numerical simulation of ice ridge breaking...sail parts. these ice ridges may affect the normal...

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

This page is intentionally left blank


“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





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


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



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


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



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


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



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 74. Breaking through ridge 1 ........................................................................................ 88



Alekseev Aleksei


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



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





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


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.



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


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]



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


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



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


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]



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


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].



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


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



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


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



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


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.



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


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



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


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.



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


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.



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


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.



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


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.



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


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



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


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 ⃗



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


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]



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


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



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


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



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


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



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


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)



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


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.



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


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



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


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



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


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



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


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



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


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



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


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.



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


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



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


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.



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


– 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]



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


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



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


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


Figure 61. Input of ice properties

Alekseev Aleksei


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.



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


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



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


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



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


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




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.



Results of simulations

Table 6. Ridge dimensions and constant ice parameters (1)

Ridge 1 Ship model №1

λ = 22


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


ice μice [-] 0.1 0.1 0.1 1.5


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




λ = 22













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


ice μice [-] 1.0 1.0 1.5


hull μship [-] 0.2 0.1 0.4


eff [-] 0.0001 0.0001 0.01

Level ice resistance Rice [N] Lindqvist formula Lindqvist formula Lindqvist formula






λ = 22













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


ice μice [-] 1.0 1.0 1.5


hull μship [-] 0.2 0.1 0.4


eff [-] 0.0001 0.0001 0.001

Level ice resistance Rice [N] Lindqvist formula Lindqvist formula Lindqvist formula


Alekseev Aleksei


Figure 74. Breaking through ridge 1





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


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.



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


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,



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


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.



REFERENCES

1. Ehle, D., 2011. Analysis of Breaking through Sea Ice Ridges for Development of a

Prediction Method. Master’s thesis. University of Duisburg Essen.

2. Ehle, D., 2012. Ships Breaking through Sea Ice Ridges. Proceedings of the Twenty-Second

International Offshore and Polar Engineering Conference.

3. Hisette Q., 2014. Simulation of Ice Management Operations. Master’s thesis. University of

Rostock.

4. Dummer, A., 2013. Numerical Simulation of Pressure Ice Ridges. Master’s Thesis.

University of Rostock.

5. Seidel, J., 2016. Numerical Prediction of the Keel Resistance of First-year Pressure Ice

Ridges on Offshore Structures. Master’s thesis. Technical University of Berlin.

6. Timco, G., et al., 1999. Comparison of Ice Load Calculation Algorithms for First-Year

Ridges. Procc. International Workshop on Rational Evaluation of Ice Forces on Structures.

pp. 88-102.

7. Hopkins, M,. 1992. Numerical Simulation of Systems of Multitudinous Polygonal Blocks.

CRREL Report.

8. Hopkins, M., 1998. Four Stages of Pressure Ridging. Journal of Geophysical Research:

Oceans, 103(C10): 21883–21891.

9. Hopkins, M., 2004. A Discrete Element Lagrangian Sea Ice Model. Engineering

Computations, 21(2/3/4):409 – 421.

10. Cundall, P. A., 1988. Formulation of a Three-dimensional Distinct Element Model.

International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts,

25(3):107 – 125.

11. Van den Berg, M., Lubbad, R., 2015. The Application of a Non-smooth Discrete Element

Method in Ice Rubble Modelling. Proc. of International Conference on Port and Ocean

Engineering under Arctic Conditions, vol.2015 - January.

12. Metrikin, I., 2014. A Software Framework for Simulating Stationkeeping of a Vessel in

Discontinuous Ice. Journal of Modeling, Identification and Control, Vol 35, No 4, pp. 211-

248.

Alekseev Aleksei


13. Yulmetov, R., 2015. Validating the Model of Iceberg Motion through Broken Ice. SAMCoT

Seminar of May 2015, Norwegian University of Science and Technology.

14. Terdiman, P., 2007. Sweep-and-prune.

15. Cundall, P. A., and Strack, O. D. L., 1979. A Computer Model for Simulating Progressive,

Large-scale Movements in Block Rock Systems. Géotechnique, 29(1):47 – 65.

16. Chen, J., 2012. Discrete Element Method for 3D Simulations of Mechanical Systems of

Non-spherical Granular Materials. Ph. D. thesis. The University of Electro-

Communications.

17. Gear, C.W., 1971. Numerical Initial Value Problems in Ordinary Differential Equations.

Prentice-Hall series in automatic computation, Prentice-Hall.

18. Schneekluth, H., 1988. Hydromechanik zum Schiffs-entwurf: Vorlesungen. Germany:

Koehler.

19. Lindqvist, G., 1989. A straightforward method for calculation of ice resistance of ships.

Proc. 10th Port and Ocean Engineering Conference under Arctic Conditions, Lulea,

POAC, Vol 2, pp 722.

20. Høyland K.V., Jensen A., Liferov P., Heinonen J., Evers K.-U., Løset S. and Määttänen M.

Physical modelling of first-year ice ridges - part I: production, consolidation and physical

properties. Proceedings of the 16th International Conference on Port and Ocean

Engineering under Arctic Conditions POAC2001. Ottawa, Ontario, Canada

21. Høyland K.V., Jensen A., Liferov P., Heinonen J., Evers K.-U., Løset S. and Määttänen M.

Physical modelling of first-year ice ridges - part II: Mechanical properties. Proceedings of

the 16th International Conference on Port and Ocean Engineering under Arctic

ConditionsPOAC2001. Ottawa, Ontario, Canada

22. International Maritime Organization Sea ice nomenclature.

23. Matuttis H.G, Chen J. (2014) Understanding the Discrete Elements Method. Singapore:

Wiley and Sons.

24. Prasanta K.S., 2014. Theory of Ships and Floating Structures. Florida Institute of

Technology.

25. Prasanta K.S., 2014. Propulsion of ships. Florida Institute of Technology.



26. Consulting Naval Architects and Marine Engineers, 2004. RV Araon Operational

Guidelines.

27. Lagrangian Discrete Element Method. Available at

http://www.cd-adapco.com/products/star-ccm®/lagrangiandem Accessed: January 2015

28. Euler angles. Available at

https://en.wikipedia.org/wiki/Euler_angles Accessed: January 2015

29. Pressure ice ridge. Available at:

https://en.wikipedia.org/wiki/Pressure_ridge_(ice). Accessed: January 2015

30. Gift wrapping algorithm. Available at

https://en.wikipedia.org/wiki/Gift_wrapping_algorithm

31. The VTK User’s Guide, Available at:

http://www.vtk.org/wp-content/uploads/2015/04/file-formats.pdf Accessed: January 2015

32. Gimbal lock, Available at:

https://en.wikipedia.org/wiki/Gimbal_lock Accessed: January 2015

https://en.wikipedia.org/wiki/Euler_angles

https://en.wikipedia.org/wiki/Pressure_ridge_(ice)

https://en.wikipedia.org/wiki/Gift_wrapping_algorithm

http://www.vtk.org/wp-content/uploads/2015/04/file-formats.pdf

https://en.wikipedia.org/wiki/Gimbal_lock

numerical simulation of ice ridge breaking...sail parts. these ice ridges may affect the normal...

Documents