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Numerical Simulation of Ice Ridge Breaking
Aleksei Alekseev
Master Thesis
presented in partial fulfillment of the requirements for the double degree:
“Advanced Master in Naval Architecture” conferred by University of Liege "Master of Sciences in Applied Mechanics, specialization in Hydrodynamics, Energetics
and Propulsion” conferred by Ecole Centrale de Nantes
developed at University of Rostock in the framework of the
“EMSHIP”
Erasmus Mundus Master Course
in “Integrated Advanced Ship Design”
Ref. 159652-1-2009-1-BE-ERA MUNDUS-EMMC
Supervisor: Prof. Robert Bronsart, University of Rostock
Reviewer: Prof. Hervé Le Sourne, ICAM
Rostock, January 2016
Alekseev Aleksei
2 Master’s Thesis developed at the University of Rostock
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Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 3
ABSTRACT
Increasing economic and industrial activities in Polar Regions require new engineering
solutions to deal with arctic hazards. One of the main challenges for vessel navigation in ice are
pressure ice ridges — sets of randomly oriented large pieces of sea ice along a line with keel and
sail parts. These ice ridges may affect the normal exploitation of ice-going vessels, subsea
pipelines, and equipment.
The objective of this master thesis was to develop and implement algorithms in a numerical
tool, capable of simulating the process of ship hull breaking through pressure ice ridge. The tool is
based on the idea to implement Discrete Element Method (DEM) and corresponding code
developed at Hamburg Ship Model Basin (HSVA) for simulation of ice ridges creation.
In the thesis the following aspects have been covered: theoretical information on pressure ice
ridges and the processes of their creation in nature and ice tank; review of available at present
methods to estimate ridge and structure interaction; general idea of DEM and its application; ridge
and hull interaction.
In the present project the author focuses on the following: modification of theoretical DEM
algorithms in order to be adopted for ridge breaking simulation; method to introduce and to treat
complex concave hull geometry with existing DEM software, taking into account adopted data
structures of three-dimensional DEM; calculation of hydrostatic properties, inertial and other
relevant characteristics of the ship hull (buoyancy, thrust, gravity, restoring forces); numerical
integration of equations of motion of ship as discrete element in order to observe realistic
performance of the vessel in an ice ridge. Interaction with level ice is not simulated but
implemented in the form of an added ice resistance based on semi-empirical formulae of Lindqvist
(1989).
The software is able to provide visualization of ship hull/ice ridge interaction, calculate ship
resistance, position, velocity, acceleration, thrust, and other relevant parameters during breaking
through an ice ridge, and simulate ramming operations corresponding to reality when ship is getting
stuck in the ridge.
The code has been validated with corresponding experimental data, provided by Hamburg Ship
Model Basin. The results have been discussed and proposals for further calibration and validation
of the existing model have been given. Finally some ideas are expressed on how to use developed
methods to simulate interaction of floating structures with other types of ice formations.
Alekseev Aleksei
4 Master’s Thesis developed at the University of Rostock
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Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 5
CONTENTS
INTRODUCTION .................................................................................................................. 15
Master’s thesis ......................................................................................................................... 16
Implementation ........................................................................................................................ 16
Part I. PRINCIPLES OF PRESSURE ICE RIDGES ......................................................... 17
1. FORMATION OF PRESSURE ICE RIDGES ............................................................ 18
1.1. Natural creation process ....................................................................................... 18
1.2. Ice ridges in ice model basin ................................................................................ 19
2. CONFIGURATION, SIZE AND SHAPE OF ICE RIDGES ..................................... 21
3. SHIP BREAKING THROUGH AN ICE RIDGE ....................................................... 22
Part II. APPLICATION OF DEM FOR ICE RIDGES SIMULATION .......................... 24
1. OVERVIEW OF AVAILABLE METHODS FOR RIDGE/STRUCTURE
INTERACTION. .......................................................................................................... 24
2. GENERAL IDEA OF DISCRETE ELEMENT METHOD ........................................ 25
3. APLLICATION OF DEM FOR ICE-RELATED PROBLEMS ................................. 26
4. OVERVIEW OF DEM ALGORITHM FOR INTERACTION OF SHIP AND ICE
RIDGE ......................................................................................................................... 28
Part III. SIMULATION OF ICE RIDGE/SHIP HULL INTERACTION ....................... 31
1. SIMULATION DOMAIN ........................................................................................... 31
2. INTRODUCING SHIP HULL GEOMETRY ............................................................. 32
2.1. Data structures of DEM ............................................................................................. 32
2.1.1. Polyhedron geometry ....................................................................................... 32
2.1.2. Computer data structures ................................................................................. 33
2.2. Ship mesh as polyhedron ........................................................................................... 34
3. INTRODUCING ICE RIDGE ..................................................................................... 35
3.1. Ridge creation as input ......................................................................................... 35
Alekseev Aleksei
6 Master’s Thesis developed at the University of Rostock
3.2. Mechanical properties and dimensions of ice pieces ............................................ 37
4. ESTIMATION OF SHIP HULL INERTIA TENSOR ................................................ 38
5. QUATERNIONS AND ORIENTATION IN SPACE ................................................. 39
5.1. Euler angles .......................................................................................................... 39
5.2. Quaternions ........................................................................................................... 41
5.3. Mesh coordinates in global reference frame ......................................................... 44
6. GRAPHICAL VIZUALIZATION ............................................................................... 45
7. BUOYANCY CALCULATION ................................................................................. 47
7.1. Varying drafts, pitch and roll angles. Displacement formula ................................... 48
7.1.1. Varying drafts, pitch and roll angles................................................................ 48
7.1.2. Displacement formula ...................................................................................... 48
7.2. Calculation of displacement. Simpson’s First Rule .................................................. 49
7.2.1. Calculation of displacement ............................................................................ 49
7.2.2. Simpson’s First Rule ....................................................................................... 49
7.3. Gift wrapping algorithm ............................................................................................ 50
7.4. Calculation of cross section area ............................................................................... 52
7.5. Buoyancy table and buoyancy moment .................................................................... 53
7.5.1. Buoyancy table ................................................................................................ 53
7.5.2. Buoyancy moment ........................................................................................... 53
8. EQUATIONS OF MOTION ........................................................................................ 54
8.1. Rectilinear degrees of freedom ............................................................................. 54
8.2. Rotational degrees of freedom .............................................................................. 54
9. PREDICTOR-CORRECTOR NUMERICAL INTEGRATOR ................................... 55
9.1. Predictor step ........................................................................................................ 55
9.2. Corrector step ....................................................................................................... 56
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 7
10. HANDLING NON-CONVEX GEOMETRIES OF SHIP HULL ........................... 57
11. BOUNDING BOXES AND NEIGHBORHOOD LIST .......................................... 60
12. VOLUME OF OVERLAP ....................................................................................... 61
13. CALCULATION OF FORCES ............................................................................... 62
13.1 Elastic force .......................................................................................................... 62
13.2 Damping force ...................................................................................................... 63
13.3 Friction force ........................................................................................................ 64
13.4 Dissipative force ................................................................................................... 64
13.5 Cohesion force ...................................................................................................... 65
13.6 Contact forces and torques ................................................................................... 65
13.7 Viscous drag force ................................................................................................ 65
13.8 Viscous drag torque .............................................................................................. 66
13.9 Gravity of ice elements ......................................................................................... 66
13.10 Buoyancy of ice elements ................................................................................. 66
13.11 Buoyancy of ship .............................................................................................. 66
13.12 Gravity of ship .................................................................................................. 67
13.13 Gravity-Buoyancy torque for ship .................................................................... 67
13.14 Propeller thrust .................................................................................................. 68
13.14.1. Propeller curve ............................................................................................. 68
13.14.2. Influence of ship hull ................................................................................... 69
13.15 Resistance in level ice ....................................................................................... 71
13.15.1. Concept ........................................................................................................ 71
13.15.2. Geometry of ship hull .................................................................................. 71
13.15.3. Crushing resistance ...................................................................................... 72
13.15.4. Breaking resistance ...................................................................................... 73
Alekseev Aleksei
8 Master’s Thesis developed at the University of Rostock
13.15.5. Submersion resistance ................................................................................. 73
13.15.6. Level ice resistance with speed.................................................................... 74
14. IMPLEMENTATION OF RAMMING ................................................................... 75
15. PROGRAM RUNNING ........................................................................................... 76
15.1. Generalities ........................................................................................................... 76
15.2. Scale of simulation, ridge dimensions and ice mechanical properties ................. 77
15.2.1. Input of ridge dimensions .............................................................................. 77
15.2.2. Input of ice rubble dimensions ...................................................................... 77
15.2.3. Input of ice properties .................................................................................... 77
15.2.4. Input of forces coefficients ............................................................................ 78
15.3. Creation of ice ridge ............................................................................................. 78
15.4. Meshing in CAD system....................................................................................... 79
15.4.1. Hull surface .................................................................................................... 79
15.4.2. Meshing ......................................................................................................... 80
15.4.3. Mesh input files ............................................................................................. 81
15.5. Ship input data ...................................................................................................... 81
15.6. Simulation ............................................................................................................. 82
15.7. Visualization and data post processing ................................................................. 83
Part IV. CODE VALIDATION ............................................................................................ 84
CONCLUSIONS AND PROPOSALS .................................................................................. 91
ACKNOWLEDGEMENTS ................................................................................................... 94
REFERENCES ....................................................................................................................... 95
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 9
List of Figures
Figure 1. Pressure ice ridges in nature [4] ................................................................................ 15
Figure 2. Ice ridges in nature [4] .............................................................................................. 17
Figure 3. Scheme of ice ridge creation in nature [26] .............................................................. 18
Figure 4. Scheme of ice ridge creation in ice tank [1] ............................................................. 20
Figure 5. Steel beam and ice blocks during ridge creation ...................................................... 20
Figure 6. Ice ridges created in model basin .............................................................................. 20
Figure 7. Main parts of an ice ridge [4] .................................................................................... 21
Figure 8. Ridge dimensions ...................................................................................................... 21
Figure 9. Phases of ship breaking through an ice ridge [3] ...................................................... 22
Figure 10. Ship before entering ridge modified from [1] ......................................................... 22
Figure 11. Ship bow starts penetrating ice ridge [1] ................................................................ 23
Figure 12. Ship middle part in contact with ridge [1] .............................................................. 23
Figure 13. Example of DEM application [27] .......................................................................... 25
Figure 14. Different ice formations .......................................................................................... 26
Figure 15.General algorithm of the software ........................................................................... 28
Figure 16. Input of ice ridge into simulation domain ............................................................... 29
Figure 17. Ship hull mesh in simulation domain ..................................................................... 29
Figure 18. Simulation domain .................................................................................................. 31
Figure 19. Entities of a polyhedron .......................................................................................... 32
Figure 20. DEM computer data structures [23]. ....................................................................... 33
Figure 21. Introduction of ship mesh into simulation domain ................................................. 34
Figure 22. Dimensions of ice ridge .......................................................................................... 35
Figure 23. Floating up technique for ridge creation ................................................................. 36
Figure 24. Dimensions of ice pieces ........................................................................................ 37
Figure 25. Classical Euler angles [28]...................................................................................... 39
Figure 26. Example of Gimbal lock [32] ................................................................................. 40
Figure 27. Axes of roll, pitch and yaw motions. ...................................................................... 42
Figure 28. Example of mesh rotation ....................................................................................... 43
Alekseev Aleksei
10 Master’s Thesis developed at the University of Rostock
Figure 29. Mesh rotated by 22.5° ............................................................................................. 43
Figure 30. Initialized simulation domain ................................................................................. 44
Figure 31. Example of .vtk visualization file ........................................................................... 46
Figure 32. Change of bow draft and pitch angle ...................................................................... 47
Figure 33. Various drafts, roll, and pitch angles ...................................................................... 48
Figure 34. Cross sections of underwater part of hull. .............................................................. 49
Figure 35. Cross section and contour line points ..................................................................... 50
Figure 36. Array of point P ...................................................................................................... 50
Figure 37. Gift wrapping algorithm ......................................................................................... 51
Figure 38. Contour line array ................................................................................................... 51
Figure 39. Example of section triangulation ............................................................................ 52
Figure 40. Buoyancy table ....................................................................................................... 53
Figure 41. Restoring buoyancy moment .................................................................................. 53
Figure 42. Discrete elements penetration of single concave mesh .......................................... 57
Figure 43. Subdivision into convex sub-meshes ...................................................................... 58
Figure 44. Multi-mesh translation ............................................................................................ 58
Figure 45. Wrong rotation of sub-meshes ................................................................................ 59
Figure 46. Proper rotation of sub-meshes ................................................................................ 59
Figure 47. Example of bounding boxes of two elements [23] ................................................. 60
Figure 48. Overlap of two discrete elements ............................................................................ 61
Figure 49. Definition of characteristic length .......................................................................... 63
Figure 50. Direction of elastic force [5] ................................................................................... 63
Figure 51. Trilinear interpolation of displacement ................................................................... 66
Figure 52. Propeller curve ........................................................................................................ 69
Figure 53. Interpolation of KT value ......................................................................................... 70
Figure 54. Description of hull form [3] .................................................................................... 71
Figure 55. Hull angles at different sections [3] ........................................................................ 72
Figure 56. Level ice resistance ................................................................................................. 74
Figure 57. Ramming operations and ramming cycle [1], [3] ................................................... 75
Figure 58. Algorithm of working with software ...................................................................... 76
Figure 59. Input of ridge dimensions ....................................................................................... 77
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 11
Figure 60. Input of rubble dimensions ..................................................................................... 77
Figure 61. Input of ice properties ............................................................................................. 77
Figure 62. Input of forces coefficients ..................................................................................... 78
Figure 63. Running NumericalRidge.exe ................................................................................. 79
Figure 64. Hull surface with convex parts ............................................................................... 80
Figure 65. Hull mesh subdivided into convex parts ................................................................. 80
Figure 66. Position of centroid of each sub mesh .................................................................... 81
Figure 67. Ship input data ........................................................................................................ 81
Figure 68. Output during simulation ........................................................................................ 82
Figure 69. Data visualization ................................................................................................... 83
Figure 70. Ship velocity and thrust charts as output of simulation .......................................... 83
Figure 71. Ship velocity (ridge 1) ............................................................................................ 85
Figure 72. Ship velocity (ridge 2) ............................................................................................ 86
Figure 73. Ship velocity (ridge 3) ............................................................................................ 87
Figure 74. Breaking through ridge 1 ........................................................................................ 88
Figure 75. Breaking through ridge 2 ........................................................................................ 88
Figure 76. Breaking through ridge 3 ........................................................................................ 88
Alekseev Aleksei
12 Master’s Thesis developed at the University of Rostock
List of Tables
Table 1. Comparison of analytical, experimental and numerical solutions ............................. 15
Table 2. Input of ice properties ................................................................................................ 37
Table 3. Range of buoyancy precalculation ............................................................................. 48
Table 4. List of forces and torques ........................................................................................... 62
Table 5. Ship data model №1 ................................................................................................... 84
Table 6. Ridge dimensions and constant ice parameters (1) .................................................... 85
Table 7. Input varied parameters (1) ........................................................................................ 85
Table 8. Ridge dimensions and constant ice parameters (2) .................................................... 86
Table 9. Input varied parameters (2) ........................................................................................ 86
Table 10. Ridge dimensions and constant ice parameters (3) .................................................. 87
Table 11. Input varied parameters (3) ...................................................................................... 87
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 13
Declaration of Authorship
I declare that this thesis and the work presented in it are my own and has been generated by
me as the result of my own original research.
Where I have consulted the published work of others, this is always clearly attributed.
Where I have quoted from the work of others, the source is always given. With the exception of
such quotations, this thesis is entirely my own work.
I have acknowledged all main sources of help.
Where the thesis is based on work done by myself jointly with others, I have made clear exactly
what was done by others and what I have contributed myself.
This thesis contains no material that has been submitted previously, in whole or in part, for the
award of any other academic degree or diploma.
I cede copyright of the thesis in favour of the University of Rostock.
Date: January 15, 2016 Signature
Alekseev Aleksei
14 Master’s Thesis developed at the University of Rostock
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Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 15
INTRODUCTION
Over the last few years there is a growing interest in the Arctic in terms of significant
hydrocarbon reservoirs, ship navigation along the Northern Sea Route and various scientific
researches. One of the main hazards for Arctic offshore structures and navigating ships are various
ice formations and particularly pressure ice ridges — sets of randomly oriented large pieces of sea
ice along a line. These ice ridges may affect the normal exploitation of ice-going vessels, subsea
pipelines, and equipment (Figure 1).
Figure 1. Pressure ice ridges in nature [4]
Nowadays still the main tool available for prediction of performance and interaction of ships
and structures in ice is model scale experiments in ice tanks. On the other hand, as in other scientific
branches numerical methods and numerical simulations for ice-related problems are gaining more
and more interest. A brief summary of advantages and disadvantages of different approaches is
presented in Table 1.
Table 1. Comparison of analytical, experimental and numerical solutions
Models tests Numerical
simulation Analytical methods
Physical
nature +++ ++ +
Cost + ++ +++
Ease of
application + ++ +++
Accuracy +++ ++ +
Alekseev Aleksei
16 Master’s Thesis developed at the University of Rostock
In this way there is growing demand and interest both in academic and industrial world for
developing numerical simulation of ice, its various formation, and ice-structure interaction.
Numerical simulation of ship breaking through an ice ridge could contribute to prediction of hull
performance at early design stage without carrying out costly and time-consuming experiments,
but providing more information and better accuracy comparing to available analytical and semi-
empirical solutions.
Master’s thesis
In this master’s thesis the author is focused on numerical simulation of ship breaking through
an ice ridge. Among numerical simulations the Discrete Element Method (DEM) is currently
considered to be a suitable tool for modelling ice-related problems, since ice blocks undertake large
independent displacements during interaction and the ice ridge itself cannot be really represented
as continuum medium. Modern commercial software dealing with DEM is usually available for
soil mechanics, rock engineering, and simulation of particles motion. But in order these to be
applied for ice problems still there is lack of physical models for ice as material. Moreover, such
software is not adapted for maritime industry and cannot be used directly for simulation of shipping
in ice sea routes.
The goal of the thesis is to develop a software that would be able to simulate the interaction
process between ship hull and ice ridge. Ice ridge characteristics (dimensions and mechanical
properties) and ship particulars serve as input for the software. As output the software will provide
corresponding data on ship performance during breaking through an ice ridge and visualization of
the process.
Implementation
The described below software is based on available code from Hamburg Ship Model Basin
(HSVA) for simulation of ice ridge creation, based on Discrete Element Method. The code is
developed in FORTRAN programming language due to its capabilities for scientific numerical
computation. In the scope of the thesis the DEM algorithms and the aforementioned software will
be extended further towards ship hull simulation with consideration of corresponding physical
phenomena.
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 17
Part I. PRINCIPLES OF PRESSURE ICE RIDGES
An ice ridge can be defined a line or a wall of broken ice forced up by pressure between
relatively large ice floes. These ridges are one of the most difficult obstacles for ice-going vessels
(Figure 2). Depending on their age and formation process, sea ice ridges can be found in many
different sizes, strength and shapes. The percentage of area covered by sea ice ridges is small in
relation to the whole ice covered area. In contrast to this, their mass can be in fact one third of the
total ice mass [29].
Figure 2. Ice ridges in nature [4]
Alekseev Aleksei
18 Master’s Thesis developed at the University of Rostock
1. FORMATION OF PRESSURE ICE RIDGES
1.1. Natural creation process
Pressure ice ridges are formed due to stresses in level ice or between ice floes driven against
each other. These stresses arise from various external factors, such as currents, wind drag and
thermal expansion. When stresses exceed certain level of strength, ice cover breaks, crushes and
bends. As a result of this, a number of ice discrete blocks appear between two ice floes or edges of
level ice cover (Figure 3).
Figure 3. Scheme of ice ridge creation in nature [26]
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 19
Some ice blocks can refreeze together. This process is called consolidation. If sea ice ridges
survive one summer, they are named second-year sea ice ridges. After lasting more than one
melting period, sea ice ridges are called multi-year sea ice ridges. Multi-year sea ice ridges mainly
differ from first-year sea ice ridges in their degree of consolidation. Consequently, the strength of
sea ice ridges increase with their age.
This thesis focuses on first-year ice ridges, in which ice blocks are poorly bounded between
each other.
1.2. Ice ridges in ice model basin
The process of ridge creation in HSVA ice tank is described in Figures 4 - 6. This process is
similar to natural creation process. First of all the layer of level ice of a given thickness is created,
using adopted freezing technique. Afterwards a steel beam with inclined faces is placed at certain
draft across the tank and fixed in order to prevent its movement. The ice cover, preliminarily cut
into stripes of certain dimensions, is pushed by main carriage against steel beam. The level ice
breaks due to bending and crushing during interaction with the beam and large number of ice blocks
is generated in front of the steel beam. This procedure is repeated several times until the ridge of a
prescribed width is formed. Finally, two rest parts of level ice cover are pushed towards each other
on top of the keel part until the ridge is closed from above. Due to the buoyancy of ice blocks the
part of level ice cover within the width of the ridge is lifted slightly above the free surface.
Alekseev Aleksei
20 Master’s Thesis developed at the University of Rostock
Figure 4. Scheme of ice ridge creation in ice tank [1]
Figure 5. Steel beam and ice blocks during ridge creation
Figure 6. Ice ridges created in model basin
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 21
2. CONFIGURATION, SIZE AND SHAPE OF ICE RIDGES
Typical ice ridge consists of three main parts: ridge keel (submerged loosely bounded ice
blocks), ridge sail (ice blocks above water surface) and consolidated layer (refrozen together ice
blocks). The graphical representation of a ridge profile is depicted in Figure 7.
Figure 7. Main parts of an ice ridge [4]
Based on the in-situ data, Timco and Burden [6] provide approximate dependencies between
various ridge dimensions such as:
Keel height Hk
Sail height Hs
Hk = 4.4 Hs
Keel width Wk
Wk = 3.9 Hk
Wk = 15.1 Hs
Sail width Ws
Cross sectional area of keel Ak
Cross sectional area of sail As
Ak = 8.0 As
Angle of keel inclination αk
αk = 26.6°
Angle of sail inclination αs
αs = 32.9° (Beaufort sea) or αs = 20.7° (temperate seas)
Figure 8. Ridge dimensions
Alekseev Aleksei
22 Master’s Thesis developed at the University of Rostock
3. SHIP BREAKING THROUGH AN ICE RIDGE
The process of ship breaking through an ice ridge can be described by a certain number of
phases, all of which are illustrated in Figure 9.
Figure 9. Phases of ship breaking through an ice ridge [3]
Phase 1.
Before entering the ridge (D), ship accelerates in
level ice and due to higher velocity the total ice
resistance is also increasing (A-B-C). When ship
reaches constant speed in level ice the ice resistance is
equilibrated with thrust from the propeller. In this
point the ship obtains the maximum kinetic energy
before entering the ridge (Figure 10).
Phase 2.
In the beginning of the phase 2 (D) the ship’s
bow starts interacting with the ridge (Figure 11).
Due to accumulation of ice blocks in the keel part, significant additional ridge resistance appears.
The ship starts to spend its accumulated kinetic energy on displacing the ice blocks around the hull.
Consequently the ship’s velocity and acceleration are decreasing.
Figure 10. Ship before entering ridge
modified from [1]
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 23
Figure 11. Ship bow starts penetrating ice ridge [1]
Phase 3.
After most of the ice blocks have been moved
around the hull, the bow part of the ship is leaving
the ice ridge. The parallel middle body now is in
contact with ice blocks. In this stage only friction
forces between hull and ice blocks create ridge
resistance, which is smaller comparing to the
Phase 2.
Phase 4.
Ship is continuing going through the ridge with
dominating friction forces. Thus the ice resistance
is relatively stable during this phase (Figure 12).
Since propeller thrust increases slightly due the smaller speed, the ship is able to start gaining its
velocity in level ice.
Phase 5.
Parallel middle body left the limits of ice ridge and the ship’s stern part now is in contact with
ice blocks. Despite of additional resistance due to presence of appendages, the total ice resistance
is decreasing, as less ice blocks are in contact with the stern part. The phase between points G and
H is a transitional part between ridge resistance and resistance in level ice.
Phase 6.
The ship left the limits of ice ridge. The resistance and thrust correspond to the level ice. Ship
is moving with constant velocity. The channel with broken ice in formed behind the hull.
Figure 12. Ship middle part in contact
with ridge [1]
Alekseev Aleksei
24 Master’s Thesis developed at the University of Rostock
Part II. APPLICATION OF DEM FOR ICE RIDGES
SIMULATION
1. OVERVIEW OF AVAILABLE METHODS FOR RIDGE/STRUCTURE
INTERACTION.
Ice ridge creation, ice ridge breaking and ice ridge/structure interaction are relatively new
research topics. There are no until so far published works on numerical simulation of ship hull
breaking through an ice ridge. The present thesis addresses this issue.
For numerical simulation of only ice ridge creation with Discrete Element Method there have
been developed following works:
• Modelling two-dimensional ridging next to solid structures with ice rubble in the
framework of «Numerical simulation of Systems of Multitudinous Polygonal Blocks» by M.
Hopkins in 1992 [7].
• A program created in the framework of the master thesis «Numerical simulation of Pressure
Ice Ridges» by A. Dummer in 2013 [4]. This work provides numerical simulation of ice ridge
creation with floating up technique, implementing three-dimensional approach based on the DEM
by P. Cundall et al [15].
• A new version of such software with floating up technique, developed at Hamburg Ship
Model Basin (HSVA) in 2015 [5], based on «Understanding the Discrete Element Method» by
H.G. Matuttis and J.Chen [23].
Naval architects and ship designers at early design stage are mostly interested if a ship has
enough capabilities to navigate in ice field, encountering ice ridges. For this issue there is existing
method developed by D. Ehle at HSVA and published in the thesis «Analysis of Breaking Through
Sea Ice Ridges for Development of a Prediction Method» in 2011 [1]. This work enables
calculating ship’s velocity and resistance during ridge breaking, taking into account the shape of
the ship’s hull and its power. The aforementioned method is based on available analytical formulae
and results of model tests in ice tank but does not allow to provide visualization of ship behavior
in the ridge and simulate different possible interaction scenarios.
As for numerical simulation of ice floes/structure interaction there are existing ongoing
research projects implementing DEM simulation of such objects like floating structures and
icebergs in ice floes [11], [12], [13].
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 25
2. GENERAL IDEA OF DISCRETE ELEMENT METHOD
DEM (also called Distinct Element Method) is a numerical computational method, which is
used for computing the motion and mutual interaction of a large number of moving objects. The
method can be programmed and applied in the fields, where big amount of individual objects move
and interact, provided the laws of interaction between them are known (possibly including friction,
hydrostatic, electrostatic, magnetic, gravitational and other types of interaction). It was first
introduced by Cundall [15] for problems in rock mechanics (Figure 13). Today DEM found its
further extension towards EDEM (Extended Discrete Element Method), taking into account
thermodynamic effects and CFD and FEM coupling. DEM finds application in such industries as:
Soil mechanics
Rock engineering
Geophysics
Mineral processing
Powder metallurgy
The domain of the simulation is
represented as a big number of discrete
elements. DEM simulation starts with
initializing particles in the simulation
domain. Along with the current positions and
velocities of the elements, their physical characteristics are used to calculate different kinds of
forces, depending on the studied problem. One of the main dominating type of forces is contact
stresses, derived from elements interaction and contacts with the domain boundaries. These forces
are then used to calculate new positions and orientations of discrete elements and their velocities
and accelerations. The equations of translational and rotational motions of elements are solved
numerically, using known differential equations solvers. The output of the simulation with DEM
allows to find positions of elements and relevant parameters (velocity of the flow, forces, etc.) The
method can be expanded further to be coupled with Finite Element Method (FEM) and
Computational Fluid Dynamics (CFD) in order to take into account more various effects, such as
deformability of discrete elements and their behavior in fluid environment.
Figure 13. Example of DEM application [27]
Alekseev Aleksei
26 Master’s Thesis developed at the University of Rostock
3. APLLICATION OF DEM FOR ICE-RELATED PROBLEMS
During observations of ice motion in the nature and in ice model tank it can be concluded that
many relevant ice formations behave as a set of individual ice pieces, moving relatively freely in
the water and interacting with each other and ship hull. Such ice formations are ice floes, rubble
ice, brash ice, ice pieces in the channel behind the ship, and finally ice ridges (Figure 14). In other
words, when there is no ice breaking process (like in level ice) the ice medium can be treated as
being discrete.
Figure 14. Different ice formations (1 — ice floes, 2 — rubble ice, 3 — brash ice, 4 — ice pieces
behind the hull, 5.1 — ice ridge (keel part), 5.2 — ice ridge (sail part))
As it can be seen from the figures, in the case of ridge the ice pieces in the keel part are loosely
bounded. This is particularly typical for first-year ice ridges, which did not go through refreezing
process. Since in this case the keel part of an ice ridge can be represented as a discrete medium and
at the same time it contributes mostly to the ship resistance during ridge breaking, one could come
to the idea of using the Discrete Element Method for simulation of keel part. On the other hand,
since ship hull behaves as a solid structure when interacting with the ridge, it can be in turn
represented as another discrete element with distinguishing features.
1 2
1
1
1
3
4 5.1 5.2
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 27
In comparison to classical applications of DEM, simulation of ice/ship interaction with ship as
a discrete element has the following particularities:
Ship hull should be floating body with corresponding treatment of the position of free
surface — the buoyancy of the hull should be considered properly;
If angles of pitch, roll and heave or draft are changing during simulation (due to interaction
with the ridge), then corresponding restoring forces and moments should appear;
Ship should possess correct value of kinetic energy before entering the ice ridge;
The thrust of the propeller should be applied to the ship and changed accordingly with the
change of velocity;
When ship’s velocity becomes very small or ship gets stuck in the ridge, then ramming
operation should be applied;
There must be no numerical ‘leakage’ — that is when ice pieces are not supposed to
penetrate inside the ship hull;
Complex ship geometry should be considered (bow and stern part, appendages and bulbous
bow, etc.)
For ice pieces as discrete elements the following features are of importance:
Appropriate mechanical characteristics of ice should be considered for the forces
calculation of theoretical Discrete Element Method;
The presence of water should be included (by introducing buoyancy forces and viscous drag
or other relevant hydrodynamic forces);
Hull-ice and ice-ice friction should be considered properly;
The interaction force between ship hull and ice pieces (elastic force) should be modelled
accordingly;
Because of big number of ice elements in the ridge, appropriate neighborhood and contact
detection algorithms should be introduced for calculation efficiency.
Alekseev Aleksei
28 Master’s Thesis developed at the University of Rostock
4. OVERVIEW OF DEM ALGORITHM FOR INTERACTION OF SHIP
AND ICE RIDGE
The proposed algorithm of DEM [23] modified for consideration of ship and ridge interaction
and implemented into developed software is outlined in Figure 15. The primary attention of this
thesis is focused on the parts of the algorithm, which are highlighted in yellow. Information on
other parts can be found in references [5].
Program start
Initialization of elements
Buoyancy calculation
Propulsion input
Predictor
Update elements
Update bounding boxes
Update neighborhood list
Compute forces and torques
Compute overlap geometry
Main loop:
time increment
Graphical output, velocity
and acceleration output
Graphical output
(initialized simulation
Program End
Corrector
Graphical output
(Buoyancy calculation)
Figure 15.General algorithm of the software
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1. Initialization of elements
In this first part prepared ice ridge must be introduced into simulation domain with number of
ice elements and its positions in space (Figure 16). The ridge can be created with the same software,
using the part responsible for ridge creation simulation. Some important physical quantities must
be defined at initialization stage (cohesion coefficient, viscous damping, friction coefficient,
densities, etc.)
Figure 16. Input of ice ridge into simulation domain
Moreover, since a ship hull is represented as another large discrete element, the geometry of
the hull must be introduced next to the ridge (Figure 17). Hull mesh can be introduced from any
relevant Computer Aided Design software (Rhinoceros, etc.), using conventional file format.
Figure 17. Ship hull mesh in simulation domain
2. Buoyancy calculation
Initialization is followed by calculation of hydrostatic characteristics of ship hull (displacement
and position of center of buoyancy) for various drafts, heel and trim angles. This calculation is
implemented outside of the main time integration loop in order to save computational time.
Alekseev Aleksei
30 Master’s Thesis developed at the University of Rostock
Subsequently in the force computation step buoyancy characteristics will be used in order to
determine buoyancy force and righting moments when position of ship changes during simulation.
3. Propulsion calculation
Then the algorithm continues with the propulsion characteristics of ship’s propeller(s) that are
introduced into simulation with a text file, taken from database of propeller tests. In the subsequent
algorithm stages propeller thrust will be calculated based on current velocity of the ship using linear
interpolation of introduced propeller curves.
4. Main loop: predictor
During predictor step equations of ship and ice blocks motions are solved numerically based
on the assumption that forces acting on discrete elements do not change. Positions and spatial
orientations of elements are calculated.
5. Update elements, update bounding boxes and neighborhood list.
These steps of the algorithm update current coordinates of vertices and equations of mesh faces
of discrete elements and ship hull in space. Also this part helps defining the elements, which might
possibly be in contact during the given time step of simulation. Thus computational time can be
reduced by avoiding calculation of interaction between elements that cannot be in contact.
6. Compute overlap geometry, forces and torques.
Then the interactions between discrete elements are calculated and based on them forces and
torques are estimated, using formulae for elastic, damping, frictional, cohesion, drag and buoyancy
forces for ice elements. Apart from that displacement and buoyancy forces of ship are interpolated
from predefined values, based on the current position and velocity. The thrust is computed from
propeller curve with current velocity of advance.
7. Corrector
During corrector step translational and rotational velocities and accelerations are estimated
from computed forces and torques. Based on these values new positions and spatial orientations
are calculated, numerically solving equations of motions both for ice pieces and ship.
8. Graphical and data output
As the software is supposed to provide visualization of ridge breaking and necessary physical
quantities regarding ship performance (first of all resistance in the ridge and velocity during
breaking) these data are written in output files at certain time steps.
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Part III. SIMULATION OF ICE RIDGE/SHIP HULL
INTERACTION
1. SIMULATION DOMAIN
The main idea of developing the DEM software is to create a numerical model of HSVA ice
tank. Thus the simulation of ship breaking through an ice ridge is going to be performed at model
scale. This is done in order to validate the developed code with existing database of model tests.
Once the numerical ice basin has been created, it can be easily expanded towards full scale
modelling by keeping appropriate values of ice and ship parameters according to Froude similitude,
which is adopted at HSVA for ice model experiments. The simulation domain with adopted
position of global origin is presented in Figure 18.
Figure 18. Simulation domain
Thus, in order to perform simulation of ice ship breaking through an ice ridge the following
information is required:
Dimensions of ice ridge (ridge length, ridge width, keel height, keel width)
Geometry (surface model) and position of ship hull before going through the ridge
Information on ship propeller (propeller characteristics/propeller curve)
Mechanical properties of ice (Young’s modulus, bending strength, density, etc.).
Ice ridge
Ship model
Free surface
Basin walls
Alekseev Aleksei
32 Master’s Thesis developed at the University of Rostock
2. INTRODUCING SHIP HULL GEOMETRY
2.1. Data structures of DEM
2.1.1. Polyhedron geometry
In the implemented algorithm of Discrete Element Method the simulation is performed with
polyhedral particles. Any polyhedron consists of faces, vertices and edges (Figure 19). To represent
a particle as polyhedron two kinds of information need to be specified: geometrical and topological.
The surface of polyhedron consists of polygons. The most universal way to represent the surface
faces is to use triangles. When coordinates of vertices are known, then topological information is
required — describing how entities of a polyhedron are connected between each other.
Figure 19. Entities of a polyhedron
Vertices
Each polyhedron is made up of a number of vertices nv. A vertex of a polyhedron is described
with its three coordinates in 3D Cartesian space V = (Vx, Vy, Vz).
Faces
Polyhedron’s vertices are connected by edges, building up faces. The number of faces is nf.
Since each face of a polyhedron is a part of the plane, it can be described, using plane equation in
the point-normal form [23]
�⃗� ∙ 𝑟 − 𝑑 = 0 (1)
In this equation �⃗� ∙ 𝑟 is a dot product between face normal �⃗� = (nx, ny, nz) and 𝑟 = (rx, ry, rz) is
a position vector of any point lying on the plane. For simplicity one of these points can be one of
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 33
the polyhedron vertices. The normal to the face can be determined from its vertices, using property
of vector cross product as
�⃗� = (𝑉2 − 𝑉1) × (𝑉3 − 𝑉1)
‖(𝑉2 − 𝑉1) × (𝑉3 − 𝑉1)‖ (2)
The distance 𝑑 from global origin to the face can be a distance to one of its vertices:
𝑑 = �⃗� ∙ 𝑉1⃗⃗ ⃗ (3)
where 𝑉1⃗⃗ ⃗ is a position vector of vertex V1.
Thus there are four parameters needed to describe a plane: (d, nx, ny, nz).
2.1.2. Computer data structures
For the simulation instead of implementing adaptive data structures with a variable number of
entities for each polyhedron (for each discrete element), we use triangulated faces of polyhedron.
This approach allows using universal data structures as presented in Figure 20.
Figure 20. DEM computer data structures [23].
As it can be seen from data structures, VERT_COORD array specifies x-, y-, and z-coordinates
of all the vertices of a polyhedron. FACE_EQUATION array defines entries of equation of each
triangular face of a polyhedron. FACE_VERTEX_TABLE contains faces topological information
— which vertices build a face. VERTEX_FACE_TABLE is inverse array of
FACE_VERTEX_TABLE and contains information which faces each vertex belongs to.
Alekseev Aleksei
34 Master’s Thesis developed at the University of Rostock
2.2. Ship mesh as polyhedron
In order to be compatible with adopted data structures of discrete elements the ship hull
geometry must be also represented as polyhedron with triangulated faces. The hull surface can be
created and represented as mesh in any CAD system, which supports conventional file format for
meshes (.obj file). This file format contains information of mesh vertices and their interconnections.
More wide spread .stl file format can also be used as initial information about hull mesh by
converting it to .obj file with appropriate software (Rhinoceros, etc.).
Figure 21. Introduction of ship mesh into simulation domain
The software reads the .obj file, calculates the number of vertices, number of faces, computes
topological information from these input data, computes entries of face equation, and stores
information with DEM data structures (Figure 21).
It is important to mention that for the purpose of using computational geometry algorithms for
overlap computation (see section 12) the ship mesh should be subdivided into convex parts and
each part should be stored in separate .obj file with origin at the ship center of gravity. More
explanations on non-convex ship hulls are presented in section 10.
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3. INTRODUCING ICE RIDGE
3.1. Ridge creation as input
The numerical ice ridge is created using the part of the same HSVA’s Discrete Element Method
software, which is responsible for ice ridge creation. The following parameters of ice ridge should
be specified (Figure 22):
Ridge length (RL)
Ridge width (RW)
Keel width (KW)
Keel height (KH)
Figure 22. Dimensions of ice ridge
In the example, presented in the figure, the keel width KW is equal to zero and thus ridge has a
triangular form of its keel part.
The ridge is created using so-called floating-up technique — the ice pieces float up to the free
surface due to buoyancy forces (Figure 23). This technique is implemented in five stages: 1 –
initialization of ice elements randomly orientated in space, 2 – floating up of the elements due to
buoyancy, 3 – preliminary shaped ice ridge, 4 – introducing pushing bars for obtaining desired
shape and size of ice ridge, 5 – exporting created ice ridge with given dimensions.
At the stage of initialization of ice pieces their dimensions (length, height, and thickness) are
determined randomly in a feasible range of values (in order not to get too big/too small ice pieces).
Before ice floats to the free surface the software prescribes random values of initial speed to the
ice pieces. When ice pieces contact with each other elastic forces occur due to mesh overlap
between them.
RL RW
KH
Alekseev Aleksei
36 Master’s Thesis developed at the University of Rostock
Figure 23. Floating up technique for ridge creation
The inclination and dimensions of artificial pushing bars are defined from keel profile, in order
to get desired shape of the ridge. When ridge is created and all ice pieces are at rest, pushing bars
start to move backwards from the ridge and finally the ridge (positions and orientations of ice
pieces) is exported from the simulation domain into output files.
The aforementioned procedure of ridge creation does nor correlate with the processes of ridge
creation in nature or in ice tank. We should notice that proper simulation of ridge creation is not of
interest, since the process of ridge creation itself does not affect the interaction with ship
hull/marine structure. Even if the process is different, it results in proper shape and configuration
of the ridge.
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“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 37
3.2. Mechanical properties and dimensions of ice pieces
When a test of a ship model in ice tank is prepared, dimensions of the ridge are defined with
the laws of Froude similitude. Based on the values of the test data from ice model tank, dimensions
of the ice pieces can be chosen as follows (Figure 24):
Average rubble length (10 cm)
Average rubble width (10 cm)
Thickness of ice piece – corresponds to the thickness of surrounding level ice
Figure 24. Dimensions of ice pieces
Apart from geometrical properties of ice ridge, mechanical properties of its ice pieces should
be introduced at initialization stage. Among those properties, recalculated to model scale, are:
Table 2. Input of ice properties
Quantity Value Units
Density of ice 900 kg/m3
Young’s modulus of ice 91000 kPa
Bending strength of ice 50 kPa
Poisson’s ration of ice 0.3 [-]
Void fraction 0.55 [-]
Friction coefficient ice-ice 1.0 [-]
Friction coefficient ice-
hull 0.1 [-]
Coefficient of cohesive
force 0.0001 [-]
Coefficient of viscous
damping 2.0 [-]
In principle, almost all the aforementioned parameters are the subject of discussion, since they
affect the values of the interaction and other forces on ice pieces and ship hull during simulation.
Once the software has been created, the analysis of influence of these parameters should be studied
in more details in order to tune the existing model.
thickness
Alekseev Aleksei
38 Master’s Thesis developed at the University of Rostock
4. ESTIMATION OF SHIP HULL INERTIA TENSOR
During simulation of ship breaking through an ice ridge the full dynamic behavior of ship is
expected when ship interacts with the ridge (roll, pitch, yaw, heave, sway and surge motions).
Though the angles of ship motions are relatively small, the rotational motion of the model takes
place. When ship penetrates the ridge the bow part can rise up because of the buoyancy of a large
number of ice elements in the ridge. This pitch angle is quite often observable during model tests.
The rotational behavior of any solid body is described with its inertia tensor. For a body with
continuous mass distribution the inertia is expressed as integral of square of radial distance of the
point mass from the reference axis multiplied by its mass:
𝐼 = ∫ 𝑟2𝑑𝑚
The second-order inertia tensor relative to chosen axis of a 3D body can be expresses via
distances to these axis as (axes can be chosen as passing through the center of gravity):
𝐼 =
(
∫(𝑦2 + 𝑧2)𝑑𝑚 −∫𝑥𝑦𝑑𝑚 −∫𝑥𝑧𝑑𝑚
−∫𝑥𝑦𝑑𝑚 ∫(𝑥2 + 𝑧2)𝑑𝑚 −∫𝑦𝑧𝑑𝑚
−∫𝑥𝑧𝑑𝑚 −∫𝑦𝑧𝑑𝑚 ∫(𝑥2 + 𝑦2)𝑑𝑚)
Denoting the components of inertia tensor in shorter form we obtain:
𝐼 = (−
𝐼𝑥𝑥 −𝐼𝑥𝑦 −𝐼𝑥𝑧𝐼𝑥𝑦 𝐼𝑦𝑦 −𝐼𝑦𝑧−𝐼𝑥𝑧 −𝐼𝑦𝑧 𝐼𝑧𝑧
)
Usually the components of ship’s inertia tensor are not known during the model test (especially
at early design stage). For this reason traditionally in ship theory the inertia tensor can be estimated,
using statistical-empirical formulae [18]:
𝐼𝑥𝑥 = 0.365𝑀𝐵2
𝐼𝑦𝑦 = 0.245𝑀𝐿2
𝐼𝑧𝑧 = 0.255𝑀𝐿2
In these formulae M is the ship’s mass, B is breadth at waterline, and L is length between
perpendiculars. Non-diagonal terms of inertia tensor are conventionally assigned to be equal zero.
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 39
5. QUATERNIONS AND ORIENTATION IN SPACE
5.1. Euler angles
In classical mechanics textbooks the rotational motion of a rigid body is usually described via
Euler's rotation equations. Those are vectorial first-order ordinary differential equations, which are
expressed in a rotating reference frame with its axes fixed to the body and parallel to the body's
principal axes of inertia. The general form of Euler's rotation equations is as follows [23]:
𝐼�̇�𝑏 + 𝜔𝑏 × (𝐼 ∙ 𝜔𝑏) = 𝜏𝑏 (4)
where I is inertia tensor, 𝜔𝑏 is angular velocity in body-fixed coordinate system, 𝜏𝑏 is applied
external torque in body-fixed coordinate system.
If we consider 3D principal 1-2-3 axes (inertia tensor I is diagonal) orthogonal body-fixed
coordinate system, then equations can be rewritten as:
𝜏1𝑏 = 𝐼1�̇�1
𝑏 − (𝐼2 − 𝐼3)𝜔2𝑏𝜔3
𝑏
𝜏2𝑏 = 𝐼2�̇�2
𝑏 − (𝐼3 − 𝐼1)𝜔1𝑏𝜔3
𝑏
𝜏3𝑏 = 𝐼3�̇�3
𝑏 − (𝐼1 − 𝐼2)𝜔1𝑏𝜔2
𝑏
(5)
Normally we will operate in global coordinate system (Figure 18). That means that torques
applied to the body 𝜏𝑏 in local coordinate system need to be transferred from torques 𝜏𝑠 expressed
in global reference frame. This can be done with rotation matrix R and classical Euler angles of
rotation, presented in Figure 25: φ (rotation around z-axis), θ (rotation around x-axis) and ψ
(rotation around new z-axis).
Figure 25. Classical Euler angles [28]
Alekseev Aleksei
40 Master’s Thesis developed at the University of Rostock
If we work with conventional Euler angles (φ, θ, ψ), then transformation of torques is done,
using rotation matrix R as:
𝜏𝑏 = 𝑅𝜏𝑠 (6)
𝑅 = (
𝑐𝑜𝑠𝜑𝑐𝑜𝑠𝜓 − 𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜓 𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜓 + 𝑐𝑜𝑠𝜑𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜓 sinθ 𝑠𝑖𝑛𝜓−𝑐𝑜𝑠𝜑𝑠𝑖𝑛𝜓 − 𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓 −𝑠𝑖𝑛𝜑𝑠𝑖𝑛𝜓 + 𝑐𝑜𝑠𝜑𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜓
𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝜃 −𝑐𝑜𝑠𝜑𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃) (7)
Denoting corresponding terms of matrix R with indices we obtain:
𝑅 = (𝑅11 𝑅12 𝑅13𝑅21 𝑅22 𝑅23𝑅31 𝑅32 𝑅33
) (8)
The Euler angles can be expressed via terms of R matrix as:
𝑐𝑜𝑠𝜃 = 𝑅33 𝑐𝑜𝑠𝜑 = −𝑅32𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜓 =𝑅23𝑠𝑖𝑛𝜃
(9)
As it can be seen, for 𝜃 = 𝜋/2 or 𝜃 = 0 equations of motions with Euler angles become
singular. This phenomena is known as “Gimbal lock” and can happen, for example, when initial
xyz and final xyz′ orientations of reference frame coincide.
Figure 26. Example of Gimbal lock [32]
When we run simulation with many discrete ice elements the likelihood of obtaining singularity
increases. This is the reason to use quaternions in numerical simulation, since it provides certain
numerical safety.
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 41
5.2. Quaternions
In numerical simulation quaternions are used to represent three-dimensional rotations.
Quaternions allow very stable numerical representation of equations of rotational motions. It is
known, that for rotation of vector in 2D space complex number representation of vector can be
used [23]. Quaternions can be described as extension of this concept for 3D applications.
In simple way, quaternion can be defined as ‘complex number with four entries’:
𝑞 = 𝑤 𝟏 + 𝑥𝐼 + 𝑦𝐽 + 𝑧𝐾 (10)
Where I, J, K are quaternion basic elements (analogy to complex i), such that:
𝐼 ∙ 𝐼 = −1 𝐽 ∙ 𝐽 = −1 𝐾 ∙ 𝐾 = −1 (11)
For the quaternions w is called as scalar part and (x, y, z) is called as vector part.
Additionally there is a unit operator 1 such that:
𝐼 ∙ 𝟏 = 𝟏 ∙ 𝐼 = 𝐼 𝐽 ∙ 𝟏 = 𝟏 ∙ 𝐽 = 𝐽 𝐾 ∙ 𝟏 = 𝟏 ∙ 𝐾 = 𝐾 (12)
Quaternion conjugate is defined similarly to complex conjugate as:
𝑞∗ = 𝑤 𝟏 − 𝑥𝐼 − 𝑦𝐽 − 𝑧𝐾 (13)
As for the complex number, the absolute value of a quaternion is defined as:
|𝑞| = √𝑞𝑞∗ = √𝑤2 + 𝑥2 + 𝑦2 + 𝑧2 (14)
A unit quaternion is denoted by 𝔮 and has an absolute value of 1:
|𝖖| = √𝖖∗𝖖 = 1 (15)
In mathematics it is proved that a vector (for example position vector of the point of ship mesh
expressed from its center of gravity) can be rotated with unit quaternion defined with its entities
as:
𝑤 = 𝑐𝑜𝑠𝜑
2𝑐𝑜𝑠
𝜃
2𝑐𝑜𝑠
𝜓
2+ 𝑠𝑖𝑛
𝜑
2𝑠𝑖𝑛
𝜃
2𝑠𝑖𝑛
𝜓
2
𝑥 = 𝑠𝑖𝑛𝜑
2𝑐𝑜𝑠
𝜃
2𝑐𝑜𝑠
𝜓
2− 𝑐𝑜𝑠
𝜑
2𝑠𝑖𝑛
𝜃
2𝑠𝑖𝑛
𝜓
2
𝑦 = 𝑐𝑜𝑠𝜑
2𝑠𝑖𝑛
𝜃
2𝑐𝑜𝑠
𝜓
2+ 𝑠𝑖𝑛
𝜑
2𝑐𝑜𝑠
𝜃
2𝑠𝑖𝑛
𝜓
2
𝑧 = 𝑐𝑜𝑠𝜑
2𝑐𝑜𝑠
𝜃
2𝑠𝑖𝑛
𝜓
2− 𝑠𝑖𝑛
𝜑
2𝑠𝑖𝑛
𝜃
2𝑐𝑜𝑠
𝜓
2
(16)
In these expressions we use the Tait–Bryan angles:
𝜑: rotation about the x-axis (Roll)
𝜃: rotation about the y-axis (Pitch)
Alekseev Aleksei
42 Master’s Thesis developed at the University of Rostock
𝜓: rotation about the z-axis (Yaw),
where the x-axis points forward, y-axis to the starboard and z-axis downwards (the same
orientation is adopted in the simulation domain — see Figure 18). It should be clarified, that these
axes originate at the body’s center of gravity — that is the point, which a 3D body rotates around
(Figure 27).
Figure 27. Axes of roll, pitch and yaw motions.
The rotation of a position vector of a point of ship mesh from position r to some new position
�̃� is then obtained with unit quaternion as [23]:
�̃� = 𝖖𝑟𝖖∗ (17)
Example of mesh rotation
To clarify the aforementioned theory let us consider example of ship mesh rotation by angle
𝜓 = 22.5° around only z-axis, passing through the center of gravity of ship (Figure 28).
This given mesh of ship hull consists of nv = 4464 vertices and nf = 3908 faces. Faces are
created between vertices. It means, that if we rotate the mesh in space, then only coordinates of the
vertices change, but topological relations between faces and vertices remain unchanged.
ψ
𝜑
𝜃
x
y
z
G
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 43
Figure 28. Example of mesh rotation
If we take only one vertex of ship mesh, located at the bow, then its position vector from the
center of gravity is 𝑟 = (𝑟𝑥, 𝑟𝑦, 𝑟𝑧). We know the values of Roll (𝜑 = 0), Pitch (𝜃 = 0) and Yaw
(𝜓 = 22.5°) angles. Thus we can compute the value of unit quaternion 𝖖. Then, using the provided
formula, we obtain the new rotated position vector of this example point.
Implementing this procedure for all the vertices nv and keeping the same topological relations
for all the faces nf, the new mesh, rotated by the angle 𝜓 = 22.5°, is obtained.
Figure 29. Mesh rotated by 22.5°
Alekseev Aleksei
44 Master’s Thesis developed at the University of Rostock
5.3. Mesh coordinates in global reference frame
So far with the help of quaternions we are able to describe the rotation and orientation of a rigid
body in 3D space around its center of gravity. But in the simulation domain all the calculations for
ship and ice discrete elements are implemented in global reference frame.
The coordinates of points of vertices of ship mesh and ice rubble can be calculated using simple
vector algebra. If we know the position vector of all the vertices of ship mesh from its center of
gravity, then the coordinates of these vertices in global reference frame can be obtained, using
vector addition as:
𝑟𝑠⃗⃗ = 𝑟𝐺⃗⃗ ⃗ + 𝑟 𝑏 (18)
where 𝑟𝑠⃗⃗ is a position vector from global origin, 𝑟𝐺⃗⃗ ⃗ is a position vector of the center of gravity
from global origin and 𝑟 𝑏 is a position vector of the mesh vertex from its center, which can be any
point of the mesh.
Thus, if the initial angles of roll, pitch and yaw of ship hull are known, which at initialization
are usually equal zero, and the position of the hull center of gravity before start of the simulation
is specified, then there is all the necessary information to locate ship in the simulation domain
(Figure 30).
Figure 30. Initialized simulation domain
Global
Origin G
𝑟𝐺⃗⃗ ⃗
Ice
ridge
Mesh
vertex
𝑟𝑏⃗⃗ ⃗ 𝒓𝒔⃗⃗ ⃗
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“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 45
6. GRAPHICAL VIZUALIZATION
One of the main advantages of the developed software is its ability to provide graphical
visualization of ship breaking through an ice ridge.
The graphical information is exported from simulation domain in .vtk files. In this software we
use so-called legacy .vtk file format.
Simple Legacy VTK file [31]
The legacy VTK file formats consist of five basic parts.
1. The first part is the file version and identifier. This part contains the single line: # vtk DataFile
Version x.x.
2. The second part is the header. The header consists of a character string terminated by end-
of-line character \n. The header can be used to describe the data and include any other
pertinent information.
3. The next part is the file format. The file format describes the type of file, either ASCII or
binary. On this line the single word ASCII or BINARY must appear.
4. The fourth part is the dataset structure. The geometry part describes the geometry and
topology of the dataset. This part begins with a line containing the keyword DATASET
followed by a keyword describing the type of dataset. Then, depending upon the type of
dataset, other keyword/data combinations define the actual data.
5. The final part describes the dataset attributes. This part begins with the keywords
POINT_DATA or CELL_DATA, followed by an integer number specifying the number of
points or cells, respectively. Other keyword/data combinations then define the actual dataset
attribute values (i.e., scalars, vectors, tensors, normal, texture coordinates, or field data).
Example of .vtk file
An example of how the data are exported into .vtk visualization file is provided in Figure 31.
These .vtk files can be opened in any appropriate viewer like ParaView, VisIt, etc.
Alekseev Aleksei
46 Master’s Thesis developed at the University of Rostock
Figure 31. Example of .vtk visualization file
Data type
Number of points
Coordinates (x, y, z)
of the points
Topological
relations
Velocities
Data name
Data type
Number of faces
Total number of
integer values to
represent a list [31]
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7. BUOYANCY CALCULATION
During simulation when ship is interacting with ice ridge its position and spatial orientation
might change. From the experiments it is known that ship’s draft and trim angle can change slightly
due to significant buoyancy force Fbk of ice rubble in the keel part of the ridge (Figure 32). In
reality when these parameters are changed corresponding restoring buoyancy forces ΔFb and
moments must appear. Thus in the simulation such forces and moments should also be taken into
account.
Figure 32. Change of bow draft and pitch angle
In principle, computation of actual buoyancy/displacement of the hull, positions of the center
of buoyancy and values of restoring forces/moments can be done at each time step during
simulation. However this approach is rather unreasonable, since buoyancy estimation requires
usage of time-consuming computational geometry algorithms. This means that most of the
computational time would be spent not on DEM, but on buoyancy calculation of the hull only.
It has been decided to estimate buoyancy of the ship hull outside of the main time-increment
loop in order to save computational time during simulation. These pre-calculated values for various
drafts, pitch and roll angles can be used later at the stage of force computation. The real in-time
buoyancy forces and moments can be interpolated between these values based on the real value of
hull draft, roll and pitch angle.
G Δθ
ΔFb
Fbk
Alekseev Aleksei
48 Master’s Thesis developed at the University of Rostock
7.1. Varying drafts, pitch and roll angles. Displacement formula
7.1.1. Varying drafts, pitch and roll angles
Precalculation of displacement and restoring moment is implemented for various drafts, pitch
and roll angles (Figure 33). Rotation of the mesh is implemented with the approach, described in
section 5. Since forces and moments during simulation are integrated between these pre-calculated
values, a proper range of draft, pitch and roll angles should be chosen, in order to cover all the
possible scenarios that might come up during simulation.
Figure 33. Various drafts (1.1-1.2), roll (2.1-2.2) and pitch (3.1-3.2) angles
In order to cover all possible configurations in the simulation with some margin, the range is
chosen as follows:
Table 3. Range of buoyancy precalculation
Minimum value Maximum value
Draft 0 T
Roll -11.25° +11.25°
Pitch -22.5° +22.5°
7.1.2. Displacement formula
As it was described in section 2, the mesh of ship hull is subdivided into a number of convex
parts nstr. For this reason the displacement Vi and the center of buoyancy CBi must be calculated
for each hull sub-mesh and the total displacement and center of buoyancy is defined as:
𝑉 = ∑𝑉𝑖
𝑛𝑠𝑡𝑟
𝑖=1
𝐶𝐵 = ∑𝑉𝑖𝐶𝐵𝑖𝑉
𝑛𝑠𝑡𝑟
𝑖=1
(19)
ΔT ΔT
∆𝜑
∆𝜑
∆𝜃
∆𝜃
1.1 1.2 2.1 2.2
3.1 3.2
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7.2. Calculation of displacement. Simpson’s First Rule
7.2.1. Calculation of displacement
For the given combination of draft, pitch and roll angles displacement can be calculated,
dividing underwater hull into certain number of cross sections and integrating the area of each
section along the length of waterline.
For the simplicity of explanations let us imagine the hull, consisting of only one convex mesh.
Then the entire displacement of the ship is only volume of the submerged part of the mesh. In order
to calculate the volumetric displacement the hull mesh can be subdivided into a number of cross
sections. If the area of each cross section is known, then these areas must be integrated in order to
compute the total volume (Figure 34).
Figure 34. Cross sections of underwater part of hull.
7.2.2. Simpson’s First Rule
Simpson’s rules are widely used in Naval Architecture for Ship Theory calculations. Simpson’s
First Rule integrates precisely the second order function by applying multipliers to groups of three
equally spaced known values of the function. As many hull curves are similar to second-order curve
representation, Simpson’s First Rule is sufficiently accurate for most of the hull-based calculations.
This rule can be applied in our software to compute displacement.
The value of the function between three equally spaced values is obtained as:
𝑓 = ℎ
3(1 ∙ 𝑓1 + 4 ∙ 𝑓2 + 1 ∙ 𝑓3) (20)
where h is the spacing value; f1, f2, f3 are the values of the function.
For our software h is the distance between equally-spaced sections and f1, f2, f3 are the sectional
areas.
Free surface
Alekseev Aleksei
50 Master’s Thesis developed at the University of Rostock
7.3. Gift wrapping algorithm
In order to obtain sectional area of one given transversal section of the ship hull mesh, the
following procedure is used.
First of all cutting plane is created at the desired position along the hull length. This cutting
plane is described by its equation f = (d, nx, ny, nz). On the other hand, all the faces of ship mesh
are also described with similar equations. This means, that usage of some computational geometry
algorithms [5] allows determining the intersection points of the mesh with this given transverse
plane (Figure 35).
Figure 35. Cross section and contour line points
Most probably after these intersection points are obtained they are not stored in ascending or
descending order. In order to get the sectional area the contour line must be determined — that is
indicating the intersection points in order to get proper line contour, as in the example of Figure
35.
To obtain this contour line the so-called ‘gift
wrapping algorithm’ or Jarvis march [30] can be used
(Figure 37). Let us imagine that array P (Figure 36)
of points coordinates is given as P = (P1, P2, … , Pn)
with some values N. As initial point of contour line
P1cl the leftmost and the uppermost point is taken —
this one in the beginning of the contour line. The next
point P2cl of the contour line is such, which has
smallest positive polar angle relative to the point P1cl as being the origin. Then for all the rest of
the points in array P point Pi+1cl is searched such, that gives the smallest angle between the lines,
Cutting plane Contour line
Figure 36. Array of point P
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which connecting Pi-1cl Picl and PiclPi . This point is the next point of the contour line. It is not
necessary to compute the angle itself, but the cosine of this angle via the properties of scalar product
of two vectors:
𝑐𝑜𝑠𝛼 =𝑃𝑖−1 𝑐𝑙 𝑃𝑖 𝑐𝑙⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ ∙ 𝑃𝑖 𝑐𝑙 𝑃𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗
|𝑃𝑖−1 𝑐𝑙 𝑃𝑖 𝑐𝑙⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗||𝑃𝑖 𝑐𝑙 𝑃𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ | (21)
The corresponding next point of the contour line must have the minimum value of cosine from
dot product (the less the cosine the more the angle). The search of the points continues until the
contour line is closed.
Figure 37. Gift wrapping algorithm
When programing gift wrapping algorithm special attention should be drawn to the following
cases:
If two vectors are collinear and have the same value of cosine, then the one of smaller length
is selected
The first index of contour should not be equal zero, which might happen because of array
initialization (Figure 38)
There must be no zero
indices in contour line
Contour line must be closed
Without these precautions significant inaccuracies
might appear in calculation of sectional area, which leads to inaccurate values of displacement and
corresponding buoyancy forces, restoring forces and moments of ship hull.
Contour line array
after gift wrapping
Initialized contour
line array
Figure 38. Contour line array
Alekseev Aleksei
52 Master’s Thesis developed at the University of Rostock
7.4. Calculation of cross section area
When the indices of contour line are known the area of cross section can be calculated. The
area is estimated, using triangulation of cross section (Figure 39):
Figure 39. Example of section triangulation
Each cross section is subdivided into a number of triangles NΔ, depending on the number of
points in the contour line. The total area of cross section is the integral of areas of each individual
triangle Ai:
𝐴 = ∑𝐴𝑖
𝑁∆
𝑖=1
(22)
The area of each triangle can be computed via properties of cross product of vectors 𝑉1𝑉2⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ and
𝑉2𝑉3⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ as:
𝐴𝑖 = 𝑉1𝑉2⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ × 𝑉2𝑉3⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗
2 (23)
Provided all the cross sectional areas are calculated, the displacement of ship for given draft,
pitch and roll can be defined as integral of sectional areas, using Simpson’s integrator.
1 2
3
V1
V2
V3
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7.5. Buoyancy table and buoyancy moment
7.5.1. Buoyancy table
After displacements and positions of the center of buoyancy are computed for all the range of
drafts, pitch and roll angles the buoyancy table for a given ship model is stored in the memory
(Figure 40).
Figure 40. Buoyancy table
Each cell of such buoyancy table contains the value of hull’s displacement. Thus displacement
is a function of three parameters in the developed software.
7.5.2. Buoyancy moment
In a similar way the buoyancy restoring moment table can be computed. The values of three
components of restoring moment can be estimated using properties of cross product as follows
(Figure 41):
�⃗⃗� = 𝐹𝑏⃗⃗⃗⃗ × 𝐺𝐵⃗⃗⃗⃗ ⃗ (24)
Figure 41. Restoring buoyancy moment
Roll
Pitch
Draft
Displacement
G
B
𝐹𝑏
𝐺𝐵⃗⃗⃗⃗ ⃗
Alekseev Aleksei
54 Master’s Thesis developed at the University of Rostock
8. EQUATIONS OF MOTION
8.1. Rectilinear degrees of freedom
Classically in mechanics equation of translational motions are expressed as Newton equations:
𝐹𝑖⃗⃗ = 𝑚𝑣�̇�⃗⃗⃗ (25)
where 𝐹𝑖⃗⃗ is a superposition of all the forces acting on the body, m mass of the body and 𝑣�̇�⃗⃗⃗ is
body’s acceleration.
8.2. Rotational degrees of freedom
As it was mentioned earlier, the 3D rotation of a rigid body can be represented via quaternions
as quaternion rotation of the position vector from global origin to body’s center of gravity. For the
equations of motion there is need for time derivatives of the quaternions and their relationship to
the angular velocity ω. These formulae, expressed in global coordinate system, can be found in
corresponding literature [23] as
𝑑𝖖
𝑑𝑡=1
2�⃗⃗� 𝖖
𝑑2𝖖
𝑑𝑡2=1
2(�⃗⃗̇� 𝖖 + �̇��⃗⃗� )
(26)
where 𝖖 is a unit quaternion of a position vector to the center of gravity of the body, �⃗⃗̇� is angular
acceleration of the body. Additional relations to the dynamic factors are as follows:
�⃗⃗� = 2�̇�𝖖∗ (27)
�̇� = 𝐽−1(�⃗� × �⃗⃗� + 𝜏 ) (28)
�⃗� = 𝐽�⃗⃗� (29)
𝐽 = 𝐴𝐽𝑏𝐴𝑇 (30)
𝐽−1 = 𝐴(𝐽𝑏)−1𝐴𝑇 (31)
Where 𝖖∗ is quaternion conjugate of unit quaternion 𝖖, �⃗� is angular momentum, 𝜏 is applied to
the body torque, Jb is inertia tensor in body-fixed reference frame, J is inertia tensor in global
coordinate system, R is rotation matrix. Rotation matrix R can be derived from components of
quaternion as:
𝑅 = (
𝖖02 + 𝖖1
2 − 𝖖22 + 𝖖3
2 2(𝖖1𝖖2 + 𝖖0𝖖3) 2(𝖖1𝖖3 − 𝖖0𝖖2)
2(𝖖1𝖖2 − 𝖖0𝖖3) 𝖖02 − 𝖖1
2 + 𝖖22 − 𝖖3
2 2(𝖖2𝖖3 + 𝖖0𝖖1)
2(𝖖1𝖖3 + 𝖖0𝖖2) 2(𝖖2𝖖3 − 𝖖0𝖖1) 𝖖02 − 𝖖1
2 − 𝖖22 + 𝖖3
2
) (32)
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9. PREDICTOR-CORRECTOR NUMERICAL INTEGRATOR
There are various existing numerical integrators for our equations of motions (Newmark solver,
Symplectic methods, Runge-Kutta solver, Euler schemes). All of them might differ in their
accuracy, stability and required CPU time. The choice of the solver is vital step for DEM
simulation.
Predictor-corrector schemes are the family of the algorithms for numerical solution of
differential equations. It belongs to the implicit Euler backward difference solver, which provides
stability of the integrator. It is known that implicit integrators require solving system of equations
at each time step, which is CPU consuming. Instead of solving systems of nonlinear equations, the
desired quantity is calculated with predictor-corrector approach [23].
At the first ‘predictor’ step the unknown value of the desired physical quantity is roughly
estimated, considering that forces on discrete elements do not change; at the second ‘predictor’
step the value is estimated more precisely, based on the computation of the forces, acting on each
discrete element.
9.1. Predictor step
Over the entire time of simulation we a looking for two variables for each discrete element: the
position vector 𝑟 and quaternion 𝖖. During the predictor step we increment time with time step dt
and estimate the new values of 𝑟 and 𝖖 for all discrete elements, assuming there are no changes in
forces and torques acting on them (let’s omit the sign of vector for a while):
𝑟 = 𝑟 + �̇�𝑑𝑡 + �̈�𝑑𝑡2
2 (33)
�̇� = �̇� + �̈�𝑑𝑡 (34)
𝖖 = 𝖖+ �̇�𝑑𝑡 + �̈�𝑑𝑡2
2 (35)
�̇� = �̇� + �̈�𝑑𝑡 (36)
As it can be seen form the structure of the formulae in our software predictor step is represented
with Taylor series.
Alekseev Aleksei
56 Master’s Thesis developed at the University of Rostock
9.2. Corrector step
The corrector step acts on the predicted coordinates after the forces and torques at given time
step are evaluated. The equations of translational and rotational motions are solved again with so-
called Gear correction coefficients, which can be found in relevant literature on predictor-corrector
solvers [17]. The values of these coefficients are c0 = 0, c1 = 0.5dt and c2 = 1.
Correction of translational motions:
�̈�𝑡 =𝐹
𝑚 (37)
Δ�̈� = �̈�𝑡 − �̈� (38)
𝑟 = 𝑟 + 𝑐0Δ�̈� (39)
�̇� = �̇� + 𝑐1Δ�̈� (40)
(41)
Correction of rotational motions:
�̈�𝑡 =1
2(�⃗⃗̇� 𝖖 + �̇��⃗⃗� ) (42)
Δ�̈� = �̈�𝑡 − �̈� (43)
𝖖 = 𝖖 + 𝑐0Δ�̈� (44)
�̇� = �̇� + 𝑐1Δ�̈� (45)
�̈� = �̈� + 𝑐2Δ�̈� (46)
In these equations the terms are computed based on the values of computed previously forces
and torques. In the aforementioned formulae we introduced the following notations:
– F — superposition of all the forces, acting on discrete element
– m — mass of the element
– �̈�𝑡 — translational acceleration at current time step
– Δ�̈� – change of translational acceleration from previous to current time step
– �̈�𝑡 — quaternion acceleration at current time step
– ∆�̈� — change of quaternion acceleration from previous to current time step.
Numerical Simulation of Ice Ridge Breaking
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10. HANDLING NON-CONVEX GEOMETRIES OF SHIP HULL
For the purpose of proper simulation running only convex meshes of discrete elements can be
used. Convex meshes are necessary, as the algorithms of overlap computation [5] can deal only
with convex bodies. The volume of overlap is required for determining elastic interaction force
between ice pieces or between ice and ship hull.
Usually modern conventional ship hulls are highly non-convex bodies. This non-convexity is
especially high in bow part (presence of bulbous bow) and stern part (appendages and propulsion
system). If ship geometry is simply introduced without considerations on its convexity, then some
physically unsound phenomena can occur during simulation. One of them is, for example,
‘numerical leakage’ — that is penetration of ice pieces inside the ship hull, which of course cannot
happen in reality.
An example of treating a simple non-convex body as just one mesh and its consequences is
presented in Figure 42. It can be seen, that if complex non-convex body is introduced into domain
as one single mesh, then some penetration of discrete elements into the body might occur, which
is irrelevant for the simulation. It is especially important to consider this phenomena for ship hull
simulation, since such penetrations cannot occur in reality.
Figure 42. Discrete elements penetration of single concave mesh
On the other hand, if each convex sub-part is introduced as one separate discrete element, then
it will disintegrate during simulation when interacting with ice ridge due to lack of interconnection
in-between the elements.
Alekseev Aleksei
58 Master’s Thesis developed at the University of Rostock
In this regard, some considerations need to be done for non-convexity of ship hull. The most
feasible approach is treating non-convex body as composite of convex meshes. The convex hull
sub-meshes should be rigidly connected between each other. As input for the software, the hull
should be subdivided into convex parts in advance (Figure 43). Each convex sub-mesh must be
saved in .obj file relative to the center of gravity of ship.
Figure 43. Subdivision into convex sub-meshes
Because of the all the aforementioned, the non-convex ship hull is presented as composite
discrete element. We should introduce means of how to translate and rotate ship hull, represented
as a set of convex meshes.
Translational motion
Translating non-convex composite discrete element is relatively easy to implement. If after
force and corrector calculations the change of ship position vector Δ𝑟⃗⃗⃗⃗ is known, then each convex
sub-mesh center Ci should be also translated in 3D space by Δ𝑟⃗⃗⃗⃗ like in the Figure 44.
Figure 44. Multi-mesh translation
G
C2
C1
C3 C4
C2
C1
C3 C4
G
Δ𝑟⃗⃗⃗⃗
Initial hull
Translated hull
Hull convex sub-meshes
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Rotational motion
After torque and corrector calculations the new values of ship quaternion is known (for the
example let us imagine it gives 15° of rotation). If each of the convex meshes would be rotated
around its center Ci with the values of ship quaternion, then the hull would disintegrate as it can be
seen in Figure 45.
Figure 45. Wrong rotation of sub-meshes
To avoid this mistake the convex sub-meshes should be rotated not around its center, but around
the center of gravity of ship with the value of ship quaternion. In other words, the position vector
𝐺𝐶𝑖⃗⃗ ⃗⃗ ⃗⃗ for each convex sub meshes should be rotated. This means all the points of ship mesh and
topological relations in input files should be expressed in relation to the center of gravity of ship.
Then proper rotations can be obtained as in Figure 46.
Figure 46. Proper rotation of sub-meshes
G
C2
C1 C3
C4
G
C2
C1 C3
C4
Initial hull
Rotated hull
G
C2
C1
C3 C4
G
C2
C1
C3 C4
Initial hull
Rotated hull ∆𝖖
Alekseev Aleksei
60 Master’s Thesis developed at the University of Rostock
11. BOUNDING BOXES AND NEIGHBORHOOD LIST
After predicted positions 𝑟 and spatial orientations 𝖖 of centres of meshes are defined,
coordinates of all the vertices of elements and position of centres of ship mesh 𝑟𝑖⃗⃗ must be
recalculated in simulation domain using properties of quaternions.
If Discrete Element Simulation is implemented with only dozens of particles then there is no
need for efficient neighborhood algorithm. In our ridge breaking simulation the number of discrete
elements can be several hundreds of ice pieces. Most of them are in contact with only neighbor
elements and there is no need to calculate interactions between non-contacting elements. In the
simulation we obtain a list of pairs of elements that can have contact based on their positions. We
use so called ‘Sweep and Prune’ algorithm, which makes use of bounding boxes.
Axis-aligned bounding boxes of all discrete elements are recomputed at each time step. These
bounding boxes help to understand if the elements that are close to each other at a given time step
could be in contact. In the Figure 47 an example of possible mutual orientation of elements and
their bounding boxes is presented.
Figure 47. Example of bounding boxes of two elements [23]
In the first case bounding boxes are not intersecting and hence the elements themselves are not
in contact. In the second and third cases bounding boxes are intersecting, but the contact between
elements is present only in the case 2. This simple approach, being a part of the ‘Sweep and Prune’
algorithm enables to decrease computational time significantly during simulation.
1
2
1
3
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12. VOLUME OF OVERLAP
For determination of elastic force of interaction between ice pieces and ship hull and ice pieces
themselves the Hertzian contact law is adopted in this simulation [23]. In order to be compatible
with Hertzian contact force formula the volume of overlap between two discrete elements needs to
be calculated (Figure 48).
Figure 48. Overlap of two discrete elements
For each pair of intersecting elements the overlap geometry between them is computed in the
following stages:
Computation of inherited vertices (originally belonging to elements vertices)
Computation of generated vertices (created due to intersection of mesh faces)
Computation of faces of overlap polyhedron
Subdivision of overlap volume into pyramids
Computation of overlap volume
Computation of contact area and centroid of overlap polyhedron
All these procedures can be implemented, using algorithms of computational geometry. In this
thesis the descriptions of the aforementioned algorithms are not presented, since their
implementation was carried out in the framework of [5] and taken as independent functions in the
code.
Calculation of overlap geometry is the most CPU-consuming part of the software and requires
usage of parallel computing in order to decrease calculation time.
Inherited vertices
Generated vertices
Alekseev Aleksei
62 Master’s Thesis developed at the University of Rostock
13. CALCULATION OF FORCES
In Discrete Element Method simulation elements are driven by different kinds of forces and
torques applied to them. Based on the calculated values of forces and torques equations of motions
are numerically integrated and new positions and orientations are obtained from the forces
calculation at a given time step. In order to simulate ship breaking through an ice ridge, apart from
classical DEM forces [23], some other types of forces and torques specific for ship need to be
considered. The list of forces and torques, computed and applied for ice and ship in the software is
presented in Table 4. In the following we provide formulae of various forces based on [23] and [5].
Table 4. List of forces and torques
Force Torque1 Application Direction
Elastic + Ice/Ship Normal
Damping + Ice/Ship Normal
Friction + Ice/Ship Tangential
Dissipation + Ice/Ship Tangential
Cohesion + Ice Normal
Viscous Drag + Ice Velocity vector
Buoyancy +
(couple)
Ice/Ship Upwards
Gravity Ice/Ship Downwards
Propeller Thrust - Ship Horizontal
Level Ice
Resistance - Ship Horizontal
13.1 Elastic force
In this simulation we use the Hertzian contact law for elastic force. It is known that the contact
elastic force is dependent on the deformation δ. It is proved [23] that instead of the value of
deformation the magnitude of overlap between two elements can be used. This approach allows to
make the assumption of non-deformable elements. For 3D applications the magnitude of the elastic
force is defined to be proportional to the volume of overlap between two discrete elements as
𝐹𝑒𝑙 =𝐸𝑉
𝑙 (47)
1 + means the force creates torque
- means the forces creates no torque
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“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 63
Where E is Young’s modulus of ice, V is volume of overlap and l is characteristic length, which
keeps consistency of SI units, defined as
𝑙 =4𝑟1𝑟2𝑟1 + 𝑟2
(48)
The quantities 𝑟1 and 𝑟2 can be defined from overlap geometry as the distances from the center
of gravity G of discrete element to the centroid C0 of overlap polyhedron (Figure 49).
Figure 49. Definition of characteristic length
The point of application of elastic force is chosen to be the centroid C0 of overlap polyhedron.
The direction of the force is weighted average of normal of triangles that form the contact area
(Figure 50).
Figure 50. Direction of elastic force [5]
13.2 Damping force
The value of normal damping force can be computed, making use of the change of overlap
volume Δ𝑉 between two successive time steps as
𝐹𝑑𝑎𝑚𝑝 = 𝛾𝑛√𝐸𝑚∗
𝑙3Δ𝑉
Δ𝑡 (49)
G1 G2
r1 r2 C0
Alekseev Aleksei
64 Master’s Thesis developed at the University of Rostock
Where 𝛾𝑛 is dimensionless damping coefficient, Δ𝑡 is a difference between time steps and 𝑚∗
is a reduced mass for two colliding elements with masses m1 and m2 defined as
𝑚∗ =𝑚1𝑚2𝑚1 +𝑚2
(50)
13.3 Friction force
In the proposed simulation the Cundall-Struck friction model is used [23], also known as a
model of ‘breaking tangential spring’. The essence of this model is consideration of friction forces
from previous time step, taking into account the fact that tangential friction cannot exceed classical
Coulomb sliding friction.
The calculation of Cundall-Struck friction force is implemented in four stages:
1. Projection of old friction force onto new tangential direction
𝑓𝑓_𝑜𝑙𝑑𝑝⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ = 𝑓𝑓_𝑜𝑙𝑑⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ − (𝑓𝑓_𝑜𝑙𝑑⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ∙ �⃗� )�⃗� (51)
2. Rescaling to the old magnitude
𝑓𝑓_𝑜𝑙𝑑𝑟⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ = |𝑓𝑓_𝑜𝑙𝑑
𝑝⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |𝑓𝑓_𝑜𝑙𝑑𝑝⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗
|𝑓𝑓_𝑜𝑙𝑑𝑝⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |
(52)
3. Vectorial addition of the new increment
𝑓𝑓_𝑛𝑒𝑤⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ = 𝑓𝑓_𝑜𝑙𝑑𝑟⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ − 𝐸𝑙𝑣𝑡⃗⃗ ⃗∆𝑡 (53)
4. Application of a cut-off of new value if friction force exceeds Coulomb sliding friction
𝜇|𝐹𝑒𝑙⃗⃗⃗⃗ ⃗|
𝑓𝑓_𝑛𝑒𝑤⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ =𝑓𝑓_𝑛𝑒𝑤⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗
|𝑓𝑓_𝑛𝑒𝑤⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |𝜇|𝐹𝑒𝑙⃗⃗⃗⃗ ⃗| (54)
13.4 Dissipative force
Dissipative tangential damping force [23] is introduced as
𝐹𝑑 = −𝛾𝑡√0.33𝐸𝑚∗ ∙ 𝑙 ∙ 𝑣𝑡 (55)
In this formula 𝛾𝑡 is non-dimensional dissipation coefficient, 𝑣𝑡 is magnitude of tangential
component of contact velocity of two elements, which is defined as
𝑣𝑡⃗⃗ ⃗ = 𝑣 − �⃗� (𝑣 ∙ �⃗� ) (56)
The contact velocity 𝑣 can be computed as
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“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 65
𝑣 = 𝑣1⃗⃗⃗⃗ + 𝜔1⃗⃗⃗⃗ ⃗ × 𝑟1⃗⃗⃗ − (𝑣2⃗⃗⃗⃗ + 𝜔2⃗⃗⃗⃗ ⃗ × 𝑟2⃗⃗ ⃗) (57)
where 𝑣𝑖⃗⃗⃗ and 𝜔𝑖⃗⃗⃗⃗ are translational and angular velocities of i discrete element.
13.5 Cohesion force
With the overlap volume and contact area A the normal cohesion can be modelled [23].
Cohesive force can be expressed as with usage of cohesion coefficient 𝑘𝑐𝑜ℎ as
𝐹𝑐𝑜ℎ = 𝑘𝑐𝑜ℎ𝐸𝐴 (58)
In fact the cohesion coefficient represents the fraction of Young’s modulus that will compete
with contact elastic force.
13.6 Contact forces and torques
All the aforementioned kinds of forces can be grouped as contact forces as they appear when
two discrete elements collide with each other. We assume that these forces are applied in the
centroid of overlap in normal and tangential directions, depending on the type of force. In this way
there is eccentricity between point of force application and center of gravity of an element, which
creates torque. Based on the contact geometry (Figure 49) the torque on i-th element can be
computed as
𝜏𝑐⃗⃗ ⃗ = 𝐹𝑐⃗⃗ ⃗ × 𝑟𝑖⃗⃗ (59)
𝐹𝑐⃗⃗ ⃗ = 𝐹𝑒𝑙⃗⃗⃗⃗ ⃗ + 𝐹𝑑𝑎𝑚𝑝⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ + 𝑓𝑓⃗⃗ ⃗ + 𝐹𝑑⃗⃗⃗⃗ + 𝐹𝑐𝑜ℎ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ (60)
13.7 Viscous drag force
In the current version of the simulation the hydrodynamic viscous drag can be simply modelled
in the form of general expression for hydrodynamic force as [5]
𝐹𝑑𝑟𝑎𝑔 = 𝐶𝑑𝜌𝑣2
2𝐴𝑑 (61)
with 𝐶𝑑 as drag coefficient and 𝐴𝑑 as average cross sectional area of ice element for drag
calculation. The direction of drag force is opposite to the direction of translational velocity of an
element.
Alekseev Aleksei
66 Master’s Thesis developed at the University of Rostock
13.8 Viscous drag torque
The torque created by viscous drag force is estimated as [5]
𝜏𝑑𝑟𝑎𝑔⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ = −0.015𝐶𝑑�⃗⃗� (62)
13.9 Gravity of ice elements
The gravity force acting on each ice discrete element with the mass 𝑚 = 𝜌𝑖𝑐𝑒𝑔 is computed as
𝐹𝑔 = 𝑚𝑔 (63)
13.10 Buoyancy of ice elements
As all the ice elements are located only in the submerged keel part of the ridge and their volume
V is calculated as for parallelepiped, the buoyancy force can be easily computed as
𝐹𝑏 = 𝜌𝑤𝑔𝑉 (64)
13.11 Buoyancy of ship
In the simulation the buoyancy force at each time step is estimated from displacement table.
Displacement is function of three variables: draft, roll and pitch angles of the ship. Let’s denote
draft as x, pitch as y and roll as z parameters. If displacement is calculated for broad range of
possible combinations of x, y and z with fine step, then the actual value of displacement D(x,y,z)
can be computed, using formula of trilinear interpolation from x1, x2, y1, y2, z1, z2 (Figure 51).
Figure 51. Trilinear interpolation of displacement
Roll
Pitch
Draft
x1
x2
y1 y2
D
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“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 67
𝐷 =𝐷(𝑥1, 𝑦1, 𝑧1)
𝐷′(𝑥2 − 𝑥)(𝑦2 − 𝑦)(𝑧2 − 𝑧) +
𝐷(𝑥1, 𝑦1, 𝑧2)
𝐷′(𝑥2 − 𝑥)(𝑦2 − 𝑦)(𝑧 − 𝑧1) +
𝐷(𝑥1, 𝑦2, 𝑧1)
𝐷′(𝑥2 − 𝑥)(𝑦 − 𝑦1)(𝑧2 − 𝑧) +
𝐷(𝑥1, 𝑦2, 𝑧2)
𝐷′(𝑥2 − 𝑥)(𝑦 − 𝑦1)(𝑧 − 𝑧1) +
𝐷(𝑥2, 𝑦1, 𝑧1)
𝐷′(𝑥 − 𝑥1)(𝑦2 − 𝑦)(𝑧2 − 𝑧) +
𝐷(𝑥2, 𝑦1, 𝑧2)
𝐷′(𝑥 − 𝑥1)(𝑦2 − 𝑦)(𝑧 − 𝑧1) +
𝐷(𝑥2, 𝑦2, 𝑧1)
𝐷′(𝑥 − 𝑥1)(𝑦 − 𝑦1)(𝑧2 − 𝑧) +
𝐷(𝑥2, 𝑦2, 𝑧2)
𝐷′(𝑥 − 𝑥1)(𝑦 − 𝑦1)(𝑧 − 𝑧1)
(65)
The value of denominator 𝐷′ is
𝐷′ = (𝑥2 − 𝑥1)(𝑦2 − 𝑦1)(𝑧2 − 𝑧1) (66)
The vertical upward buoyancy force is then computed as
𝐹𝑏 = 𝜌𝑤𝑔𝐷 (67)
13.12 Gravity of ship
The vertical downward gravity force of the ship is computed from volumetric displacement,
calculated for initial position of ship in simulation domain, as
𝐹𝑔 = 𝜌𝑤𝐷0𝑔 (68)
In the software avoidance of introducing ship’s mass as input parameter has been done on
purpose, since small differences in input and computed values of displacement could induce small
initial heave motion.
13.13 Gravity-Buoyancy torque for ship
The restoring moment from couple of gravity and buoyancy forces is estimated again with
trilinear interpolation formula taking corresponding values of pre-computed torques from torque
table, which has the same dimensions as buoyancy table. Each single entry in this table is defined
as (Figure 41)
�⃗⃗� = 𝐹𝑏⃗⃗⃗⃗ × 𝐺𝐵⃗⃗⃗⃗ ⃗ (69)
Alekseev Aleksei
68 Master’s Thesis developed at the University of Rostock
13.14 Propeller thrust
13.14.1. Propeller curve
In our DEM simulation the ship is represented as a large discrete element with some special
characteristics compared to ice discrete elements. One of the striking features is its propulsion
system, which gives an additional horizontal force component during the simulation.
The forces and moments produced by propeller are usually defined as some non-dimensional
coefficients based on the tests in cavitation tunnel. Such propeller characteristics are unique for the
given geometrical profile of the propeller (P/D pitch ratio, skew, etc.). Marine propellers are
conventionally described with three numbers (J — advance ratio, KT — thrust coefficient, KQ —
torque coefficient):
𝐽 = 𝑉𝐴𝑁𝐷
𝐾𝑇 =𝑇
𝜌𝑁2𝐷4 𝐾𝑄 =
𝑄
𝜌𝑁2𝐷5 (70)
where
– VA is the velocity of advance
– N is the revolution rate
– D is the propeller diameter
– T is the propeller thrust
– Q is the propeller torque
The open water efficiency (without presence of the hull) is defined as the ratio of thrust power
PT and delivered power PD as
𝜂0 =𝑃𝑇𝑃𝐷=
𝑇𝑉𝐴2𝜋𝑄𝑁
=𝐾𝑇𝐾𝑄
𝐽
2𝜋 (71)
Propeller open water characteristics are usually given either in the form of propeller curve
(Figure 52) or in tabular form from propeller experiments. This software operates with non-
dimensional propeller characteristics as provided from HSVA propeller database.
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“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 69
Figure 52. Propeller curve
13.14.2. Influence of ship hull
The aforementioned propeller curve is applicable in open water — without presence and
influence of ship hull. When propeller is located behind the ship hull the conditions of the flow
around propeller change. Due to hull influence the propeller is advancing with the velocity VA lower
than the ship speed V:
𝑉𝐴 = (1 − 𝑤) ∙ 𝑉 (72)
where w is wake fraction. Wake fraction is normally estimated with empirical formulae
expressed through parameters of hull shape.
Apart from wake influence, there is also influence of high pressure over the stern due to
propeller rotation. This creates some augment of resistance and means that the resistance of towed
model and required thrust are not the same. In this way the total thrust delivered by the propeller
for the entire ‘hull-propeller system’ can be expressed as:
𝑇 = (1 − 𝑡) ∙ 𝑇𝑝𝑟𝑜𝑝 (73)
where t is a thrust deduction factor, which can be determined from empirical formulae.
In the software the input propulsion parameters are:
Propeller open water characteristics
Wake fraction
Thrust deduction factor
The thrust delivered by the propeller during simulation depends on the current velocity of
advance and can be computed from the propeller curve. The procedure of propeller thrust
computation is as follows:
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.2 0.4 0.6 0.8 1 1.2
J
KT 10KQ Efficiency
Alekseev Aleksei
70 Master’s Thesis developed at the University of Rostock
1. Calculation of actual velocity of advance
𝑉𝐴 = (1 − 𝑤)𝑉 (74)
2. Calculation of advance ratio as
𝐽 = 𝑉𝐴𝑛𝐷
(75)
where n is revolution rate and D is propeller diameter.
3. Interpolation of KT value from the propeller curve (Figure 53)
𝐾𝑇 = 𝐾𝑇1 + (𝐾𝑇2 − 𝐾𝑇1)𝐽 − 𝐽1𝐽2 − 𝐽1
(76)
Figure 53. Interpolation of KT value
4. Calculation of thrust value
𝑇 = 𝐾𝑇𝜌𝑤𝑛2𝐷4 (77)
5. Consideration of thrust deduction factor t for actual value of thrust
𝑇𝑛𝑒𝑡 = (1 − 𝑡)𝑇 (78)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.2 0.4 0.6 0.8 1 1.2J
KT 10KQ Efficiency
J1, KT1
J1, KT1
KT
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“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 71
13.15 Resistance in level ice
13.15.1. Concept
When ship is breaking through an ice ridge it interacts not only with the ridge itself, but also
with level ice. Generally propeller thrust can only compensate level ice resistance, while ridge
breaking is performed thanks to the vessel’s kinetic energy. Thus the consideration of level ice
resistance is vital in order to get realistic behavior of ship.
In general the level ice resistance depends on the geometry of ship hull (which defines its ability
to break the ice), properties of ice (thickness, friction and strength) and the speed of the ship.
In this software the simulation of motion in level ice is not modelled and thus level ice
resistance is taken into account with improved version of Lindqvist ice resistance model [19]. This
theory assumes that total level ice resistance for a given velocity of the ship can be subdivided into
crushing resistance, breaking resistance and submersion resistance as:
𝑅 = 𝑅𝑐 + 𝑅𝑏 + 𝑅𝑠 (79)
13.15.2. Geometry of ship hull
The geometry of the hull is described with certain angles of hull as depicted in Figure 54. Here
ϕ is stem angle, α is waterline angle, ψ is angle between vertical and normal to the ship surface at
the stem. If these angles are taken over the entire beam of the ship, then some averaged values can
be introduced as �̅�, �̅�, and �̅�.
Figure 54. Description of hull form [3]
The usage of proposed by Lindqvist mean angles has been modified by HSVA in order to
improve consideration of ship hull geometry. The idea is to use the angles, measured in five
sections over the beam of the hull: at the centerline, and in four sections at 1
4𝐵,
2
4𝐵,
3
4𝐵,
15
16𝐵,
where B is the hull beam (Figure 55). Then average angles of each section can be obtained as semi-
sum of angles between previous and next sections (zero index corresponds to angles at centerline):
Alekseev Aleksei
72 Master’s Thesis developed at the University of Rostock
𝜙1/4𝑎𝑣 =
𝜙0 + 𝜙1/4
2, 𝛼1/4
𝑎𝑣 =𝛼0 + 𝛼1/4
2, 𝜓1/4
𝑎𝑣 =𝜓0 + 𝜓1/4
2
𝜙2/4𝑎𝑣 =
𝜙1/4 + 𝜙2/4
2, 𝛼1/4
𝑎𝑣 =𝛼1/4 + 𝛼2/4
2, 𝜓1/4
𝑎𝑣 =𝜓1/4 + 𝜓2/4
2
𝜙3/4𝑎𝑣 =
𝜙2/4 + 𝜙3/4
2, 𝛼1/4
𝑎𝑣 =𝛼2/4 + 𝛼3/4
2, 𝜓1/4
𝑎𝑣 =𝜓2/4 + 𝜓3/4
2
𝜙15/16𝑎𝑣 =
𝜙3/4 +𝜙15/16
2, 𝛼1/4
𝑎𝑣 =𝛼3/4 + 𝛼15/16
2, 𝜓1/4
𝑎𝑣 =𝜓3/4 +𝜓15/16
2
(80)
Figure 55. Hull angles at different sections [3]
13.15.3. Crushing resistance
Before ice sheet is broken by hull, the edge of ice is crushed. Lindquist’s formula for crushing
resistance is proposed as
𝑅𝑐 = 𝐹𝑣
𝑡𝑎𝑛𝜙0 + 𝜇𝑐𝑜𝑠𝜙0𝑐𝑜𝑠𝜓0
1 − 𝜇𝑠𝑖𝑛𝜙0𝑐𝑜𝑠𝜓0
(81)
In the formula μ is friction coefficient between ice and hull (can be taken as 0.1) and the value
of crushing force 𝐹𝑣 =1
2𝜎𝑏𝐻𝑖𝑐𝑒
2 , where 𝜎𝑏 is bending strength of ice and 𝐻𝑖𝑐𝑒 is thickness of level
ice.
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“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 73
13.15.4. Breaking resistance
The contribution of breaking resistance at i-section of ship hull is estimated as
𝑅𝑏,𝑖 = 𝑘𝜎𝑏𝐵
4
𝐻𝑖𝑐𝑒2
𝐿𝑐𝑢𝑠𝑝2(𝑡𝑎𝑛𝜓𝑖
𝑎𝑣 +𝜇𝑐𝑜𝑠𝜙𝑖
𝑎𝑣
𝑠𝑖𝑛𝛼𝑖𝑎𝑣𝑐𝑜𝑠𝜓𝑖
𝑎𝑣)(1 +1
𝑐𝑜𝑠𝜓𝑖𝑎𝑣) (82)
This expression contains the following terms:
– 𝑘 = 3
64 is calculation factor
– 𝐿𝑐𝑢𝑠𝑝 is the length of the cusp, assumed to be 1/3 of the characteristic length of ice
𝐿𝑐𝑢𝑠𝑝 =1
3(
𝐸𝐻𝑖𝑐𝑒3
12(1 − 𝜈2)𝜌𝑤𝑔)
14
(83)
– E, ν and σb are respectively the Young’s modulus, Poisson’s ratio and bending strength of
ice
– B is ship’s beam
The total bending resistance is then
𝑅𝑏 =∑𝑅𝑏,𝑖
4
𝑖=1
(84)
13.15.5. Submersion resistance
The submersion resistance is divided by Lindqvist into two parts:
– Rp – resistance due to loss of potential energy
𝑅𝑝 = (𝐻𝑖𝑐𝑒Δ𝜌𝑤𝑖 + 𝐻𝑠𝑛𝑜𝑤𝑒𝑓𝑓
Δ𝜌𝑤𝑠)𝑔𝐵𝑇𝐵 + 𝑇
𝐵 + 2𝑇 (85)
– Rf – resistance due to friction
𝑅𝑓 = (𝐻𝑖𝑐𝑒Δ𝜌𝑤𝑖 + 𝐻𝑠𝑛𝑜𝑤𝑒𝑓𝑓
Δ𝜌𝑤𝑠) ∙ 𝑔𝜇(𝐴𝑢 + 𝐴𝑓𝑐𝑜𝑠 �̅�𝑐𝑜𝑠�̅�) (86)
In these formulae
– Δ𝜌𝑤𝑖 = 𝜌𝑤 − 𝜌𝑖 is the difference of densities of water and ice
– Δ𝜌𝑤𝑠 = 𝜌𝑤 − 𝜌𝑠 is the difference of densities of water and snow
– 𝐻𝑠𝑛𝑜𝑤𝑒𝑓𝑓
is effective thickness of snow (for the case of numerical ice basin is neglected)
– 𝐴𝑓 and 𝐴𝑢 are approximated bow and bottom areas, covered by ice
𝐴𝑓 = 𝐵𝑇√1
𝑠𝑖𝑛2 �̅�+
1
𝑡𝑎𝑛2 �̅� 𝐴𝑢 = 𝐵 (𝐶𝑜𝑣𝑏𝐿𝑝𝑝 −
𝑇
𝑡𝑎𝑛 �̅�−
𝐵
4𝑡𝑎𝑛 �̅�) (87)
Alekseev Aleksei
74 Master’s Thesis developed at the University of Rostock
– T and Lpp are ship’s draft and length between perpendiculars
– Covb is the fraction of bottom area covered by ice
13.15.6. Level ice resistance with speed
When ship is propagating in level ice with some velocity the ice resistance is also increasing.
The total resistance can be computed as
𝑅𝑖𝑐𝑒 = (𝑅𝑐 + 𝑅𝑏) ∙ (1 + 1.4𝑉
√𝑔𝐻𝑖𝑐𝑒) + 𝑅𝑠 (1 + 9.4
𝑉
√𝑔𝐿) (88)
Example of resistance calculation
The level ice by Lindqvist formula has been computed for some model of ship hull (with the
typical dimensions of model for tests in ice tank) for the range of velocities between zero and 1 m/s
and presented in Figure 56.
Figure 56. Level ice resistance
0
50
100
150
200
250
300
350
400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Ric
e [N
]
V [m/s]
Rice [N]
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 75
14. IMPLEMENTATION OF RAMMING
When ship is moving through an ice ridge the dimensions of the latter can be so large that ship
is not able to break through the ridge at first attempt. Physically this means that the amount of
ship’s kinetic energy plus propeller thrust at full speed in level ice is not enough to overcome the
resistance produced by the ridge. In this case the speed of ship is decreasing rapidly and ship gets
stuck. When this happens, ramming operation mode should be implemented by the captain — in
other words the ship is going into reverse direction until it reaches some sufficient distance from
the ridge and accelerates forward again, breaking the ridge. Such ramming mode is implemented
until the ship finally breaks through the ridge. The number of this ramming cycles can vary from
2-3 rams even up to 10-11 rams for very large ice ridges.
An example of ramming test data and ram cycle can be seen in Figure 57. The stages of
ramming are as follows: 1-2 — acceleration in level ice, 2-3 — drop of the speed due to ridge
resistance, 3-4 — thrust inverse and moving to back to the original position.
Figure 57. Ramming operations and ramming cycle [1], [3]
In the software the ramming is implemented in the following way:
1. Ship accelerates from initial position in the simulation domain
2. If velocity at some time step is going below 0.01 m/s then the direction of propeller thrust
in inversed. The calculation of level ice resistance is turned off, as if the ship was moving
in an ice free channel.
3. When ship reaches its original position, the thrust in inversed again and ship starts to
accelerate towards the ridge
4. The calculation of level ice resistance is activated when ship reaches the stop-position of
the previous ram.
Alekseev Aleksei
76 Master’s Thesis developed at the University of Rostock
15. PROGRAM RUNNING
15.1. Generalities
The software is presented as executable file NumericalRidge.exe that can be compiled for 32
or 64 Bits operating system. The interaction with the user is implemented with text input files,
subdivided into corresponding groups for ice properties, ridge, ship mesh and hull particulars. The
output of the program is implemented in different folders for data visualization .vtk files and for
simulation output data .csv files (ship velocity, thrust, acceleration, position in domain, etc…).
In order to be able to work with this simulation tool the user should have access to any CAD
system, supporting meshing options for pre-processing and graphical data visualizer for post
processing. The author has used Rhinoceros CAD system with combination of ParaView graphical
visualizer.
General algorithm of working with the software is illustrated in Figure 58.
Figure 58. Algorithm of working with software
All the calculations, and hence input and output of the software are implemented in SI units. In
the next sections we provide short information on each of the phases.
Scale of
simulation
Ridge
dimensions &
ice mechanical
properties
Creation of ice
ridge
Working with
CAD system
Ship data
input Simulation
Graphical
visualization
Data post
processing
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15.2. Scale of simulation, ridge dimensions and ice mechanical properties
Before starting to work with the software one should define the scale of simulation, as it
determines the dimensions of simulation domain. Since the software has been developed and tested
with the concept of ‘numerical ice basin’ at current stage it is advisable to work at model scale as
in, for example, HSVA ice tank. The corresponding dimensions of the full-scale ridge should be
divided by scaling factor.
15.2.1. Input of ridge dimensions
The input information on ridge dimensions should be specified in the first part of the file
SimulationSettings.txt as depicted in Figure 59.
Figure 59. Input of ridge dimensions
15.2.2. Input of ice rubble dimensions
The input information on ridge dimensions should be specified in the second part of the file
SimulationSettings.txt as depicted in Figure 60.
Figure 60. Input of rubble dimensions
15.2.3. Input of ice properties
The input information on ice properties should be specified in the third part of the file
SimulationSettings.txt as depicted in Figure 61.
Figure 61. Input of ice properties
Alekseev Aleksei
78 Master’s Thesis developed at the University of Rostock
In this input:
– ‘sigma_b’ is bending strength of ice
– ‘nu’ is Poisson’s ratio of ice
15.2.4. Input of forces coefficients
For the purpose of program validation and future study of influence of various parameters the
input for coefficients of forces calculation (see also Section 13) is defined in the fourth part of the
file SimulationSettings.txt as depicted in Figure 62.
Figure 62. Input of forces coefficients
In this input:
– ‘coh_coeff’ is the coefficient of cohesive force
– ‘damp_viscous is the coefficient of viscous damping
– ‘gamma_n’ is the coefficient of normal damping force
– ‘gamma_t’ is the coefficient of tangential dissipation force
– ‘mu’ is the friction coefficient between ice pieces
15.3. Creation of ice ridge
Once the parameters for ridge creation have been defined, the simulation of ridge creation can
be started executing NumericalRidge.exe from command window by choosing option ‘1) New
ridge’ as in Figure 63. When simulation has been finished the output “Simulation successfully
ended” appears on the screen.
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“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 79
Figure 63. Running NumericalRidge.exe
15.4. Meshing in CAD system
15.4.1. Hull surface
The surface of ship hull can be exported from any shipbuilding or general-purpose CAD
system, provided that exchange file formats are available. For example, one of the most popular
way to represent the surface is to create .iges file, which contains a set of smaller individual
surfaces, building up the whole ship surface. As it was mentioned earlier, convex parts should be
extracted separately. Example of surface of ship hull, subdivided into convex parts, is presented in
Figure 64.
Alekseev Aleksei
80 Master’s Thesis developed at the University of Rostock
Figure 64. Hull surface with convex parts
15.4.2. Meshing
Each of the convex parts should be meshed separately. Following aspects should be considered
with caution:
There should not be any discontinuities in mesh, which might come from the meshing
software
Mesh should be triangulated
The main interacting bow parts and bulbous bow/wedge are better to be meshed with
‘Convex Hull’ plug-ins, if applicable
The center of each sub mesh for elastic force computation can be defined as geometrical
centroid of a given convex part
High aspect ratios of mesh edges are better to be avoided
Figure 65. Hull mesh subdivided into convex parts
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15.4.3. Mesh input files
Each of the convex sub meshes is exported into separate .obj file with the name
‘shipmesh1.obj’, ‘shipmesh2.obj’, etc. The following exporting options should be chosen:
Saving objects as polygon mesh
Not exporting objects names
Not exporting layer/group names (relevant for the meshing software)
The origin of the global coordinate system should be located at the ship’s center of gravity, so
that coordinates of sub-meshes vertices are saved in ship-related reference frame.
Apart from .obj file the position of the centroid of each sub mesh should be stored as it was in
simulation domain. These positions (Figure 66) are written in files with the name ‘structure1.txt’,
‘structure2.txt’, etc. Indices r(1), r(2), r(3) correspond to x-, y-, and z-coordinates respectively.
Figure 66. Position of centroid of each sub mesh
15.5. Ship input data
The following input data for ship need to be specified (Figure 67):
Figure 67. Ship input data
Alekseev Aleksei
82 Master’s Thesis developed at the University of Rostock
Number of convex sub meshes n_struct
Distance from keel to the center of gravity KG
Angle of orientation in space and corresponding axis
Length over waterline, beam, depth of ship
Mass of the hull
Position in the simulation domain
Initial velocity
15.6. Simulation
After all the input data have been prepared, simulation can be started by running again
NumericalRidge.exe and choosing option ‘3) Model test’. During execution following parameters
of simulation are output on the screen at each time step (Figure 68), so that user can see the current
status:
– Gravity and interpolated value of buoyancy force
– Current position in simulation domain
– Current values of angles of ship orientation in space
– Current velocity and advance ratio
– Propeller thrust
– Breaking, crushing, submersion and total ice resistance
Figure 68. Output during simulation
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15.7. Visualization and data post processing
After simulation has been finished the results can be visualized with ParaView software. A
created set of .vtk files, which are stored in ‘output’ folder in the root directory of the software,
needs to be loaded to ParaView. The data are depicted with color scheme, corresponding to the
velocity of discrete elements. With the usage of ParaView tools animation videos can be produced
in order to get real-time simulation visualization (Figure 69).
Figure 69. Data visualization
Apart from visualization of ship breaking through the ridge, all relevant parameters of
simulation can be plotted against time thanks to corresponding .csv files. These data serve for
further analysis and validation with experimental data (Figure 70).
Figure 70. Ship velocity and thrust charts as output of simulation
Alekseev Aleksei
84 Master’s Thesis developed at the University of Rostock
Part IV. CODE VALIDATION
The simulation of ship breaking through an ice ridge can be compared with experimental data.
Here we consider experimental results provided by Hamburg Ship Model Basin. During model
tests the following parameters are tracked as time series:
– Propeller torque
– Propeller thrust
– Propeller revolutions
– Velocity of carriage
– Position of the carriage
– Surge, Sway, Heave, Roll, Pitch, Yaw of model
Model tests have been performed for ice ridges of various dimensions with the ship model,
whose particulars are given in Table 5.
Table 5. Ship data model №1
Ship model №1
λ = 22
Quantity Units Full scale Model scale
Length between
perpendiculars Lpp [m] 126.6 5.754
Beam B [m] 23 1.045
Draft T [m] 7.5 0.340
Mass M [kg] 15555∙103 1460
Propeller diameter D [m] 5.17 0.235
Number of propellers Pnbr [-] 2
Range of advance ratio J [-] [0 ÷ 1.05]
Range of thrust coefficients KT [-] [-0.022 ÷ 0.44]
Hull mean angles �̅�, �̅�, �̅� [deg] 29.0, 32.9, 52.54
Hull stem angles 𝜙0, 𝛼0,𝜓0 [deg] 24.0, 85.0, 24.1
Hull angles 1/4 Beam 𝜙1/4, 𝛼1/4,𝜓1/4 [deg] 22.0, 29.0, 39.8
Hull angles 2/4 Beam 𝜙2/4, 𝛼2/4,𝜓2/4 [deg] 31.0, 23.0, 57.0
Hull angles 3/4 Beam 𝜙3/4, 𝛼3/4,𝜓3/4 [deg] 36.0, 17.0, 68.1
Hull angles 15/16 Beam 𝜙15/16, 𝛼15/16,𝜓15/16 [deg] 32.0, 11.0, 73.7
In order to perform an analysis of influence if input parameters on ship performance several
simulations have been carried out with selected variable parameters, whereas other were fixed. The
simulations results have been compared with experimental data in terms of the velocity of ship
breaking through an ice ridge.
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Results of simulations
Table 6. Ridge dimensions and constant ice parameters (1)
Ridge 1 Ship model №1
λ = 22
Quantity Units Full scale Model scale
Ridge length L [m] [NA] 3.137
Ridge width W [m] 30 1.364
Keel height Hk [m] 6.9 0.314
Level ice thickness Ice_T [m] 0.913 0.0415
Porosity porosity [-] 0.55 0.55
Density of ice ρ [kg/m3] 900 900
Visouse damping coefficient damp_viscous [-] [NA] 2.0
Normal damping coefficient gamma_n [-] [NA] 0.2
Tangential dissipation coefficient gamma_t [-] [NA] 0.2
Table 7. Input varied parameters (1)
Input varied parameters Ship model №1
λ = 22
Quantity Units Test 1 Test 2 Test 3 Test 4
Young’s modulus of ice Eice [Pa] 9.09∙104 9.09∙106 5.0∙103 9.09∙105
Friction coefficient ice-
ice μice [-] 0.1 0.1 0.1 1.5
Friction coefficient ice-
hull μship [-] 1.0 1.0 1.0 0.4
Cohesion coefficient coh_co
eff [-] 0.0001 0.0001 0.0001 0.001
Level ice resistance Rice [N] Lindqvist
formula
Lindqvist
formula 210
Lindqvist
formula
Figure 71. Ship velocity (ridge 1)
Alekseev Aleksei
86 Master’s Thesis developed at the University of Rostock
Table 8. Ridge dimensions and constant ice parameters (2)
Ridge 2 Ship model №1
λ = 22
Quantity Units Full scale Model scale
Ridge length L [m] [NA] 3.137
Ridge width W [m] 38 1.727
Keel height Hk [m] 8.5 0.386
Level ice thickness Ice_T [m] 0.913 0.0415
Porosity porosity [-] 0.55 0.55
Density of ice ρ [kg/m3] 900 900
Visouse damping coefficient damp_viscous [-] [NA] 2.0
Normal damping coefficient gamma_n [-] [NA] 0.2
Tangential dissipation coefficient gamma_t [-] [NA] 0.2
Table 9. Input varied parameters (2)
Input varied parameters Ship model №1
λ = 22
Quantity Units Test 1 Test 2 Test 3
Young’s modulus of ice Eice [Pa] 2.73∙105 9.09∙106 9.09∙106
Friction coefficient ice-
ice μice [-] 1.0 1.0 1.5
Friction coefficient ice-
hull μship [-] 0.2 0.1 0.4
Cohesion coefficient coh_co
eff [-] 0.0001 0.0001 0.01
Level ice resistance Rice [N] Lindqvist formula Lindqvist formula Lindqvist formula
Figure 72. Ship velocity (ridge 2)
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Table 10. Ridge dimensions and constant ice parameters (3)
Ridge 3 Ship model №1
λ = 22
Quantity Units Full scale Model scale
Ridge length L [m] [NA] 3.137
Ridge width W [m] 40 1.727
Keel height Hk [m] 6.3 0.386
Level ice thickness Ice_T [m] 0.983 0.0447
Porosity porosity [-] 0.55 0.55
Density of ice ρ [kg/m3] 900 900
Visouse damping coefficient damp_viscous [-] [NA] 2.0
Normal damping coefficient gamma_n [-] [NA] 0.2
Tangential dissipation coefficient gamma_t [-] [NA] 0.2
Table 11. Input varied parameters (3)
Input varied parameters Ship model №1
λ = 22
Quantity Units Test 1 Test 2 Test 3
Young’s modulus of ice Eice [Pa] 9.09∙104 9.09∙106 9.09∙106
Friction coefficient ice-
ice μice [-] 1.0 1.0 1.5
Friction coefficient ice-
hull μship [-] 0.2 0.1 0.4
Cohesion coefficient coh_co
eff [-] 0.0001 0.0001 0.001
Level ice resistance Rice [N] Lindqvist formula Lindqvist formula Lindqvist formula
Figure 73. Ship velocity (ridge 3)
Alekseev Aleksei
88 Master’s Thesis developed at the University of Rostock
Figure 74. Breaking through ridge 1
Figure 75. Breaking through ridge 2
Figure 76. Breaking through ridge 3
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 89
Discussions
The validation results obtained allow to make the following preliminary conclusions about
adopted model and its consistency with experimental data:
1. Ridge № 1.
In the test №1 simulation has demonstrated that the rate of decrease of ship velocity
was lower than in the experiment. The first guess of possible explanation of it was the
lack of elastic force between ship and ice. Thus for this reason in the test №2 the
stiffness of the model was increased artificially by introducing higher elastic modulus
of ice. This measure allowed do decrease the velocity significantly, but still the goal
minimum of velocity was not reached in the simulation. Test №4, which was
implemented with significantly higher values of friction and cohesion, revealed that the
goal minimum velocity of ridge breaking was attained. However the rate of change of
velocity in the beginning of interaction was rather high and the gap between
experimental and simulation curves was larger. Test №3 with constant resistance of
level ice, which corresponds to the level ice velocity of the ship, has demonstrated better
alignment of experimental and simulation curves but the minimum ship velocity was
not reached.
2. Ridge № 2
The analysis of ridge breaking in these tests shows again that the values of input
parameters taken for test №3 lead to overestimation of ridge resistance and the rate of
change of velocity was higher than expected. On the other hand usage of intermediate
values of stiffness with Young’s modulus in combination with higher friction
coefficient leads to rather good alignment of experimental and simulation curves.
Accordingly intermediate values of parameters in the test №2 resulted in intermediate
position of simulation curve between test №1 and test №3.
3. Ridge №3
Test №1 with coefficients values of friction, cohesion and elastic forces again
demonstrates inability of the ridge to make the ship to decelerate accordingly. Increase
of elastic modulus of ice in the test №2 helped to increase ridge resistance but not to the
Alekseev Aleksei
90 Master’s Thesis developed at the University of Rostock
considerable extent. Finally, when cohesion and friction forces have been increased the
more realistic correspondence of the curves was obtained, though there is still lack of
correspondence between velocities in the end of interaction time.
4. In all the aforementioned cases velocity of the ship decreases when ship hull is in
contact with ice ridge. Thus, as it was expected from simulation tool, the numerical ice
ridge creates significant additional resistance when interacting with ship. As it can be
seen the rate of velocity change depends on the parameters of coefficients of forces
calculations to a great extent.
In general this first brief validation of the numerical tool has proved that numerical simulation
of ship breaking through an ice ridge can be done with DEM as it provides realistic behavior of
ship and ice pieces during interaction. However in order to get truly working numerical tool the
calibration of the created model of forces calculation in the software is needed. Such calibration
requires analysis of much more test cases and study of influence of all the aforementioned
parameters. Apart from those not to forget other forces and parameters, which were not tested in
this code validation part (tangential dissipation, normal damping, viscous drag, etc).
Such calibration would require running a lot of simulations with different configurations and
various input parameters and remains out of the scope of this master’s thesis.
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CONCLUSIONS AND PROPOSALS
Conclusions
The purpose of the present thesis was to develop a numerical tool, capable of simulating the
process of ship breaking through an ice ridge. This goal has been successfully achieved by creating
a software in FORTRAN programming language. This software became the first known attempt to
apply numerical simulation for ice ridge breaking.
The numerical solver is based on implementation of Discrete Element Method representing a
ship hull as a set of convex discrete elements. The software has been designed in such way, so that
visualization of ship breaking through an ice ridge is the main output of the program. Apart from
that, all the relevant parameters of ridge and hull interaction can be exported. Such parameters are
ship position and orientation, velocity and acceleration, thrust, etc. The software provides high
flexibility in terms of modelling different interaction configurations and initial conditions such as:
dimensions of the ridge, ship’s velocity, position and orientation before breaking, parameters of
forces calculation, etc.
In the first part of the thesis general information on the nature of ice ridges and the processes
of their creation in nature and ice tank is given. It is followed by review of available today DEM
solutions for analysis of interaction of ice with solid structures. Thereupon the idea on how to use
Discrete Element Method for ship breaking through an ice ridge is introduced.
The main part of the thesis has been dedicated to practical implementation of this idea. The
pre-existing DEM software for ridge creation has been modified and adapted for ship breaking
through the ridge. Thus there has been invented procedures of how to import and treat complex
non-convex ship hull geometries in simulation domain. Then some striking features of ship as a set
of discrete elements have been introduced and their practical realization in the software has been
clarified.
Significant attention has been drawn on how to implement the ship hull as floating body in
DEM simulation. For this some computational algorithms, vector algebra and numerical integrators
have been used in order to estimate buoyancy characteristics of the hull. The concept of pre-
calculated buoyancy table has been introduced in order to save computational resources during
time integration of ridge breaking.
Alekseev Aleksei
92 Master’s Thesis developed at the University of Rostock
In order to carry out validation with experimental data apart from ridge breaking the level ice
resistance is needed. This has been introduced based on Lindqvist ice resistance theory, modified
by Hamburg Ship Model Basin. This approach enabled to take into account the influence of ship
hull geometry and ice mechanical properties of ice for consideration of level ice resistance.
Finally in the last part the validation of the numerical simulation with experimental data has
been carried out. Discussions and suggestions of the obtained results have been provided for future
calibration of the model. In general it can be said that more in-depth validation is required and the
software and forces calculation need to be tuned in order to get more precise solutions.
Proposals
Calibration of the model has not been performed entirely, since the developed software is the
first prototype and such calibration was out of the scope of the master’s thesis. Thus intensive
verification is still needed. Apart from that, the software can be improved in many aspects:
– Consideration of level ice resistance can be improved by substituting semi-empirical
adapted Lindqvist formula with other methods, like pre-sawn ice with breaking
formulations or breaking and splitting, etc.
– The contact forces calculation can be improved in terms of forces coefficients and their
models
– The hydrodynamic forces on ice pieces are better to be computed with more reliable
methods (more accurate estimation of drag, added masses of ice pieces, etc.)
Apart from physics of the problem, computational speed could be improved by:
– introducing parallel computation in the parts of the software, in which it has not been
implemented (buoyancy calculation, neighborhood algorithm, partly in forces calculation)
– using more advanced programming techniques in CPU consuming overlap computation
– defining the list of faces of the ship mesh, which are really in contact by avoiding search of
the intersections between non-neighboring faces of ship and ice elements
On the whole the work, performed in the scope of the thesis, has demonstrated that the idea of
introducing DEM modelling as a simulation tool for ice ridge breaking is a feasible approach. As
long as one deals with the ice/structure interaction processes, in which there is no pure ice breaking,
Numerical Simulation of Ice Ridge Breaking
“EMSHIP” Erasmus Mundus Master Course, period of study September 2014 - February 2016 93
then DEM simulations can provide sound results. This concept along with the described software
can be expanded further towards modelling ship navigation in various ice formations such as brash
ice, ice floes and moving in an ice channel behind an icebreaker. Moreover, possible consideration
of other physical effects and coupling with such tools as Finite Element Analysis and
Computational Fluid Dynamics could provide powerful numerical solutions. In the light of current
interests in Arctic research and development such tools could cover a wide range of simulation of
shipping in various ice conditions.
Alekseev Aleksei
94 Master’s Thesis developed at the University of Rostock
ACKNOWLEDGEMENTS
First and foremost, I would like to express my gratitude to the head of the Arctic Technology
Department of HSVA Dipl.-Ing Peter JOCHMANN for providing me this interesting and challenging
topic and to all the HSVA employees, who guided me through the research and experiments.
I would like to give special thanks to M.Sc. Quentin Hisette for his personal thorough supervision
and reediness to extend a helping hand during writing the thesis. Without his extensive
recommendations and support the achievements of this project would have been impossible.
I also address my thanks to Mr. J. Seidel for spending his time by providing me clear explanations
of the subject.
I would also like to thank M.Eng. I. Svistunov for providing valuable references and his personally
taken photos from Arctic Expeditions.
Finally I would like to express appreciation to my supervising Professor R. Bronsart from the
University of Rostock for his interest in the subject and making sure that the work was progressing
well. At the same time I convey thanks to my external reviewer Prof. H. Le Sourne from ICAM.
This thesis was developed in the frame of the European Master Course in “Integrated Advanced
Ship Design” named “EMSHIP” for “European Education in Advanced Ship Design”, Ref.:
159652-1-2009-1-BE-ERA MUNDUS-EMMC.
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