Employment of a Multi-Material ALE approach
using nonlinear soil models to simulate large
deformation geotechnical problems
vorgelegt von
M.Sc.
Montaser Bakroon
ORCID: 0000-0002-2564-4000
an der Fakultät VI – Planen Bauen Umwelt
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften
– Dr.-Ing. –
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Univ.-Prof. Dr.-Ing. habil. Yuri Petryna
Gutachter: Univ.-Prof. Dr.-Ing. Frank Rackwitz
Gutachter: Univ.-Prof. Dr.-Ing. Jürgen Grabe
Tag der wissenschaftlichen Aussprache: 14. Juli 2020
Berlin 2021
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Preface by the Author
The current thesis is the summary of my work at the Chair of Soil Mechanics and
Geotechnical Engineering at the Technische Universität Berlin the period of 2015 -
2020. The work contributes to the further improvement of numerical approach in
simulation of complex geotechnical problems.
I would like to express my deepest appreciation to all those who made it possible
for me to finish my PhD work through their constant support, both academically
and spiritually. First, I like to thank my supervisor Prof. Frank Rackwitz who has
guided me through the scientific path with all its complications and impediments. His
continuous support and interest in my work has led me to the point for which I am
within my deepest feelings grateful. I would also like to thank Prof. Jürgen Grabe
from Technische Universität Hamburg-Harburg for his interest in my work, for being
the second reviewer of my thesis and also for his fruitful and constructive remarks.
I would like to extend my thanks to Dr. Daniel Aubram who has closely followed up
my work. His remarks and comments based on his vast knowledge and experience in
the field of numerical methods have always been very informative and instructive. My
thanks are also extended to my colleague and friend M.Sc. Reza Daryaei with whom I
worked closely during my research. The discussions and comments we had has deeply
improved the final outcomes.
Moreover, I am indebted to my colleagues at the chair. I want to thank M.Sc. Fabian
Remspecher and Dr. Le Viet Hung for providing us the experimental results from the
lab. I thank also Dr. Ralph Glasenapp, Dr. Marcel Ney, and M.Sc. Christian Carow
for their support. My thanks are also dedicated to all other members of the Chair
whose support cannot be really described by words. I extend my gratitude to German
Academic Exchange Service (DAAD) for financial support during my study.
The last but not the least, I am deeply indebted to my parents and my family, specially,
my wife, Basmalla and our children, Sara, Mohammed, Abdalaziz, Zaina, and Leen for
their patience, understanding, and encouragement throughout my PhD study.
Montaser Bakroon
Berlin, March 2020
iv PREFACE BY THE AUTHOR
Abstract
Numerical simulation of geotechnical installation problems, specifically offshore pile
installation problems pose several challenges including the treatment of large defor-
mation using the conventional finite element method (FEM), capturing the non-linear
behavior of the soil, and the presence of pore water in the porous medium. In this
work, a clear and straightforward approach for modeling large deformation problems
is developed which addresses the three aforementioned considerations.
During the last decades, efforts have been made to tackle the problems associated with
large deformation during the simulation of pile installation. In this thesis, the Multi-
Material Arbitrary Lagrangian-Eulerian (MMALE) is employed to address the large
deformation problem. Briefly, the method consists of three sub-steps, a Lagrangian
step, a remeshing step, and a remapping step, which are performed sequentially owing
to the operator-split scheme. The advantage of the MMALE with a special focus on
the remeshing step is discussed thoroughly using various benchmarks.
Beside a robust element formulation, a sophistical constitutive equation is required to
capture the soil realistic behavior in large strains due to penetration. The mechanical
behavior of granular materials like sand is highly nonlinear due to the presence of an
evolving internal structure formed by the grains. The strength and stiffness are gener-
ally a function of the stress and density state and the loading history. A constitutive
equation based on hypoplastic framework is chosen and defines evolution equations for
the effective stress, void ratio, and the so-called intergranular strain tensor suitable
for simulating cyclic loading effects. Interfaces are implemented in two hydrocodes to
employ hypoplastic constitutive equation.
Additionally, the presence of water is inevitable in offshore projects and needs to be
considered in the numerical evaluation. Hence, a simplified coupled formulation is
introduced to the developed code to consider the presence and effects of pore water in
the soil behavior.
Finally, the approach is verified and validated by various analytical and experimen-
tal geotechnical benchmarks, respectively. Also, the method is used to study the pile
buckling during pile installation under different boundary conditions.
Keywords: Multi-Material Arbitrary Lagrangian-Eulerian; Large deformations; Cou-
pled formulation; Simplified u-p; Pile installation; Granular material; Hypoplastic con-
stitutive equation; Pile buckling; Remeshing methods;
vi ABSTRACT
Zusammenfassung
Die numerische Simulation von geotechnischen Installationsproblemen, beispielsweise
von Offshore-Pfählen, ist mit mehreren Herausforderungen verbunden, darunter die
Behandlung großer Verformungen mit der Finite-Elemente-Methode (FEM), die Erfas-
sung des nichtlinearen Verhaltens des Bodens und die Berücksichtigung des Vorhanden-
seins von Porenwasser im Boden. In den letzten Jahren wurden einige Anstrengungen
unternommen, um diese Fragestellungen bei der Simulation von Pfahlinstallationspro-
blemen in geeigneter Weise zu behandeln.
In der vorliegenden Arbeit wird ein Ansatz zur Modellierung großer Verformungsproble-
me entwickelt, der die zuvor genannten Punkte bercksichtigt. Als Grundlage wird eine
Multi-Material Arbitrary Lagrangian-Eulerian (MMALE) Methode verwendet, um die
großen Verformungen bei der Pfahlinstallation numerisch zu simulieren. Das Verfah-
ren besteht aus drei Teilschritten - einem Lagrange-Schritt, einem Remeshing-Schritt
und einem Remapping-Schritt, die aufgrund des Operator-Split-Schemas nacheinander
durchgefhrt werden. Der Vorteil der MMALE Methode mit besonderem Fokus auf den
Remeshing-Schritt wird anhand verschiedener Benchmarks diskutiert. Neben einer ro-
busten Elementformulierung ist eine nichtlineare Stoffgesetzgleichung erforderlich, um
das Bodenverhalten bei großen Verformungen realitätsnah zu erfassen. Das mechanische
Verhalten von körnigen Materialien wie Sand ist vor allem aufgrund des Vorhandenseins
seiner inhärenten von den Körnern gebildeten Struktur stark nichtlinear. Die Festigkeit
und die Steifigkeit eines Sandbodens sind im Allgemeinen eine Funktion des Spannungs
und Dichtezustands und der Belastungsgeschichte. Es wird eine auf der Hypoplastizi-
tät basierende Konstitutivgleichung gewählt, die Evolutionsgleichungen für die effek-
tive Spannung, die Porenzahl und den sogenannten intergranularen Dehnungstensor,
der zur Simulation von zyklischen Belastungen geeignet ist, enthält. über Schnittstel-
len wird dieses Stoffgesetz in zwei FE-Programme implementiert. Darüber hinaus wird
eine vereinfachte gekoppelte Formulierung zur Simulation von Porenwasserdruckent-
wicklung in den Code implementiert. Die Modellansätze werden durch verschiedene
analytische und experimentelle geotechnische Benchmarks verifiziert und validiert. Das
entwickelte numerische Modell wird schließlich verwendet, um das Verhalten eines offe-
nen Stahlrohrpfahls während der Einbringung unter verschiedenen Randbedingungen
zu untersuchen.
Stichwörter: Multi-Material Arbitrary Lagrangian-Eulerian (MMALE) Methode; Gr-
oße Verformungen; Gekoppelte Formulierung; Vereinfachter u-p Ansatz; Pfahlinstalla-
tion; Sand; Körniges Material; Hypoplastzität; Remeshing-Methoden
viii ZUSAMMENFASSUNG
This work is dedicated to my family and friends
Structure
This thesis is a cumulative dissertation composed of six chapters four of which have
been published and listed in Tab. 1.
Table 1: Publications comprising this dissertation
Reference Status
Paper 1
(Chapter 3)
Bakroon, M., Daryaei, R., Aubram, D., and Rackwitz,
F. (2020). "Investigation of mesh improvement in
multi-material ALE formulations using geotechnical
benchmark problems." International Journal of
Geomechanics,
https://doi.org/10.1061/(ASCE)GM.1943-5622.0001723
Published
Paper 2
(Chapter 4)
Bakroon, M., Daryaei, R., Aubram, D., and Rackwitz,
F. (2018). "Numerical evaluation of buckling in steel
pipe piles during vibratory installation." Soil Dynamics
and Earthquake Engineering, 122, 327-336.
https://doi.org/10.1016/j.soildyn.2018.08.003
Published
Paper 3
(Chapter 5)
Bakroon, M., Daryaei, R., Aubram, D., and Rackwitz,
F. (2020). "Implementation of a locally undrained
formulation to simulate pile installation in saturated
granular soil."
in prepara-
tion
Paper 4
(Appendix A)
Bakroon, M., Daryaei, R., Aubram, D., and Rackwitz,
F. (2017). "Arbitrary Lagrangian Eulerian Finite
Element Formulations Applied to Geotechnical
Problems." Numerical Methods in Geotechnics, J.
Grabe, ed., BuK! Breitschuh & Kock GmbH, Hamburg,
Germany, 33-44.
Published
xiv Structure
Structure xv
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Contents
Preface by the Author iii
Abstract v
Zusammenfassung vii
Structure vii
Contents xvi
List of Figures xxi
List of Tables xxvii
Notation xxix
1 Introduction 1
1.1 Motivation .................................. 1
1.2 Objectives .................................. 2
1.3 Structure of the work ............................ 4
2 MMALE method and Hypoplastic material model 5
2.1 Foreword ................................... 5
2.2 Element formulation: Lagrangian, CEL, and MMALE .......... 6
2.2.1 Review of Implicit/Explicit schemes ................ 6
2.2.2 Theory of CEL and MMALE ................... 12
2.2.3 History of CEL and MMALE Applications ............ 14
2.2.4 Evaluation of CEL and MMALE performance using two bench-
marks ................................ 14
2.2.5 Results and discussion ....................... 17
2.3 Constitutive equation: Hypoplasticity framework ............ 27
2.3.1 Foreword .............................. 27
2.3.2 Literature .............................. 28
2.3.3 Theory ................................ 28
2.3.4 Verification of the Hypoplastic constitutive equation model with
one element test ........................... 31
xviii CONTENTS
2.3.5 Application of the Hypoplastic constitutive equation in AbaqusR
:
Pile penetration ........................... 34
2.3.6 Application of the Hypoplastic material model in LS-DYNAR
:
Sand column collapse ........................ 36
2.4 Conclusion .................................. 37
3 Investigation of mesh improvement in MMALE 41
3.1 Introduction ................................. 42
3.2 Details of MMALE and CEL ....................... 44
3.2.1 Operator splitting .......................... 45
3.2.2 Remeshing step (Mesh smoothing algorithms) .......... 46
3.2.3 Volume-weighted smoothing .................... 47
3.2.4 Laplacian or Simple average smoothing .............. 48
3.3 Equipotential smoothing .......................... 48
3.3.1 Remapping step ........................... 49
3.3.2 Soil-structure coupling ....................... 51
3.4 Numerical Examples ............................ 51
3.4.1 Strip footing ............................. 52
3.4.2 Sand column collapse ........................ 60
3.4.3 Soil cutting by blade ........................ 65
3.5 Summary and Conclusions ......................... 70
4 Buckling in steel pipe piles 73
4.1 Introduction ................................. 74
4.1.1 Motivation .............................. 74
4.1.2 Previous research in numerical pile buckling analysis ...... 75
4.2 Numerical model .............................. 77
4.2.1 Description of the MMALE method ................ 77
4.2.2 Description of the model ...................... 77
4.2.3 Verification of shell element formulation ............. 78
4.2.4 Validation against experimental results .............. 81
4.3 Parametric study of pile buckling during penetration .......... 83
4.3.1 Reference model ........................... 83
4.3.2 Effect of pile imperfection and soil heterogeneity ......... 85
4.3.3 Results and discussion ....................... 87
4.4 Conclusion and outlook ........................... 91
5 Simplified u−pformulation 93
5.1 Introduction ................................. 94
5.2 Methodology ................................ 95
5.2.1 The MMALE numerical approach ................. 95
5.2.2 The Hypoplastic constitutive equation .............. 97
5.2.3 Pile-soil interaction ......................... 98
5.2.4 The u−pformulation ....................... 99
5.2.5 Code implementation ........................ 100
5.3 Validation of the implemented approach ................. 101
CONTENTS xix
5.4 Verification of the implemented approach ................. 104
5.5 Back-calculation of scaled model tests ................... 111
5.5.1 General remarks of the numerical model ............. 111
5.5.2 Definition of the driving load for the case of impact driving ... 113
5.5.3 Results and discussion ....................... 113
6 Conclusions and Outlook 121
6.1 Conclusion .................................. 121
6.2 Outlook ................................... 125
7 Acknowledgments 127
A ALE in AbaqusR
Vs. LS-DYNAR
129
A.1 Introduction ................................. 130
A.2 Numerical methods description ...................... 130
A.2.1 Lagrangian approach ........................ 130
A.2.2 Eulerian approach ......................... 131
A.2.3 Arbitrary Lagrangian Eulerian approach ............. 131
A.3 Numerical model description ........................ 132
A.4 Results ................................... 133
A.4.1 Model verification .......................... 134
A.4.2 Mesh size sensitivity in ALE .................... 134
A.4.3 ALE gradient remapping ...................... 138
A.4.4 Effect of time step size ....................... 138
B Remapping methods 141
B.1 Donor cell scheme .............................. 141
B.2 Van Leer scheme .............................. 142
B.3 Momentum advection ............................ 142
B.3.1 Element center projection ..................... 142
B.3.2 Half-Index-Shift (HIS) ....................... 144
C Hypoplastic implementation guidelines 147
C.1 Introduction to documentation ....................... 147
C.2 Software requirement for generating the user-defined subroutines in LS-
DYNAR
................................... 148
C.3 Generating the lsdyna.exe file ....................... 148
C.4 LS-DYNAR
UMAT interface implementation ............... 149
C.5 Invoking user defined keyword in *.k file ................. 150
Bibliography 161
xx CONTENTS
List of Figures
2.1 Schematic view of the strip footing problem [Hill, 1950] ......... 10
2.2 Finite element mesh. 2D model for implicit Lagrangian and explicit
Lagrangian analysis (Left), 3D model for explicit CEL model (Right) .10
2.3 Normalized punch pressure vs. penetration depth for different mesh den-
sities of the strip footing problem (left) and normalized punch pressure
vs. penetration depth for implicit Lagrange FEM, explicit Lagrange
FEM and explicit CEL methods for the strip footing problem (right) .11
2.4 Velocity field for implicit Lagrange FEM, explicit Lagrange FEM and
explicit CEL method after a 0.5 m punch ................. 12
2.5 Mesh distortion comparison for implicit Lagrange FEM, explicit La-
grange FEM and explicit CEL method after a 0.5 m punch ....... 13
2.6 Schematic view of the pipeline displacement problem [Merifield et al.,
2009] ..................................... 16
2.7 Mesh configuration of the model ...................... 16
2.8 Mesh distortion using Lagrangian method (left) and MMALE (right) .17
2.9 Comparison of vertical resistance from different numerical methods and
analytical result ............................... 18
2.10 Comparison of horizontal resistance from different numerical methods
and analytical result ............................ 18
2.11 Comparison of MMALE and CEL interface reconstruction ....... 19
2.12 Velocity vectors of sand movement during vertical pipe displacement of
a) 0.25Db) 0.5Dand horizontal pipe displacement of c) 0.25Dand e)
0.5D..................................... 20
2.13 Final deformed shape of soil and computational mesh using MMALE
and CEL methods, more element concentration is observed in MMALE 21
2.14 Final velocity vectors of sand movement after enforcement of both ver-
tical and horizontal pipe displacement .................. 22
2.15 Pipeline response during penetration and lateral displacement ..... 22
2.16 Schematic view of the sand column problem ............... 23
2.17 Initial configuration of the sand column collapse model ......... 24
2.18 Mesh deformation at an intermediate stage of sand column collapse using
a classical Lagrangian method ....................... 25
2.19 CEL mesh (above) and MMALE mesh (below) at the end of calculation 25
2.20 Comparison of free surface at different time stations for MMALE and
CEL with experimental results ....................... 26
xxii LIST OF FIGURES
2.21 (a) Schematic of the Oedometer test; (b) FE-mesh and boundary con-
ditions .................................... 32
2.22 Void ratio vs. vertical stress curve for oedometric compression test using
the Hypoplastic UMAT ........................... 33
2.23 (a) Schematic of the triaxial test (left), FE-mesh and boundary condi-
tions (right) ................................. 33
2.24 Deviatoric stress vs. axial strain for Triaxial compression test using the
Hypoplastic UMAT ............................. 33
2.25 Pile penetration model schematic diagram(left);Void ratio for CEL model
distributions at different penetration depths, z/D = 5.0 (Middle); z/D
= 8.5 (Right) ................................ 35
2.26 Comparison of the measured and predicted load-displacement curves of
shallow penetration test PP-26-H[23] ................... 35
2.27 Final numerical result of the sand shape using the Hypoplastic UMAT 37
2.28 Comparison of numerical results of sand column shape and distance with
experimental measurements ........................ 38
2.29 Void ratio for the Hypoplastic UMAT soil material ........... 38
3.1 Schematic diagram of different grid-based approaches comparing the
remeshing step effects on grid distortion level ............... 43
3.2 Flowchart of the operator split scheme applied to the CEL and MMALE
calculation steps ............................... 45
3.3 The initial arrangement of the arbitrary node K in a grid in 2D (left)
and 3D (right) used to illustrate the smoothing/remeshing methods de-
scribed in Eq. 3.3.6 ............................. 47
3.4 Comparison of different smoothing/remeshing algorithms based on the
achieved grid quality improvement (the numbers in the squares repre-
sents the Jacobian distortion index in percent), the elements colored
with red have an element quality less than 90% ............. 50
3.5 Numerical mesh configuration of the strip footing problem [Bakroon
et al., 2017b] ................................ 54
3.6 Comparison of the punch pressure curves obtained from the Lagrangian,
SALE, CEL, and MMALE with the analytical solution ......... 55
3.7 (a) Mesh distortion and (b) velocity field after 0.5 m of strip footing
penetration for different numerical methods. ............... 56
3.8 The effective plastic strain after 0.5 m penetration for CEL (left) and
MMALE (right) ............................... 57
3.9 MMALE time optimization achieved by changing the number of La-
grangian cycles in strip footing problem with 2.5-cm mesh element size 58
3.10 Change in the normalized contact area during the simulation as a crite-
rion to investigate leakage ......................... 59
3.11 The amount of material passed through the Lagrangian part (flux/leakage)
during the simulation ............................ 59
3.12 Relative comparisons of computations cost between CEL and MMALE
with their corresponding advection (The results are normalized accord-
ing to those of CEL for each case) ..................... 60
LIST OF FIGURES xxiii
3.13 Normalized kinetic energy and kinetic energy loss during the simulation
for MMALE and CEL (the values are normalized with respect to the
maximum value of kinetic energy loss curve for CEL) .......... 61
3.14 Initial configuration of the numerical model for the case of CEL and
MMALE; the model size is 1.65x1.2 m but only the mesh of the detail
A is shown .................................. 62
3.15 Mesh deformation for Lagrangian simulation of sand column collapse .63
3.16 (a) Final shape of the flowed soil as well as the mesh distortion in the
sand column collapse for CEL (top) and MMALE (bottom), (b) Soil
interface reconstruction in CEL (top) and MMALE (bottom), the con-
tours represent the volume fraction of the soil in the elements; the results
correspond to the detail B and not the whole model ........... 64
3.17 Comparison of the runout distance from the numerical models and the
experimental measurements in the sand column collapse problem .... 64
3.18 Comparison of the normalized kinetic energy loss during advection for
the sand column problem (the values are normalized with respect to the
maximum value of CEL curve) ....................... 65
3.19 Soil particle trajectory, (b) Comparison of the displacement between
several particles obtained from CEL and MMALE ............ 66
3.20 Schematic view of the soil cutting problem ................ 67
3.21 Mesh distortion during the soil cutting using the SALE method .... 68
3.22 Mesh distortion and soil deformation using CEL (above) and MMALE
(below) methods in the soil cutting problem ............... 68
3.23 Schematic of the assumed conditions in the soil cutting problem for
deriving an analytical solution [McKyes, 1985] .............. 69
3.24 Comparison of the induced horizontal and vertical forces on the blade
obtained from MMALE and CEL methods with the analytical solution
in the soil cutting problem ......................... 69
3.25 Comparison of the internal and kinetic energy curves of the soil cutting
problem ................................... 70
4.1 Schematic diagram of the (a) isometric view, (b) side view, (c) planar
view of the one-quarter numerical model configuration with (d) vibratory
load history curve .............................. 79
4.2 Benchmark model configuration under uniform axial compression ... 80
4.3 Resulting buckling modes using different element formulations ..... 81
4.4 Penetration depth vs. time curve obtained from the numerical model
and experimental measurement ...................... 82
4.5 Isolines of the induced (a) vertical and (b) horizontal stress in the soil,
and (c) the corresponding loading at 8.98 sec for the validation model .82
4.6 Mean strain contour plots in pile after 0.65 m penetration for the refer-
ence model .................................. 84
4.7 Isolines of the induced (a) vertical and (b) horizontal stress in the soil,
and (c) the corresponding loading at 8.98 sec for the reference model .84
4.8 Schematic diagram of initial pile section compared to a perfect circle .85
xxiv LIST OF FIGURES
4.9 Schematic of initial pile geometry from different views which illustrates
the out-of-straightness ........................... 86
4.10 Schematic diagram of initial pile geometry from different views illustrat-
ing flatness ................................. 87
4.11 (a) The planar view and (b) the cross section of the model illustrating
the location of the applied heterogeneity (rigid sphere) in the soil ... 88
4.12 Comparison of the imperfect piles with the reference model based on the
(a) vertical displacement (b) lateral displacement, and (c) the internal
energy .................................... 89
4.13 (a) Contours of induced mean infinitesimal strain in the imperfect piles
and the reference model and (b) the pile tip cross section compared to
its initial .................................. 90
5.1 Schematic diagram of MMALE approach compared to the classical La-
grangian FEM [6] .............................. 97
5.2 Flowchart of the calculation process inside the user-defined subroutine . 102
5.3 Schematic of the developed FE model (left), and the respective boundary
conditions (right) of the triaxial test ................... 103
5.4 Comparison of the FE model and experiment results of drained triaxial
compression test of Berlin sand (a) Deviator stress and (b) Void ratio
vs. strain increment ............................. 103
5.5 Comparison of FE model and experiment results of undrained triaxial
compression test of Berlin sand with 100 kPa confinement stress (a)
Deviator stress vs. mean stress, (b) Deviator stress vs. strain increment,
and (c) excess pore water pressure vs. strain increment ......... 105
5.6 Comparison of FE model and experiment results of undrained triaxial
compression test of Berlin sand 500 kPa confinement stress (a) Deviator
stress vs. mean stress, (b) Deviator stress vs. strain increment, and (c)
excess pore water pressure vs. strain increment ............. 106
5.7 Schematic of the pile penetration problem using the (a) classical La-
grangian and (b) MMALE element formulation ............. 107
5.8 Comparison of the induced lateral and vertical effective stress and excess
pore water pressure in the pile penetration problem in loose Mai Liao
sand at the pile tip depth of 2 m; the positive pore water pressure values
correspond to the compression ....................... 110
5.9 Numerical model configuration of the pile driving experiment (axisym-
metric boundary conditions are applied accordingly) ........... 112
5.10 a) Load application curve of the impact driving b) acceleration history
of the pile induced by one impact blow .................. 114
5.11 Comparison of the pile tip depth and pile tip displacement curves from
the drained and undrained simulation and the experiments [Madsen
et al., 2012; Qiu and Grabe, 2011] ..................... 115
5.12 Comparison of the deformed soil shape obtained from the (a) drained
and (b) undrained simulation and (c) the experiment at the pile tip
depth of 3D. The bottom boundary of all models and the side boundary
of the experiment are cropped ....................... 115
LIST OF FIGURES xxv
5.13 Comparison of the movement of the soil regime for the case of (a) drained
and (b) undrained simulation at the pile tip depth of 3D ........ 116
5.14 Induced horizontal stress in the soil profile at the pile tip depths of 2D
and 3Dat three lateral distances, 0.5D,1.0D, and 1.5Dfrom the pile
shaft obtained from the drained and undrained simulation ....... 118
5.15 Generated PWP at the pile depth of the a) 2D and b) 3D obtained from
the undrained simulation; positive values indicate compression while the
negative values indicate suction ...................... 119
A.1 FE model Initial configuration (left), Material deformation in a La-
grangian analysis (middle) and an Arbitrary Lagrangian Eulerian anal-
ysis ALE (right) ............................... 131
A.2 Geometry and boundary conditions assigned to strip footing problem
[Bakroon et al., 2017a] ........................... 132
A.3 Numerical mesh configuration of the strip footing problem ....... 135
A.4 Comparison of punch pressure results for Lagrangian, and ALE with
analytical solution ............................. 135
A.5 Normalized punch pressure vs. penetration depth for different mesh
densities of the strip foot-ing problem analysed by AbaqusR
...... 136
A.6 Normalized punch pressure vs. penetration depth for different mesh
densities of the strip foot-ing problem analysed by LS-DYNAR
..... 136
A.7 a) Initial mesh configuration, b) Lagrangian deformed mesh calculated
by AbaqusR
and LS-DYNAR
, ALE mesh deformation in c) AbaqusR
,
d) LS-DYNAR
............................... 137
A.8 a) Initial mesh configuration, gradient ALE mesh deformation in b)
AbaqusR
, c) LS-DYNAR
.......................... 138
A.9 Effect of gradient mesh on the accuracy of pressure results ....... 139
A.10 Time step effect for the strip footing problem analysed by ALE method 139
B.1 Illustration of donor cell advection algorithm (a-c) described in Eq.
B.1.1-B.1.2 and van Leer (d-f) advection algorithms described in Eq.
B.2.1-B.2.2 in one direction. The horizontal axis depicts the node coor-
dinates, and the vertical axis represents the arbitrary solution variable.
(a) Initial state variable distribution after the Lagrangian step, (b) Node
coordinates after rezoning step, (c) New state variable distribution in el-
ement after transport for donor cell method. (d) Initial state variable
value distribution and auxiliary lines for distribution calculation, (e) The
calculated piecewise distribution, (f) New state variable distribution in
element after transport for van Leer method ............... 143
B.2 Illustration of a) element center projection and b) half-index shift method
for momentum advection .......................... 145
xxvi LIST OF FIGURES
List of Tables
1 Publications comprising this dissertation ................. xiii
2.1 Material parameters for the soil ...................... 9
2.2 Calculation time comparison for MMALE and CEL for pipeline displace-
ment problem ................................ 21
2.3 Mohr-Coulomb properties for the sand column collapse model [Solowski
and Sloan, 2013] ............................... 25
2.4 Calculation time comparison for MMALE and CEL for sand column
collapse problem .............................. 26
2.5 The required parameters of hypoplastic material model with intergran-
ular strain .................................. 31
2.6 Hypoplastic parameters of Hochstetten sand for Oedometer test [Niemu-
nis and Herle, 1997] ............................. 32
2.7 Hypoplastic parameters of Hochstetten sand for triaxial test [Niemunis
and Herle, 1997] ............................... 33
2.8 Hypoplastic parameters for the used soil model. ............. 35
2.9 Equivalent Hypoplastic properties of the sand based on the Mohr-Coulomb
material model ............................... 36
3.1 Comparison criteria and their purpose for the numerical examples ... 53
4.1 General properties of the pile used in benchmark models ........ 78
4.2 Mohr-Coulomb material constants for Berlin sand [Schweiger, 2002] .. 78
4.3 Comparison of the resulting critical buckling stress under axial pressure 81
4.4 Elastoplastic properties of the pile used in a parametric study ..... 83
4.5 Properties of the oval pile section ..................... 85
5.1 Hypoplastic material constants for Berlin sand ............. 101
5.2 Triaxial test series .............................. 104
5.3 Hypoplastic material constants for the Mai-Liao sand [Cudmani, 2001] 108
5.4 Additional constants for simulating the soil-fluid mixture ........ 108
5.5 Parameters associated with the static pile force in the case of impact
driving .................................... 111
5.6 Hypoplastic material constants for Berlin sand [Le, 2015] ........ 113
A.1 Material parameters for the soft soil .................... 133
A.2 Time step comparison for Lagrangian and ALE model .......... 134
A.3 Analysis time comparison for different mesh sizes of ALE model .... 137
A.4 Analysis time comparison for different mesh sizes of ALE model .... 138
A.5 Effect of different time step sizes on calculation time .......... 140
C.1 UMAT interface variables ......................... 149
C.2 Stress/strain assignment order in LS-DYNAR
UMAT .......... 149
C.3 *MAT_USER_DEFINED_MATERIAL_MODELS keyword required
input parameters for Hypoplastic model ................. 150
C.4 Parameters input order within the UMAT ................ 150
Symbols and Notations
Latin Letters
Aarea
csound speed in the corresponding material
ccohesion
cflatness value
cwave speed
csoil shear strength
cddilatational wave speed
C fourth order tensor of the soil stiffness
dinner pile diameter
dcutting depth
dmin minimum diagonal of the shell element
Dpipe diameter
Dmax longest oval diameter
Dmin shortest oval diameter
Drrelative density of the soil
Dstrain/stretching rate tensor
evoid ratio
Eelastic Modulus
Eppotential energy
Ekkinetic energy
fbbarotropy factor
fe,fdpycnotropy factors
Fapplied load vector
h height of the drop
Hhorizontal resistance force
Iinternal force vector
Isecond-order unit tensor
khydraulic conductivity
xxx NOTATION
Kstiffness matrix
Kbulk modulus
Kffluid bulk modulus
Ktcurrent tangent stiffness matrix
KW
0bulk modulus of the water
Lscharacteristic length of the element
L(T,e) fourth-order tensor associated with the linear part of the behavior
m drop mass
mminimum mass of the pile and soil
mT,mRadditional constants account for changes in loading direction
Mmass matrix
Mfourth order tensor depending on the stress tensor, T
nporosity
NcV coefficient for vertical strength
NcH coefficient for horizontal strength
NswV coefficient for vertical strength for self-weight term
NswH coefficient for horizontal strength for self-weight term
N(T,e) second-order tensor related to the nonlinear part of the behavior
ppore water pressure
pmean effective stress
Pmean total stress
Ptotal force per unit width
Pfac stiffness factor
qc,a CPT tip resistance
qult maximum punch pressure
S saturation degree
Ssource term
t impact duration
Δt time step
u displacement of the soil skeleton
u displacement vector
uincrement of displacement
˙
uvelocity
¨
uacceleration
vivelocity at node i
Vvertical resistance force
NOTATION xxxi
Greek Letters
αintegration parameters
βintegration parameters
γintegration parameters
γLam´eelastic constants
γsubmerged unit weight of soil
γfunit weight of the fluid
˙
Tco-rotational Jaumann stress rate
δintergranular strain
δpile alignment with respect to the vertical axis
˙
urate of strain of the undrained solid-fluid mixture
˙
strain rate
ηreduction factor due to energy dissipation during impact
λparameter approximating the amount of heaving
λout-of-roundness
μLam´eelastic constants
νpoisson ratio
ρmaterial density
ρχadditional constant account for changes in loading direction
σyyield Stress
σtotal stress
σeffective stress
ϕfield variable
Φflux function
ϕfriction angle
Φflux function
χconstant used for smoothing the weighting factor
ψ(i+1/2) auxiliary parameter for element center i+1/2
ψvelocity at each element node in one dimension
ˆwnormalized penetration depth
ωmax maximum element eigen value
w0ovality coefficient
Operators
×cartesian product; direct product of sets
xxxii NOTATION
⊗tensor product; dyadic product
NOTATION xxxiii
Abbreviations
ALE Arbitrary Lagrangian-Eulerian
CEL Coupled Eulerian Lagrangian
CPU Central Processing Unit
DIN Deutsches Institut für Normung eV
EVF Element Void Fraction
FE finite element
FEM finite element method
LDFEM Large Deformation finite element method
LSTC Livermore Software Technology Corporation
MMALE Multi-Material Arbitrary Lagrangian-Eulerian
MPP massively parallel processing
MUSCL Monotonic Upwind Scheme for Conservation Laws
PC Personal Computer
PFEM Particle finite element method
PWP Pore Water Pressure
RAM Random Access Memory
RITSS Remeshing and Interpolation Technique with Small Strain
SALE Simplified Arbitrary Lagrangian-Eulerian
UMAT User-defined Material model
VUMAT Vectorized User-defined Material model
xxxiv NOTATION
Chapter 1
Introduction
1.1 Motivation
Concerning the evaluation of geotechnical problems, one may choose an evaluation
method among several possible investigation methods. The first method is to use
the available theoretical solutions, if any, for the current problem. Alternatively, one
can conduct experimental tests capturing the site conditions. The last but not the
least, is to simulate the problem using numerical techniques. The numerical method
has gained an increasing attention due to its potential advantages including ensuring
reliable results using low cost and resources.
Some of the main challenges in numerical simulation of geotechnical problems are the
choice of a numerical approach, the choice of a constitutive equation for the soil mate-
rial, and consideration of water in the soil pores. In the case of the first challenge, one
of the most widely used numerical approach is the Finite Element Method (FEM) [Be-
lytschko et al.,2000] which is still used in many geotechnical problems and is available
in almost every commercial software in geotechnical engineering [Dassault Systèmes,
2016;Hallquist,2017]. Common application problems include calculation of the sta-
bility of retaining walls or a soil depot and bearing capacity of the shallow foundation.
Such problems are considered as small deformation problems since further deformation
results in loss of stability/serviceability of the building/structure. Such geotechnical
problems can be properly analyzed by using conventional Lagrangian FEM.
However, in case of specific problems, such as geotechnical installation problems, the
conventional Lagrangian FEM shows considerable shortcomings, since the soil un-
dergoes significant deformation. These problems are generally referred to as large-
deformation problems. Therefore, in order to calculate the bearing capacity of piles,
one solution is to ignore the installation process, i.e. the pile is assumed to be wished-in.
Yet, the wished-in-place assumption may not be generally realistic. The pile enforces
huge distortion in the neighboring soil regime. This distortion influences the soil den-
sity and stress distribution, which in turn affects the contact between soil and pile. The
soil behavior is therefore affected by the installation of pile which can result in inaccu-
1
2CHAPTER 1. INTRODUCTION
rate pile bearing capacity prediction [Budhu,2010]. Therefore, an advanced numerical
approach is required to simulate such large deformation problems.
In the case of the second challenge, the choice of a constitutive equation for the soil
material, a robust constitutive equation is required to capture the complex nonlinear
mechanical behavior of granular materials, such as sandy soils, which is one of the
key aspects in geotechnical analysis and design. The mechanical behavior of granular
materials like sand is highly nonlinear due to the presence of an evolving internal
structure formed by the grains [Kolymbas,2012]. The strength and stiffness is generally
a function of the stress and density state and the loading history. The accuracy of
numerical simulations is in a large part influenced by the choice of the material model.
In the case of the third challenge, consideration of water in the soil pores, most cases of
geotechnical problems associate with the presence of water which cannot be neglected.
Concerning the change in the loading speed, drainage condition, etc., the soil-water
mixture can exhibit significantly different behavior. In the numerical simulation field,
considering the presence of water is usually referred to as a coupling scheme where
the solution of soil and water are done simultaneously [Zienkiewicz and Shiomi,1984].
There are a handful of coupling scheme available in the literature. The difference
between these schemes are the assumption made in the water-soil behavior, resulting
in a simpler equation, potentially lower computation cost, and reasonable accuracy.
Having the three challenges addressed, one can achieve a robust numerical technique
which is capable of simulating most of the pile installation problems under different
boundary conditions. In the following sections, the devised approach to address the
three aspects are discussed.
1.2 Objectives
This study aims at tailoring an advanced numerical approach to simulate the geotechni-
cal large deformation problems. The main focus is to simulate the pipe-pile installation
problem, as it is one the popular foundation types [Madsen et al.,2012], and at the
same type cumbersome to model.
During the last few decades, efforts have been made to develop methods that overcome
the shortcoming of classical approaches. Some of the most promising approaches which
rely on a computational mesh include the Coupled Eulerian-Lagrangian (CEL) method,
the Simplified or Single-Material Arbitrary Lagrangian-Eulerian (SALE) method, and
the Multi-Material Arbitrary Lagrangian-Eulerian (MMALE) method. The intuition
behind these methods was to exploit the advantages of both Lagrangian and Eulerian
methods [Belytschko et al.,2000;Benson,1992a].
The traditional formulations of FEM utilize either the Lagrangian viewpoint or the
Eulerian viewpoint. In the Lagrangian FEM, the mesh velocity coincides with the
material velocity, meaning that if the soil deforms, the mesh deforms accordingly. After
a specific amount of deformation, elements may encounter huge distortion which causes
the solution not to converge or leads to inaccurate results. In addition, excessively
deformed elements may cause problems if contact constraints have to be enforced. In
1.2. OBJECTIVES 3
the Eulerian formulation, on the other hand, the mesh is fixed, and the material moves
independently through the mesh. Despite the advantage of handling large deformations
and vorticity, Eulerian methods require special techniques for treating path-dependent
material behavior and tracking material interfaces.
In this work the CEL, SALE, and MMALE methods are investigated and compared
as candidates for employment in the numerical approach. The investigation consists of
comparing their performance in simulating various simple but challenging problems.
Concerning the constitutive equations of soils, conventional approaches are based on
the elastoplasticity approach. By using this theory, a yield surface and plastic surface
must be defined, which means that in a specific range of admissible states the material
behaves purely elastic. In an elastic state, the induced strain is reversible by unloading,
which is generally not the case for granular soils. The class of constitutive equations
based on the so-called hypoplasticity concept, on the other hand, do not distinguish
between elastic and plastic states and have been proven more accurate in simulating the
complex behavior of granular materials under cyclic loading and over a wide range of
stress and density states with only one set of parameters for a specific granular material
and incorporating state parameters such as void ratio [Bakroon et al.,2017a;Dijkstra
et al.,2011;Pucker and Grabe,2012;Qiu et al.,2011]. Besides, by removing extra
definitions regarding yield and plastic surface, such models are easier to implement
[Kolymbas,2012].
As a part of this work, it is intended to present an approach to realistically predict
granular material behavior. To this purpose, an advanced material model based on the
Hypoplasticity concept is adopted, since it defines evolution equations for the effective
stress, void ratio, and the so-called intergranular strain tensor.
The pile installation procedure occurs relatively fast, i.e. the pore water in the soil
has no time to dissipate. This condition is referred to as a locally undrained condition.
Under such conditions, it is possible to simplify the general coupled formulation. In
this work, the theoretical background of the devised coupling scheme is presented.
Afterward, the scheme is included in the numerical approach along with the numerical
element formulation and the constitutive equation. In summary, the following points
are conducted and discussed in this work:
1. Comparing and investigating two advanced numerical approaches, Coupled Euler-
ian-Lagrangian (CEL) and Multi-Material Arbitrary Lagrangian Eulerian (MM-
ALE) considering large deformation geotechnical problems.
2. Implementing the hypoplastic material model which is a powerful constitutive
model for soils to predict granular material behavior to be utilized for simu-
lation of large deformation geotechnical problems into two commercial codes,
AbaqusR
and LS-DYNAR
along with verification and validation of the imple-
mentation.
3. Implementing and adapting a simplified u-p formulation to the Hypoplastic con-
stitutive material model, to enhance modeling of saturated soil, followed by the
validation of the implemented model and the simulation of the soil-structure
interaction problems.
4. Applying the tailored numerical approach to realistic geotechnical problems. The
case studies mainly include the evaluation of pile buckling during installation and
back-calculation of an experimental test.
1.3 Structure of the work
This study follows the following structure. Chapter 2gives an overview of different
numerical approaches to simulate large deformation geotechnical problems including
the evaluation of their performance in simulating various benchmark problems. Also,
the theoretical background of the hypoplsatic material model is presented. Then the
validation and verification of the material model using element tests are presented.
Details of the numerical description of CEL and MMALE methods such as operator
splitting, remeshing, and remapping steps, and soil-structure coupling are described
in Chapter 3. Moreover, the advantages of the remeshing step in MMALE is high-
lighted via back-calculation of various benchmark problems. Chapter 4describes the
implemented coupled formulation scheme, the simplified u-p formulation, in the code
as well as its performance in small-scale experiments. Subsequently, the application
of the suggested numerical approach in small- and large-scale problems is discussed in
Chapter 5. Finally, concluding remarks and outlook are provided in Chapter 6.
Chapter 2
Theoretical Background of the
MMALE Method and
Implementation of the Hypoplastic
Material Model Interface
2.1 Foreword
Installation processes in geotechnical engineering, referred to as large deformation prob-
lems, are considered important to be studied. However, analysis of such problems faces
many challenges, as the large soil deformations need special techniques for numerical
simulation. Large deformations may cause severe distortion of the computational mesh,
which may stop the analysis in early stages. The dynamic analysis of a non-linear large
deformation Geotechnical problems using the standard Lagrange finite element method
(FEM) can experience numerical difficulties including contact problems, convergence
and large mesh deformations. Moreover, the extremely non-linear behavior of the soil
material consequently involves non-convergence issues. Hence, several variants of the
Finite Element Method (FEM) have been developed to overcome these problems [Ben-
son,1992a;Aubram et al.,2017]. Among these, the Arbitrary Lagrangian-Eulerian
(ALE), is argued to overcome the disadvantages of the classical Lagrangian approach
in solid mechanics while showing good agreement with experimental results. The ALE
method is classified into two groups of methods, Simplified ALE (SALE) and Multi-
Material ALE (MMALE). One can also consider the known method Coupled Eulerian-
Lagrangian (CEL) as a special case of MMALE which is explained in Chapter 3.
The chapter is divided into two main sections. First, the theoretical background of
MMALE and the CEL methods are presented along with both classical explicit and
implicit FEM. The methods are then evaluated using two large deformation benchmark
problems. The limitation of different FEM formulations (implicit Lagrange, explicit
Lagrange) against (CEL) method, in non-linear dynamic large deformation problem is
shown.
6CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
Second, the fromulation of the hypoplastic constituve equation is thoroughly described.
Afterward, the implementation of the hypoplastic user material interface in two com-
mercial codes, AbaqusR
and LS-DYNAR
are discussed in detail including verification
and validation, and back-calculation of two large deformation benchmark problems.
2.2 Element formulation: Lagrangian, CEL, and
MMALE
2.2.1 Review of Implicit/Explicit schemes
The implicit solution method depends on the state of the FE model at the time point
under study [Taylor et al.,1995;Kutts et al.,1998]. In other words, if the FE model
is updated from tto Δt + 1 then the equilibrium will be satisfied for this sate at time
Δt + 1, while the explicit solution method will satisfy the equilibrium at time Δt +1
depends on the data of the model at time t.
The explicit solution method has the advantage to minimize the processing time and
memory requirements [Doweidar et al.,2010]. Many more advantages have the explicit
solution method over the Implicit solution method for large deformations in geotechni-
cal problems. The computational time in the explicit method is linearly proportional
to model size, while in the implicit method the time increases quadratically with the
model size as will be discussed in the equations below. The explicit dynamic FEM
solves equations without iterations in contrast to implicit methods which involves iter-
ations to satisfy a convergence at each increment, that leads to a time/CPU consuming
solution. Implicit and explicit FEM generally implement different procedures to update
the stress and state variables of a material in the computational model.
The stability limit of the explicit method is bounded by a maximum time increment,
which must be less than the speed of sound to cross the smallest element size of the
model. This limitation is excluded from the implicit scheme and it solves the dynamic
quantities at time Δt + 1 based not only on the information at time Δt but also at
time Δt + 1. Implicit scheme gives acceptable, accurate results for the same models
solved by explicit scheme using time increments 10 or 100 times the time increments
Δt used in explicit scheme, but the response prediction will deteriorate as the time
step size, Δt, increases relative to the period, T, of typical modes of response. The
implementation of large deformations, contact constraints, and sliding are very easy in
the explicit method.
Implicit solution method
The implicit method uses a backward Euler operator [Hilber and Hughes,1978], which
updates the FE model from tto Δt+1 then the equilibrium will be satisfied for this sate
at time Δt + 1, while the explicit solution method will satisfy the equilibrium at time
Δt + 1 depends on the data of the model at time t. That means the explicit solution
method has the advantage to minimize the processing time and memory requirements.
The explicit dynamic FEM solves equations without iterations in contrast to implicit
2.2. ELEMENT FORMULATION: LAGRANGIAN, CEL, AND MMALE 7
methods which involves iterations to satisfy a convergence at each increment, that
leads to a time CPU consuming solution. Implicit and explicit FEM generally imple-
ment different procedures to update the stress and state variables of a material in the
computational model.
The time integration of the dynamic problem uses the operator defined by Hilber and
Hughes [1978]. Which is used to control the numerical damping. The implicit procedure
uses suitable root finding technique of a full Newton-Raphson solution method [Sun
et al.,2000]:
Δu(i+1)=Δu(i)+K−1
t·F(i)−I(i)(2.2.1)
Where Ktis the current tangent stiffness matrix, Fthe applied load vector, Ithe
internal force vector, uis the increment of displacement, and Δt is the time step. For
as implicit method, the algorithm is defined by Sun et al. [2000]:
M¨
u(i+1) +(1+α)Ku(i+1) −αKu(i)=F(i+1) (2.2.2)
Where Mis the mass matrix, Kthe stiffness matrix, Fthe vector of applied loads
and u the displacement vector:
u(i+1) =u(i)+Δt ˙
u(i)+Δt21
2−β¨
u(i)+β¨
u(i+1) (2.2.3)
and
˙
u(i+1) =˙
u(i)+Δt (1 −γ)¨
u(i)+γ¨
u(i+1)(2.2.4)
where ˙
uis velocity and ¨
uis acceleration.
with
β=1
41−α2,γ=1
2−α, −1
2≤α≤0 (2.2.5)
where α,γ,and βare the integration parameters. For certain values of γand βthe
integration scheme can be rendered unconditionally stable. The parameter α=−0.05
is chosen by default in AbaqusR
as a small damping term to avoid numerical noise [Sun
et al.,2000].
Explicit solution method
The explicit dynamics method is based on the application of an explicit time integration
rule with the use of diagonal mass matrices for the elements. The central difference
scheme is applied to the equations of motions of the body:
8CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
˙u(i+1/2) =˙u(i−1/2) +1
2Δt(i+1) +Δt(i)¨u(i)(2.2.6)
¨
u(i)=M(−1) ·F(i)−I(i)(2.2.7)
(i),i−1
2,and i+1
2(2.2.8)
The superscript (i), (i−1/2),and (i+1/2) refer to the increment value and mid-
increment values, respectively. By knowing the values of ˙u(i−1
2)and ¨u(i)from previous
increment, the central difference operator is explicit in that the kinematic state can
be advanced [Sun et al.,2000]. The explicit method is simple and does not require
iterations or a tangent stiffness matrix. In order to further increase computational
efficiency, the element mass matrix is lumped to become diagonal. This leads to an
inexpensive matrix inversion in Eq. (8). The value of the time increment has to be
limited by:
Δt ≤2
ωmax
(2.2.9)
where ωmax is the maximum element eigen value. A practical method to implement
the above limit is:
Δt = min Le
cd(2.2.10)
where Leis the characteristic minimum element length in the model, cdis the dilata-
tional wave speed, and can be calculated by the following equation:
cd=λ+2μ
ρ(2.2.11)
where γand μare the Lamé elastic constants and ρis the material density. In a quasi-
static analysis, the time increment used for an explicit scheme is much smaller than
with the implicit scheme for an equivalent problem. The sizes of the elements should
be regular in order to obtain efficient analysis results. Otherwise, a single element may
increase the time of the analysis for the whole model [Dassault Systèmes,2016]. In a
quasi-static analysis, it is impractical to run the model with its real time scale, as it
will be very large. There are a number of ways to overcome this issue. A mass scale
is one of a used style practice. According to the equations (10) and (11), the time
increment is proportional to the square root of the density. That is, if the density
increased by a factor a2the time will be reduced by a. However, the changes must
not increase the internal forces in order not to alter the solution. In the quasi-static
simulation, controlling the internal forces has an important rule. The internal forces
must not affect the mechanical response in order not to produce increase values for
2.2. ELEMENT FORMULATION: LAGRANGIAN, CEL, AND MMALE 9
Table 2.1: Material parameters for the soil
E [kPa] c [kPa] ν[-]
2980 10 0.49
the internal forces which is not real. Kutts et al. [1998] recommend that the internal
energy should not increase more than 5% of the kinetic energy. So that the dynamic
effects will be reduced and can be neglected.
Comparison of Explicit and Implicit Lagrangian in a benchmark (Strip foot-
ing)
The first example is a strip footing problem which uses Tresca model to investigate
and compare between the previous numerical analysis methods. The strip footing is a
problem where material deformations are large and a closed-form analytical solution is
available. This problem will be used also in section 3.4.1 to evaluate other numerical
formulations. Three computational models using different numerical analysis methods
are compared: implicit Lagrange FEM, explicit Lagrange FEM, and the explicit Cou-
pled Eulerian-Lagrangian method. The results of the pressure under the footing will be
compared to the analytical solution done by Hill [1950]. Hill processed a billet which
held in a container and hollowed out by punch. He regarded the problem as a plane
strain problem in order to simplify the solution. The container is smooth, so the sides
of the material will be fixed only in the horizontal direction and the bottom face will
be fixed in the vertical direction. The punch assumed as a rigid body with no relative
displacement base and smooth sides.
For the ratio of base over soil width = 0.5, the maximum punch pressure for this
problem is 2c1+1
2π[Hill,1950] where cin this model is the soil shear strength. The
soil material parameters used in the problem are shown in Tab. 2.1, where cis the soil
cohesion, vis poison ratio and Eis the modulus of elasticity. The Tresca constitutive
model is adopted.
As illustrated in Fig. 2.1, the strip footing is assumed rigid with 2 m width and 1 m
height, the soil is4mx4m.Astheproblem is plain strain, in the implicit and explicit
analysis, a two dimensional model will be used. For the soil a 4-node bilinear plane
strain quadrilateral with reduced integration and hourglass control (AbaqusR
element
type CPE4R) will be used. For the explicit CEL analysis a three dimensional model will
be used as shown in Fig. 2.2. The available 3D element is an 8-node linear Eulerian brick
with reduced integration and hourglass control (AbaqusR
element type EC3D8R). The
CEL mesh has a depth of one element and realizes plane strain boundary conditions
to be comparable with the two dimensional models. The rigid body elements were
meshed by a 4-node 3D bilinear rigid quadrilateral (R3D4).
The model can be simplified by using a symmetry plane, only half of the model will
be modeled as shown in Fig. 2.2. Symmetric boundary conditions are imposed on the
plane of symmetry by prescribing fixed condition in the normal direction of the side of
these planes.
10 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
Figure 2.1: Schematic view of the strip footing problem [Hill,1950]
Figure 2.2: Finite element mesh. 2D model for implicit Lagrangian and explicit La-
grangian analysis (Left), 3D model for explicit CEL model (Right)
2.2. ELEMENT FORMULATION: LAGRANGIAN, CEL, AND MMALE 11
Figure 2.3: Normalized punch pressure vs. penetration depth for different mesh densi-
ties of the strip footing problem (left) and normalized punch pressure vs. penetration
depth for implicit Lagrange FEM, explicit Lagrange FEM and explicit CEL methods
for the strip footing problem (right)
In the CEL model two regions should be defined, the first region includes a soil domain
in which the soil properties first assigned to, the second region is a void layer which
includes no material at the beginning of the simulation but after that it allows the soil
material to move in due to the heaving of the soil. Theses void elements have an EVF
= 0 at the beginning of the simulation [Dassault Systèmes,2016] and this percentage
reaches 1 which means that this element is full of material. It should/may be noted
that these void elements have neither strength nor stiffness in order not to affect the
solution results.
A general contact is applied for the CEL model. Rough contact (no slip) condition
between footing base and soil is used for the implicit and explicit Lagrangian models
while, smooth contact is applied between footing side and soil.
To examine the appropriate mesh density for the strip footing problem, four mesh
widths where selected (5, 10, 15 and 25cm). Fig. 2.3 shows that the mesh coarseness
relates the pressure under the footing. The pressure-punch curve is plotted for the
four mesh densities which analyzed using an explicit CEL analysis. The pressure in
all models increases in the 5 cm punch. However, it continues to increase relatively
for the 10, 15 and 25 cm meshes, only for the 5 cm mesh model it remains constant
after 5 cm of punch. It is clearly shown that the 5 cm mesh width is the appropriate
and accepted mesh density, and it will be chosen for the implicit Lagrangian, explicit
Lagrangian and explicit CEL comparison. The comparative analysis between implicit,
explicit and CEL are shown in Fig. 2.3. The plastic stress is reached after 5 cm of
punch for all models but for implicit, the pressure continues to increase as the punch
increases. It is good to notice that the explicit Lagrangian and CEL models are in good
agreement until 30 cm of punch but, after that, the explicit model starts to increase
due to the excessively distortion to the mesh elements besides the edge of the footing
as in Fig. 2.4. The explicit CEL model has a constant pressure regardless the punch
increasing.
12 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
Figure 2.4: Velocity field for implicit Lagrange FEM, explicit Lagrange FEM and
explicit CEL method after a 0.5 m punch
The fluctuating of pressure in the explicit Lagrangian model is due to the basics of the
dynamic explicit method which deals with each step of loading by itself and without
comparing the results with whole simulation time which results in load oscillations.
The implicit analysis results are smooth but increasing due to the upward soil motion
block as shown in Fig. 2.4.
The velocity field in implicit and explicit models illustrates the concentration of stresses
around the footing edge which known as singular plasticity point [Qiu and Grabe,2011].
These singularities move the soil down, then to the sides away from the side of the foot-
ing. In explicit CEL method the velocity field shows a regular distribution of pressure
under the footing which is indicated clearly in Fig. 2.5. The material moves down,
then to the sides and after that to the top which is similar to the results of Qiu and
Grabe [2011] and Aubram [2013]. In conclusion, this strip footing problem shows that
the explicit CEL method is appropriate to the large deformation soil problems due to
the stability and robustness of results.
2.2.2 Theory of CEL and MMALE
There are many FE methods that solve geotechnical problems such as the classical
Lagrangian method, where the mesh elements distort as the material deforms. This
limitation hinders the feasibility of classical FEM when it comes to modeling large soil
deformation problems. New methods were developed by researchers which enables the
material to flow through mesh elements. This feature enables the modeling of large de-
formation problems. Coupled Eulerian-Lagrangian (CEL) method and Multi-Material
Arbitrary Lagrangian Eulerian (MMALE) method are among the trending methods for
simulating large deformation geotechnical problems. These methods efficiently handle
large deformation problems, owing to the imposed non-alignment condition of the com-
2.2. ELEMENT FORMULATION: LAGRANGIAN, CEL, AND MMALE 13
Figure 2.5: Mesh distortion comparison for implicit Lagrange FEM, explicit Lagrange
FEM and explicit CEL method after a 0.5 m punch
putational mesh and the material [Aubram et al.,2015;Doweidar et al.,2010;Harewood
and McHugh,2007]. They have been previously applied to turbulent problems to sim-
ulate fluid and gas flow and provide acceptable results. Its application in geotechnical
engineering is still limited and require further investigation.
Concurrent to CEL studies, several works were done in applying the ALE method to
geotechnical problems. In a series of work done by Aubram [2013] and Aubram et al.
[2015], an advanced SALE formulation is implemented in Ansys, and its performance
is evaluated by simulating shallow penetration into the sand. A good agreement be-
tween numerical results and experimental measurements was observed. In addition,
Bakroon et al. [2017b] thoroughly investigated the performance of SALE, Lagrangian
explicit and implicit methods regarding computation costs, mesh size optimization,
and time step size effect by using a benchmark model. The efficiency of SALE method
in the aspect of both accuracy and mesh improvement was proved (see Appendix A).
Nevertheless, regarding extremely large deformations SALE showed limitations. Conse-
quently, studies focused on applying the MMALE to geotechnical problems. Previously,
MMALE has been applied in the particular case of geotechnical problems such as an un-
derground explosion, (see for example [Daryaei and Eslami,2017]), where the soil was
merely considered as a medium for shock wave transmission and was not studied thor-
oughly. In addition, the installation processes have not been considered. Recently, the
theoretical background and considerations regarding geotechnical installation process
using MMALE have been outlined by Aubram [2016]; Aubram et al. [2017]. Further-
more, a study conducted by Bakroon et al. [2018a] assessed the feasibility of CEL and
MMALE methods in geotechnical large deformation problems against SALE and clas-
sical Lagrangian methods. It was concluded that MMALE method could be considered
as a promising candidate for solving complex large deformation problems. A detailed
description of CEL and MMALE methods are presented in Chapter 3and Appendix
B.
14 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
2.2.3 History of CEL and MMALE Applications
During the last decade, CEL [Qiu and Grabe,2012;Qiu et al.,2011;Wang et al.,
2015] and other approaches [Aubram et al.,2015;Aubram,2014;Konkol,2013] became
popular for solving geotechnical problems. Qiu et al. [2009], for example, applied CEL
to a strip footing, an anchor plate problem, and other applications. They conclude
that the CEL method is most suitable for the geotechnical applications involving very
large soil deformations.
CEL has been extensively evaluated in the context of geotechnical problems. For
example, a study done by Wang et al. [2015] compared the performance of CEL with
other numerical methods for large deformation geotechnical problems. Another study
thoroughly evaluated the CEL compatibility with a complex soil material model and
the results were compared to an experimental test [Bakroon et al.,2018b].
There are various applications of CEL in literature. One of the initial application of
CEL in geomechanics can be found in work done by Qiu et al. [2011] where three
numerical benchmarks were used to assess CEL in treating large deformations. It was
argued that CEL is well suited for large geotechnical problems. Later, more studies
were carried out in applying CEL to geotechnical problems [Qiu et al.,2009;Hilber and
Hughes,1978;Sun et al.,2000]. A comprehensive and thorough study was conducted
by Wang et al. [2015] concerning three different numerical approaches, including CEL.
They argued that CEL can be used as one of the innovative methods in treating large
deformation problems.
On the other hand, application of ALE to geotechnical problems is rather new and
mostly uses a simplified mesh formulation (so-called simplified or single-material ALE).
Moreover, current ALE research is focused on rather technical aspects of the method
[Barlow et al.,2016;Aubram,2013;Aubram et al.,2017], hence further studies are
required concerned with its application in geotechnical engineering. A recent work
done by Aubram et al. [2017] compares simplified ALE with the classical FEM using
a benchmark case where an analytical solution is available. They conclude that ALE
provides more accurate and stable results when applied to large deformation geotech-
nical problems. In the following section, two benchmarks are used to evaluate the CEL
and MMALE performance against classical Lagrangian methods.
2.2.4 Evaluation of CEL and MMALE performance using two
benchmarks
The performance of two advanced numerical methods using Multi-Material Arbitrary
Lagrangian-Eulerian (MMALE) and Coupled Eulerian-Lagrangian (CEL) formulations
is studied. The evaluation is based on two large deformation benchmark cases which
classical pure Lagrangian methods cannot model. Applications of CEL in the literature
have proven its efficiency and robustness in modeling large deformation geotechnical
problems. MMALE is an enhanced version of CEL in which the computational mesh
can be rezoned in an arbitrary way so that mesh nodes are concentrated in areas of
interest. This form of solution adaptivity provides more data in regions undergoing
large deformations compared to the fixed mesh in CEL methods. MMALE has gained
2.2. ELEMENT FORMULATION: LAGRANGIAN, CEL, AND MMALE 15
popularity in the field of fluid dynamics. In this section, the applicability of MMALE to
geomechanical problems is investigated with regard to accuracy and robustness. Two
geomechanical problems, pipeline displacement and sand column collapse, are analyzed
for this purpose. It can be concluded that MMALE handles such large deformation
problems more efficiently than CEL.
Pipepiles
Pipelines are one of the key components in offshore industrial projects. Pipes are ini-
tially placed on the seabed. After installation, the pipe penetrates the soil due to its
own weight. Moreover, the varying thermal effects of the pipe induces a lateral force
resulting e.g. in a lateral movement. Calculating combined horizontal and vertical
resistance of the soil against pipe movement can lead to a more optimized and safe
design. There is a large amount of literature concerned with various aspects of em-
bedded pipeline behavior in seabed in the field of theoretical, physical, and numerical
modeling [Aubram,2013;Merifield et al.,2009].
The vertical and horizontal resistance force is usually calculated based on bearing
capacity theory for a shallow embedded footing [Skempton,1951]. The equations are
modified to take the problem conditions into account such as soil heaving, buoyancy,
shape of pipe etc.
The schematic view of the problem is illustrated in Fig. 2.6. After a pipe with diameter
Dpenetrates to a depth w, the soil starts to heave to a width of Bheave with the height
of Hheave. This increases the lateral resistance of the soil which can be taken into
consideration to reach an optimum design.
The analytical equations calculating the horizontal and vertical resistance forces are
presented in Equations 2.2.12 and 2.2.13 (cf. [Merifield et al.,2009]). These equations
consist of two terms. The first term is attributed to the undrained soil strength while
the second term considers the self-weight effects of the soil.
V
D=NcV su+NswV γw(2.2.12)
H
D=NcH su+NswH γw(2.2.13)
NcV =aˆwb=5.3ˆw0.25 (2.2.14)
NcH =cˆwd=2.7ˆw0.64 (2.2.15)
NswV =1
2ˆw1+1
λ×
⎡
⎢
⎣
sin−14ˆw(1 −ˆw)
2−(1 −2ˆw)ˆw(1 −ˆw)⎤
⎥
⎦
(2.2.16)
NswH =ˆw
2+4
λ×⎡
⎢
⎣
sin−14ˆw(1 −ˆw)
2ˆw(1 −ˆw)−(1 −2ˆw)⎤
⎥
⎦(2.2.17)
16 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
Idealized heave
γ‘, S
D‘
B* = λD‘ /2
w = w
D
H
V
Pipe
D
P
W
h
Soil
B
w
A
A
h*
WIP
WIP
heave
heave
heave
heave
heave
heave
sub
contact
u
Figure 2.6: Schematic view of the pipeline displacement problem [Merifield et al.,2009]
D=0.6m
Filled soil elements
Void elements
Pipe
V
H
1m
8m
R=4m
2.35m
2.35m
1.2m
Figure 2.7: Mesh configuration of the model
where V= vertical resistance force, H= horizontal resistance force; ˆw= normalized
penetration depth ( ˆw=w/D); λ= parameter approximating the amount of heaving;
γ= submerged unit weight of soil; D= pipe diameter; NcV and NcH = coefficient
for vertical and horizontal strength, respectively; and NswV and NswH = coefficient for
vertical and horizontal strength for self-weight term, respectively (see Fig. 2.7).
As suggested by Merifield et al. [2009] the values of 3 and 1.6 are used for λin vertical
and horizontal force, respectively, as well as the corresponding coefficients a,b,c, and
din Equations 3 and 4.
In this problem, a pipe is placed above the soil which represents the seabed. The soil
is considered to be fully saturated with average shear strength, su=1.5 kPa in overall
depth. The submerged unit weight of the soil is γ=6kN/m3. The Young’s modulus
Eof the soil is calculated by E/su= 500. The elastoplastic material model with Tresca
yield criterion was employed. Due to significant strength difference between pipe and
soil, the pipe was considered as a rigid part. No friction was considered between pipe
and soil (smooth surface).
The pipe is moved in vertical direction until depth of 1.0Dto simulate the embedment
of the pipe. The velocity rate of 0.01 m/sec was assigned to ensure quasistatic loading
conditions. The vertical resistance force was calculated and compared to analytical
equations.
To compare the horizontal resistance of the pipe with analytical solution, ten models
were developed, where the pipe was displaced horizontally at different embedment
2.2. ELEMENT FORMULATION: LAGRANGIAN, CEL, AND MMALE 17
Large mesh
deformations
Inaccurate heaving
behavior
Lagrangian MMALE
Void elements
Pipe
Eulerian filled elementsLagrangian elements
Figure 2.8: Mesh distortion using Lagrangian method (left) and MMALE (right)
depths from 0.1Dto 1.0Dwith intervals of 0.1D. No vertical displacement was allowed
at this stage.
MMALE and CEL methods are used for numerical simulation. The mesh configuration
of the model is shown in Fig. 2.7. The minimum element size was 0.04 m which
increased at the boundaries to 0.15 m resulting in total number of 10,900 elements. A
void layer of 1 m height was defined above the soil layer to allow soil heaving simulation.
The model was considered as a 2D problem, however, 3D solid elements were used since
no 2D Eulerian elements are available. The thickness in normal direction is considered
as one element.
2.2.5 Results and discussion
A Lagrangian model was first adopted. The quality of mesh elements is reduced drasti-
cally after significant penetration as shown in Fig. 2.8. Hence, the results were consid-
ered unreliable. This emphasizes on a need for more advanced models for this problem.
Subsequently, MMALE and CEL methods were used for the simulation. Both methods
converged to solution after 1.0Dpenetration. Vertical and horizontal resistance forces
of CEL and MMALE are checked against analytical equations in Fig. 2.9 and Fig. 2.10,
respectively. The vertical resistance force for both CEL and MMALE are in a good
agreement with analytical equations. In Fig. 2.9, it is shown that both methods provide
acceptable results. The CEL method gives stiffer behavior than MMALE. After about
0.2m penetration, both results from the MMALE and CEL with coarse mesh, start
experiencing oscillation, while the MMALE with fine-mesh is smoother. It can be
argued that due to coarser mesh, less coupling nodes area available which causes the
oscillation. Nevertheless, this oscillation does not cause significant errors.
For calculated horizontal force in Fig. 2.10, both methods give higher values at low
penetration depths, but lower values at higher depths in comparison to analytical
equation. This can be attributed to complex mechanism of heaving and its effect on
the resistance force mentioned in Merifield et al. [2009].
By using the same model configuration, MMALE captures a better mesh resolution
for areas of interest than CEL. This argument is supported by Fig. 2.11 where CEL
18 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
0
1
2
3
4
5
6
7
8
9
0 0.2 0.4 0.6 0.8 1
Vertical resistance, H/D (kPa)
Normalized vertical displacement, w/D
CEL−Fine mesh
MMALE−Fine mesh
MMALE−Coarse mesh
Analytical solution, (Merifield et al., 2009)
Figure 2.9: Comparison of vertical resistance from different numerical methods and
analytical result
0
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
Horizontal resistance, H/D (kPa)
Normalized horizontal displacement, w/D
CEL
MMALE
Analytical solution, (Merifield et al., 2009)
Figure 2.10: Comparison of horizontal resistance from different numerical methods and
analytical result
2.2. ELEMENT FORMULATION: LAGRANGIAN, CEL, AND MMALE 19
More surface smoothing
CEL MMALE
MMALECEL
Mesh rezoning
Figure 2.11: Comparison of MMALE and CEL interface reconstruction
and MMALE interfaces are compared together using the initial mesh element size.
MMALE provides a smoother interface than CEL. Hence, it is possible to achieve an
acceptable accuracy with increasing the element size in MMALE.
The velocity vectors of soil after application of vertical displacement are shown in
Fig. 2.12a and Fig. 2.12b. The velocity vectors of the horizontal displacement after
reaching 0.5Dpenetration are shown in Fig. 2.12c and Fig. 2.12d. The arrow at center
of the pipe shows its movement direction.
The velocity field shows clearly which part of the soil regime undergoes significant
movement. This movement is due to the shear band mechanism appearing due to
excessive pipe movement and soil softening.
In Fig. 2.12a and Fig. 2.12b, the soil flow regime is distinguished by dense arrows. This
is similar to failure mechanism of a strip footing in general soil mechanics theory. In
Fig. 2.12c and Fig. 2.12d, a new shear zone is developed. At both displacement modes
the velocity field is uniform which is a criterion for stability of the numerical methods.
A more realistic model has been developed to account for simultaneous displacement of
pipe in horizontal and vertical direction. Similar to previous model, the pipe is initially
placed above the soil. Then the pipe moves vertically with the rate of 0.01 m/s into
the soil until the depth of 0.5Dfor simulation of partial embedment. Subsequently, the
horizontal displacement was applied with the same rate of 0.01 m/s. During horizontal
movement, 60% of the obtained maximum vertical force in the last phase was main-
tained to model the self-weight of the pipe and its containing fluid. Vertical movement
is allowed during this stage.
The model was solved with both CEL and MMALE. To reduce the number of irrele-
vant affecting variables, the simulations were conducted under same configurations and
conditions on a conventional personal computer with 4-core CPU with 3.2 GHz.
Fig. 2.13 shows the final deformed shape for both CEL and MMALE. The mesh in
MMALE model is significantly concentrated around the pipe, which is also the inter-
ested area of study. In addition, due to mesh concentration, more nodes are available
which enhances coupling with Lagrangian elements leading to more accurate results.
20 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
(b)
(a)
(c)
(d)
Figure 2.12: Velocity vectors of sand movement during vertical pipe displacement of
a) 0.25Db) 0.5Dand horizontal pipe displacement of c) 0.25Dand e) 0.5D
2.2. ELEMENT FORMULATION: LAGRANGIAN, CEL, AND MMALE 21
CEL
MMALE
Figure 2.13: Final deformed shape of soil and computational mesh using MMALE and
CEL methods, more element concentration is observed in MMALE
Besides, more Eulerian elements at coupling interface reduce the possibility of leakage.
In Fig. 2.14 the velocity field of the soil is shown. The arrow in the middle of the
pipe shows the pipe movement direction. Compared to Fig. 2.15, more soil volume is
displaced. Results obtained from the model agrees well with similar tests available in
the literature (Dutta et al. 2015).
Fig. 2.15 shows a comparison of horizontal force for MMALE and CEL. Both results
converge to a similar value with negligible differences.
Furthermore, the calculation time for both CEL and MMALE is considered as a com-
parison criterion. As illustrated in Tab. 2.2, MMALE was about 35% faster than CEL.
Although MMALE has one more step in calculation process (e.g. remeshing step),
Table 2.2: Calculation time comparison for MMALE and CEL for pipeline displacement
problem
Numerical method Calculation time h:min:sec
CEL 02:38:43
MMALE 01:42:14
22 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
Figure 2.14: Final velocity vectors of sand movement after enforcement of both vertical
and horizontal pipe displacement
1.8
2
2.2
2.4
2.6
2.8
3
0 0.1 0.2 0.3 0.4 0.5
Horizontal resistance, H/D (kPa)
Normalized horizontal displacement, u/D
MMALE
CEL
Figure 2.15: Pipeline response during penetration and lateral displacement
2.2. ELEMENT FORMULATION: LAGRANGIAN, CEL, AND MMALE 23
di
i
h
h∞
∞x
d
Figure 2.16: Schematic view of the sand column problem
which increases calculation cost in comparison to CEL, it is not necessary to perform
it at each calculation step. Hence, the remeshing and remapping step can be performed
after several Lagrangian steps without affecting the results, which leads in decreasing
computational time.
Sand column collapse
Collapse of sand column has been extensively studied as an experimental test and
benchmark or numerical methods verification. The benchmark will be further used
in section 3.4.2. Conventionally, a sand specimen is deposited inside a container. As
shown in Fig. 2.16, at least a side of the container is released abruptly which allows
the sand to flow on the surface. Then, the corresponding parameters such as runout
distance, slope angle, etc. are studied; see [Lube et al.,2005] for further information.
Lube et al. [2005] carried out several experiments on two dimensional sand columns.
The results of this experiment are used as a benchmark case for numerical assessment
of MMALE and CEL. The evaluation parameters are the runout distance and sand
column height.
The configuration of the numerical model which is obtained from the experiment is
shown in Fig. 2.17. The column of sand is at rest until one side of the container is
removed to let the soil flow by its own weight. The initial width and height of the soil
column is di= 0.0905 m and hi= 0.635 m leading to height to width aspect ratio a=
7. In the experiment, the depth of the soil in direction normal to flow is 0.2 m. It was
reported that in this direction no relative difference in runout distance was observed.
Therefore, it is possible to model the experiment in two dimensions. However, due
to lack of 2D Eulerian elements in the commercial hydrocodes used, a 3D model with
depth of 1 m was developed consisting of hexahedral elements with 1-point integration.
The Mohr-Coulomb material model is used, and the surface friction angle is assumed
equal to the internal friction angle of the sand. Material properties are summarized in
24 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
63.5 cm 65 cm
120 cm
9.5 cm
Rigid body with friction of 0.6
Void
Elements Filled with soil
Figure 2.17: Initial configuration of the sand column collapse model
Tab. 2.3 based on a research by Solowski and Sloan [2013]. It should be noticed that
the density was assumed by the authors as an average value of sand density (Tab. 2.3).
In reasonable range of sand density, the effect of this parameter was observed to be
negligible.
For both MMALE and CEL, a void region should be defined to allow the soil to flow
in this region after the collapse has started. The gravity acceleration is taken as 9.806
m/s2. The total calculation time of the problem is 2 seconds. Rigid parts are employed
to model the container and the flowing surface. The container is assumed smooth and
frictionless. The gate is released after in-situ stresses are initialized.
Again, the problem was first modeled using the classical Lagrangian approach. In
Fig. 2.18, the deformed shape of sand clearly shows the inability of the method for
such large material deformations. At about 30% of the calculation time, the mesh
quality is significantly reduced. Consequently, the time step size decreased drastically.
Even if the termination time was reached, the resulting mesh size would have made
the results unreliable due to excessive mesh distortion. In contrast to the Lagrangian
method, both CEL and MMALE simulations reached a converged solution. This is
due to the implemented advection technique which enables the calculation of sand
motion independently of mesh deformation. In Fig. 2.19, the final soil shape is shown.
Mesh element size was initially taken as 15 mm for both MMALE and CEL. Despite
convergence, the initial CEL model results in a poorly resolved free surface of the
collapsing sand column. Hence, the mesh element size was refined to 7.5 mm. The sand
column shape after flow was also evaluated in terms of its measured runout distance
and height. This is shown in Fig. 2.20 for different times.
Owing to the remeshing feature in MMALE, a mesh density at the free surface com-
parable to that of the fixed CEL mesh could be reached using a coarser initial mesh.
Clearly, the mesh can be adapted to material deformations during remeshing, which
renders MMALE computationally less expensive than CEL at comparable accuracy.
The computation time for CEL and MMALE using the mesh of Fig. 2.20 are summa-
rized in Tab. 2.4.
2.2. ELEMENT FORMULATION: LAGRANGIAN, CEL, AND MMALE 25
Lagrangian solution
Highly distorted elements
Bad mesh distribution
Figure 2.18: Mesh deformation at an intermediate stage of sand column collapse using
a classical Lagrangian method
(b)
(a)
The mesh is fixed in space
CEL model
MMALE model
The mesh is remapped
Figure 2.19: CEL mesh (above) and MMALE mesh (below) at the end of calculation
Table 2.3: Mohr-Coulomb properties for the sand column collapse model [Solowski and
Sloan,2013]
Parameter Value
Density (kg/m3) 1,600
Friction Angle (◦)31
Dilatancy angle (◦)1
Cohesion (kPa) 0.01
Poisson ratio 0.3
Elastic Modulus (kPa) 840
26 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
experimental MMALE CEL
0
0
200
200
0
200 200
0
200 200 200
400
400
600
0 200 400 600
800
0 200 400
0 200 4000 200 4000 200
600 800
1000
mm mm
mm
mmmmmm
mm
mm
mm
mm
mm
mm
2.0 s
0.42 s 0.5 s
0.33 s0.25 stime = 0.17 s
Figure 2.20: Comparison of free surface at different time stations for MMALE and
CEL with experimental results
Table 2.4: Calculation time comparison for MMALE and CEL for sand column collapse
problem
Numerical method Calculation time h:min:sec
CEL 00:05:13
MMALE 00:03:24
2.3. CONSTITUTIVE EQUATION: HYPOPLASTICITY FRAMEWORK 27
2.3 Constitutive equation: Hypoplasticity frame-
work
2.3.1 Foreword
In addition to an advanced element formulation, an advanced soil material model is
required to predict the progressive behavior of the soil. The material model is imple-
mented into two codes, AbaqusR
and LS-DYNAR
. An Oedometer test and triaxial test
was simulated using the Hypoplastic material model which were compared with von
Wolffersdorff (1996) for validation and verification [24]. Afterward, two advanced ele-
ment formulations, CEL and MMALE, were used along with the Hypoplastic material
model to resolve large deformation problems. In such numerical approaches, the mesh
is not aligned with the material, which is a different scheme than what is conducted in
classical FEM. This scheme alleviates the huge mesh distortion issue in large deforma-
tion problems. Therefore, the study of compatibility and stability of the Hypoplastic
material model with these two finite element methods is evaluated.
In computational geomechanics, the numerical simulation of soil-structure-interaction
problems where the soil material undergoes large deformations has become an active
area of research [Aubram,2013;Wang et al.,2015]. Classical finite element meth-
ods (FEM) based on a Lagrangian formulation suffer from mesh elements distortion
which may deteriorate accuracy or even stop the solution at early stages of the cal-
culation. Novel techniques and advanced numerical methods have been developed to
resolve these issues associated with the Lagrangian approach. These methods have
been proven a powerful and reasonably accurate alternative to experimental and ana-
lytical solutions. The Arbitrary Lagrangian-Eulerian (ALE) and the Coupled Eulerian-
Lagrangian (CEL) are two of such methods.
Capturing the complex nonlinear mechanical behavior of granular materials, such as
sandy soils, is one of the key aspects in geotechnical analysis and design. The accuracy
of numerical simulations is in a large part influenced by the choice of the material model.
Moreover, geotechnical problems are often characterized by large deformations, hence
there is an increasing interest for combining advanced soil material models with suitable
and robust element formulations in order to reach a convergent and reliable solution.
Conventional approaches to constitutive modeling of soils are based on elastoplasticity.
By using this theory, a yield surface and plastic surface must be defined, which means
that in a specific range of admissible states the material behaves purely elastic. In
an elastic state, the induced strain is reversible by unloading, which is generally not
the case for granular soils. The class of constitutive equations based on the so-called
"hypoplasticity" concept, on the other hand, does not distinguish between elastic and
plastic states and have been proven more accurate in simulating the complex behavior
of granular materials under cyclic loading and over a wide range of stress and density
states with only one set of parameters for a specific granular material and incorporating
state parameters such as void ratio [Dijkstra et al.,2011;Bakroon et al.,2017a;Pucker
and Grabe,2012;Qiu et al.,2011]. Besides, by removing extra definitions regarding
yield and plastic surface, such models are easier to implement [Kolymbas,2012]. The
first hypoplastic constitutive model was proposed by Kolymbas [1977]. Since then,
28 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
various extensions have been developed. One of the most comprehensive hypoplastic
constitutive equations has been developed by Gudehus [1996] and calibrated by Bauer
[1996]. Later, von Wolffersdorff [1996] improved the mathematical formulation of this
model. Finally, Niemunis and Herle [1997] resolved the associated problems in strain
accumulation in cyclic loading by defining a new state variable called intergranular
strain. The hypoplastic model with intergranular strain is also used in this study. This
material model is extensively used in numerical simulations of geotechnical problems,
and different implementations in various finite element codes are available [Gudehus
et al.,2008].
2.3.2 Literature
Several studies have been done using advanced element formulations with hypoplastic
material models. Dijkstra et al. [2011] modelled pile installation in saturated soil
using ALE method. In his study, an elastic pile is fixed and the soil flows around
the pile. The computational model handled the large soil deformations induced by
pile installation and provided comparable results. Qiu et al. [2011] used a so-called
Coupled Eulerian-Lagrangian (CEL) method and the hypoplastic material model to
simulate displacement of a strip footing, pile installation, and ship grounding; the CEL
method can be considered as a variant of an MMALE method where the solution is
remapped onto the original, i.e. Eulerian mesh. The results were satisfactory for such
problems. Another study was conducted by Pucker and Grabe [2012] to study the
affecting parameters on the rotary pile installation. The numerical results were used
to explain the soil state after drilling which was measured at the site.
2.3.3 Theory
The hypoplastic material model used is the one developed by von Wolffersdorff [1996]
which is considered as the enhanced version of the constitutive model introduced by
Gudehus [1996] and Bauer [1996]. A further modification was applied by Niemunis
and Herle [1997] which will be described later in this section. The hypoplastic concept
employs a closed form expression relating the co-rotational Jaumann rate of effective
stress of the grain skeleton to the stretching, with no distinction between elastic and
plastic deformation. Some of the main assumptions are defined as follows [Niemunis
and Herle,1997]:
1. The mechanical behavior of the granular material is completely determined by
the effective stress, T, and void ratio, e.
2. Grains (soil particles) are assumed rigid and permanent during the whole process
(i.e. no crushing compression or abrasion of grains). Therefore, the deformation
is attributed to change in void ratio or rearrangement of grain contacts only.
3. Loading rate effects are not considered.
4. Homogeneity of the soil is maintained at homogenous boundary conditions (e.g.
no shear localization).
2.3. CONSTITUTIVE EQUATION: HYPOPLASTICITY FRAMEWORK 29
5. The void ratio, i.e. the ratio of the pore volume and solid volume, is constrained
by three limiting void ratios. Two upper and lower void ratios, eiand edcor-
responding to minimum and maximum density, respectively. The third limiting
parameter is the void ratio at critical state, ec.
6. A granular hardness parameter, hs, is defined to adapt the limiting void ratio
parameters based on the current mean pressure.
Hypoplasticity predicts the nonlinear and inelastic behavior of granular materials quite
well. The constitutive model can detect some of the main interesting soil properties
like dilatancy, that is, the increase void ratio due to shear loading. Accordingly, the
dependency on the void ratio of the soil allows for realistic simulation of compaction
processes. The novelty of hypoplastic constitutive model is that the soil behavior under
loading and unloading condition is defined by a single incrementally nonlinear equation,
unlike elastoplastic material models. This is done by considering the strain path and
the current void ratio. The hypoplastic constitutive equation takes the form [Bakroon
et al.,2018a]:
˙
T=L:D+N||D|| (2.3.1)
where ˙
Tdenotes the co-rotational Jaumann stress rate, Dis the strain/stretching rate
tensor, L(T,e) is a fourth-order tensor associated with the linear part of the behavior
and N(T,e) is a second-order tensor related to the nonlinear part of the behavior.
The accuracy and performance of hypoplasticity is mainly dependent on L(T,e) and
N(T,e) which are defined by Eq. (2.3.2) and (2.3.3):
L=fbfe
ˆ
T:ˆ
Ta2F
a2
I+ˆ
T:ˆ
T(2.3.2)
N=fdfbfe
ˆ
T:ˆ
Ta2F
aˆ
T+ˆ
T∗(2.3.3)
where
a=√3(3−sin ϕc)
2√2 sin ϕc
(2.3.4)
fb=ei0
ec0βhs
n
1+ei
ei−trT
hs1−n
×3+a2−a√3ei0−ed0
ec0−ed0α−1
(2.3.5)
fd=e−ed
ec−edα
(2.3.6)
fe=ec
eβ(2.3.7)
30 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
F=
1
8tan2ψ+2−tan2ψ
2+√2tan ψ cos 3θ−1
2√2tan ψ (2.3.8)
tan ψ=√3ˆ
T∗(2.3.9)
cos 3θ=−√6tr ˆ
T∗·ˆ
T∗·ˆ
T∗
ˆ
T∗:ˆ
T∗3/2(2.3.10)
ˆ
T=T
tr T(2.3.11)
ˆ
T∗=ˆ
T−1
3I(2.3.12)
ei
ei0
=ec
ec0
=ed
ed0
= exp −−trT
hsn(2.3.13)
Where fbis the barotropy factor and feand fdare the pycnotropy factors. The
barotropy factor relates the void ratio to mean pressure while the pycnotropy fac-
tors consider densification effects. The definition of other parameters appeared in Eq.
(2.3.4)-(2.3.13) are listed in Tab. 2.5.
The hypoplastic material model in the form of Eq. (2.3.1) accurately predicts granular
material behavior under monotonic loads and sufficiently large strains. However, under
cyclic loads excessive strain accumulation can be observed. By introducing a so-called
intergranular strain tensor together with a refined small strain stiffness formulation,
the reference model could be improved [Niemunis and Herle,1997]. The improved
constitutive model is rewritten as:
˙
T=M(T,e,δ):D(2.3.14)
Where Mis the fourth order tensor depending on the stress tensor, T, void ratio, e,
and the intergranular strain Δwhich represents the stiffness that is calculated from
both hypoplastic tensors defined by Eq. (2.3.2) and (2.3.3), L(T,e) and N(T,e). The
difference in this case, is the dependence of Mon two parameters, the normalized
intergranular strain value, ρ, and its direction, ˆ
δ, with the strain rate ˆ
δ:D.ρand ˆ
δ
are defined in Eq. (2.3.15) and Eq. (2.3.16), respectively.
ρ=(||δ||/R) (2.3.15)
ˆ
δ≡δ/||δ|| for δ =0
0for δ =0 (2.3.16)
2.3. CONSTITUTIVE EQUATION: HYPOPLASTICITY FRAMEWORK 31
Table 2.5: The required parameters of hypoplastic material model with intergranular
strain
Hypoplastic model parameters Intergranular strain parameters
Constant Description Constant Description
ϕc[◦] Critical friction angle RMaximum intergranular strain
hs[MPa] Granular hardness mRStiffness multiplication factor
nExponent
ed0Min. void ratio mTStiffness multiplication factor
ec0Critical void ratio
ei0Max. void ratio at zero pressure χSmoothing constant
αExponent βrSmoothing constant
βExponent
The general equation determining the stiffness value Mtakes the form:
M=[ρχmT+(1−ρχ)mR]L+ρχ(1 −mT)L:ˆ
δˆ
δ+ρχNˆ
δfor ˆ
δ:D>0
ρχ(mR−mT)L:ˆ
δˆ
δfor ˆ
δ:D≤0
(2.3.17)
Where χis a constant used for smoothing the weighting factor ρχ, and mTand mR
are additional constants which account for changes in loading direction upon unload-
ing/reloading.
2.3.4 Verification of the Hypoplastic constitutive equation mo-
del with one element test
The details regarding the introduction of an interface in case of LS-DYNAR
are de-
scribed in Appendix C.
To verify and validate the performance of the UMAT subroutine of the hypoplastic
model with intergranular strain, an oedometer test as well a triaxial test were simu-
lated using a single element. These basic tests were also simulated by Niemunis and
Herle [1997] and von Wolffersdorff [1996]. The implemented UMAT has the feature of
switching the intergranular strain on and off. This point makes it possible to compare
the implemented model with both references.
Oedometer test
An oedometer test is a common geotechnical test where the soil specimen is placed
inside a rigid container and loaded vertically. The lateral displacement of the specimen
is constrained by the container to simulate one dimensional compression. It is also
possible to apply cyclic loading during the test. In this case, the soil particles are
rearranged during repeated loading and unloading, leading to an accumulative com-
paction. Realistic prediction of accumulated compaction by numerical simulation is
not an easy task, but the results produced by the hypoplastic model with intergranular
32 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
Table 2.6: Hypoplastic parameters of Hochstetten sand for Oedometer test [Niemunis
and Herle,1997]
ϕc[◦]hs[MPa] ne
d0ec0ei0αβm
RmTRχβ
r
33 1000 0.25 0.55 0.95 1.05 0.25 1 0.5 2.0 1 ×10−46.0 0.5
strain are reasonable. The schematic figure of the oedometer test as well as its finite
element model equivalent is demonstrated in Fig. 2.21. The soil used for the simulation
is Hochstetten sand whose hypoplastic parameters are summarized in Tab. 2.6. The
initial void ratio of the sand is chosen as e0=0.695, and vertical stress is increased
monotonically until a maximum value of σ1= 1 MPa is reached. Subsequently, a cyclic
load varying between 0.5 MPa and 0.8 MPa is applied.
Fig. 2.22 shows the resulting void ratio vs. vertical stress curve obtained from the
compression test. The curve from the implemented UMAT is compared with the one
obtained from Niemunis and Herle [1997] in Fig. 2.22. It can be observed that the
implemented UMAT provides perfectly consistent results according to reference models.
Triaxial test
Another common geotechnical test is the triaxial test, where the soil specimen is placed
inside a cylindrical chamber filled with water to simulate confining stress present in-
situ. The concept is to apply different stresses on vertical and horizontal directions on
the soil. Initially, the specimen is subjected to equal pressure in all directions (consoli-
dation stage). Subsequently, the vertical stress is increased to evaluate specimen shear
strength. From triaxial test, fundamental soil parameters are obtained such as shear
strength parameters. The test can be performed under different drainage and sample
preparation conditions. However, the focus of this study is on dry sand. Fig. 2.23
shows the schematic of a triaxial test as well as its numerical model equivalent. The
material parameters for triaxial test are listed in Tab. 2.7. The only difference between
Tab. 2.7 and Tab. 2.7 is the βfactor which is taken here as 1.75.
The initial void ratio is taken as e0=0.695 with initial isotropic stress σ1=σ2=σ3=
100 kPa. and the maximum deviator stress is σ1= 300 kPa. Fig. 2.24 shows that the
strain vs. stress curve is in good correlation to the reference model by Niemunis and
Herle [1997].
Figure 2.21: (a) Schematic of the Oedometer test; (b) FE-mesh and boundary condi-
tions
2.3. CONSTITUTIVE EQUATION: HYPOPLASTICITY FRAMEWORK 33
Figure 2.22: Void ratio vs. vertical stress curve for oedometric compression test using
the Hypoplastic UMAT
Figure 2.23: (a) Schematic of the triaxial test (left), FE-mesh and boundary conditions
(right)
Table 2.7: Hypoplastic parameters of Hochstetten sand for triaxial test [Niemunis and
Herle,1997]
ϕc[◦]hs[MPa] ne
d0ec0ei0αβm
RmTRχβ
r
33 1000 0.25 0.55 0.95 1.05 0.25 1.75 5.0 2.0 1 ×10−46.0 0.5
Figure 2.24: Deviatoric stress vs. axial strain for Triaxial compression test using the
Hypoplastic UMAT
34 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
2.3.5 Application of the Hypoplastic constitutive equation in
Abaqus R
: Pile penetration
Two example applications of the hypoplastic material in conjunction with the implicit
Lagrange, explicit Lagrange and CEL methods are presented. The first example is a
strip footing problem which uses Tresca model to investigate and compare between
the previous numerical analysis methods. The second example simulates a single pile
penetration into a subsoil, this test shows that the user subroutine hypoplastic model
is acceptable for modelling granular material. By comparing the results, it is concluded
that the CEL method using hypoplastic soil material is suitable for large deformation
geotechnical problems.
In this contribution a user subroutine for granular soil material behavior is developed
based on hypoplasticity which was implemented in the AbaqusR
/explicit package.
Accordingly, the explicit user subroutine version is verified by comparing the results
with implicit version utilizing basic element tests (Oedometer and Triaxial tests).
Simulations of geotechnical problems often require fine mesh models and advanced
constitutive equations for the soil. However, the increase of mesh elements in con-
junction with non-linear soil models leads to a time/CPU consuming solution. This
calls for efficient numerical methods that often use explicit algorithms to advance the
solution in time. The available hypoplasticity subroutine is in a form that can be used
with implicit numerical methods. Therefore, one objective of the present research is to
reformulate this implicit subroutine to be applicable with the explicit methods.
The explicit user subroutine version is verified by comparing the results with implicit
version utilizing basic element tests (Oedometer and Triaxial tests). Three example
applications of the hypoplastic material in conjunction with the implicit Lagrange,
explicit Lagrange and CEL methods will be presented. The second example is a pile
penetration problem using hypoplastic soil material which investigated and compared
to experimental results, this test shows that the user subroutine hypoplastic model is
acceptable for modelling granular material. This model simulates the pile penetration
process into granular material using CEL method with regard to the hypoplastic ma-
terial model parameters shown in Tab. 2.8. Displacement control provides numerical
convenience, better stability, fewer iterations and represents physical reality [Arslan
and Sture,2008].
The schematic diagram of the problem shown in Fig. 2.25 shows a layer of void elements
which allows the material to flow after the heaving of the soil occurs. The CEL method
allows the flow of the material through the mesh elements without any distortion of
the mesh. In Fig. 2.25 illustrates the void ratio distribution along the pile shaft which
can be shown clearly the contraction along the pile shaft if formed as the typical
experimental results. A good correlation between the results of the CEL model and
the experiments carried out at TU-Berlin [Aubram,2013], see Fig. 2.26.
2.3. CONSTITUTIVE EQUATION: HYPOPLASTICITY FRAMEWORK 35
Table 2.8: Hypoplastic parameters for the used soil model.
ϕc[◦]hs[MPa] ne
d0ec0ei0e0αβm
RmTRχβ
r
31.5 76.5 0.29 0.48 0.78 0.9 0.546 0.13 1.0 5.0 2.0 1 ×10−46.0 0.5
Figure 2.25: Pile penetration model schematic diagram(left);Void ratio for CEL model
distributions at different penetration depths, z/D = 5.0 (Middle); z/D = 8.5 (Right)
Figure 2.26: Comparison of the measured and predicted load-displacement curves of
shallow penetration test PP-26-H[23]
36 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
Table 2.9: Equivalent Hypoplastic properties of the sand based on the Mohr-Coulomb
material model
ϕc[◦]hs[MPa] ne
d0ec0ei0e0αβm
RmTRχβ
r
33 10 0.25 0.55 0.95 1.05 0.695 0.25 1.5 5.0 2.0 1 ×10−42.0 0.5
2.3.6 Application of the Hypoplastic material model in LS-
DYNA R
: Sand column collapse
Problems in soil mechanics and geotechnical engineering are often characterized by
large deformations and complex material behavior. For example, the mechanical be-
havior of granular materials like sand is highly nonlinear due to the presence of an
evolving internal structure formed by the grains. The strength and stiffness is gen-
erally a function of the stress and density state and the loading history. While LS-
DYNAR
has proved to be among the most robust hydrocodes for modelling large
deformations and dynamic problems, it currently does not provide material models
capturing granular material behavior over a wide range of stress and density states
under monotonic and cyclic loads with only one set of parameters for a specific gran-
ular material and incorporating state parameters such as void ratio. Therefore, an
advanced soil mechanical constitutive equation based on the hypoplasticity framework
has been implemented in LS-DYNAR
using the UMAT interface in conjunction with
the *USER_DEFINED_MATERIAL_MODEL keyword. The implemented hypoplas-
tic model defines evolution equations for the effective stress, void ratio, and the so-called
intergranular strain tensor suitable for simulating cyclic loading effects. Previously, the
model has been successfully implemented in other hydrocodes. However, in comparison
to other hydrocodes, LS-DYNAR
employs more advanced element technology, such as
the Multi-Material Arbitrary Lagrangian Eulerian (MMALE) formulation, to simulate
large deformation problems. The motivation of this work was to combine the hypoplas-
tic material model with MMALE to study more realistically problems in soil mechanics
and geotechnical engineering. The implementation is validated by comparing to results
of oedometer and triaxial laboratory tests. It is shown that the UMAT is able to sim-
ulate soil behavior under cyclic loading and undrained conditions. The combination of
the UMAT with MMALE is evaluated using the example of a sand column collapse.
The numerical results are in good agreement with experimental test results and can be
seen as a promising starting point for further applications.
In this study, the implementation of the hypoplastic material model into LS-DYNAR
is verified and validated using basic benchmarks, oedometer and triaxial element tests.
Subsequently, a real-size experiment is simulated using the implemented material model
in conjunction with the available MMALE technology.
A numerical model similar to the one described in 2.2.5 (Fig. 2.16 and Fig. 2.17) is used
where the soil is simulated with hypoplastic material model. The hypoplastic parame-
ters are estimated to fit the Mohr-Coulomb parameters using numerical simulations of
triaxial tests with same conditions described in the previous example. The equivalent
hypoplastic properties of the soil are shown in Tab. 2.9. The problem is simulated by
both Mohr-Coulomb material and the hypoplastic UMAT.
2.4. CONCLUSION 37
Figure 2.27: Final numerical result of the sand shape using the Hypoplastic UMAT
In Fig. 2.27, the final shape of the sand column after collapse is shown. It is observed
that an advanced material model in conjunction with an advanced element formulation
gives a more realistic behavior. In Fig. 2.28, the runout distances obtained from Mohr-
Coulomb and hypoplastic models are compared with the experimental results. Both
the runout distances and the heights of the sand column predicted by hypoplasticity
in different time stamps are closer to the experimental measurements. In addition,
the sand column shape during the collapse has a more realistic form compared to the
experimental results.
In the UMAT implementation of the hypoplastic model, the void ratio is a history
variable which is stored at each time step. In Fig. 2.29, the void ratio variation is
shown after reaching a stable state. By noting the value of 0.695 as the initial void
ratio, the left bottom corner compacted due to overburden weight (the green region),
while the sliding layers experienced loosening (the red region) as expected. The sudden
change in void ratio value at interface region is attributed to the averaged void ratio
value for mixed elements of soil and void. Since void ratio of the void elements are
zero, the resulting averaged void ratio value becomes smaller than their neighboring
elements. Therefore, the values obtained in the interface should be evaluated with care.
2.4 Conclusion
In the first part of the chapter, the performance of two numerical analysis approaches
tailored for large deformation problems, CEL and MMALE, was evaluated using two
example applications. These examples cannot be solved using classical Lagrangian
methods since the latter stop at early stages or provide unacceptable results. For the
first problem addressing lateral pipeline displacement an analytical solution is avail-
able. On the other hand, for the second problem of sand column collapse, experimental
measurement is available. Therefore, it was possible to thoroughly investigate both
methods and compare their results. Both methods provided comparable results within
acceptable calculation time which proves their efficiency and robustness. One of the
major differences between MMALE and CEL lies in the remeshing resp. rezoning step.
In CEL the mesh is rezoned to its original configuration, while in MMALE the mesh
38 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
Experiment (Lube et al. 2005)
0
200 200
400
0
200
0
200
0.33 s0.25 stime=0.17s
0.50 s
0.42 s
2.0 s
mm
mm
mm
mm
200
mm
200
mm
0 200 400 600
0 200
0 200mm
400
0 200 400mm
600mm
800
0 200 400
0 200 400mm
600 800mm
1000mm
Hypoplastic UMAT
Mohr-Coulomb
Figure 2.28: Comparison of numerical results of sand column shape and distance with
experimental measurements
0.611 _
0.680 _
0.750 _
0.820 _
0.890 _
0.471 _
0.541 _
Figure 2.29: Void ratio for the Hypoplastic UMAT soil material
2.4. CONCLUSION 39
is rezoned to an arbitrary mesh, including the Eulerian (fixed) or Lagrangian mesh as
limit cases. The utilized rezoning technique in MMALE has several advantages. At
the same mesh size, MMALE interface resolution is generally higher in comparison
to CEL. Moreover, an MMALE mesh provides a natural form of solution adaptivity,
meaning that mesh density and resolution is increased in areas of interest. On the
other hand, it is possible to use coarser meshes in MMALE simulations than in CEL
simulations at comparable accuracy in order to reduce calculation times. Addition-
ally, for problems with structural (Lagrangian) parts, MMALE provides more coupling
nodes which increases the robustness of the model and decreases the problem of mate-
rial leakage in CEL methods. The findings highlight that the Multi-Material Arbitrary
Lagrangian-Eulerian method is suitable for simulation of large deformations, and it can
be considered as a promising tool for modelling more complex geotechnical problems.
In the second part of the chapter, the hypoplastic material model has been imple-
mented to two hydrocodes AbaqusR
and LS-DYNAR
capture the nonlinearities and
special characteristics of mechanical soil behavior. The implementation of the UMAT
is verified using two principal single-element numerical benchmarks: cyclic oedometer
test and cyclic triaxial test. The results obtained are identical to those reported in the
literature [Niemunis and Herle,1997].
Afterward, the hypoplastic model implementation was tested in conjunction with the
advanced MMALE element formulation in form of simulation of two large deformation
problems.
In case of AbaqusR
, a pile penetration problem was modelled. The load-displacement
curve was in a good agreement with previously simulated resutls. The compaction and
dilatancy can be visualized clearly in the pile penetration problem.
A sand column collapse experiment is simulated to evaluate the performance of the
hypoplastic model with the advanced MMALE element formulation. By comparing
the deformed shape and the runout distances at several time stations, it was observed
that the implemented UMAT performs well in capturing complex soil behavior.
By considering the obtained results and its advantages, the implemented Hypoplastic
UMAT can be considered as a promising material model for realistic simulations of
complex nonlinear soil behavior under large deformations. It is concluded that the
hypoplastic material model can simulate the special behaviour of the soil, for example
dilatancy and contactancy.
40 CHAPTER 2. MMALE METHOD AND HYPOPLASTIC MATERIAL MODEL
Chapter 3
Investigation of mesh improvement
in multi-material ALE formulations
using geotechnical benchmark
problems
This chapter is the accepted version of the following publication:
Bakroon, M., Daryaei, R., Aubram, D., and Rackwitz, F. (2020). Investigation of mesh
improvement in multi-material ALE formulations using geotechnical benchmark prob-
lems. International Journal of Geomechanics, https://doi.org/10.1061/(ASCE)GM.1943-
5622.0001723
c
2020. This accepted manuscript is made available under the CC-BY-NC-ND 4.0
license. license http://creativecommons.org/licenses/by-nc-nd/4.0/
Abstract
Two of the mesh-based numerical approaches suitable for geotechnical large deforma-
tion problems, the multi-material ALE (MMALE) and the Coupled Eulerian-Lagrangi-
an (CEL) methods are investigated. The remeshing step in MMALE is claimed to hold
advantages over CEL, but its effects on application problems are not studied in detail.
Hence, the possible capabilities and improvements of this step are studied in three large
deformation geotechnical problems with soil-structure interaction. The problems are
validated and verified using experimental and analytical solutions, respectively. By us-
ing the remeshing step in MMALE, a smoother material interface, lower remap related
errors, and better computation cost are achieved.
42 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
3.1 Introduction
Small deformation geotechnical problems can be adequately analyzed by using conven-
tional Lagrangian FEM. However, such an approach exhibits considerable shortcomings
when the soil undergoes significant deformation. Examples include pile penetration,
soil cutting, slope failures, and liquefaction events. Hence, efforts were made to de-
velop methods that simulate the numerical problems associated with large material
deformation.
There are various methods to handle such numerical problems which can be categorized
into two classes, point-based and mesh-based methods (here only methods derived from
continuum mechanics assumption are considered). Examples of point-based methods
are material point method (MPM) [Brackbill and Saltzman,1982] and smoothed par-
ticle hydrodynamics (SPH) [Gingold and Monaghan,1977], whereas classical FEM
(small-strain Lagrangian), Eulerian, ALE, and CEL methods are listed as mesh-based
methods [Aubram et al.,2015]. Concerning methods that rely on a computational
mesh, the most promising approaches include the Coupled Eulerian-Lagrangian (CEL)
method and the Arbitrary Lagrangian-Eulerian (ALE) method, which is chosen for this
study. The latter can be subdivided into Simplified ALE (SALE) and Multi-Material
ALE (MMALE) methods. These methods are popular in fluid dynamics yet not well-
known and extensively used in the context of geomechanics. Therefore, the motivation
of this paper is to evaluate the possible advantages of MMALE over CEL in case of
large deformation geotechnical problems.
Two categories of ALE are generally distinguished, based on a number of materials that
might be present in a single element (Fig. 3.1). Simplified ALE (SALE) approaches
resolve material boundaries (free surfaces or material interfaces) in a Lagrangian way
using edges and faces (in 3D) of the computational mesh. Therefore, each mesh element
is filled with only one material. Unlike SALE, MMALE allows multiple materials to be
defined in each element such that material boundaries can flow through the mesh. This
method reconstructs the interfaces between multiple materials, making it is suitable to
model more complicated and large deforming problem. Fig. 3.1 provides a schematic
comparing all the methods discussed in the present study.
There are various applications of CEL in literature concerned with large deformation
problems in geomechanics and geotechnical engineering, e.g., [Bakroon et al.,2018b;
Heins and Grabe,2017]. One of the earliest works is that done by Qiu et al. [2011],
where three numerical benchmarks were used to assess CEL. It was argued that CEL
is well suited for large geotechnical problems. Similar conclusions were drawn in a
comprehensive and thorough study conducted by Wang et al. [2015] concerning three
different numerical approaches, including CEL.
Concurrent to CEL studies, several works were done in applying the ALE method to
geotechnical problems. One of the earliest works in application of such similar methods
in geotechnical engineering is the remeshing and interpolation technique with small
strain, RITSS method developed by Hu and Randolph [1998b]. In this method, after 10-
20 steps of simple infinitesimal strain incremental analysis a rezoning step is performed.
Since then, this method is subjected to many improvements and applications such as
3.1. INTRODUCTION 43
Large Deformations
Grid-based Numerical methods
in this study
Lagrangian
method
Small
Deformations
Non-Lagrangian methods
Rezone to
arbitrary mesh
Lagrangian Step
Remap/Advection step
Rezone to
original mesh
MMALESALE CEL
Single material
elements
Multiple material
elements
Geotechnical Problem Type
Figure 3.1: Schematic diagram of different grid-based approaches comparing the
remeshing step effects on grid distortion level
44 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
inclusion of an h-adaptivity rezoning [Hu and Randolph,1998a] which is then used to
simulate pullout test [Song et al.,2008]. Similarly, in a series of works done at the
university of Newcastle such as [Nazem et al.,2008;Sabetamal et al.,2014], an ALE
method with coupled formulation was developed to simulate problems such as offshore
large deformation problems.
In a work done by Aubram et al. [2015], an advanced SALE formulation is implemented,
and its performance is evaluated by simulating shallow and pile penetration into the
sand. A good agreement between numerical results and experimental measurements
was observed.
On the other hand, Bakroon et al. [2018b] assessed the feasibility of SALE in large
geotechnical deformation problems. It was concluded that for extremely large problems,
the SALE exhibits shortcomings, unlike MMALE which can converge to a solution.
Therefore, MMALE was suggested to be considered as an alternative approach to
SALE for solving complex large deformation problems. Consequently, studies focused
on applying the MMALE to geotechnical problems.
The structure of this study is as follows. In section 3.2, details of the numerical imple-
mentation of CEL and MMALE algorithms such as operator splitting, remeshing, and
remapping steps, and soil-structure coupling are described. Section 3.4 presents three
numerical examples to investigate the performance of CEL and MMALE, including a
discussion of the results. Concluding remarks are provided in section 3.5.
3.2 Details of MMALE and CEL
The original CEL method was developed by Noh [1964]. In this method, the material
regions are treated as Eulerian, while the region boundaries are defined as polygons
which are then approximated by Lagrangian meshes overlapping the Eulerian mesh.
The Eulerian mesh is fixed throughout the analysis. Some commercial codes imple-
mented variants of the original CEL approach. In the particular CEL method used in
this study, a Lagrangian step is first conducted which solves the physics of the problem
by using a mesh which deforms with the material. In the case of the pure Lagrangian as
well as the Lagrangian step in SALE, MMALE, and CEL, employed in this work, the
updated Lagrangian (UL) [Belytschko et al.,2000;Hallquist,2017] is used. Concern-
ing the utilized objective stress rate, the Jaumann rate is used [LSTC,2015;Hallquist,
2017].
After performing the Lagrangian step, the mesh is rezoned to its initial configuration to
maintain mesh quality (rezoning/remeshing step). Subsequently, the solution is trans-
ported from the deformed mesh to the updated/original mesh (remapping/advection
step). This method is different than the CEL method developed by Noh [1964] where
the Eulerian solution is not divided into a rezone and remap step [Benson,1992a].
The Arbitrary Lagrangian-Eulerian (ALE) method has been developed by Hirt et al.
[1974a] and Trulio and Trigger [1961] to address the mesh distortion issue attributed
to classical Lagrangian approaches. In each ALE calculation cycle, similar to CEL, the
general strategy is to perform a three-step scheme consisting of a Lagrangian step, a
3.2. DETAILS OF MMALE AND CEL 45
Lagrangian Step
Mesh deforms
with material
Remap Step
MMALE elements
Filled elements
Rezoned
Elements
A
dvected Material
Interface reconstruction
Figure 3.2: Flowchart of the operator split scheme applied to the CEL and MMALE
calculation steps
remeshing (rezone) step, and a remapping step. After the Lagrangian step, the rezone
step relocates the nodes of the mesh in such a way that mesh distortion is reduced.
Unlike CEL, however, the updated mesh is not necessarily identical to the original
mesh but could be obtained through the application of a smoothing algorithm [Donea
et al.,2004]. Finally, the remapping step transfers the solution variables from the old
onto the new (rezoned) mesh.
The focus of this paper is to evaluate the remeshing step in MMALE and CEL as the
main distinguishing factor between these methods. The general solving strategy has
been discussed in section 3.1, which is also available in the literature [Benson,1992a].
Therefore, the remeshing step, as well as some other features of MMALE and CEL,
are described in this section.
3.2.1 Operator splitting
Generally spoken, operator splitting is a strategy to divide a complicated equation into
a sequence of simpler equations [Benson,1992a]. Operator splitting can be used to
solve the general Eulerian conservation equation:
∂ϕ
∂t +∇·Φ=S(3.2.1)
Where ϕis the field variable, Φis the flux function, and Sis the source term. This
equation can be solved whether in one step [Donea et al.,1982;Bayoumi and Gadala,
2004] or alternatively in multiple steps where the equation is broken up into a series of
less complicated equations, i.e., into a Lagrangian term (∂ϕ
∂t =S) and an Eulerian term
(∂ϕ
∂t +∇·Φ=0)[Benson,1992a]. The schematic view of operator splitting is drawn
in Fig. 3.2.
46 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
3.2.2 Remeshing step (Mesh smoothing algorithms)
The main difference between CEL and ALE (SALE and MMALE) emerges when one
compares the remeshing (rezoning) step in both methods. In case of remeshing step in
CEL, the new mesh is trivially the original mesh at the beginning of the calculation,
while in ALE, the remeshing step is performed by using mesh smoothing algorithms
that produce a new, less distorted mesh based on the deformed mesh of the Lagrangian
step. The new mesh is not necessarily the original mesh of CEL.
To define a robust rezoning algorithm, two criteria must be satisfied. First, the qual-
ity of the grid elements must be maintained. Second, the grid should be focused on
zones with a rapid variation of material flow to reduce computational errors, which is
referred to as the adaptivity control criterion. While these goals seem easy to achieve,
they expose a challenge in the derivation of a robust rezoning algorithm. If one consid-
ers quality maintenance as the only important factor, then accuracy in areas of high
variations will be lost, since pretty similar sizes will be assigned to rezoned grid ele-
ments. Algorithms developed merely on this criterion may be strongly dependent on
mesh quality, which may not provide a unique solution. Weighting each criterion is
therefore difficult, and it may be problem dependent [Knupp et al.,2002].
Rezoning/smoothing techniques can either change the nodal connectivity, such as h-
adaptivity where new elements are generated, or keep the nodal connectivity and only
relocate the nodes such as r-adaptivity method where the node position are relocated
to obtain a smoother mesh [Di et al.,2007].
The focus here is to study those smoothing methods where the nodal connectivities
are not changed. Such rezoning algorithms can be divided into different groups, each
having its advantages and drawbacks. Coordinate- or grid-based algorithms can be
applied to the gird, locally or globally. In local coordinate-based algorithms, the nodes
are moved based on local criteria [Donea et al.,2004;Benson,1989].
For example, based on neighboring element areas around the node, a ratio of mini-
mum to the maximum area as well as the maximum cosine value of the vertex angles
connecting this node to other nodes is calculated. By these two values, the movement
requirement of the node will be determined [Benson,1989].
The shortcoming of this method is that it is based on ad hoc quality measures, which
means this class of problems is only applicable to a specific group of problems. In
addition, there is no guarantee that the resulting mesh is unfolded [Knupp et al.,
2002].
An example of a global smoothing algorithm is the one developed by Brackbill and
Saltzman [1982], where they modified the Winslow algorithm [Winslow,1963]. Extra
terms were added to make the smoothing algorithm stronger. However, the coefficients
of such terms are assigned somewhat arbitrary and without a clear guide. In addition,
this method is independent of the Lagrangian grid, which makes the resulting mesh,
far from the Lagrangian mesh. To resolve this issue, an iterative approximate solution
is used. However, it is not guaranteed if the resulting grid is unfolded. Besides, there
is no theory to specify the number of iterations by the user [Knupp et al.,2002].
3.2. DETAILS OF MMALE AND CEL 47
K
α
α
αα
K
L
L
L
L
L
L
E
E
E
E
E
E
E
E
EE
E
E
L
L
L
L
E
EE
E
Figure 3.3: The initial arrangement of the arbitrary node K in a grid in 2D (left) and
3D (right) used to illustrate the smoothing/remeshing methods described in Eq. 3.3.6
There are numerous studies in remeshing techniques, but to the knowledge of the
authors, this step is the least developed aspect of ALE methods. A short description
of the three popular methods will be provided.
3.2.3 Volume-weighted smoothing
To better clarify the smoothing methods, Fig. 3.3 was drawn where the arbitrary node
K, is supposed to be rezoned (relocated). Variables subscripted with Greek letters refer
to element variables while subscripts with capital letters refer to local node numbering
within an element. Also, the letter A is an arbitrary letter corresponding to the nodes
of each element adjacent to node K. Therefore in case of the 2D mesh in Fig. 3.3,A
canbeLorE,orK.
In volume weighted smoothing, the new position of the node is determined by using
the volume of each neighboring element sharing that node. The method is illustrated
by Eq. (3.2.2) and (3.2.3).
First, the nodal coordinates of each element adjacent to node K, are averaged using
Eq. (3.2.2) to obtain the coordinate xA(the point is marked with red cross in Fig.
3.3). The parameter, N, corresponds to numbers of element nodes, which can be four
or eight for two- and three dimensions, respectively.
The new position of the node K, x∗
K, is then obtained by the volume-weighted averaging
as in Eq. (3.2.3) using the volume of each adjacent element, Vα, and the total number
of adjacent elements, nadj [Ghosh and Kikuchi,1991]:
xα=1
N
N
A=1
xA(3.2.2)
x∗
K=nadj
α=1 Vαxn
α
nadj
α=1 Vα
(3.2.3)
48 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
3.2.4 Laplacian or Simple average smoothing
In this method, the new position of the node K, x∗
K, will be simply defined based on the
averaged position of the Nnodes, xα, directly connected to K (nodes L in Fig. 3.3).
This means that four nodes are considered in two dimensional quadrilateral meshes
and six nodes in three dimensional hexahedral meshes. The new location of node K is
thus calculated by,
x∗
K=1
N
N
j=1
xj(3.2.4)
3.3 Equipotential smoothing
This method is more complicated than the previous methods and is intended to smooth
the whole mesh or a part of it globally. The equipotential method is based on the
solution of the Laplace equation (Eq. 3.3.1) associated with the logical, generally
curvilinear coordinates representing the grid lines in structured meshes [Winslow,1963].
The concept is to solve Eq. 3.3.1 for the Cartesian coordinates of the mesh lines, that is
x(ξi), (i=1,2,3) instead of the curvilinear coordinates ξ=(ξ1,ξ
2,ξ
3), resulting in Eq.
(3.3.2). In this method, all the element faces which share the node K are considered
in the calculation (Nodes L and E in Fig. 3.3). Therefore, in two dimensions, eight
nodes will be studied while in three dimensions, eighteen nodes will be studied (Fig.
3.3). For more information regarding the calculation process, the reader is advised to
see [Souli et al.,2000].
c∇2ξ= 0 (3.3.1)
γ1∂ξ1ξ1x+γ2∂ξ2ξ2x+γ3∂ξ3ξ3x+2β1∂ξ1ξ2x+2β2∂ξ1ξ3x+2β3∂ξ2ξ3x= 0 (3.3.2)
where
γi=∂ξix2
1+∂ξix2
2+∂ξix2
3i=1,2,3 (3.3.3)
β1=(∂ξ1x·∂ξ1x)(∂ξ3x·∂ξ3x)−(∂ξ1x·∂ξ2x)∂ξ3x2(3.3.4)
β2=(∂ξ2x·∂ξ1x)(∂ξ1x·∂ξ3x)−(∂ξ1x·∂ξ2x)∂ξ3x1(3.3.5)
β3=(∂ξ3x·∂ξ2x)(∂ξ1x·∂ξ2x)−(∂ξ3x·∂ξ1x)∂ξ2x2(3.3.6)
To investigate quantitatively the effectiveness of each smoothing method, a simple
numerical model was developed, as shown in Fig. 3.4. The model consists of nine
elements where the upper right node is subjected to a displacement in both horizontal
and vertical directions. The left lateral and the lower edge of the model is fixed. An
elastic material model is assumed. After displacement, the deformed mesh is evaluated
based on the so-called Jacobian distortion index ranging from 0 to 1. This index
describes the deviation of the element from its ideal rectangular form. A value close
3.3. EQUIPOTENTIAL SMOOTHING 49
to 1 indicates an element whose shape is close to its ideal form, while a value of 0
indicates a heavily distorted element [Plaxico et al.,2009]. In Fig. 3.4 the distortion
index is shown in percentage.
Without using any smoothing method, representing a purely Lagrangian mesh, the
deformation is significant in the upper right element and its three adjacent elements.
On the other hand, by using the smoothing methods, the distortion is decreased. In this
simple example, all smoothing methods provided acceptable results. Another model
was also developed where further displacement was applied. In the upper right element,
a non-convex element was obtained, and none of the smoothing methods could handle
the non-convex element and provided a folded mesh.
Indeed, the present example is too simple to study the performance of each smoothing
method thoroughly. The smoothing methods will be later discussed using a benchmark
model in section 3.3.
3.3.1 Remapping step
After generating a new grid, the solution variables have to be transferred to the new
mesh. There are several methods to remap the solution from the Lagrangian mesh
onto the new mesh [Margolin and Shashkov,2003;Benson,1992a]. Because the mesh
topology does not change in both ALE and Eulerian methods, the remap can be stated
as an advection problem which can be solved using conservative finite difference or finite
volume methods. In such advection algorithms, the difference between the reference
and the rezoned grid is interpreted as volume flux, that is, the change of element/cell
volume equals the sum of in- and outfluxes across the cell boundary. The updated
value of cell-centered solution variables is then determined by calculating the influx
and outflux of this variable in each cell using the information of the adjacent cells.
Conventionally, each advection algorithm is applied in one coordinate direction and
then extended to two or three dimensions using the operator-split technique [Benson,
1992a;Souli and Benson,2013].
Another group of remapping algorithms treats the intersection of the reference and
rezoned grid as polygons or polyhedra [Margolin and Shashkov,2003;Kucharik and
Shashkov,2012;Berndt et al.,2011].
One of the main differences between these two concepts is the way to treat mixed/multi-
material cells. When using advection algorithms, the mixed cells are treated differently
than the pure cell, while in intersection-based remapping, both pure and mixed cells are
treated alike. For more information about the remapping method based on polygons
and polyhedra, the reader is referred to [Margolin and Shashkov,2003;Kucharik and
Shashkov,2012;Berndt et al.,2011;Chazelle,1989,1994].
The current remapping algorithms used in geotechnical engineering are mostly based on
advection algorithms. A more detailed description regarding the most utilized advec-
tion algorithms, namely the first-order accurate donor cell and second-order accurate
Van Leer (MUSCL) scheme is available in the literature [Benson,1992a].
50 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
Initial mesh
Applied
Displacement
45o
Initial mesh
Deformed mesh
Mesh quality<90%
Volume
95 98 96
89
91
98
96 89 42
Equipotential
96 90 45
90
97
96
96 96 96
Laplacian
95 90 46
90
99
96
96 96 95
Lagrangian
24
72
97
91 61 72
97
9187
Figure 3.4: Comparison of different smoothing/remeshing algorithms based on the
achieved grid quality improvement (the numbers in the squares represents the Jacobian
distortion index in percent), the elements colored with red have an element quality less
than 90%
3.4. NUMERICAL EXAMPLES 51
3.3.2 Soil-structure coupling
Almost all problems in geotechnical engineering are characterized by soil-structure-
interaction and contact between different materials. Multi-material elements in CEL
or MMALE naturally handle contact without contact elements or algorithms [Benson
and Okazawa,2004].
These elements use the same velocity for all materials, which is a manifestation of
the no slip contact condition in mixture theory. However, in many soil-structure-
interaction problems, like pile penetration, interfacial slip, and frictional contact play
an important role. Moreover, in many situations, the soil undergoes large deformations
while deformation of the structure is moderate. Coupling between Lagrangian and non-
Lagrangian parts becomes necessary in such cases.
A penalty contact scheme is utilized in most codes owing to its simplicity and ro-
bustness. As a simple description, the penalty method applies springs between nodes
of Lagrangian and the Eulerian parts. These springs have seeds and anchors. The
seeds are attached to the Lagrangian nodes, while anchors are attached to the Eulerian
nodes. In practice, it is better to have more nodes in the Lagrangian part interface, to
ensure that at least one Eulerian node is tracked by one Lagrangian node. The spring
forces are calculated based on the relative penetration of master and slave parts, and
the calculated contact spring stiffness.
3.4 Numerical Examples
In this section, three application problems are presented which exhibit specific chal-
lenges in numerical simulation. Such classical examples are crucial for comparison of
different numerical methods since they have a reduced number of complexities. These
examples are modeled using MMALE and CEL, and the corresponding results are com-
pared. The comparison includes the calculation time, and the effect of mesh density
on it, accuracy in terms of leakage, interface, and energy loss, which will be described
during the section. Tab. 3.1 lists the comparison criteria and their specific purpose for
each numerical example discussed in this section.
For all simulations mentioned in this study, the calculations were carried out in the
commercial code, LS-DYNAR
, on a server with two 2.93GHz quad-core Intel CPU
X5570 processors and 48 GB of RAM.
A short description of the element technology and time stepping is provided for com-
pleteness. For SALE, 1-point ALE elements are used while for MMALE and CEL,
1-point reduced integration elements are used. Among the various smoothing meth-
ods, equipotential smoothing for the MMALE simulations is applied. This smoothing
algorithm is commonly used and provides more stable results compared to other meth-
ods. For the advection step, van Leer method is chosen over donor cell since it benefits
from second-order accuracy [Benson and Okazawa,2004].
52 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
Most CEL and ALE methods use explicit schemes to advance the solution in time. In
explicit methods, to maintain stability and acceptable accuracy, an appropriate time
step size must be assigned. The critical time step can be estimated by
Δte=Ls
c(3.4.1)
where Lsis the characteristic length of the element, and cis the sound speed in the
corresponding material. Determining a suitable time step size is crucial in geotechnical
applications. In MMALE and CEL methods, the maximum time step size is also
restricted by the advection algorithm: the distance of material transport should be less
than one element.
3.4.1 Strip footing
The strip footing problem is a well-known benchmark. In this problem, the soil under-
goes significant deformation, which challenges the classical Lagrangian methods.
Problem Description
In this problem, large soil deformations are induced by displacement-controlled pene-
tration of a rigid footing. The resulting pressure under the footing can be verified with
the analytical solution provided by Hill [1950] using plasticity theory. The footing is
initially placed above a container filled with soil. The problem is modeled as plane
strain, the lateral boundary nodes of the soil are fixed in the horizontal direction, and
the bottom nodes are fixed in the vertical direction. The footing is assumed rigid with
smooth (zero friction) sides and a perfectly rough (no slip) base.
Fig. 3.5 illustrates the initial and boundary conditions of the problem. The strip
footing and the soil dimensions are 2 ×1 m and 4 ×4 m, respectively. Only half of
the symmetric problem is modeled. The Tresca failure criterion is adopted according
to which plastic deformations occur when shear stresses reach the value c=10kPa,
the undrained shear strength of the soil. The Poissons ratio and the Youngs modulus
are assigned as ν=0.49 and E= 2980 kPa, respectively. For the ratio of footing base
over soil width = 0.5, the maximum punch pressure for this problem can be calculated
from qult =(2+π)cu[Hill,1950].
Numerical model consideration
The problem is analyzed using four different methods: Lagrangian, SALE, CEL, and
MMALE. The element size in the uniform mesh is 5 cm, with a total number of elements
of 3200. The initial mesh configuration is shown in Fig. 3.5. The footing in all models is
simulated as a rigid body. Frictionless penalty contact between the sides of the footing
and the soil is defined. To assess the dependency of results to mesh size, several models
with different element sizes were analyzed in another work [Bakroon et al.,2017b].
3.4. NUMERICAL EXAMPLES 53
Table 3.1: Comparison criteria and their purpose for the numerical examples
Application Criterion Purpose Ref. No.
Strip footing
(section 3.4.1)
•Induced pressure
under the footing
Quantitative comparison with an
analytical solution
Fig. 3.6
•Mesh distortion Qualitative comparison of mesh quality
maintenance
Fig. 3.7a
•Velocity field in
the soil
Qualitative comparison of the uniformity
in the velocity field
Fig. 3.7b
•Effective plastic
strain
Qualitative comparison according to
engineering judgment
Fig. 3.8
•Number of
Lagrangian cycles in
MMALE
Calculation time optimization without
deterioration in the results
Fig. 3.9
•Contact area Quantitative comparison with the ideal
contact area
Fig. 3.10
•Flux/Leakage Quantitative comparison with ideal zero
leakage
Fig. 3.11
•Relative
computation cost
Evaluation of remeshing and advection
effects
Fig. 3.12
•Mesh density Evaluation of the effects concerning the
increase in the calculation time
Fig. 3.12
•Energy loss Quantitative comparison with zero energy
loss
Fig. 3.13
Sand column
(section 3.4.2)
•Mesh distortion Qualitative comparison of mesh quality
maintenance
Fig. 3.15
•Interface
reconstruction
Qualitative comparison of improvement in
interface reconstruction
Fig. 3.16
•runout distance Quantitative comparison with
experimental measurement
Fig. 3.17
•Energy loss Quantitative comparison with zero energy
loss
Fig. 3.18
•Particle
trajectories
Quantitative comparison of soil particle
flow and evaluation of methods in
capturing complex material movement
Fig.
3.19a
•Calculation time Evaluation of the effect of remeshing in the
reduction of calculation time
Section
3.4.2
Soil cutting
(section 3.4.3)
•Mesh distortion Qualitative comparison of mesh quality
maintenance
Fig. 3.21
Fig. 3.22
•Induced vertical
and horizontal forces
on the blade
Quantitative comparison with an
analytical solution
Fig. 3.24
•Internal and
kinetic energy time
histories
Qualitative comparison of the convergence
of the results; verification of the steady
state condition
Fig. 3.25
•Calculation time Evaluation of the effect of remeshing in the
reduction of calculation time
Section
3.4.3
54 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
Lagrangian and SALE CEL and MMALE
1 m
1 m
Rigid
Body
4 m
2 m
Lagrangian
Elements 4 m
2 m
2 m
Eulerian
Elements
Void Elements
Rigid
Body
1 m
1 m
Figure 3.5: Numerical mesh configuration of the strip footing problem [Bakroon et al.,
2017b]
The models were solved using SALE method. Compared to the analytical solution, the
optimum mesh size for this problem was reported to be 5 cm. Therefore, 5 cm mesh
size is chosen for all the simulations of this problem.
Results
The methods are compared based on pressure results and computation time. A La-
grangian model is also developed to highlight the huge mesh distortion. Fig. 3.6 shows
the pressure results under the footing versus penetration depth for Lagrangian, SALE,
CEL, and MMALE compared to the analytical solution. By using the Tresca failure cri-
terion, the pressure should reach a constant value after small penetration. Considering
the accuracy of results, the Lagrangian and SALE solution differ from the analytical
result by approximately 15% and 10%, respectively. The observed inaccuracy in case of
the Lagrangian and SALE can be attributed to several points. The resulting pressure
from CEL and MMALE curves follow the same trend as the analytical result, unlike
the curves obtained from the Lagrangian and SALE method. It should be noted that
initial results included noises which are inevitable in the explicit formulation [Dassault
Systèmes,2016]. One may argue that the error is caused due to the element locking
[Heisserer et al.,2007].
3.4. NUMERICAL EXAMPLES 55
0
1
2
3
4
5
6
7
8
0 0.1 0.2 0.3 0.4 0.5
Normalized Punch Pressure p/c [−]
Penetration Depth [m]
Analytical solution π+2
Lagrangian
SALE
CEL
MMALE
Figure 3.6: Comparison of the punch pressure curves obtained from the Lagrangian,
SALE, CEL, and MMALE with the analytical solution
It should be noted that the reduced integration elements are used, which overrules the
possibility of element locking. Another possible reason may be the proximity of the
boundaries. Comparing the results obtained from the MMALE and CEL and their
accurate results, this argument cannot be valid for this problem. Considering the
MMALE and CEL results, the distorted element near the corner should be the cause
of this problem.
The resulting deformation for Lagrangian, SALE, CEL, and MMALE analysis is shown
in Fig. 3.7a. During the Lagrangian solution, the mesh is heavily distorted under the
corner of the footing and above. Nevertheless, the simulation continued until the ter-
mination time. By using SALE, the overall mesh distortion is alleviated. By using
different rezoning methods (e.g., volumetric, equipotential, etc.), different meshes are
obtained, but no change in pressure results are observed. In SALE, there are still
problems associated with areas around the footing corner where the material encoun-
ters significant deformation. These elements are still distorted even with the applied
rezoning step. In CEL and MMALE, however, since the material can flow through the
mesh, this issue is appropriately addressed. In CEL, the initial mesh is maintained
while in MMALE, a new arbitrary mesh is generated.
The instantaneous material velocity field at 0.5 m penetration depth is plotted in Fig.
3.7b. The results of the Lagrangian simulation show a sharp change of the velocity
distribution near the lateral boundary of the footing. This is somewhat reduced when
using SALE. When using CEL and MMALE, the velocity field is almost uniform in all
regions, indicating that the soil particles are moving smoothly counterclockwise from
the bottom of the footing to the side and then to the top.
In Fig. 3.8 the effective plastic strain after penetration is shown, which represents the
failure pattern of the soil. Despite the identical pressure results shown in Fig. 3.6,
the MMALE provides a clear failure line under the footing. However, CEL underesti-
mates the failure line by providing a discontinued line. This can be attributed to two
improvements done by MMALE. First, more elements are present in the failure area.
56 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
SALE - Equipotential MMALECELSALE - VolumetricLagrangian
SALE - Equipotential MMALECELSALE - VolumetricLagrangian
Figure 3.7: (a) Mesh distortion and (b) velocity field after 0.5 m of strip footing
penetration for different numerical methods.
Second, less advection is conducted in MMALE due to remeshing, which avoids loss in
accuracy caused by advection.
The performance of each method also is assessed with regard to computation time.
The Lagrangian method requires the least computation time among all methods, while
the SALE required the most, about three times more than the classical Lagrangian
method. The underlying reason is that in SALE two additional steps, remeshing and
remapping, are included in the calculation. Another affecting parameter is the distor-
tion of the elements in areas around the corner of the footing since the minimum time
step is controlled by those deformed elements. The simple idea behind the implemented
smoothing algorithms reduces mesh quality in such non-convex regions instead of im-
proving it, i.e., the smoothing algorithms become unstable. The CEL and MMALE
methods solve the problem much faster than SALE because mesh quality is easily
3.4. NUMERICAL EXAMPLES 57
Figure 3.8: The effective plastic strain after 0.5 m penetration for CEL (left) and
MMALE (right)
maintained. In other words, the minimum time step size did not change significantly
during the calculation, unlike SALE. Compared to calculation time obtained from CEL,
MMALE is about 40% faster in spite of an additional rezoning sub-step.
The resulting calculation times above for MMALE were based on the optimal set of
solution parameters. By using the default settings, a new mesh is generated, and
the solution is remapped after each Lagrangian step, which increases calculation time
significantly. In many situations, however, the magnitude of deformation obtained
after a time increment is small enough to perform several Lagrangian cycles before
executing one rezoning and remapping cycle without affecting results considerably. On
the other hand, if the number of Lagrangian cycles before a rezoning and remapping
cycle is increased, the magnitude of element distortion may reduce the size of the
critical time step, which results in more computation cost. Hence, to reach a minimum
computation time, an optimum number of Lagrangian cycles should be assigned. This
optimum number is problem dependent, and no predetermination can be made.
To optimize the computation cost for the strip footing example, six models are devel-
oped where the number of Lagrangian cycles before a remap and rezone cycle varies,
ranging from 1 to 30 Lagrangian cycles. To highlight the effect of a number of La-
grangian cycles on calculation time, the mesh size was reduced to 2.5 cm, resulting
in 12800 elements. The corresponding calculation times in minutes are drawn in Fig.
3.9. With the default configuration of MMALE (1 Lagrangian cycle per each rezone
and remap cycle), the computation cost is about 70 minutes while assigning 10-20 La-
grangian cycles; it is reduced by 70%. For a large number of Lagrangian cycles, on the
other hand, reduction of the critical time step through mesh distortion becomes more
pronounced, hence calculation time increases.
58 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
0
10
20
30
40
50
60
70
80
0 5 10 15 20 25 30
Calculation Time (minutes)
Number of lagrangian cycles each remeshing step
Trial runs
Trendline
Figure 3.9: MMALE time optimization achieved by changing the number of Lagrangian
cycles in strip footing problem with 2.5-cm mesh element size
In this example, by changing the number of Lagrangian cycles, up to 5% change in
pressure results was observed. However, for each problem, the accuracy of the results
should be checked since they may be affected by a number of Lagrangian cycles.
To investigate this point further, the calculated contact area of the pile with the soil is
shown in Fig. 3.10. In penalty contact method, the contact force is calculated based
on the force required to avoid the penetration of the two distinct parts. Generally, this
constraint is not adequately maintained and one part penetrates or leaks inside the
other part. In the case of excessive leakage, the contact force will not be accurately
computed. To quantitatively investigate this matter, the parameter contact area is
used. Theoretically, the value of the contact area should be maintained as of what
is calculated at the beginning of the simulation since during the simulation, only the
bottom side of the footing is in contact. If this value is increased, it means that leakage
has occurred and some of the elements in the second row of the footing has come into
contact. In the case of CEL, an increase of 20% in the contact area is observed. On the
other hand, by increasing the number of Lagrangian steps to 50, a significant leakage
occurs. Nevertheless, values below this number are providing an acceptable range of
leakage. This criterion can be hence used as a limiting factor for a proper number of
Lagrangian steps.
In addition, one can see the amount of leakage using a parameter referred to as flux,
which indicates the volume of material passed through the Lagrangian part, in this
case, the footing. A high value of flux indicates that a significant volume of material
has passed through the Lagrangian part, and therefore, the errors attributed to leakage
are significant. This introduces inaccuracies in the simulation. The computed value of
flux is shown in Fig. 3.11 for both MMALE and CEL. As the simulation continues,
the cumulated volume leaked through the Lagrangian footing increases with a faster
rate for CEL, which indicates a possibly less accurate result for this method.
The effect of mesh size on computation cost for MMALE and CEL is illustrated in
Fig. 3.12 for various cases where the mesh is refined up to 8 times. In addition,
3.4. NUMERICAL EXAMPLES 59
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1
Normalized contact area A/Ai
Normalized time T/Ti
CEL
MMALE 1−Lag. step
MMALE 10−Lag. step
MMALE 25−Lag. step
MMALE 50−Lag. step
Ideal contact area
Figure 3.10: Change in the normalized contact area during the simulation as a criterion
to investigate leakage
0.0
0.5
1.0
1.5
0 1 2 3 4 5
Flux (m3) x10-3
Time (sec)
Flux − MMALE
Flux − CEL
Figure 3.11: The amount of material passed through the Lagrangian part (flux/leakage)
during the simulation
60 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
0
20
40
60
80
100
1x1x1 2x2x1 4x4x1 8x8x1
Relative computaion cost %
Mesh density
CEL
Advection CEL
MMALE
Advection MMALE
Figure 3.12: Relative comparisons of computations cost between CEL and MMALE
with their corresponding advection (The results are normalized according to those of
CEL for each case)
the corresponding computation time of advection for each method is drawn. The
computation cost of CEL model is normalized to 1 for each case. The remaining
computation times (MMALE, advection in MMALE and CEL) are relatively drawn.
In all cases, the MMALE is about 20-40% faster. However, the trend is not linear,
i.e., in the case of one-fourth of the original size, the computational gain is the least.
In all cases of CEL, more than 40% of the time is spent on advection whereas in case
of MMALE it is less than about 30%. The underlying reason is the remeshing step,
which reduces the advection calculation by providing a mesh which follows the material
deformation pattern.
In the context of the numerical modeling, it is desired to keep the mesh as Lagrangian
as possible since the advection procedures introduce errors in the calculation, one of
which is the loss of kinetic energy during the advection. Typically, the momentum is
preferred over the kinetic energy to be conserved during the advection to maintain the
monotonicity of the solution. Maintaining both the momentum and kinetic energy is
not possible as it invalidates the monotonicity conditions. This leads to kinetic energy
loss during the simulation [Souli and Benson,2013].
To compare the performance of MMALE and CEL regarding this matter, the kinetic
energy and the loss of kinetic energy are shown in Fig. 3.13. The use of remeshing
results in a reduction of energy less to almost one-fourth of one calculated by CEL. In
the case of kinetic energy curves, the one obtained from CEL is oscillating, which may
indicate some instabilities in the method compared to the smooth curve of MMALE.
3.4.2 Sand column collapse
The collapse of the sand column on a rigid horizontal plane is an experimental test
which has various engineering applications such as determining the angle of repose. In
the context of geotechnical engineering, this problem can simply represent problems
3.4. NUMERICAL EXAMPLES 61
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
Normalized energy/energy loss
Time (sec)
Kinetic energy − MMALE
kinetic energy − CEL
Kinetic energy loss − MMALE
Kinetic energy loss − CEL
Figure 3.13: Normalized kinetic energy and kinetic energy loss during the simulation
for MMALE and CEL (the values are normalized with respect to the maximum value
of kinetic energy loss curve for CEL)
such as a landslide. In such tests, a column of sand is held in a container, and the
holding gate is suddenly released, allowing the sand to collapse by its own weight. For
further information regarding sand column theories and experiments see [Lube et al.,
2007;Staron and Hinch,2007;Doyle et al.,2007].
Problem Description
An experimental study performed by Lube et al. [2005] has been chosen as a reference
model to analyze the robustness of numerical methods. The experimental results of
runout distance and height of the sand column are compared to the obtained numerical
values. This problem has been extensively used for performance evaluation of numerical
methods such as the work done by Solowski and Sloan [2013].
In the experiment, the sand column is placed in a rectangular container. Then, one
side of the rectangular container is lifted fast to impose the 2D flow condition. The
initial width of the soil column is di= 0.0905 m with a height to the width aspect ratio
(height to width) of 7. The depth of the test soil in a direction normal to flow is 0.2
m. The friction of the horizontal plane (flowing surface) is equal to internal friction of
the sand.
Numerical model consideration
Fig. 3.14 shows the initial configuration of the numerical model. A uniform mesh
with an element size of 15 mm is used for the MMALE and CEL simulations. Purely
Lagrangian and SALE models were also developed for reasons of comparison. All the
models are three-dimensional, defining a slice with one element in a direction normal
to the plane. The CEL and MMALE models contain a void region defined to let
the soil material flow to these elements after the collapse starts, unlike SALE model
where no void elements are needed. Elements with 1-point integration are used, and
62 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
0.635 m
y
x
0.0905 m
P5
x
P4
x
P2
x
P1x
P3x
Rigid body with friction of 0.6
Void
Elements Filled with soil
Detail A
Detail A Model
1.65 m
1.2 m
Figure 3.14: Initial configuration of the numerical model for the case of CEL and
MMALE; the model size is 1.65x1.2 m but only the mesh of the detail A is shown
Mohr-Coulomb is chosen as the material model. Unfortunately, no data regarding the
properties of the test sand are reported by Lube et al. [2005].
Therefore, the soil properties are assumed as follows, the density, ρ= 1600 kg/m3,
the friction angle, ϕ=33
◦, the dilatancy angle of ψ= 0, the cohesion, c=0.01 kPa,
the Poissons ratio, ν=0.3, and the elastic modulus, E= 840 kPa,. The gravity
acceleration is 9.806 m/s2. The left boundary (wall of the container in the experiment)
was modeled using a frictionless rigid body part which was removed after the stresses
were initialized. The bottom surface was modeled by a rigid body part as well, having
tangential penalty friction equal to soil internal friction angle. The runout distance, as
well as the height of the sand column, were measured at different times and compared
to numerical results.
Results
To express the shortcomings of the classical simple based formulations against multi-
material based formulations, the problem was also simulated with SALE methods. In
this case, the mesh became highly distorted, and the calculation stopped. The mesh
clearly tracked the material particles, which can be justified by the concentration of
mesh elements as shown in Fig. 3.15. Due to local rezoning inside the material domain,
the mesh quality is to some extent uniform, but elements are severely stretched in the
horizontal direction due to the constraints imposed by the material boundary on the
remeshing capability. Therefore, after reaching approximately 15% of the calculation
time, the time step size decreased significantly so that the calculation could not be
continued.
In the case of both CEL and MMALE, simulation continued until the final runout
distance of the sand column was reached because of the advection technique, i.e., the
material can flow through the mesh. Fig. 3.16 shows that the remeshing capability of
MMALE concentrates the mesh in areas of interest, i.e., where the free surface of the
sand is located. The newly generated mesh takes the trend of the material movement
and deformation. Hence, the resulting interface is smooth, which is not the case when
3.4. NUMERICAL EXAMPLES 63
Highly distorted elements
Equally mesh distribution
Figure 3.15: Mesh deformation for Lagrangian simulation of sand column collapse
using the CEL method. The difference in concentration of mesh nodes also affects the
final shape of the collapsed sand column, i.e., the final interface of MMALE is curved,
whereas the interface of CEL is almost linear. The advantage of MMALE over CEL is
also highlighted in Fig. 3.16, where the volume fraction of sand is plotted. In elements
completely filled with sand, the volume fraction equals one, which is represented by
blue color. Void elements are drawn in red color, and those elements intersected by
the free surface are partially filled with sand, thus have a volume fraction between
zero and one. MMALE produces an almost smooth interface, whereas the interface
obtained with CEL has a stepped shape and is more diffusive. The diffusion thickness
of the interface obtained from CEL is about three times more than the one of MMALE.
The difference can be attributed to errors caused by remapping. In advection based
remapping methods, only principal directions (normal to element edges) are considered
for calculating the advection, neglecting the advection in diagonal directions. Through
the MMALE rezoning capability, the element directions are to some extent adjusted
to flow directions which results in less remapping errors due to diagonal advection.
Moreover, the total advected material volume using an MMALE mesh is usually smaller
than for a comparable CEL mesh because the difference between the rezoned mesh and
the mesh after the Lagrangian step is reduced.
To compare both methods with the experimental measurements, Fig. 3.17 is plotted,
which draws the shape of the sand regime at several times measured during the ex-
periment and calculated by numerical simulations. During the whole simulation, the
obtained runout distance from CEL is underestimated, which becomes more evident
at the further stages of the simulation. On the other hand, the MMALE provides a
good agreement in the runout distance with the experiment. Also, at later stages of
the simulation, there is a difference in a sand shape calculated by each method. The
final sand shape predicted by MMALE is closer to the experimental values than with
CEL.
By evaluating the kinetic energy loss during advection in Fig. 3.18, similar to the
strip footing problem, the CEL results in about four times more energy loss than
MMALE. This may explain the underestimated runout distance calculated by CEL
which highlights the role of the remeshing in addressing the issues associated with
complex and high speed deformation problems.
Nevertheless, the height of the final deformed shape is underestimated, which can be
attributed to the employed material model. In any case, the fact that the remeshing
64 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
0.333 _
0.500 _
0.667 _
0.833 _
1.000 _
Volume Fraction
0.000 _
0.167 _
CEL
MMALE
CEL
MMALE
Smooth surface reconstruction
Rough surface reconstruction
t = 22 mm
t = 7 mm
(a)
(b)
Detail B
Model
1.65 m
1.2 m
Figure 3.16: (a) Final shape of the flowed soil as well as the mesh distortion in the sand
column collapse for CEL (top) and MMALE (bottom), (b) Soil interface reconstruction
in CEL (top) and MMALE (bottom), the contours represent the volume fraction of the
soil in the elements; the results correspond to the detail B and not the whole model
Experimental model Lube et al.
MMALE
CEL
0
0
200
200
200
0
200 200
400
400
600 800
0 200 400
0 200 400 mm0 200 mm 600 800 mm
1000 mm
mm
mm
mm
mm
2.0 s
0.5 s
0.25 s
time = 0.17 s
Figure 3.17: Comparison of the runout distance from the numerical models and the
experimental measurements in the sand column collapse problem
3.4. NUMERICAL EXAMPLES 65
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Normalized kinetic energy loss
Time (sec)
Energy loss − MMALE
Energy loss − CEL
0.2
0.4
0.6
0.8
1
Figure 3.18: Comparison of the normalized kinetic energy loss during advection for the
sand column problem (the values are normalized with respect to the maximum value
of CEL curve)
step devised in MMALE improved the accuracy, the interface resolution, and the overall
deformed shape is highlighted in this problem.
In Fig. 3.19, the location of several material points tracked through the simulation
is drawn. In case of ALE, the displacement of any point would be averaged from the
displacement of its neighboring mesh nodes in the element containing the point during
the Lagrangian step. In the vertical direction, unlike the horizontal direction, both
methods predict the same position. The location of the points near the right side of
the column changes more notably. The maximum variation between the calculated
positions is attributed to point P4 with almost 30 cm difference. In this point, the
change in both horizontal and vertical direction is extreme and in the diagonal direction
of the initially generated Eulerian mesh. By close observation of the final mesh of
the MMALE, it is observed that the elements are arranged in a way to capture the
movement of the sand column in this direction. Concerning the fact that a considerable
amount of particles undergoes such movements, the MMALE may be a better choice
over CEL for this problem.
3.4.3 Soil cutting by blade
Soil cutting tests are conventionally used to design cutting blades. Such problems can
also be a good indicator of the ability of a numerical approach to treating material
separation, which is similar to the case of pile installation. Different semi-empirical
relations are available in the literature for predicting the horizontal and vertical cutting
force of the blade [McKyes,1985].
However, these relations are often too simple to deliver acceptable results because the
complexity of real soil behavior is not adequately modeled [Onwualu,1998]. Moreover,
conducting parametric studies using experiments is costly and time-consuming.
Since the material is split during cutting, i.e., new free surfaces are generated, this
test is considered as a challenging large deformation problem. In a purely Lagrangian
66 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Y axis (m)
X axis (m)
P1 − MMALE
P2 − MMALE
P3 − MMALE
P4 − MMALE
P5 − MMALE
P1 − CEL
P2 − CEL
P3 − CEL
P4 − CEL
P5 − CEL
P5
Initial soil position
P4
P3
P1 P2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Y displacement (m)
Time (sec)
MMALE
CEL
P5
P1
P2
P3
P4
Figure 3.19: Soil particle trajectory, (b) Comparison of the displacement between sev-
eral particles obtained from CEL and MMALE
3.4. NUMERICAL EXAMPLES 67
Void elements
Filled elements
1 m
3 m 1 m
Rigid Blade
0.75 m
0.2 m
1.7 m
1 m
45°
Figure 3.20: Schematic view of the soil cutting problem
simulation, this would mean that the mesh elements must be separated from each other
during the blade progression. Efforts have been made to model such problems using
advanced numerical techniques. An application similar to soil cutting by the blade is
the penetration of a hollow pile, where the soil is cut by installing the pile.
Problem description
The test consists of a cutting blade with an inclination angle of 45◦, which passes
through a body of clay, as shown in Fig. 3.20. The horizontal component of the
cutting blade velocity is initialized from 0 up to 0.04 m/s in the course of two seconds
to avoid instant loading, which induces shock load. Afterward, the velocity is kept
constant until the end of the solution. The total simulation time is 24 seconds.
Numerical model consideration
The soil model used in the simulation is assigned as an elastic-plastic material employ-
ing the von-Mises failure criterion, which has a density, ρ= 2000 kg/m3, the cohesion
c= 50 kPa, the Poissons ratio ν=0.25, and the elastic modulus of E= 1000 kPa.
The parameters are taken from the example in Peng et al. [2017] with some modifi-
cations. The cutting blade is modeled as a rigid body to minimize the dependency of
the model to the blade. The interaction between soil and cutting blade is assigned as
a frictionless contact. A uniform mesh size, as shown in Fig. 3.20 was used with a size
of 0.02 m. The model thickness in a perpendicular direction to the plane is 0.05 m.
A rather large area of void elements around the elements filled with soil is required to
allow the material to flow through the mesh during the cutting process.
Results
As a first step, the problem has been analyzed using the SALE method. In this
method, the mesh deforms significantly, and the solution terminates only after the
short amount of time since the elements cannot get out of the way of the cutting blade
(Fig. 3.21). Consequently, it is not possible to handle such problems using SALE or
Lagrangian methods. By contrast, the results obtained with both CEL and MMALE
are reasonable. Fig. 3.22 shows the material deformation after cutting approximately
0.9 m of the soil. It can be seen that these methods pose no restrictions concerning the
topological changes in the material domain (material separation) as cutting proceeds.
68 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
Figure 3.21: Mesh distortion during the soil cutting using the SALE method
Void elements
Void elements
MMALE method
Filled elements
Filled elements
Mesh
rezoning
0.91 m
0.91 m
CEL method
Figure 3.22: Mesh distortion and soil deformation using CEL (above) and MMALE
(below) methods in the soil cutting problem
The amount of material penetration into cutting blade elements (so-called material
leakage) is limited and can be neglected.
To verify the performance of both methods, a closed-form analytical solution suggested
by McKyes [1985] is presented in eqs. (3.4.2)-(3.4.4). FVand FH, therein are the
required vertical and horizontal forces, respectively, to cut the soil. The problem is
considered as plane strain. In addition, the tool is considered as smooth and rigid
[McKyes,1985].
P=cdcot ϕ
sin α1 + sin ϕ
1−sin ϕe(2α−π) tan ϕ−1+qd1 + sin ϕ
1−sin ϕe(2α−π) tan ϕ
sin α(3.4.2)
FH=P
sin(α+ϕ)+cd cot α(3.4.3)
FV=P
cos(α+ϕ)−cd (3.4.4)
3.4. NUMERICAL EXAMPLES 69
Figure 3.23: Schematic of the assumed conditions in the soil cutting problem for de-
riving an analytical solution [McKyes,1985]
1500
1000
500
0
500
1000
0 5 10 15 20
Force (N)
Time (sec)
Fv MMALE
Fv Analytical solution
FH Analytical solution
FH MMALE
Fv CEL
FH CEL
McKyes, 1985
Figure 3.24: Comparison of the induced horizontal and vertical forces on the blade
obtained from MMALE and CEL methods with the analytical solution in the soil
cutting problem
Where Pis the total force per unit width, cis the cohesion, and dis the cutting
depth. Other parameters are shown in Fig. 3.23. Using the c=50kPa,d=0.25 m,
ϕ0◦,α=45
◦,q= 0 kPa, and considering the model width of 0.05 m, the forces are
calculated as FH= 893 N and FV= 356 N.
Fig. 3.24 shows the vertical and horizontal forces induced on the cutting blade for both
CEL and MMALE, as well as the analytical solution. By assigning the same material
model, both methods converge to a similar value. Compared to the analytical solution,
the horizontal and vertical forces from both methods are in good agreement.
As a verification measure, internal and kinetic energy were checked. As a rule of thumb,
the kinetic energy of the deforming material should not exceed the range of 5% to 10%
of internal energy during the simulation [Dassault Systèmes,2016].
The internal energy in both MMALE and CEL converge to the same value (Fig. 3.25);
however, in CEL, a sudden jump is observed. Also, a sudden increase is observed
in kinetic energy in CEL. Considering the quasi-static condition of the problem, it is
unlikely that such sudden variations possibly occur during the simulation. Therefore, it
can be argued that MMALE provides more stable and smoother results. Nevertheless,
70 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
0.1
1
10
100
1000
0 5 10 15 20
Energy (J)
Time (sec)
Kinetic CEL
Internal CEL
Kinetic MMALE
Internal MMALE
Figure 3.25: Comparison of the internal and kinetic energy curves of the soil cutting
problem
the tolerance for internal to kinetic energy ratio is still in the range of 5% for both
methods.
In this problem, the same mesh size is used in both methods. Due to the quasi-static
condition applied to the model, the amount of distortion at each time step is limited,
which makes it possible to increase the number of a Lagrangian cycle per rezone step in
MMALE. The optimized computation cost of MMALE was then almost half of CEL.
3.5 Summary and Conclusions
In this research, the effect of the remeshing step in MMALE is evaluated and compared
against CEL, a particular case of MMALE where no remeshing is performed. The
evaluation is based on the calculation cost optimization, accuracy, and stability. Three
large deformation problems were presented and discussed, for which experimental or
analytical results are available. By using the remeshing step, the following points were
observed in those problems:
•Computation cost optimization can be performed by modifying a number of
Lagrangian cycles before a rezone and remap cycle. Therefore, in these cases
about 20 - 40% reduction in calculation time, can be achieved. This is not the
case in CEL, as shown in the strip footing and soil cutting problem.
•Using the MMALE, a better accuracy can be achieved compared to the CEL,
for instance in the example of a sand column collapse, the error in the predicted
runout distance calculated by MMALE was 2% while in the case of CEL it was
about 20%.
•Due to the consideration of the material motion, the remeshing step helps to
reach a better resolution of the material interface, as shown in the example of
a sand column collapse where the diffusion thickness of the interface was three
times less than CEL.
3.5. SUMMARY AND CONCLUSIONS 71
•Owing to the remeshing step in MMALE less remap-related errors, including en-
ergy loss during advection and material leakage which deteriorate the simulation
results, are produced, and better stability is achieved since less volume is trans-
ported during the remap step. In the case of the strip footing about 70% less
energy loss and 30% less leakage was observed.
Finally, it can be concluded that MMALE is suitable, though the highly sophisticated
numerical method for applications in geotechnical engineering involving large material
deformations and topological changes of the material domain.
The problems discussed here were modeled using simple material constitutive equa-
tions. Further investigations are required to assess the performance of more complex
material models in conjunction with MMALE. Moreover, the multi-phase simulation,
such as the inclusion of pore water pressure has not been performed using MMALE ele-
ment formulation. Further studies regarding problems with various drainage conditions
are needed.
72 CHAPTER 3. INVESTIGATION OF MESH IMPROVEMENT IN MMALE
Chapter 4
Numerical evaluation of buckling in
steel pipe piles during vibratory
installation
This chapter is the accepted version of the following publication:
Bakroon, M., Daryaei, R., Aubram, D., and Rackwitz, F. (2018). Numerical evalua-
tion of buckling in steel pipe piles during vibratory installation. Soil Dynamics and
Earthquake Engineering, 122, 327-336. https://doi.org/10.1016/j.soildyn.2018.08.003
c
2020. This accepted manuscript is made available under the CC-BY-NC-ND 4.0
license. license http://creativecommons.org/licenses/by-nc-nd/4.0/
Abstract
The buckling of steel pipe piles during installation is numerically studied. Generally,
numerical simulation of installation processes is challenging due to large soil deforma-
tions. However, by using advanced numerical approaches like Multi-Material Arbitrary
Lagrangian-Eulerian (MMALE), such difficulties are mitigated. The Mohr-Coulomb
and an elastic-perfectly plastic material model is used for the soil and pile respec-
tively. The pile buckling behavior is verified using analytical solutions. Furthermore,
the model is validated by an experiment where a pipe pile is driven into sand us-
ing vibratory loading. Several case scenarios, including the effects of heterogeneity
in the soil and three imperfection modes (ovality, out-of-straightness, flatness) on the
pile buckling are investigated. The numerical model agrees well with the experimen-
tal measurements. As a conclusion, when buckling starts, the penetration rate of the
pile decreases compared to the non-buckled pile since less energy is dedicated to pile
penetration given that it is spent mainly on buckling.
74 CHAPTER 4. BUCKLING IN STEEL PIPE PILES
4.1 Introduction
4.1.1 Motivation
Instability due to buckling is one of the dominant pile failure modes which leads to
a sudden increase in pile deformation. Buckling is often observed in slender struc-
tures subjected to axial compressive force [Timoshenko and Gere,1961]. The topic
of buckling is broad and actively discussed in the field of structural and mechanical
engineering. However, the focus of this study is to evaluate pile buckling during in-
stallation processes concerning heterogeneity in the soil and pile imperfections. During
installation, the pile penetrates the soil which provides to some extent lateral support.
Therefore, the embedded part of the pile behaves differently compared to the upper
part of the pile which is not yet laterally supported. In most studies of pile buckling,
only one of these conditions is assumed, i.e., either a pile with no lateral supports or a
completely embedded pile. Hence, the buckling evaluation during installation process
under semi-embedded pile condition is the motivation of this study. Semi-embedded
piles are frequently observed in offshore geotechnical engineering.
Generally speaking, buckling is classified into two main groups: Global buckling, where
the pile deforms in a way similar to Euler’s buckling problem, and local buckling, where
the pile deformation occurs in the cross-section and is usually localized [Bhattacharya
et al.,2005;DIN EN 1993-1-6:2007,2007].
The global buckling phenomenon is characterized by defining a critical stress/load
which depends on the slenderness ratio (Length/radius of gyration) and structure stiff-
ness. Moreover, the critical stress can be considered as a state dividing two types of
equilibrium, i.e., before reaching critical stress, the problem is in stable equilibrium,
whereas after passing the critical stress, an unstable equilibrium is reached. In geotech-
nical engineering design codes for pile performance, the general buckling is considered
using the slenderness ratio as well as the soil and pile stiffness [DIN EN 1993-1-6:2007,
2007].
The local buckling, on the other hand, is usually characterized as localized damage to
the pile which can occur at any stage of pile installation or operation. Local buckling
can occur due to several reasons such as pile imperfections, impact with obstacles, forces
induced by the soil, etc. Local buckling which is often encountered at the pile tip may
cause increased driving resistance, pile deviation from its longitudinal axis which in
turn decrease its bearing capacity, and changed pile response to lateral loads due to
change in section modulus. By further penetration, the pile may collapse [Aldridge
et al.,2005]. Also, Chajes [1974] reported that a small initial imperfection in a column
results in less load-carrying capacity than Euler load.
In practice, one can refer to Rennie and Fried [1979] as one of the first works which
mentioned and addressed pile buckling during pile installation. Another example of
local/pile tip buckling during pile installation is the Goodwyn-A platform construc-
tion project in Australia where many piles were severely crushed during installation
[Kramer,1996]. It was reported that the major reason for pile collapse was the damage
propagation starting from the pile tip. A similar case was observed at Valhall water
4.1. INTRODUCTION 75
injection platform where five out of eight skirt piles were damaged during installation
[T. Alm,2004].
In analytical studies, there are numerous researches regarding pile buckling. Timo-
shenko and Gere [1961] derived empirical solutions to calculate the critical buckling
stress for cylindrical shells as well as effects of fabrication inaccuracies. Young et al.
[2002] listed a comprehensive database of different structural shapes under various
loading conditions including cylindrical shells. In another study, Aldridge et al. [2005]
compared the soil and pile stiffness to consider soil-structure interaction, and con-
cluded that in order to have progressive buckling, the soil must be stiffer than the pile.
Otherwise the pile deformation will spring back elastically.
The buckling effect in pile bearing capacity has also been experimentally evaluated.
Singer et al. [2002] compiled various experiments with their theoretical background on
buckling evaluation of thin-walled structures as well as state of the art in experiments.
A recent study was also conducted by Vogt et al. [2009] where they investigated mi-
cropile performance in soft clay. Based on the experimental results, they developed a
mathematical model which accounts for soil structure interaction as well as the pile
imperfection.
4.1.2 Previous research in numerical pile buckling analysis
The performance of the numerical simulation of pile buckling can be evaluated from
various viewpoints, including simulation of large deformations in the soil, the capability
of detecting buckling in a pile during installation, and effect of pile imperfections on
driving performance. Geotechnical installation processes, like pile driving and vibro-
replacement, generally involve large deformations and material flow which pose sim-
ulation challenges when conventional numerical methods are used. Moreover, during
the installation process, the pile can be subjected to extreme loads due to soil resis-
tance which induces irreversible deformations to the pile, which results in poor pile
performance. Finally, the pile shape may contain imperfections due to fabrication in-
accuracies or transportation effects, which causes non-uniform stress distribution and
can result in reduced performance.
Numerical methods have been increasingly used in recent decades to study pile buckling
problems. These studies can be divided into two groups, namely those done in the field
of structural engineering, where the concern is the structural response of the pile, and
those done in geotechnical engineering where the soil-structure interaction effects on
pile stability are evaluated.
The field of structure buckling, especially the buckling of shell structures, is vast.
However, the literature review is limited to studies regarding pipe piles. In a general
report done by Schmidt [2000], the advances in buckling studies of shell structures
including the underlying theory, design code criteria, and use of numerical models are
compiled. In the recent decade, several types of research focused on the effects of
imperfection in pile strength reduction such as [Hilburger et al.,2006;Edlund,2007;
Ning and Pellegrino,2015].
76 CHAPTER 4. BUCKLING IN STEEL PIPE PILES
Generally, numerical studies of pile buckling in geotechnical engineering can be cat-
egorized according to how soil-structure interaction is taken into account. A group
of numerical models uses a system of non-linear lateral springs to capture the pile
confinement due to the soil, which is commonly referred to as “p-y” method. Such
methods employ complex equations for springs to capture realistic soil behavior. The
“p” term refers to lateral soil pressure per unit length of pile, while the term “y” refers
to lateral deflection [API,2003]. The “p-y” method has been used for various geotech-
nical problems concerning pile buckling, such as evaluation of pile buckling embedded
in liquefiable soils [Bhattacharya et al.,2009;Dash et al.,2010], buckling in partially
embedded piles [Budkowska and Szymczak,1997;Zhou et al.,2014], and buckling of
piles with initial imperfections [Erbrich et al.,2011].
Another group of numerical models defines the surrounding soil as elements whose
interaction with the pile is defined through a contact model. Most of these models
consider structural elements as “wished-in-place”, meaning that the pile installation
process has no effect on stress distribution; see for instance [Feng et al.,2013;Jesmani
et al.,2014].
This assumption was mostly made to avoid huge mesh distortion issues in numerical
simulation of the installation process. The wished-in-place assumption may not be
generally realistic with regard to confinement realization of soil presence due to two
reasons. First, the soil is disturbed by the installation process, for example, soil den-
sification during hammering. Second, deformations may be induced in a pile during
installation, whose effects on the pile bearing capacity is discussed in Kirsch et al.
[2015]. Therefore, the wished-in-place assumption can result in an overestimated pile
bearing capacity.
Consequently, a new approach is required which can reduce the assumptions made
in previous methods. It is believed that numerical approaches specialized for large
deformation analysis can be a good candidate for such studies.
In this study, a novel numerical approach is presented which enables studying pile
buckling during installation in the soil. By using this method, the complex behavior of
soil structure interaction is captured. Various complexities can be taken into consid-
eration such as initial pile imperfection. To the knowledge of the authors, numerical
evaluation of buckling behavior of perfect or imperfect piles during installation in soil
considering soil-structure interaction has not been studied in literature so far.
The structure of the paper is as follows: in section 4.2, the developed numerical model is
discussed along with the employed material models. The model is then validated using
both empirical equations and experimental results. In section 4.3, a parametric study
is conducted to investigate the heterogeneity effect of the soil as well as the effects of
initial imperfections in installation performance. Subsequently, the concluding remarks
are presented and discussed. In section 4.3.3, the results are summarized and discussed.
Finally, an outlook for future works is presented.
4.2. NUMERICAL MODEL 77
4.2 Numerical model
4.2.1 Description of the MMALE method
Pile installation is considered as a large deformation problem, a group of problems
whose numerical analysis via the conventional numerical approaches is often challenging
[Bakroon et al.,2017b;Aubram et al.,2015;Bakroon et al.,2017a]. Efforts are made
to improve the current available numerical techniques to treat this particular type of
problems. Concerning methods that rely on a computational mesh, one of the most
promising approaches is the Multi-Material Arbitrary Lagrangian-Eulerian (MMALE)
method [Benson,1992a].
The general strategy of MMALE is to generate a mesh usually non-aligned with ma-
terial boundaries or material interfaces. This may give rise to so-called multi-material
elements containing a mixture of two or more materials. A material-free or void mesh
zone must be introduced which holds neither mass nor strength. Such zones are nec-
essary for non-Lagrangian calculations to catch material flow into initially unoccupied
(i.e., void) regions of the physical space. After performing one or several Lagrangian
steps, the mesh is rezoned to its initial configuration to maintain mesh quality (re-
zoning/remeshing step). A new arbitrary mesh is developed which is different from
the initial mesh configuration. Subsequently, the solution is transported from the de-
formed mesh to the updated/original mesh (remapping/advection step). The sub-steps
are not performed in parallel but in a sequential routine using operator-splitting tech-
nique. For more information regarding the MMALE, the reader is referred to Benson
[1992a]; Aubram et al. [2017]; Aubram [2016].
The MMALE application in geotechnical problems is limited, although it is popular
in simulation of the soil in other fields such as an underground explosion, where the
soil is considered as a medium for transmitting shock waves [Daryaei and Eslami,
2017]. A recent study conducted by Bakroon et al. [2018b] assessed the feasibility
of MMALE in realistic geotechnical large deformation problems in comparison with
classical Lagrangian methods. It was concluded that MMALE could be considered as
promising candidates for solving complex large deforming problems. The applicability
of MMALE in conjunction with a complex soil material model was also investigated in
another work done by Bakroon et al. [2018a].
4.2.2 Description of the model
In this section, a description of modeling considerations using MMALE technique in
LS-DYNAR
/Explicit is presented. A model is developed, where a pile is installed in
the soil using vibratory force. All the models in the following sections use the same
configurations discussed here. The model configurations in isometric, side, and planar
view are shown in Fig. 4.1 (a-c), respectively. The load history curve of the vibratory
force is depicted in Fig. 4.1d.
The pile has 1.5 m height, 0.2 m diameter, and 0.005 m thickness which is modeled using
the conventional Lagrangian element formulation with reduced integration point and
a uniform element size of 2 cm (3000 elements). An elastic-perfectly plastic material
78 CHAPTER 4. BUCKLING IN STEEL PIPE PILES
Table 4.1: General properties of the pile used in benchmark models
Density Elastic Modulus Yield Stress Poisson ratio Thickness Radius
ρ(kg/m3)E(MPa) σy(MPa) ν t (m) R(m)
7850 2.1E5 250 0.3 0.005 0.1
Table 4.2: Mohr-Coulomb material constants for Berlin sand [Schweiger,2002]
Density Friction Dilatancy Cohesion Poisson Elastic Modulus
ρ(kg/m3) angle φangle ψ c (MPa) ratio ν E (MPa)
1900 35◦1◦0.001 0.2 20
model based on von Mises failure criterion is used for the pile with properties listed
in Tab. 4.1. A mesh with 2 m height and 1 m radius with the one-point integration
MMALE element formulation is generated. A gradient mesh, ranging from 0.6−8
cm element width is used in the horizontal direction, whereas a uniform mesh in the
vertical direction with 2.5 cm is considered (367,200 elements). The mesh is filled
with the soil up to the height of 1.8 m. A void domain with 0.2 m height, which has
neither mass nor strength, is defined above the soil material to enable the soil to move
to this domain after penetration starts. To avoid additional complexities regarding
the soil material model, the Mohr-Coulomb constitutive equation is adopted, whose
corresponding material constants for Berlin sand are estimated and listed in Table 4.2.
The initial stress in the soil is defined with assigning the gravity acceleration as 10
m/s2.
The equipotential smoothing technique is applied where the computational grids are
rearranged to maintain the mesh quality [Winslow,1963]. For the advection step, the
2nd-order accurate van Leer method is chosen [van Leer,1997].
For installation processes, the pile is characterized by using the Lagrangian formulation
whereas the soil in MMALE is defined by using the Eulerian formulation. Coupling
thus becomes necessary between the Lagrangian and Eulerian meshes. To define the
coupling between pile and soil, penalty contact is defined with a tangential friction
coefficient of 0.1. The pile head is fixed against horizontal movements. The lateral
sides of the soil are constrained against movements in a direction perpendicular to
their faces, while fixity in all directions is applied to the bottom of the soil.
The process of numerical model validation is presented which is divided into two main
parts, verification of the pile element formulation, and validation against experimental
results. The first part focuses mainly on differences in results obtained from the shell
and solid element formulation using three benchmark tests to achieve a realistic pile
behavior. The second part deals with the soil-structure interaction model as well as
the performance of MMALE by comparing to experimental measurements.
4.2.3 Verification of shell element formulation
The pile behavior in the numerical model depends on various parameters including the
element formulation (shell or solid element), mesh size, and a number of integration
4.2. NUMERICAL MODEL 79
Figure 4.1: Schematic diagram of the (a) isometric view, (b) side view, (c) planar view
of the one-quarter numerical model configuration with (d) vibratory load history curve
80 CHAPTER 4. BUCKLING IN STEEL PIPE PILES
XY
Z
Axial load
φ=0.2m
t=0.005 L=0.25m
Figure 4.2: Benchmark model configuration under uniform axial compression
points (reduced or full integration). A benchmark model is chosen to evaluate the pile
behavior under uniform axial compression. Both solid and shell elements with reduced
integration are used. A full integration shell element is also used for comparison. The
pile has the properties listed in Tab. 4.1.
This benchmark model investigates one of the possible forms of buckling in cylindrical
pipe piles which occurs due to an applied uniform axial load. Theoretically, the critical
uniform axial stress value which causes buckling, σcr, is calculated by Timoshenko and
Gere [1961]:
σcr =Et
R3(1 −ν2)(4.2.1)
A numerical model is developed to evaluate the axial critical stress using different
element formulations. The model configuration is shown in Fig. 4.2. A pile with
a length of 0.25 m is generated which is under a distributed axial compressive load
with a total magnitude of 100 kN. It should be noted that this load does not play a
role in eigenvalue calculation and is only used to show the load application direction.
Initially, a mesh size of 0.005 m is chosen which provides reasonably accurate results.
The pile is fixed in the bottom while the top surface is fixed in horizontal directions (X
and Y). The comparison criterion is the least eigenvalue which is subsequently used
to calculate the critical buckling stress. The corresponding critical stress is compared
with the empirical Eq. (4.2.1). The buckling mode determined from each element
formulation is shown in Fig. 4.3. as well as their corresponding eigenvalues and critical
axial stress as listed in Tab. 4.3. The shell elements provide an accurate result with
about 5% difference while the solid element significantly underestimated the critical
buckling stress compared to the empirical equation. The difference between reduced
and full integration shell element is negligible.
4.2. NUMERICAL MODEL 81
Table 4.3: Comparison of the resulting critical buckling stress under axial pressure
Eigenvalue Critical stress
(MPa)
Empirical equation [Timoshenko and Gere,1961] 199.6 6.35E3
Reduced integration shell 189.4 6.03E3
Full integration shell 189.2 6.02E3
Reduced integration solid 20.1 0.64E3
XY
Z
Shell Elements
Reduced Integration
Displacement
Magnitude %
0.0
8.7
17.4
26.1
34.9
43.6
52.3
61.0
69.7
78.4
87.2
95.9
100.0
Shell Elements
Full Integration
Solid Elements
Reduced Integration
Figure 4.3: Resulting buckling modes using different element formulations
4.2.4 Validation against experimental results
The proposed numerical model is validated by back-calculating an experimental test
carried out at the laboratory of the Chair of Soil Mechanics and Geotechnical Engi-
neering at Technische Universität Berlin (TU Berlin). The test set-up consists of a
half-cylindrical pile with 1.5 m length, 0.005 m thickness, and 0.2 m outer diameter
as well as a chamber with three rigid steel walls and one glass panel. The pile is
fixed in the horizontal direction via pile guides to ensure penetration along the glass
panel. A vibratory motor produces the driving force of 1670 N with the frequency of
23 Hz. The imposed dead load on the pile is about 410 N. The chamber is filled with
Berlin sand. Two displacement sensors are mounted on the pile to measure vertical
pile displacement.
A quarter model is developed based on the descriptions in Section 4.2.2. Here, to make
the model independent of pile properties, the pile is modeled rigid. Fig. 4.4 shows
the resulting displacement curve obtained from the numerical model compared with
experimental measurement. To focus on the evaluation of the penetration trend, the
penetration depth is normalized by its maximum value. Initially, the penetration rate
is significant due to less soil resistance and confining pressure. By further penetration,
the resulting force normal to the pile skin increases considerably, leading to an increase
of the frictional force. Therefore, the penetration rate decreases. The same trend
was observed in the experiment. Hence, the numerical model captures the penetration
trend accurately enough.
82 CHAPTER 4. BUCKLING IN STEEL PIPE PILES
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9
Normalized penetration, z/zmax
Time (sec)
Experiment − TU Berlin
Numerical model
Figure 4.4: Penetration depth vs. time curve obtained from the numerical model and
experimental measurement
-2.9e+04 _
-2.5e+04 _
-2.1e+04 _
-1.7e+04 _
-1.3e+04 _
-4.1e+04 _
-3.7e+04 _
-3.3e+04 _
-9.0e+03 _
-5.0e+03 _
-1.0e+03 _
0.0 _
(b) Horizontal stress (c)
Stress (Pa)
(a) Vertical stress
1.3m
Stress concentrations
under pile tip
−500
0
500
1000
8.94 8.96 8.98 9.00
Force (N)
Time (sec)
Picked time
Compression
Tension
End of loading
x
Figure 4.5: Isolines of the induced (a) vertical and (b) horizontal stress in the soil, and
(c) the corresponding loading at 8.98 sec for the validation model
Fig. 4.5 shows vertical and horizontal stress contours at the last compression force
cycle of the vibratory loading curve. The contours can be used to determine the size
of the influenced area. By evaluating the vertical stress distribution contours (Fig.
4.5a), it can be observed that the areas around the pile are influenced and disturbed
during the installation. A relatively large vertical stress is seen in the soil under the
pile tip. At a depth of about 1.3 m from the soil surface, the contours become almost
linear, indicating the vibratory force influence region which is reasonably far from the
boundary. The horizontal stress as shown in Fig. 4.5b is relatively large around the
pile. In areas far enough from the pile, the lateral stress in the soil reaches its in-situ
value, which verifies that the boundary distance is far enough from the dynamic source
to have substantial effects. Based on the above arguments, it can be said that the
numerical model captures the expected behavior of the soil during the pile penetration
reasonably.
4.3. PARAMETRIC STUDY OF PILE BUCKLING DURING PENETRATION 83
Table 4.4: Elastoplastic properties of the pile used in a parametric study
Density Elastic Modulus Yield Stress Poisson ratio
ρ(kg/m3)E(MPa) σy(MPa) ν
7850 2.1E3 250 0.3
4.3 Parametric study of pile buckling during pene-
tration
In this section, the pile buckling phenomenon during installation is evaluated by us-
ing the MMALE computational model described in Section 4.2.2. In section 4.3.1,
a reference model is developed as a comparison basis. In section 4.3.2, the buckling
phenomenon is studied under two distinct conditions, various imperfect piles, and het-
erogeneous soil. In section 4.3.3, the results are presented and discussed.
The pile in both the experiment and the numerical model, did not exhibit buckling for
this amount of penetration, using realistic parameters. Therefore, in order to reach
significant buckling in this small container after a short amount of penetration a low
elastic modulus was assigned to the pile. By using a relatively less elastic modulus,
a larger calculation time step was also reached in the numerical model. The reduced
elastic modulus of the pile is listed in Tab. 4.4, which corresponds to 1% of the elastic
modulus used in the model in section 4.2. In all the calculations, the harmonic vertical
force applied to the pile head is drawn in Fig. 4.1d.
The models presented in this study, are small scale, compared to the problem size
encountered in practice. The reason behind choosing this model dimension was due to
the experiment container size, with which the numerical model was validated. In the
parametric study, the numerical model is still maintained to avoid modifications in any
part of the model. For instance, by scaling up the model size, one has to assign a new
loading magnitude.
Nevertheless, it is possible to expand the numerical model to adapt to practical geotech-
nical applications using realistic values for both soil and pile. To this extent, the model
should be scaled up. To do so, one can use larger element sizes while keeping their
aspect ratio.
4.3.1 Reference model
In this model, a perfect cylindrical pile is installed in the soil using a vibratory load.
This model is developed as a comparison basis to the models where the pile holds
initial imperfection or the soil contains heterogeneity. The comparison criteria are
the mean strain, internal energy, load-penetration curve, and pile lateral displacement.
The mean strain is calculated as one-third of the strain tensor trace and defined based
on the infinitesimal theory. The internal energy is the work done to induce strain in a
unit volume of the solid part which can be used here to evaluate the accumulated strain
in a pile during installation. The lateral displacement curve is obtained by averaging
the nodal displacements of all nodes in the pile.
84 CHAPTER 4. BUCKLING IN STEEL PIPE PILES
Figure 4.6: Mean strain contour plots in pile after 0.65 m penetration for the reference
model
-2.9e+04 _
-2.5e+04 _
-2.1e+04 _
-1.7e+04 _
-1.3e+04 _
-4.1e+04 _
-3.7e+04 _
-3.3e+04 _
-9.0e+03 _
-5.0e+03 _
-1.0e+03 _
0.0 _
Stress (Pa)
(c)
−500
0
500
1000
8.94 8.96 8.98 9.00
Force (N)
Time (sec)
Picked time
End of loading
Tension
Compression
x
(b) Horizontal stress(a) Vertical stress
1.3m
Stress concentrations
under pile tip
Figure 4.7: Isolines of the induced (a) vertical and (b) horizontal stress in the soil, and
(c) the corresponding loading at 8.98 sec for the reference model
With the same model configuration of section 4.2.2, the pile did not suffer any signif-
icant buckling until about 8 seconds of the simulation which corresponds to 0.65 m
penetration. Therefore, all the models with imperfections and soil heterogeneity are
compared for this duration.
The mean strain contour plots, as well as the deformed pile tip section, are shown in
Fig. 4.6. It is observed that the induced strain in a pile is less than 0.05% which is
negligible. Also, the pile tip holds its initial cross-section with minimal deformations.
Fig. 4.7 shows the horizontal stress distribution in the soil at the final stage. The stress
contours are almost symmetric, yet different of what was observed in the validation
model in Fig. 4.5. The underlying reason is believed to be caused by the applied
changes in the pile, i.e. change of the pile property from a rigid to an elastoplastic
behavior and reduction of its Young’s modulus.
4.3. PARAMETRIC STUDY OF PILE BUCKLING DURING PENETRATION 85
Table 4.5: Properties of the oval pile section
Dmax(cm) Dmin (cm) w0(cm) Timoshenko
19.8 18.4 0.35
Oval pile section
Circular pile section
Dmin
Dmax
Figure 4.8: Schematic diagram of initial pile section compared to a perfect circle
4.3.2 Effect of pile imperfection and soil heterogeneity
Initial imperfections are somewhat unavoidable in piles which may have been caused
by fabrication tolerances or mishandling in transportation [Jardine,2009]. Although
small, the imperfection cannot be neglected since it is proved to have an influence on
pile buckling [Nadeem et al.,2015]. Currently, a handful of considerations are avail-
able in design codes for the imperfections based on fabrication tolerance [DIN EN
1993-1-6:2007,2007;MSL Engineering Limited,2001]. The codes are mainly based
on empirical equations which don’t usually capture the complex condition of instal-
lation process which is encountered in practice. The numerical models studied above
considered an initially perfect circular section with no imperfections.
To evaluate the imperfection effects on the pile performance, three models are proposed
where the pile is modeled initially as imperfect. These include piles with the oval cross-
section, flat side, and with out-of-straightness in length. In the following subsections,
the characteristics of each imperfection are described. Subsequently, the results from
each model are compared and discussed.
Scenario 1: Oval-shaped pile
One of the most common types of imperfection in cylindrical steel pipe piles is Ellip-
ticity resp. out-of-circularity where the pile takes the shape of an oval. The schematic
diagram of an oval-shaped pile compared to a perfect circular pile is shown in Fig. 4.8.
In comparison to a circle, an oval-shaped pile may buckle more during penetration.
According to Timoshenko and Gere [1961], ovality is defined as w0=(Dmax −Dmin)/4,
where Dmax and Dmin correspond to the longest and shortest oval diameter, respec-
tively.
To investigate ovality effects on buckling, a model is developed where an initial out-of-
circularity is applied to the pile. The oval pile properties are listed in Tab. 4.5.
86 CHAPTER 4. BUCKLING IN STEEL PIPE PILES
Top section
Bottom section
3D Side Top
δ
δ
Figure 4.9: Schematic of initial pile geometry from different views which illustrates the
out-of-straightness
Scenario 2: Out-of-straightness pile
Another form of imperfection is out-of-straightness where the pile top and bottom
axis are not on the same vertical axis. This shift in cross-section over length causes
different behavior in soil compared to the straight pile. In case of this imperfection,
more pressure can be induced on neighboring soil regime since the pile tends to push
the soil further to the side. Therefore, the pile can be prone to buckling due to the
unbalanced state.
The schematic diagram of this imperfection is shown in Fig. 4.9 where the pile align-
ment with respect to the vertical axis differs by the amount of δ. The reduction is
applied gradually, starting from the pile head to its tip. The assigned values δfor this
model is 1.2 cm.
Scenario 3: Flat Pile
This form of imperfection is also common which can be caused during transportation
or storage. From the cross-section view, the pile section is deformed, and a part of the
curvature is flattened. Fig. 4.10 shows an example of a pile of flatness. The flatness is
defined by two parameters, out-of-roundness, λ, and flatness, c. The parameters cand
λcan be related to each other by the following formula, c=2
√λD [MSL Engineering
Limited,2001]. A model is developed where the flatness value of c=7.8cm is assigned
which corresponds to λ=0.8 cm.
Scenario 4: Heterogeneous soil
In this section, a model is presented, where the pile hits a rigid sphere which somewhat
represents a heterogeneity in the soil. This condition which in practice can represent a
4.3. PARAMETRIC STUDY OF PILE BUCKLING DURING PENETRATION 87
D=19 cm
Flatness at one side
3D Side Top
λ
λ
c
Figure 4.10: Schematic diagram of initial pile geometry from different views illustrating
flatness
boulder in the soil, can cause early buckling in a pile and thus affecting its performance
during installation. This concept is not rare in the literature. A similar study was
conducted by Holeyman et al. [2015] where a Boulder-soil-pile model was simulated
using 1-D wave equation theory to study soil and boulder failure mechanisms. The
goal of the following model is to study the buckling propagation in a pile with further
penetration in the soil after it hits the boulder. The rigid sphere with an assumed
diameter of 10 cm is located inside the soil at a depth of 25 cm below the soil surface
and 8 cm away from the center of the pile (see Fig. 4.11). The sphere is assumed fixed
in all directions to avoid any extra effects which can be induced by the soil -boulder
interaction.
Several further conditions can be included in the model such as the definition of a
non-rigid heterogeneity with/without a different geometry. This, however, introduces
additional complexities to the model which is not the focus of this study.
4.3.3 Results and discussion
Results
The four models (ovality, flatness, straightness, and heterogeneity) are compared in Fig.
4.12 to the reference model using the criteria mentioned in the previous section. First,
the penetration rate of the models is investigated. The reference model penetrated 0.65
m without suffering any significant strain. For the four scenarios, the final penetration
is less than the reference model. In case of heterogeneity, the penetration rate decreases
after it hits the boulder at 1.5 seconds. After about 2 seconds, the penetration rate
of the model with an oval cross-section starts to decrease. The penetration rate of the
88 CHAPTER 4. BUCKLING IN STEEL PIPE PILES
Figure 4.11: (a) The planar view and (b) the cross section of the model illustrating the
location of the applied heterogeneity (rigid sphere) in the soil
other two imperfect piles, namely the piles with flatness side and out-of-straightness,
decrease after about 4 seconds.
Concerning the fact that the same driving force was used for all models, it can be
concluded that the same energy is applied to all the piles. Therefore, according to the
energy conservation law, the driving energy must have been spent on other phenomena
such as lateral displacements, additional strains and/or buckling in a pile. Therefore,
to assess this point, the lateral displacement, as well as the internal energy of each pile,
is compared in Fig. 4.12b and Fig. 4.12c, respectively.
It is observed that at the same time when the penetration curve differs from the ref-
erence model, the corresponding lateral displacement and internal energy of the piles
start to increase drastically. The lateral displacement of the pile is limited and is main-
tained after a specific amount of penetration, which can be attributed to the strong soil
resistance. Thus, the remaining driving energy must have been spent on buckling. By
comparing Fig. 4.12b and Fig. 4.12c, this point becomes clear where after the lateral
displacement reaches an almost constant value, the internal energy starts to grow sig-
nificantly. The curves in Fig. 4.12c are cut to the value of 42 J. Also, a decrease in the
internal energy value is observed after significant jumps for some models. This can be
attributed to the induced elastic strains in the pile which after further penetration the
pile springs back elastically. The possibility of the occurrence of this behavior has also
been reported by Aldridge et al. [2005]. As a result, it can be argued that the driving
energy for the pile installation is reflected in the model mainly in three different forms,
lateral and horizontal displacement, and pile buckling.
The induced mean infinitesimal strain which is defined as one-third of the strain tensor
trace, as well as the pile section deformation, are shown for each pile in Fig. 4.13.
The pile shapes correspond to the time stamps, where maximum internal energy was
recorded. In comparison to the reference model, a relatively significant strain/buckling
is sustained by the piles in the models. Most of the strain is accumulated at the
4.3. PARAMETRIC STUDY OF PILE BUCKLING DURING PENETRATION 89
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0 0 1 2 3 4 5 6 7 8
Vertical displacement (m)
Time (sec)
Reference model
Ovalitiy
Flatness
Straightness
Boulder
0
10
20
30
40
50
Energy (J)
0
0.05
1
1.5
2
Horizontal displacement (m) /100
x
Excessive buckling
x
xxx
Figure 4.12: Comparison of the imperfect piles with the reference model based on the
(a) vertical displacement (b) lateral displacement, and (c) the internal energy
90 CHAPTER 4. BUCKLING IN STEEL PIPE PILES
Figure 4.13: (a) Contours of induced mean infinitesimal strain in the imperfect piles
and the reference model and (b) the pile tip cross section compared to its initial
pile tip which points out the damage starting point. Furthermore, the progressive
pile deformation after further penetration is non-symmetric. In addition, each model
exhibits a different buckling mode due to the different initial imperfection. Also, it is
observed that the cross sections of the imperfect piles tend to take the forms similar to
the so-called peanut-shape as reported in the literature [Aldridge et al.,2005;Kramer,
1996].
Concerning the above discussion, it is argued that the proposed numerical model cap-
tures the complex site conditions such as the effect of soil resistance, pile imperfection,
and heterogeneity during installation processes. In addition, the model provides reli-
able measures to assess pile buckling
A brief discussion regarding plugging
In examples above, the soil inside the pile moved as a block along with pile as it was
driven. This phenomenon is referred to as plugging which closely affects the pile bearing
capacity and also the installation performance. To determine the plugging occurrence
and its effect on pile penetration resistance, there are several equations available in the
literature [Yu and Yang,2012]. For instance, Jardine et al. [2005] derived two equations
based on the inner pile diameter, d, CPT tip resistance, qc,a, and relative density of
the soil, Dr, to determine if plugging occurs:
d≥2.0(Dr−0.3) or d≥0.03qc,a (4.3.1)
If one of the two equations is fulfilled, the pile is considered unplugged.
In the current numerical model and the experiment, the corresponding values of Drand
qc,a for Berlin sand are estimated to be 0.75 and 6.3 MPa, respectively [Röhner,2010].
4.4. CONCLUSION AND OUTLOOK 91
Hence, by having d=0.19 m, none of the conditions above are satisfied, indicating
that the plugging may occur during the pile penetration. In both the numerical model
and the experiment, plugging was observed. Nevertheless, the plugging is a wide area
of research, and therefore it requires further and more focused investigations.
4.4 Conclusion and outlook
The focus of this study is to evaluate pile buckling during installation processes con-
cerning heterogeneity in the soil and pile imperfections. A novel MMALE numerical
approach, with an efficient soil-structure interaction scheme, was employed to improve
the numerical analysis of pile buckling which omits simplifications used in previous
studies. To capture a realistic buckling behavior of the pile, shell element types with
reduced integration points were used which provided a more accurate result than solid
elements.
It was observed that the pile rigidity and stiffness play an important role in soil stress
distribution during installation. In case of a rigid pile, the stress distribution was
different than what was observed in the elastoplastic pile. The underlying reason
can be the pile deformation during the installation. This highlight the importance of
consideration of pile deformation during the installation.
In addition, effects of various complex conditions, imperfections in pile geometry and
heterogeneities in the soil, were investigated and compared to a reference model where
the pile holds perfect cylindrical shape with no heterogeneity in the soil. Each scenario
exhibited a different buckling mode. Before a significant buckling could be observed,
the penetration rate started to decrease. At this time, both lateral displacement and
internal energy started to grow. The out-comes of this work show that the driving
energy of pile installation can be spent on other phenomena such as pile buckling and
lateral soil displacement. Consequently, less penetration will be observed. In addition,
the initial imperfection not only accelerates the buckling process but also changes the
buckling mode of the pile.
Although a small-scale model was employed in this study, the proposed numerical
approach can be used in large-scale problems. Hence, this approach can help engineers
to study the sensitivity of numerous variables, such as pile thickness, diameter, and/or
different soil conditions in reaching a cost-effective pile design without encountering
buckling during the installation process.
The presented work focused on a specific area, i.e. pile imperfection and soil hetero-
geneity. There are numerous affecting parameters on pile buckling which cannot be
summarized in one study. Following points may also be considered in future works:
•A more realistic soil material model taking into account the various drainage
conditions
•Possible effects of various dynamic loading types (hammer or vibratory) using
suitable shock-absorbing boundaries
•Soil heterogeneity such as the presence of a lens or multiple layers
92 CHAPTER 4. BUCKLING IN STEEL PIPE PILES
•Pile bearing capacity evaluation considering phenomena such as plugging
Chapter 5
Implementation of a locally
undrained formulation to simulate
pile installation in saturated
granular soil
This paper is in preparation:
Bakroon, M., Daryaei, R., Aubram, D., and Rackwitz, F. (2020). Implementation of a
locally undrained formulation to simulate pile installation in saturated granular soil.
Abstract
The Multi-Material Arbitrary Lagrangian-Eulerian (MMALE) in conjunction with the
hypoplastic material model has shown its capabilities in the evaluation of large defor-
mation problems such as pile installation processes. The presence of water is inevitable
in offshore projects and needs to be considered in the numerical evaluation. Hence,
a coupled formulation is introduced into the MMALE for modeling undrained con-
dition in saturated granular soils which is derived from the u−pformulation. The
model is validated and verified against experimental triaxial tests and other similar
numerical implementation, respectively. Afterward, a pile installation test is simulated
using the drained and undrained formulation. The process of pile installation in satu-
rated granular soil is neither perfectly drained nor undrained, yet, by using the drained
and undrained numerical simulations, one can predict the soil behavior for the two ex-
tremes of a problem and use them to reach a better judgment for the real soil behavior.
Keyword: Locally undrained formulation, Pile installation, Multi-Material Arbitrary
93
94 CHAPTER 5. SIMPLIFIED U−PFORMULATION
Lagrangian-Eulerian, Hypoplastic material, Saturated granular soil, User-defined ma-
terial subroutine (UMAT)
5.1 Introduction
For a handful of offshore projects, specially windfarms installed at shallow depths (<25
m), the monopiles are generally favored as the foundation. According to a report in
2012, about 75% of the wind parks were built on monopiles [Madsen et al.,2012]. This
type of foundation requires no seabed preparation and is easily fabricated.
On the other hand, the evaluation and monitoring of the pile installation process exhibit
challenges since there is no visual access to the pile at greater depths. In addition, it
is a known fact that the soil undergoes disturbance and change in stress distribution
which may favorably or adversely affect the final pile resistance. As an alternative
evaluation method, a robust numerical approach can be used to evaluate soil behavior
during the installation.
The efforts in the numerical simulation of large deformation problems such as pile
installation, which are classified as Large Deformation Finite Element (LDFE) anal-
ysis, start from many years ago and covers a variety of topics including the choice of
the proper element formulation, constitutive equation, etc. One of the current active
fields in the development of the numerical methods is the introduction of a coupled
formulation to the method to capture the soil-water/fluid mixture behavior.
Concerning the implementation of a coupled formulation in LDFE methods, the work
done by Qiu and Grabe [2012] should be mentioned where the undrained condition
was introduced for large deformation problems involving cohesive materials. The Cou-
pled Eulerian-Lagrangian (CEL) element formulation available in the commercial code
AbaqusR
was used which is limited to 3D elements. Similarly, Monforte et al. [2017]
introduced a computation framework for the saturated porous medium into the Parti-
cle Finite Element Method (PFEM), where three one-phase mixed formulations were
implemented. He concluded that for the PFEM, the best candidate for soil mixture
would be the Displacement-Jacobian-Pressure, u−θ−pformulation. Concurrently, a
theoretical framework of a homogenous equilibrium mixture model based on the hybrid
mixture theory has been developed for geomechanical multi-material flow with a focus
on the Arbitrary Lagrangian Eulerian (ALE) method in the work of Aubram [2016].
In the past several years, the works of the authors focused on the development of
a robust numerical approach capable of simulating the pile installation problems ad-
dressing difficulties such as large element deformation and large strain calculation. As
a summary, the developed model includes the employment of a sophisticated element
formulation in solving large geotechnical problems, the Multi-Material Arbitrary La-
grangian Eulerian (MMALE), which is originally developed for problems in the field
of fluid dynamics [Trulio and Trigger,1961], however, it has also shown promising
results in geotechnical applications [Bakroon et al.,2018b]. More information about
this technique will be presented in section 5.2.1. The employed element formulation is
5.2. METHODOLOGY 95
available both as 3D and 2D formulation which enhances the computation efficiency of
the simulated problems.
In addition, a constitutive equation based on the hypoplasticity concept is later added
to the model to predict the non-linear behavior of the granular materials. More in-
formation will be presented in section 5.2.2. The combination of the MMALE and
hypoplastic material model has been previously used and evaluated in the works of the
authors [Bakroon et al.,2018b;Daryaei et al.,2019]. Nevertheless, the models were
simulated with the drained condition, i.e. the effects of pore water were ignored.
As the goal of current and previous works is to develop a full-scale numerical approach
applicable also for offshore problems, where the presence of water is important, a cou-
pled formulation is added to the aforementioned model. In a work done by Zienkiewicz
and Shiomi [1984], a set of equations for coupled systems based on the equation of mo-
tion in porous media [Biot,1941] are presented. Among the influencing parameters in
these equations, the soil and fluid displacement have a significant effect and therefore,
possible simplifications based on soil and fluid motion can be derived [Biot,1941].
Here, a simplified form of the so-called u−pformulation [Zienkiewicz and Shiomi,
1984], where uis the displacement of the solid skeleton and pis the pore pressure in
the fluid, has been implemented and compared against experimental measurements.
The simplified formulation calculates the pore water pressure build-up using the vol-
umetric strain while ignoring the pore water pressure dissipation, assuming that the
installation process happens fast enough, that the undrained condition can be reason-
ably considered. Compared to the general numerical solution for saturated soil, the
simplified u−pformulation has lower computational costs.
The structure of the work is as follows. In section 5.2, the technical aspects of the
element formulation, the constitutive equation, the soil-structure interaction, and the
implemented simplified u−pformulation are presented. Then, the validation and
verification of the model are discussed in sections 5.3 and 5.4, respectively. Afterward,
a pile installation model test is back-calculated and the results are shown and discussed
in section 5.5. Finally, the conclusion of this study and outlook on future works are
presented.
5.2 Methodology
In this section, details regarding the employed MMALE element formulation, Hypoplas-
tic constitutive equation, soil-structure interaction are described. Afterward, the con-
cept of the coupled formulation, the so-called u−pformulation, as well as a description
of the implementation procedure in the code subroutine, UMAT, are discussed.
5.2.1 The MMALE numerical approach
The MMALE technique is as an advanced mesh-based numerical formulation benefiting
from the advantages of both classical Lagrangian and Eulerian schemes in the Finite
Element Method (FEM). In the Lagrangian scheme, the mesh nodes are fixed to the
96 CHAPTER 5. SIMPLIFIED U−PFORMULATION
material particles, resulting in mesh movement and deformation in accordance with
the material particles. Concerning the large deformation problems such as pile instal-
lation, this method shows considerable shortcomings such as large distortion, solution
divergence, or unreliable results. In the Eulerian scheme, on the other hand, the mesh
is fixed, which results in the independent movement of the material through the mesh.
To this extent, the solution must be transported/advected to the initial mesh after
each calculation step. Several considerations should be made to ensure a reasonable
accuracy in the Eulerian scheme, such as techniques to treat path-dependent material
behavior and track material interfaces [Benson,1992a].
In the MMALE, having inspired from the two previous viewpoints, the grid deforms
as in the classical Lagrangian formulation, followed by a solution transport to a new
mesh. The new mesh, however, is not similar to the initial mesh, rather it resembles
a less distorted mesh which somewhat conforms to the material deformation [Aubram,
2013;Bakroon et al.,2017b].
In the case of MMALE, the conservation equation, ∂φ
∂t +∇·Φ=S, is computationally
expensive to solve directly, where φis the field variable, Φis the flux function, and Sis
the source term. Thus, the operator splitting technique, which is a method to simplify
a complicated equation by breaking it into a sequence of simpler equations [Benson,
1992a;Aubram,2013], is employed which turn the equation into a Lagrangian term
∂φ
∂t =Sand a Eulerian term (∂φ
∂t +∇·Φ=0)[Benson,1992a;Margolin and Shashkov,
2003].
In the employed MMALE, the Lagrangian step is solved using the updated Lagrangian
method. In the Eulerian step, two substeps emerge: the remeshing and remapping step.
In the remeshing step, a new mesh is generated and in the remapping step, the solution
is transported from the previous mesh to the new mesh. The development of a new mesh
is based on two criteria; maintaining the quality of the mesh elements, and focusing on
the zones with a rapid variation of material flow. By doing so, computational errors
are reduced. Although these goals seem simple to satisfy, they pose challenges in the
development of a robust remeshing algorithm. For instance, by considering only the
quality maintenance, there will be accuracy loss in areas of high variations, since pretty
similar sizes will be assigned to the new mesh elements [Knupp et al.,2002].
There are several methods to remap the solution from the Lagrangian mesh onto the
new mesh [Benson,1992a;Donea et al.,2004]. In the case of geotechnical engineering
applications, one can use the advection-based remap algorithms, where the element-
centered solution variables are updated based on the in- and outfluxes across the ele-
ment boundary [Benson,1992a;Souli and Benson,2013].
One of the robust advection methods is the Van Leer algorithm. In this method, a
piecewise linear function is defined inside each element to redistribute the initial state
variable value over the length of the element. The van Leer algorithm is monotonic,
conservative, second-order accurate, however, it is relatively expensive in view of the
computation time. Also, distorted elements can cause some errors, causing the algo-
rithm actually to become less accurate [Benson,1992a;van Leer,1997].
5.2. METHODOLOGY 97
Figure 5.1: Schematic diagram of MMALE approach compared to the classical La-
grangian FEM [6]
In Fig. 5.1, the possible advantages of the MMALE method over the classical La-
grangian FEM is depicted in a schematic form. The mesh in MMALE is distorted
similar to the Lagrangian mesh, yet owing to the independent material movement
through the mesh, the solution can continue further. To ensure free material move-
ment in MMALE, a material-free or void zone should generally be defined within the
mesh which takes no mass nor strength. During the simulation, the materials may
move to this region.
5.2.2 The Hypoplastic constitutive equation
In the case of pile installation, the soil undergoes large deformation whose behavior
is generally non-linear. At this stage, the mechanical behavior of soils, especially the
granular soils are very complex. More specifically, the driving process induces another
difficulty in the soil behavior prediction due to their highly dynamic loading nature, i.e.
98 CHAPTER 5. SIMPLIFIED U−PFORMULATION
the soil may experience both loosening and compaction during the driving at different
areas. This highlights the importance of a robust constitutive equation.
The constitutive equations based on the hypoplasticity concept are shown to captures
such complex soil behavior from the beginning of the loading, such as dilatancy and
contractancy, and do not distinguish between elastic and plastic deformation. Also, the
hypoplastic constitutive model is popular for its simplicity since it uses a single incre-
mentally nonlinear equation to predict the soil behavior under loading and unloading
steps. The stress rate of the granular material, ˙
T, is determined by the effective stress,
T, intergranular strain, σ, and the void ratio, e[Niemunis and Herle,1997]:
˙
T=M(T,e,δ):D(5.2.1)
The void ratio in the Eq.(5.2.1) is governed by the minimum, maximum, and critical
void ratio, ei,ed, and ec, respectively.
In this work, the hypoplastic constitutive equation developed by Niemunis and Herle
[1997] which is the improved version of the hypoplastic equation developed by von
Wolffersdorff [1996] is used. The improvement includes addressing the previous issues in
the prediction of the accurate strain accumulation during the cyclic loading [Niemunis
and Herle,1997]. The constitutive equation is implemented in LS-DYNAR
in a previous
work of the authors and has shown good agreement with the various experiments
[Bakroon et al.,2018a].
5.2.3 Pile-soil interaction
For the simulated soil-structure interaction in the models, the penalty contact scheme is
used for normal direction, while the Coulomb scheme is devised in tangential direction
[Hallquist,2017]. The contact force is generally measured based on the arbitrary pen-
etration of the interacting parts, e.g. pile and soil. By assuming springs with specific
stiffness, k, in principal directions, the force is calculated, assuming that it is originally
caused by the spring compression with an amount of u. The energy equation is then
modified by adding the extra term (i.e. a penalty term) as follows [Wriggers,2006]:
Π=Ep+Ek+1
2kΔu2(5.2.2)
Where Epand Ekare potential and kinetic energy, respectively.
It is clear that the choice of the interface stiffness plays a crucial role and should
be approximated with care. Conventionally, the stiffness value takes the same order
of magnitude as the stiffness of the interface elements normal to the interface. For
instance, the following equation can be used in the case of 2Dshell elements:
kc=pfac ×A×K
dmin
(5.2.3)
Where A,K, and dmin correspond to the area, bulk modulus and minimum diagonal
of the shell element. Pfac is a stiffness factor.
The equation above works fine assuming that the materials hold similar bulk moduli
[Hallquist,2017]. However, if a hard object penetrates a very soft material, undesired
5.2. METHODOLOGY 99
effects such as excessive penetration may occur. Since the soil has a relatively smaller
bulk modulus, it is believed that such case applies to the installation problem. There-
fore, a different equation is used based on the recommendation of the code [Hallquist,
2017]:
kc=0.5×pfac ×m
Δt2(5.2.4)
Where mis the minimum mass of the pile and soil and Δt is the time step. Again,
a stiffness factor, Pfac, can be assigned to tune the stiffness, however, this has to be
done carefully as it may cause some instability issues [Hallquist,2017].
5.2.4 The u−pformulation
There are a variety of methods to simulate the interaction of soil-fluid mixture, most of
which are presented and listed in Zienkiewicz and Shiomi [1984]. The general system
of equations is referred to as a full mixed, u−p−U, formulation where uis the
displacement of the soil skeleton, pis the pore fluid pressure, and Uis the relative
pore fluid displacement. By assuming the pile installation as medium speed problem,
one may omit a couple of negligible terms and therefore utilize the u−p, formulation
[Zienkiewicz and Shiomi,1984]. Here, a special case of u−pformulation based on the
descriptions in Aubram [2019] is described and implemented.
Generally speaking, the governing equation of the porous medium is the momentum
balance of the saturated porous medium:
div σ= 0 or div σ=∇p(5.2.5)
where σis the total stress, σis the effective stress, and pis the pore water pressure.
According to Terzaghi’s principle of effective stress
˙
σ=˙
σ−˙pIresulting in ˙
P=˙p+˙p(5.2.6)
where Iis the second-order unit tensor, p=−1
3trσis the mean effective stress,
P== −1
3trσis the mean total stress, tr s=I:sis the trace of any second-order
tensor s, : denotes double contraction, and a superposed dot denotes the material
time derivative.
In addition, by assuming the infinitesimal strain, the general constitutive equation
takes the form of Eq. (5.2.7):
˙
σ=C:˙
(5.2.7)
˙
=1
2∇˙
u+(∇˙
u)T(5.2.8)
Where ˙
is the strain rate, uis the displacement of the soil, and Cis the fourth order
tensor of the soil stiffness. In the case of saturated soil, the applied force is sustained
by both the soil and the pore water.
Using the trace of the strain rate, i.e. tr ˙
=v, the volumetric strain can be calculated
and therefore the bulk modulus of the soil skeleton, K, can be defined as:
˙p=−Kvwhere K=1
9I:C:I(5.2.9)
100 CHAPTER 5. SIMPLIFIED U−PFORMULATION
The total volume change in the soil-fluid mixture should equal the volume change in
the fluid due to fluid compaction, ˙v1from Eq. (5.2.11), and pore water dissipation,
˙v2from Eq. (5.2.12), due to continuity. Subsequently, the Eq. (5.2.13) is obtained:
˙v=˙v1+˙v2(5.2.10)
˙v1=−n
Kf
˙p(5.2.11)
˙v2=k
γf∇2p(5.2.12)
˙v=−n
Kf
˙p+k
γf∇2p(5.2.13)
Where nis the porosity, kis the hydraulic conductivity, and Kfand γfare the bulk
modulus and unit weight of the fluid, respectively. Eq. (5.2.13) is referred to as the
storage equation in the literature [Aubram,2019].
Using Equations (5.2.6), (5.2.9), and (5.2.13) one obtains:
˙p=S
K˙
P+cp∇2p(5.2.14)
where 1
S=1
K+n
Kf
and cp=Sk
γf
(5.2.15)
Eq. (5.2.14) along with Eqs. (5.2.5)-(5.2.7) form the general coupled formulation for
the unknowns uand pwhich is referred to as u−pformulation.
One may assume that the installation process occurs so fast, that a locally undrained
condition applies. Hence, div q= 0 leading to ˙v=˙v1. Consequently, by substituting
(5.2.11)into(5.2.7) the following equation is obtained:
˙
σ=C+Kf
nI⊗I:˙
u=Cu:˙
u(5.2.16)
Where ˙
uis the rate of strain of the undrained solid-fluid mixture. The tensor product
I⊗Irepresents a fourth-order unit tensor.
5.2.5 Code implementation
The MMALE formulation used in this study is the one available in the commercial
code, LS-DYNAR
. The hypoplastic material model has been previously implemented
in the code via a user-defined subroutine, UMAT, in the framework of the explicit
solver as described in the work of the authors [Bakroon et al.,2018a]. The u−p
formulation is implemented as an interface which calls the hypoplastic UMAT. In Fig.
5.2, the calculation procedure in the subroutine is drawn as a flowchart. Initially, the
total stress, σ, and strain increment, ˙, as well as the material variables, are sent from
the code to the interface. At the beginning of the solution, the hydrostatic pore water
pressure, uhyd is generated while the excess pore water pressure, un, is assigned to zero.
Subsequently, the effective stress, σ, is calculated based on the given σ, and the uhyd.
5.3. VALIDATION OF THE IMPLEMENTED APPROACH 101
Afterward, σis sent to the hypoplastic UMAT where they are updated. Meanwhile,
the bulk modulus of the soil-fluid mixture, Km, is updated to account for the presence
of the pore water. The updated excess pore water pressure, un+1, is calculated using
the volumetric strain increment,
v, obtained from the code. Finally, the updated
total stress, σn+1, and bulk modulus, Km, is sent back to the code.
In the next solution steps, the effective stress is calculated based on the given total
stress from the code, and the calculated excess and hydrostatic pore water pressure
from the previous step. The effective stress is sent to the Hypoplastic UMAT to be
updated.
5.3 Validation of the implemented approach
The implemented numerical approach is validated by back-calculating a series of drained
and undrained triaxial tests for the Berlin sand [Rackwitz,2003]. The corresponding
material constants for the hypoplastic material model are listed in Tab. 5.1. The
conditions of the test series are listed in Tab. 5.2. For the drained simulation, three
samples, sand A, B, and C, with different hypoplastic material constants are tested.
On the other hand, for the case of the undrained simulation, one sand type, (sand M
in Table 5.1), is used while the cell pressure and the relative density are varied. The
test is back-calculated using a one-element 2D axisymmetric formulation with configu-
rations shown in Fig. 5.3. The results of the drained tests are summarized in Fig. 5.4
where the deviatoric stress and the change in the void ratio are plotted against strain
increment. In Fig. 5.4a the numerical model reaches the peak stress earlier than the
test and is somewhat overestimated. After that, the curves from the numerical model
and the test converge at the end of the test. As shown in Fig. 5.4b, the numerical
model starts to diverge from the experiment at about 5% strain. In all experiment
cases, the model underestimates the dilatancy up to 6% which corresponds to the test
with the sand sample C. Compared to other samples, the sand sample C has the max-
imum grain skeleton stiffness value, hs, which may have caused this difference. The
results of the undrained triaxial tests in the form of stress path, deviatoric stress, and
excess PWP for cell pressures of 100 and 500 kPa are shown in Fig. 5.5 and Fig. 5.6,
respectively. Compared to the experiment, the numerical results in Fig. 5a and Fig.
5.6a do not initially match the experiment, nevertheless they reach a good agreement
at the critical state line. Additionally, by comparing the numerical results for different
Table 5.1: Hypoplastic material constants for Berlin sand
Sand φc[◦]hs[GPa] n ed0ec0ei0α β
Berlin sand A 32 3.73 0.20 0.46 0.75 0.9 0.14 1
Berlin sand B 31 6.65 0.26 0.48 0.81 0.97 0.12 1
Berlin sand C 32 10.7 0.24 0.53 0.84 1 0.12 1
General Berlin sand "M" 31.5 3* 0.35 0.4 0.59 0.71 0.13 1
*The actual value of granular hardness, hs, is 10 GPa. This value is reduced to 30% due to
numerical stability at low-stress levels.
102 CHAPTER 5. SIMPLIFIED U−PFORMULATION
Begin
Obtain the state
variables icnluding
total stress σ and strain
rate ϵ from the code
calculate effective
stress
σ' = σ - u
hyd
-u
n
update effective
stress σ'
calculate and update the
corresponding bulk modulus
of the soil fluid mixture
Km = Ks + Kw / n
Calculate volumetric strain rate
ϵ
v
= ϵ
1
+ ϵ
2
+ ϵ
3
calculate excess PWP
u
n+1
= u
n
- ϵ
v
* K
w
/ n
update total stress
σ = σ' + u
hyd
+u
n+1
Assign the updated bulk modulus to the
material
return the state variables including total
stress σ to the code
Finish
Is it start of the
calculation?
No
Yes
Initialize Hydrostatic PWP, uhyd
Obtain initial effective stresses, σ'
Assign excess PWP un=0
Hypoplastic UMAT subroutine
.
....
.
Figure 5.2: Flowchart of the calculation process inside the user-defined subroutine
5.3. VALIDATION OF THE IMPLEMENTED APPROACH 103
Figure 5.3: Schematic of the developed FE model (left), and the respective boundary
conditions (right) of the triaxial test
0
50
100
150
200
250
300
350
400
450
0 5 10 15 20
Strain %
Sand A
Sand B
Sand C
Sand A FEM
Sand B FEM
Sand C FEM
q=σ1−σ3 (kPa)
(a)
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0 5 10 15 20
Void ratio e (−)
Strain %
Sand A
Sand B
Sand C
Sand A FEM
Sand B FEM
Sand C FEM
(b)
Figure 5.4: Comparison of the FE model and experiment results of drained triax-
ial compression test of Berlin sand (a) Deviator stress and (b) Void ratio vs. strain
increment
104 CHAPTER 5. SIMPLIFIED U−PFORMULATION
Table 5.2: Triaxial test series
Test No. Initial void ratio e0σ
c[kPa] Drainage condition
Berlin sand A 0.470 100
DrainedBerlin sand B 0.480 100
Berlin sand C 0.535 100
TX558 0.589 100
Undrained
TX552 0.533 100
TX559 0.566 500
TX554 0.506 500
relative densities at the same cell pressure, the curves seem to follow the same path
until around 180 and 600 kPa for cell pressures of 100 and 500 kPa, respectively.
According to Fig. 5.5b and Fig. 5.6b, the numerical model overestimates the effective
stress compared to the experiment in all cases. The simulations with 500 kPa cell
pressure reach acceptable agreement after 10% strain while the same cannot be said
for the simulations with 100 kPa cell pressure.
In the case of predicted excess PWP in Fig. 5.5c and Fig. 5.6c, the numerical simu-
lations with 500 kPa cell pressure are close to the experimental measurements despite
the overestimated peak PWP. On the other hand, less agreement is observed for the
case of 100 kPa cell pressure.
As an outcome, the numerical model seems to match the experiment better in case of
the relatively higher confinement pressures.
5.4 Verification of the implemented approach
To produce a benchmark model for the purpose of verification, a pile penetration
problem is back-calculated using the already available built-in coupled formulation in
the code, which can only be used with the classical Lagrangian element formulation
but not with the MMALE element formulation. Three models are developed to this
extent. The first two models are developed with the classical Lagrangian formulation,
one of which is using the built-in coupled formulation in the code while the other
one employs the implemented UMAT. The third model utilizes the MMALE element
formulations with the implemented UMAT. The problem geometry, as well as other
numerical considerations, are shown in Fig. 5.7.
The numerical model is simulating a pile penetration into loose-saturated sand. The
pile has 0.15 m radius, 6 m length, and is modeled as a rigid part, which penetrates the
soil with rate of 0.2 m/s. Due to the limitations in the built-in coupled formulation,
the problem could not be modeled as axisymmetric. Therefore, three dimensional solid
elements are used in the problem with plane-strain boundary conditions. This may not
play an important role in the problem since the target is to compare the implementation
and not to back-calculate any experiment. Owing to the symmetry of the problem, a
half-model is used. The mesh of the soil elements varies from 0.05-0.3 m, resulting in a
total number of about 7000 elements. The size of the pile mesh is generally maintained
as 0.025 m which is half of the soil element size to maintain a robust soil-structure
5.4. VERIFICATION OF THE IMPLEMENTED APPROACH 105
0
200
400
600
800
1000
1200
0 100 200 300 400 500 600 700 800 900
TX558 Exp.
TX552 Exp.
TX558 FEM
TX552 FEM
q=σ1−σ3 (kPa)
p’=(σ1
’+2*σ3
’ )/3 (kPa)
(a)
0
200
400
600
800
1000
1200
0 5 10 15 20
Strain %
TX558 Exp.
TX552 Exp.
TX558 FEM
TX552 FEM
q=σ1−σ3 (kPa)
(b)
−500
−400
−300
−200
−100
0
100
200
0 5 10 15 20
Excess PWP, u kPa
Strain %
TX558 Exp.
TX552 Exp.
TX558 FEM
TX552 FEM
(c)
Figure 5.5: Comparison of FE model and experiment results of undrained triaxial
compression test of Berlin sand with 100 kPa confinement stress (a) Deviator stress
vs. mean stress, (b) Deviator stress vs. strain increment, and (c) excess pore water
pressure vs. strain increment
106 CHAPTER 5. SIMPLIFIED U−PFORMULATION
0
500
1000
1500
2000
2500
3000
0 500 1000 1500 2000 2500
TX559 Exp.
TX554 Exp.
TX559 FEM
TX554 FEM
q=σ1−σ3 (kPa)
p’=(σ1
’+2*σ3
’ )/3 (kPa)
(a)
0
500
1000
1500
2000
2500
3000
0 5 10 15 20
Strain %
TX559 Exp.
TX554 Exp.
TX559 FEM
TX554 FEM
q=σ1−σ3 (kPa)
(b)
−800
−600
−400
−200
0
200
400
600
0 5 10 15 20
Excess PWP, u kPa
Strain %
TX559 Exp.
TX554 Exp.
TX559 FEM
TX554 FEM
(c)
Figure 5.6: Comparison of FE model and experiment results of undrained triaxial
compression test of Berlin sand 500 kPa confinement stress (a) Deviator stress vs. mean
stress, (b) Deviator stress vs. strain increment, and (c) excess pore water pressure vs.
strain increment
5.4. VERIFICATION OF THE IMPLEMENTED APPROACH 107
10 m
6 m
10 m
Void
Elements
Filled
Elements
Pile r=0.15m
0.5 m
Pile r=0.15m
Soil
surface
9.5 m
6 m
10 m
Pile r=0.15m
0.5 m
Pile r=0.15m
Gap 1mm
Soil
surface
(a) (b)
Figure 5.7: Schematic of the pile penetration problem using the (a) classical Lagrangian
and (b) MMALE element formulation
108 CHAPTER 5. SIMPLIFIED U−PFORMULATION
Table 5.3: Hypoplastic material constants for the Mai-Liao sand [Cudmani,2001]
φc[◦]hs[MPa] n ed0ec0ei0α β mRmTR χ βr
31.5 32 0.32 0.57 1.04 1.2 0.4 1 5 2 1×10−46.0 0.5
Table 5.4: Additional constants for simulating the soil-fluid mixture
ρsoil [kg/m3]ρwater [kg/m3]Kw(N/m2) Soil porosity, npor
2000 1000 1×1080.48
interaction.
The water table and the zero stress level are assigned at the soil surface. A hydrostatic
pore water pressure distribution is initialized over the soil depth. The lateral stresses
in the soil are initialized assuming K0=0.5. The conventional boundary conditions are
assumed, i.e. fixities in normal directions for both the bottom and lateral sides of
the mesh. No outflow boundary is required to define since an undrained condition is
assumed.
The complex soil behavior is realized using the hypoplastic material model as mentioned
in section 5.2.2. The corresponding material constant parameters are listed in Tab. 5.3.
The initial void ratio of einitial =0.95 is assigned, which reflects the relative density
of ID=0.2. In the case of modeling a pore-fluid mixture, the following parameters in
Tab. 5.4 are used. The bulk modulus of the water is KW
0=2.09 GPa, which builds
up extremely high excess PWP when the soil is considered as saturated. Concerning
the fact that a very small amount of air can be present in the soil, the bulk modulus of
the water-air mixture, Kwcan be reduced using the equation Eq. 5.4 [Koning,1963]:
KW=S
KW
0
+1−S
p−1
(5.4.1)
Where Sis the saturation degree, KW
0is the bulk modulus of the water, p is the ab-
solute fluid pressure, which is 101 kPa. Note that by assuming a small air content in
the soil pores, say 0.1% (S=0.999), the bulk modulus of the water-air mixture can be
reduced to Kw=1×108Pa (see (17), which is about 20 times smaller than the one in
case of a fully saturated soil (Kw=2×109Pa).
In the case of the classical Lagrangian formulation, such large material deformation
problems pose some challenges, for instance, huge mesh distortion which leads to early
simulation termination. Thus, a so-called "zipper-method" is employed to tackle this
issue, where a small gap (say 1 mm) between the axis of symmetry and soil elements
is devised to allow soil elements to slide along the pile [Cividini and Gioda,1988]. As
a result, the soil elements tend to move more laterally which in turn reduces the mesh
distortion. To avoid gap closure, a frictionless rigid wall is defined as a lateral boundary
condition on the soil elements near the axis of symmetry. To ensure sliding of the soil
elements and avoid their compression, the pile is initially wished-in to the depth of 0.5
m. In Fig. 5.7a the details of the methods are depicted.
5.4. VERIFICATION OF THE IMPLEMENTED APPROACH 109
In the case of MMALE, a Eulerian mesh is simply drawn which is generally similar
to the case of the Lagrangian model. Unlike the zipper method, neither gap nor any
wished-in assumption is required, since the large material deformation can be easily
handled in the MMALE method. Thus, the pile can be placed on the top of the soil
surface. On the other hand, a material-free or void region is defined above the soil to
allow the complex material movement during the pile penetration. In this problem,
about 1400 void elements are added to the top of the soil surface. In Fig. 5.7b the
details of the methods are depicted.
A similar soil-structure interaction approach based on the penalty contact scheme is
used for both classical Lagrangian and MMALE model. Assuming that the soil near the
pile liquefies during the pile penetration, a frictionless tangential contact is assigned.
Fig. 5.8 shows the results of the three developed models consisting of induced lateral
and vertical effective stress and excess pore water pressure distribution. The results
are shown at the pile depth of 2 m.
In all cases, the vertical effective stress has decreased underneath the pile tip and
around the shaft, which can be attributed to the soil liquefaction. Immediately un-
derneath this zone, an increase in the vertical effective stress value is observed. In the
case of the built-in coupled scheme, a higher compression value is observed compared
to the implemented one with the classical Lagrangian element formulation.
In the case of the lateral effective stress, a decrease around the pile shaft and under the
pile tip is observed in all cases. The area takes the form of a wedge which extends up
to the soil surface. The liquefied zone can also be observed up to a small depth below
the pile tip, followed by a zone where a higher lateral effective stress is generated.
In the case of the induced excess pore water pressure, two zones are observed; a com-
pression zone around the pile followed by a suction zone underneath the pile tip. Some
differences are observed between the implemented and the built-in coupled formulation.
The depicted compression zone in the case of the built-in scheme is less uniform than
the implemented scheme. Also, the compression and suction area are smaller in the
built-in scheme compared to the implemented scheme with the Lagrangian formulation.
Concerning the comparison of the MMALE with the zipper method, the results are
generally in good agreement with each other, yet some differences between the results
are noticed. Despite the similarity of the soil surface deformation, the width of the sur-
face heaving is greater in the case of the zipper method compared to the one obtained
from MMALE. In return, the height of the surface heaving in MMALE is higher. In
the case of the effective lateral stress, the width of the liquefied zone is greater in the
zipper method. A similar trend is observed in the case of excess pore water pressure.
The underlying reason may be the difference between the nature of the methods, i.e.
in the zipper method, the soil is somewhat constrained to slide or move more laterally,
whereas in MMALE such a constraint is not applied and the complex movement of the
soil such as particle rotation can be captured.
110 CHAPTER 5. SIMPLIFIED U−PFORMULATION
Lagrangian
Implemented coupled formulation in UMAT
MMALE
Implemented coupled formulation in UMAT
-33333 _
-25000 _
-16667 _
-8333 _
0 _
-50000 _
-41667 _
σ'z (N/m2)
-16666 _
-12500 _
-8333 _
-4166 _
0 _
-25000 _
-20833 _
σ'x (N/m2)
-3333 _
5000 _
13333 _
21667 _
30000 _
-20000 _
-11667 _
Excess PWP (N/m2)
Lagrangian
Built-in coupled formulation
Figure 5.8: Comparison of the induced lateral and vertical effective stress and excess
pore water pressure in the pile penetration problem in loose Mai Liao sand at the pile
tip depth of 2 m; the positive pore water pressure values correspond to the compression
5.5. BACK-CALCULATION OF SCALED MODEL TESTS 111
Table 5.5: Parameters associated with the static pile force in the case of impact driving
Pile outer
diameter
[m]
Pile
height
[m]
Pile
thickn.
[m]
Half-pile
mass [kg]
Pile
density
[kg/m3]
Mounting
+ Motor
[kg]
Total static
load [kg]
Impact
driving
0.2 1.5 0.004 19.26 10220 26.02+22.10
= 48.12
67.38
5.5 Back-calculation of scaled model tests
In this section, the implemented approach is used to back-calculate a pile-penetration
test done at the laboratory of the Chair of Soil Mechanics and Geotechnical Engineering
at Technische Universit¨at Berlin (TU Berlin) by Le et al. [2019]. The tests consist of
a pile installation tests using the impact driving. The driving has continued until the
pile reached a depth of 0.87 m.
A container consisting of three rigid steel walls and one glass panel is filled with the
Berlin sand. A half pipe-pile with 1.5 m length, 0.004 m thickness, and 0.2 m outer
diameters, is placed in the container which is constrained in the horizontal direction
using pile guides. Several measurements are done during the test including the pile
penetration and induced stresses at different locations [Le et al.,2019].
5.5.1 General remarks of the numerical model
Owing to the capabilities of the newly implemented approach, it is possible to model
the experiment using the MMALE formulation with the axisymmetric condition which
reduces the computation time significantly. Additionally, it is expected that MMALE
captures the complex movement of the material in such sophisticated problems. This
is less likely possible using the built-in coupled scheme as it is limited to the three-
dimensional classical Lagrangian formulations. Fig. 5.9 shows the numerical model
configuration as well as the geometry.
In the test, drainage condition is applied at the bottom of the container; however,
due to the highly dynamic nature of the driving force one can assume that a locally
undrained condition is occurring around the pile shaft. Hence, the test is neither fully
undrained nor drained.
Nevertheless, it is currently possible to simulate a fully drained and undrained condition
for the aforementioned test using the available and the implemented approach, respec-
tively. Subsequently, it may be possible to reach the real test condition by evaluating
the respective results of the simulated tests.
In the numerical model, the pile is assumed as rigid. Lateral constraints are applied
to ensure the vertical movement of the pile. The weight of the motor and mounting
are realized as a box on the pile head. The size of the box is defined in such a way to
capture the exact applied weight in the experiment as summarized in Tab. 5.5.
A 2D area-weighted axisymmetric MMALE formulation is used for the soil elements.
The mesh size varies from 0.004-0.04 m. The minimum mesh element size is governed
by the pile thickness to ensure a robust soil-structure interaction. Above the soil, a
112 CHAPTER 5. SIMPLIFIED U−PFORMULATION
F
140 cm
150 cm
20 cm
D= 20 cm, t = 0.4 cm
85 cm
Soil
Elements
Void
Elements
Motor + Mounting
Mass
Figure 5.9: Numerical model configuration of the pile driving experiment (axisymmetric
boundary conditions are applied accordingly)
material-free or void area is defined to enable the material to flow inside. A similar
mesh size is chosen for this area as well. Conventional fixities are applied to the soil
boundaries as shown in Fig. 5.9.
Due to numerical instability at low confinement stress regions (area near the surface),
a very small surcharge, 0.2 kPa, is applied on the soil surface. The effect of this amount
is found to be negligible and minimal compared to the case of no-surcharge condition
regarding the penetration results.
The hypoplastic material model is used with the material constants listed in Tab. 5.6
[Le,2015]. The initial void ratio of the soil is assigned as e0=0.465 corresponding to
a relative density of ID=0.75. In the case of the drained simulation, the buoyant mass
density is applied ρ= 1098 kg/m3, whereas in the case of the undrained simulation
the saturated mass density is applied, ρ= 2098 kg/m3. The soil-structure interaction
is defined using the penalty contact scheme with a tangential friction coefficient of 2/3
tan ϕ=0.4.
5.5. BACK-CALCULATION OF SCALED MODEL TESTS 113
Table 5.6: Hypoplastic material constants for Berlin sand [Le,2015]
φc[◦]hs[MPa] n ed0ec0ei0α β mRmTR χ βr
31.5 230* 0.3 0.391 0.688 0.791 0.13 1 4.4 2.2 1×10−46.0 0.2
*The actual value of granular hardness, hs, is 2300 MPa. This value is reduced by 10% due
to low-stress soil state
5.5.2 Definition of the driving load for the case of impact driv-
ing
Conventionally, the dynamic force for impact is reported as energy per blow, however,
the energy cannot be realized directly in the numerical model. The force for each blow
can, however, be approximated using the Eq. (5.5.1) suggested by Al-Kafaji [2013]:
Fd=πηm√2gh
2t=π×0.765 ×22.1×√2×9.81 ×0.28
2×0.01 =6.224 kN (5.5.1)
where ηis the reduction factor due to energy dissipation during impact, mis the drop
mass, his the height of the drop, and tis the impact duration. The values in the
equation above are taken from the specification and measurements in the experiments
done at TUB.
The total number of blows is reported to be 177. The interval between each blow is
significantly long. In the numerical model, the intervals between each blow should be
decreased to reach a suitable computation cost. However, the blow intervals should
not be placed too close as well. In principle, the optimum interval may be determined
by evaluating the duration, in which the pile acceleration varies significantly. In other
words, the next blow should be applied at the time, at which the pile is at a steady
state. Fig. 5.10 shows the acceleration history of the pile due to one impact. In
this case, the interval of 0.15 seems to be large enough to avoid overlapping in the
acceleration results of the pile.
5.5.3 Results and discussion
Fig. 5.11 compares the pile penetration obtained from both simulations with the
experimental measurements. It has been observed that the experimental measurement
lies between the two cases which may be referred to as the extremes. In the experiment,
the soil condition is not perfectly drained nor undrained rather a combination of these
conditions. To highlight this point, the penetration curves of both simulations are
averaged and compared with the measurement. Interestingly, the resulting penetration
curve agrees well with the experimental measurement. The soil surface deformation at
the pile tip depth of 3D, with Dbeing the pile diameter is shown in Fig. 5.12. The
most significant difference between the two cases is the length of the soil entrapped
inside the pile (in the literature, this is conventionally referred to as the soil plug). In
the case of the drained simulation, the pile is punching through the soil whereas a high
plugging is observed in the case of the undrained simulation. This may be attributed
to the increased stiffness of the soil mixture which is introduced by considering the
high bulk modulus of the water. This affects the overall soil behavior as well as the
114 CHAPTER 5. SIMPLIFIED U−PFORMULATION
0
2000
4000
6000
8000
10000
0 0.05 0.1 0.15 0.2 0.25
Force (N)
Time (sec)
Force
(a)
(a)
−80
−60
−40
−20
0
20
0 0.1 0.15 0.2 0.3 0.4 0.5
Acceleration (m/sec2)
Time (sec)
Acceleration
High
variation
Low
variation
(b)
(b)
Figure 5.10: a) Load application curve of the impact driving b) acceleration history of
the pile induced by one impact blow
5.5. BACK-CALCULATION OF SCALED MODEL TESTS 115
0
1
2
3
4
5 0 20 40 60 80 100 120 140 160 180
Normalized pile tip depth (z/D)
Number of blows
Experiment
Numerical−Drained
Numerical−Undrained
Numerical−average
Evaluation depth 1
Evaluation depth 2
Figure 5.11: Comparison of the pile tip depth and pile tip displacement curves from
the drained and undrained simulation and the experiments [Madsen et al.,2012;Qiu
and Grabe,2011]
Figure 5.12: Comparison of the deformed soil shape obtained from the (a) drained
and (b) undrained simulation and (c) the experiment at the pile tip depth of 3D. The
bottom boundary of all models and the side boundary of the experiment are cropped
116 CHAPTER 5. SIMPLIFIED U−PFORMULATION
a) Drained Simulation b) Undrained Simulation
0.0024_
0.0033_
0.0043_
0.0052_
0.0062_
0.0071_
0.0005_
0.0014_
0.0081_
0.0090_
0.0100_
Total Velocity
(m/sec)
Figure 5.13: Comparison of the movement of the soil regime for the case of (a) drained
and (b) undrained simulation at the pile tip depth of 3D
contact formulation since the contact takes into account the total bulk modulus and
not the one for the soil skeleton. Additionally, the total stress is considered in contact
formulation. Therefor, it may not reflect the reality as the pile interacts only with the
soil.
Although, it seems that the soil deformation in the experiment is captured by the
drained simulation reasonably accurate, yet by referring to the penetration curves in
Fig. 5.11, this conclusion cannot be completely correct.
In order to investigate the differences further, Fig. 5.13 is presented where the velocity
field of the soil regime is depicted. In both cases and under this model configuration, the
soil regime is experiencing a rotational movement. The center of rotation, however,
is further away from the pile shaft in case of the drained simulation. In the region
near the pile shaft, the soil is moving downward with the pile in case of the drained
simulation, unlike the case of the undrained where the soil regime is moving in the
upward direction. Also, the velocity magnitude of the soil regime outside the pile is
relatively higher in the case of the undrained simulation.
The horizontal effective stress distribution of the pile at the depth of 2Dand 3Dare
plotted in Fig. 5.14. Generally, the effective horizontal stress in the case of drained
simulation is more than the undrained case. In the case of the drained simulation, the
peak horizontal stress value is observed near the pile tip. On the other hand, the peak
lateral stress in the case of undrained simulation occurs above the pile tip. The peak
value, in this case, is significantly more compared to the case of the drained simulation
5.5. BACK-CALCULATION OF SCALED MODEL TESTS 117
and descends as distancing from the pile shaft. As stated earlier, the two simulations
provide the extremes of the problem. To reach the realistic site condition one may use
a sort of interpolation between the results obtained from both cases. To this extent,
the average value of the horizontal stress is plotted in Fig. 5.14 which shows an overall
increase of the horizontal stress after pile installation compared to the initial state in
the soil profile.
Fig. 5.15 shows the generated PWP at the pile tip depth of 2Dand 3D. At both
depths, a compression region is developed under the pile tip up to a depth of 2Dfrom
the pile tip. Inside the pile, a high compression zone is observed. Also, a suction zone
is noticed along the pile shaft. At top of the pile shaft, the suction zone forms a line
which makes an angle of 35◦with the vertical line. The measured angle is close to the
critical state friction angle of the soil which is φc=31.5◦. However, at the pile tip
depth of 3D, this angle reaches the value of 45◦. After some distances from the pile
shaft the zone shrinks and takes the form of a line which extends horizontally up to
the boundary. A correlation may be made with the curves of the undrained simulation
shown in Fig. 5.14, where a peak stress value is observed above the pile tip at the same
position where the suction is occurring.
As a summary, one can see the notable differences between the drained and undrained
simulations. Each of them provides an extreme case where full drainage and no drainage
condition apply. The differences shown here were with regard to the penetration depth,
induced horizontal stress, and the soil deformation shape.
Conclusion
A simplified u−pformulation was introduced via a user-defined hypoplastic material
model, UMAT, to the MMALE method to take into account the presence of water in
the porous medium. The simulation of undrained complex dynamic problems such as
offshore pipe-pile installation is hence possible.
The study consists of two parts. In the first part, the implemented UMAT was validated
using a series of triaxial experiments. Afterward, the UMAT was verified against an
already available explicit coupled formulation in a commercial code using a classical
Lagrangian formulation.
In the second part, two numerical models were developed, assuming drained and
undrained conditions to simulate a pipe-pile penetration problem using the impact
driving. Several points were made concerning the differences observed.
In the case of the drained simulation, the soil simply punched through the soil, unlike
the undrained simulation where the soil inside the pile plugged. Also, the soil regime
movement was different, i.e. the soil was following a rotational movement in case of
the drained simulation whereas in the case of the undrained simulation most of the
soil moved upward. Moreover, the horizontal stress distribution in the soil obtained
from the undrained simulation showed a peak value above the pile tip which was not
observed in the drained simulation. Around this zone, a high negative PWP was also
observed.
118 CHAPTER 5. SIMPLIFIED U−PFORMULATION
0
1
2
3
4
5
6
7 0 5 10 15 20 25 30
Pile tip
z/D
Horizontal stress (kN/m2)
Station at 0.5 D
pile tip depth at 3 D
Drained
Undrained
Average
0
1
2
3
4
5
6
7 0 5 10 15 20 25 30
Pile tipPile tipPile tip
z/D
Horizontal stress (kN/m2)
Station at 1.5 D
pile tip depth at 3 D
Drained
Undrained
Average
0
1
2
3
4
5
6
7 0 5 10 15 20 25 30
Pile tipPile tip
z/D
Horizontal stress (kN/m2)
Station at 1.0 D
pile tip depth at 3 D
Drained
Undrained
Average
0
1
2
3
4
5
6
7 0 5 10 15 20 25 30
Pile tipPile tip
z/D
Horizontal stress (kN/m2)
Station at 1.0 D
pile tip depth at 2 D
Drained
Undrained
Average
0
1
2
3
4
5
6
7 0 5 10 15 20 25 30
Pile tipPile tipPile tip
z/D
Horizontal stress (kN/m2)
Station at 1.5 D
pile tip depth at 2 D
Drained
Undrained
Average
0
1
2
3
4
5
6
7 0 5 10 15 20 25 30
Pile tip
z/D
Horizontal stress (kN/m2)
Station at 0.5 D
pile tip depth at 2 D
Drained
Undrained
Average
Figure 5.14: Induced horizontal stress in the soil profile at the pile tip depths of 2D
and 3Dat three lateral distances, 0.5D,1.0D, and 1.5Dfrom the pile shaft obtained
from the drained and undrained simulation
b) PWP at 3Da) PWP at 2D
-9.00 _
-6.00 _
-3.00 _
0.00 _
3.00 _
6.00 _
-15.0 _
-12.0 _
9.00 _
12.0 _
15.0 _
PWP (kPa)
~35°
~45°
Figure 5.15: Generated PWP at the pile depth of the a) 2D and b) 3D obtained
from the undrained simulation; positive values indicate compression while the negative
values indicate suction
Nevertheless, these two simulations capture the two extremes of what may occur during
the pile installation. In reality, this system is neither fully saturated nor drained but
something in between. But, by simulating both drained and undrained simulations,
one may draw boundaries of what may occur in the problem. In this case, for instance,
the measured pile penetration from the experiment lies between the two numerical
simulations. In other words, the undrained simulation showed less penetration while
the drained simulation showed more penetration compared to the test. Subsequently,
an interpolation, in this case as a rough estimation the arithmetic average, can be
utilized to approximate quantitative variables such as horizontal stress distribution in
the soil profile. However, special care should be taken regarding the choice of contact
formulation since the default contact uses total stress in its formulation. Consequently,
the proposed numerical approach facilitates a more accurate evaluation of such complex
problems.
Concerning the points mentioned above, a more comprehensive coupled formulation
considering the hydraulic permeability of the soil may be required to reach the real
soil behavior as it seems that it plays a crucial role in the soil behavior during the
pile installation. On the other hand, the formulation must be cost-efficient to avoid
excessive computation cost.
120 CHAPTER 5. SIMPLIFIED U−PFORMULATION
Chapter 6
Conclusions and Outlook
6.1 Conclusion
The results of the work presented here has led to the development of a numerical
method to simulate large deformation problems with a special focus on the pile driving
applications. The numerical approach is tailored to tackle several main challenges,
three of which are considered. The first challenge involves the choice of a robust ele-
ment formulation, MMALE. The second challenge is the employment of a sophisticated
constitutive equation for the nonlinear behavior of the soil, the hypoplastic material
model. The last but not the least, is the utilization of a cost-effective coupled scheme
for multi-phase materials, the so-called u−pformulation.
In this work and for the first time, all the three points above are combined to develop
a comprehensive method to facilitate the numerical simulation of large-deformation
problems. A very popular yet challenging problem is the pile installation. Previously,
the classical FE methods were used where several assumptions had to be made and
therefore it was only applicable to a certain set of piles, for instance closed-ended piles
with inclined pile tip. In the developed method, almost any type of pile, such as pipe
piles, can be used without any assumptions and limitations in pile geometry.
The first part of the thesis involves the choice of a robust element formulation whose
necessity is not new and have been already noted in the literature. Nevertheless,
this point is again highlighted in Appendix A, where the shortcoming of classical La-
grangian methods and advantages of the advanced methods are presented, evaluated,
and discussed in form of benchmark problems. Issues including contact loss, element
distortion, accuracy loss, and divergence appears in case of the classical Lagrangian
method. In contrast, the chosen advanced element formulation, ALE, alleviates the
limitations above.
A class of ALE formulation, the MMALE method, is studied and proves to be suitable
for the utilization in large deformation problems. MMALE is similar to the currently
practiced CEL, yet with a significant difference which is the generation of an arbitrary
computational grid unlike CEL where the original grid is always maintained. In chap-
ter 3, the theory of both MMALE and CEL are thoroughly presented and discussed.
122 CHAPTER 6. CONCLUSIONS AND OUTLOOK
In both methods, a three sub-step scheme is used. First, a Lagrangian step is per-
formed causing the mesh to distort due to material deformation. This is followed by
a rezoning/remeshing step which draws a new non-/less-distorted mesh. Finally, the
solution is transported form the distorted to the new mesh. The main difference be-
tween these two methods emerges in the rezoning/remeshing step, where the MMALE
method constructs a new arbitrary mesh which is neither the original mesh (like CEL)
nor the distorted mesh (like the Lagrangian method). The significance and possible
effects of the rezoning are evaluated by simulating various benchmark problems, the
strip footing, the sand column collapse, and the soil cutting problem.
In the case of strip footing, the pressure under the footing matches closely with the
empirical formulation. A smoother contour can be obtained in case of MMALE. In
addition, the calculation cost can be decreased up to an optimized value without sig-
nificant accuracy loss. Moreover, the calculation errors in the form of leaked materials
and contact loss are reduced using the MMALE method.
In the sand column problem, the final shape of the sand after collapse is compared to
the experimental measurement. In MMALE, the computational mesh rezones in a way
to capture the complex movement of the soil. As a result, more elements are focused
on the area where the sand column is moving. This results in a smooth soil interface
in the case of MMALE unlike the CEL where a staggered/jagged interface is obtained.
Consequently, a rough estimation of the material is calculated using CEL in contrast
to the MMALE. Additionally, The run-out distance, which is the distance that the
sand moved horizontally from the initial position, is checked at different time stamps.
Results from the MMALE and the experiment matched better than the CEL. The
underlying reason for the underestimated run-out distance by CEL may be attributed
to the relatively more kinetic energy loss during the simulation compared to MMALE.
Also, material points are tracked less accurately in CEL compared to MMALE.
The soil cutting problem is also simulated for both MMALE and CEL. Although it
is loosely related to geotechnical engineering applications, this problem challenges the
numerical formulations by their ability to capture material separation. The material
separation can occur in installation of open-ended piles. This problem does not con-
verge using classical Lagrangian method and stops at early stages of calculation. The
MMALE method, on the other hand, treated the material separation soundly and cal-
culated the vertical and horizontal force on the blade accurately. A similar observation
can be made in the case of the CEL as the large deformation is treated well and the
vertical/horizontal forces on the blade are close to the empirical equations. The energy
curves in the case of CEL, however, showed sudden increase/decrease during the simu-
lation whereas the curves obtained from MMALE are relatively smooth. The MMALE
method shows potential advantages compared to the CEL such as:
•Mesh adaptation to the high variating area determined by the material movement
•Better material interface resolution due to rezoning
•Optimization of computation cost by determining the frequency of remeshing
•Possibility of using coarser mesh grids
6.1. CONCLUSION 123
The numerical approach is then used to back-calculate pile installation problem in both
small- and large-scale in chapter 4. In the case of small-scale tests, complicated driving
loads, vibratory and impact loading, are used. The pile penetration curve is in a good
agreement with the experimental measurement. Also, the soil surface deformation as
well as plugging matched closely with the test.
An interesting application of the developed method is the evaluation of pile deformation
during the driving, especially the pile tip buckling. Previously, the evaluation and
probability of buckling was done using empirical methods. With the development of
the numerical model, it is now possible to evaluate this phenomenon numerically for
the first time.
In this study, a pile with elastoplastic material properties are driven inside the soil.
Moreover, the imperfections applied on the pile geometry to evaluate their possible
effects. Also, an inhomogeneity is introduced inside the soil. The model can handle
the complex conditions of the problem including the post-buckling interaction of the
soil and pile. Additionally, with changing the initial pile geometry condition, each case
of the pile buckling exhibited different behavior. Compared to the ideal rigid pile, a
lower penetration depth is achieved under similar conditions. This is attributed to the
fact that the driving energy is spent on other phenomena such as buckling.
The second part of the thesis deals with the implementation of an advanced hypoplas-
tic constitutive equation and its conjunction with the MMALE developed in the first
part. In the code used for the MMALE formulation, no proper constitutive equation
is available which can contribute to a more realistic prediction of stress- and density-
dependent behavior of granular soil. To this extent, the hypoplastic constitutive equa-
tion is introduced in both hydrocodes, AbaqusR
and LS-DYNAR
. The implementation
procedure of the constitutive equation is described in Appendix C. The advantages of
the hypoplastic constitutive equation are listed as follows:
•The nonlinear behavior of the soil is captured.
•The dilatancy and contractancy can be adequately predicted.
•Soil parameters such as void ratio is available.
•The so-called ratcheting problem due to cyclic loading is addressed.
The implemented hypoplastic model is verified using single-element tests. Results are
in good agreement with those from the original implementation of the developers. In
addition, the hypoplastic equation is evaluated in conjunction with the MMALE ele-
ment formulation. A large deformation benchmark problem, the sand-column collapse
test, is simulated for this purpose. The results including the run-out distance are close
to the experimental measurements.
The numerical models mentioned above are simulated under drained condition which
is still not applicable in problems involving the presence of pore water in the soil. In
the third part of the work, a coupled scheme is utilized which enables capturing the
complex behavior of soil-fluid mixture.
124 CHAPTER 6. CONCLUSIONS AND OUTLOOK
There are a variety of coupled schemes available in the literature, however, some of
which are computationally too expensive to be used in problems such as pile installa-
tion. Therefore, with several assumptions and simplifications, one may reach a cost-
efficient coupled scheme without much of loss in accuracy.
As mentioned in Chapter 5, by assuming that the acceleration of water is small com-
pared to soil, the corresponding terms in the governing equations are omitted. The
method is referred to as u−pformulation since all unknowns are determined using the
soil displacement, u, and pore water pressure, p. Moreover, since the installation occurs
fast, it can be reasonably assumed that the pore water would not dissipate during the
procedure and therefore the u−pformulation can be simplified by omitting the soil
permeability. The method is referred to as a simplified u−pformulation.
The suggested simplified u−pformulation is evaluated using various geotechnical
benchmark problems, from a simple element tests to sophisticated and large-scale pile
installation problems which tackle the aspects discussed in this work. The single el-
ement triaxial tests used to verify and validate the hypoplastic constitutive equation
are now used with the simplified u−pformulation. The results are generally in good
agreement. The pore water pressure generation is comparable with the experimental
tests.
The simplified formulation has also been checked against currently available coupled
schemes of LS-DYNAR
in a CPT problem. Three models are developed. The first
model used the coupled scheme of LS-DYNAR
with the Lagrangian/explicit formula-
tion. In the second model, the simplified u−pwith the Lagrangian/explicit formulation
is used. The zipper method is used in both cases to simulate the penetration. In the
third model, the simplified formulation is used in conjunction with MMALE. Results
from the simplified formulation are in good agreement with those from the already
available coupled scheme in LS-DYNAR
.
By using the simplified u−pformulation:
•The soil-fluid mixture behavior can be modeled under locally undrained condi-
tion.
•Owing to the assumptions made, the governing equations are simplified and the
calculation cost is reduced.
•In conjuction with the MMALE and Hypolastic constitutive equation, a compre-
hensive numerical approach is obtained.
Although the simplified u−pformulation does not simulate the pore water pressure
dissipation but with combining the drained and undrained simulation, one may get an
insight of the possible outcome of the simulation under semi-drained condition.
The results of the work has led to facilitation and further improvement of the numerical
method application in geotechnical engineering field with a special focus on pile instal-
lation problems. Using the described method, several main challenges of the numerical
simulation of large deformation problems such as pile installation is addressed. Also,
the saturated soil behavior can be reasonably predicted. Additionally, by implemen-
6.2. OUTLOOK 125
tation of the aforementioned method using the axisymmetric element formulation, a
significant reduction in the computational cost is achieved.
6.2 Outlook
There are a handful of aspects which are assumed or idealized in the numerical approach
which can be improved by further research. For instance, the pore water dissipation
after the pile installation changes the stress state in the soil which cannot be simulated
with the current numerical approach. This may be important in case of a long pause
between pile and superstructure installation. Moreover, the installation procedure may
not be applied on an unsaturated soil. A scheme can be hence introduced to simulate
the unsaturated soils.
Also, the employed contact scheme considers the total stress for calculation of tangential
forces, whereas in reality it should be only the effective stress which should be applied
for the calculation. With the presence of a robust contact interface scheme, addressing
the aforementioned issue is possible.
In addition, applying the method on multi-layered soils is costly using the conventional
computational resources. Owing to the massively parallel processing (MPP), One may
migrate the method to high computing platforms to address this simulation challenge.
Yet, the procedure is not straightforward as the calculation techniques, communications
between processing units, etc. are different and some modifications shall be necessary.
Finally, despite the introduced complexities in the numerical benchmark problems, it
is still possible to introduce more complexity to the problems, such as employing the
hypoplastic constitutive equation on pile buckling problem. This may provide a better
insight regarding the pile installation procedure as a more robust material model is
used.
The variety of the problems which can be solved using the presented numerical ap-
proach makes it attractive for numerical simulation field in geotechnical engineering.
The method can be utilized to derive more specific insight regarding the installation
problems and can lead to a more cost-efficient pile design.
126 CHAPTER 6. CONCLUSIONS AND OUTLOOK
Chapter 7
Acknowledgments
The author is thankful for the financial support obtained from Deutscher Akademischer
Austauschdienst (DAAD) with grant number 91561676
128 CHAPTER 7. ACKNOWLEDGMENTS
Appendix A
Arbitrary Lagrangian-Eulerian
Finite Element Formulations
Applied to Geotechnical Problems
This chapter is the accepted version of the following publication:
Bakroon, M., Daryaei, R., Aubram, D., and Rackwitz, F. (2017). Arbitrary Lagrangian
Eulerian Finite Element Formulations Applied to Geotechnical Problems. Numerical
Methods in Geotechnics, J. Grabe, ed., BuK! Breitschuh & Kock GmbH, Hamburg,
Germany, 33-44.
c
2020. This accepted manuscript is made available under the CC-BY-NC-ND 4.0
license. license http://creativecommons.org/licenses/by-nc-nd/4.0/
Abstract
The Arbitrary Lagrangian Eulerian (ALE) method is an explicit numerical formula-
tion which has become a standard tool to solve large deformation problems in solid
mechanics. In this study, a strip footing problem has been modeled to evaluate the
competences of ALE in the context of geotechnical engineering. This evaluation is done
by applying ALE into a previously analytically solved problem. Moreover, ALE has
been investigated in two commercial codes: AbaqusR
and LS-DYNAR
. This includes
a detailed comparison regarding the ALE remapping method, mesh size optimization
and sensitivity analysis, time step size, computation time, accuracy, and stability of
results. Results from ALE solution showed a good accuracy of both codes compared
with analytical solution. In addition, it was observed that automatic time step deter-
mination of the codes is accurate and by decreasing the time step size no significant
improvement is observed. ALE approved its efficiency in solving large deformation
geotechnical problems.
130 APPENDIX A. ALE IN ABAQUSR
VS. LS-DYNAR
A.1 Introduction
Geotechnical processes such as structural installations impose a large deformation on
the soil. Modelling such large deformation has been one of the main focuses of research
in recent decades. During the last decade, numerical methods have been increasingly
employed to study soil behavior and characteristic in various geotechnical problems.
In comparison to theoretical and experimental solutions, numerical methods showed
reliable and accurate results considering complex soil behavior. Currently there are
many presented and implemented calculation algorithms in commercial codes. One of
the most consistent methods is the Finite Element Method (FEM). There are numerous
approaches in FEM such as Lagrangian, Eulerian, and ALE.
Conventionally Lagrangian methods are used for small deforming problems. For large
deformation problems, the Lagrangian approach encounters large mesh distortion which
causes stability issues. A Eulerian method can treat large deformations by letting the
material to flow through fixed mesh elements, which is called advection. This allows
the material to undergo large deformation without any mesh distortion, making the
solution to continue. The limitation of the traditional Eulerian methods used in fluid
dynamics is that they are not able to treat situations where different materials interact
or the materials possess path-dependent behavior. ALE methods use a generalized
formulation and capture the advantages of both Eulerian and Lagrangian methods for
simulating large deformation problems. Therefore, the ALE approach is particularly
suited for geotechnical applications.
Aubram et al. [2015] modelled shallow penetration and pile penetration problems into
sand by implementing ALE into the commercial FEM code Ansys. A good agreement
between numerical results and experimental measurement was observed. Dijkstra et al.
[2011] modelled full phase pile installation using ALE. In his study, an elastic pile
is fixed and the soil flows around the pile. The model handled large deformation
induced by pile installation and provided comparable results. Konkol and Bałachowski
[2016] used AbaqusR
to compare Lagrangian and ALE method regarding a pile jacking
simulation. ALE provided better results in comparison to Lagrangian method.
A.2 Numerical methods description
A.2.1 Lagrangian approach
Conventionally Lagrangian algorithm is used to assess soil behavior under static, quasi-
static, and dynamic loads which usually induce deformations. In this formulation, each
individual nodes of mesh are attached to material particles, meaning that they move
with soil particles as they deform. This method naturally maintains free surfaces and
interfaces [Belytschko et al.,2000;Wriggers,2008]. Moreover, advanced soil material
models can be implemented almost straightforward by this method [Das,2008]. There
are cases of Lagrangian method application in large deforming problems available in
literature [Dijkstra et al.,2011;Hong et al.,2015]. However, they are not generally well-
suited for these problems, since they usually terminate at early stages due to extreme
A.2. NUMERICAL METHODS DESCRIPTION 131
Remapped
mesh
Advected
material
Distorted
mesh
Initial configuration ALE methodLagrange method
Lagrange
Part
Figure A.1: FE model Initial configuration (left), Material deformation in a Lagrangian
analysis (middle) and an Arbitrary Lagrangian Eulerian analysis ALE (right)
mesh distortion. Even if convergence occurs, low quality mesh after huge displacement
arises, making the results unreliable [Aubram,2014].
A.2.2 Eulerian approach
Eulerian algorithm is a widely used method for large deformation problems such as
fluid dynamics. The mesh is fixed to its place and particles move freely inside the
mesh. After each solution step, advection phase is carried out, where the material
is transferred between mesh elements. Eulerian mesh is better suited for large and
turbulent deformations, such as gas and fluid flow problems. However, Eulerian codes
are expensive in view of computation time due to advection step. Furthermore, the
precision of this method is lower than Lagrangian view as a result of advection step
[Benson,1992a;Aubram et al.,2017].
A.2.3 Arbitrary Lagrangian Eulerian approach
To overcome the shortcoming of these two methods and employ their advantages, the
Arbitrary Lagrangian Eulerian (ALE) method has been developed by Hirt et al. [1974b].
Fig. A.1 shows the difference between Lagrangian and ALE approach. In each ALE
solution step, the general strategy is to perform three substeps which are sequentially
arranged as Lagrangian step, rezone or remap step, and the advection step. In the La-
grangian step the mesh deforms as the material deforms. The rezone step interpolates
the mesh into a modified mesh to obtain a new better localization for the distorted
mesh. The advection step lets the material to flow to the new modified mesh. This
is done by transporting the material state variables form the last step to the new
mesh. There are numerous mesh updating algorithms for determining the suitable
redistributing of the new mesh [Aubram et al.,2015;Donea et al.,2004].
132 APPENDIX A. ALE IN ABAQUSR
VS. LS-DYNAR
4 m
4 m
Smooth
2 m
No relative
displacement
P
Figure A.2: Geometry and boundary conditions assigned to strip footing problem
[Bakroon et al.,2017a]
A.3 Numerical model description
The strip footing is a problem where material deformations are large and a closed-form
analytical solution is available. Two computational models using different numerical
analysis methods are compared: Lagrangian FEM and ALE method. Furthermore, two
commercial codes LS-DYNAR
and AbaqusR
are used for this simulation. The results
of the pressure under the footing are compared to the analytical solution done by Hill
[1950]. Hill processed a billet which is held in a container and hollowed out by punch.
He regarded the problem as a plane strain problem in order to simplify the solution.
The container is smooth, so the sides of the material are fixed only in the horizontal
direction and the bottom face are fixed in the vertical direction. The punch is assumed
as a rigid body with no horizontal displacement relative to the soil. Lateral sides of
the punch is assumed smooth, meaning that the soil will slip along the sides. Fig. A.2
illustrates the problems.
As illustrated in Fig. A.2, the strip footing is assumed as a rigid part with 2 m width
and 1 m height, the soil is 4 m ×4 m. Only half of the problem is modeled due
to simplicity. Symmetric boundary conditions are imposed on the plane of symmetry
by prescribing fixed condition in the normal direction. The plane strain condition is
applied.
For the ratio of base over soil width = 0.5, the maximum punch pressure for this
problem can be calculated with qult =2c(1 + 1
2π) where cis the soil shear strength
[Hill,1950].
The soil material parameters used in the problem are shown in Tab. A.1, where v
is Poisson ratio and Eis the modulus of elasticity. The Tresca constitutive model is
adopted.
A.4. RESULTS 133
Table A.1: Material parameters for the soft soil
E[kPa] c[kPa] v[-]
2980 10 0.49
In AbaqusR
, a 4-node bilinear plane strain quadrilateral with reduced integration and
hourglass control (AbaqusR
element type CPE4R) is used. The rigid body elements
were meshed by a 2-node 2D linear rigid link (R2D2).
In LS-DYNAR
, 1-point ALE element were used (ELFORM 5 in *SECTION_SOLID
keyword). The rigid body was modelled with the reduced 4-point element formulation
(ELFORM 1). This is an efficient formulation which is applicable to general cases.
For both softwares the rigid footing nodes were tied to soil’s surface nodes to reduce
model dependency on contact. Frictionless tangential penalty contact between the
lateral side of the footing and the top soil surface is defined to allow sliding between
soil and foundation.
Among various smoothing methods, equipotential algorithm as described in Winslow
[1967] was used for both LS-DYNAR
and AbaqusR
. This smoothing algorithm is more
commonly used and provides more stable results [Dassault Systèmes,2016]. In this
smoothing method, by solving Laplace equations the new mesh is drawn. For advection
step, there are two methods available in LS-DYNAR
: Donor cell and Van Leer. Van
Leer method is preferred over donor cell since it benefits from second order accuracy
and is more stable [Jonsson et al.,2015]. Therefore, Van Leer advection method was
used.
A.4 Results
In this section, the comparison between Lagrangian and ALE method is presented.
The same problem was used by Qiu et al. [2011] to compare the results of implicit,
explicit Lagrangian, and CEL formulation. The results of this study were compara-
ble with values calculated by Qiu et al. [2011]. Afterwards, a thorough investigation
for a 2D Lagrangian and ALE performance comparison was made such as mesh size
optimization, ALE remapping step, analysis time comparison, and effect of time step
size. The simulations are carried out using a server at TU Berlin with two 2.93 GHz
quad-core Intel CPU X5570 and 48 GB of RAM. Due to limited number of elements
in the model, only one core of the CPU was used to scale up the computation time.
The utilized versions of codes were LS-DYNAR
R9.1.0 and AbaqusR
2017.
The comparison is conducted based on pressure results and computation time. In the
following sub-sections, different criteria and considerations were studied and discussed.
134 APPENDIX A. ALE IN ABAQUSR
VS. LS-DYNAR
Table A.2: Time step comparison for Lagrangian and ALE model
2D Lagrangian 2D ALE
AbaqusR
00:04:32 00:13:30
LS-DYNAR
00:04:45 00:15:49
A.4.1 Model verification
Two Lagrangian models were developed to verify the geometry and boundary conditions
in both codes. The element size was taken as 5 cm, with total number of elements of
3200. The initial mesh configuration is shown in Fig. A.3.
Fig. A.4 shows the pressure results under the footing versus penetration depth for
both codes compared to analytical solution done by Hill (1950). The pressure is ob-
viously increasing as penetration continues. In AbaqusR
, the pressure increase rate is
higher than in LS-DYNAR
for Lagrangian solution. This difference can be attributed
to the definition of hourglass in both codes. In AbaqusR
the default hourglass method
is a viscoelastic approach. In contrast, for reduced integrated elements, as stated by
Hallquist [2017], LS-DYNAR
recommends stiffness hourglass method proposed by Hal-
lquist [2017]; Belytschko and Bindeman [1993]. Nevertheless, computation time of both
codes were pretty similar as shown in Tab. A.2. Considering the accuracy of results,
it can be stated that this model predicts the expected behavior of the problem.
ALE models were compared to Lagrangian models for both codes in order to verify the
improvement in results and to emphasize on the effects of remapping step. In these
models, the same geometry and boundary conditions as of Lagrangian method were
applied. In Fig. A.4, it is clearly observed that pressure results follow the same trend
as analytical results unlike what was observed in Lagrangian. This accuracy is achieved
in expense of more computation time (Tab. A.2).
It should be noted that initial results contained noises which is inevitable in explicit
formulation [Dassault Systèmes,2016]. Simple average smoothing procedure was ap-
plied to the results. The following diagrams from both codes have a limited noise after
smoothing was applied.
A.4.2 Mesh size sensitivity in ALE
To assess the dependency of results to mesh size, several models with different element
sizes were developed. Five models were analyzed with element sizes of 2.5, 5, 10, 12.5,
and 20 cm. The pressure results for AbaqusR
and LS-DYNAR
models are illustrated
in Fig. A.5 and Fig. A.6, respectively. The results are also compared to analyti-
cal solution. AbaqusR
shows unstable pressure results with increasing the mesh size,
whereas LS-DYNAR
shows a stable pressure behavior. By changing the mesh size the
accuracy of model in AbaqusR
differed slightly, unlike LS-DYNAR
where the difference
between results were noticeable. Regarding the analysis time, as shown by Tab. A.3 by
reducing the mesh, computation time increases nonlinearly. Moreover, by comparing
computation time between AbaqusR
and LS-DYNAR
, a slight difference is no-ticed. In
Fig. A.5 and Fig. A.6 it is also noticed that decreasing mesh size after 5 cm to 2.5 cm
A.4. RESULTS 135
Initial mesh configuration
1 m
1 m
Rigid
Body
4 m
2 m
ALE Elements
Figure A.3: Numerical mesh configuration of the strip footing problem
0
1
2
3
4
5
6
7
8
0 0.1 0.2 0.3 0.4 0.5
Normalized Punch Pressure p/c [-]
Penetration depth [m]
Ls-Dyna, Explicit 2D
Abaqus, Explicit 2D
Ls-Dyna, ALE 2D
Abaqus, ALE 2D
Analytical solution π+2
Figure A.4: Comparison of punch pressure results for Lagrangian, and ALE with ana-
lytical solution
136 APPENDIX A. ALE IN ABAQUSR
VS. LS-DYNAR
0
1
2
3
4
5
6
7
0 0.1 0.2 0.3 0.4 0.5
Normalized Punch Pressure p/c [-]
Penetration depth [m]
20 cm mesh
12.5 cm mesh
10 cm mesh
5 cm mesh
2.5 cm mesh
Analytical solution π+2
Figure A.5: Normalized punch pressure vs. penetration depth for different mesh den-
sities of the strip foot-ing problem analysed by AbaqusR
0
1
2
3
4
5
6
7
0 0.1 0.2 0.3 0.4 0.5
Normalized Punch Pressure p/c [-]
Penetration depth [m]
20 cm mesh
12.5 cm mesh
10 cm mesh
5 cm mesh
2.5 cm mesh
Analytical solution π+2
Figure A.6: Normalized punch pressure vs. penetration depth for different mesh den-
sities of the strip foot-ing problem analysed by LS-DYNAR
gives the same pressure results. Hence, the effects of further reduction are negligible,
indicating that the optimum mesh size is reached, which is 5 cm. For purpose of better
illustration, this model is called the reference model.
Fig. A.7 shows the resulting deformation for reference model obtained from Lagrangian
and ALE analysis in LS-DYNAR
and AbaqusR
. It can be seen that both codes improve
the mesh under the edge of footing and its lateral side which encounters large mesh
distortion. There are however, some differences among the results from AbaqusR
and
LS-DYNAR
. It is also observed that AbaqusR
can handle the boundary elements better
than LS-DYNAR
.
A.4. RESULTS 137
Table A.3: Analysis time comparison for different mesh sizes of ALE model
Mesh size AbaqusR
LS-DYNAR
2.5 cm 02:46:00 02:48:10
5cm 00:13:30 00:15:49
10 cm 00:01:17 00:01:38
12.5 cm 00:00:55 00:00:53
20 cm 00:00:30 00:00:22
ALE mesh - Ls-DynaALE mesh - AbaqusInitial mesh configuration Lagrangian -
Abaqus, Ls-Dyna
Figure A.7: a) Initial mesh configuration, b) Lagrangian deformed mesh calculated by
AbaqusR
and LS-DYNAR
, ALE mesh deformation in c) AbaqusR
, d) LS-DYNAR
138 APPENDIX A. ALE IN ABAQUSR
VS. LS-DYNAR
Initial configuration ALE mesh - Abaqus ALE mesh - Ls-Dyna
1 m
1 m
1 m
3 m
2 m
10 cm mesh size
5 cm mesh size
Figure A.8: a) Initial mesh configuration, gradient ALE mesh deformation in b)
AbaqusR
, c) LS-DYNAR
Table A.4: Analysis time comparison for different mesh sizes of ALE model
Mesh size AbaqusR
ALE 2D LS-DYNAR
ALE 2D
5 cm (reference model) 00:13:30 00:15:49
5-10 cm 00:10:17 00:09:16
A.4.3 ALE gradient remapping
To further enhance the model regarding reducing computation costs, the soil was di-
vided into two parts with different element sizes. As the elements near the footing
play a crucial role in determining the pressure values, smaller element sizes were as-
signed to the first meter of top soil layer. This model is compared with the refer-
ence model. Fig. A.8 shows the deformed ALE mesh after penetration. It is no-
ticed that AbaqusR
handles the remapping technique different than with LS-DYNAR
.
LS-DYNAR
handles remeshing for each zone individually while AbaqusR
handles the
remeshing globally. Also, a reduction in analysis time is achieved by using this strat-
egy as presented in Tab. A.4. It should also be noted that no difference in pressure
values were reported by both codes as appears in Fig. A.9.
A.4.4 Effect of time step size
Conventionally, time step size in commercial softwares are calculated as follows:
Δte=Ls
c(A.4.1)
Where Lsis the characteristic length, and cis the wave speed in the corresponding ma-
terial. Determining the correct time step size is critical in geotechnical simulations. In
commercial codes the equation above is used to assign the time step size automatically.
A.4. RESULTS 139
0
1
2
3
4
5
6
7
0 0.1 0.2 0.3 0.4 0.5
Normalized Punch Pressure p/c [-]
Penetration depth [m]
Ls-Dyna reference mesh
Ls-Dyna gradient mesh
Abaqus reference mesh
Abaqus gradient mesh
Analytical solution π+2
Figure A.9: Effect of gradient mesh on the accuracy of pressure results
0
1
2
3
4
5
6
7
0 0.1 0.2 0.3 0.4 0.5
Normalized Punch Pressure p/c [-]
Penetration depth [m]
Ls-Dyna Automatic
Ls-Dyna 1e-5
Ls-Dyna 1e-6
Abaqus Automatic
Abaqus 1e-5
Abaqus 1e-6
Analytical solution π+2
Figure A.10: Time step effect for the strip footing problem analysed by ALE method
In this section, the effect of time step size is studied regarding the accuracy and analysis
time.
To do so, the model was analyzed with different time step sizes: automatic, 1e−5, and
1e−6seconds. As shown in Fig. A.10, the difference between the results are negligible.
On the contrary, by using 1e−5and 1e−6the calculation time increases significantly
(Tab. A.5). In AbaqusR
model, for the case where 1e−6time step size was assigned,
after 82% of punch the analysis stopped.
Therefore, it can be argued that the automatic time step size determination in LS-
DYNAR
and AbaqusR
provides reliable results.
140 APPENDIX A. ALE IN ABAQUSR
VS. LS-DYNAR
Table A.5: Effect of different time step sizes on calculation time
Mesh size AbaqusR
LS-DYNAR
Automatic 00:13:30 00:21:18
1e-5 00:33:34 02:46:10
1e-6 03:02:23 (not converged) 17:58:30
Conclusion
In this research a strip footing problem, for which an analytical solution is available,
has been simulated by two commercial codes AbaqusR
and LS-DYNAR
. ALE method
capabilities in geotechnical applications were evaluated by comparing the results to
Lagrangian method. Although the Lagrangian solution converged, the mesh quality
reduced drastically after app. 25 cm of penetration. In contrast, by using remapping
feature in ALE, the level of mesh distortion was limited, and therefore the accuracy
and reliability of the results was increased.
In order to study the mesh size effect, five mesh sizes where used: 2.5, 5, 10, 12.5, and 20
cm. The AbaqusR
results showed an accurate behaviour in comparison to the analytical
solution with noticeable fluctuations. On the other hand, the LS-DYNAR
results shows
stable solutions but in expense of losing accuracy.
Automatic time step size allocations feature in both codes were compared to smaller
manual max. time step assignment. It was observed that there is no significant im-
provement in results by decreasing time step sizes. This shows the robustness of auto-
matic time step size technique implemented in both AbaqusR
and LS-DYNAR
. Com-
putation time was also evaluated between AbaqusR
and LS-DYNAR
. In case of using
large mesh sizes, no major difference was observed. However, by using finer meshes the
computation cost of AbaqusR
was less compared to LS-DYNAR
. Comparing pressure
values with empirical solution, the error of AbaqusR
was less than 1% while the error
of LS-DYNAR
was less than 5%.
Appendix B
Remapping methods
B.1 Donor cell scheme
The donor cell algorithm is a simple first-order accurate Godunov method applied to
the advection equation. This algorithm satisfies the monotonicity and conservative
requirements. The concept of a donor cell is to assume that the distribution of the
initial state variable over an element is constant and illustrated by the value of the
element center, φn
(i+1/2). To avoid multiple values at interfaces, the state variable value
at interface is determined by Eq. B.1.1. The new state variable for the next step,
φ(n+1)
(i+1/2), is calculated based on the current value of the cell, φn
(i+1/2), the state variable at
cell interfaces, φinand φn
(i+1). The calculation is done through the volumetric averaging
which accounts for the flux at both interfaces, fn
iand fn
(i+1), and the current volume of
the cell, fn
(i+1/2). The graphical illustration of the donor cell method for one dimensional
advection is presented in Fig. B.1.
In spite of fast calculation time, the donor cell scheme is only first-order accurate and
strongly diffusive and dispersive. However, strong diffusion hides the dispersion errors.
Excessive reduction in the value of the variable which contains low-speed, high frequen-
cies is also another shortcoming of donor cell scheme. This remapping scheme does
not provide acceptable results in pure Eulerian simulations but may provide reason-
able results in ALE provided that the rezoned mesh is kept as Lagrangian as possible
[Benson,1992b;Souli and Benson,2013].
φn
i=⎧
⎨
⎩
φn
i−1
2
fn
i>0
φn
i+1
2
fn
i≤0(B.1.1)
φn+1
i+1/2=φn
i+1/2fn
i+1/2+φn
ifn
i−φn
i+1 fn
i+1
fn
i+1/2+fn
i−fn
i+1
(B.1.2)
142 APPENDIX B. REMAPPING METHODS
B.2 Van Leer scheme
The Van Leer algorithm is a higher-order Godunov method where a piecewise linear
function is defined to redistribute the initial state variable value inside each element.
Van Leer replaces the piecewise constant distribution in the donor cell scheme with a
linear or higher-order interpolation function. An example of such functions is presented
in B.2.1, where a linear distribution of state variable, φ, is defined based on the values
of the adjacent elements. Moreover, an element level constraint B.2.2 should be applied
to ensure conservation. Both the function and the number of adjacent elements can
be modified to reach another forms of distribution. A simple graphical description for
one dimensional advection using van Leer method is presented in Fig. B.1.
φ(x|i:i+1)=φi+1
2+φi+3/2−φi−1/2
xi+3/2−xi−1/2x−xi+1/2(B.2.1)
xi+1
xi
φ(x)dx =φi+1/2x−xi+1/2(B.2.2)
The van Leer scheme (MUSCL) is monotonic, conservative, second-order accurate, at
the expense of more computation time. Due to averaging function among neighboring
cells, this method can be only applicable for rectangular elements. Although this
method is believed to be second-order accurate, distorted elements can still introduce
some errors, and the scheme actually becomes less accurate [Benson,1992a;van Leer,
1997;Souli and Benson,2013].
B.3 Momentum advection
Both previous methods are applicable to map cell-centered solution variables. However,
there are variables, such as velocity, which is located at nodes. Transportation of
velocity is crucial since it is used to calculate the momentum. To resolve this issue,
two approaches are applicable. In the first approach, a so-called dual mesh method can
be used to generate an auxiliary mesh, whose cell-centers are located at the nodes of the
reference mesh. The second approach uses the cell-centered remapping algorithms to
transport the node-centered variables by defining auxiliary parameters. In this latter
method, the general idea is to construct a relationship between node-centered and
cell/element-centered variables. By using any of these approaches, it is possible to use
the first or second order cell-centered remapping algorithm described in the previous
sections [Amsden and Hirt,1973]. The following is focused on the nodal advection
methods based on auxiliary parameters.
B.3.1 Element center projection
Amsden and Hirt [1973] used momentum to transport the nodal velocities. The mo-
mentum is transported using the cell-centered remapping algorithms (donor cell or
van Leer). The updated velocity is back-calculated from the change in the updated
momentum values. The calculation procedure is shown in Eqs. B.3.1-B.3.3, using the
notation presented in Fig. B.1a.
B.3. MOMENTUM ADVECTION 143
Figure B.1: Illustration of donor cell advection algorithm (a-c) described in Eq. B.1.1-
B.1.2 and van Leer (d-f) advection algorithms described in Eq. B.2.1-B.2.2 in one
direction. The horizontal axis depicts the node coordinates, and the vertical axis
represents the arbitrary solution variable. (a) Initial state variable distribution after
the Lagrangian step, (b) Node coordinates after rezoning step, (c) New state variable
distribution in element after transport for donor cell method. (d) Initial state variable
value distribution and auxiliary lines for distribution calculation, (e) The calculated
piecewise distribution, (f) New state variable distribution in element after transport
for van Leer method
144 APPENDIX B. REMAPPING METHODS
ΔPi−1/2=pi−1fi−1−pifi(B.3.1)
Pn+1
i=Pn
i+1
2ΔPi−1
2+ΔPi+1
2(B.3.2)
vn+1
i=Pn+1
i
Mn+1
i
(B.3.3)
Where
pi+1/2=ρi+1/2¯vi+1/2(B.3.4)
¯vi+1/2=1
2(vi+vi+1) (B.3.5)
Pi=Mivi(B.3.6)
Pi+1/2=Mi+1/2¯vi+1/2(B.3.7)
In the equations above, P(i+1/2), is the element momentum, Piis the nodal momentum,
p(i+1/2) is the specific momentum, viis the nodal velocity, v(i+1/2) is the mean element
velocity, fis the flux, and Mis the mass.
This method is easy to implement since few extra parameters are required to be defined.
However, it is dispersive, and monotonicity is violated in areas near discontinuities
[Amsden et al.,1980;Benson,1992a].
B.3.2 Half-Index-Shift (HIS)
The Half-Index-Shift (HIS) algorithm was developed by Benson [1992a] to address
the dispersion and monotonicity problems associated with the former method. Two
auxiliary element-centered parameters are defined which relate to node-centered state
variables through Eq. B.3.8. Subsequently, ψ1
(i+1/2) and ψ2
(i+1/2) are transported using
the element centered remapping methods. The updated velocity will be calculated by
Eq. B.3.9.
ψ1
i+1/2=viψ2
i+1/2=vi+1 (B.3.8)
vi=1
2×Mi+1
2ψ1
i+1
2
+Mi−1
2ψ2
i−1
2
Mi+1
2+ΔMi−ΔMi+1 (B.3.9)
Where M is the mass, viis the velocity at node i, and ψ(i+1/2) is the auxiliary parameter
for element center i+1/2. The superscript on the ψdenotes the velocity at each element
node in one dimension. An illustration of the parameters is shown in Fig. B.2.
B.3. MOMENTUM ADVECTION 145
Figure B.2: Illustration of a) element center projection and b) half-index shift method
for momentum advection
146 APPENDIX B. REMAPPING METHODS
Appendix C
Guidelines and Documentation for
Implementation of Hypoplasticity
User Material Subroutine UMAT in
LS-DYNA R
C.1 Introduction to documentation
The goal of this documentation is to provide a complete and unambiguous guideline
for the implementation of a user-defined subroutine through LS-DYNAR
hydrocode.
This documentation is applied for a hypoplastic material model used for capturing the
behavior of course materials as sand. It is good to notice that the original code for this
subroutine is implemented in the form of a UMAT subroutine in AbaqusR
/Implicit, so
this Interface also can employ an implicit UMAT version to Explicit. This documen-
tation comes with the following files:
1. dyna21.f which is needed to include the interface with required UMAT.
2. Shared library-LS-DYNA-nmake.zip which should be extracted to be used for
generation of the lsdyna.exe file
3. Oedometer.k source file for the one element test used for validation.
4. Triaxial.k source file for the one element test used for validation.
5. This documentation file as Pdf.
To implement a user material subroutine in LS-DYNAR
, you need to modify a file
called dyna21.f. This file includes up to 10 subroutines that can be implemented
in LS-DYNAR
, which can be found in the attached folder with this documentation.
This modification should be done with a corresponding compatible software system
mentioned in section C.2.
In this implementation, the subroutine umat43 will be modified to include the im-
plemented interface which calls the hypoplastic UMAT mentioned before. For more
148 APPENDIX C. HYPOPLASTIC IMPLEMENTATION GUIDELINES
information regarding the UMAT implementation, the reader is advised to read LSTC
[2015]. The next step is to generate a lsdyna.exe file which is described in details in
section C.3. For more discussion about LS-DYNAR
interface a describtion is brought
in section C.4.
To invoke the implemented subroutine in a model problem, the user should define a part
in the keyword input deck that uses *MAT_USER_DEFINED_MATERIAL_MODELS
with appropriate input parameters.
C.2 Software requirement for generating the user-
defined subroutines in LS-DYNA R
Depending on the LS-DYNAR
version that it is installed on the PC, one can choose
the compatible compiling software for invoking the user-defined material model.
Two compatible versions can be used:
•First group
LS-DYNAR
9.1
Intel Parallel Studio XE 2013
Visual studio 2012
•Second group
LS-DYNAR
8.1
Intel FORTRAN 11.1
Visual studio 2008
C.3 Generating the lsdyna.exe file
To implement a user-defined interface in LS-DYNAR
, a new executable file should
be replaced with the default installed lsdyna.exe. The object files and multiple source
routines required for this step is made available by LSTC for both Windows R
or Linux
platforms. In the following steps, the generation of lsdyna.exe file is explained:
1. The interface with the hypoplastic UMAT code should be placed in file dyna21.F
which is attached to this documentation.
2. Using the command prompt of the Intel Composer XE 2012 with Intel R
64 visual
studio 2012.
3. Reaching the shared library folder provided by DYNAmore support and is at-
tached to this documentation in the Shared library-LS-DYNA-nmake.zip
4. Using the command nmake.exe
5. Then the generated file will be found in the same shared library folder
C.4. LS-DYNAR
UMAT INTERFACE IMPLEMENTATION 149
Table C.1: UMAT interface variables
Input variables for UMAT Description
sig(6) stresses in previous time step
eps(6) strain increments
epsp effective plastic strain in previous time step
hsv(*)†history variables in previous time step excluding plastic strain
cm(*)†material constants array
dt1 current time step size
tt current time
temper current temperature
failel flag indicating failure of an element
capa the transverse shear correction factor for shell elements
crv(lq1,2,*) array representation of curves defined in the keyword deck
etype character string that equals solid, shell, or beam
†The * denotes to a user-defined array size. the maximum size of history variable array and
material constants array is by default limited to 142 and 48, respectively.
Table C.2: Stress/strain assignment order in LS-DYNAR
UMAT
Stress direction Stress array index Strain direction Strain array index
σ11 sig(1) 11 eps(1)
σ22 sig(2) 22 eps(2)
σ33 sig(3) 33 eps(3)
σ12 sig(4) 12 eps(4)
σ23 sig(5) 23 eps(5)
σ31 sig(6) 31 eps(6)
6. Replacing the lsdyna.exe in the installed LS-DYNAR
folder ...\LSDYNA\program
with the new generated file in the previous step
C.4 LS-DYNA R
UMAT interface implementation
The implemented interface subroutine is placed in subroutine UMAT43 which is found
in dyna21.f file. This interface calls the hypoplastic UMAT which will be placed all
together in subroutine UMAT43.
The Subroutine UMAT43 has a heading as follows (cm, eps, sig, epsp, hsv, dt1, capa,
etype, tt, temper, failel, crv). The description of these parameters are listed in Tab.
C.1.
The numbering order for assigning the stress/strain indices in stress/strain array in
LS-DYNAR
UMAT interface follows the method illustrated in Tab. C.2.
150 APPENDIX C. HYPOPLASTIC IMPLEMENTATION GUIDELINES
Table C.3: *MAT_USER_DEFINED_MATERIAL_MODELS keyword required in-
put parameters for Hypoplastic model
Parameter Description Value
RO Mass density Problem-dependent
MT User material type 43
LMC Length of material constant array (number of material constants to be input) 16
NHV Number of history variables to be stored 16
IORTHO Orthotropic flag 0 (Non-orthotropic)
IBULK Address of bulk modulus in material constants array Problem-dependent
IG Address of shear modulus in material constants array Problem-dependent
IVECT Vectorization flag 0 (non-vectorized)
Table C.4: Parameters input order within the UMAT
cm(1)=phi_deg cm(9)=beta
cm(2) not used cm(10)=m_R
cm(3)=hs cm(11)=m_T
cm(4)=en cm(12)=r_uc
cm(5)=ed0 cm(13)=beta_r
cm(6)=ec0 cm(14)=chi
cm(7)=ei0 cm(15) not used
cm(8)=alpha cm(16)=e0
C.5 Invoking user defined keyword in *.k file
The keyword *MAT_USER_DEFINED_MATERIAL_MODELS in the LS-DYNAR
finite element model set up, is used to define the user material parameters. The required
parameters for this subroutine are listed in Tab. C.3 and Tab. C.4.
In LS-DYNAR
, defining bulk and shear modulus as an input is compulsory for every
material model. However, the hypoplastic material model does not take an initial value
for such parameters as input, since such parameters are calculated internally. Never-
theless, it is possible to calculate the initial bulk modulus with available hypoplastic
parameters using Eq. (C.5.1).
K=1
3
hs
n1+ 1
ep3ps
hs1−n
(C.5.1)
Where Kis the bulk modulus and epis the void ratio at given skeleton pressure ps[3].
Other parameters will be in Tab. C.3.
During the calculation, LS-DYNAR
calls the corresponding UMAT subroutine based
on the used element type (2D plane strain, plane stress, 3D, etc.) for state variable
update. The subroutine usually takes the following form:
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160 APPENDIX C. HYPOPLASTIC IMPLEMENTATION GUIDELINES
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