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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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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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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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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(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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(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] E. Oñate and R. Owen (eds.), Particle-Based Methods, Computational Methods in Applied Sciences 25, DOI 10.1007/978-94-007-0735-1_3, © Springer Science+Business Media B.V. 2011
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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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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