Contact modelling in the Material Point Method MSc Thesis Ivaylo Pantev Delft University of Technology CONTACT MODELLING IN THE MATERIAL POINT METHOD MSC THESIS by Ivaylo Pantev in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering, Geo-Engineering specialisation, at the Delft University of Technology, to be defended publicly on Thursday October 27, 2016 at 16:00 PM. Student number: 4418204 Project duration: January, 2016 – October, 2016 Thesis committee: Dr P.J. Vardon, TU Delft chairman Prof. Dr M. A. Hicks, TU Delft Prof. Dr Ir. L. J. Sluys, TU Delft J.L. Gonzalez Acosta, MSc, TU Delft PREFACE The following work marks the end of my academic career to date and the start of a new chapter in life. It represents the culmination of two wonderful years in Delft, a time to never be forgotten. The topic of this thesis topic embodies a core belief held throughout my studies - never to stray from a challenge. I have consistently found tackling a difficult, broad or novel problem has been the best way to learn and improve. This thirst for knowledge influenced my decision to pursue an MSc at TU Delft, one of the finest technical universities in the world. It is my hope that such challenges will never be too far in the future and that they will provide opportunities for continuous development throughout my career. Here I would like to express my gratitude towards the graduation committee. To my supervisor, Dr Phil Vardon for always remaining positive, even when I was not. You managed to provide inspi- ration and nurture a deep interest in the topic and numerical modelling in general. I am grateful to Prof. Michael Hicks and Prof. Bert Sluys for being ever present, bringing their humbling experience and providing a critical perspective during our meetings. Not least, Leon Acosta was always avail- able for a chat, brainstorming or debugging session - our conversations made me strive to look for solutions and improve. Here I would also like to thank Noor Pruijn, for setting a stellar example with her work on related issues in MPM and exchanging ideas throughout the thesis. Credit is also due to Bin Wang who was instrumental to the MPM research in the Geo-Engineering section and the start of this project. I owe a substantial amount to my fellow graduate students and friends in Delft, for forming a tight-knit and supportive community. Being part of a truly multicultural environment has enriched us all, taught us humility and patience among many other things, creating lasting ties across the globe. For all the friendships that started here, thank you. Finally, I would like to thank my parents and family without whom none of this would have been possible. I am eternally grateful for their unconditional support, timely guidance, wisdom and for giving me the freedom to pursue my own path. This is all one could ever wish for and comprises a debt that can never be repaid. Ivaylo Pantev Delft, October 27th 2016 iii ABSTRACT In recent years, the Material Point Method has emerged as a promising alternative to the Finite Element Method for solving problems involving localised deformations, large displacements or ro- tations, fracture, contact/impact problems and others. FEM has been a staple of engineering disci- plines both in academia and industry for decades, however its shortcomings make it less suitable for certain applications in geotechnical engineering. MPM, bridges the gap between Finite Elements and meshless methods by using a reference background grid and moving material points to rep- resent the discretised material. This gives it the capacity to faithfully reproduce behaviours which would be difficult to capture without involved procedures with other presently available methods. In this work, methods of modelling contact between deformable continua were explored in the context of the Material Point Method with the goal of producing usable code applicable to geotech- nical engineering scenarios, thus supplementing the work of the Geo-Engineering section at TU Delft. A dynamic, explicit computational framework was developed in the FORTRAN programming language allowing efficient modelling of multi-body interactions using different contact conditions. The influence of decisions regarding the contact logic, definition of surface normals and variable update procedures were compared. Coupling between FEM and MPM was explored for solving problems involving regions with small and large deformations. The work presented here aims to extend the applicability of the Material Point Method to soil-structure interaction problems. In addition to the contact algorithms explored, an alternative method for material point gener- ation using Centroidal Voronoi Tessellations was explored. This allowed for improved distribution of material points over domains of generic shape, while achieving results similar to a FEM-based approach, which uses Gauss Points. Placing material points at the integration points brings the in- herent disadvantage that complex domain shapes cannot be easily discretised using a structured grid. The ability of the two methods to reproduce a uniform density field was compared and the novel approach was implemented for a contact problem - an elastic collision between two disks. Results showed a the potential of the CVT-based scheme for use in further analyses. v CONTENTS Abstract v List of Figures xi List of Tables xiii 1 Introduction 1 1.1 Problem Statement......................................2 1.1.1 Methodology.....................................2 1.1.2 Code development goals...............................3 1.2 Report Structure.......................................3 2 Background and Methodology5 2.1 The Finite Element Method.................................5 2.1.1 General Procedure..................................5 2.1.2 Convergence and Stability Criteria.........................6 2.1.3 Contact in FEM....................................7 2.2 The Material Point Method.................................8 2.2.1 Governing Equations.................................9 2.2.2 Solution Procedure.................................. 10 2.2.3 Updating Material Point Properties......................... 13 2.2.4 Implicit and Explicit Integration........................... 13 2.3 Contact in the Material Point Method........................... 14 2.3.1 Detection....................................... 14 2.3.2 Resolution....................................... 14 2.4 Coupling with FEM...................................... 15 2.5 Outstanding Issues in MPM................................. 16 2.5.1 The Explicit MPM code................................ 16 3 Contact Algorithms 17 3.1 Overview of contact algorithms............................... 17 3.1.1 Standard MPM.................................... 17 3.1.2 The Multi-Velocity Field-Based Algorithm..................... 17 3.1.3 Notable developments................................ 18 3.1.4 Other approaches................................... 20 3.2 Contact Detection...................................... 21 3.3 Contact Geometry...................................... 22 3.3.1 Point-To-Point.................................... 22 3.4 Multi Velocity Field...................................... 23 3.4.1 Procedure....................................... 23 3.4.2 Interaction with FEM domain............................ 25 3.5 Point-To-Segment...................................... 25 3.5.1 Procedure....................................... 25 3.5.2 Alternative...................................... 27 vii viii CONTENTS 3.6 The Contact Normal and Tangent.............................. 27 3.6.1 The Normal...................................... 28 3.6.2 Nodal support array................................. 28 3.6.3 Gradient of Nodal Mass............................... 29 3.6.4 Gradient of Density.................................. 29 3.6.5 Geometric Approach................................. 30 3.6.6 The Tangent...................................... 31 3.6.7 Implementation and Comparison.......................... 32 3.6.8 Domain Boundary Treatment............................ 33 4 Material Point Generation 35 4.1 Cell Subdivision....................................... 35 4.2 Tessellations......................................... 36 4.3 Comparison.......................................... 36 4.4 Application.......................................... 40 4.5 Conclusion.......................................... 41 5 Implementation and Benchmark Problems 43 5.1 Dynamic Explicit MPM Formulation............................ 43 5.1.1 Initial Stress Assignment............................... 43 5.1.2 Energy calculation.................................. 44 5.1.3 Simple Motions.................................... 44 5.1.4 Axial Vibration of a Bar................................ 44 5.2 Contact Problems...................................... 46 5.2.1 Elastic Block Collision................................ 46 5.2.2 Elastoplastic Block Collision............................. 49 5.2.3 CVT implementation................................. 49 5.2.4 Block sliding on inclined plane........................... 50 5.2.5 Interaction Between MPM and FEM Domains................... 52 6 Conclusions and Recommendations 55 6.1 General Conclusions..................................... 55 6.2 Conclusions regarding the Multi-Mesh Algorithm..................... 55 6.2.1 Distance between contacting bodies........................ 56 6.2.2 Normal calculation.................................. 56 6.2.3 Stability.......................................