Continuum Modelling using the Discrete Element Method. Theory and Implementation in an Object-Oriented Software Platform M. Santasusana E. Oñate Publication CIMNE Nº-381, September 2012 Continuum Modelling using the Discrete Element Method. Theory and Implementation in an Object-Oriented Software Platform M. Santasusana E. Oñate Publication CIMNE Nº-381, September 2012 International Center for Numerical Methods in Engineering Gran Capitán s/n, 08034 Barcelona, Spain Abstract The Discrete Element Method is a relatively new technique that has nowadays and intense research in the field of numerical methods. In its first conception, the method was designed for simulations of dynamic system of particles where each element is considered to be an independent and non-deformable entity that interacts with other particles by the laws of the contact mechanics and moves following the second Newton’s law. This first approach for the DEM has obtained excellent results for granular media simulations or another discontinuous- like case. The existing challenge nowadays for the DEM is to be able to simulate the behaviour on a continuous media discretized by a mesh of particles ruled by the equations of the DEM. Although there exist more adequate methods to solve the continuous problem as they are the different variants of the Finite Element Method, the DEM is expected to have a better behaviour when the failure of the media occurs; in terms of tracking the evolution of the fracture locally between the elements of the discretization and also the post-fractural behaviour of the material. Nowadays, there are several DEM codes that try to solve this problem although there is no one which can assure an accurate solution applicable universally to any case. The objective of the present work is to develop calculation software for the Discrete Element Method included in the platform for numerical methods KRATOS, which is developed in CIMNE. One of the goals of the so-called DEM-Application is to be able to reproduce a wide set of engineering problems; not only the discrete ones such as the excavation or agroalimentary applications but also to reproduce the continuous media, simulating compression test for concrete or asphalt samples for instance. In addition it is desired that the application permits the coupling with another methods, particularly the Finite Element Method. In order to do this, the present work includes the study of all the advances and ideas that, globally in the numerical method field and particularly in CIMNE, have been discussed to give other approaches and to keep improving and developing the to the Discrete Element Method.1 Title: Continuum modelling using the Discrete Element Method. Theory and implementation in an object- oriented software platform. Author: Miquel Santasusana Isach Supervisors: Eugenio Oñate Ibáñez de Navarra, Miguel Ángel Celigueta Jordana. Key words: Discrete Element Method, KRATOS, CIMNE, continuum simulation, C++ programming. I II Resum El Mètode dels Elements Discrets és un mètode relativament nou el qual avui dia és objecte d’una intensa recerca en el món dels mètodes numèrics. Originalment el mètode fou concebut per a la simulació de sistemes dinàmics de partícules on cada element és considerat com a entitat independent i indeformable que interacciona amb les altres seguint les lleis del contacte mecànic i es mou segons la segona llei de Newton. Aquest primer plantejament sobre el M.E.D. a tingut molts bons resultats per a simulacions de medis granulars o qualsevol assimilable a un medi discontinu. El repte actual per al DEM és ésser capaç de simular també el comportament d’un medi continu discretitzat per una malla de partícules que interaccionin segons les lleis del M.E.D. Tot i existir mètodes molt més adequats a resoldre aquest problema com son les variants del Mètode dels Elements Finits, el M.E.D. promet tenir un millor comportament a l’hora de seguir l’evolució de la fractura a nivell local entre els elements de la discretització i el comportament post fracturat del material. Actualment, existeixen molts programes de Elements Discrets que intenten resoldre aquest problema sense haver-n’hi cap que asseguri una solució acurada aplicable a nivell universal i amb versatilitat. L’objectiu de la tesina és desenvolupar un programa de càlcul d’Elements Discrets inclòs en la plataforma per mètodes numèrics KRATOS, desenvolupada al CIMNE. Un dels objectius de l’anomenada DEM-Application és poder reproduir un ampli conjunt de problemes d'enginyeria; no tan sols els que són merament “discrets” com el medis granulars, tal i com poden ser l’excavació o aplicacions agroalimentàries, sinó també la simulació del medi continu, com ara reproduir provetes de formigó, asfalt, etc. Paral·lelament es desitja que l’aplicació permeti el càlcul acoblat amb altres mètodes, en particular amb el Mètode dels Elements Finits. Per dur-ho a terme, en la tesina, s’ha estudiat tot el conjunt d’avenços i idees que, en el món dels mètodes numèrics a nivell global i a CIMNE en particular, es plantegen per donar altres punts de vista, millorar i continuar desenvolupant el Mètode dels Elements Discrets.2 Títol: Modelització del medi continu mitjançant el Mètode dels Elements Discrets. Teoria i implementació en una plataforma de programació orientada a objectes. Autor: Miquel Santasusana Isach Supervisors: Eugenio Oñate Ibáñez de Navarra, Miguel Ángel Celigueta Jordana. Paraules clau: Mètode dels Elements Discrets, KRATOS, CIMNE, simulació del continu, programació C++. III IV Table of Contents ABSTRACT ............................................................................................................................................... I RESUM ................................................................................................................................................... II TABLE OF CONTENTS ............................................................................................................................. V LIST OF FIGURES ....................................................................................................................................IX INTRODUCTION AND OBJECTIVES .......................................................................................................... 1 PART I: THE DISCRETE ELEMENT METHOD ............................................................................................. 3 1. OVERVIEW OF THE METHOD ......................................................................................................... 3 1.1. BRIEF HISTORY OF THE DISCRETE ELEMENT METHOD............................................................................... 3 1.2. INTRODUCTION AND GENERAL ASPECTS OF THE DEM FORMULATION ......................................................... 3 1.2.1. Preliminary steps: ............................................................................................................... 5 1.2.2. Contact Search ................................................................................................................... 6 1.2.3. Evaluation of Forces ........................................................................................................... 6 1.2.4. Integration of Motion Equations ........................................................................................ 7 2. DEM THEORY DISCUSSION ............................................................................................................. 8 2.1. CONTACT DETECTION ....................................................................................................................... 8 2.1.1. Buffer Zone ....................................................................................................................... 10 2.1.2. Bounding Box/Sphere representation .............................................................................. 11 2.1.3. Brute Force Search Method .............................................................................................. 12 2.1.4. Static Cell Search (grid-based method) ............................................................................ 13 2.1.5. Dynamic Cell Search (grid-based method) ....................................................................... 14 2.1.6. No binary Search Method ................................................................................................. 16 2.1.7. Tree-based algorithms ..................................................................................................... 17 2.1.8. Local contact resolutions .................................................................................................. 19 2.2. CONSTITUTIVE MODELLING OF THE CONTACT ...................................................................................... 27 2.2.1. Normal interaction forces ................................................................................................ 28 2.2.2. Absolute position method and incremental method ........................................................ 29 2.2.3. Relative importance of the accuracy on the stiffness value ............................................. 30 2.2.4. Indentation permitted ...................................................................................................... 31 2.2.5. Gain of energy .................................................................................................................. 31 2.2.6. Numerical damping and physical damping .....................................................................