Formulation of a Dynamic Material Point Method (MPM) for Geomechanical Problems
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Formulation of a Dynamic Material Point Method (MPM) for Geomechanical Problems Von der Fakultät für Bau– und Umweltingenieurwissenschaften der Universität Stuttgart zur Erlangung der Würde eines Doktors der Ingenieurwissenschaften (Dr.-Ing.) genehmigte Abhandlung, vorgelegt von ISSAM K. J. AL-KAFAJI aus Bagdad, Irak Hauptberichter: Prof. Dr.-Ing. Pieter A. Vermeer Mitberichter: Prof. Dieter F. Stolle, P. Eng. Mitberichter: Prof. Dr.-Ing. habil. Christian Moormann Tag der mündlichen Prüfung: 13. März 2013 Institut für Geotechnik der Universität Stuttgart 2013 c Issam AL-Kafaji All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, without the permission in writing of the author. Keywords: Dynamics, large deformation, material point method, two-phase media, pile driving Printed by Ridderprint BV, The Netherlands, 2013 ISBN 978-90-5335-705-7 (D93 - Dissertation, Universität Stuttgart) Preface The application of numerical analyses in geotechnical engineering is substantial, but most usually quasi-static, small-deformation problems are analysed. Following pio- neers in structural engineering, the Lagrangian finite element method (FEM) has been introduced into geotechnical engineering, rather than the Eulerian FEM, or resembling methods, as typically used in fluid mechanics for the analyses of flow problems with corresponding large-displacements. In recent years coupled Eulerian Lagrangian FEM has been applied successfully for solving dynamical large-deformation problems, e.g. for simulating pile driving, but it is difficult to extend this method to soil-fluid interac- tion problems as typical in branches of geotechnical engineering. For this reason, Issam K.J. Al-Kafaji has chosen to focus his research on the so-called Material Point Method (MPM), which may be conceived as an advancement of FEM. At the beginning of this study Issam has set himself very ambitious goals, i.e. the extension of MPM for application to large-deformation problems in geomechanics, in- cluding pile driving in sand using an advanced highly non-linear constitutive model. Secondly, the modeling of two-phase soil with full consideration of dynamic generation and dissipation of pore pressures has been carried out. These goals have been reached in a convincing way. In addition the candidate has performed an extensive survey of relevant literature and is nicely putting his study in a wider context of leading edge, worldwide research in geomechanics. As a consequence, his research has already at- tracted good attention in the international research community. To my judgment this well-written study is of finest quality and finally Issam passed his PhD examination with the highest possible grade, i.e. ’with Honor’. The study was enabled with support of the ’German Academic Exchange Service’. On arriving at the University of Stuttgart Issam had to learn basics of both MPM and German. So the first thing he did was to follow language courses. For getting familiar with MPM he went to the University of Stellenbosch in South-Africa, where Dr. Corné Coetzee taught him the essentials of this method. I am also indebted to Corné for his continuing support of this PhD study. This support was embedded in the Geo-Install IAPP project, being funded by the European Commission within the framework of Marie Curie FP7. Within the framework of the Geo-Install project Issam was enabled to spend a year at ’Deltares’ in Delft, The Netherlands. This gave him the experience of working in an industry and of course in again another country. No doubt, Issam had to adjust himself to different ways of working and together with his family he had to get used to different places of living. This cannot always be easy, but to me it would seem that he mostly enjoyed it. During many years it was a great pleasure to me to work with Issam. This was not only because of his fast progress in science and his abilities as a team player, but also be- cause of his fine personality. I know that I share these feelings towards him with others Preface who advised him, i.e. Dr. Corné Coetzee and Professor Dieter Stolle from McMaster’s University in Hamilton, Canada. Pieter A. Vermeer Delft, March 2013 Acknowledgments I would like to express my greatest gratitude to my supervisor Professor Pieter Vermeer. I am proud that he gave me the opportunity to be one of his PhD students. I really appreciate his patience, continuous support, guidance and encouragement to conduct high quality research. Without Professor Vermeer this thesis would not have been com- pleted. Likewise, I would like to acknowledge with much appreciation the ultimate help of my co-supervisor Professor Dieter Stolle. I consider myself lucky that I had the op- portunity to cooperate with him during his sabbatical in Stuttgart. I feel indebted to Dr. Corné Coetzee and Professor Zdzisław Wi˛eckowski for the continuous support, advise and feedback. I also wish to thank Professor Bernhard Westrich for the daily discussions and his high interest in my research. I would like to convey thanks for the German Academic Exchange Service for funding the research and completeness of this thesis. I am also indebted to Deltares for the expert advise throughout my research period. Here, I would like to thank Dr. John van Esch for his support and assistance in the research during my stay in Deltares. Special thanks to my colleagues in the institute of geotechnical engineering, Stuttgart University for the wonderful time and ultimate support. Here, I thank Thomas Benz, Maximilian Huber, Annette Lächler, Martino Leoni, Axel Möllmann, Ruth Rose, Nadja Springer, Sven Möller, Bernd Zweschper, Ayman Abed, Lars Beuth, Syawal Satibi, Lia Daniela Zaman, Markus Wehnert, Marcus Schneider, Peter Strolle, Reinhold Mößner, Fursan Hamad, Josef Hintner and Stephan Ries. My thanks extend to Professor Minna Karstunen, Dr. Paul Hölscher, Dr. Peter Fokker, Dr. Alexander Rohe, Dr. Sascha Henke, Shuhong Tan, Phuong Nguyen, Wiep Hellinga and Oday Ibrahim. I would like to thank all my colleagues in Baghdad University. Special thanks to Pro- fessor Somer Nacy and Professor Adnan Nagy for their continuous encouragement. Thanks to my parents, my wife, my brothers, my sisters and my daughters. Without them this thesis would not have been completed. Issam Al-Kafaji Stuttgart, March 2013 To the memory of my father... Declaration I hereby declare that this thesis is the result of my own research effort except where otherwise indicated. I have only used the resources that are listed in the list of references. Issam Al-Kafaji Stuttgart, March 2013 Contents 1 Introduction 1 1.1 Basicobjectives .................................. 1 1.2 Thesislayout.................................... 3 2 Large deformation: A review 5 2.1 Thefiniteelementmethods ........................... 5 2.1.1 LagrangianversusEulerianFEM. 6 2.1.2 Combined Lagrangian-Eulerian methods . 8 2.2 Meshlessmethods................................. 8 2.3 Mesh-basedparticlemethods . 9 2.3.1 Particle-in-cellmethod. 9 2.3.2 Fluid-implicitparticlemethod . 10 2.3.3 Materialpointmethod . 10 3 Mathematical and numerical models: FEM 19 3.1 Mathematicalmodel ............................... 19 3.1.1 Motionandkinematics. 20 3.1.2 Constitutiverelation ........................... 21 3.1.3 Conservationlaws ............................ 22 3.1.4 Boundaryandinitialconditions . 23 3.2 Weakformofmomentumandtraction. 25 3.3 Space discretization of the virtual work . 26 3.3.1 Numericalspaceintegration . 29 3.3.2 Lumped-massmatrix .......................... 30 3.4 Timediscretization ................................ 32 3.4.1 Single degree-of-freedom spring-mass system . ..... 34 3.4.1.1 Effect of frequency content on the iteration procedure . 36 3.4.1.2 Conservation properties of the scheme . 38 3.4.1.3 Stability requirements and critical time step . 41 3.4.1.4 A predictor for the first iteration . 42 3.4.2 TimeintegrationofFEMequilibrium . 44 3.4.2.1 Critical time step for discrete systems . 47 3.5 Numericalexamples ............................... 48 3.5.1 Spring-masssystem ........................... 48 3.5.2 Tensionbar ................................ 53 3.6 Concludingremarks ............................... 55 i Contents 4 The dynamic material point method 57 4.1 BasicconceptofMPM .............................. 57 4.2 DiscretizationinMPM .............................. 59 4.3 Initializationofparticles . 59 4.4 MPMsolutionprocedure............................. 63 4.4.1 Initialization of equations of motion . 63 4.4.2 Solving the equations of motion . 65 4.4.2.1 Identical Lagrangian algorithm . 65 4.4.2.2 Modified Lagrangian algorithm . 67 4.4.2.3 Overall solution algorithm for a single time step . 70 4.5 BoundaryconditionsinMPM . 72 4.5.1 Zero kinematic and traction boundary conditions . 72 4.5.2 Nonzero kinematic and traction boundary conditions . 72 4.5.2.1 Concept of moving mesh . 73 4.6 Mitigating error associated with grid-crossing for quasi-static problems . 75 4.6.1 Attempts to reduce the grid-crossing error . 77 4.6.2 GaussintegrationinMPM. 78 4.6.3 Numerical example on MPM and mixed integration . 80 4.7 Treatment of spatially unbounded domains using absorbing boundaries . 82 4.7.1 Formulation of boundary dashpots . 82 4.7.1.1 Integration of the dashpots matrix . 85 4.7.2 Extending the boundary dashpots to Kelvin-Voigt elements . 86 4.7.2.1 Stability of Kelvin-Voigt element . 88 4.7.2.2 Choosing the spring