levee into a downstream area, typically the river that was dammed. Spillways release flood waters so that the water does not overtop and damage, or even destroy, the dam. As flow through a spillway involves complex free surface behaviour SPH is a very attractive method for such modelling. Figure 11 shows flow into a spillway from four open gates using approximately 500,000 fluid particles with a resolution of 25 mm. Figure 11a shows the initial release as the front just leaves the spillway

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Fig. 10 SPH simulation of a Ticonderoga-class cruiser travelling at 20 knots in a 6 m swell. In (a) the bow of the ship is completely above the sea surface and is about to “slam” down onto it. In (b) the bow has dipped significantly after the slamming event and has dug into the next wave in the swell, leading to green water washing over the deck

structure. The steady state flow is reached after 2.5 min of the opening of the gates and is shown in Figure 11b.

3.4 Dam Wall Collapse under Earthquake Loading Dam failures are catastrophic and create significant economic loss and loss of human life. A common cause of failure is the cracking that results from seismic load, uneven settling of foundations, and thermal and residual stresses. Fracture of the Koyna dam subjected to earthquake-type motion is modelled using an elastic brittle SPH formulation [21], and the fracture pattern is shown in Figure 12.

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Fig. 11 Discharge of water from a dam and flow down the spillway

Fig. 12 Fracture pattern of the dam subjected to base motion in the horizontal and vertical directions (coloured by damage with blue being no damage and red being fully fractured)

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The base of the dam is subjected to fluctuating loads in the horizontal and vertical directions to simulate the ground motion during an earthquake. The cracks originate normal to the wall surface, because the tangential stress is approximately equal to the maximum principal stress near the free surface. As the cracks propagate towards the interior, they branch one or more times to produce pairs of cracks in each branch. These new cracks propagate towards the surfaces of the dam structure because this provides the path of least resistance to fracture. During the periodic dynamic loads, the interaction of the impinging stress waves with the rarefaction compression waves reflected from the free surfaces slows the normal propagation of cracks. So as the crack front approaches the free surface it decelerates causing bending of the advancing crack front. This is observed for the uppermost crack on the left face and the lowermost crack on the right face in Figure 12b. This phenomenon is known as ‘crack arrest’. The patterns are qualitatively similar to those observed in practice [22].

3.5 Fracture of a Structural Column during Projectile Impact Brittle fracture of slender structures under impact has importance in many applications, such as high-speed weapons systems, impact resistant buildings, and extreme geophysical events. Here we use SPH to investigate the collision of a high speed projectile with a stationary column. Figure 13 shows the fracture of a stationary concrete column when hit by a steel projectile travelling from right to left at a velocity of 125 m/s. Figure 13a shows the onset of fragmentation. Two distinct regions can be identified based on the level of fragmentation: the completely damaged central region with debris/fine particles and the larger fragments above and below this debris zone. The debris cloud erupts horizontally from the left face of the column, leading to its catastrophic failure (Figure 13b). The front of the debris cloud is flat with projections near the ends, and the rear is conical in shape due to the inclined primary fracture planes. Near the top and bottom ends of the column, there are regions of low stress surrounded by cracks (non-red regions in Figure 13b) which show the secondary fracture planes that become fragmentation boundaries.

3.6 Excavation Excavation is an important part of mining and construction. The digging of rocks by moving machinery is well modelled using DEM [23]. Figure 13 shows the filling sequence for an ESCO bucket with non-spherical particles ranging in size from 100 to 300 mm. As the bucket moves towards the dragline, the lip and teeth bite into the overburden, producing an initially thin stream of particles flowing into the bucket. As it fills, the resistance to shear of the material already in the bucket needs to be overcome in order for the material to be pushed up into the back of the bucket. A pile

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Fig. 13 Fracture pattern of the column and fragment distribution after collision for an elastic projectile (coloured by damage with blue-red corresponding to damage range 0–1)

of particles forms in front of the bucket that is of comparable height to the particles in the bucket. This pile is bulldozed along in front of the bucket. Its size increases until the resistance to shear of this pile exceeds the resistance of the material inside the bucket causing it to flow and increase in volume and therefore height. Once the bucket is substantially filled the drag process is complete. The front cables shorten, the bucket lifts and the rock is removed. The drag cycle for this rock material takes 12 s.

4 Geo-Hazard and Extreme Geophysical Flows Geo-hazard or extreme flow events are abrupt, large scale motions of particulate solids and/or fluids. They can generate significant loss of life and economic damage.

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Fig. 14 Progress in the filling and lifting of an ESCO dragline bucket for a non-spherical rock overburden with a 100 mm bottom particle size

They include landslides, debris flows, flooding induced by extreme rain and dam collapse, storm surge, tsunamis, pyroclastic flows and volcanic lava flows. Computational modelling of extreme rock and fluid flow events, such as landslides and dam collapses, can provide increased understanding of their post-initiation course. This can provide valuable insight into opportunities for designing mitigation strategies and to enable more informed management of such disaster scenarios. For the prediction of fluid based geo-hazards such as dam breaks SPH is ideally suited to predicting the resulting complex, highly three dimensional, free surface flows. These flows involve splashing, fragmentation, and interaction with complex topography and engineering structures [24]. For landslides, which are collision dominated flows of rocks, DEM provides an ideal method for prediction [5, 6, 24–26].

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Fig. 15 Landslide from a collapsing mountain peak with particles coloured by velocity

4.1 Landslide Figure 8 shows the collapse of the mountain peak into a valley. The particles are coloured by their speed with blue being stationary and red being 50 m/s or higher. The initial peak was converted into a mound of non-spherical particles that are then free to flow on the underlying topography. The mass of rock in the landslide is 12 million tonnes. This represents a volume of 3.04 million m3 and consists of 245,000 particles. The particle diameters are between 2.0 m and 10.0 m with a mean diameter of 2.4 m. They collapse forwards and down two side valleys producing a left and right branch of the landslide, which are separated by a series of small peaks. By 20 s, the three branches all merge to create a single large flow of particles down the central valley. By 26 s, the supply of new material from the original peak location has slowed and the main landslide is moving at its peak speed and is approaching the valley floor where it comes to rest.

4.2 Flooding from Dam Wall Collapse The St Francis Dam, located in the San Francisquito Canyon, about 15 km north of Santa Clarita, California, failed on 12 March 1928 and at least 450 people were killed in the resulting floods. The dam wall was 57 m high, 213 m long and at the time of failure contained 47 million m¸s of water. Here we show an SPH prediction of the scenario involving instantaneous collapse of the sections of the dam wall that failed. Within 10 s the leading water has collapsed and travelled around 200 m into the canyon. In the shallow valley just beyond the dam wall the flood front has a parabolic shape and is deeper and faster in the middle. By 1 min the flood front has travelled 0.5 km from the dam wall and has reached the opposite side of the valley. The water speed across most of the valley floor is around 20 m/s. By 2 min

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(see Figure 16) the water stretches across a region approximately 1.2 km wide in the main valley. The SPH method is able to capture important 3D flow structures as water flows from the dam breach and criss-crosses the valley downstream resulting in hydraulic jumps that are generated by irregularity in the real surface topography. The momentum of the flood water presses the water up against the right wall of the canyon which records the highest flood levels. The water then flows along this wall until it separates from the sharp bend in the foreground of this frame. The water then flows back across the now narrow canyon to the left wall where it is again reflected creating a smaller hydraulic jump diagonally across the valley. Water begins to enter the large flatter side valley on the left in the foreground. The flooding of two of the earlier side valleys is now well advanced. Figure 16 shows the flooding at 4 min after the failure. The predicted arrival time for the flood front at a power generation station using SPH is 3.5 min, which is close to the observed value of 4.5 min (taking account of the time needed to flood the station following the arrival of the water).

4.3 Tsunami A tsunami is one or more waves generated in a body of water by an impulsive disturbance that vertically displaces the water. Earthquakes, landslides, volcanic eruptions and explosions can generate tsunamis. Tsunamis can savagely attack coastlines, causing devastating property damage and loss of life. The extent of the damage depends on the strength of the tsunami wave its angle of attack on the coastline and the bathymetry of the near shore region. Thus, the prediction of these waves can prove to be extremely useful in minimising the loss of life and property. Here SPH is used to predict the impact of a tsunami wave as it approaches a coastline. Approximately 3 million fluid and 700,000 boundary particles are used. The fluid resolution is 3.5 m. The boundary has a resolution of 7 m. A tsunami wave with a speed of 30 m/s and a wave height of 45 m is shown approaching the coast in Figure 17a.The incident wave indundates the valleys and travels inland about 1 km in the first minute. Figure 17b shows the water middle stages of the inundation process. Here the leading water in the valleys is still travelling inland, but water closer to the coast is already flowing back into the ocean. The return wave that has been reflected off the shore line has significant structure reflecting the topographic complexity of the coast line.

5 Conclusions DEM, SPH and their combination have been shown to successfully simulate fluid and collisional based particulate flows, multiphase flows (fluid-particulate and fluidbubble flows), elastoplastic deformation and elastic-brittle failure of solids. Consequently they can be used to model challenging applications such as industrial

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Fig. 16 Flooding following the collapse of the St Francis dam at (top) 2 min, and (bottom) 4 min after the collapse

processing, civil, marine and coastal engineering, fluid-structure interaction and extreme geophysical flows. The methods provide robust tools that can be used to understand complex flow processes and as part of design and optimisation of processes and equipment and for risk and mitigation strategy evaluation.

Acknowledgements The modelling of the banana screen was carried out under the auspice and with the financial support of the Centre for Sustainable Resource Processing, which is

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Fig. 17 Incoming linear tsunami wave (top) and aftermath one minute later (bottom) with retreating waves and inundation of valleys and low lying areas. Fluid is coloured by speed with red being high (30 m/s) and dark blue being stationary

established and supported under the Australian Government’s Cooperative Research Centres Program. The authors wish to thank their collaborators at ETRI (South Korea) for their contribution to the SPH-DEM bubble modelling.

References 1. Cundall, P.A., Strack, O.D.L., A discrete numerical model for granular assemblies. Geotechnique, 29:47–65, 1979. 2. Walton, O.R., Numerical simulation of inelastic frictional particle-particle interaction (Chapter 25). In: Particulate Two-phase Flow, M.C. Roco (ed.), pp. 884–911, 1994. 3. Campbell, C.S., Rapid granular flows. Annual Review of Fluid Mechanics 22:57–92, 1990. 4. Haff, P.K., Werner, B.T., Powder Technology 48:239, 1986. 5. Cleary, P.W., Large scale industrial DEM modelling. Engineering Computations 21:169–204, 2004. 6. Cleary, P.W., Industrial particle flow modelling using DEM. Engineering Computations 26:698–743, 2009.

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7. Monaghan, J.J., Simulating free surface flows with SPH. Journal of Computational Physics 110:399–406, 1994. 8. Cleary, P.W., Prakash, M., Ha, J., Stokes, N., Scott, C., Smooth particle hydrodynamics: Status and future potential. Progress in Computational Fluid Dynamics 7:70–90, 2007. 9. Cleary, P.W., Sinnott, M.D., Assessing mixing characteristics of particle mixing and granulation devices. Particuology 6:419–444, 2008. 10. Prakash, M., Cleary, P.W., Noui-Mehidi, M.N., Blackburn, H., Brooks, G., Simulation of suspension of solids in a liquid in a mixing tank using SPH and comparison with physical modeling experiments. Progress in Computational Fluid Dynamics 7:91–100, 2007. 11. Cleary, P.W., Sinnott, M.D., Morrison, R.D., Separation performance of double deck banana screens – Part 1: Flow and separation for different accelerations. Minerals Engineering 22:1218–1229, 2009. 12. Cleary, P.W., Sinnott, M.D., Morrison, R.D., Separation performance of double deck banana screens – Part 2: Quantitative predictions. Minerals Engineering 22:1230–1244, 2009. 13. Cleary, P.W., Recent advances in DEM modelling of tumbling mills. Minerals Engineering 14:1295–1319, 2001. 14. Cleary, P.W., Ha, J., Ahuja, V., High pressure die casting simulation using smoothed particle hydrodynamics. International Journal on Cast Metals Research 12:335–355, 2000. 15. Cleary, P.W., Ha, J., Prakash, M., Nguyen, T., 3D SPH flow predictions and validation for high pressure die casting of automotive components. Applied Mathematical Modelling 30:1406– 1427, 2004. 16. Cleary, P.W., Prakash, M., Ha, J., Novel applications of SPH in metal forming. Journal of Materials Processing Technology 177:41–48, 2006. 17. Cleary, P.W., Pyo, S.H., Prakash, M., Koo, B.K., Bubbling and frothing liquids. ACM Transaction on Graphics 26, Article No. 97, 2007. 18. Cleary, P.W., Monaghan, J.J., Conduction modelling using smoothed particle hydrodynamics. Journal of Computational Physics 148:227–264, 1999. 19. Cleary, P.W., Modelling confined multi-material heat and mass flows using SPH. Applied Mathematical Modelling, 22:981–993, 1998. 20. Cleary, P.W., Rudman, M., Extreme wave interaction with a floating oil rig: Prediction using SPH. Proc. CFD 9:332–344, 2009. 21. Das, R., Cleary, P.W., Effect of rock shapes on brittle fracture using smoothed particle hydrodynamics. Theoretical and Applied Fracture Mechanics 53:47–60, 2010. 22. Lee, O.S., Kim, D.Y., Crack-arrest phenomenon of an aluminum alloy. Mechanics Research Communications 26:575–581, 1999. 23. Cleary, P.W., The filling of dragline buckets. Mathematical Engineering in Industry 7:1–24, 1998. 24. Cleary, P.W., Prakash, M., Smooth particle hydrodynamics and discrete element modelling: Potential in the environmental sciences. Philosophical Transactions of the Royal Society of London A 362:2003–2030, 2004. 25. Cleary, P.W., Campbell, C.S., Self-lubrication for long run-out landslides: Examination by computer simulation. Journal of Geophysical Research 98(B12):21911–21924, 1993. 26. Campbell, C.S., Cleary, P.W., Hopkins, M.A., Large scale landslide simulations: Global deformation, velocities and basal friction. Journal of Geophysical Research 100(B5):8267–8283, 1995.

Parallel Computation Particle Methods for Multi-Phase Fluid Flow with Application Oil Reservoir Characterization John R. Williams, David Holmes and Peter Tilke

Abstract This contribution presents a strategy for programming mechanics simulations including particle methods on multi-core shared memory machines.

1 Introduction 1.1 Oil Reservoir Characterization Estimation of porous media properties such as absolute and relative permeability are key to managing oil and gas recovery. Understanding the behavior of fluids as they flow through porous media is important to a variety of contemporary problems in earth science and engineering. To complement traditional displacement type experiments on rock core samples [2–6], numerical techniques are used widely for both explicit parameter determination, and as research tools to probe complex physical phenomena, not easily observed in experiments. Here we describe an SPH formulation and validation tests, which can model multi-phase fluid flow through the rock matrix at the pore scale. Early work into reservoir simulation involved numerical tests on idealized and statistical reconstructions of reservoir rock [7–13], but later, application of X-ray John R. Williams Massachusetts Institute of Technology, 77 Massachusetts Av., Cambridge, MA 02139, USA; e-mail: [email protected] David Holmes James Cook University, Angus Smith Drive, Douglas, Queensland 4811, Australia; e-mail: [email protected] Peter Tilke Schlumberger-Doll Research Center, 1 Hampshire St, Cambridge, MA 02139-1578, USA; e-mail: [email protected]

E. Oñate and R. Owen (eds.), Particle-Based Methods, Computational Methods in Applied Sciences 25, DOI 10.1007/978-94-007-0735-1_4, © Springer Science+Business Media B.V. 2011

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micro-tomography on core samples [14–17] has provided researchers with voxelized representations of actual rock geometries on which to base more highly accurate and case specific models. Examples of numerical techniques used to determine properties from X-ray CT images include the random walk method (determining permeability from its relationship to diffusion [18–22]), the finite difference method (both fluid flow and electrical diffusion [23]), the finite element method (both fluid flow and electrical diffusion [24–26]) pore network models developed with realistic dimensions and connectivity (single-phase [22–27], two-phase [28, 29]), and the lattice-Boltzmann method (single-phase [19–20, 26, 30, 31], multi-phase [30, 32– 34]). Due to its ability to explicitly represent multi-phase wettability and capillary forces, the lattice-Boltzmann method [35, 36] provides the most detail on grain scale flow of conventional numerical methods. There are, however, limitations to latticeBoltzmann regarding solution robustness (related to statistical ‘tuning’ parameters) and the method’s inability to account for electrical and chemical phenomena that can have important cross-relationships with flow. Instead, we favor an alternative particle based method. Smooth particle hydrodynamics (SPH) is a mesh-free Lagrangian particle method first proposed for astrophysical problems by Lucy [37] and Gingold and Monaghan [38] and now widely applied to fluid mechanics problems [39–44] and continuum problems involving large deformation [44, 45] or brittle fracture [46]. As a Lagrangian particle method (see also dissipative particle dynamics (DPD) [47, 48]), fluid mass in SPH is advected with each particle. In multi-phase problems, phase interfaces are addressed intrinsically by this mass advection and properties like surface tension, wettability and capillary forces can be included using pair-wise inter-particle forces, analogous to the molecular forces driving such phenomena in reality [43, 49–52]. In our experience, SPH is less sensitive to small corrections in model parameters than lattice-Boltzmann, in-part due to the inherent robustness of a method with direct analogy to molecular physics. Additionally, it has been shown that the generality of the SPH formulation accommodates the inclusion of a variety of physical phenomena with a minimum of effort (miscible flows, chemical transport and precipitation [43, 52–54], thermal problems [39, 42, 55–59] and electrical/magnetic fields [39, 60, 61]). There is a computational price for managing free particles when compared to grid based alternatives. However, in many circumstances this expense can be justified by the versatility with which such a variety of multi-physics phenomena can be included. Additionally, new parallel hardware architectures such as multi-core [62] are removing many of the barriers which have traditionally limited the practicality of high resolution numerical techniques like SPH.

1.2 Multi-Core Parallel Computing Multi-core machines can increase the speed at which applications execute. In particular, on board data access is more than 10,000 times faster than cross machine

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access. However, new parallel programming challenges are introduced because each core can address all of the main memory, leading to potential memory access conflicts, such as race conditions and deadlock. Some software architects address this by using a process on each core leveraging the operating system which guarantees each process runs in its own isolated memory space. The penalty for this isolation is the time consuming task of cross process communication, which requires object marshalling and then re-instantiation of the object in the new memory space. Here we show that a large class of computational physics problems, including “particle” simulations, can be decomposed into orthogonal compute tasks that can be executed safely in parallel threads within a single process on multi-core machines. A new task management algorithm called H-Dispatch [62] is developed that allows optimal use of memory by matching the task size to the available L3 cache, while optimizing the CPU usage by employing a “hungry” task pull strategy rather than the common push strategy. The technique is demonstrated on SPH problems and it is shown that an optimal task size exists. If the task size is too small adding more cores can actually slow down execution because the problem becomes dominated by messaging latency. However, when the task size is increased an optimal speedup is attained. It is shown that a near linear speedup is attained on a 24-core machine. It is noted that the algorithm is quite general and can be a applied to a wide class of computational tasks on heterogeneous architectures involving multi-core and GPGPU hardware. One solution that ensures memory isolation is to run a separate MPI [63] process on each core. The operating system then ensures an isolated memory space for each process. Data is then shared across processes(cores) by sending MPI message requiring object marshalling and un-marshalling. The problem of memory conflicts is avoided but the cross-core/cross-process communication overhead is significant, on the order of 10,000 machine cycles. So if the time to access a variable in memory is say A cycles then we will now incur A + 10,000 cycles to access that same variable in another process. We note that not all data needs to be communicated in this way, and in computational mechanics problems only “ghost region data” is shared across process boundaries. However, in 3D calculations the ghost regions can be roughly 50% of the unknowns. In essence, the MPI strategy turns each core into an information island, with information transfer being limited by the speed at which MPI messages can be marshaled and delivered across processes. While this is around 100 times faster than cross-machine MPI messages, this is still relatively slow compared to sharing main memory between the cores. An alternative strategy, which allows memory sharing across cores, is to share a single process across all cores, but use separate threads of execution on each core. In order to avoid memory contention “thread safety” must now be managed explicitly by the programmer. “Thread safe” programming can be complex even for the best programmers and the non-deterministic nature of running multiple threads makes detection of race conditions difficult. However, there are specific classes of problem where thread safety can be guaranteed. Indeed, this is the basis of OpenMP [64] and Cilk++ [65] that break “for loops” into parallel execution.

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We show below that in a large class of computational physics problems, including “particle” simulations, we can decompose the problem into orthogonal compute task that share memory but execute “safely” in parallel on multi-core machines. In the next sections we detail an SPH formulation for fluid flow, its validation and testing and its implementation on a multi-core architecture. The problem of managing 3D space to ensure orthogonal compute tasks and the problem of Ghost Regions are addressed.

2 Computational Physics Using Particle Methods 2.1 Overview Mesh based numerical methods have been the cornerstone of computational physics for decades. Here, integration points are positioned according to some topological connectivity or mesh to ensure compatibility of the numerical interpolation. Examples of Eulerian mesh based methods include finite difference (FD) [66] and the lattice Boltzmann method (LBM) [30, 34–36, 67], while Lagrangian examples include the finite element method (FEM) [68]. While powerful for a wide range of problems, mesh limitations for problems involving large deformation and complex material interfaces has led to significant developments in meshless and particle based methodologies [44–62, 69]. For such methods, integration points are positioned freely in space, capable of advection with material in a Lagrangian sense. For methods like molecular dynamics (MD) [70] and the discrete element method (DEM), such points represent literal particles, atoms and molecules for MD and discrete grains for DEM [71–73], while for methods like dissipative particle dynamics (DPD) [74] and smooth particle hydrodynamics the particle analogy is largely figurative. For such methods, particles provide positions at which to enforce a partition of unity (Figure 1). By partitioning unity across the particles, continuity can be imposed without a defined mesh, allowing such methods to represent a continuum in a generalized way. In Eulerian mesh based methods like FD and LBM, continuity is inherently provided by the static mesh, while for Lagrangian mesh based approaches like FEM, continuity is enforced through the use of element shape functions. The partition of unity imposed on mesh-free particle methods can be seen to be a generalization of shape functions for arbitrary integration point arrangements. From Li and Liu [44]: . . . meshfree methods are the natural extension of finite element methods, they provide a perfect habitat for a more general and more appealing computational paradigm – the partition of unity.

The advantage of partition of unity methods is that any expression related to a field quantity can be imposed on the continuum. Where for a bounded domain – in Euclidean space, a set of nonnegative compactly supported functions, φ(xj ), sums to unity (Figure 1), i.e.

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Fig. 1 Partition of unity constructed from basis functions

n

φ(xj ) ≡ 1 on

(1)

j =0

Correspondingly, the value of some field function, f (xi ), at the point xi in space can be determined from its value at all other points via f (xi ) =

n

φ(xj )f (xj )

(2)

j =0

The function f (xi ), can be related to any physical field expression; hydrodynamic, mechanical, electrical, chemical, magnetic etc. Such versatility is a key advantage of meshfree particle methods. We shall now derive the equations for fluid flow.

2.2 SPH for Fluid Flow By discretizing the fluid volume into a finite number of disordered integration points or ‘particles’, any function, such as density or velocity, can be approximated by the summation interpolant fi =

n mj j =0

ρj

fj φ(ri − rj , h)

(3)

where smoothing length h is generally set as the initial particle spacing, mj and ρj are the mass and density of particle j at position rj , and the fraction mj /ρj ac-

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Fig. 2 The support domain and smoothing function in 2 dimension for some particle a

counts for the approximate volume of space each particle represents so as to maintain consistency between the discrete expression (5), and the continuous field that it represents. Correspondingly, the gradient of f is given ∇fi =

n mj j =0

ρj

fj ∇i φ(ri − rj , h)

(4)

Figure 2 illustrates a smoothing function for a single integration point in space, a. Authors such as Tartakovsky and Meakin [43, 50] and Hu and Adams [51] have suggested a variation to (5) and (6) where a particle number density term, ni is used where ni = ρi /mi and then fi =

n fj j =0

∇fi =

nj

n fj j =0

nj

φ(ri − rj , h)

(5)

∇i φ(ri − rj , h)

(6)

Applying this to the particle number density itself ni =

n

φ(ri − rj , h)

j =0

and similarly, mass density of each particle is given by

(7)

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ρi = mi ni = mi

n

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φ(ri − rj , h)

(8)

j =0

This expression conserves mass exactly, much like the summation density approach of conventional SPH [19]. Use of a particle number density variant of the SPH formulation is typically motivated by the need to accommodate multiple fluid phases of significantly differing densities. Use of (7) and (8) eliminates the artificial surface tension effects observed by Hoover [40] and removes density discrepancies which would otherwise manifest at phase interfaces. The multi-phase formulation used in this chapter follows that presented by Tartakovsky and Meakin [43, 50]. Determination of particle velocity is achieved through discretization of the Navier–Stokes conservation of linear momentum equation. In this work, a modified version of the expression provided by Morris et al. [41] and used by Tartakovsky and Meakin [43] has been used, where n dviα Pj ∂φij 1 Pi + 2 =− dt mi n2i nj ∂riα j =0

+

β β n ri − rj ∂ϕij 1 µi + µj . β + Fiα (viα − vjα ) β β mi ni nj |r − r |2 ∂r j =0

i

j

(9)

i

where Pi is the pressure, µi is the dynamic viscosity, vi is the particle velocity and Fi is the body force applied on particle i. Indices α and β refer to vector components and, corresponds to an Einstein’s summation on the right of the expression. An equation of state proposed by Morris and co-workers [41] has been used to determine particle pressure at each time step via Pi = c2 (ρi − ρ0 )

(10)

where ρ0 is the fluid reference density while c is the artificial sound speed. Following Morris et al. [41], the artificial sound speed term, c, should be chosen according to ρ0 V02 ρ0 νV0 ρ0 L0 |F | 2 c ≈ Max , , (11) ρ L0 ρ ρ where ν is the kinematic viscosity ν = µ/ρ0 , V0 and L0 are the velocity and length scales and |F | is the magnitude of body force per unit mass, and ρ is the maximum allowed amount of density fluctuation (generally chosen as being around 1%) meaning that c will scale with the degree of incompressibility of the system. In this work, we integrate the differential rate equation (9) using a conventional Leapfrog [35] numerical integration scheme. A stable solution can be achieved by enforcing the following conditions on the time step length [19, 16, 36] t ≤ 0.125

h2 , ν

t ≤ 0.25

h , 3c

t ≤ 0.25 min(h/(3|Fi |))1/2

(12)

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where |Fi | is the magnitude of the force on a particle. We use a quintic spline kernel function following Morris [41] such that, given R = |ri − rj |/ h, then ⎧ (3 − R)5 − 6(2 − R)5 + 15(1 − R)5 0≤R<1 ⎪ ⎪ ⎪ ⎪ ⎨ (3 − R)5 − 6(2 − R)5 1≤R<2 W (R, h) = αd × for 5 ⎪ (3 − R) 2≤R<3 ⎪ ⎪ ⎪ ⎩ 3≤R 0 (13) where αd = 120/ h, αd = 7/478πh2 , αd = 3/359πh3 in 1, 2 and 3 dimensions respectively.

2.3 Testing and Verification To verify the accuracy of the developed SPH code, simulations of several well defined one, two and three-dimensional flow problems were carried out. Results from these simulations are detailed in Holmes et al. [62]. As an example we briefly describe a three dimensional test that has direct application to fluid flow through rock cores. Ordered sphere packings have been used extensively within the literature as an idealized three-dimensional porous medium. Authors such as Hasimoto [75], Zick and Homsy [76] and Sangani and Acrivos [77] have each presented well verified results for flows through simple cubic, body- and face-centered cubic arrays of spheres with porosities ranging up to the close-touching limits of the spheres. In the case of spheres fluid flow will continue in sphere packs well past the point where sphere radii exceed the close touching limit (Figure 3). Authors such as Larson and Higdon [78] and Roberts and Schwartz [79] have used such model geometries to represent consolidated porous media. In this work, we have tested the performance of the developed SPH code for three-dimensional flow using a simple cubic array of spheres with sphere radii up to, and past, the sphere close touching limits as per Larson and Higdon [78] (Figure 3). Again, symmetry of the periodic system facilitated the reduction of the model to the representative three-dimensional unit cell shown in the right part of Figure 3. Periodic boundaries were enforced in all three model dimensions and a center-to-center sphere distance, d, of 1 × 10−3 m was chosen. The fluid was assigned the properties of water (ρ = 103 kgm3 , ν = 10−6 m2 s−1 ). A variety of solid volume fractions, φ, were used in the simulations following the work of Larson and Higdon [78], where φ=

Vsolid Vcell

(14)

In all simulations, flow was driven from rest by a constant body force of F = 0.049 ms−2 and an artificial sound speed of c = 0.07 ms−1 was chosen. During

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Fig. 3 Cubic packing of overlapping spheres.

model definition, SPH particles were initially arranged on a uniform hexagonal close packed grid and particles positioned inside the bounds of solid spherical grains were designated as being boundary particles.

2.4 Characterization of Flow Flow through the cubic array of consolidated spheres has been characterized in terms of a friction coefficient, K, and the permeability, k. For the case of a spherical grain in infinite dilution, the drag force, Fd , can be expressed via the Stokes– Einstein form, Fd = 6πµrU . Using this term to non-dimensionalize the sphere pack drag, we establish the friction coefficient used by authors such as Zick and Homsy [76] and Larson and Higdon [78] as K=

Fd 6πµrU

(15)

The force and velocity terms were determined from simulation results and permeability was determined through Darcy’s law with the dimensionless form, k 1 Vcell = (16) K 6πrd 2 d2 Tests of the mesh sensitivity are reported by Holmes et al. [62] and it was found that approximately 30 particles should span a circular pore throat. The results for dimensionless permeability are plotted below and show good agreement with Sangani and Acrivos.

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Table 1 Friction coefficient for various values of solid volume fraction for flow through a cubic array of consolidated spheres

Table 1 shows the results for the friction coefficient for various volume fractions and shows excellent agreement with those predicted by Larson and Higdon even up to high solid volume fractions, where the pore throats to pore volume ratio is large. Friction coefficient results are plotted in Figure 4, while results for dimensionless permeability are plotted in Figure 5.

3 Application of SPH to Pore Scale Physics Oil reservoirs are extremely difficult to characterize because in a reservoir extending kilometers only a tiny volume of the rock can be sampled. Furthermore, if the rock is sampled directly by taking a core it is very difficult to maintain the in-situ conditions when the sample is retrieved. Re-creation of in-situ conditions in the laboratory is both time consuming and expensive. Indirect sampling of the rock, using seismic, electromagnetic, acoustic and other means can complement the laboratory tests but often they too are inconclusive. Thus, many researchers believe numerical modeling is an essential tool to give us a better understanding of rock physics. The goal is to predict from digital rock images macroscopic properties such as porosity, absolute permeability, relative permeability, electrical conductivity and elastic properties.

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Fig. 4 Friction coefficient

Fig. 5 Dimensionless permeability

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Given micro-CT scans of a rock sample, the digital image data is first segmented to detect the boundaries between the rock matrix and the pore space as shown in Figure 5. We note that we can use our partition of unity approach to segment the image. Since the digital image data gives us a voxelized sample of the rock we can directly use the same SPH code to interpolate the data and determine the boundary between rock and pore space. f (x) =

n

ak φ(x − k)

(17)

k=0

where ak is the voxel value at x = k. This process is illustrated in Figure 6 and the workflow for generating a numerical model in Figure 7. Fluid flow in the range of low Reynolds numbers experiences a no-slip flow condition at solid boundary surfaces. For the permeable rock applications of interest in this paper, flow will occur in this low Reynolds number range and so no-slip boundary conditions must be enforced to accurately reproduce the appropriate flow profiles. The method developed in our previous work by Holmes et al. [80] uses imposed artificial velocities at boundary particles to create antisymmetry in the velocity field at boundary surfaces. This allows complex pore geometries to be simulated as shown below. The developed SPH simulator has been used to analyze flow though model geometries derived from X-ray CT images of 23.6% porosity Berea sandstone as shown in Figure 6. Figures 8 and 9 show typical results of multi-phase flow applied to pore scale analysis of an idealized oil reservoir; tests that can be repeated on the digital rock geometries.

4 Parallel Computation on Multi-Core 4.1 Parallel Algorithms and the Ghost Region Issue In computational mechanics problems involving parallel processing we divide the problem into a number of tasks which can be “scattered” out to the various processors and executed simultaneously. In cross machine computing we divide the unknowns into non-overlapping domains. However, there is spatial coupling of unknowns across domains so each domain must keep a copy of the “ghost region” belonging to its neighboring domains (Figure 10). Updating unknowns from time step N to step N + 1 within a domain then proceeds in parallel. Only unknowns “belonging” to the domain are updated at this stage. Once the time step is complete the “ghost regions” are then updated by sending MPI messages from one machine to the other. We note that there needs to be a synchronization point that ensures all machines have finished the update on their own domains before messages updating

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Fig. 6 Digital image segmentation using particles as interpolation functions to directly generate SPH particles

Fig. 7 Workflow for numerical model generation

the “ghost regions” are sent. Such synchronizations generally mean that computations on every machine must halt. Synchronization points in coordinating parallel computing tasks are critical as we shall see below. If an MPI process is launched on each core then the computational process is almost identical to that described above for cross machine computation. The only difference is that the MPI messages can be optimized for in-machine communication. In the Microsoft.NET, environment marshalling and un-marshalling objects across AppDomain boundaries (somewhat equivalent to Linux process boundaries) allows approximately 100,000 messages per second, so that each message takes

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Fig. 8 SPH simulation of flushing of oil from rock pores

Fig. 9 SPH multi-phase fluid simulation (a) water-rock non-wetting, (b) water-rock wetting

roughly 10,000 machine cycles. Additionally, problem size must be very large to reduce the fraction of ghost points needing to be communicated to manageable levels (Figure 11). In the case of shared memory there are no “ghost regions” since any unknowns from neighboring domains may be read directly from memory. However, we must now devise a strategy for ensuring that writing does not corrupt data being read. One method is to provide two memory slots for each variable, one for vn and one for vn+1 . Using this strategy only one synchronization point is necessary at the end of the time step. Typically, the domain boundaries are minimized so that the “ghost regions” are as small as possible and message passing is minimized (see relationship in Figure 11).

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Fig. 10 Illustration of ghost regions in numerical simulation

Fig. 11 Data communication fraction in typical cluster computing problem

However, using our shared memory strategy there is no penalty involved in dividing the problem up into smaller domains (Figure 12). Indeed, there is a benefit in doing this because we can now optimize the task size to match the underlying hardware,

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Fig. 12 Contrast of domain decompositon in cluster and multcore computing

Fig. 13 (a) Speed-up versus number of cores, (b) efficiency versus number of cores

particularly the L3 cache size. On a 24 core machine we have shown that we get better efficiency in breaking the problem into hundreds of smaller domains (see Figure 12). Figure 13 shows the performance of the H-Dispatch strategy compared to MPI and traditional scatter-gather. When using modern languages such as Java and C#, we need also to minimize garbage collection because all cores must be stopped while the heap is re-mapped. We achieve this by essentially managing memory on each core explicitly. We allocate a block of memory for each core at the start of the computation and hold it until the end of the computation. This is achieved by allocating a master thread on each core. The thread “pulls” tasks from a single dispatcher queue as fast as it can. The task size is such that it can be mapped into the memory allocated. Furthermore, load balancing across cores is ensured no matter if one core runs slower than another. Indeed, even if a core “fails” by not responding within some given period of time, we can resubmit the task to the queue and it will execute on another core. The end of the time-step occurs when there are no more tasks in the queue.

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4.2 Spatial Hashing in Particle Methods The central idea behind spatial hashing is to overlay some regular spatial structure over the randomly positioned particles e.g. an array of equally sized cells. We can then perform spatial reasoning on the cells rather than on the particles themselves. Spatial hashing assigns particles to cells or ‘bins’ based on a hash of particle coordinates. The numerical expense of such an algorithm is O(N) where N is particle number [15, 16]. A variety of programmatic implementations of hash cell methods have been used (for example linked list). In this work a dictionary hash table is used where a generic list of particles is stored for each cell and indexed based on an integer key unique to that cell, i.e. key = (k × ny + j ) × nx + i where nx an ny are the total number of cells in the x and y dimensions and i, j and k are the integer cell coordinates in the x, y and z dimensions. The cells also provide a means for defining task packages to the cores. By assigning a number of cells to each core we assure “task orthogonality” in that each core is operating on different set of particles. Traditional software applications for shared memory parallel architectures have utilized locks to avoid thread contention. We note that a core may “read” the memory of particles belonging to surrounding cells but may not update them. We execute a single loop in which global memory for both previous and current field values (νtn and vtn+1 ) is stored for each particle. Gradient terms can then be calculated as functions of values in previous memory, while updates are written to the current value memory, in the same loop. In parallel, minimizing the frequency of so called synchronization points has advantages for performance and we utilize a “rolling memory algorithm” that allows such previous and updated terms to be maintained without needing to replace the former with the latter at the end of each step and thus, ensuring a single synchronization per step. In a typical SPH simulator, two operations must be done on particles in each cell per step, separated by a synchronization, the first to determine the particle number density of each particle, and the second to perform the field variable updates. Interacting particles must be known for each of these two stages. Using a standard SPH formulation, the performance of two structure variations can be compared in a case study. In structure A, interacting particles are determined in the first stage for all cells and stored in a global list for use in the second. In structure B, interacting particles are determined as needed in each stage for each cell, i.e. twice per cell per time step. Recalculation of interacting particles means they need only be kept in a local thread list that is overwritten with each newly dispatched cell. While differences in execution memory are to be expected of the two code versions, the differences in execution time are more surprising. For low core counts (< 10 cores), as would be expected, the single search variant (A) solves more quickly than the double (B) due to less computations. After this point, however, the double search (B) is shown to provide marked improvements in speed over (A) (up to 50%). This can be attributed to better cache blocking of the second approach and the significantly smaller amount of data experiencing latency when being loaded from RAM to cache. The fact that such performance gains only manifest when more than 10 cores

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Fig. 14 (a) Memory vs. number of particles, (b) speed-up vs. number of cores

are used, suggests that for less than 10 cores, RAM pipeline bandwidth is sufficient to handle a global interaction list.

5 Conclusions The verification of SPH as an accurate analysis tool for single-phase flows has been detailed and its extension to multi-phase flows and complex pore geometries demonstrated. The ability of SPH to simulate multiple fluid phases with accurate expression of surface tension and interfacial properties such as wettability and contact angle, make the method a powerful numerical tool for geo-numerics problems. Since flow in reservoir rock typically occurs in the range of low Reynolds number, the enforcement of no-slip boundary conditions is an important factor in simulation. Using the no-slip boundary conditions we show that SPH can handle the degree of complexity of boundary surfaces characteristic of an actual permeable rock sample. We present a parallel numerical simulation framework, which allows parallel implementation of a wide range of numerical methods in a multi-core, shared memory, environment. We use a novel domain decomposition methodology that takes optimal advantage of the shared memory architecture and allows for dynamic load balancing of the cores.

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The Particle Finite Element Method for Multi-Fluid Flows S.R. Idelsohn∗, M. Mier-Torrecilla, J. Marti and E. Oñate

Abstract This paper presents the Particle Finite Element Method (PFEM) and its application to multi-fluid flows. Key features of the method are the use of a Lagrangian description to model the motion of the fluid particles (nodes) and that all the information is associated to the particles. A mesh connects the nodes defining the discretized domain where the governing equations, expressed in an integral form, are solved as in the standard FEM. We have extended the method to problems involving several different fluids with the aim of exploiting the fact that Lagrangian methods are specially well suited for tracking any kind of interfaces.

1 Introduction Particle methods aim to represent the behavior of a physical problem by a collection of particles, where each particle moves accordingly to its own mass and the internal and external forces applied on it. All physical and mathematical properties are attached to the particle itself and not to the elements as in finite element methods (FEM). For instance, physical properties like viscosity or density, physical variables like velocity, temperature or pressure and also mathematical variables like gradients or volumetric deformations are assigned to each particle and they represent an average of the property around the particle position. Particle methods are advantageous to treat discrete problems such as granular materials but also to treat continuous problems with internal interfaces in multi-fluid flows, frictional contact in fluid-solid interactions or free surfaces with breaking

S.R. Idelsohn · M. Mier-Torrecilla · J. Marti · E. Oñate CIMNE International Center for Numerical Methods in Engineering, 08034 Barcelona, Spain; e-mail: [email protected] ∗

ICREA Research Professor at CIMNE.

E. Oñate and R. Owen (eds.), Particle-Based Methods, Computational Methods in Applied Sciences 25, DOI 10.1007/978-94-007-0735-1_5, © Springer Science+Business Media B.V. 2011

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waves. Particles are associated the different materials and thus the interfaces can be easily followed. The most relevant characteristic of particle methods is that there is not a specified solution domain. The problem domain is defined by the particle positions. In order to evaluate the interacting forces between particles any classical approximation method may be used, including FEM, finite difference, meshless methods, etc. This means that a particle method may be used with or without a mesh, depending on the method chosen to evaluate the interacting forces. Particle methods can be roughly classified in two types: (a) those based on probabilistic models, such as molecular dynamics, direct simulation Monte Carlo and lattice-gas automata; and (b) those based on deterministic models, such as SPH or other meshless methods, particle-mesh hybrid methods and the Particle Finite Element Method. The first class of methods represents macroscopic properties as statistical behavior of microscopic particles, so that a huge number of particles should be simulated for a long time to obtain accurate average values, while the second class of methods relies on the macroscopic Navier–Stokes equations. This contribution is devoted to the Particle Finite Element Method and its application to multi-fluid flows. We introduce the basics of the method in Section 2, present the multi-fluid flow governing equations and their treatment with PFEM in Section 3, and illustrate the capabilities of the method in several examples in Section 4.

2 Particle Finite Element Method The Particle Finite Element Method (PFEM) [20, 29] is a numerical technique for modeling and analysis of complex multidisciplinary problems in fluid and solid mechanics involving thermal effects, interfacial and free-surface flows, and fluidstructure interaction, among others. PFEM is a particle method in the sense that the domain is defined by a collection of particles that move in a Lagrangian manner according to the calculated velocity field, transporting their momentum and physical properties (e.g. density, viscosity). The interacting forces between particles are evaluated with the help of a mesh. Mesh nodes coincide with the particles, so that when the particles move so does the mesh. On this moving mesh, the governing equations are discretized using the standard finite element method. The possible large distortion of the mesh is avoided through remeshing of the computational domain. Due to the fact that all the hydrodynamical information is stored in the nodes, remeshing does not introduce numerical diffusion. A robust and efficient Delaunay triangulation algorithm [13] (see Figure 1) allows frequent remeshing. This gives the method excellent capabilities for modeling large displacement and large deformation problems. The particles are used to generate a discrete domain within which the integral form of the governing differential equations are solved. An algorithm is needed to define the boundary contours from the collection of particles. PFEM uses the alpha-

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Fig. 1 Delaunay triangulation of a cloud of nodes

(a)

(b)

(c)

Fig. 2 Alpha-shape: (a) collection of nodes, (b) Delaunay triangulation of the convex hull, (c) mesh after alpha-shape

shape technique [6] to recognize the external boundary after the Delaunay triangulation of the domain convex hull (see Figure 2): All nodes defining an empty sphere with a radius r(x) larger than αh(x) are considered to be boundary nodes. h(x) is the distance between two neighboring nodes and the parameter α is chosen so that α 1. Large values of α result in the convex hull of the collection, while small values return a boundary constituted just by the nodes. The error in the boundary surface definition is proportional to h. One of the advantages of the alpha-shape technique is the easy way to determine when particles separate from the fluid domain, as may happen in free surface problems (e.g. splashing). The method is based on the following features: • the information is particle-based, i.e. all the geometrical and mechanical information is attached to the nodes, • the Lagrangian point of view for describing the motion, • the equations are discretized and solved on a finite element mesh that is constructed at each time step, • the boundaries of the domain are defined via the alpha-shape technique. The use of a Lagrangian formulation eliminates the standard convection terms present in Eulerian formulations. These convection terms are responsible for nonlinearity, non-symmetry and non-self-adjoin operators, which require the introduction of stabilization terms to avoid numerical oscillations. All these problems are absent in the Lagrangian formulation. Only the nonlinearity due to the unknown of

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Fig. 3 PFEM solution steps illustrated in a simple dam break example. As the gate of the dam is removed the water begins to flow. (a) Continuous problem (b) Step 1, discretization in cloud of nodes at time t n ; (c) Step 2, boundary and interface recognition; (d) Step 3, mesh generation; (e) Step 4, resolution of the discrete governing equations; (f) Step 5, nodes moved to new position for time t n+1

the final particle position remains. The resulting systems of equations are solved with a symmetric iterative scheme, such as the conjugate gradient method. Linear shape functions (P1 /P1 ) are used for all unknowns. This equal order approximation for both the velocity and the pressure variables does not satisfy the inf-sup condition and therefore pressure stabilization is required. A typical solution with the PFEM involves the following steps (illustrated in Figure 3): 1. The starting point at each timestep is the cloud of nodes in the fluid and solid domains. 2. Identification of the external boundary and the internal interfaces. The alphashape method is used for boundary definition. 3. Discretization of the domain with a finite element mesh generated by Delaunay triangulation. 4. Solution of the Lagrangian governing equations of motion for the fluid domain together with the boundary and interface conditions. Computing the relevant state variables at each timestep: velocities, pressure, temperature, and concentration. 5. Moving the mesh nodes to a new position in terms of the time increment and the velocity field computed in step (4). 6. Back to step (1) and repeat the solution process for the next timestep. Thus PFEM combines the advantages of particle methods (namely only the “wet” domain considered, it is appropriate for changing domains, allows fluid fragmentation, tracks interfaces accurately, and does not introduce numerical diffusion when solving convection) with the accuracy of the finite element method.

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Up to now, the method has been successfully applied to naval and coastal engineering [4, 20, 23, 28, 29], fluid-structure interaction [14–16, 21, 35], melting of polymers in fire [30], excavation problems [3], forming processes [8, 27] and multifluid flows [17, 18].

3 Multi-Fluid Flows The simultaneous presence of multiple fluids with different properties in external or internal flows is found in daily life, environmental problems, and numerous industrial processes. Examples are fluid-fuel interaction in enhanced oil recovery, blending of polymers, emulsions in food manufacturing, rain droplet formation in clouds, fuel injection in engines, and bubble column reactors, to name only a few. Although multi-fluid flows occur frequently in nature and engineering practice, they still pose a major research challenge from both theoretical and computational points of view. In the case of immiscible fluids, the dynamics of the interface between fluids plays a dominant role. The success of the simulation of such flows will depend on the ability of the numerical method to model accurately the interface and the phenomena taking place on it, such as the surface tension. The origin of the surface tension force lies in the different intermolecular attractive forces that act in the two fluids, and the result is an interfacial energy per area that acts to resist the creation of new interface, so that the interface behaves like a stretched membrane trying to minimize its area. The main difference between a single-fluid (homogeneous) flow and a multi-fluid (heterogeneous) one is the presence of internal interfaces. In addition to the wellknown difficulties in the simulation of homogeneous flows (namely the coupling of pressure and velocity through the incompressibility constraint, the need of the discretization spaces to satisfy the inf-sup condition, and the non-linearity of the governing equations), numerical methods for multi-fluid flows face the following challenges: 1. Accurate definition of the interface position. The interface separating the fluids needs to be tracked accurately without introducing excessive numerical smoothing. 2. Modeling the jumps in the fluid properties across the interface. Large jumps of fluid density and viscosity across the interface need to be properly taken into account in order to satisfy the momentum balance at the vicinity of the interface. 3. Modeling the discontinuities of the flow variables across the interface. Velocity and pressure may be discontinuous across the interface under certain conditions. 4. Modeling the surface tension. Since surface tension plays a very important role in the immiscible interface dynamics, this force needs to be accurately evaluated and incorporated into the model.

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Fig. 4 Two-fluid flow configuration

The computation of multi-fluid flows requires to solve, besides the governing equations in each fluid, all the physical phenomena the interface may be subject to, such as surface tension, thermal diffusion, chemical diffusion, phase change or chemical reactions, and to model interface topology changes like breakup or coalescence.

3.1 Governing Equations Let ⊂ Rd , d ∈ {2, 3}, be a bounded domain containing two different fluids (see Figure 4). We denote time by t, the Cartesian spatial coordinates by x = {xi }di=1 , and the vectorial operator of spatial derivatives by ∇ = {∂xi }di=1 . The evolution of the velocity u = u(x, t) and the pressure p = p(x, t) is governed by the Navier–Stokes equations: ρ

du = ∇ · σ + ρg in × (0, T ) dt dρ + ρ∇ · u = 0 in × (0, T ) dt

(1a) (1b)

where ρ is the density, σ the Cauchy stress tensor, g the vector of gravity acceleration, and dφ/dt represents the total or material derivative of a function φ. The constitutive equation for a Newtonian and isotropic fluid takes the form 1 (2) σ = −pI + 2µ D − εV I 3 with I the identity tensor, µ the dynamic viscosity, D = 12 (∇u + ∇T u) the strain rate tensor, and εV = ∇ · u the volumetric strain rate. Let int (t) be the interface that cuts the domain in two open subdomains, ¯+ ∪ ¯ − , and int = + (t) and − (t), which satisfy: + ∩ − = ∅, = + − + − ¯ = ∂ ∩ ∂ . In each subdomain, the physical properties are defined ¯ ∩ as: + + ρ if x ∈ + (t) µ if x ∈ + (t) ρ = ρ(x, t) = , µ = µ(x, t) = (3) − − ρ if x ∈ (t) µ− if x ∈ − (t)

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If density and viscosity are assumed to remain constant in each fluid (i.e. fluids are incompressible, immiscible, and isothermal), we have that dρ/dt = 0 and dµ/dt = 0. Consequently, we have on the one side that εV = ∇ · u = 0, this is the mass conservation equation for incompressible flows; and on the other side, that (d/dt) int = 0. This latter consequence means that interfaces are material surfaces, which move with the fluid velocity u, and therefore, they are naturally tracked in Lagrangian formulations.

Boundary and interface conditions In order for the Navier–Stokes problem (1) to be well posed, suitable boundary conditions need to be specified. On the external boundary ∂ = D ∪ N , such that

D ∩ N = ∅, we consider the following: u = u¯ σ · n = σ¯ n

on D

(4)

on N

(5)

u¯ is the prescribed velocity, n the outer unit normal to N , and σ¯ n the prescribed traction vector. A Neumann boundary N with σ¯ n = 0 is called free surface. On the internal interfaces int , the coupling conditions are [1]: [[u]] = 0

on int

(6)

[[σ ]] · n = γ κn

on int

(7)

with n now the unit normal to int , γ the surface tension coefficient, κ the interface curvature, and [[φ]] = φ + − φ − represents the jump of a quantity φ across the interface. Equation (6) expresses the continuity of all velocity components. The normal component has to be continuous when there is no mass flow through the interface, and the tangential components have to be continuous when both fluids are viscous (µ+ , µ− > 0), similar to a no-slip condition. Equation (7) expresses that the jump in the normal stresses is balanced with the surface tension force. This force is proportional to the interface curvature and points to the center of the osculating circle that approximates int . The surface tension coefficient γ is assumed constant and its value depends on the two fluids at the interface.

3.2 Discontinuities at the Interface Discontinuities at the interface can be of two types: • C 0 discontinuity, when the flow variable has a kink (i.e. the gradient has a jump), and

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(a)

(b)

(c) Fig. 5 Flow discontinuities for: (a) density jump, (b) viscosity jump, and (c) surface tension

• C −1 discontinuity, when the flow variable itself has a jump. Differences in density at the interface cause a kink in the hydrostatic pressure profile, leading to a jump in the pressure gradient, and then to a C 0 discontinuity in the pressure field (Figure 5a). Differences in viscosity lead to discontinuous components of the strain rate tensor D, and therefore to a C 0 discontinuity of the velocity field at the interface (Figure 5b): t · [[σ ]] · n = 0

=⇒

µ+

∂un ∂ut + ∂n ∂s

+

− µ−

∂un ∂ut + ∂n ∂s

−

=0

(8)

with ∂s = t · ∇ the tangential derivative. Both differences in viscosity and the presence of surface tension cause a C −1 discontinuity in the pressure field (Figures 5b and 5c), as shown in [17]: n · [[σ ]] · n = γ κ

=⇒

p+ − p− = 2(µ+ − µ− )

∂un − γκ ∂n

(9)

Notice that even in the case of γ = 0, pressure is discontinuous when µ+ = µ− .

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(a)

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(b)

Fig. 6 (a) Moving mesh adapted to the interface, and (b) fixed mesh, where interface moves through the elements

3.3 Interface Description A major challenge in the simulation of interfaces between different fluids is the accurate description of the interface evolution. The location of the interface is in general unknown and coupled to the local flow field which transports the interface. It is essential that the interface remains sharp along time and is able to fold, break and merge. In the past decades a number of techniques have been developed to model interfaces in multi-fluid flow problems, each technique with its own particular advantages and disadvantages. Comprehensive reviews can be found in e.g. [2, 36, 37, 40]. The main classification of interface descriptions is regarding the reference frame adopted (see Figure 6). In the moving mesh methods, the mesh is deformable and adapted to the interface, which is explicitly tracked along the trajectories of the fluid particles. Examples are methods based on the Arbitrary Lagrangian-Eulerian (ALE) formulation [12, 33], the deformable-spatial-domain/stabilized space-time deformation (DSD/SST) method [38, 39], or the fully Lagrangian formulation such as in [10, 34] and the Particle Finite Element Method [4, 19, 20, 22]. On the other hand, fixed mesh methods use a separate procedure to describe the position of the interface. They can be further grouped in front-tracking methods, which use massless marker points to follow the fluid interface while the Navier– Stokes equations are solved on a fixed mesh [9, 41], and front-capturing methods, which introduce a new variable ψ in the model to describe the presence or not of a fluid in a position of the domain. The most extended front-capturing methods are the Volume-of-Fluid, originally developed by Hirt [11], and the Level Set method by Osher et al. [31].

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(a) Interface across elements

(b) Nodal interface

Fig. 7 Possible interface representations in PFEM: (a) interface across elements for miscible fluids, (b) nodal interface (with interfacial nodes in black) for immiscible fluids

3.4 PFEM for Multi-Fluid Flows One of the features of particle methods is that all the physical properties are attached to the nodes instead of to the elements. The mesh is frequently updated and hence, it is difficult to keep physical properties at the element level. Multi-fluid flows can have a jump in the fluid properties of several orders of magnitude. One must decide where does the internal interface between two different fluids occur. The typical solution for a particle method would be to have the interface inside the elements sharing particles with different densities so that, at the element integration point k, density takes the mean value ρk =

nv 1 ρa nv a=1

(where nv is the number of nodes of the element). We call this possibility interface across elements (Figure 7a). Another possibility is to impose that the interface between different materials is described by element edges. This is called nodal interface (Figure 7b). For the nodal interface one must accept that elements sharing particles with two different densities have one or the other particular density value. Now, the density at the element integration point k takes the value + ρ if k ∈ + (10) ρk = ρ − if k ∈ − Both possibilities have advantages and disadvantages. Interfaces across elements are more stable as they do not change much when remeshing is performed but on the other hand, nodal interfaces are more accurate because they allow to represent exactly the jumps in the physical properties and the discontinuities of the flow variables, as illustrated in Figure 8.

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(a) Interface across elements

(b) Nodal interface Fig. 8 Density and pressure representations for the different interface definitions: in the interface across elements (a), standard linear elements cannot represent accurately the pressure weak discontinuity; while in the nodal interface (b) the representation is exact

We mainly focus on immiscible multi-fluid flows to exploit the fact that Lagrangian methods are able to track interfaces in a natural and accurate way. For this purpose, we use the nodal interface, and since in this representation the interface is described by mesh nodes and element edges, it is a well-defined curve and the information regarding its location and curvature is readily available. The interface nodes carry the jump of properties (e.g. density, viscosity), maintaining the interface sharp without diffusion along time. Furthermore, it is straightforward to impose the on the interface and to treat any number of fluids. Therefore in PFEM the interface is tracked accurately without introducing numerical smoothing (challenge 1). Regarding the modeling of the jumps in the fluid properties across the interface (challenge 2), while in fixed mesh methods typically the interface is considered to have a finite thickness and the fluid properties change smoothly and continuously from the value on the one side of the interface to the value on the other side, PFEM treats the interface in a sharp manner, so that it is clear which property value is valid at each point. Regarding the modeling of the discontinuities in the flow variables across the interface (challenge 3), in fixed mesh methods where the physical properties have been smoothed, functions are continuous across the interface and thus not appropri-

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Fig. 9 Pressure profiles when using continuous and discontinuous representations

ate for the approximation of discontinuous variables. When the physical properties are modeled sharp, the elements cut by the interface require a special treatment in order to be able to represent the discontinuities. Gravity dominated flows will require “enrichment” of the pressure approximation, and viscosity dominated flows will require “enrichment” of the velocity approximation. On the contrary, C 0 discontinuities need no special attention when the interface is aligned with the mesh, as the kinks in the solution are automatically represented. Only C −1 discontinuities need some attention in PFEM. In particular, the pressure field has been made double-valued at the interface, i.e. pressure degrees of freedom have been duplicated (p+ , p− ) in the interface nodes [17]. The pressure discontinuity caused by the jump in viscosity and/or surface tension (Eq. (9)) is thus optimally approximated. Figure 9 shows that the use of continuous pressure representations may introduce errors in the incompressibility condition. Moreover, stabilization is needed in incompressible flows when interpolation spaces for velocity and pressure do not satisfy the inf-sup condition. Many stabilization procedures have been proposed in the literature, such as the StreamlineUpwind/Petrov-Galerkin, Galerkin Least-Squares, Finite Calculus, or Orthogonal Sub-Scale methods. Those that include the projection of the pressure gradient need to be modified when density changes at the interface to take into account the variation of the hydrostatic pressure gradient. In PFEM the pressure gradient projection is modeled discontinuous to take into account the jump in density. For details, refer to [18, 24]. Regarding the modeling of the surface tension force fst = γ κn at immiscible interfaces (challenge 4), this force is naturally incorporated in the weak form of the momentum equation in the finite element method. There are several ways to calculate the curvature κ from the information of the interface location. The one we follow in PFEM is based on the osculating circle of a curve, which is defined as the circle that approaches the curve most tightly among all tangent circles at a given point. From the radius of the osculating circle, the quantity κn required for fst is calculated as (see Figure 10): n=

R , |R|

κn =

R |R|2

(11)

Details on the accuracy of the surface tension computation have been given in [26]. The accuracy of the curvature calculation is improved by refining the mesh close to the interface. By means of a distance function, we prescribe an element size which is fine at the interface and coarse far away. This allows us to use arbitrarily

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Fig. 10 Calculation of the osculating circle at node x

fine meshes without increasing the total number of elements to impractical values as it would be the case with a uniform mesh.

3.5 Combustion Problem Within the flows labeled as multi-fluid there are the fluid-combustible interaction problems mentioned before. These problems are important for materials such as the polymers, which are characterized by their strength, low cost, and easy processability, with applications ranging from packaging to injection of molded parts or structural components. The behavior of polymers in fire is therefore of great interest due to their role in the ignition and stages of fire growth. of these problems is due some materials with characteris as the strength, low cost, and easy processability has been used as polymers, with applications ranging from packaging to injection of molded parts to structural components. Their behaviour in fire is therefore of considerable interest because they play an important role in the ignition and stages of fire growth. When a polymer is heated, it starts to be pyrolysed while ejecting volatile gases [5]. The fluid above the solid fuel provides a region where the combustion reaction can take place. Once the pyrolysis products are released from the surface, they are able to mix with the surrounding air while being heated from the nearby flame. The rate of fuel consumption depends on both the reaction rate and the speed at which fuel and oxidized are mixed. Once heated to ignition, oxidation of the gaseous fuel leads to the generation of combustion products. The intermediates and final products of the combustion are a complex function of the reaction rates and local conditions. Solving the combustion problem means to solve for the flow together with the chemical species. Therefore the Navier–Stokes equations (1) apply for the multispecies multi-reaction gas but they require the following additional equations: ρC

dT = ∇ · (κ∇T ) dt

in × (0, T )

(12)

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dYk (13) = wk + ∇ · (Dρ∇Yk ) in + × (0, T ) dt where T = T (x, t) is the temperature, Yk = Yk (x, t) the mass fraction of species k, C the heat capacity, κ the thermal conductivity, wk the source term of specie k and D the diffusion coefficient. The coupling between temperature and velocity will be considered by introducing the Boussinesq approximation. The mass fractions Yk are defined by ρ

Yk =

mk m

(14)

where mk is the mass of species k present in a given volume + and m is the total mass of gas for this volume. Obviously, the sum of mass fractions must be (15) Yk = 1 for k = 1 to N, where N is the number of species in the reacting mixture. Let + (t) represent the fluid and − (t) the combustible. In each subdomain, the physical properties are defined as + + κ if x ∈ + (t) C if x ∈ + (t) , κ(x, t) = , C(x, t) = − − C if x ∈ (t) κ − if x ∈ − (t) + D if x ∈ + (t) (16) D(x, t) = 0 if x ∈ − (t) and the boundary condition at the interface is the following: −ρD

∂YF = f (T )(1 − YF ) ∂n

(17)

In the present study, the polymer/air reactive system is modeled as a simplified one-step chemical reaction between the fuel (F ) and the oxidizer (O) F + sO → (1 + s) Products

(18)

where s is the stoichiometric ratio [32]. These species are identified by their mass fractions YF , YO and YP . Species reaction rates wk are all related to the single-step reaction rate [42] 1 (19) wm = −B 2 YF YO e−(E/R T ) T where B, E and R are appropiate constants and the temperature T . So the oxidizer and product reaction rates are linked to the fuel reaction rate wO = (s)wm

(20)

wP = (1 + s)wm

(21)

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and the heat realease per unit of volume from combustion is therefore scaled according to Q = −wm H in + (22) where H is the heat of combustion. The value of Q is introduced as a source term in Eq. (12). The effect of gasification can be introduced by adding a (nonlinear) energy loss term in Eq. (12). This term represents the energy that migrates from the system to the gas due to the gasification of a part of the material during the heating process. The gasification heat flux has the following form: qgas = H εv

in −

(23)

with H being the heat of degradation and εv = f (T ), where f (T ) expresses the relation between the volume variation due to the temperature εv and the temperature itself. In our work the following Arrhenius function has been chosen [7]: f (T ) = −ρAe−E/RT

(24)

The computed mass loss has to be included in the problem to ensure that the volume variation of the sample is correctly modeled. This term is positive in + because it represents the conversion rate from solid to gases due to evaporation, devolatization and heterogeneous reactions, and thus negative in − .

4 Numerical Examples The Particle Finite Element Method described here has been tested in several multifluid flow problems (such as the two-fluid sloshing [18], extrusion of viscous fluids [17] and bubble rise [26]), and fluid-structure interaction problems where the structure is modeled as a viscous fluid [15, 16]. In this section we show the capability of PFEM to handle interfaces with changing topology in flows with surface tension and to model combustion flows. Topology changes in multi-fluid flows can be divided into two classes: (a) Films that fragment. If a bubble approaches a flat surface or another bubble, the fluid in between must be squeezed out before the bubbles are sufficiently close so that the film becomes unstable to attractive forces and fragment. (b) Threads that break. A long and thin cylinder of fluid will generally break by the Plateau–Rayleigh instability in the region where the cylinder becomes sufficiently thin so that surface tension pinches it into two. We have simulated an example of each class: (a) the breakup of a bubble in Section 4.1 and (b) the breakup of an injected fluid in Section 4.2. The last numerical example deals about melting and combustion of a candle (Section 4.3).

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(a) t = 0 s

(b) t = 1.5 s

(c) t = 2.5 s

(d) t = 3.5 s

(e) t = 4.5 s

(f) t = 5.5 s

(g) t = 6.0 s

(h) t = 6.5 s

Fig. 11 Bubble breakup

4.1 Bubble Breakup The problem consists in a bubble rising in a liquid column as illustrated in Figure 11a [26]. We consider a rectangular domain (0, 1) × (0, 2) with a flat interface at y = 1 and a circular bubble centered at (0.5, 0.5) and radius equal to 0.25. The physical properties of the fluids are: ρ1 = 1000, ρ2 = 100, µ1 = 10, µ2 = 1, g = 0.98, and γ = 24.5. The bubble rises due to buoyancy, approaching the flat interface. The film of heavy fluid that separates the two regions of light fluid becomes thinner and thinner until it fragments and the regions fuse (Figure 11). Whereas in the physical reality the fragmentation of the film is caused by attractive forces at the microscopic scale (forces which are usually not included in the continuum description), in our simulations fragmentation is caused by a connectivity change at the interface, as illustrated in Figure 12. One of the main difficulties we face in our Lagrangian approach is the connectivity changes introduced by the remeshing process. In general, these reconnections may alter the equilibrium at the interface, slow down convergence and affect mass

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(a) t n

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(b) t n+1

Fig. 12 Connectivity change that produces breakup at fluid films spanned by just one mesh element

(a) t = 5.965 s

(b) t = 5.98 s

(c) t = 6.0 s

(d) t = 6.125 s

Fig. 13 Pressure field at breakup (variable scale ranges in legend)

conservation. Thus, in interfacial flows it is essential to avoid them. We are using an unconstrained Delaunay triangulator which does not allow to fix connectivities. Therefore, to ensure that a specific connectivity remains, we refine long interfacial edges and remove nodes too close to the interface. Unfortunately, this strategy would preclude the possibility of breakup, as the interface could elongate endlessly. In the way PFEM defines interfaces, it is possible to have fluid regions spanned by just one element layer (Figure 12). The breakup criterium we have implemented in PFEM is to permit connectivity changes in elements where all nodes lie at the interface. In this way, a thin fluid thread can stop elongating and fragment. Breakup is then

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Fig. 14 Flow evolution for uj = 0.025 m s−1 (F r = 0.6, Re = 44)

dependent on the mesh resolution, that is, it happens when the thickness of the film is similar to the mesh resolution of the interface. This is not a drawback specific of PFEM, breakup is mesh dependent in front-capturing methods as well. For example, in the level set method, two interfaces are described as two different zero contours of the same level set function, and these interfaces will automatically merge once they get close enough, relative to the spatial resolution of the mesh where the level set function is defined. The pressure field at breakup is shown in Figure 13. The different pressure values inside and outside the bubble equilibrate after breakup, what occurs at t = 5.97 s.

4.2 Negatively Buoyant Jet Negatively buoyant jets consist in a dense fluid injected vertically upward into a lighter fluid. The jet momentum is continually being decreased by buoyancy forces until the vertical velocity becomes zero. The jet then reaches its maximum penetration height, reverses direction and flows back. This problem has been experimentally and numerically studied in [25]. We consider the injection of water (ρw = 1000 kg m−3 , µw = 10−3 Pa s) through a nozzle in the base of a tank containing oil (ρo = 900 kg m−3 , µo = 200 × 10−3 Pa s).

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Fig. 15 Flow evolution for uj = 1 m s−1 (F r = 28, Re = 1310)

For negatively buoyant jet flows with very low Froude (F r, inertia vs. buoyancy) and Reynolds (Re, inertia vs. viscosity) numbers, the injected fluid reaches an almost constant maximum height, as shown in Figure 14. At higher velocities at the nozzle (i.e. higher F r and Re numbers), the jet begins to oscillate between a maximum and a minimum height, and over a certain threshold, instabilities at the interface cause the jet to break into droplets (see Figure 15).

4.3 Candle Combustion The problem considered here is a two-dimensional burning rod inside a closed contanier, as illustred in Figure 16a. For simplicity, we will refer to this object as a “candle". The dimensions of the candle are 50 cm high by 5 cm thick. From time t = 0 to t = 10 s, the temperature at the candle top is set to 950 K. In the solid phase, the processes of heating and gasification take place. Simultaneously, in the gas phase chemical processes are initiated, and temperature, fuel and oxidizer concentration gradients develop. The pysical properties are ρ1 = 1 kg m−3 , ρ2 = 1170 kg m−3 , µ1 = 0.001 Pa s, µ2 = 106 Pa s, D = 10−5 m2 s−1 . Figures 16 to 18 show snapshots of the temperature evolution and the flame zone in time for all configurations. Notice that as the flame grows (see Figures 17 and 18) the combustion takes place in a larger area. Finally, the flame is extinguished, first in the configuration in

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(a) t = 0 s

(b) t = 3 s

(c) t = 10 s

(d) t = 63 s

Fig. 16 Temperature evolution in the combustion of a vertical (bottom-up) candle

Figure 16 and later in the other examples, when reactions stop due to the particles of non-oxygenated air returning to the combustion zone. In the next example, the material properties for the candle are the same as in the previous example. The temperature increases in the candle due to heat of combustion, and the viscosity decreases by several orders of magnitude as a function of temperature [30]. This induces the melting and flow of the candle material in the heated zone. The melt flows down along the heated face of the sample and drips onto the surface below. Figures 19 and 20 show the progressive melting of the candle exposed to the heat from combustion, along with the change of the flame shape. The dripping material transports the flame (see Figure 19c) and continues burning on the surface below. After some seconds, the candle falls down and the flame is extinguished.

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(a) t = 12 s

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(b) t = 75 s

Fig. 17 Temperature evolution in the combustion of a vertical (top-down) candle

(a) t = 7 s

(b) t = 41 s

Fig. 18 Temperature evolution in the combustion of a horizontal candle

5 Summary and Conclusions The Particle Finite Element Method (PFEM) has been used to solve the incompressible Navier–Stokes equations for heterogeneous fluid flows (such as the rising bubble, extrusion of materials with different viscosity and candle combustion with melting and dripping). The results show the ability of the method to deal with problems from the simple case of fluids with a single interface to the case of strong mixed fluids with multiple interfaces. Problems with a big difference between the two materials were also performed without showing any instability.

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(a) t = 5 s

(b) t = 10 s

(c) t = 12 s

(d) t = 25 s

Fig. 19 Melting and dripping of a candle: temperature evolution

(a) t = 28 s

(b) t = 57 s

(c) t = 105 s

Fig. 20 Melting and dripping of a candle: viscosity evolution

Acknowledgements M. Mier-Torrecilla thanks the Catalan Agency for Administration of University and Research Grants (AGAUR), the European Social Fund and CIMNE for their support.

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Support from the European Commission and the European Research Council through the “Real Time Computational Techniques for Multi-Fluid Problems” project is also gratefully acknowledged.

References 1. Batchelor, G., An Introduction to Fluid Dynamics. Cambridge University Press, 1967. 2. Caboussat, A., Numerical simulation of two-phase free surface flows. Archives of Computational Methods in Engineering 12:165–224, 2005. 3. Carbonell, J., Modeling of ground excavation with the Particle Finite Element Method. Ph.D. Thesis, Universitat Politècnica de Catalunya, Barcelona (Spain), 2009. 4. Del Pin, F., Idelsohn, S., Oñate, E., Aubry, R., The ALE/Lagrangian Particle Finite Element Method: A new approach to computation of free-surfaces flows and fluid-object interactions. Computers and Fluids 36(1):27–38, 2007. 5. Drysdale, D., An Introduction to Fire Dynamics. Wiley Interscience, 1985. 6. Edelsbrunner, H., Mücke, E., Three-dimensional alpha shapes. ACM Transactions on Graphics 13:43–72, 1994. 7. Fernandez-Pello, A., Flame spread modeling. Combustion Science and Technology 39:119– 134, 1984. 8. Gonzalez-Ferrari, C., El Método de los Elementos Finitos de Partículas: Aplicaciones a la pulvimetalurgia industrial. Ph.D. thesis, Universitat Politécnica de Catalunya, 2009. 9. Harlow, F., The Particle-in-Cell computing method for fluid dynamics. Methods in Computational Physics 3:313–343, 1964. 10. Hirt, C., Cook, J., Butler, T., A Lagrangian method for calculating the dynamics of an incompressible fluid with free surface. Journal of Computational Physics 5:103–124, 1970. 11. Hirt, C., Nichols, B., Volume of Fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics 39:201–225, 1981. 12. Hughes, T., Liu, W., Zimmermann, T., Lagrangian-Eulerian finite element element formulation for incompressible viscous flows. Computer Methods in Applied Mechanics and Engineering 29:239–349, 1981. 13. Idelsohn, S., Calvo, N., Oñate, E., Polyhedrization of an arbitrary 3D point set. Computer Methods in Applied Mechanics and Engineering 192:2649–2667, 2003. 14. Idelsohn, S., Del Pin, F., Rossi, R., Oñate, E., Fluid-structure interaction problems with strong added-mass effect. International Journal for Numerical Methods in Engineering 80:1261– 1294, 2009. 15. Idelsohn, S., Marti, J., Limache, A., Oñate, E., Unified Lagrangian formulation for elastic solids and incompressible fluids: Application to fluid-structure interaction problems via the PFEM. Computer Methods in Applied Mechanics and Engineering 197:1762–1776, 2008. 16. Idelsohn, S., Marti, J., Souto-Iglesias, A., Oñate, E., Interaction between an elastic structure and free-surface flows: Experimental versus numerical comparisons using the PFEM. Computational Mechanics 43:125–132, 2008. 17. Idelsohn, S., Mier-Torrecilla, M., Nigro, N., Oñate, E., On the analysis of heterogeneous fluids with jumps in the viscosity using a discontinuous pressure field. Computational Mechanics, 2009 (in press). 18. Idelsohn, S., Mier-Torrecilla, M., Oñate, E., Multi-fluid flows with the Particle Finite Element Method. Computer Methods in Applied Mechanics and Engineering 198:2750–2767, 2009. 19. Idelsohn, S., Oñate, E., Del Pin, F., A Lagrangian meshless finite element method applied to fluid-structure interaction problems. Computers and Structures 81(8–11), 655–671, 2003. 20. Idelsohn, S., Oñate, E., Del Pin, F., The Particle Finite Element Method: A powerful tool to solve incompressible flows with free-surfaces and breaking waves. International Journal for Numerical Methods in Engineering 61(7):964–989, 2004.

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21. Idelsohn, S., Oñate, E., Del Pin, F., Calvo, N., Fluid-structure interaction using the Particle Finite Element Method. Computer Methods in Applied Mechanics and Engineering 195(17– 18):2100–2123, 2006. 22. Idelsohn, S., Storti, M., Oñate, E., Lagrangian formulations to solve free surface incompressible inviscid fluid flows. Computer Methods in Applied Mechanics and Engineering 191(6– 7):583–593, 2001. 23. Larese, A., Rossi, R., Oñate, E., Idelsohn, S., Validation of the Particle Finite Element Method (PFEM) for simulation of free surface flows. Engineering Computations 25:385–425, 2008. 24. Mier-Torrecilla, M., Numerical simulation of multi-fluid flows with the Particle Finite Element Method. Ph.D. Thesis, Technical University of Catalonia, 2010. 25. Mier-Torrecilla, M., Geyer, A., Phillips, J., Idelsohn, S., Oñate, E., Numerical simulations of negatively buoyant jets in an immiscible fluid using the Particle Finite Element Method. Journal of Fluid Mechanics, 2010 (submitted). 26. Mier-Torrecilla, M., Idelsohn, S., Oñate, E., A pressure segregation method for the Lagrangian simulation of interfacial flows. International Journal for Numerical Methods in Fluids, 2010 (submitted). 27. Oliver, J., Cante, J., Weyler, R., González, C., Hernández, J., Particle finite element methods in solid mechanics problems. In: Computational Plasticity, Vol. 1, pp. 87–103, Springer Verlag, 2007. 28. Oñate, E., Idelsohn, S., Celigueta, M., Rossi, R., Advances in the Particle Finite Element Method for the analysis of fluid-multibody interaction and bed erosion in free surface flows. Computer Methods in Applied Mechanics and Engineering 197:1777–1800, 2008. 29. Oñate, E., Idelsohn, S., Del Pin, F., Aubry, R., The Particle Finite Element Method: An overview. International Journal of Computational Methods 1(2):267–307, 2004. 30. Oñate, E., Rossi, R., Idelsohn, S., Butler, K., Melting and spread of polymers in fire with the Particle Finite Element Method. International Journal for Numerical Methods in Engineering 81:1046–1072, 2009. 31. Osher, S., Sethian, J., Fronts propagating with curvature dependant speed: algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics 79:12–49, 1988. 32. Poinsot, T., Veynante, D., Theorical and Numerical Combustion. Edwards, 2001. 33. Ramaswamy, B., Kawahara, M., Arbitrary Lagrangian-Eulerian finite element method for the analysis of free surface fluid flows. Computational Mechanics 1:103–108, 1986. 34. Ramaswamy, B., Kawahara, M., Lagrangian finite element analysis applied to viscous free surface fluid flow. International Journal for Numerical Methods in Fluids 7:953–984, 1987. 35. Rossi, R., Ryzhakov, P., Oñate, E., A monolithic FE formulation for the analysis of membranes in fluids. International Journal of Space Structures 24:205–210, 2009. 36. Scardovelli, R., Zaleski, S., Direct Numerical Simulation of free-surface and interfacial flow. Annual Reviews of Fluid Mechanics 31:567–603, 1999. 37. Shyy, W., Computational Fluid Dynamics with Moving Boundaries. Taylor & Francis, 1996. 38. Tezduyar, T., Behr, M., Liou, J., A new strategy for finite element computations involving moving boundaries and interfaces. The deforming-spatial-domain/space-time procedure: I. The concept and preliminary numerical tests. Computer Methods in Applied Mechanics and Engineering 94:339–351, 1992. 39. Tezduyar, T., Behr, M., Mittal, S., Liou, J., A new strategy for finite element computations involving moving boundaries and interfaces. The deforming-spatial-domain/space-time procedure: II. Computation of free-surface flows, two-liquid flows and flows with drifting cylinders. Computer Methods in Applied Mechanics and Engineering 94:353–371, 1992. 40. Unverdi, S., Tryggvason, G., Computations of multi-fluid flows. Physica D: Nonlinear Phenomena 60:70–83, 1992. 41. Unverdi, S., Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows. Journal of Computational Physics 100:25–37, 1992. 42. Xie, W., DesJardin, P., An embedded upward flame spread model using 2D direct numerical simulations. Combustion and Flame 156:522–530, 2009.

On Material Modeling by Polygonal Discrete Elements B. Schneider, G.A. D’Addetta and E. Ramm

Abstract The contribution gives an overview on a discrete element model with polygonal particles in a two-dimensional setting allowing the simulation of granular as well as quasi-brittle material response. It briefly describes the basic formulation for geometry and applied contact models in normal and tangential direction, supplemented by friction on a background plate. Special emphasis is put on modeling of cohesion; three different models with an increasing complexity are introduced, namely an overlay brittle beam lattice, a beam with damage and an interface model. Homogenization of the discrete particle response is utilized deriving variables like stresses and strains for an interpretation in the context of classical and micropolar continua. Several numerical examples for different loading scenarios are added, among them the simulation of a quasi-brittle material sample with a heterogeneous microstructure. In addition conceptual small scale experiments with regular particles of steel nuts have been performed; results from tests and simulations for samples with and without cohesion are compared.

1 Introduction The notion of “particle method” is not unique and might even cause confusion, for example when it is used for special discretization concepts of continuum mechanics problems like the Smoothed Particle Hydrodynamic Method (SPH) or the Particle Finite Element Method (PFEM). In the present contribution it is used in the classical sense of the Discrete Element Method (DEM), an area becoming a very successful discipline of its own with applications in many different fields, see the reviews [2– 5, 20].

B. Schneider · G.A. D’Addetta · E. Ramm Institute of Structural Mechanics, Stuttgart University, Pfaffenwaldring 7, 70550 Stuttgart, Germany; e-mail: {schneider, gad, ramm}@ibb.uni-stuttgart.de

E. Oñate and R. Owen (eds.), Particle-Based Methods, Computational Methods in Applied Sciences 25, DOI 10.1007/978-94-007-0735-1_6, © Springer Science+Business Media B.V. 2011

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Fig. 1 Geometry of contact

The present discrete element model describes the mechanical response of materials consisting of separable solid particles on a mesoscopic scale. These materials maybe granular like sand, soil or powder. However the particles can also be initially glued together like in concrete or ceramics and may disintegrate during loading. These materials are sometimes denoted as cohesive-frictional or quasi-brittle. In case the cohesion either initially does not exist or disappears during loading the particles interact only by contact. The geometry of particles can be modeled by smooth contours like circles (2D)/spheres (3D), ellipses/ellipsoids and superellipses/superellipsoids or with sharp corners as polygons/polyhedrons, with triangles/tetrahedrons being a special case. They also may be composed of several elementary geometrical solids. It is obvious that the complexity of the geometrical description varies depending on the number of parameters involved; it comes without saying that the simplest model applying circles/spheres with the radius as the only parameter is by far the most often version in particle simulations.

2 Basic Discrete Element Method In the present study we concentrate on the two-dimensional case applying polygonal particles; although the 2D-model is a coarse simplification of a real material sample it allows to study the mechanical complexity of polygons since particles often exhibit shapes with sharp corners. The applied DEM model with convex polygonal particles is based on the original work of Tillemans and Herrmann [21]. It utilizes a special form of Voronoi tessellation denoted as vectorizable random lattice for mesh generation developed in [19]. The modified version applied in this study has already been described in detail in [7–10], so that the model is only briefly outlined in the sequel. The particles are assumed to be rigid and unbreakable and can interact by contact as well as cohesive forces. The three degrees of freedom of each particle – two translations of the center of mass and one rotation around it – are summarized in the generalized displacement ug = [uT , φ]T = [u1 , u2 , φ3 ]T , see Figure 1. The balance of linear and angular momentum of each individual particle yield the equations of motion

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Mg x¨ g = fg + ˆfg .

(1)

Mg and x¨ g are the generalized diagonal mass matrix and particle acceleration ⎡ ⎤ ⎡ ⎤ M 0 0 x¨1 x¨ g = ⎣ x¨2 ⎦ . Mg = ⎣ 0 M 0 ⎦ ; 0 0 ϕ¨

(2)

M and express the particle mass and mass moment of inertia, respectively. x¨1 and x¨2 are the accelerations in e1 - and e2 -direction and ϕ¨ is the angular acceleration. The right hand side comprises the forces and torques from particle interaction fg = [fT , m]T and from external loads ˆfg = [ˆfT , m] ˆ T , for example gravity. fg is composed g,ct with nct contacting particles and a contribution of a contribution from contact f from cohesion fg,ch with nch bonded particles fg = fg,ct + fg,ch =

nct j =1

g,ct

fj

+

nch

g,ch

fj

.

(3)

j =1

Contact and cohesive forces are introduced in the subsequent sections. For all particles of a sample the equations of motion (1) result in a coupled system of ordinary differential equations; they are numerically integrated by a predictorcorrector scheme introduced in [14] and used also in [1]. We use an explicit version with a predictor and one corrector step which takes into account time derivatives of the displacement up to fifth order; for details of the implemented version of the time integration, see [7].

3 Models for Contact The contact search is in general the most time consuming part in particle dynamics and needs special attention from efficiency point of view, for reviews on various algorithms confer [1, 25]. In general these schemes can be classified into body- and spatial-based categories. In the present study a multilevel approach on the basis of a spatial-based linked cell algorithm is adopted for the preparation of neighborhood lists, compare [1]. This process is described in detail in [7, 21]. When two particles contact each other a small overlap Ao of the rigid bodies is allowed resulting in repulsing contact forces as indicated in Figure 1. The contact force is split into a normal and a tangential component in n- and t-direction ct ct ct fct = fct n + ft = fn n + ft t .

(4)

They are applied at the midpoint of the contact line bo , denoted the contact point. Due to the lever lct they also give rise to a torque mct around the center of mass.

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3.1 Normal Direction For the normal force two alternative models are investigated both consisting of an elastic and a viscous part. The first one was introduced in [21] as (1) E A o n + Meff γn vn n . (5) fct n =− dc (1)

En is the elastic “contact stiffness” which plays the role of a penalty parameter. Ao is the overlapping area, see Figure 1. γn denotes the viscous damping coefficient and vn is the relative velocity in normal direction. dc describes a characteristic length and Meff is the effective mass of two contacting particles i and j 1 1 1 = + ; dc dc,i dc,j

Meff =

Mi Mj . Mi + Mj

(6)

dc,i and dc,j are the diameters of equivalent circles having the same area as the two particles. The second model combines an elastic force proposed in [13] with a viscous force (2) fct (7) n = − En Ao bo + Meff γn vn n . The elastic part can be derived from a potential and is therefore energy conserving. bo is length of the contact line as indicated in Figure 1.

3.2 Tangential Direction Again two different models have been used for the tangential force. The first one [21] selects the minimum of a frictional and a viscous force # $ ct fct (8) t = −sgn(vt ) min µt fn , Meff γt |vt | t . µt is the friction coefficient and γt the viscous parameter. vt denotes the relative velocity in tangential direction. For the second version an elasto-plastic model with Coulomb frictional yield limit similar to the one in [6] has been chosen el fct t = −Et ut t ;

ct F = fct t − µt fn ≤ 0 .

(9)

uel t describes the relative elastic displacement component in tangential direction. F is the yield function again governed by the frictional parameter µt . This model is able to reproduce sticking as well as sliding friction.

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Fig. 2 Overlay of beam lattice and beam deformations by particle movements

3.3 Contact with Background Plate In view of model tests described in Section 6 we also discuss the movement of a particle ensemble on a background plate. For this the force between each particle and the background medium has to be modeled in addition to the interaction between particles. Analogous to the tangential contact model (9) an elasto-plastic model with Coulomb friction is applied at each point on the particle surface contacting the background plate el τ ct b = −Eb ub ;

ct F = ττ ct b − µb |σb | ≤ 0 .

(10)

σbct is the normal stress between the particle surface and the background plate, for example resulting through gravity loads of the particles. The distributed shear stresses τ ct b are integrated numerically to a resulting force and torque acting at the center of the particle.

4 Models for Cohesion In the next step, in addition to the contact forces, cohesive or adhesive forces are introduced if the particles initially stick together or are bonded in their initial state. Cohesion is a complex phenomenon on a fine scale; in the present study we concentrate on models on a coarse scale as a compromise between sufficient realism and efficiency. We will discuss three different models with increasing complexity and expense.

4.1 Brittle Beam In this most simplistic representation cohesion is modeled on the structural level. A beam lattice consisting of small strain shear beams (Timoshenko beams) as an overlay model [16–18, 21] is introduced connecting the centers of mass of all neighbored particles as sketched in Figure 2. The beams are rigidly connected to the particles on both ends and do not possess any mass. Their stiffness can be derived from the elastic potential for beams representing cohesive forces

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⎡

ch

L

= 0

⎤T ⎡

1 εax ⎦ ⎣ ⎣ κ 2 γ %

⎤⎡

⎤

0 EAch 0 εax 0 EI ch 0 ⎦ ⎣ κ ⎦ dL , γ 0 0 GAch Q &' ( Cel

(11)

where εax , κ, γ are the axial strain, curvature and transverse shear strain, respectively. The elastic material matrix Cel contains the axial stiffness EAch , bending stiffness EI ch and shear stiffness GAch Q with Young’s modulus E, cross sectional area A, moment of inertia I and shear area AQ . From Eq. (11) the usual stiffness expression for a Timoshenko beam element can be derived g g,ch fi ch ui = K (12) g,ch ij ug . fj j It yields the generalized forces fg,ch derived from the generalized displacements ug of the particles. Each beam deforms with the six degrees of freedom of the particles as shown in Figure 2 (right) leading to additional forces and torques on both particles i and j caused by vectors containing the generalized beam end forces. The individual beam behaves initially elastic until a cracking criterion is satisfied

εax εmax

2 +

max(|φi |, |φj |) = 1. φmax

(13)

Cracking is allowed only if the axial strain εax is positive and therefore only applied under tension. Concerning the bending, only the maximum value of the end rotations φi or φj is monitored. εmax and φmax are given threshold values for breaking combined to the interaction or failure criterion (13). The beam connections representing cohesion are checked after each time step considering the updated coordinates of the centers of mass of the particles. Overstressed beams break and are removed from the calculation. The present beam model represents an extremely brittle failure and sudden loss of cohesion in the sense of a “tension cut-off” criterion. It might however be extended by a softening type of gradual failure. It has to be kept in mind that the location for monitoring the cohesion is transferred from the contact zone to the centers of the particles. Despite of the mentioned limitations it can be stated that it is a very simple, inexpensive and easy to implement model. It is also well known that pure lattice models have been successfully applied for the simulation of quasi-brittle materials.

4.2 Beam with Damage In order to reproduce less brittle, i.e. more ductile failure behavior, a second still efficient model is introduced. It keeps the shear beam as structural model which de-

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Fig. 3 (a) Damage evolution and (b) damage surface

forms with the degrees of freedom of the particles; however in this case the failure of the beam is transferred to the cross section at the middle between two particles. The material model accounts for a successive degradation of the cohesive connection in the sense of a strain resultant criterion. For this purpose the elastic beam element with zero mass is enhanced by a damage model with softening instead of using the above introduced brittle failure criterion. The degradation of E is represented by the usual factor (1 − d) where d is the isotropic damage parameter yielding the elasto-damage material matrix ⎤ ⎡ 0 0 (1 − H d)EAch ⎦ ; H = 0 for εax ≤ 0 . 0 (1 − d)EI ch 0 Ced = ⎣ 1 for εax > 0 0 0 (1 − d)GAch Q (14) The Heaviside function H is inserted in order to represent the unilateral character for the axial stiffness, only present in tension but not in compression. The evolution of damage is defined by linear softening exemplarily depicted in Figure 3a for axial elongation ⎧ ⎪ : ε ∗ ≤ ε0 ⎪0 ⎨ ε m ε ∗ − ε0 d= ε ∗ (t) = max ε˜ (τ ) . (15) : ε0 < ε ∗ < εm ; ⎪ 0≤τ ≤t ε∗ εm − ε0 ⎪ ⎩1 : ε∗ ≥ ε m

ε∗ is the largest strain level reached at each time. The maximum strain level is denoted by εm when the beam section is completely damaged (d = 1) and fails. The damage mode is characterized by a combined axial strain-bending-shear failure at the middle between two particles, defined by the equivalent strain ε˜ =< εax + |κmid |

h > +|γmid| . 2

(16)

The damage surface is displayed in Figure 3b. The measure combines the strains from axial elongation εax , curvature κmid and shear deformation γmid at the middle cross section; h is the height of the beam. As indicated by the Macaulay bracket for the first part only tensional strains are considered. They are evaluated at the outermost fiber (marked by a cross in Figure 4) where the material is stretched most;

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Fig. 4 Strains of beam

Fig. 5 Numerical integration with reference edge

the strain distribution following beam theory is displayed in Figure 4. According to the assumed kinematics the shear strain γ is constant across the height so that only its absolute value enters the equivalent strain; it is the only variable which defines the failure under compression. This damage model exploiting the kinematics of a beam at the middle between two particles is able to reproduce a more ductile failure behavior for cohesion being still numerically inexpensive.

4.3 Softening Interface The third and most advanced model refines the previous version. A continuous connection between two particles in a sense of a zero-thickness interface element is assumed. The model adopts again the kinematics with a linear displacement field across the contact line as in the previous beam model (“plane sections remain plane”) however allows a complex stress distribution after successive debonding of the interface. This material degradation at each point of the interface is described by small strain, non-associated plasticity coupled in normal and tangential direction by a two-surface Mohr–Coulomb yield function [24]. The evolving stress distribution is numerically integrated at a finite number of points along the interface height h, computing the resulting force and torque on the particle similar to stress resultants in a beam, see Figure 5. The particle i has been chosen as master; its bond has been selected as reference edge. The local coordinate system n-t defining the components of the displacement field as well as the normalized coordinate ξ for integration is fixed along this edge. Having this discretization

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Fig. 6 Idealization of interface by springs in (a) undeformed and (b) deformed stage

Fig. 7 (a) Yield surface and (b) plastic potential for interface in biaxial stress plane

in mind one could idealize the model as a series of non-linear springs coupled in normal and tangential direction as depicted for an undeformed (with zero thickness) and deformed stage in Figure 6. In the elastic regime uncoupled stiffnesses kn and kt are assumed for the springs in both directions. This interface model has been documented in detail in [7, 8, 11]; here only some basic ingredients are mentioned. Figure 7 displays the yield surface and the plastic potential in the biaxial σt -σn plane, defined by the functions f1 /f2 and g1 /g2 , remax the angles γ and ϕ as well spectively. Besides the limit values for the stresses σn/t as ψ = ϕ control the shape of failure envelopes. In these diagrams κ denotes the 1 an softening variable; three stages are marked from the initial undamaged state , 2 to the completely softened state . 3 The intermediate partially damaged situation distinction between yield function and plastic potential is necessary because certain geomaterials do not obey the normality condition. p p The softening driven by the plastic displacements un and ut at the integration points (“springs”) is controlled by predefined fracture energies Gf,n and Gf,t . Two different softening evolutions for tension and shear are introduced, see Figures 8a and b. The linear softening functions describe the uncoupled cases when either a pure tension or a pure shear failure mode governs the degradation of the interface. A simultaneous coupled tensile and shear softening is treated as linear combination of both modes as depicted in Figure 8c for the extreme case of κ = 1.

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Fig. 8 Evolution of softening for (a) tension, (b) shear and (c) a combination thereof

The explicit mathematical model for the interface material is described in [7, 8], where also the numerical realization is explained. It follows the usual small strain plasticity concept, applying return algorithms for the time integration with an elastic predictor and a plastic corrector. When all integration points have reached complete softening the bond is detached. The standard contact as described in Section 3 comes into play once two initially bonded particles with broken bond again contact each other.

5 Homogenization 5.1 Motivation Homogenization is a common procedure in order to determine the response of heterogeneous materials on a macroscale. It is well known that these concepts are based on the existence of a clear scale separation, also known as Hashin’s MMM (MicroMeso-Macro) principle of homogenization [15]. In the present context it means that there is a distinct scale difference between the particles (microscale) and the entire structure (macroscale) allowing establishing a Representative Volume Element (RVE) on an intermediate mesoscale; this is expressed by the condition D d δ as indicated in Figure 9 for an arbitrary particle ensemble. In contrast to the usual application of RVEs determining constitutive relations and material parameters for a macroscopic analysis we concentrate on homogenization of the fluctuating response during the non-linear discrete element analysis. The resulting mechanical quantities such as stresses and strains allow a better interpretation of the overall response in terms of continuum mechanics and in particular of the influence of distinct material parameters. A key question is related to the definition and size of an RVE. On the one hand it should not be too large being still able capturing local effects like evolving shear bands. On the other hand it should be big enough avoiding small scale fluctuations, see Section 5.5.

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Fig. 9 Structure consisting of particle ensembles (RVE)

Fig. 10 Representative Volume Element (RVE): (a) definition and (b) base particles

5.2 Representative Volume Element (RVE) A particle ensemble with N particles in diameter, denoted as RVEN, is cut out around each so-called base particle in the center of the RVE for which the homogenized variables are determined, see Figure 10a. These are all particles inside the structure leaving out the particles close to its edge (Figure 10b) because the RVEs are not complete and not representative anymore. The size of the RVE defined at the beginning is not changed during the non-linear analysis. The variables are plotted on the base particle for which they are calculated.

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Fig. 11 Boundary of RVE with force f

5.3 Averaging Procedure 5.3.1 Stresses According to the scale separation argument d/D is sufficiently small so that the volumetric contributions to the balance equations may be neglected compared to the surface contributions. It means that only the forces f of the boundary particles will enter the averaged balance equation, see Figure 11. Based on this assumption the balance of momentum reads divT = 0 ;

0=

divT dv = R

t da = ∂R

n∂R

fi .

(17)

i=1

T is the Cauchy stress tensor within the RVE volume R; t is the traction vector on the RVE boundary ∂R. The integral is discretized and replaced by the sum of forces f at the RVE boundary acting from outside the RVE on the individual particles. n∂R is the number of particles on the entire boundary. The volume average of the stresses T taken over the RVE yields 1 T = T dv (18) V R where TT = div(xM ⊗ T) − %xM ⊗&'divT( .

(19)

=0

xM

is the position vector from the RVE center to the boundary particle center. Following Eq. (17) the last term in Eq. (19) vanishes. Inserting (19) into (18) leads to 1 1 1 TT = (xM ⊗T)n da− xM ⊗ divT dv = xM ⊗%&'( Tn da = TT . V ∂R V R V ∂R &' ( % t =0 (20)

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Fig. 12 Displacement field at RVE boundary

Using Eq. (17) this yields 1 T = V

n∂R 1 t ⊗ x da = fi ⊗ xM i . V ∂R M

(21)

i=1

Again the integral has been replaced by the sum over all boundary particles in the sense of the discrete element concept.

5.3.2 Strains The same concept of homogenization can be applied to kinematic variables. For the average strain we start from the discrete displacements of boundary particles from which a continuous polygonal displacement field can be developed, see Figure 12. This linear displacement field is integrated along the RVE boundary and inserted into the averaged strain expression 1 ε sym dV εε sym = V R 1 1 = (u ⊗ n + n ⊗ u) dA (22) V ∂R 2 n ∂R 1 1 ∆wi [(ui + ui+1 ) ⊗ ni + ni ⊗ (ui + ui+1 )] . = V 4 i=1

The procedure ends in a summation along all boundary particles leading to the strain field projected on the base particle. In a similar way other variables can be obtained, for example the spatial velocity gradient. It is also mentioned that this consistent homogenization satisfies the balance of energy, known as the Hill condition.

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Fig. 13 Boundary of RVE with force f and moment m

5.4 Extension to Higher Order Continua In order to capture the microstructure of a material in the neighborhood of a material point higher order continua have been introduced in the past. If these neighborhood influences are taken into account the homogenization of the response for the particle ensemble can be extended to these theories allowing deriving higher order dynamic and kinematic variables. Here we concentrate on a micropolar (Cosserat) continuum and briefly mention the possibilities for a gradient continuum.

5.4.1 Micropolar Continuum The key assumption in this case is that the forces on the outer surfaces of the boundary particles are transferred to the center of the elements producing extra moments in addition to the forces already seen above, see Figure 13 compared to Figure 11. The extra moments m are due to the contact forces fct as well as the contributions mch from beams or interfaces representing cohesion with the n˜ ct and n˜ ch particles outside the RVE, respectively m = mct + mch =

n˜ ct j =1

ct lct j × fj +

n˜ ch

mch j .

(23)

j =1

lct is the so-called branch vector, see Figure 1. In the averaging process two moments can be identified, namely the continuum ¯ of moment or moment of the stress distribution M and the couple stress tensor M the Cosserat theory # $ 1 ¯ n ⊗ xM da ¯ M+M M + M = V ∂R n∂R (24) 1 M (xM × f + m ) ⊗ x = i i i i V i=1

with

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Fig. 14 Particle ensemble in reference and actual configuration

¯ = M

n∂R 1 mi ⊗ x M i . V

(25)

i=1

It is worth mentioning that moments on a mesoscale are necessary transferring the particle concept through the homogenization procedure into an enhanced continuum theory, in the present case a macroscopic Cosserat method. If the moments are not taken into account, the couple stresses vanish and the stress tensor becomes symmetric. In a similar process average values for kinematic variables like rotations and curvatures or additional quantities like energies can be derived. Further details for the homogenization process are given in [7, 12].

5.4.2 Gradient Continuum In this case the key idea is an enhanced kinematic description of an RVE via a Taylor series approximation for the difference vector of two particles, see Figure 14. In other words higher gradients of deformations are taken into account in the homogenization procedure. In this way the intrinsic statistical feature of the heterogeneous material is automatically included. For the homogenization of the kinematics the position vector of a target particle N is expanded in a vector Taylor series around the base particle M xN t = +

1 M 1 ∂xM 1 ∂ 2 xM t t + : ∆x0 ⊗ ∆x0 xt + · ∆x 0 M 2 0! 1! ∂(xM 2! ) ∂(x 0 0 ) .. 1 ∂ 3 xM t ˜ . ∆x0 ⊗ ∆x0 ⊗ ∆x0 + w M 3 3! ∂(x0 )

. ˜. = xM F · ∆x0 + G : ∆x0 ⊗ ∆x0 + K .. ∆x0 ⊗ ∆x0 ⊗ ∆x0 +w t +% &' ( approx ∆xt

(26)

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Fig. 15 Biaxial test: (a) initial and (b) deformed structure

In the present equation the series is truncated after terms of third order leaving the ˜ Equation (26) intrinsically contains the higher order terms in the residual vector w. deformation gradient F and the second and third deformation gradient tensors G and K. All deformation tensors are evaluated in the sense of an RVE and represent average quantities. For the sake of brevity, the reader is referred to [7] where further details of the homogenization procedure are given.

5.5 Size of RVE With the following example we want to discuss the above mentioned dilemma between a too big and a too small RVE size. It is a biaxial test with about 2500 polygonal particles under constant side pressure q = 1 kN/cm2 and a constant increase of velocity of the top and bottom plate (Figure 15). The size of the sample is 64 cm × 40 cm. The contact models (5) and (8) were used with the material data: Young’s modulus En(1) = 100 kN/cm2 , density = 5 g/cm3 , γn = 105 , µt = 0, γt = 0; the time increment is ∆t = 1 · 10−6 s. No cohesion is assumed. In the initial and the deformed structure the vertical center line is marked easing the identification of the particles after deformation. In the right picture two evolving shear zones are indicated clearly visualized by the two offsets of the marked particles. Figure 16 shows the overall response in form of a normalized load-displacement relation. After a certain load is reached shear bands evolve resulting in a sudden brittle failure. This part is zoomed out in the figure indicating three stages, marked 1 to . 3 In Figure 17 the vertical stress normalized to the average stress (“load”) by obtained by averaging via RVEs is shown. The vertical stresses are evaluated by homogenization along the marked vertical center line from the bottom to the top.

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Fig. 16 Biaxial compression test: normalized load-displacement diagram

1 to 3 for RVE3, RVE7, RVE13 and RVE39 (at Fig. 17 Influence of size of RVE: three stages each stage from left to right)

Four different sizes of RVE ranging from a very small RVE3 to an extremely 1 to . 3 large one RVE39 are investigated for the three loading stages It can be clearly recognized that the small RVE shows distinct fluctuations even anticipating 2 The bigger the RVE the more shear banding with diminishing stresses in stage . the stresses are smoothed out along the line, so no local effects can be seen anymore. Evaluating other samples it turned out that the size RVE5 represents a sufficient compromise between too small (undesired fluctuations) and too large (too much smoothing) RVEs.

6 Examples 6.1 Samples without Cohesion 6.1.1 Biaxial Compression Test of Granular Material In contrast to the example discussed in Section 5.5 with a dense material, a porous granular material is investigated first followed by a wide biaxial compression test. The mesh generation follows the principles producing a random lattice via Voronoi tessellation from which a dense particle sample without pores is derived [19], 1 In the next step all particles are randomly scaled down by see Figure 18 (stage ).

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Fig. 18 Construction of porous packing with random scaling by 30–60%

Fig. 19 (a) Initial and (b) deformed state of porous biaxial test

2 a given value (stage ); they also may be rotated by a statistically measure. All particles are positioned into a box applying gravity load through the DEM method3 ology. When all particles come to rest (stage ) the top of the sample is cut getting 4 a smooth boundary (stage ). The concept is applied to the porous sample shown in Figure 19 with the same data as the previous example in Section 5.5. In the deformed state no clear shear band could be observed, however a zoom indicates a distinct thickness change of the zone marked in dark grey. A comparison of both zoomed details reveals the clear densification of the sample. Next the height to width ratio of dense specimen, investigated in Section 5.5, is changed from 1 : 1.6 to 1 : 0.37 without modifying the test set-up. Now the rectangular block with 38 cm × 102 cm consists of 3800 particles. After vertical loading distinct zig-zag shear bands evolve (Figure 20) which are also clearly indicated by the horizontal strains ε11 in Figure 21.

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Fig. 20 Wide biaxial compression test: deformation at late stage of the simulation

Fig. 21 Wide biaxial compression test: horizontal strains at early (left) and late (right) stage of the simulation

Fig. 22 Experiment with steel nuts

6.1.2 Uniaxial Compression of Model Material The following conceptual test has been chosen in order to investigate the quality of the present DEM on regular particles without a geometrical bias. A rectangular sample of 217 hexagonal steel nuts (17 cm × 19 cm) lying on a background steel plate has been tested under uniaxial compression, see experimental setup in Figure 22. One loading platen is moved to the front by a constant low velocity vˆ2 = −6.2 · 10−4 m/s whereas the other one is fixed. For the simulation the geometry of the nuts with a hole and slightly round corners has been approximated by sharp edged solid hexagons. Contact in normal (7) and tangential (9) direction as well as with the background plate (10) is modeled. The data for the model without cohesion are experimentally verified: = 3.3 g/cm3 , µt = 0.29 and µb = 0.26; (2) the stiffnesses are chosen as En = 1.0 · 102 N/cm4 , Et = 2.9 · 102 N/cm2 , Eb = 8.4 · 102 N/cm3 and the time increment as ∆t = 1 · 10−5 s. In order to break symmetry of the sample the normal contact stiffness of the particle besides the center at the upper row is reduced by 10%.

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Fig. 23 Comparison of experiment and simulation

In Figure 23 results from experiment and simulation are compared at different time instants. The grey scale constitutes the velocity in e2 -direction. Both cases initially show a diffuse lateral expansion followed by the evolution of distinct shear bands. The mode shown at t = 22.4 s is considerably changed until t = 41.2 s. This change is inherent in both, the experiment as well as the DEM simulation.

6.2 Samples with Cohesion 6.2.1 L-shape Test The L-shape specimen is mimicked after a concrete benchmark experimentally tested in [26]. The data of the model material are taken from Section 5.5 using the contact models (5) and (8), however in this case cohesion is modeled by a brittle beam lattice described in Section 4.1; the beams have a Young’s modulus of E = 1000 kN/cm2 ; the threshold values for cracking are εmax = 0.03 and φmax = 3◦ . The specimen is loaded by a uniform displacement of the boundary particles at the right edge (Figure 24a). The load-displacement diagram is shown in Figure 24b demonstrating that the failure initiated at the reentrant corner is extremely brittle. Although the data have not been fitted to the tests the crack patterns shown in Figure 25 fit the experimental results qualitatively very well.

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Fig. 24 (a) L-shape specimen and (b) load-displacement diagram

1 4 Fig. 25 L-shape specimen: crack evolution –

6.2.2 Uniaxial Compression of Cohesive Model Material We use the same hexagonal steel nuts as in Section 6.1.2 however bond them together by standard glue. In this case the sample stands between two glass panes in the testing device, see Figure 26a. The friction between particles and panes is so low that it could be neglected. Again a prescribed velocity vˆ2 = −3.3 · 10−5 m/s is applied to the upper row of particles leaving the other degrees of freedom free to move. For the present study the cohesion between particles (adhesion through glue) is modeled by the beam with damage, confer to Section 4.2. The data are = 3.3 g/cm3 , E = 28 N/cm2 , ε0 = 1.1 · 10−3 , εm = 2.3 · 10−1 ; the time increment is chosen as ∆t = 1 · 10−4 s. Figure 26b compares the particle samples with 1 and 22 particles for experiment and simulation in the undeformed configuration

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Fig. 26 (a) Experiment with glued steel nuts and (b) comparison of experiment and simulation at 1 and deformed 2 stage undeformed

2 the figure also shows the detached glue connections in stage 2 deformed stage ; where the respected beams of the overlay lattice have already reached the descending softening branch and are damaged. The horizontal beams are mainly elongated whereas the diagonal ones are essentially sheared. The pattern of localization is identical to that of the simulation. For three experiments the load-displacement diagrams are displayed in Figure 27 and supplemented by the smooth curve from simulation applying the beam with damage. Although geometry and material of the nuts do not differ very much from each other the scatter of the results for all three experiments is clear; the deviation is caused by the rather primitive gluing process. On the other hand the simulation represents a kind of overall mean response having a lower failure load though. The rather smooth transfer into the post-critical regime in the experiments is due to the more ductile failure of the glue under shear compared to the material model applied in the analysis. Increasing the strains ε0 and εm at beginning and end of softening (see Figure 3a) leads to the expected results, namely an increase of failure load with a steeper descend for ε0 and increased energy dissipation for higher values of εm without a change of the peak load.

6.2.3 Modeling of Microstructure for Concrete under Uniaxial Compression For a detailed investigation of concrete the microstructure has to be modeled differentiating between stiffness and strength of aggregates, matrix and their bond layer, see Figure 28a. According to van Mier et al. [22] the ratios for stiffnesses and strengths for all three phases are

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Fig. 27 Load-displacement diagram: three experiments with glued particles (thin lines) and simulation (thick line)

(b)

kn

(m)

kn

= 0.1 ;

(a)

kn

kn(m)

max,(b)

(b)

kt

(m)

kt

= 0.4 ;

(a)

= 2.8 ;

kt

kt(m)

= 2.8 ;

σn/t

max,(m)

σn/t max,(a) σn/t

max,(m) σn/t

= 0.25 = 100 .

The cohesive connections between grains are stiffer and have higher strength than those between matrix particles. The connections at the bond layers between grain and matrix particles are in turn even less stiff and fail at lower loading. The elastoplastic interface described in Section 4.3 is adopted modeling the cohesion for all three materials. The lattice shown in Figure 28b is only added for visualization of the respective bond condition during loading. For the matrix following properties are chosen: normal and tangential stiffnesses (m) (m) (m) = 600 kN/cm2 ; fracture energies Gf,n = 4.996 · kn = 2000 kN/cm2 , kt

10−4 kN/cm2 , Gf,t = 5.988 · 10−3 kN/cm2 . The yield stresses are statistically dis(m)

tributed for each interface about ±10% the average values σnmax,(m) = 0.04 kN/cm2 , σtmax,(m) = 0.12 kN/cm2 . The shape parameters are ϕ = 26.6◦ , γ = 10◦ , ψ = 0◦ and the density is = 2.5 g/cm3 . The contact models (5) and (8) have been used with the contact stiffness En(1) = 100 kN/cm2 . Viscous damping as well as friction are set to zero (γn = γt = 0; µt = 0) and the time step is chosen as ∆t = 5 · 10−7 s.

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Fig. 28 Microstructure with matrix, bond and aggregate

Fig. 29 Failure evolution during simulation sim∗1

Applying this microstructure enhanced discrete element model uniaxial displacement driven compression tests are simulated, see the evolution of simulation sim∗1 in Figure 29. In Figure 30b the black lines mark the locations where the cohesive interfaces are completely eliminated, i.e. κ = 1. The cracks run primarily along the aggregate boundaries as it is typical for concrete. In Figure 30a the nominal stresses are plotted versus the nominal strain representing a load-displacement diagram. Besides simulation sim∗1 the average sim∗i of seven simulations with different statistically

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4 Fig. 30 (a) Load-displacement diagram and (b) eliminated interfaces of sim∗1 at stage

varied samples are shown and compared to three experimental results performed with different loading platens by Vonk [23]. The ascending branches as well as peak loads are very well reproduced by the simulation; there is some deviation in the post-critical regime though.

7 Conclusions The response of quasi-brittle materials using polygonal particles is characterized by more realism but also larger complexity and effort compared to circular particles. Besides contact search modeling of cohesion is a further parameter increasing the expense of the simulation. We have discussed three models for cohesion with an increasing effort. The lattice model of brittle beams is extremely efficient but results in a sudden failure mechanism. On the other side of the spectrum the interface model is the most realistic version; however it is also the most time consuming model. The beam with damage is a compromise between the two other models. Its quality has been checked also in relation to experimental results obtained from conceptual tests using conventional steel nuts as particles; by this the geometrical bias usually inherent in a sample of particles is diminishing. The tests have been performed for ensembles of particles which are loosely connected or glued together representing cohesion (adhesion). It turned out that the agreement between experiments and simulations are qualitatively excellent and quantitatively sufficient considering the substantial scatter of the material parameters.

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Acknowledgement The authors are indebted for the financial support of the German Research Foundation (DFG) within the research project “Fragmentierung kohäsiver Reibungsmaterialien mit diskretem Partikelmodell” under grant no. RA 218/22-1.

References 1. Allen, M.P., Tildesley, D.J., Computer Simulation of Liquids. Oxford University Press, Oxford, 1987. 2. Bi´cani´c, N., Discrete element methods. In: Encyclopedia of Computational Mechanics: Volume 1: Fundamentals, Stein, E., de Borst, R., Hughes, T.J.R. (eds.), pp. 311–337. Wiley, Chichester, 2004. 3. Bi´cani´c, N., Discrete element methods. In: The Finite Element Method for Solid and Structural Mechanics, 6th edn., Zienkiewicz, O.C., Taylor, R.L., pp. 245–277. Elsevier, Oxford, 2005. 4. Cundall, P.A., A discontinuous future for numerical modelling in geomechanics. Geotech. Eng. 149:41–47, 2001. 5. Cundall, P.A., Hart, R.D., Numerical modelling of discontinua. Eng. Comput. 9:101–113, 1992. 6. Cundall, P.A., Strack, O.D.L., A discrete numerical model for granular assemblies. Géotechnique 29:47–65, 1979. 7. D’Addetta, G.A., Discrete models for cohesive frictional materials. Ph.D. Thesis, Bericht Nr. 42, Institut für Baustatik, Universität Stuttgart, Germany, http://elib.unistuttgart.de/opus/volltexte/2004/1943/pdf/diss_gad.pdf, 2004. 8. D’Addetta, G.A., Ramm, E., A microstructure-based simulation environment on the basis of an interface enhanced particle model. Granul. Matter 8:159–174, 2006. 9. D’Addetta, G.A., Kun, F., Herrmann, H.J., Ramm, E., From solids to granulates – Discrete element simulations of fracture and fragmentation processes in geomaterials. In: Continuous and Discontinuous Modelling of Cohesive Frictional Materials, Vermeer, P.A., Diebels, S., Ehlers, W., Herrmann, H.J., Luding, S., Ramm, E. (eds.), Lecture Notes in Physics, vol. 586, pp. 231–258. Springer, Berlin, 2001. 10. D’Addetta, G.A., Kun, F., Ramm, E., On the application of a discrete model to the fracture process of cohesive granular materials. Granul. Matter 4:77–90, 2002. 11. D’Addetta, G.A., Schneider, B., Ramm, E., Particle models for cohesive frictional materials. In: Computational Modelling of Concrete Structures, Proceedings of EURO-C 2006, Meschke, G., de Borst, R., Mang, H., Bi´cani´c, N. (eds.), Mayrhofen, Austria, 27–30 March 2006, pp. 269–280. Taylor & Francis, London, 2006. 12. Ehlers, W., Ramm, E., Diebels, S., D’Addetta, G.A., From particle ensembles to Cosserat continua: Homogenization of contact forces towards stresses and couple stresses. Int. J. Solids & Struct. 40:6681–6702, 2003. 13. Feng, Y.T., Owen, D.R.J., A 2D polygon/polygon contact model: Algorithmic aspects. Eng. Comput. 21:265–277, 2004. 14. Gear, C.W., Numerical Initial Value Problems in Ordinary Differential Equations. PrenticeHall, Englewood Cliffs, 1971. 15. Hashin, Z., Analysis of composite materials – A survey. ASME J. Appl. Mech. 50:481–505, 1983. 16. Herrmann, H.J., Hansen, A., Roux, S., Fracture of disordered, elastic lattices in two dimensions. Phys. Rev. B 39:637–648, 1989. 17. Kun, F., Herrmann, H.J., A study of fragmentation processes using a discrete element method. Comput. Methods Appl. Mech. Eng. 138:3–18, 1996.

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18. Kun, F., D’Addetta, G.A., Herrmann, H.J., Ramm, E., Two-dimensional dynamic simulation of fracture and fragmentation of solids. Comput. Assist. Mech. Eng. Sci. 6:385–402, 1999. 19. Moukarzel, C., Herrmann, H.J., A vectorizable random lattice. J. Stat. Phys. 68:911–923, 1992. 20. Pöschel, T., Schwager, T., Computational Granular Dynamics: Models and Algorithms. Springer, Berlin, 2005. 21. Tillemans, H.-J., Herrmann, H.J., Simulating granular solids under shear. Phys. A 217:261– 288, 1995. 22. van Mier, J.G.M., Schlangen, E., Vervuurt, A., van Vliet, M.R.A., Damage analysis of brittle disordered materials: Concrete and rock. In: Mechanical Behaviour of Materials, Proceedings of the ICM7, Bakker, A. (ed.), pp. 101–126. Delft University Press, Delft, 1995. 23. Vonk, R., Influence of boundary conditions on softening of concrete loaded in compression. Report TUE/BKO 89.14, Faculteit Bouwkunde, Technische Universiteit Eindhoven, The Netherlands, 1989. 24. Vonk, R., Softening of concrete loaded in compression. Ph.D. Thesis, Technische Universiteit Eindhoven, The Netherlands, 1992. 25. Williams, J.R., O’Connor, R., Discrete element simulation and the contact problem. Arch. Comput. Methods Eng. 6:279–304, 1999. 26. Winkler, B.J., Traglastuntersuchungen von unbewehrten und bewehrten Betonstrukturen auf der Grundlage eines objektiven Werkstoffgesetzes für Beton. Ph.D. Thesis, Institut für Baustatik, Festigkeitslehre und Tragwerkslehre, Leopold-Franzens-Universität Innsbruck, Austria, 2001.

Discrete Numerical Analysis of Failure Modes in Granular Materials Luc Sibille, Florent Prunier, François Nicot and Félix Darve

Abstract The question of failure for geomaterials, and more generally for nonassociative materials, is revisited through the second-order work criterion defining, for such media, a whole domain of bifurcation included in the plastic limit surface. In a first theoretical part of the chapter, relations between the vanishing of the second-order work, the existence of limit states and the occurrence of failures characterized by a transition from a quasi-static pre-failure regime to a dynamic post-failure regime, are presented and illustrated from discrete element computations. Then boundaries of the bifurcation domain and cones of unstable loading directions are given in fully three-dimensional loading conditions for a phenomenological incrementally non-linear relation, and in axisymmetric loading conditions for a numerical discrete element model. Finally, conditions for the triggering and the development of failure inside the bifurcation domain are described and emphasized from direct simulations with the discrete element method for proportional stress loading paths.

Luc Sibille Laboratoire GeM – Université de Nantes, ECN, CNRS – IUT de St-Nazaire, BP 420, 44606 St-Nazaire Cedex, France; e-mail: [email protected] Florent Prunier LGCIE, INSA-Lyon, Université de Lyon, 69621 Villeurbanne, France; e-mail: [email protected] François Nicot Unité ETNA, Cemagref de Grenoble, BP 76, 38402 St-Martin-d’Hères Cedex, France; e-mail: [email protected] Félix Darve Laboratoire 3S-R – INPG, UJF, CNRS, BP 53, 38041 Grenoble Cedex 9, France; e-mail: [email protected]

E. Oñate and R. Owen (eds.), Particle-Based Methods, Computational Methods in Applied Sciences 25, DOI 10.1007/978-94-007-0735-1_7, © Springer Science+Business Media B.V. 2011

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1 Introduction The question of failure is central in engineering and it has been tackled from theoretical, experimental and numerical points of view. This question is delicate, particularly for geomaterials (soils, rocks, concretes) because of the non-associative character of their plastic strains, which implies the existence of a whole domain of bifurcations in the stress space and not only a single plastic limit surface concentrating all kinds of failures as for associative materials. From a theoretical point of view the existence of such a bifurcation domain was clearly established since the works by Hill [9]. Indeed, roughly speaking, a nonassociative material has a non-symmetrical elasto-plastic matrix. Thus the secondorder work criterion, linked to vanishing values of the determinant of the symmetric part of this constitutive matrix, can be satisfied before the plastic limit condition, linked to vanishing values of the determinant of the constitutive matrix itself. The existence of failure states before the plastic limit has been also proven by Rice [16] since for a non-associative material the determinant of the acoustic tensor (criterion for shear band formation) can vanish in the hardening regime (i.e. before the plastic limit condition). In the first part of this paper, the theoretical framework is recalled by emphasizing the link between second-order work, kinetic energy and limit states. Indeed failure is due to the existence of some limit states (classically limit stress states). Besides, failure is associated to some bursts of kinetic energy at the transition from the quasi-static pre-failure regime to the dynamic post-failure regime. Thus, the first part is devoted to clarify the link between these three notions. Then a phenomenological elasto-plastic relation (an incrementally non-linear model) applied to 3D loading conditions allows to show the boundary of the bifurcation domain in 3D and the instability cones, where the second-order work is taking negative values. From an experimental point of view, some failure states strictly inside Mohr– Coulomb’s surface have been observed since many years typically for given loading paths. The most classical ones giving rise to a diffuse mode of failure are the undrained triaxial compressions on a loose sand for axially force controlled tests and the drained triaxial tests with a constant deviatoric stress for a constant injection rate of water inside the sample (see, for example, [5]). Localized failures have been also observed repeatedly experimentally in the hardening regime on dense sands [6]. From a numerical viewpoint, the modelling of failure modes with a numerical investigation of all the mechanical/geometrical details is a difficult task with the finite element method, essentially because such a numerical method is typically not well adapted to the description of bifurcation states. This is the reason why this paper is devoted to a numerical investigation of failure inside granular materials through a discrete element method. This method has indeed the great advantage to simulate failure in a very realistic and natural way. Thus, the main part of this paper is devoted to the analysis of failure inside a cubical specimen of 10,000 spheres. It will be observed first that, in close agreement with experiments and theory, there exists a stress bifurcation domain strictly inside Mohr–Coulomb’s surface and that some instability cones gather the stress directions where the second-order work is taking

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negative values. Besides, elasto-plastic theory shows that there are three necessary and sufficient conditions for an effective failure: 1. the stress state has to be inside the bifurcation domain, 2. the loading direction must belong to the current instability cone, 3. the (energy conjugate) loading variables must be mixed ones (i.e. some stresses and strains). The capacity of discrete element method to check these three necessary and sufficient conditions and to simulate all the features of diffuse failure (exponentially growing strains, burst of kinetic energy, decreasing intergranular stresses, not any localization pattern) will be remarkably illustrated in the last part of the chapter.

2 Theoretical Background Considering a soil specimen, failure can be localized (the kinematic field experience a discontinuous aspect) or diffuse (no localization pattern is visible). In this chapter, we focus on the diffuse failure of soil specimens, related to an exponential increase in strain rates, with outbursts in kinetic energy. That corresponds to the transition from a quasi-static regime (the system reaches an equilibrium state under the external loading) toward a dynamic regime (the internal stress inside the system can no longer balance the external loading, leading to dynamic effects). Thus, this section investigates in which conditions kinetic energy of a soil specimen, initially in equilibrium after a given loading path, may increase (passing from zero to a strictly positive value) over an infinitesimal loading.

2.1 Kinetic Energy and Second-Order Work For this purpose, let us consider a system of volume Vo , initially in a configuration Co . After a loading history, the system is in a strained configuration C and occupies a volume V , in equilibrium under a prescribed external loading. An external stress distribution f acts on the current boundary ( ) of the material. The instantaneous change in the system, in the equilibrium configuration C at time t, is governed by the following energy conservation equation that includes dynamic effects: ∂(δui ) fi δui dS − σij dV (1) δEc (t) = ∂xj

V where δEc represents the system’s current change in kinetic energy related to the incremental displacement field δu, and σ is the Cauchy stress tensor. Equation (1) represents the Eulerian form of the energy conservation, since all variables are given

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with respect to the current evolving configuration. The notation δX represents the ˙ incremental change of any variable X, equal to Xδt. It is of course more convenient to come back to the initial configuration Co , which is fixed. This transformation can be operated on both integrals of Eq. (1), leading to the Lagrangian form of energy conservation [11]: ∂(δui ) δEc (t) = Fi δui dSo − ij dVo (2) ∂Xj

o Vo where denotes the Piola–Kirchoff stress tensor of the first type and o is the Vo boundary. and F are, respectively, the transformed quantities of σ and f through the bijection mapping the material points from the current configuration to the reference configuration. Time differentiation of Eq. (2) gives, after some algebra [10]: ∂(δui ) 2 Ec (t + δt) = δFi δui dSo − δij dVo (3) ∂Xj

o Vo Following Hill’s definition [9], W2 =

δij Vo

∂(δui ) dVo ∂Xj

(4)

denotes the global second-order work of the system, associated with the incremental change (δij , δ(∂ui /∂Xj )). Both incremental quantities δ(∂ui /∂Xj ) and δij are related through the constitutive equation. ) In Eq. (3), the boundary integral o δFi δui dSo represents the external loading applied to the system, through the control parameters.) In case of a constant loading (or “dead forces”, after Hill), the boundary integral o δFi δui dSo vanishes, and Eq. (3) reads: (5) 2 Ec (t + δt) = −W2 Thus, the occurrence of outburst in kinetic energy is directly related to the vanishing of the second-order work. Assuming that geometrical changes can be omitted over the incremental evolution considered, (4) can be rewritten as follows: W2 = δσi δεi dVo = w2 dVo (6) Vo

Vo

Moreover if we restrict henceforth the analysis to the context of the material point scale, the second-order work is a quadratic form associated with the symmetric part K s of the tangent stiffness matrix: w2 = δσi δεi = Kijs δεi δεj

(7)

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Vectorial notations are used in Eq. (6), where (3 × 3) tensors σ and ε were replaced with six components vectors σ and ε. Equation (7) shows that the second-order work varies with the direction of δε. Because of its symmetry, all eigenvalues of K s are real. When the eigenvalues are strictly positive, w2 is strictly positive for any direction of δε. w2 first vanishes in the same time as K s admits a nil eigenvalue (in that case, det K s = 0). As a consequence, the existence of outburst in kinetic energy is strongly related to the spectral properties of K s . In homogeneous laboratory tests, some components (or linear combinations of components) of strain and stress are imposed, and the response is computed by means of the conjugate variables. Let us exemplify in two-dimensional conditions, by considering proportional strain paths. The incremental axial strain is constant, and both incremental lateral and axial strains are proportional: λ δε1 + δε2 = 0 and δε1 = const.

(8)

To investigate the response along this loading path, the variation of the variable δσ1 − λ δσ2 in terms of ε1 is analyzed. Starting from the constitutive relation: δσ1 δε1 =K (9) δσ2 δε2 it can be shown that:

δσ1 − λ δσ2 = K11 − λ(K12 + K21 ) + λ2 K22 δε1

(10)

Noting that:

K11 − λ(K12 + K21 ) + λ K22 2

K11 K12 = [1 − λ] K21 K22

1 −λ

(11)

the second-order work w2 (λ) associated with the incremental strain direction

1 → − reads: uε = −λ w2 (λ) = K11 − λ(K12 + K21 ) + λ2 K22 (δε1 )2

(12)

Finally, it follows that δσ1 − λ δσ2 =

w2 (λ) δε1

(13)

When w2 (λ) = 0, then δ(σ1 − λ σ2 ) = 0. As a consequence, the vanishing of the second-order work along the incremental → 1 strain direction − uε = −λ is related to the existence of a maximum for the curve giving the evolution of the variable σ1 − λ σ2 in terms of ε1 . The maximum of σ1 − λ σ2 corresponds to a proper limit state.

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This is the generalization of the plastic limit condition, observed for example at the peak of the axial stress, for drained triaxial tests. As the lateral stress is assigned to remain constant (δσ2 = 0), the second-order work is vanishing at the axial stress peak: δσ1 = 0 implying that δσ1 δε1 + δσ2 δε2 = 0. The plastic limit condition is a particular example of limit state since both incremental stress components are nil. In the proportional strain loading path discussed above, one incremental stress term (σ1 − λ σ2 ) passes through a maximum, whereas a strain term (λ ε1 +ε2 ) is imposed constant: this is a mixed condition. If the loading is defined with only strain terms, the kinematics on the boundary of the specimen is prescribed, preventing any outburst in kinetic energy from occurring. No failure is therefore visible. In conclusion, the occurrence of failure mode requires three conditions: 1. One eigenvalue of K s is negative or nil (the mechanical state belongs to the bifurcation domain [4]). 2. The incremental loading direction considered is associated with a negative or nil value of the second-order work. 3. A mixed loading condition has to be applied, involving at least one stress term.

2.2 DEM Investigation for Proportional Strain Loading Paths The two-dimensional example discussed above considering proportional strain paths, is illustrated here through numerical experiments based on the discrete element method (DEM) [1]. However, the granular assembly considered below constitutes a three-dimensional sample loaded in axisymmetric conditions with respect to direction ‘1’ (σ2 = σ3 and ε2 = ε3 ). In these conditions, the proportional strain loading defined in Eq. (8) writes: λ δε1 + 2 δε3 = 0

and δε1 = positive const.

(14)

The granular assembly considered has a cubical shape and is composed of about 10,000 spheres [19]. The inter-particle interaction at contact points is modeled, in the normal direction to the tangent contact plane, by a purely elastic behaviour (characterized by a stiffness kn ). For the direction included in the tangent contact plane the relation is elastic perfectly plastic (characterized by a stiffness kt and a friction angle ϕc ). Simulations were performed with the code SDEC developed by Donzé and Magnier [7]. In the following of this section we consider a dense numerical sample E1 (characteristics are given in Table 1) exhibiting a dilatant behaviour during classical triaxial compressions. In Figure 1a the change of variable σ1 − λσ3 is shown in terms 1 ofε1 for λ = 1.21 → − and corresponding to the incremental strain direction u = −1.21 . For this strain ε

−1.21

direction, the evolution of σ1 − λσ3 presents a maximum [5] and the second-order work plotted in Figure 1c vanishes at this maximum as shown by Eq. (13). The

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Table 1 Characteristics of the numerical samples Sample

kn /ds ∗ (MPa)

kt /kn

ϕc (deg)

Void ratio e

Coordination number z

E1 E3

356 356

0.42 0.42

35.0 35.0

0.618 0.693

4.54 4.42

∗d s

represents the sphere diameter 60

30

50

20

40 q (kPa)

σ1 − λ σ3 (kPa)

40

10 0

−20

10

full strain control mixed control 0.5

1 ε1 (%)

1.5

0 0

2

(a)

0.025 Kinetic energy (J)

−3

40

60 p (kPa)

80

100

120

0.03

100 w2 (J . m )

20

(b) 200

0 −100 −200 −300 0

(c)

30 20

−10

−30 0

full strain control mixed control

full strain control mixed control

0.02 0.015 0.01 0.005

0.5

1 ε1 (%)

1.5

2

0 0

0.5

1 ε1 (%)

1.5

2

(d)

Fig. 1 (a), (b), (d) Comparison of simulated responses for a proportional strain loading path between a full strain control (λ δε1 + 2 δε3 = 0; δε1 > 0) and a mixed control (λ δε1 + 2 δε3 = 0; δσ1 − λ δσ3 > 0); (c) vanishing of the second-order work during the full strain control. (p = (σ1 + 2σ3 )/3; q = σ1 − σ3 )

control parameters of the loading are, in this case, defined with only strain terms (Eq. 14), and the kinetic energy of the sample (equal to the sum of the kinetic energy of all particles) stays low1 (Figure 1d). The simulation can be carried on until reaching the total quasi-static liquefaction (see the vanishing of stresses in Figure 1b) where the kinetic energy also vanishes. We consider now the case where the control parameters are mixed and defined by: 1

As the discrete element method is a dynamic method, all evolutions of the granular assembly, even quasi-static, imply production of kinetic energy.

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λ δε1 + 2 δε3 = 0 and δσ1 − λ δσ3 = positive const.

(15)

The response of the sample is compared with the previous one in Figures 1a, b and d. The maximum of σ1 − λσ3 , corresponding to the vanishing of W2 , cannot be exceeded. Moreover, while the maximum of σ1 − λσ3 is approached, an outburst of kinetic energy is developing (as explained by Eq. 5) and sample never gets back to an equilibrium state. This response corresponds to a very sudden failure of the sample. Actually, in Figure 1a, the peak of σ1 − λσ3 is slightly exceeded because failure has began to develop slightly before it, with a sharp increase of kinetic energy. Hence the sample response switches from a quasi-static regime to a dynamic regime where contribution of inertial terms in the stress state are not anymore negligible. This dynamic response explains also the non-vanishing of the shear stress q = σ1 − σ3 in Figure 1b.

3 Cones of Unstable Loading Directions, Bifurcation Domain 3.1 Basic Concepts As seen in relation (7) expression of w2 reads w2 = δεi Kijs δεj or equivalently when K is invertible w2 = δσi Sijs δσj

(16)

with S = K −1 . Writing the constitutive relation in principal axes in threedimensional conditions: ⎤ ⎡ 1 ν31 ν21 ⎡ ⎤ ⎡ ⎤ − − E E E δε1 ⎢ ν112 1 2 ν323 ⎥ δσ1 ⎥ ⎣ ⎦ ⎣δε2 ⎦ = ⎢− ⎣ E1 E2 − E3 ⎦ δσ2 δε3 δσ3 − ν13 − ν23 1 E1

E2

E3

Equation (16) can be developed as follows: δσ 2 δσ12 δσ 2 ν21 ν12 + 2 + 3 − + δσ1 δσ2 − · · · E1 E2 E3 E1 E2 ν23 ν31 ν13 ν32 + + δσ3 δσ2 − δσ1 δσ3 = 0 E3 E3 E1 E2 The left-hand side of this equation is a quadric, with neither constant terms nor terms of degree one according to δσi . Consequently, the solution is an elliptical

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Fig. 2 Solutions of equation: α1 X 2 + α2 Y 2 + α3 Z 2 = 0

cone if the quadric is not degenerated. The real nature of this solution depends on the positiveness of det(S s ). By calling (α1 , α2 , α3 ) the eigenvalues of S s , the four possible solutions are displayed in Figure 2. Through these results, the directional nature of the second-order criterion is established. Nevertheless the previous development holds only for incrementally linear materials that is to say for elastic behaviour. In fact, for elasto-plastic materials the constitutive relation is at least incrementally piece-wise linear with a linear relation in plastic loading regime, and an other linear relation in unloading regime. That is why the above discussion has to be made in a given tensorial zone.2 Moreover, it is necessary to verify that solutions belong geometrically to the tensorial zone considered and to cut them (i.e. to keep only the part of the cone include in the tensorial zone for example). Furthermore, if we make the assumption that eigenvalues of S s are strictly positive at the virgin state and are evolving continuously with the loading parameter, solutions appear sorted like presented in Figure 2. First, det(S s ) > 0, and no nonzero solutions exist. Second det(S s ) = 0, there is only one unstable loading dir2 A tensorial zone is a domain of the loading space in which the incremental constitutive relation is linear [3].

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ection. Third, det(S s ) < 0, an elliptical cone of unstable loading directions appears. Finally, det(S s ) might vanish again and unstable loading directions would be included between the intersection of the two planes. Nevertheless we have never observed such solution with the constitutive models used. Hence it is now possible to define the limit of the bifurcation domain. This limit is the surface gathering all mechanical states for which only one unstable loading direction exist. With the assumptions of positiveness of the eigenvalues at the virgin state and of their continuous evolution with the loading parameter, the limit of the bifurcation domain is given by the following relation for incrementally piece-wise linear model [15]: min

i=1,...,n

#

$ det(S s )⊂Zi = 0

with

ui ⊂ Zi

(17)

with n the number of tensorial zones of the constitutive model, ui the eigenvector corresponding to the vanishing eigenvalue, and Zi the tensorial zone considered. Eventually, same analysis can be performed in the strain rate space without any restriction. In fact it can be proved that [14] det(S s ) det(K s ) = # $2 det(S)

(18)

As a consequence both determinant vanish at the same time.

3.2 Illustration We propose now to illustrate remarks made above. Numerical results, are displayed with the constitutive models of Darve [2] and with the discrete element method.

3.2.1 Phenomenological Constitutive Relations Without going into details of Darve’s models, we just recall that they are not based on the classical concepts of elasto-plasticity. Decomposition of the strain in an elastic and plastic part is not assumed, and no plastic potentials are defined. According to the first model, the non-linear relation which links strain rate to stress rate is directly described by an incrementally non-linear relation. The second model is a simplification of the first one, and becomes incrementally piece-wise linear with eight tensorial zones. In principal axes, these models are written as follows: ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ δσ12 δε1 δσ1

1 ⎥ ⎢ ⎣δε2 ⎦ = 1 N + + N − ⎣δσ2 ⎦ + N + − N − ⎣δσ22 ⎦ (19) 2 2 δσ 2 δε3 δσ3 δσ3

Discrete Numerical Analysis of Failure Modes in Granular Materials

octolinear model

nonlinear model 1000

σ1 (kPa)

1000

σ1 (kPa)

197

500 0 0

500 0 0

500 1000

σ2 (kPa)

0

500

1000

500

1000

σ2 (kPa)

σ3 (kPa)

500

0

(a)

1000

σ3 (kPa)

(b) bifurcation domain for the non-linear model σ1/p 0

non-linear octo-linear Mohr-Coulomb σ /p 2

σ /p

0

3

0

(c) Fig. 3 Limit of the bifurcation domain plotted in the 3D stress space for constitutive models of Darve

and

⎡

⎤ ⎤ ⎡ ⎤ ⎡ δε1 δσ1 |δσ1 |

⎣δε2 ⎦ = 1 N + + N − ⎣δσ2 ⎦ + 1 N + − N − ⎣|δσ2 |⎦ 2 2 |δσ3 | δε3 δσ3 ⎡

with

1 E1±

N±

⎢ ⎢ ± ⎢ ν = ⎢− 12± ⎢ E1 ⎣ ν± − 13± E1

⎤ ν± ν± − E21± − E31± ⎥ 2 3 ± ⎥ ν32 ⎥ 1 − ±⎥ E2± E3 ⎥ ⎦ ν± − 23± 1± E2

(20)

(21)

E3

Limits of the bifurcation domain, given by Eq. (17) for the incrementally piecewise linear relation, are displayed in Figure 3 for both models. As a remark, the incrementally non-linear model can be seen as incrementally piece-wise linear with an infinity of tensorial zones. Thus, with a numerical effort, an approximation of the bifurcation limit has been displayed [15].

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dσ1

dσ1

dσ3

dσ3 p =600 kPa 0

p0=600 kPa dσ2 q=494 kPa η=0.646 dσ1

q=640 kPa η=0.787

dσ2 dσ1

dσ3

dσ3

p =600 kPa

p0=600 kPa q=1544 kPa

0

dσ2 q=1494 kPa η=1.361

dσ2

a)

η=1.385

b)

Fig. 4 3D instability cones for a dense sand of Hostun. Figure 4a presents the cones obtained with the octo-linear model. The planes represent the limit between the 8 tensorial zones, the meshes correspond to the analytical solution of Equation (16) and the point clouds the solution obtained with the numerical method. Figure 4b shows results obtained with the non-linear model using the numerical method. po is the initial confining pressure, q = σ1 − σ3 , η = q/p

(a)

(b)

Fig. 5 Definition of stress probes (a), and strain responses (b), in the axisymmetric plane of stress increments and strain increments, respectively

Then, instability cones for stress-strain states situated beyond the bifurcation limit are plotted. Figure 4 presents the 3D cones obtained with Darve’s constitutive models [13]. The method used to draw these cones of unstable stress directions consist to realize stress probes [8], as presented in Figure 5a in axisymmetric conditions for simplicity’s sake. At different stress states along the loading path (presently a drained − → triaxial path), a small stress increment δσ , with a constant norm, is applied from

Discrete Numerical Analysis of Failure Modes in Granular Materials Sample E1

Sample E3 η = 0.0 η = 0.74 η = 0.78 η = 0.82

90 120

60

150

180

0

330

210 300

η = 0.19 η = 0.35 η = 0.46 η = 0.64

90 120

60

150

30

240

199

30

180

0

330

210 300

240

270

270

Fig. 6 Circular diagrams of the normalized second-order work at a confinement pressure σ3 = 100 kPa, for samples E1 and E3. The little circle in dash line represent the zero values

the considered stress state, in all directions of the stress increment space. For each stress direction, the corresponding strain response is computed (Figure 5b), giving access to the value of the second-order work. Cones of instability gather, for a given stress state, all stress directions for which w2 ≤ 0. In three-dimensional conditions the principle is the same, but stress probes describe a sphere instead of a circle (Figure 5a).

3.2.2 Discrete Element Model In the same way as with the phenomenological constitutive relations, stress probes have been performed with the DEM along drained triaxial compressions, in axisymmetric conditions only [18]. Two samples are considered here, the dense and dilatant sample E1 (already used for proportional strain loading paths in Section 2.2), and a looser and essentially contractant sample E3 (see characteristics in Table 1). Figure 6 presents circular diagrams of the normalized second-order work w2n computed from stress probes for a confinement σ3 = 100 kPa and different deviatoric stress levels η = q/p. In such diagrams, an arbitrary constant value c is added to the polar value of w2n in order to have ∀ α,

w2n (α) + c > 0 ,

(22)

where α is the stress probe direction (see Figure 5a) and w2n is defined as − → − → δσ · δε w2n = − → − → . δσ δε

(23)

A dashed circle is drawn in the circular diagrams to represent vanishing values of w2n . Outside the dashed circle w2n is positive, inside it is negative. As for the octo-

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Sample E3

800 lom bc rite

rio

n

800

rio bc

rite

400 300

Mo

500 σ1 (kPa)

1

σ (kPa)

500

hr−

Mo

Co u

lom

600

h r−

600

n

700

Co u

700

400 300

ine

e

cl

ti sta ro yd

200

ic

100 0 0

lin

o

dr

Hy

200

H

t sta

100

100

200 2

1/2

300

σ (kPa) 3

400

500

0 0

100

200 300 1/2 2 σ3 (kPa)

400

500

Fig. 7 Synthesis of cones of unstable stress directions in the axisymmetric plane of stresses; full circles represent stress probes for which no vanishing or negative values of w2 were found

linear model and the non-linear model (Figure 4), for sufficiently high values of η a set of stress directions for which w2 ≤ 0 are found. They form a cone of unstable stress directions. For the loosest sample E3 cones open for much lower deviatoric stress level η. It is characteristic of a loose sand where the bifurcation domain is wider than for a dense sand, as shown in [4]. As the computational cost to simulate stress probes is quite important, the number of stress probes has been voluntarily limited and we were not able to find the stress state where the first unstable stress direction appears, i.e. the exact limit of the bifurcation domain. A synthesis of stress probes performed and cones of instability found is presented in Figure 7. One can see a domain strictly included inside the Mohr–Coulomb criterion where cones of instability exist.3 This domain constitutes the bifurcation domain and corresponds qualitatively to the bifurcation domain plot in the threedimensional stress space in Figure 3.

3

For the densest sample E1 and the highest confining pressure no cone of instability were found for the tested stress state. This may be due to a compaction of the sample at relatively high pressure, increasing its density and consequently reducing the bifurcation domain [17].

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3.3 Case of the Proportional Strain Loading Path In this subsection we consider again proportional strain loading paths presented in Section 2.1 through relation (8). We just extend this description to take into account three-dimensional space as follows: ⎧ const. ∈ R − {0} ⎨ δε1 = const. λ1 δε1 + δε2 = 0 λ1 ∈ R (24) ⎩ λ2 δε2 + δε3 = 0 λ2 ∈ R It can be verified that axisymmetric conditions are obtained with λ2 = −1, plane strain condition with λ1 = 0 or λ2 = 0, and an undrained axisymmetric triaxial test can be simulated with λ2 = −1 and λ1 = 1/2. For such strain path the second-order work reads w2 = δε1 (δσ1 − λ1 δσ2 + λ1 λ2 δσ3 ) + (λ1 δε1 + δε2 ) (δσ2 − λ2 δσ3 ) + (λ2 δε2 + δε3 ) δσ3 and consequently, because of condition (24), vanishes at an extremum of (σ1 − λ1 σ2 + λ1 λ2 σ3 ). To illustrate this, we have considered the following loading program with the incrementally non-linear model. First a drained triaxial path under an initial confining pressure p0 = 200 kPa is followed until reaching the state σ1 = 298 kPa, σ2 = σ3 = 200 kPa, ε1 = 0.292% and ε2 = ε3 = −0.079%. This state is located just before reaching the bifurcation domain limit. From this state, a proportional strain path (24) is applied. In the present case λ1 = 0.0249 and λ2 = −40.1. The response of this loading path is given in Figure 8. Along this loading path, three instability cones have been computed, the first one before reaching the σ1 − λ1 σ2 − λ2 σ3 peak, the second one at the peak and the third one after the peak. Figure 9 presents this path with the cones in the 3D strain space. Through Figures 9 and 10, the following comments can be made. Before reaching the first cone, the loading path is outside the bifurcation domain, no bifurcation and failure can occur. At the first cone, the loading path just come in the bifurcation domain. Nevertheless, loading path is outside the cone, as a consequence no instability can occur at this state along this path. At the second cone, the loading path goes through the peak of σ1 − λ1 σ2 + λ1 λ2 σ3 , as a consequence w2 vanishes and the loading path is just tangent to the instability cone. The path becomes unstable. Effective failure occurs if σ1 − λ1 σ2 + λ1 λ2 σ3 is driven. In our case, ε1 is driven, then the peak can be dropped in. After the peak, w2 < 0, then the loading path is unstable. At the third cone, path is clearly inside the cone. It is worth noting that all stress-strain states presented are inside the plasticity limit. Therefore, concept of limit state and flow rule is generalized here for stress-strain states situated inside the bifurcation domain but strictly before the plasticity limit condition. The simulation of failure with the discrete element method presented in Section 2.2, was done in axisymmetric conditions. However conclusions given here in

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Fig. 8 Response to the drained triaxial path until σ1 = 298 kPa, σ2 = σ3 = 200 kPa, ε1 = 0.292% and ε2 = ε3 = −0.079% (point A); then response to the proportional strain path (Eq. 24) defined by λ1 = 0.0249 and λ2 = −40.1

three-dimensional conditions hold in axisymmetric conditions. Hence, in Figure 1 failure occurs at the peak of σ1 − λσ3 , because, even if the loading path is inside the bifurcation domain before this peak, the strain direction defined by the loading program is included in the cone of unstable strain directions, for the first time, at the peak of σ1 − λσ3 .

4 From Limit States to Failure Occurrence The objective of this section is to confirm through DEM simulations, that inside the bifurcation domain displayed in Section 3.2.2, the occurrence of failure requires: the loading directions to belong to the cone of unstable loading directions (associated to the current stress state); and the control loading parameters to be mixed (involving strain and stress terms).

Discrete Numerical Analysis of Failure Modes in Granular Materials

203

Fig. 9 Loading path in the 3D strain space with three instability cones computed before, at, and after the σ1 − λ1 σ2 − λ2 σ3 peak

4.1 Mixed Loading Parameters Concerning the control mode of the loadings, stress probes presented in Section 3 were conducted with only stress control parameters (δσ1 = cst1 and δσ3 = cst3, with constants cst1 and cst3 chosen to impose the desired direction α). For such full stress controls, no failure can occur before the plastic limit condition [12], representing limit stress states. That is why we have been able to simulate these stress probes, where numerical samples recovered an equilibrium state after each probe direction, whatever the sign of the second-order work computed. Let us now reconsider these stress probes as proportional stress loading paths where the ratio between δσ1 and δσ3 can be defined by: √ δσ1 − λ δσ3 = 0 where: λ = 2 sin α/ cos α for α ∈ [180 deg; 270 deg[ (25) The energy conjugated variables are thus δσ1 − λ δσ3 = 0 and δε1 on one hand, and δσ3 and λ δε1 + 2 δε3 on the other hand. For λ = 1, the stress direction is α = 215.3 deg and corresponds to a path where the deviatoric stress is constant δq = 0; in addition, δε1 + 2 δε3 = δεv . For sample E3, this latter stress path is included in the cone of instability for η = 0.46, as displayed in Figure 11. Figure 12 shows for this stress probe direction, totally controlled with stress parameters, that there is no outburst of kinetic energy as expected, and that the change of the incremental volumetric strain δεv presents a minimum (maximum of dilatancy). This

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0.2 0.1 0 −0.1 −0.2 −0.3 −0.4

−0 5

Fig. 10 Zoom of the three cones of Figure 9 sample E1; η = 0.82 sample E3; η = 0.46

90 120

60

150

30

180

0 W2 = 0

α = 200°

Fig. 11 Comparison of the normalized second-order work between samples E1 and E3, for the confinement pressure σ3 = 100 kPa

330

210 q = cst. α = 215.3° 240 α = 220° α = 230°

300 270

α = 250°

extremum of volumetric strain corresponds to a limit state. The proportional stress path characterized by λ = 1 can also be followed by controlling the sample with

Discrete Numerical Analysis of Failure Modes in Granular Materials

205

160 140 120

−1

x 10

−0.5

100 0

80

δεv (%)

σ1 or σ3 (kPa)

−3

total stress control

60 control with δq=0 and δεv < 0

40 σ3

20 0

σ1

0

0.02

0.04 0.06 Simulation time (s)

0.08

0.5 1 1.5 2 0

control with δq = 0 and δεV < 0 total stress control

0.002

0.004 0.006 δε1 (%)

0.008

0.01

−3

10

x 10

v

control with δ q = 0 and δε < 0

Kinetic energy (J)

8

6

4

2 total stress control 0 0

0.02

0.04 0.06 Simulation time (s)

0.08

Fig. 12 Comparison of responses of sample E3 along a constant deviatoric stress direction, controlled either with stress parameters only or with mixed (stress and strain) parameters

mixed control parameters4 defined by δq = 0 and δεv = negative const.

(26)

Hence, a constant dilatancy rate is imposed. Response of the sample is presented in Figure 12. Until the minimum of δεv , the kinetic energy stays low (quasi-static response) and the sample followed the imposed path. Then, when the limit state constituted by the minimum of δεv is reached, this latter cannot be exceeded, stresses vanish suddenly5 and an outburst of kinetic energy occurs highlighting the transition to a dynamic response. This simulated response corresponds to a proper failure that can be seen as a sudden liquefaction. Consequently, failure of the sample has been triggered by the mixed mode of control, whereas there was no characteristic feature of failure for a total stress control. 4

To ensure such a mixed control mode (in stress and strain) with control parameters defined as linear combinations of principal stress or strain components, a specific algorithm is run every time step of the DEM cycle; details can be found in [17, 20]. 5 Due to the dynamic response of the sample, stress components do not vanish all together.

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L. Sibille et al. Table 2 Stress directions and belonging to cones of instability for samples E1 and E3

α (deg)

λ

∈ cone of unstable stress directions ? sample E1 (η = 0.82) sample E3 (η = 0.46)

200 220 230 250

0.515 1.19 1.69 3.89

No No Yes No

No Yes Yes (close to the limit of the cone) No

4.2 Stable and Unstable Loading Directions In Section 4.1 only one stress loading direction, belonging to the cone of instability for sample E3, has been considered. We investigate now a wider range of stress directions to compare responses of samples E1 and E3 along each direction, and in particular for directions included in cones of instability and for others excluded. Figure 11 displays the normalized second-order work computed at η = 0.82 for sample E1 and η = 0.46 for E3. The two cones of instability, corresponding to the two samples, are not superimposed. For the loosest sample E3, the cone is more opened and includes lower values of α than for the densest sample E1 [20]. As shown by Darve et al. [4], this difference is typically related to the difference in porosity of the two samples. In the following four stress directions are considered (α = 200, 220, 230 and 250deg), their belonging to cones of instability is summarized in Table 2. To be able to trigger the failure along these stress directions, mixed loading control parameters are chosen as detailed in Section 4.1. δσ1 − λ δσ3 = 0 is imposed to prescribe the desired stress direction (see Table 2 for values of λ). However, instead of imposing a change of the parameter λ δε1 + 2 δε3 , the latter is kept constant, i.e. λ δε1 + 2 δε3 = 0 (see footnote 4). If simulations are run in these conditions there is no evolution of samples, they stay at their initial mechanical state maintained by these two control parameters. A perturbation is necessary to conclude about the stability of the mechanical state considered with the control parameter chosen. Since simulations are performed without gravity, some particles float in the pores of samples and are not involved in the contact force network, at the equilibrium state considered. The sample is perturbed by imposing an instantaneous velocity in a random direction on eight floating particles. Samples are virtually split into eight sub-parallelepipeds, each perturbed particle is chosen randomly in each subparallelepiped respectively. The velocity imposed to each particle is computed such that the value of kinetic energy provided is equal for each particle. The perturbation corresponds to a total external input of kinetic energy of 10−5 J. This input is small compared with the maximum value of kinetic energy “naturally” developed for fully stress controlled probes: 10−4 J. It is worth noting that responses simulated with such a perturbation are totally similar to responses that would be obtained by imposing a change of the control parameter involving strain terms, i.e. λ δε1 + 2 δε3 < 0,

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207

α = 200 deg; λ = 0.515 120

E3

100

0.005

0 0

60

E1

E3

40

E1

perturbation

80

3

0.01

σ (kPa)

0.015 perturbation

Kinetic energy (J)

0.02

20

0.02 0.04 0.06 Simulation time (s)

0

0.08

0

0.02 0.04 0.06 Simulation time (s)

0.08

α = 220 deg; λ = 1.19 120

0.005

0 0

80 60

3

0.01

σ (kPa)

E3

E1 E3

20

E1 0.02 0.04 0.06 Simulation time (s)

40

perturbation

100

0.015 perturbation

Kinetic energy (J)

0.02

0.08

0

0

0.02 0.04 0.06 Simulation time (s)

0.08

Fig. 13 Responses of samples E1 and E3 to a perturbation in kinetic energy, for the stress directions α = 200 and 220 deg

as shown in [17, 18] (see also [5] for proportional strain loading paths). It is simply another way to test the stability of samples. Figures 13 and 14 present the simulated responses of samples E1 and E3 in terms of time evolutions of the kinetic energy and of radial stress σ3 . Changes of the axial stress σ1 are very close to σ3 changes, and are thus not displayed. Arrows in diagrams indicate the time of application of the perturbation. •

•

•

Stress directions α = 200 and 250 deg are not included in cones of instability for both samples. For these directions, after the application of the perturbation, the kinetic energy fluctuates slightly but finally vanishes, and both samples recover an equilibrium at a stress state close to the initial one. The stress direction α = 220 deg is included in the cone of instability for the sample E3 but not for the sample E1. The failure, highlighted by the outburst of kinetic energy and the vanishing of stresses, is observed only for E3, whereas E1 is almost unaffected by the perturbation. The stress direction α = 230 deg is included in the cone of instability of both samples E1 and E3. For both samples failure occurs as shown by the sudden vanishing of stresses. However for E3 the burst of kinetic energy has not com-

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L. Sibille et al. α = 230 deg; λ = 1.69 0.02

120

0.005

σ (kPa)

0 0

60

3

0.01

80

E3

40

perturbation

100

0.015 perturbation

Kinetic energy (J)

E1

20

0.02 0.04 0.06 Simulation time (s)

0

0.08

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Fig. 14 Responses of samples E1 and E3 to a perturbation in kinetic energy, for the stress directions α = 230 and 250 deg

pletely developed. This intermediate result can be explained by the proximity of the tested direction, α = 230 deg, to the boundary of the cone. These results show clearly that, even if the mechanical state of the sample is controlled with mixed parameters (as defined in Section 4.1), failure occurs only when the loading direction is included in the cone of instability. In other words, failure can be triggered only if the loading direction is associated with a nil or negative value of the second-order work.

5 Conclusion Three approaches have been considered in this chapter to investigate what is failure of granular media samples: theoretical considerations, application of phenomenological rate-independent constitutive relations, and – in a more detailed manner – simulations of failure by a discrete element model.

Discrete Numerical Analysis of Failure Modes in Granular Materials

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1. The theoretical developments have provided a firm basis to explain the link between second-order work and failure. Indeed when the second-order work is taking a zero (or negative) value in a given stress direction, if a proper arbitrarily small additional load is applied in this direction, a burst of kinetic energy appears leading to a dynamic regime of deformations, which is typical of failure. 2. The 3D phenomenological analysis has leaded to the existence of a bifurcation domain, preserving the conical structure of Mohr–Coulomb criterion, and of instability cones, which can be not unique and have an elliptical cut in 3D. 3. The discrete element method is able to describe failure in a very natural way without any ad-hoc ingredient. This is the essential reason why this method has been extensively used to investigate failure in granular media. Bifurcation domain and instability cones have been obtained as conjectured by the elastoplastic theory. Moreover, the three necessary and sufficient conditions for an effective failure have been successfully checked: • • •

the stress state has to belong to the bifurcation domain, the actual stress direction has to be inside an instability cone, the proper loading parameters have necessary to be mixed ones.

The observed failure modes are characterized by bursts of kinetic energy, exponentially growing strains, and decreasing stresses. This is basically a kind of generalized liquefaction. Because no strain localization pattern has been observed (and moreover the strain localization criterion is not satisfied at these stress-strain states), these failure modes have been called “diffuse” [4]. These various features and conclusions have been also observed in some laboratory experiments performed in drained and undrained conditions on Hostun sand [5].

References 1. Cundall, P.A., Strack, O.D.L., A discrete numerical model for granular assemblies. Geotechnique, 29(1):47–65, 1979. 2. Darve, F., Flavigny, E., Meghachou, M., Yield surfaces and principle of superposition revisited by incrementally non-linear constitutive relations. International Journal of Plasticity, 11(8):927–948, 1995. 3. Darve, F., Labanieh, S., Incremental constitutive law for sands and clays. simulations of monotonic and cyclic tests. International Journal for Numerical and Analytical Methods in Geomechanics, 6:243–275, 1982. 4. Darve, F., Servant, G., Laouafa, F., Khoa, H.D.V., Failure in geomaterials: Continuous and discrete analyses. Computer Methods in Applied Mechanical Engineering, 193(27–29):3057– 3085, 2004. 5. Darve, F., Sibille, L., Daouadji, A., Nicot, F., Bifurcations in granular media: Macro- and micro-mechanics approaches. Comptes Rendus Mécanique, 335(9–10):496–515, 2007. 6. Desrues, J., Shear band initiation in granular materials: Experimentation and theory. In Geomaterials Constitutive Equations and Modelling, F. Darve (ed.), pp. 283–310. Elsevier, Taylor and Fancis Books, 1990.

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7. Donzé, F.V., Magnier, S.A. Spherical discrete element code. In Discrete Element Project Report 2, Laboratory GEOTOP, Université du Québec à Montréal, 1997. 8. Gudehus, G., A comparison of some constitutive laws for soils under radially symmetric loading and unloading. In Proceedings 3rd Numererical Methods in Geomechanics, Aachen, 2–6 April. A.A. Balkema, Rottedam, 1979. 9. Hill, R., A general theory of uniqueness and stability in elastic-plastic solids. Journal of the Mechanics and Physics of Solids, 6:239–249, 1958. 10. Nicot, F., Darve, F., A micro-mechanical investigation of bifurcation in granular materials. International Journal of Solids and Structures, 44:6630–6652, 2007. 11. Nicot, F., Darve, F., Khoa, H.D.V., Bifurcation and second order-work in granular materials. International Journal for Numerical and Analytical Methods in Geomechanics, 31:1007– 1032, 2007. 12. Nova, R., Controllability of the incremental response of soil specimens subjected to arbitrary loading programmes. J. Mech. Behav. Mater., 5(2):193–201, 1994. 13. Prunier, F., Laouafa, F., Darve, F., 3D bifurcation analysis in geomaterials, investigation of the second order work criterion. European Journal of Environmental and Civil Engineering, 13(2):135–147, 2009. 14. Prunier, F., Laouafa, F., Lignon, S., Darve, F., Bifurcation modeling in geomaterials: From the second-order work criterion to spectral analyses. International Journal for Numerical and Analytical Methods in Geomechanics, 33:1169–1202, 2009. 15. Prunier, F., Nicot, F., Darve, F., Laouafa, F., Lignon, S., 3D multi scale bifurcation analysis of granular media. Journal of Engineering Mechanics (ASCE), 135(6):493–509, 2009. 16. Rice, J.R., The localization of plastic deformation. In Theoretical and Applied Mechanics, W.T. Koiter (ed.), pp. 207–220. North-Holland Publishing Compagny, Delft, 1976. 17. Sibille, L., Modélisations discrètes de la rupture dans les milieux granulaires. PhD Thesis, Institut National Polytechnique de Grenoble, 2006. 18. Sibille, L., Donzé, F.V., Nicot, F., Chareyre, B., Darve, F., From bifurcation to failure in a granular material, a DEM analysis. Acta Geotechnica, 3(1):15–24, 2008. 19. Sibille, L., Nicot, F., Donzé, F.V., Darve, F., Material instability in granular assemblies from fundamentally different models. International Journal for Numerical and Analytical Methods in Geomechanics, 31:457–481, 2007. 20. Sibille, L., Nicot, F., Donzé, F.V., Darve, F., Analysis of failure occurrence from direct simulations. European Journal of Environmental and Civil Engineering, 13:187–201, 2009.

Homogenization of Granular Material Modeled by a 3D DEM C. Wellmann and P. Wriggers

Abstract Within this contribution the mechanical behavior of dry frictional granular material is modeled by a three-dimensional discrete element method (DEM). The DEM uses a superquadric particle geometry which allows to vary the elongation and angularity of the particles and therefore enables a better representation of real grain shapes compared to standard spherical particles. To reduce computation times an efficient parallelization scheme is developed which is based on the Verlet list concept and the sorting of particles according to their spatial position. The macroscopic mechanical behavior of the particle model is analyzed through standard triaxial tests of periodic cubical samples. A technique to accurately apply stress boundary conditions is presented in detail. Finally, the triaxial tests are used to analyze the influence of the sample size and the particle shape on the resulting stress-strain behavior.

1 Introduction The mechanical behavior of dry frictional granular material exhibits various complex effects like e.g. high pressure sensitivity, plastic deformation nearly from the onset of loading, dilatancy and the tendency to form shear bands. To capture these effects in an appropriate macroscopic stress-strain relation is an awkward problem. Attempts often lead to solutions based on a huge number of material parameters whose proper determination is an awkward problem for its own. However, the complex macroscopic behavior is an immanent result of the relatively simple microstructure of the material, which is described by the grain shape and size, the intergranular friction and the deformation of grains through contact forces. Modeling the microstructure using discrete element models can give insight into the micro-mechanisms C. Wellmann · P. Wriggers Institute of Continuum Mechanics, Appelstr. 11, 30167 Hannover, Germany; e-mail: {wellmann, wriggers}@ikm.uni-hannover.de

E. Oñate and R. Owen (eds.), Particle-Based Methods, Computational Methods in Applied Sciences 25, DOI 10.1007/978-94-007-0735-1_8, © Springer Science+Business Media B.V. 2011

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Fig. 1 Examples of superquadric particles with radius parameters r1 = r2 = r3 /2 and exponents (a) 1 = 2 = 0.5, (b) 1 = 2 = 1, (c) 1 = 2 = 1.5

a)

b)

c)

yielding the effects shown on the macroscale. Here this approach is used to analyze the influence of some microstructural parameters on the macroscopic behavior. The granular material is modeled by a three-dimensional discrete element method (DEM) that uses a superquadric particle geometry, which allows to vary the particle elongation and angularity. Because of the complex particle shape the contact detection process is computationally expensive. Hence, an efficient parallelization scheme for shared memory architectures will be shown which is based on the concept of Verlet lists and a spatial sorting of the particles. The macroscopic behavior is determined through triaxial tests of randomly generated particle samples. To get rid of spurious boundary effects periodic cubical samples are used. The parameters of the DEM are adapted to a reference material, Leighton Buzzard sand size fraction B. The influence of the sample size is analyzed using four sample sizes. Next the influence of the particle elongation and angularity on the macroscopic behavior is analyzed by a number of tests. Finally, the numerical results are compared to experimental results drawn from the literature. In the next section the main features of the DEM are outlined. Thereafter the parallelization scheme will be introduced in Section 3. Section 4 covers the techniques used to perform accurate stress-controlled triaxial tests of periodic samples including the random sample generation process. Finally, the results of the analyses are presented in Section 5 which is followed by the conclusion.

2 Discrete Element Method In order to be able to represent real grain shapes more accurately the DEM uses superquadric [7] particle geometries. A superquadric is defined by five parameters through the explicit equation / x1 2/1 x2 2/1 1 2 x3 2/2 + + = 1. F (x) = r1 r2 r3

(1)

Homogenization of Granular Material Modeled by a 3D DEM

rimax

Fig. 2 Two-dimensional sketch of adjacent particles enclosed by bounding spheres. If the dotted circles intersect, the pair (Pi , Pj ) is added to a Verlet list

Pi dverlet 2

213

rjmax Pj dverlet 2

The ri are the radius parameters and control the size of the particle in the three orthogonal directions. The exponents i control the angularity where i = 0 yields a cuboid, i = 1 yields an ellipsoid and i = 2 yields an octahedron, compare Figure 1. For inter-particle contacts the Hertzian contact theory [11] is applied. Hence, particles are assumed to behave linear elastic and the small overlap of contacting particles is considered as an elastic deformation at the contact point. The elastic repulsive contact force is given by f N = γ E ∗ δ 3/2

with

1 − ν12 1 − ν22 1 = + , E∗ E1 E2

(2)

where δ is the overlap distance, γ is a function of the principal radii of curvature of the contacting surfaces, and νi and Ei are Poissson’s ratio and Young’s modulus of the particles. For the tangential contact force the solution of Mindlin [16] is used combined with the Coulomb friction law. For the integration of the particle equations of motion an explicit central difference method is used for the translational part and a momentum conserving fourth order Runge–Kutta method for the rotational part [17]. For a more detailed description of the DEM the reader is referred to [24].

3 Parallelization Discrete element simulations dealing with a reasonable number of particles undergoing reasonably large deformations have high computational demands. Usually memory requirements are uncritical compared to the large computation times which result from the need of small integration time steps and the contact detection performed in each step. This is especially true if non-spherical particles are used for which the interparticle contact check becomes non-trivial like e.g. for the superquadric particles. To reduce this problem and exploit the possibilities of modern multi-core machines an efficient parallelization scheme for shared memory architectures was developed and implemented using the OpenMP standard [1]. The scheme is based on a split of the contact detection process into a global part and a local part and the use of so-called Verlet neighbor lists [23]. For the Verlet neighbor list

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update particle states v i += m1i f i dt xi += v i dt

check if verlet-lists have to be updated xi − xvi ≥ dverlet 2 no

yes

update verlet lists xvi = xi ﬁnd all Pj with xj − xi − rimax − rjmax ≤ dverlet

Fig. 3 Flowchart of a single time step within a DEM code using the Verlet list concept

check for contact and compute contact forces for all Pj in verlet-list of Pi if (Pi, Pj ) in contact compute contact force f ij update resultant forces f i += f ij , f j -= f ij

Fig. 4 Hourglass filled with about 105 superquadric particles. The particles are colored according to their center’s spatial position

concept each particle is enclosed by a bounding-sphere. The idea is to store each particle Pj whose bounding-sphere distance to particle Pi is smaller than a user defined Verlet distance dverlet in a list of potential contact partners of Pi , see Figure 2. As the particles move in time the bounding-sphere distances change. However, as long as no particle center has moved more than dverlet /2, particle pairs that are not stored in a list cannot come into contact and hence the lists need no update. The detailed contact check then is only performed for the potential pairs in the list. The performance of this scheme crucially depends on the choice of the Verlet distance dverlet, where a smaller dverlet yields shorter neighbor lists but more frequent updates. For tight packages of superquadrics undergoing slow deformations and optimum is found at dverlet ≈ 5 × 10−2 r¯ with r¯ the average particle radius.

Homogenization of Granular Material Modeled by a 3D DEM

10

5

Fig. 5 Two-dimensional sketch of spatial sorting method. The particle centers are sorted into grid cells and the cells are ordered by moving along the smallest grid dimension first. The order of particles in the same cell is arbitrary

3

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particle index

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C2 40000 C

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0 0

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particle index

(a)

80000

100000

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20000

40000

60000

80000

100000

particle index

(b)

Fig. 6 (a) Verlet lists for random particle order. (b) Verlet lists after spatial sort of particles

Using the Verlet list concept a time step in the DEM code is depicted in Figure 3. The update of the particle states and the check if the Verlet lists have to be updated are trivial to parallelize since the computation for one particle is independent from the others. The update of the Verlet lists itself is more difficult to parallelize but has only small influence on the overall performance since the update is performed unfrequently. Hence, it will not be discussed here. The most expensive and therefore most important part regarding parallelization is the detailed contact check and contact force computation. Here each potential pair (Pi , Pj ) from the lists is checked for contact and the contact force is added to the resulting forces of Pi and Pj . The problem with parallelizing this part is the risk of data race conditions if one core is working on (Pi , Pj ) and another one on (Pi , Pk ) and both try simultaneously to update the resulting force of Pi . Forcing the cores to update the forces one by one by placing the corresponding code in an OpenMP critical section is not an option, since the performance reduces dramatically due to the high number of potential contact pairs and the relatively low cost for each pair. This can be avoided by sorting the particles according to their spatial position. Figure 4 shows a sample of about 105 superquadric particles flowing through an hourglass. The corresponding Verlet lists can be visualized in a two-dimensional plot by placing a dot at (i, j ) with i < j for each pair (Pi , Pj ). For unsorted particles this yields a plot like it is shown in Figure 6a, where the upper left triangle

C. Wellmann and P. Wriggers 1

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2 4 8

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# verlet updates [103]

efficiency

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(a)

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(b)

Fig. 7 (a) Efficiency for 2, 4 and 8 cores. (b) Number of Verlet updates and particle resorts

is randomly filled. Now the particles are sorted by placing their centers into a threedimensional grid and moving along the smallest dimension first, medium second and largest last, compare Figure 5. The resulting Verlet lists in Figure 6b show a band structure which can be exploited to distribute the particle pairs to the cores like it is depicted in Figure 6b: Let nc be the number of cores available. Then the set of pairs is divided into nc equisized (same number of pairs) chunks C I and each of this is subdivided into two equisized chunks CJI . If the maximum particle index involved in CJI −1 is smaller than the minimum index in CJI , CJI −1 and CJI are independent and can be processed in parallel without the risk of a data race condition. Depending on the bandwidth of the Verlet list structure more than two sub-chunks might be required to fulfill this criterion. Even the number of chunks C I might have to be reduced if the bandwidth is too large or the number of particles too low. As long as the Verlet lists remain unchanged the chunks and sub-chunks remain valid and require no update. The cost of the update is negligible using standard search and sort algorithms. As the particles move over time the bandwidth of the Verlet list structure might grow making a resorting of the particles necessary. As it is required very unfrequently the cost of the resort is also negligible especially if one takes into account that sorting improves the data locality of the code and therefore yields an additional performance benefit. Figure 7a shows the efficiency attained for the hourglass case shown in Figure 4 when using two, four or eight cores. The corresponding number of Verlet update and resort operations is shown in Figure 7b. It can be seen that the highest efficiency is reached in the beginning of the simulation where the system is rather static and few Verlet updates and resorts are required. As particles reach the bottom of the hourglass and the system becomes more dynamic the efficiency drops to values of approximately 0.94 for two cores, 0.84 for four cores and 0.65 for eight cores.

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L2

Pi

217

Gi (α)

d(α)

f ji

f ij

Fig. 8 Two-dimensional sketch of periodic unit cell with particles Pi and ghost particles Gi . Due to periodicity each boundary contact exists two times on opposite sides of the unit cell

Pj xji

e2

L1 e1

4 Periodic Triaxial Tests The macroscopic or bulk mechanical behavior of the material modeled by the DEM is analyzed by tests on random sample packages. Considering that the size of these packages should be much larger than the typical particle size these packages will be called representative volume elements (RVEs) in the following. The contact forces and particle displacements resulting from a DEM simulation are transferred to volume averaged stresses and strains which are used to characterize the macroscopic behavior.

4.1 Periodic RVEs A problem with RVE tests of DEM samples is the accurate application of strain or stress boundary conditions. The simplest solution is to enclose the sample by rigid walls or dummy particles so that the displacements at the boundary can be prescribed. However, this scheme yields artificial boundary effects due to the particlewall or particle-dummy contacts. Simulations of this type were recently done with spherical particles in [8], with sphere-clusters in [20] and with convex polyhedral particles in [5]. To avoid artificial boundary effects periodic cubical samples can be used where the particles at one side are in contact with particles at the opposite side so only particle-particle contacts appear. Schemes of this type were used in e.g. [3, 22] with spherical particles, in [4] with ellipsoid-like particles and in [18] with ellipsoids. Figure 8 shows a two-dimensional sketch of a periodic cubical sample. All particle centers are restricted to lie within the box specified by the RVE dimensions Li . The contact at boundaries is realized by introducing so-called ghost particles Gi which are copies of the particles Pi inside the box displaced by

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σ1

L3

Fig. 9 Application of stress boundary conditions. The inner boundaries are fixed and the outer boundaries are considered as walls loaded by the RVE averaged principal stress σi and the applied principal stress σi

A1

L2 e3

L1

e2

tw

e1

unit cell displacement vectors d(α) with d(α) =

3

αi Li ei .

(3)

i=1

Of course only those ghost particles need to be considered which are in contact with a particle inside the RVE. Furthermore, each boundary contact appears twice on opposite sides of the RVE. Hence, α is restricted to the set S + of 13 vectors S + = {(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1),

(4)

(1, −1, 0), (1, 0, −1), (0, 1, −1), (1, 1, −1), (1, −1, 1), (−1, 1, 1)} .(5) Within the Verlet list concept the ghost particles can be handled like normal particles so that the set of ghost particles only needs to be updated when the Verlet lists are updated. Additionally, particles which left the RVE on one side are reentered at the opposite side. Contact forces from pairs (Gi , Pj ) are applied to the real particles Pi and Pj . The real particle states are updated by the time integration schemes and afterwards the ghost particle states are updated using the displacement vectors d(α).

4.2 Boundary Conditions To apply boundary conditions to the periodic RVE the RVE dimensions Li can be varied over time. The volume averaged principal strains are given by

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219

Li (t) − Li (t0 ) , Li (t0 )

(6)

i (t) =

where the RVE configuration at t = t0 is chosen as reference configuration. For the volume averaged stress tensor the standard formula reads (compare e.g. [6]) σ =

1 xj i ⊗ f j i , V

(7)

j i∈B

where V = L1 L2 L3 is the RVE volume and B is the set of contacts between particles inside and outside the RVE. Considering that B consists only of particleghost contacts and that each of these contacts appears twice on opposite sides we can deduce σ = =

1 V 1 V

# $ # $ x j i ⊗ f j i + x j i − d(α) ⊗ −f j i

(8)

j i∈B +

d(α) ⊗ f j i ,

(9)

j i∈B +

where B + is the set of contacts between real particles and ghost particles. Now the application of strain boundary conditions is accomplished by simply specifying the evolution of the RVE dimensions Li (t). However, to be able to model standard tests on granular materials like e.g. the triaxial test stress boundary conditions are required. For this purpose an adaptive dimension control similar to the scheme used in [22] is used. The inner boundaries of the RVE are fixed and the outer boundaries are considered as walls of thickness tw and density ρ, compare Figure 9. The inner side of the walls is loaded by the RVE averaged stress while on the outer side a user specified stress σi (t) is applied. Denoting the averaged stress in the principal directions by σi (t) the equation of motion of a wall reads ρ tw Ai L¨ i = Ai (σi − σi ) 1 ⇔ L¨ i = (σi − σi ) . ρ tw

(10) (11)

Furthermore, to simulate quasi-static stress controlled tests it is necessary to control the rate of deformation of the RVE. To measure the dynamical effects in the system the rate of deformation of the RVE can be compared to an intrinsic inertial time of the system yielding the dimensionless inertia parameter [10] m ¯ , (12) I = ˙ ¯ dp where m ¯ is the average particle mass, d¯ the average particle diameter and p the pressure inside the sample. The static limit is defined as I → 0 and it was shown in [2] that for I < 10−4 dynamical effects are small enough to consider the simulation

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as quasi-static. Hence, the stress applied to an RVE wall σi (t) will be controlled by specifying a target stress σ˜ i and a corresponding inertia parameter I . This yields the strain rate d¯ p ˙i = I , (13) m ¯ where p is the volume average pressure. Specifying an approximate elastic modulus E for the sample we can write the evolution equation and discrete update formula for the applied stress σ˙ i = E ˙i sign (σ˜ i − σi ) σi (t + ∆t) = σi (t) + σ˙ i ∆t .

(14) (15)

This together with equation (11) defines the stress controlled dimension control. To reduce oscillations in the wall movement it is useful to include a damping term in equation (11) 1 E (16) L¨ i = L˙ i , (σi − σi ) − 2 ζ ρ tw ρ tw Li so that ζ = 1 yields a critically damped system. As typical control parameters we chose ρ to be the particle density and tw as a small fraction of the average particle radius tw ≈ 0.1 r¯ . The RVE dimensions and dimension velocities are updated in each time step until the averaged principal stress reaches the target stress. Herein each dimension is controlled separately so that it is possible to use strain controlled boundary conditions in one direction and stress controlled in the other.

4.3 Random Sample Generation It is well known that the way of generating test samples of granular material has large influence on their resulting mechanical behavior. Regarding dry frictional materials the goal for laboratory testing is to produce uniform samples which are typically characterized by their solid fraction , i.e. the ratio of the particle volume to the overall volume of the sample. This is then compared to the minimum and maximum values min and max determined by standardized methods to characterize the sample as loose, medium dense or dense. Hence, a numerical algorithm is desirable, which, given the particle geometries, is able to produce samples of certain density. Here this is accomplished by a three-step algorithm: 1. Given the RVE size and the particle geometry parameters particles are randomly ˜ is reached. Next, all particles are generated until a user specified solid fraction shrinked by the same factor to give a user specified 0 . The shrinked particles

Homogenization of Granular Material Modeled by a 3D DEM

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0.72 0.71 0.7

Φ

0.69 Φmax≈ 0.72 Φmin≈ 0.64

0.68 0.67 0.66 0.65 0.64 0

0.2

0.4

0.6

0.8

1

µ

(a)

(b)

Fig. 10 (a) Solid fraction reached for different friction coefficients. (b) RVE with ghost particles

are placed in the RVE box by random sequential addition so that no overlap occurs. 2. In an iterative process the particles are translated, rotated and scaled in order to ˜ In each iteration a loop over all particles in random order is performed. reach . For each particle the distances to its nearest neighbors are determined. Then the particle is either translated or rotated in a way to maximize the minimum distance. Afterwards the particle is scaled. The decision if the particle is translated or rotated and the amount of translation, rotation and scaling is chosen randomly but in a way that excludes particle overlaps. 3. In the last step the particle package is compressed using stress controlled boundary conditions until a certain pressure is reached. The final solid fraction can be controlled by the inter-particle friction µ where a smaller µ yields a higher . ˜ = 0.55 and µ ∈ Typical values used for the sample generation are 0 = 0.26, [0, 1]. Figure 10a shows the relation between the final solid fraction reached and the inter-particle friction used in the final step of the packing algorithm. Figure 10b shows a generated sample with ghost particles.

5 Results For the first test series the parameters of the discrete element model were adapted to a real reference material. For this purpose we chose Leighton Buzzard Sand size fraction B for the following reasons: •

The size of the grains lie in a relatively narrow range between 0.6 and 1.18 mm which is favorable for the performance of the DEM code.

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Table 1 Material parameters of Leighton Buzzard sand fraction B from different references Ref.

ρ min g/cm3

max

[13] [15] [21] [25]

2.65 2.65 2.65 2.65

0.66 0.66 0.68 0.66

• • •

0.57 0.56 0.56 0.56

d10 mm

d50 mm

d60 mm

d60 /d10 Shape

0.64

0.78 0.8 0.84

0.81

1.27 1.3

rounded-subrounded rounded subangular-subrounded rounded

The grain shape is described as rounded to subrounded which can be better represented by the superquadric geometry than very angular grains. The grain material is silica and the grains show a high resistance against crushing which is favorable since we are not able to model particle breakage. The sand is widely used in research for laboratory testing so that sufficient reference data exists to check our model.

Table 1 lists characteristic values of the reference sand measured by different groups. The values of d10 , d50 and d60 describe the grain size distribution. Regarding the grain shape, however, there exist no exact and meaningful standardized characterization method. Hence, the grain shape is categorized by visual inspection and comparison to reference charts which is rather insufficient for adapting the geometry parameters of the DEM model. A more detailed analysis of the shape was done in [9] using an automated imaging method. Here approximately 1 500 grains were spread on a flat plate and pictures were taken from the top. Assuming that the smallest particle dimension is oriented normal to the plate the maximum inscribed and the minimum circumscribed circle of a particle were determined by an image analysis software. The diameters of these circles are referred as the large and intermediate dimension L and I of the particle. Then the volume of the particle was deduced from its mass and the small dimension of the particle S was derived from the volume under the assumption that the particle volume equals the volume of an ellipsoid of principal dimensions L, I and S. The resulting average particle dimensions are listed in Table 2. For the random particle generation process we chose the radius parameters ri of the superquadrics from Gaussian distributions with the mean taken from Table 2 and the standard deviation taken as 20% of the mean but with the restriction ri ∈ [0.25 mm, 0.75 mm]. The angularity parameters i could only be determined by visual comparison of the superquadrics and the grains which are described as rounded-subrounded. A uniform distribution from the interval [0.6, 1.2] was chosen. For the elastic parameters typical values for silica were chosen from the literature as E = 50 GPa and ν = 0.2. The friction between dry and wet particles was analyzed in [12, 19] yielding a value of µ = 0.24 as inter-particle friction coefficient.

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Table 2 Average grain dimensions for Leighton Buzzard sand fraction B from [9] L¯

I¯

S¯

1.14 mm

0.79 mm

0.61 mm

2.8

0.73

2.6

0.72

2.4

0.71

2.2 solid fraction

0.7

2 σ1 / σ3

µc = 0.0 µc = 0.001 µc = 0.005 µc = 0.01 µc = 0.02 µc = 0.05 µc = 0.1 µc = 0.2 µc = 0.5

1.8 µc = 0.0 µc = 0.001 µc = 0.005 µc = 0.01 µc = 0.02 µc = 0.05 µc = 0.1 µc = 0.2 µc = 0.5

1.6 1.4 1.2 1 0.8 0

5

10

15 -ε1, %

(a)

20

25

0.69 0.68 0.67 0.66 0.65 0.64

30

0

5

10

15 -ε1, %

20

25

30

(b)

Fig. 11 (a) Principal stress ratio vs. compressive strain for samples generated with different friction µc in the compression step. (b) Corresponding solid fractions

5.1 Packing Density To study the influence of the packing density on the macroscopic mechanical behavior samples of different solid fractions were generated using the algorithm described in Section 4.3 and the geometry parameters from above. Cubical RVEs with an initial edge length of L = 10 mm were used corresponding to a number of particles of about 1 700. The different solid fractions reached at a hydrostatic pressure of p = 1 kPa are shown in Figure 10. The resulting maximum and minimum of max ≈ 0.72 and min ≈ 0.64 lie above the values reported in Table 1 of max ≈ 0.66 and min ≈ 0.56. This is because first, the superquadrics are only an approximation of the real grain shapes and second, the packing algorithm does not model the standardized methods for determination of max and min in the laboratory since this would be an enormous computational effort. The macroscopic mechanical behavior of the samples is studied by means of standard triaxial tests: The samples are first compressed to a reference pressure of p = 25 kPa. From this reference state the principal stresses σ2 = σ3 = −p are kept constant using the adaptive stress control from Section 4.2 and the sample is compressed in the 1-direction using a strain rate corresponding to an inertia parameter of I = 10−4 . Figure 11 shows the resulting principal stress ratio and evolution of the solid fraction versus the compressive strain. The results are in good qualitative agreement with tests on the reference material: Initially loose packings show the slowest increase of stress ratio which reaches a constant value after about 10 to 15% axial strain. The dense packages show a very steep initial increase and reach a

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C. Wellmann and P. Wriggers Table 3 Data of RVE size test series L, mm number of samples avg. number of particles

10 10 1 700

15 8 6 000

20 5 14 000

30 4 45 000

3 2.2 2.5 2 2 εv, %

σ1 / σ3

1.8 1.6

1.5 1

1.4 0.5

σ3 = 25 kPa σ3 = 50 kPa σ3 = 100 kPa σ3 = 200 kPa

1.2 1 0

5

10 -ε1, %

(a)

15

σ3 = 25 kPa σ3 = 50 kPa σ3 = 100 kPa σ3 = 200 kPa

0 20

0

5

10 -ε1, %

15

20

(b)

Fig. 12 (a) Principal stress ratio vs. compressive strain for different reference pressures. (b) Corresponding volumetric strains

maximum shear strength which raises with the initial solid fraction. Afterwards they show a decrease to a steady value which is independent of the initial solid fraction. The solid fraction curves show that the dense packages behave dilatant nearly from the onset of compression while the solid fraction of the loose packages initially increases. After about 5% strain all samples show dilatant behavior and tend towards a unique value of ≈ 0.65 at a strain of about 25%.

5.2 RVE Size To analyze the influence of the RVE size triaxial tests with cubical RVEs of four different sizes were done, see Table 3. For each size a number of samples were generated over which the results are averaged. Each sample was generated with µ = 0.15 in the compression phase yielding an initial solid fraction of ≈ 0.67 corresponding to a relative density of 37% or a loose to medium dense state. To additionally check the influence of the reference pressure tests with σ2 = σ3 = 25, 50, 100, 200 kPa were performed for the smallest RVE size. The results are shown in Figure 12 in terms of the ensemble average values and related standard deviations. Obviously, neither the stress ratio nor the volumetric strain of the particle model depends on the reference pressure. Hence, tests on the larger RVEs were only done for one reference pressure. For evaluation of the test series two characteristic values are considered: First, the final shear strength characterized by the principal stress ratio averaged over the range −1 ∈ [0.17, 0.2] where it is more or less constant,

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0.24 2.3 - dεv / dε1 , dε1 = 10% - 15%

0.22

σ1 / σ3

2.25

2.2

2.15

0.2

0.18

0.16

0.14 2.1 0.12 10

15

20 sample size, mm

(a)

25

30

10

15

20 sample size, mm

25

30

(b)

Fig. 13 (a) Final stress ratio vs. sample size. (b) Volumetric strain rate vs. sample size

compare Figure 12. Second, the dilatancy, i.e. the ratio −∆v /∆1 averaged over the range −1 ∈ [0.1, 0.15]. The resulting mean values and standard deviations for the different RVE sizes are plotted in Figure 13. The stress ratio increases monotonically from 2.15 for L = 10 mm to 2.3 for L = 30 mm while the corresponding standard deviation decreases. However, the increase suggests that a sample size of L = 30 mm, which is equivalent to a number of 45 000 ≈ 353 particles, is not sufficient to be really representative. The evolution of the dilatancy gives a similar picture. It increases monotonically from 0.175 at L = 10 mm to 0.208 at L = 30 mm accompanied by a decreasing standard deviation.

5.3 Particle Shape Finally, the influence of the particle shape on the shear strength and dilatancy was studied. Therefore different geometry parameters were used in the sample generation process than those adapted to the Leighton Buzzard sand. The superquadric particle geometry basically enables two kinds of shape variation. First, it is possible to vary the elongation of the particles, i.e. change their shape from rather spherical to e.g. rather cylindrical. Second, the angularity of the particles can be varied from round to rather octahedral or cubical, compare Figure 1. The first variation was realized by introducing a shape parameter α which is the maximum allowed ratio of two orthogonal particle dimensions ri /rj . Hence, in the particle generation process the radius parameters ri of a particle were initially chosen uniformly distributed from the interval [1, α]. Afterwards the ri were scaled so that the sphere equivalent radii of the particles, i.e. the radius of a sphere having the same volume as the particle, were uniformly distributed in [0.5 mm, 1.0 mm] and therefore having a similar size distribution as the Leighton Buzzard sand. For the second shape variation a shape parameter β was introduced and the exponent parameters of the superquadrics were chosen uniformly distributed in the interval [1 − β, 1 + β].

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(a)

(b)

Fig. 14 (a) Particle sample with α = 4 and β = 0.1. (b) Particle sample with α = 1 and β = 0.4

σ1 / σ3

-∆ εV / ∆ ε1

2.3 2.2 2.1 2 1.9 1.8 1.7 0

0.1

β 0.2

0.3

1.75 1.8 1.85 1.9 1.95

0.4 1 2

(a)

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2

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3

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4

α

2.05 2.1 2.15 2.2 2.25

0.3 0.2 0.1 0 -0.1 0

0

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β 0.2 0.05

0.3

0.4 1 0.1

1.5

2

0.15

2.5

3

3.5

4

α

0.2

(b)

Fig. 15 (a) Final stress ratio vs. shape parameters α and β. (b) Corresponding dilatancy averaged over the range of −1 ∈ [0.1, 0.15]

For the triaxial tests cubical samples with L = 30 mm were used yielding approximately 8 000 particles per sample. Tests were done for all combinations of α ∈ {1, 1.5, 2, 3, 4} with β ∈ {0, 0.1, 0.2, 0.3, 0.4}. Two samples with different shape parameter combinations are shown in Figure 14. The elasticity parameters and inter-particle friction were chosen like in the previous tests and the results are again evaluated in terms of the final shear strength and dilatancy. Figure 15 shows their evolution over the tested shape parameter ranges. The final principal stress ratio σ1 /σ3 varies from about 1.75 for spherical particles (α = 1, β = 0) to about 2.2 for the most elongated (α = 4) and angular (β = 0.4) particle sample corresponding to an increase of 25%. Herein the elongation has a stronger influence than the angularity and the most significant changes happen in the range between α = 1 and α = 2 while for higher elongation values the increase in shear strength is rather small. Similar results were presented in [4, 14] where the

Homogenization of Granular Material Modeled by a 3D DEM Φ0

avg. coordination number

7 6.5 6 5.5 5 4.5

0.68 0.66 0.64 0.62 0.6 0

0.61

227

0.1

0.62

β

0.2

0.63

0.3 0.64

0.4 1 0.65

(a)

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2

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4

α

0.68

0

4.5

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β 0.2 5

0.3

0.4 1 5.5

1.5

2.5

2

6

3

3.5

4

α

6.5

(b)

Fig. 16 (a) Initial solid fraction. (b) Final average coordination number

shear strength of ellipsoid like particles was found to be higher than that of spherical particles. The influence of the angularity is biggest for the smallest elongation values where there is a moderate increase of shear strength as β increases. The dilatancy shows a similar dependency on the shape parameters: The smallest values with nearly isovolumetric shearing are reached by spherical particles and the highest strain ratios with ∆V /∆1 ≈ 0.2 are reached by the most elongated and angular particles. Again the influence of the elongation is ahead of the angularity’s influence with the biggest changes between α = 1 and α = 2. To find an explanation of this behavior two characteristic parameters of the particle packages were analyzed. First, the solid fraction of the particle sample before shearing, i.e. when the sample is under hydrostatic pressure. Second, the average coordination number, i.e. the number of contacts of a particle, in the final stage of the triaxial test. The results are shown in Figure 16. Notably, the initial solid fraction shows a similar dependency on the shape parameters as the shear strength and dilatancy with spherical particles reaching the smallest density and the most elongated and angular particles building the closest packing. Hence, it can be deduced that a closer particle packing possesses a higher shear strength and behaves more dilatant during shearing. Note that this effect is not to be mixed up with that depicted in Figure 11 since here all packages were generated using the same inter-particle friction and we only consider the final shear strength at −1 ≈ 0.2. The average coordination number mainly depends on the particle elongation and is increasing from about 4.8 for α = 1 to 6.55 at α = 4 which is an increase of about 25%. Compared to this the influence of the angularity is negligible. To some extent the increasing coordination number might explain the increasing shear strength and dilatancy since the additional contacts can support additional loads and resist the relative movement of adjacent particles yielding a reduced mobility and therefore higher volumetric strains during shearing. However, opposed to the shear strength and dilatancy the coordination number continuously increases between α = 1 and α = 4 so that additional effects seem to arise in the range between α = 2 and α = 4 which dilute the increase of shear strength and dilatancy. To connect these

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0.16

α= 1 α =1.5 α= 2 α= 3 α= 4

0.5

α= 1 α =1.5 α= 2 α= 3 α= 4

0.14 0.12 0.1 f(θ)

f( ||f|| )

0.4

0.3

0.08 0.2 0.06 0.1

0.04

0

0.02 0

0.5

1

1.5 ||f||, N

2

2.5

3

0

0.2

(a)

0.4

0.6

0.8 θ

1

1.2

1.4

1.6

(b)

Fig. 17 (a) Distribution of the norm of contact forces for samples with varying α and β = 0.1 at the final stage of the triaxial test. (b) Corresponding distribution of the angle between the contact forces and the loading direction

effects with changes in the microstructure the distribution of contact forces within samples with varying α and β = 0.1 were analyzed in the final stage of the triaxial test. Figure 17 shows the distribution of the norm of the contact forces and the angle between the contact forces and the loading direction θij = cos−1

|f ij · e1 | f ij

.

(17)

While the distribution of the norm of contact forces seems to be rather independent of the elongation, the orientation of the contact forces changes towards the loading direction (θ = 0) as the particles become more elongated. Here, as for the shear strength and the dilatancy, the main changes happen between α = 1 and α = 2. Hence, the shear strength and dilatancy might be more connected to the orientation of contact forces than on the very number of contact forces. Altogether, it can be summarized that the bulk behavior shows a strong dependency on the particle geometry especially in the vicinity of spherical particles (α = 1, β = 0). The influence of the elongation clearly exceeds that of the angularity in which the biggest changes take place between α = 1 and α = 2. The increase in shear strength can be explained by an increase of the average coordination number and a reorientation of the contact forces. The dilatancy seems to be strongly correlated with the shear strength where a higher shear strength is accompanied by higher volumetric strains during shear. Finally, the DEM results are compared to experimental results of triaxial tests on Leighton Buzzard sand with three different relative densities taken from [21], see Figure 18. Here the DEM results with the geometry parameters adapted to the Leighton Buzzard sand and the largest RVE size are used. While the volumetric strain shows a good quantitative agreement the stress ratio resulting from the DEM simulation reaching a final value of about 2.3 is much smaller than the experimental values of about 3.7. Regarding the above shape ana-

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229

3.5 3

4

2.5

3.5

εv, %

σ1 / σ3

2 3 2.5

1.5 1

2

0.5

Sim., Rd = 37%, σ3 = 25 kPa Exp., Rd = 29%, σ3 = 25 kPa Exp., Rd = 35%, σ3 = 50 kPa Exp., Rd = 25%, σ3 = 100 kPa

1.5 1 0

5

10 - ε1, %

(a)

15

Sim., Rd = 37%, σ3 = 25 kPa Exp., Rd = 29%, σ3 = 25 kPa Exp., Rd = 35%, σ3 = 50 kPa Exp., Rd = 25%, σ3 = 100 kPa

0 -0.5 20

0

5

10 - ε1, %

15

20

(b)

Fig. 18 (a) Comparison of the stress ratio from DEM simulations and laboratory experiments. (b) Corresponding volumetric strain. Experimental results are taken from [21]

lysis this difference lies beyond the range which could be reached by a variation of the superquadric geometry parameters. Hence, it seems that the superquadric shape, even though it reaches considerably higher shear strengths than the spherical, is still insufficient to obtain a reasonable quantitative agreement with experiments on the reference material. This is believed to be mainly due to the fact that while superquadrics can represent elongated and angular particles, they are still restricted to be convex. Hence, contacts between adjacent particles are always restricted to a single point which excludes the transmission of torques. Indeed, simulations using simple non-convex particles like glued spheres [20] show a remarkable increase in shear strength compared to simple spheres. The same was shown for packings of convex polyhedra [5], which can transfer torques over edge or face contacts. Hence, a logical step towards a more realistic model would be the introduction of a more realistic particle shape where the simplest possibility is to use clusters of the already implemented superquadrics. However, to adapt the additional geometric degrees of freedom a more detailed description of real grain shapes would be required which is not available at the moment.

6 Conclusion Within this contribution a discrete element model was introduced whose main features are the superquadric particle geometry and the application of the Hertzian contact model. Hence, the only non-geometric DEM parameters are the elastic constants of the particles and the inter-particle friction. The superquadric geometry on the other hand enables the variation of the elongation and angularity of the particles. To handle the computational effort of the DEM simulations a parallelization scheme was developed that reduces the computation times on shared memory architectures in an effective way. The scheme is based on the use of Verlet neighbor lists and a

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spatial sorting of the particles and can be easily implemented into standard DEM codes. Next, a homogenization scheme based on periodic cubical unit cells was introduced and a method to accurately apply strain and stress boundary conditions for quasi-static tests was developed. Furthermore, an algorithm was depicted that is capable to generate random particle packages of varying density. These methods were then applied to perform numerical triaxial tests whereat the parameters of the DEM model were adapted to Leighton Buzzard sand size fraction B. An analysis of the influence of the RVE size showed that the variation between different samples decreases with sample size but that the resulting macroscopic behavior is still not converged even for samples as large as 45 000 particles. In the next step the influence of the particle shape was studied by varying the elongation and angularity of the particles. Here it was shown that the influence of the elongation on the shear strength and dilatancy of the sample packages exceeds that of the angularity. Especially in the vicinity of spherical particles there is a considerable increase in shear strength and dilatancy with elongation. Finally, the simulation results were compared to laboratory experiments on Leighton Buzzard sand. A good quantitative agreement for the dilatancy behavior was found but on the other hand the shear strength of the numerical samples was much smaller than in the experiments. This was argued to be the result of a still insufficient particle geometry, which excludes the existence of multiple contacts between adjacent particles and hence the transmission of torques. Consequently, a logical step to a more realistic particle model would be the introduction of enhanced particle shapes like e.g. superquadric clusters. This, however, would necessitate an improved description of real grain shapes that is not available at the moment.

References 1. The OpenMP API specification for parallel programming. http://openmp.org. 2. Agnolin, I., Roux, J.N., On the elastic moduli of three-dimensional assemblies of spheres: Characterization and modeling of fluctuations in the particle displacement and rotation. Int. J. Solids Struct. 45(3–4):1101–1123, 2008. 3. Antony, S.J., Kruyt, N.P., Role of interparticle friction and particle-scale elasticity in the shearstrength mechanism of three-dimensional granular media. Phys. Rev. E 79(3), 2009. 4. Antony, S.J., Kuhn, M.R., Influence of particle shape on granular contact signatures and shear strength: New insights from simulations. Int. J. Solids Struct. 41(21):5863–5870, 2004. 5. Azema, E., Radjai, F., Saussine, G., Quasistatic rheology, force transmission and fabric properties of a packing of irregular polyhedral particles. Mech. Mater. 41(6):729–741, 2009. 6. Bardet, J.P., Vardoulakis, I., The asymmetry of stress in granular media. Int. J. Solids Struct. 38(2):353–367, 2001. 7. Barr, A.H., Superquadrics and angle-preserving transformations. IEEE Comput. Graphics Appl. 1(1):11–23, 1981. 8. Belheine, N., Plassiard, J.P., Donze, F.V., Darve, F., Seridi, A., Numerical simulation of drained triaxial test using 3D discrete element modeling. Comput. Geotech. 36(1–2):320–331, 2009. 9. Clayton, C.R.I., Abbireddy, C.O.R., Schiebel, R., A method of estimating the form of coarse particulates. Geotechnique 59(6):493–501, 2009.

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10. da Cruz, F., Emam, S., Prochnow, M., Roux, J.N., Chevoir, F., Rheophysics of dense granular materials: Discrete simulation of plane shear flows. Phys. Rev. E 72(2), 2005. 11. Hertz, H., Über die Berührung fester elastischer Körper (On the contact of elastic solids). Journal für die reine und angewandte Mathematik 92:156–171, 1882. 12. Ishibashi, I., Perry, C., Agarwal, T.K., Experimental determinations of contact friction for spherical glass particles. Soils Found. 34(4):79–84, 1994. 13. Kingston, E., Clayton, C.R.I., Priest, J., Best, A., Effect of grain characteristics on the behaviour of disseminated methane hydrate bearing sediments. In: Proceedings of the 6th International Conference on Gas Hydrates. 2008. 14. Lin, X., Ng, T.T., A three-dimensional discrete element model using arrays of ellipsoids. Geotechnique 47(2):319–329, 1997. 15. Lings, M.L., Dietz, M.S., An improved direct shear apparatus for sand. Geotechnique 54(4):245–256, 2004. 16. Mindlin, R.D., Compliance of elastic bodies in contact. J. Appl. Mech. 16:259–268, 1949. 17. Munjiza, A., The Combined Finite-Discrete Element Method. Wiley, 2004. 18. Ouadfel, H., Rothenburg, L., An algorithm for detecting inter-ellipsoid contacts. Comput. Geotech. 24(4):245–263, 1999. 19. Rowe, P.W., The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. R. Soc. London, Ser. A 269(1339):500–527, 1962. 20. Salot, C., Gotteland, P., Villard, P., Influence of relative density on granular materials behavior: DEM simulations of triaxial tests. Granular Matter 11(4):221–236, 2009. 21. Schnaid, F., A study of the cone-pressuremeter test in sand. Ph.D. Thesis, University of Oxford, 1990. 22. Thornton, C., Numerical simulations of deviatoric shear deformation of granular media. Geotechnique 50(1):43–53, 2000. 23. Verlet, L., Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard–Jones molecules. Phys. Rev. 159(1):98–103, 1967. 24. Wellmann, C., Lillie, C., Wriggers, P., Comparison of the macroscopic behavior of granular materials modeled by different constitutive equations on the microscale. Finite Elem. Anal. Des. 44(5):259–271, 2008. 25. White, D.J., Bolton, M.D., Displacement and strain paths during plane-strain model pile installation in sand. Geotechnique 54(6):375–397, 2004.

Some Consideration on Derivative Approximation of Particle Methods Hitoshi Matsubara, Shigeo Iraha, Genki Yagawa and Doosam Song

Abstract In this paper, the accuracy of the derivative approximation of the particle methods is discussed. Especially, we show that the issue of decreasing accuracy on a boundary area in the SPH method is due to the lack of the boundary integration. Through some numerical examples, the convergence of error norm of energy obtained by the SPH and the MPS methods is studied.

1 Introduction The advantage of the particle methods lies in the fact that no cost is needed in mesh generation contrary to the usual finite element methods [1], although there have been several studies to remove this bottleneck of the finite element methods. It is well known that the particle or the node based methods [2–6] have the following merits [7]: (a) Simulations of very large deformations can be easily handled. (b) The data structure of particle or node discretization can be linked easily with the CAD database. (c) The adaptive refinement procedure can be easily controlled. (d) The mesh-free discretization provides accurate representation of the geometry of the objects to be solved. Hitoshi Matsubara · Shigeo Iraha Department of Civil Engineering and Archtecture, University of the Ryukyus, 1, Senbaru, Nishihara, Okinawa 903-0213, Japan; e-mail: {matsbara, iraha}@tec.u-ryukyu.ac.jp Genki Yagawa Center for Computational Mechanics Research, Tokyo University, 2-36-5, Hakusan, Bunkyo-ku, Tokyo 112-0001, Japan; e-mail: [email protected] Doosam Song School of Architecture, Sungkyunkwan University, 300 Cheoncheon, Suwon 440-746, Korea; e-mail: [email protected]

E. Oñate and R. Owen (eds.), Particle-Based Methods, Computational Methods in Applied Sciences 25, DOI 10.1007/978-94-007-0735-1_9, © Springer Science+Business Media B.V. 2011

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Lucy [8] and Gingold and Monaghan [9] originated the smoothed particle hydrodynamics (SPH) for astrophysical computations, which is a purely Lagrangian description and has been extended to solve a wide range of issues in physics or engineering fields. Today, SPH is being extensively applied to simulations of supernovas, collapse as well as formation of galaxies [7], also SPH has been extended to treat the incompressive flow problems or the solid mechanics [11]. Concerning the accuracy improvements of SPH, Belytschko et al. [12] examined corrected first derivative approximations, where the proposed approximations restored the accuracy in various benchmark problems, but establishment of the complete approximation was not reached because of the lack of integrability. On the other hand, Libersky et al. [13,14] introduced ghost particles to reflect a symmetrical surface boundary condition. Recently, Chen et al. [15–17] proposed a simple corrective kernel approximation by applying the kernel estimate to the Taylor series expansion. By using these modified numerical techniques, the derivative approximation of SPH is improved dramatically. However, the problem of the accuracy issue on the boundary or the edge of the object is not completely solved even if we use the above techniques. A similar particle approach is called the moving particle semi-implicit (MPS) method [18–21], which is based on the Lagrangian description. In this method, the differential operators such as the gradient and Laplacian in the governing equations are previously-prepared in an intuitive manner. The general accuracy in this method remains to be clarified. As discussed in the above, the accuracy of the derivative approximation of the particle methods is one of the most important issues, and the accuracy on the boundary or the edge remains to be solved in particular. One of the reasons might be due to the fact that we cannot calculate boundary integrals precisely because there is no definite or exact boundary in the particle methods. In this study, we study the accuracy of the derivative approximation for SPH and MPS methods. Particularly, concerning the accuracy at the boundary or the edge of the body to be solved, we theoretically examine the discretization formulation especially for SPH. Then, the present paper aims to discuss the influence of number of particles or particle distributions on the accuracy of SPH and MPS method through some numerical examples.

2 Formulation and Some Remarks on Particle Methods In the so-called Smoothed Particle Hydrodynamics or SPH method, the continuum is assumed to be a collection of particles, and its density, velocity, stresses etc. are evaluated through the particle-wise manner. Its interpolation is based on the following concept of kernel estimate: f (x) ∼ f (x$ )w(x − x$ )dx$ , x ∈ # (1) = #

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where f (x) is an a vector function of the three-dimentional position vector x and w(x − x$ ) is the kernel function selected to be zero beyond the interaction area. The approximation for spatial derivatives is obtained by substituting ∇ · f (x) for f (x) in Equation (1) as follows: ∼ ∇ · f (x$ )w(x − x$ )dx$ (2) ∇ · f (x) = #

The divergence of the above equation is given as [22] # $ ∇ f (x$ )w(x − x$ ) dx$ − f (x$ )∇ · w(x − x$ )dx$ ∇ · f (x) ∼ = #

#

(3)

The first term of the right-hand side of Equation (3) can be converted by means of the divergence theorem into an integral over the surface of the domain of integration as follows: # $ ∇ f (x$ )w(x − x$ ) dx$ = f (x$ )w(x − x$ )ds (4) #

s

The surface integral becomes zero because the kernel function w(x − x$ ) is zero on the boundaries. Therefore, the approximation for spatial derivatives of f (x) is defined as ∼ (5) ∇ · f (x) = − f (x$ )∇ · w(x − x$ )dx$ #

Based on the above-mentioned theoretical background, we can obtain the following equations as the SPH discretization: f (x) =

N mj j =1

∇ · f (x) = −

ρj

f (xj )w(x − xj )

N mj j =1

ρj

f (xj )∇ · w(x − xj )

(6)

(7)

where N is the number of particles in an interaction area, mj is the mass of particle j and ρj is defined as N mk w(x − xk ) (8) ρj = k=1

On the other hand, a modified kernel [15–17], which is resulted by dividing Equations (6) and (7) by the sum total of kernel function, has been proposed in order to improve the accuracy of discretization as shown in the following equations:

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Fig. 1 Interaction area around particle i (h: radius of interaction area)

N !

f (x) =

mj ρj

j =1

f (xj )w(x − xj )

N ! j =1 N !

∇ · f (x) = −

j =1

(9) mj ρj

mj ρj

N ! j =1

w(x − xj )

f (xj )∇ · w(x − xj ) (10) mj ρj

∇ · w(x − xj )

In the SPH method, Equations (6), (7), (9) and (10) are directly employed for calculations. It is here noted that these equations can be used only when the conversion from Equation (3) to Equation (5) is valid. In other words, we should be careful if these are applied to the continuum with a boundary. Or this conversion is valid only in cases where there are no physical boundaries in the interaction area as shown in Figure 1a, and we cannot apply this conversion to the interaction area with physical boundaries as shown in Figure 1b. There will be some effects on the accuracy of results in the continuum with physical boundaries, although the effects are not an issue in such applications as the astro-dynamics, which treat unbounded body. Thus, in the interaction area with physical boundary as shown in Figure 1b, we must employ the following equation: f (x$ )∇ · w(x − x$ )dx$ (11) ∇ · f (x) ∼ = f (x$ )w(x − x$ )ds − s

#

As shown in the above equation, the boundary integration is an indispensable term in the interaction area with physical boundary, and we need the boundary mesh in order to perform this integration. In the Moving Particle Semi-implicit or MPS method [18], the gradient vector of an arbitrary particle i is defined as the weighted averaging “gradient model” between

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Fig. 2 Evaluation of gradient at particle i in MPS method (h: radius of interaction area)

the particles in the interaction area. Let us assume that particle i and particle j has positional vectors ri and rj , and physical value φi and φj , respectively. φj can be approximated by the Taylor expansion around the particle i as follows: # $ φj = φi + ∇φ|ij · rj − ri (12) # $ ∇φ|ij · rj − ri = φj − φi

(13)

Dividing both sides of Equation (12) by the absolute value of a relative positional vector between particles i and j , the following equation is obtained: # $ rj − ri φ − φi = j (14) ∇φ|ij · rj − ri rj − ri By multiplying the unit vector in the direction of a relative positional vector of particles i and j by both sides of Equation (14), we obtain # $ # $ $# $ # r j − ri φj − φi rj − ri rj − ri = ∇φij = ∇φ|ij · (15) rj − ri rj − ri rj − ri 2 The right-hand side of Equation (15) is an element of relative positional vector in the “gradient model” as shown in Figure 2, and the gradient vector of particle i is modeled as the following equation by using the particles, which are in interaction area, # $# $ $ # d φj − φi rj − ri ∇φi = w rj − ri (16) n0 rj − ri 2 j =i where d is the number of dimension, n0 the density of particle number and w(rj − ri ) the weighting function. It is considered in this method that the accuracy will be deteriorated when the distribution of particles is arranged in an irregular pattern. We need to study this issue theoretically and numerically as well.

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Fig. 3 Timoshenko beam model; particles are regularly located as 8 × 2, 16 × 4, 24 × 6, 32 × 8, 40 × 10, 48 × 12 and 56 × 14

3 Numerical Examples 3.1 Energy Norm in Elasticity Field Both SPH and MPS are applied to the bending problem of the Timoshenko’s cantilever beam in elasticity [23], where the length is 8, the height 2 and the width 1 as shown in Figure 3. This model includes parabolic variation of applied shear traction at x = L with essential boundary condition at x = 0 to match the exact solution [24]. In this example, the theoretical displacement fields are given by the following equations: D2 Py 2 · (6L − 3x) x + (2 + ν) y − (17) u=− 6EI 4 P D2 x 2 2 v= · 3νy (L − x) + · (4 + 5ν) + (3L − x) x 6EI 4

(18)

We assume the plane stress condition with the Poisson’s ratio ν of 0.3 and the Young’s modulus E of 1.0e + 3 as the material constants. As shown in Figure 3, we prepare 7 symmetrically arranged nodal patterns: 8 × 2, 16 × 4, 24 × 6, 32 × 8, 40 × 10, 48 × 12, and 56 × 14. Here, enforcing the theoretical values given by Equations (17) and (18) on the particles, the strain energy values are calculated. In order to study the convergence, the standard deviation of the error norm in energy or SE is defined as follows: * + n +1 SE = , (19) (|ei | − |e¯i |)2 n i=1

-

with |ei | =

# $T # $ εi − εiexact σi − σiexact

(20)

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Fig. 4 Standard deviation of energy norm versus numbers of particles, where “SPH1” means the results with Equations (6) and (7), “SPH2” Equations (9) and (10), and “Moving Least Squares method” based on 1st order polynomial approximation

where εi and σi are the numerical results of strain and stress on particle i, respectively, and εiexact and σiexact the theoretical values of the same, respectively. The latter is given as Py εx = − (21) (L − x) EI νP y · (L − x) (22) εy = EI 2 P D εxy = − y2 (23) (1 + ν) EI 4 It is noted here that the stresses of Equation (20) are calculated by using the stressstrain relationship in elasticity, σ = Dε (D: stress-strain matrix), as usual. In this chapter we employ the biquadratic B-Spline function as the weighting function for the SPH method, which is given as ⎡ 3 4 ⎤ 2 rj rj rj 5 ⎣ ⎦ (0 ≤ rj ≤ h) + 8 − 3 1 − 6 w(rj ) = πh2 h h h (24) where h is the radius of the support of the weighting function, and |rj | the distance between two points x and xj . Regarding the MPS method, the following inverse power weighting functions is chosen: −α (α = 1) (25) w(rj ) = rj Figure 4 shows the analytical results of the standard deviation of the norm in energy versus the numbers of particles, where “SPH1” means the case employing

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Fig. 5 Comparisons of strain distributions; 56 × 14 particles with regular arrangement (the figures in the right-hand side are the zoomed ones)

Equations (6) and (7) as the SPH discretization, whereas “SPH2” means the case of Equations (9) and (10). The results of moving least squares method [25] based on the 1st order polynomial approximation are also shown in this figure. From this figure, we can see that, although the accuracy of SPH2 is better than that of SPH1, the convergence characteristic of these methods is not satisfactory even if the numbers

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Fig. 6 Two kinds of particle distributions in case of 3721 particles

of particles increase. We consider that this result is due to the fact that the boundary integration is neglected as discussed in Section 2. On the other hand, the accuracy and convergence of MPS is better than those of SPH1 and SPH2, but worse than those of the Moving least squares method. Then, we examine the distribution of εxy in the SPH and MPS method in case of 56 × 14 particles arrangement as shown in Figure 5. It is seen from this figure that, as for the results by all the method except SPH1, some error is observed on the boundaries x = 8.0 and 0.0. Particularly, the results of SPH2 methods diverge around these points and those of SPH1 are not agreeable with the exact solution at the whole domain.

3.2 Strain Distributions in Complicated Displacement Field Let us apply the SPH and the MPS methods to the complex displacement field described as (26) u = sin x 2 + y 2 (27) v = cos x 2 + y 2 In this example, Equations (26) and (27) are employed as the given displacement vector at the particles, and the calculated strains on the particles are compared with theoretical ones given as follows: x · cos x 2 + y 2 (28) εxx = x2 + y2

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Fig. 7 Comparison of strain distributions; 3721 particles with regular arrangement

εxy

y · sin x 2 + y 2 εyy = − x2 + y2 y · cos x 2 + y 2 x · sin x 2 + y 2 = − x2 + y2 x2 + y2

(29)

(30)

In this chapter, the SPH and the MPS methods based on the regular particle pattern and the random one are compared as shown in Figure 6. Figure 7 shows the calculated distributions of strains (εxx , εyy , εxy ) by the SPH and the MPS methods in case of the regular particle pattern. From this figure, the distributions of strains are in good agreement with the exact results in the inner area x = 9.67 or 1.0 except for SPH1, whereas the results of SPH1 shows deviations from the theoretical ones over the whole domain and the results of SPH2 show

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Fig. 8 Comparisons of shear strain distributions; 3721 particles with random arrangement

some deviations from the theoretical ones on the boundary area x = 10.0. On the other hand, the results of the MPS method are generally in good agreement with the exact ones even on the boundary area. Figure 8 shows the calculated distributions of strain εxy by the SPH and the MPS methods in case of random particle pattern. From this figure, it is seen that the SPH1 gives deviated solutions from the theoretical ones in general, while the SPH2 does large errors only on the boundaries. On the other hand, the results of the MPS are relatively in good agreement with the theoretical ones, but there are also some errors on the boundaries.

4 Conclusion Discussed in the present paper is the accuracy of the derivative approximation of the particle methods. Especially shown is that the lack of accuracy on the boundary area in the SPH method is due to the lack of the proper boundary integration. In the numerical examples, we show that, although the accuracy of the SPH2 (modified version of the original SPH) is better than that of the SPH1 (original SPH), the accuracy of both methods is not converged even if the number of particles increases. Furthermore, in the examples of complicated strain fields, we show that,

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in case of the random particle pattern, the distributions of strains by the SPH and the MPS methods, especially those of εxy , are not corresponding to the exact ones. From the above, it is concluded that more study is needed to re-examine the discretization procedure of the SPH and the MPS methods, especially concerning the discretization of boundaries of the analysis domain.

Acknowledgements We are grateful to Professor Nakaza and Mr. Iribe, the Disaster Prevention Research Center for Island Regions in University of the Ryukyus for their important contributions to the particle theory.

References 1. Zienkiewicz, O.C., Taylor, R.L., Finite Element Method, 5th ed., Vol. 1, ButterworthHeinemann, Oxford, 2000. 2. Oñate, E., Idelsohn, S.R., Celiguetaa, M.A., Rossia, R., Advances in the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free surface flows, Computer Methods in Applied Mechanics and Engineering, 197(15):1777–1800, 2008. 3. Yagawa, G., Node-by-node parallel finite elements, a virtually meshless method, International Journal for Numerical Methods in Engineering 60(1):69–102, 2004. 4. Matsubara, H., Yagawa, G., Convergence studies for Enriched Free Mesh Method and its application to fracture mechanics, Interaction and Multiscale Mechanics: An International Journal 2(3):277–293, 2009. 5. Yagawa, G., Matsubara, H., Enriched free mesh method: An accuracy improvement for nodebased FEM, Computational Plasticity 7:207–219, 2007. 6. Tian, R., Matsubara, H., Yagawa, G., Advanced 4-node tetrahedrons, International Journal for Numerical Methods in Engineering 68(12):1209–1231, 2006. 7. Li, S., Liu, W.K., Meshfree Particle Methods, Springer, Heidelberg, 2007. 8. Lucy, L.B., A numerical approach to the testing of the fission hypothesis, The Astronomical Journal 82:1013–1024, 1977. 9. Gingold, R.A., Monaghan, J.J., Smoothed particle hydrodynamics: Theory and application to non-spherical stars, Mon. Not. R. Astr. Soc. 181:375–389, 1977. 10. Li, S, Liu, W.K., Meshfree and particle methods and their applications, Applied Mechanics Review 55(1):1–34, 2002. 11. Monaghan, J.J., Simulating free surface flows with SPH, Journal of Computational Physics 110:399–406, 1994. 12. Belytschko, T., Krograuz, Y., Organ, D., Fleming, M., Krysl, P., Meshless methods: An overview and recent developments, Computer Methods in Applied Mechanics and Engineering 139:3–47, 1996. 13. Libersky, L.D., Petschek, A.G., Smooth particle hydrodynamics with strength of materials. In Advances in the Free-Lagrange Method, H.E. Trease, M.J. Fritts, W.P. Crowley (eds.), Lecture Notes in Physics, Vol. 395, pp. 248–257, Springer, 1993. 14. Randles, P.W., Libersky, L.D., Smoothed particle hydrodynamics: Some recent improvements and applications, Computer Methods in Applied Mechanics and Engineering 139:375–408, 1996.

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15. Chen, J.K., Beraun, J.E., Jih, C.J., An improvement for tensile instability in smoothed particle hydrodynamics, Computational Mechanics 23:279–287, 1999. 16. Chen, J.K., Beraun, J.E., Jih, C.J., Completeness of corrective smoothed particle method for linear elastodynamics, Computational Mechanics 24:273–285, 1999. 17. Chen, J.K., Beraun, J.E., Carney, T.C., A corrective smoothed particle method for boundary value problems in heat conduction, International Journal for Numerical Methods in Engineering 46:231–252, 1999. 18. Koshizuka, S., Oka, Y., Moving-particle semi-implicit method for fragmentation of incompressible fluid, Nuclear Science and Engineering 123:421–434, 1996. 19. Chikazawa, Y., Koshizuka, S., Oka, Y., Numerical analysis of three dimensional sloshing in an elastic cylindrical tank using moving particle semi-implicit method, Computational Fluid Dynamics 9:376–383, 2001. 20. Koshizuka, K., Nobe, A., Oka, Y., Numerical analysis of breaking waves using the moving particle semi-implicit method, International Journal for Numerical Methods in Fluids 26(7):751–769, 1998. 21. Iribe, T., Fujisawa, T., Koshizuka, S., Reduction of communication between nodes on largescale simulation of the particle method, Transactions of JSCES, No. 20080020, 2008. 22. Swegle, J.W., Attaway, S.W., Heinstein, M.W., Mello, F.J., Hicks, D.L., An analysis of smoothed particle hydrodynamics, SANDIA Report SAND93-2513, 1994. 23. Timoshenko, S.P., Goodier, J.N., Theory of Elasticity, McGraw-Hill, New York, 1979. 24. Augarde, C.E., Deeks, A.J., The use of Timoshenko’s exact solution for a cantilever beam in adaptive analysis, Finite Elements in Analysis and Design 44:595–601, 2008. 25. Lancaster, P., Salkauskas, K., Surfaces generated by moving least squares methods, Mathematics of Computation 37(155):141–158, 1981.

Discrete Element Modelling of Rock Cutting Jerzy Rojek, Eugenio Oñate, Carlos Labra and Hubert Kargl

Abstract This paper presents numerical modelling of rock cutting processes. The model consists of a tool-rock system. 3D geometry is considered in the model. The rock is modelled using the discrete element method, which is suitable to study problems of multiple material fracturing like that of rock cutting. The paper presents brief overview of the theoretical formulation and calibration of the discrete element model by simulation of the unconfined compressive strength (UCS) and indirect tension (Brazilian) tests. Numerical examples illustrate the paper. Rock cutting processes typical for underground excavation using both roadheader and TBM cutting tools are simulated. Numerical results are compared with the available experimental data.

1 Introduction A variety of rock-cutting technologies is used in civil as well as in mining engineering. Figures 1a and 2a show machines performing rock cutting in underground excavation, a roadheader and a tunnel boring machine (TBM), respectively. Roadheaders excavate the rock by means of conical point attack picks (Figure 1b) mounted on a rotating cutterhead supported by a boom which is independently movable in the vertical and horizontal direction. In the excavation with TBMs, the rock is

Jerzy Rojek ´ Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-106 Warsaw, Poland; e-mail: [email protected] Eugenio Oñate · Carlos Labra International Center for Numerical Methods in Engineering (CIMNE), Universidad Politécnica de Cataluña, Campus Norte UPC, 08034 Barcelona, Spain; e-mail: {onate, clabra}@cimne.upc.edu Hubert Kargl Sandvik Mining and Construction GmbH, Zeltweg, Austria; e-mail: [email protected]

E. Oñate and R. Owen (eds.), Particle-Based Methods, Computational Methods in Applied Sciences 25, DOI 10.1007/978-94-007-0735-1_10, © Springer Science+Business Media B.V. 2011

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(a)

(b) Fig. 1 Roadheader: (a) rock excavation with a roadheader, (b) typical design of a point attack pick

cut by means of cutter discs (Figure 2b) installed on a rotating cutter head, which is pressed against the tunnel face. The basic physical phenomenon occurring during rock cutting is desintegration of the rock under mechanical action of a cutting tool. Design of cutting tools and setting parameters of cutting operations requires knowledge about the cutting process. Cutting force is one of the main factors characterizing a cutting process. Theoretical evaluation of the cutting force is not an easy task. Simple analytical models, like those developed by Evans [3] or by Nishimatsu [8], can provide an approximate estimation of cutting forces only. Numerical simulation can provide valuable information about the cutting phenomenon. Numerical methods based on the continuum models, like finite element methods, have serious problems in modelling discontinuities of the material occurring during rock cutting [5]. The discrete element method takes into account all

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(a)

(b) Fig. 2 TBM: (a) general view of a TBM cutterhead, (b) TBM disc cutters

kinds of discontinuities and material failure characterized with fracture and therefore is a suitable tool to study rock cutting [4, 11, 12].

2 Numerical Model of Rock Cutting A system consisting of a tool and rock sample is considered in the model (Figure 3). The rock material is represented as a collection of spherical (in 3D) or cylindrical (in 2D) discrete elements interacting among themselves with contact forces. The tool is considered a rigid body with a surface discretised with triangular facets. The tool-rock interaction is modelled assuming Coulomb friction model. A numerical model of rock cutting has been developed within the authors’ own implementation of the discrete element method (DEM) in the computer program DEMPack [9, 10].

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Fig. 3 Geometrical scheme of a rock cutting model

Fig. 4 Motion of a discrete element

3 Discrete Element Method Formulation The translational and rotational motion of rigid spherical or cylindrical elements (particles) is governed by the standard equations of rigid body dynamics. For the i-th element (Figure 4) we have mi u¨ i = Fi ,

(1)

Ji ω˙ i = T i ,

(2)

where ui is the element centroid displacement in a fixed (inertial) coordinate frame X, ωi – the angular velocity, mi – the element mass, Ji – the moment of inertia, Fi – the resultant force, and T i – the resultant moment about the central axes. The form of the rotational equation (2) is valid for spheres and cylinders (in 2D) and is simplified with respect to a general form for an arbitrary rigid body with the rotational inertial properties represented by a second order tensor. Vectors Fi and T i are sums of:

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Fig. 5 Contact interaction between two discrete elements

• • •

all forces and moments applied to the i-th element due to external load, Fiext and T iext , respectively, contact interactions with neighbouring spheres Fijcont , j = 1, . . . , nci , where nci are the number of elements being in contact with the i-th discrete element, damp damp forces and moments resulting from external damping, Fi and T i , respectively nci damp ext Fijcont + Fi , (3) Fi = Fi + j =1 nci

T i = Tiext +

damp

scij × Fijcont + T i

,

(4)

j =1

where scij is the vector connecting the centre of mass of the i-th element with the contact point with the j -th element (Figure 5). Equations of motion (1) and (2) are integrated in time using the central difference scheme. The time integration operator for the translational motion at the n-th time step is as follows: Fn u¨ ni = i , (5) mi n+1/2

u˙ i

n−1/2

= u˙ i

+ u¨ ni t ,

n+1/2

= uni + u˙ i un+1 i

t .

(6) (7)

The first two steps in the integration scheme for the rotational motion are identical to those given by Eqs. (5) and (6): ω˙ ni = n+1/2

ωi

T ni , Ji

n−1/2

= ωi

+ ω˙ ni t .

The vector of incremental rotation θ i is calculated as

(8) (9)

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θ i = ωi

(10)

t ,

If necessary it is also possible to track the total change of rotational position of particles [1]. Explicit integration in time yields high computational efficiency of the solution for a single step. The disadvantage of the explicit integration scheme is its conditional numerical stability imposing the limitation on the time step t. The time step t must not be larger than a critical time step t cr t ≤ t cr

(11)

determined by the highest natural frequency of the system νmax t cr =

2 . νmax

(12)

Exact determination of the highest frequency νmax would require solution of the eigenvalue problem defined for the whole system of connected rigid particles. The maximum frequency of the whole system can be estimated as the maximum of natural frequencies νie of subsets of connected particles surrounding each particle e, cf. [2]: D D νmax ≤ νmax , where νmax = max νie (13) i,e

elements1

cont

The contact force between two F can be decomposed into normal and tangential components, Fncont and Ftcont , respectively + Ftcont = Fncont n + Ftcont , F cont = Fcont n

(14)

where n is the unit vector normal to the particle surface at the contact point. The contact forces Fncont and Ftcont are obtained using a constitutive model formulated for the contact between two rigid spheres. In the present work rock materials are modelled using elastic perfectly brittle model of contact interaction, where we assume initial bonding for the neighbouring particles. These bonds can be broken under load allowing us to simulate initiation and propagation of material fracture. Contact laws for the normal and tangential direction for the elastic perfectly brittle model are shown in Figure 6. When two particles are bonded the contact forces in both normal and tangential directions are calculated from the linear constitutive relationships: (15) Fncont = kn un , Ftcont = kt ut ,

(16)

where Fncont – normal contact force, Ftcont – tangential contact force, kn – interface stiffness in the normal direction, kt – interface stiffness in the tangential direction, un – normal relative displacement, ut – tangential relative displacement. 1

In the next part of this section indices denoting the elements will be omitted.

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(a)

(b) Fig. 6 Force-displacement relationships for the elastic perfectly brittle model: (a) in the normal direction, (b) in the tangential direction

Cohesive bonds are broken instantaneously when the interface strength is exceeded in the tangential direction by the tangential contact force or in the normal direction by the tensile contact force. The failure (decohesion) criterion can be written as Fncont ≤ Rn , (17) Ftcont ≤ Rt ,

(18)

where Rn – interface strength in the normal direction, Rt – interface strength in the tangential direction. In the absence of cohesion the normal contact force can be compressive only (Rn ≤ 0) and tangential contact force can be nonzero due to friction Ftcont = µ|Fncont |

(19)

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if Rn < 0 or zero otherwise. The friction force is given by Eq. (19) expressing the Coulomb friction law, with µ being the Coulomb friction coefficient. A quasi-static state of equilibrium of the assembly of particles can be achieved by application of adequate damping. Damping is necessary to dissipate kinetic energy. damp damp and T i in equations (3) and (4) in the present work are Damping terms Fi of non-viscous type and are given by damp

Fi

= −α t Fiext + Ficont damp

Ti

= −α r Ti

u˙ i , u˙ i

ωi , ωi

(20) (21)

where αt and α r , are respective damping constants for translational and rotational motion.

4 Determination of Rock Model Parameters The discrete element model can be regarded as a micromechanical material model, the contact model parameters being micromechanical parameters. Assumption of adequate micromechanical parameters yield required macroscopic rock properties, the most important being the Young modulus E, Poisson’s coefficient µ, compressive strength σc and tensile strength σt . For the elastic-brittle model of interaction between discrete elements described in Section 3 we have the following set of constitutive parameters: kn – contact stiffness in the normal direction, kt – contact stiffness in the tangential direction, Rn – interface strength in the normal direction, Rt – interface strength in the tangential direction, µ – Coulomb friction coefficient, α t – damping coefficient for translational motion, αr – damping coefficient for rotational motion. Determination of the model parameters is the key issue in the use of the discrete element method.

4.1 Dimensionless Micro-Macro Relationships In the present work the micromechanical parameters have been determined using the methodology developed by Huang [4] based on the combination of the dimensional analysis with numerical simulation of the standard laboratory tests for rocks,

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unconfined compression test and Brazilian test. Dimensional analysis is based on the Buckingham π theorem, which states that any physically meaningful functional relationship of N variables (Q1 , Q2 , . . . , QN ) can be expressed equivalently by a function of N − r dimensionless parameters (π1 , π2 , . . . , πN−r ), where r is the number of primary dimensions (minimum independent dimensions required to specify the dimensions of all the relevant parameters), and N − r is the maximum number of independent parameters [7]. Here we will search functions defining the macroscopic material parameters: Young’s modulus E, Poisson’s ratio ν, compressive strength σc and tensile strength σt in terms of microscopic parameters: kn , kt , Rn , Rt , µ, αt , αr . Macroscopic properties can also depend on other parameters, like particle size characterized by the average radius r, material density ρ, porosity of the particle assembly n. The set of the parameters can be completed with geometrical parameters represented by the specimen size L (due to possible scale effect) and loading velocity V . Thus, the number of relevant parameters N is 12. We have three primary dimensions involved: mass, length, time (r = 3). We can assume there are nine independent parameters. The set of parameters is not unique and can be modified by taking into account some other parameters that can influence macroscopic properties. In [13] the minimum and maximum element radii, rmin and rmax , respectively, have been included to the relevant parameters, in order to better consider the influence of the element size distribution on macroscopic properties. To some extent, this influence is taken in our formulation by the porosity n which depends on the size distribution, the wider size distribution the lower porosity in the discrete element model can be achieved. Having in mind there are alternative approaches, our procedure is based on the following √ set of nine independent parameters: {kn r/Rn , Rt /Rn , kt /kn , n, r/L, µ, α t , α r , V / kn /ρ}. Since the material properties will be studied under √ quasi-static conditions, the set of parameters can be reduced by removing V / kn /ρ, α t and α r . Further on, assuming that the element size r is small compared to macroscopic dimension L (r L), we can neglect the influence of the parameter r/L. The friction coefficient µ has influence mainly on the post-failure material behaviour, so we can omit it in the relationships for elastic constants and strength parameters. The set of relevant dimensionless parameters is reduced to the following one: {kn r/Rn , Rt /Rn , kt /kn , n}. Assuming that the elastic constants are determined in the range in which the failure is not initiated yet, in the relationships for elastic constants we can consider only two dimensionless parameters: {kt /kn , n}. Since constitutive relationships for 2D are given for the depth (the third dimension) of 1 m, the parameters for 2D have different meaning and dimensions from those for 3D. Therefore we have to consider separately the cases of 2D and 3D. The behaviour of the discrete element model in 2D and 3D is also different – this is another reason why the dimensionless relationships for 2D and 3D are different.

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4.1.1 Dimensionless Micro-Macro Relationships for 2D Problems Following [4] the following dimensionless functional relationships linking macroscopic and microscopic parameters have been postulated for the 2D discrete element model: E 2D kT = E ,n , (22) kn kn kT ν = 2D ,n , (23) ν kn RT kT σc r = 2D , ,n , (24) c Rn Rn k n RT kT σt r = 2D , , n . (25) t Rn Rn k n The specific form of the dimensionless relationships (22)–(25) have been obtained from the results of numerical simulations of the unconfined compression test (UCS) and Brazilian tests. The results of a simulation of the UCS test are presented in Figure 7 in the form of failure evolution with distribution of stresses in the direction of loading. The material sample of 50 × 50 mm represented by an assembly of randomly compacted 4979 discs of radii 0.262–0.653 mm (average radius 0.465 mm) has been generated using the high density sphere packing algorithm developed in [6]. Compaction of the particle assembly has been characterized by a porosity n of 13%. The stress-strain curve obtained in the analysis (Figure 8) can be used to determine the Young modulus E and compressive strength σc . The simulation provides the value of the Poisson ratio ν, as well. The cylindrical specimen of the diameter 50 mm for the simulation of the Brazilian test has been obtained by trimming adequately the specimen used in the UCS modelling. The failure mode with distributions of averaged stresses in the direction normal to the loading is shown in Figure 9. The failure in the form of splitting along the diameter parallel to the loading predicted in simulation agrees very well with the experimental observations. The stress distribution is in a very good agreement with theoretical solutions [14]. The force-time curve obtained in the simulation is plotted in Figure 10. Taking the maximum force Pmax we find the tensile strength as: σt =

2Pmax πLD

(26)

Simulations of both the UCS and Brazilian test have been performed for the dimensionless parameter kt /kn in the range from 0 to 2, assuming Rt /Rn = 1. The curves representing the dimensionless relationships (22) and (23) are plotted in Figure 11, the curves corresponding to the relationships (24) and (25) are shown in Figure 12.

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(a) = 0.0014 s

(b) = 0.0016 s

(c) = 0.0018 s

(d) = 0.0020 s

Fig. 7 Simulation of unconfined compression test – failure evolution with distribution of stress along the loading direction

Fig. 8 Simulation of unconfined compression test – stress-strain curve

4.1.2 Dimensionless Micro-Macro Relationships for 3D Problems The methodology developed for 2D models has been extended in this work on the three-dimensional discrete element modelling. Analogically to Eqs. (22)–(25) the following dimensionless functional relationships for 3D discrete element models have been postulated:

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(a) = 0.0010 s

(b) = 0.0014 s

(c) = 0.0016 s

(d) = 0.0018 s

Fig. 9 Simulation of Brazilian test – failure of the rock sample with distribution of stress in the direction normal to the loading

Fig. 10 Simulation of the Brazilian test – load-time curve

kT ,n , kn 3D kT ,n , ν = ν kn

Er = 3D E kn

(27) (28)

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Fig. 11 Dimensionless relationships between the microscopic parameters and macroscopic elastic constants: (a) relationship for Young’s modulus, (b) relationship for Poisson’s ratio

Fig. 12 Dimensionless relationships between the microscopic parameters and (a) compressive strength, (b) tensile strength.

RT kT , ,n , Rn kn 2 σt r 3D RT kT = t , ,n . Rn Rn kn

σc r 2 = 3D c Rn

(29) (30)

The specific form of the dimensionless relationships have been obtained from the results of numerical simulations of the laboratory tests. Results of the numerical simulation of the UCS and Brazilian tests are shown in Figure 13. The failure obtained in simulation is similar to the failure observed in the experiments. The simulations have been performed for the dimensionless parameter kt /kn in the range from 0 to 1, assuming Rt /Rn = 1. The relationships (27), (28) and (29) obtained from the simulations of the UCS test are plotted in Figures 14 and 15a. The rela-

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(a) (b) Fig. 13 Results of the numerical simulation of the laboratory tests for rocks: (a) unconfined compression test, (b) Brazilian test

Fig. 14 Elastic dimensionless parameters as functions of kt /kn

tionship (30) obtained from the numerical simulations of the Brazilian test is given in Figure 15b.

5 Simulation of Rock Cutting The discrete element model presented above has been applied to simulation of rock cutting. Two laboratory tests of rock cutting have been analysed, the first one consists of rock cutting with a single roadheader pick and the other one is the linear cutting test with a TBM disc cutter.

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Fig. 15 Dimensionless compressive strength parameters as a function of kt /kn : (a) for compression, (b) for tension

(a)

(b)

Fig. 16 Laboratory rock cutting test: (a) the cutting test rig, (b) rock cutting process (laboratory of Sandvik Mining and Construction GmbH, Zeltweg, Austria)

5.1 Simulation of Rock Cutting with a Single Roadheader Pick Scale-one cutting tests with a single roadheader pick are performed on the cutting testrig (Figure 16) built in the laboratory of SANDVIK Mining and Construction (Zeltweg, Austria) to study cuttability of specific rocks and performance of cutting tools. Cutting of a sandstone block by a rotating roadheader pick was chosen for numerical analysis. Mechanical properties of the rock have been determined experimentally and are the following: Young modulus E = 18690 MPa, compressive strength σc = 127 MPa and tensile strength σt = 12 MPa.

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Fig. 17 2D numerical simulation of rock cutting: (a) numerical model, (b) failure mode during rock cutting

5.1.1 2D Simulation of the Rock Cutting Test A numerical model developed for simulation of the rock cutting test is shown in Figure 17a. The rock specimen is discretized using 30,750 cylindrical elements of radii r = 1–1.5 mm. Using the dimensionless relationships (22)–(24) the following set of microscopic parameters has been determined for the rock under consideration: kn = 1.61129·1010 Pa, contact stiffness in the tangential direction kt = 0.3222·1010 Pa, Coulomb friction coefficient µ = 0.839 and cohesive bond strengths in the normal and tangential direction, Rn = Rt = 0.29 · 105 N/m. The model parameters were verified by simulations of the UCS and Brazilian tests using specimens of similar characteristics as those of the rock specimen. The values of 118 MPa and 16.8 MPa were obtained for the compressive and tensile strengths, respectively. These values were accepted as satisfactorily agreeing with the experimental results. The model of rock cutting was supplemented with the parameters of the rocktool interaction and global damping. For the rock-tool interaction the following set of parameters has been assumed: kn = ks = 5 · 1010 MPa, µ = 0.5. Non-viscous damping has been assumed taking the damping factors α nvt = α nvr = 0.2. Figure 17b shows the rock failure mode obtained in the simulation. A satisfying accordance with the failure observed in the labratory test can be watched. Figure 18 shows variation of the cutting force obtained in the numerical simulation. The numerical cutting force is compared with the average experimental value. As it can be seen in Figure 18 the mean cutting force from the numerical analysis agrees quite well with the average experimental force (about 7000 kN).

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Fig. 18 2D numerical simulation of rock cutting – cutting force variation

Fig. 19 3D numerical model of rock cutting

5.1.2 3D Simulation of the Rock Cutting Test All the three components of a cutting force can be calculated using a threedimensional model of the rock cutting test. The geometrical model created is shown in Figure 19. The material sample has been discretized using 71,200 spherical particles with average radius of 1.02 mm. The discrete element assembly has been generated using the high density sphere packing algorithm [6]. The tool was assumed rigid and its surface was discretized with a fine mesh of triangular facets representing accurately a complex tool tip geometry. The micromechanical parameters for the rock considered were found with help of dimensionless relationships given in Figures 14a and 15a. First, the ratio between

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Fig. 20 Numerical simulation of the laboratory rock cutting test

the contact stiffness in the tangential and normal direction has been assumed 0.4, since for this value the brittle failure of rock in 3D simulations of the UCS and Brazilian test has been correctly reproduced. From the curve given in Figures 14a the contact stiffness in the normal direction kn = 2.6 · 107 N/m has been obtained, then we have the contact stiffness in the tangential direction kT = 1.04 · 107 . The value of cohesive bond strengths in the normal Rn can be calculated from the plots in Figures 15a or 15b. The results obtained from these two plots are slightly different, Rn = 117 N from the plot in Figure 15a vs. Rn = 90 N from Figure 15b. Since the failure in cutting of sandstone is of brittle character and splitting of chips is mainly due to tensile stresses, the value Rn = 90 N according to the indirect tensile test simulation results has been adopted. Similar value for the shear bond strength has been taken, RT = 86 N. The results of numerical simulation are shown in Figure 20. Splitting of chips typical for brittle rock cutting can be seen. The three components of cutting forces obtained in simulation are plotted in Figure 21. Numerical forces are compared with experimental average measurements. Quite a good agreement can be observed.

5.2 Simulation of the Linear Cutting Test The linear cutting test has been simulated. Figure 22 shows the model geometry, consisting of the disc cutter and a rock sample. Only the area of the cutter ring interacting directly with the rock is considered. A rock sample with dimensions of 400 × 150 × 50 mm is represented by an assembly of randomly generated and densely compacted 40449 spherical elements of radii ranging from 0.8 to 6.0 mm. The granite properties are assumed in the simulation, appropriate DEM parameters being evaluated. The disc cutter is treated as a rigid body and discretized with

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Fig. 21 Rock cutting forces – comparison of numerical results with experimental average values

Fig. 22 Model geometry of the linear cutting test – model geometry

triangular facets. The parameters describing the disc interaction with the rock are as follows: contact stiffness modulus kn = 10 GPa, Coulomb friction coefficient µ = 0.8. The velocity of the disc cutter is assumed to be 10 m/s. Figure 23 shows the cutter disc during cutting. Normal contact force history is shown in Figure 24. Numerical results have been compared with experimental ones provided by Herrenknecht AG. A good agreement between the numerical and average experimental values is clearly seen.

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Fig. 23 Simulation of the linear cutting test – rock failure with distribution of macroscopic stresses (lowest principal stress – maximum compressive stress) 1200 TBM data Simulation Simulation − mean 1000

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6 Concluding Remarks The three-dimensional discrete element model of rock cutting is capable to represent correctly complexity of a rock cutting process. A good qualitative and quantitative

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agreement of numerical results with experimental measurements has been found out in the validation of the model developed in the present work. The discrete element model developed can be employed in the design of rock cutting tools and processes.

Acknowledgement The authors acknowledge partial funding by the EU project TUNCONSTRUCT (contract no. IP 011817-2).

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