ADAPTIVE COHESIVE VOLUMETRIC FINITE ELEMENT METHOD .ADAPTIVE COHESIVE VOLUMETRIC FINITE ELEMENT METHOD

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  • ADAPTIVE COHESIVE VOLUMETRIC FINITE ELEMENT METHODFOR DYNAMIC FRACTURE SIMULATIONS

    BY

    MARIUSZ ZACZEK

    B.S., University of Illinois at Urbana-Champaign, 1999

    THESIS

    Submitted in partial fulfillment of the requirementsfor the degree of Master of Science in Aeronautical and Astronautical Engineering

    in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2001

    Urbana, Illinois

  • c

    Copyright by Mariusz Zaczek, 2001

  • To my mother

    iii

  • ACKNOWLEDGMENTS

    First and foremost, I would like to thank the Center for Simulation of Advanced Rockets

    (CSAR) who has sponsored my research over the past two years. I would also like to express

    my appreciation and gratitude to my advisor, Prof. Philippe Geubelle. Without his advice, support

    and understanding I would have never been able to finish this thesis. I am especially grateful to him

    for spending countless hours in helping with this research and in taking the time to answer many

    of my questions. Thank you also to Dhirendra Kubair, Spandan Maiti, Jason Kamphaus and the

    various other members of the Structures and Solid Mechanics Group who have been great friends

    and have helped me tremendously over my time here. In addition, I would also like to thank Prof.

    Ricardo Uribe who has allowed to expand my horizons by working on various robotic projects as

    part of the Advanced Digital Systems Laboratory.

    Lastly, I would like to thank my mother who has always believed in me and encouraged me to

    be the best that I can and never forget to smile. Without her I would have never had the energy to

    work so hard.

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  • TABLE OF CONTENTS

    CHAPTER PAGE

    1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    2 METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Review of the Cohesive/Volumetric Finite Element Scheme . . . . . . . . . . . . . 6

    2.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Finite Element Implementation . . . . . . . . . . . . . . . . . . . . . . . 112.1.3 Stability and Mesh Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    2.2 Nodal Time Step Subcycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Dynamic Cohesive Node/Element Insertion . . . . . . . . . . . . . . . . . . . . . 19

    2.3.1 Geometry and Database Management . . . . . . . . . . . . . . . . . . . . 192.3.1.1 1-D Insertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1.2 2-D Insertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    2.3.2 Cohesive Element Stability and System Equilibrium . . . . . . . . . . . . 262.3.2.1 Cohesive Damping . . . . . . . . . . . . . . . . . . . . . . . . 282.3.2.2 Cohesive Element Pre-Stretching . . . . . . . . . . . . . . . . . 30

    2.3.3 Insertion Region Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.3.1 Bounding Box Approach . . . . . . . . . . . . . . . . . . . . . 332.3.3.2 Stress-based Selection Approach . . . . . . . . . . . . . . . . . 35

    2.4 Parallel Implementation using Charm++ . . . . . . . . . . . . . . . . . . . . . . 372.4.1 Mesh Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4.2 Computational Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.3 Structure of Standard Charm++ FEM Framework . . . . . . . . . . . . . 412.4.4 Parallel Structure of the Dynamic Insertion Code . . . . . . . . . . . . . . 44

    3 1-D ANALYSIS AND RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2 Multi-Time Step Nodal Subcycling Results . . . . . . . . . . . . . . . . . . . . . 493.3 Dynamic Cohesive Node Insertion Results . . . . . . . . . . . . . . . . . . . . . . 54

    3.3.1 Blind Cohesive Node Insertion Results . . . . . . . . . . . . . . . . . . . 553.3.2 Damping of Blind Insertion . . . . . . . . . . . . . . . . . . . . . . . . . 583.3.3 Dynamic Insertion with Pre-Stretch . . . . . . . . . . . . . . . . . . . . . 60

    v

  • 3.3.4 Combined Insertion with Subcycling . . . . . . . . . . . . . . . . . . . . . 633.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

    4 2-D ANALYSIS AND RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1 Multi-Time Step Subcycling Results . . . . . . . . . . . . . . . . . . . . . . . . . 67

    4.1.1 Equal Subcycled to Non-Subcycled Region Ratio of 1:1 . . . . . . . . . . 694.1.2 Unequal Subcycled to Non-Subcycled Region Ratio . . . . . . . . . . . . 704.1.3 Multi-Time Step Nodal Subcycling Observations . . . . . . . . . . . . . . 73

    4.2 Dynamic Cohesive Element Insertion . . . . . . . . . . . . . . . . . . . . . . . . 754.2.1 Insertion Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    4.2.1.1 Blind Insertion . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2.1.2 Damping of Blind Insertion . . . . . . . . . . . . . . . . . . . . 784.2.1.3 Insertion with Pre-Stretch . . . . . . . . . . . . . . . . . . . . . 814.2.1.4 Insertion Analysis Observations . . . . . . . . . . . . . . . . . . 83

    4.2.2 Insertion Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2.2.1 Bounding Box Insertion . . . . . . . . . . . . . . . . . . . . . . 844.2.2.2 Stress Based Insertion in L-Angle Specimen . . . . . . . . . . . 894.2.2.3 Stress-based Insertion in Vertical Interface Specimen . . . . . . 944.2.2.4 Stress-based Insertion in Angled Interface Specimen . . . . . . . 964.2.2.5 Insertion Interval Selection . . . . . . . . . . . . . . . . . . . . 1024.2.2.6 Dynamic Insertion Combined with Subcycling . . . . . . . . . . 105

    4.3 Parallelization Using Charm++ . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

    5 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . 1125.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2 Recommendations for Future Research . . . . . . . . . . . . . . . . . . . . . . . . 115

    REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

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  • LIST OF FIGURES

    Figure Page

    1.1 Illustration of the fracture process associated with the Titan IV SRMU grain col-lapse accident (taken from Chang et al., (1994)). . . . . . . . . . . . . . . . . . . . 2

    2.1 CVFE concept showing one 4-node cohesive element between two linear-straintriangular volumetric elements. The cohesive element is shown in its deformedconfiguration. In its undeformed configuration is has no thickness and the adjacentnodes are superposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    2.2 Bilinear cohesive failure law for the pure tensile or mode I (t 0, left) and pureshear or mode II (n 0, right) cases. An unloading and reloading path is alsoshown in the mode I case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    2.3 Coupled cohesive failure model described by Equation 2.4; variation of normal(top) and shear(bottom) cohesive tractions with respect to normal (n) and tangen-tial (t) displacement jumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    2.4 Time step defined by: (a) element size or (b) element type. . . . . . . . . . . . . . 142.5 Subcycling region distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Time step assignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.7 (a) Standard 1-D mesh. (b) 1-D mesh with inserted cohesive node. . . . . . . . . . 202.8 2-D Cohesive element representation. . . . . . . . . . . . . . . . . . . . . . . . . 212.9 2-D cohesive element insertion: (a) proposed edge for cohesive insertion, (b) in-

    serted cohesive element, (c) criss-crossed cohesive element. . . . . . . . . . . . 212.10 Connectivity update of nodes and elements. . . . . . . . . . . . . . . . . . . . . . 222.11 Common 2-D insertion cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.12 Illustrative example of three cohesive element insertions using Cases #2 and #3 in

    Figure 2.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.13 Illustration of insertion Case #5 in Figure 2.11. . . . . . . . . . . . . . . . . . . . 262.14 1-D blind insertion test problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 272.15 1-D blind insertion test problem: evolution of the displacement jump across

    the cohesive element (i.e., between nodes 2 & 4 in Figure 2.14) resulting from acohesive element insertion at time 0, 1000t (33 3 s), 2000t (66 6 s). . . . . . . . 28

    2.16 Schematic representation of a damped 1-D cohesive element. . . . . . . . . . . . . 28

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  • 2.17 Effect of cohesive damping: evolution of the displacement jump across the cohe-sive element for the simple 1-D test problem shown in Figure 2.14 and resultingfrom blind cohesive element insertion with damping at time 0, 1000t (33 3 s)and 2000t (66 6 s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    2.18 1-D cohesive element pre-stretching concept. . . . . . . . . . . . . . . . . . . . . 302.19 1-D cohesive element pre-stretching concept, with the pre-stretch applied equally

    on the two nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.20 2-D separation contributions from neighboring cohesive elements. . . . . . . . . . 332.21 1-D test problem separation