Axisymmetrical topology

optimization of an FPSO main

bearing support structure

Master thesis

Ing. E. van Vliet, 4094654

Faculty of Mechanical, Maritime and Materials Engineering (3ME)

Department of Ship Structures and Hydromechanics

E.vanVliet-2@student.tudelft.nl

Axisymmetrical topology optimization of an

FPSO main bearing support structure

v2.1

E. van Vliet

Committee:

Prof. Dr. Ir. M. L. Kaminiski (chairman)

Dr. Ir. A. Romeijn

Dr. Ir. M. Langelaar

Ir. J. van Nielen

Ir. R. ten Have

03-03-2015

Delft University of Technology

Faculty of Mechanical, Maritime and Materials Engineering (3ME)

Department of Ship Structures and Hydromechanics

Contents

Abstract 6

Prologue 9

Symbols and acronyms 13

1 Introduction 17

1.1 A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 FPSO fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Thesis subject and goals 21

2.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Problem approach . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 A step toward topology optimization . . . . . . . . . . . . . . . . 26

2.3.1 Optimization potential . . . . . . . . . . . . . . . . . . . . 26

2.3.2 Why topology optimization? . . . . . . . . . . . . . . . . 28

2.4 Goals and boundaries . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Mathematical modeling 35

3.1 Formulation of main problem . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Objective function . . . . . . . . . . . . . . . . . . . . . . 36

3.1.2 Relative stress constraints . . . . . . . . . . . . . . . . . . 37

3.1.3 Load preservation constraints . . . . . . . . . . . . . . . . 38

3.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.1 MMA approximation . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 Lagrangian duality . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Topological derivatives . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Objective derivative . . . . . . . . . . . . . . . . . . . . . 45

3.3.2 Suppressing intermediate densities using SIMP . . . . . . 45

3.3.3 Constraint derivatives . . . . . . . . . . . . . . . . . . . . 46

3.3.4 Analysis of penalization . . . . . . . . . . . . . . . . . . . 48

3

3.3.5 Derivatives for arbitrary orientation (DAO) . . . . . . . . 48

4 Test model verification 52

4.1 Arbitrary orientation with stress and displacement constraint . . 53

4.2 Displacement and relative stress constraints . . . . . . . . . . . . 53

4.3 Coupled design spaces with relative constraints . . . . . . . . . . 58

5 Application to main bearing support structure 61

5.1 Finite element modeling . . . . . . . . . . . . . . . . . . . . . . . 62

5.1.1 Element description . . . . . . . . . . . . . . . . . . . . . 62

5.1.2 Modeling and discretization . . . . . . . . . . . . . . . . . 63

5.1.3 Main bearing modeling . . . . . . . . . . . . . . . . . . . 65

5.2 General programming for all loadcases . . . . . . . . . . . . . . . 67

5.2.1 Topological derivatives . . . . . . . . . . . . . . . . . . . . 67

5.2.2 Displacement decomposition . . . . . . . . . . . . . . . . . 71

5.2.3 Coefficient ratios . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.4 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2.5 Initial feasibility . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.6 Asymptotal increase . . . . . . . . . . . . . . . . . . . . . 75

5.2.7 Ansys-Matlab coupling . . . . . . . . . . . . . . . . . . . . 76

5.3 Additional programming for bilateral loadcases . . . . . . . . . . 78

6 Results 79

6.1 Axisymmetric loadcase . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2 Bilateral loadcase . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7 Conclusions 90

7.1 Program stability and convergence . . . . . . . . . . . . . . . . . 90

7.2 Resulting structure . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.3 Increased main bearing loads . . . . . . . . . . . . . . . . . . . . 95

7.4 Evaluation of set goals . . . . . . . . . . . . . . . . . . . . . . . . 96

8 Recommendations and contingencies 98

8.1 Constraints for failure modes . . . . . . . . . . . . . . . . . . . . 98

8.2 Constraints in multiple radial planes . . . . . . . . . . . . . . . . 99

8.3 Incorporate pre-existing structure . . . . . . . . . . . . . . . . . . 99

8.4 Lower bearing modeling . . . . . . . . . . . . . . . . . . . . . . . 100

8.5 Contact elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.6 Radial basis functions . . . . . . . . . . . . . . . . . . . . . . . . 101

8.7 Removal of obsolete elements . . . . . . . . . . . . . . . . . . . . 101

8.8 Pre-tension adjustment . . . . . . . . . . . . . . . . . . . . . . . . 102

Index 106

4

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Abstract

Keywords: structural mechanics, topology optimization, relative constraints, method

of moving asymptotes, convex approximation, primal-dual Newton method, Lagrangian

duality, axisymmetric structures, offshore structures, FPSO, solid isentropic material

with penalization, large diameter bearings, harmonic elements, finite element method,

linear static analysis, bilateral loadcases.

Unevenly distributed loads on rollers within an FPSO turret main bearing causes

increased wear, failure and possible down-time of the weathervaning system.

They are caused by the relative deformation between the inner and outer bear-

ing ring. In what manner these deform depends on the geometry of the bearing

itself, but also in large part on the supporting structure: the turret casing and

the turntable. Bluewater Energy Services (BES) is interested in a structural so-

lution to this problem. However, structural problems with these type of relative

constraints are far from esoteric (the industry has shown attempted solutions

of both stiff and flexible nature) and till date, the conventional approach is to

launch an iterative design process between the company and the bearing man-

ufacturer, starting at some chosen initial design. Since this initial design itself

was not designed for -or possibly already had problems with- roller load distri-

bution, one cannot be guaranteed a solution, let alone an optimal design.

This thesis proposes a new perspective on this problem using topology opti-

mization. By combining the method of moving asymptotes (MMA), structural

axisymmetry and relative constraints, a program is written which can determine

a feasible, optimal solution in an iterative fashion. This solution represents

the topology of the structures adjacent to the main bearing (i.e. part of the

turntable and part of the turret casing) such that the rollers remain uniformly

loaded. By incorporating a class of harmonic elements, the user is not limited

to mere axisymmetric loadcases, but can also apply any preferred type of bilat-

eral loadcase. Effects of stress concentration, buckling and fatigue are neglected.

The resulting algorithm proved to be stable and showed convergence in all con-

6

straints (relative and absolute alike) and the demand for a solid and isotropic

solution using the SIMP method. This ensures the calculated topological deriva-

tives provide accurate information as required by MMA. A few iterations of a

bilateral loadcase are shown in figure 1 as an example of how the algorithm dis-

cards elements from the design domain. Two types of loadcases were examined:

1. An axisymmetrical loadcase where only the gravitational load of the turntable

and risers are modeled. In this model there is no tangial structural re-

sponse.

2. A bilateral loadcase where, besides gravitational loads, overturning and

transverse loads are applied. In this model tangial structural response

does occur.

In both types of loadcases (i.e. axisymmetric and bilateral) the converged so-

lutions showed a lot of structural similarities, among which a tubular structure

increasing the torsion resistance of the inner bearing ring (see figures 2a and

2b). Additional numerical models show that, indeed, this type of geometry is

optimal when trying to limit torsional deformation. The support for the outer

ring is different in both cases due to the difference in nature between the load-

cases.

The product of this thesis is an optimization program that is capable of han-

dling multiple relative constraints in an axisymmetrical structure subjected to

non-axisymmetrical loadcases. Further research has to be done in order to pro-

vide a more accurate and definite topology, in which the two most important

steps are the modeling of the lower bearing and the inclusion of constraints in

multiple radial planes (using derivative convolution) such that the main bear-

ing is evaluated at multiple key points. These steps are not too difficult to

overcome, but, due to time imitations, they are outside the scope of this the-

sis. More elaborate contingencies include subjecting the model to stress and

fatigue constraints and model the main bearing rollers with contact elements.

By incorporating these constraints, topology optimization can be considered a

serious candidate for solving problems of relative nature, and one can find this

relativity in other places than just the main bearing; also the lower bearing and

the swivelstack will probably be interesting subjects. Since topology optimiza-

tion is a rather unknown concept within the offshore industry, a company such

as BES might stand a lot to gain in expanding and utilizing this type of knowl-

edge, especially since there are clear indications that conventional engineering

has difficulty dealing with relative constraints.

7

Figure 1: Iteration 2, 9, and 40 of the bilateral loadcase. Be-

tween the turntable on the left and the casing on the right we

see a solid black square which represents the main bearing.

(a) Solution of axisymmetrical loadcase. (b) Solution of bilateral loadcase.

Figure 2: A close up view of the axisymmetric and bilateral

solution x(κ). Clearly, the solution shows to favor a tubular

structure around the inner main bearing ring. The solid square

in the center represents the main bearing itself (which remains

untouched by the algorithm).

8

Prologue

Whether it is common or not to write a prologue for a master thesis I don’t

know, but studying has become a significant part of my life, so I guess I’m

pretty much obliged to. I got the call from Bluewater only after I already half-

committed my graduation to a third party. If they would invite me that same

day, I told them, I would be willing to reconsider. So that day we started what

is now marking the end to a long and rather strenuous student carrier. ‘What

better way to end with some fun, axisymmetrical topology optimization?’, I

hear you think. And indeed you are correct -sir- as long as, while programming,

stoicism outlasts your error-induced rancor. There is no guarantee it will, not

without the constant support of friends and family, but I’ll save that for the

end. One exception though, and it’s a cliche but it doesn’t count when it’s true:

Thank you, for years and years of continued unconditional support and love,

representing altruism at its finest

mom and dad

Much of what you are holding right now is math, so if that doesn’t quite spike

your interest, in the following I will try to explain this thesis in plain English.

Structures need to be designed, then build. If you want to sit, you design a

chair. If you want to fish, you design a fishing pole. A chair is stiff, a fishing

pole is flexible. If a chair were flexible you will find a large portion of yourself

laying on the floor rather than sitting in the chair. If you design a stiff fishing

pole it will break in half and you will sob inconsolably. There are, however,

structures that do not intrinsically demand either stiffness or flexibility. They

need a bit of both, or a lot of both, or some specific ratio, or... well, you don’t

really know, that’s the point. In a sense, common human intuition sometimes

falls short when designing these structures, just as it does in our structure: the

‘FPSO main bearing support structure’.

I’m not going to tell you what it is or what it looks like, and Google probably

won’t help you much either. You only need to know that it has a problem, and

that that problem get worse when you make the structure bigger. The thing is:

9

we really want to make it bigger though, and as a starting point we take an old,

smaller design and make it bigger. Then, to fix our emerging problem, we make

slight modifications to the new and bigger design. Sounds familiar? That’s

because your body was ‘designed’ in somewhat the same fashion, by means of

evolution.

When one mentions evolution, the assumption usually is that it comes up with

the best answer to all of natures’ hardships and competitions. In a lot of cases,

yes, it does. In others... not quite. Conveniently, a whole string of these bad

designs is mentioned in the ‘Greatest Show on Earth’ by R. Dawkins. I could

have chosen the human eye as an example 1, but consider that one homework.

Instead, let’s look at a giraffe.

In mammals, the ‘recurrent laryngeal’ nerve (RLN) is a nerve that connects the

larynx 2 to the more central ‘vagus’ nerve which runs up to the brain. The

RLN branches off the vagus nerve just above the aorta, loops around it and

heads back up to join the larynx. This means that, in humans, the RLN makes

a small detour of a several centimeters rather than moving from point A to B

in a straight line. Compared to the whole complexity of your body, one might

brush this off as being fairly insignificant. But when we examine the giraffe

we see a RLN that extended as far as 5 meters in length! Not only a waste of

resources but also an unnecessary increased risk of damaging the nerve, not to

mention the 10 meters (down and up the full length of the neck) signals have to

travel when sent by the brain. Perhaps, a consequence could be that “despite

possession of a well developed larynx and a gregarious nature, the giraffe is able

to utter only low moans or bleats”, as mentioned by Dawkins.

Somewhere down the evolutionary tree humans and giraffes share a common

ancestor, a creature that did not look anything like a human or a giraffe but did

function as an initial ‘design’ 3 for both. As the neck of the ancestor steadily

grew, the RLN, which was stuck around the aorta, had no option but to extend

along with it. The current giraffe might still function satisfactory as an animal,

but it will never be an optimum since all an engineer has to do to improve the

design is replace the idiotic 5 meter nerve with a simple 30 odd centimeter one.

Evolution cannot do so; it can only move forward with what is already there,

missing potential optima that might have been exploited. Hypothetically, if

trees were to continuously extend in length over time, there comes a point that

the giraffe reaches a physical boundary from whereon it can no longer keep up.

1Whose design can be compared to installing an electric socket on the wrong side of the

wall, then drill a hole through that same wall to feed a wire through in order to plug it in.2The organ commonly known as the ‘voice box’, considering its link to sound production3In context I chose to call it design. The ancestor itself was, of course, never actually

‘designed’ but also came about by means of natural selection.

10

This boundary could be the limit on neck weight, or maybe it could actually be

RLN which at a certain length impedes the animals functionality. In this newly

developed evolutionary situation, perhaps a monkey appears to be the optimum

solution in reaching the top of the tree. The ancestor turned out to be flawed

as an initial design for a giraffe.

Enough of this biological digression. Where am I going with all of this? Well,

by taking an existing structure, enlarge and subsequently modify it to adhere to

certain characteristics, engineers are basically following the same evolutionary

path. But, as we have seen in the giraffe, this might in some aspects result in

absurd and wasteful designs. Also, it will probably make you hit a ceiling earlier

than needed, meaning that there comes a point that, no matter what you do, you

cannot improve performance without a complete overhaul. To an engineer this

should at least raise some suspicion when applying this empirical strategy, es-

pecially when the purpose of, or demand on a structure slowly changes overtime.

Let’s circumvent this evolutionary design path. Let’s switch off human intuition.

The number of ways to design a structure is near infinite, and our brains do not

have the capacity to evaluate even a fraction of them. A computer, although

also limited, hugely extends this capacity, and it doesn’t have any interference

from intuition or a bias towards a certain configuration. Letting a computer

design a structure for you is a special branch within engineering and mathe-

matics called ‘topology optimization’ 4. Basically, you start off with a volume

consisting of a large number of small blocks (in this thesis around 12,000 +)

and tell the computer to start removing blocks without the structure falling

apart. Consider it a very elaborate game of Jenga, but there are a few differ-

ences. For one, besides the structure not falling apart we might add additional

demands, such as how much or in what manner it may deform under pressure;

and two, the computer may not only remove blocks but also add them when

needed. We let the computer do the designing for us, and as a snack, we get to

glance inside the ‘thoughts’ of a computer, see what structures it is considering

before discarding them and moving on to yet other, better designs. We do not

anticipate the computer’s decisions. If we could, we might as well design the

structure ourselves and avoid all these programming efforts; they would have

become obsolete. The appealing thing, to me, is that no human can tell what

the outcome will be. In a sense, we’ve created a self-governing brain5; a brain

whose sole purpose is to come up with the best structure given the most com-

plex of demands. There has been substantial development in this field, and the

4Derived from the Greek word ‘topo’ meaning place. In this context in means where to

put material, or how to distribute it throughout ‘Euclidian’ space.5In case you’re wondering what such a brain looks like on paper, skip to the appendices B

and C

11

flexibility of algorithms in describing different scenarios and demands is ever

growing. Hopefully, this thesis contributes to that effort.

I’d like to add one more thing before moving on to the thesis. There is a group

of people who had to hear me complain about (trivial) things the past four

(or far more) years: Robert Wouters, Joran van Aart, Thijs Muskens, Jolanda

Jacobs, Henk-Jan Bosman, Erik Verboom, Julian de Kat, Elodie Mendels, Pien

Minnen and Malte Verleg. Thanks for listening; I probably would’ve gone nuts

by now if it wasn’t for you.

12

Symbols and acronyms

Latin symbols

Symbol Description Defined

a Mode number

b Mode number

B Set containing all link180 elements modeling

main bearing rollers

figure 5.2

B Strain-displacement matrix (5.6), (3.28)

c Contribution factor; transforming global to lo-

cal displacements

(5.3)

e Amplitude vector for physical harmonic dis-

placements

(??)

^e Amplitude vector for virtual harmonic displace-

ments

(??)

E Elasticity vector; containing Young’s modulus

all elements within Θ

(3.20)

E Elasticity matrix; describing the deformation of

an element under load

(??)

Eg General elasticity (210 GPa)

f0 Objective function (3.2)

fm Constraint functions with m ∈ N and m 6= 0 (3.6), (3.7)

fρ Dimensionless unit load in DOF number ρ (3.22)

g Lagrangian dual fuction (3.12)

g Local displacement vector (5.4)

h Double harmonic vector (??)

H Harmonic primitives vector (??)

i Constraint number6

I(n) Identity matrix with size n

j Element number

k Iteration number

6unless otherwise specified

13

k Elemental stiffness matrix

K Global stiffness matrix

L Roller length vector (5.2)

L Lagrangian function (3.10)

m Total number of constraint functions

n Total number of design variables (number of el-

ements in Ω)

p Penalization factor (3.20)

p Set of mode numbers used in physical loadcase

P Stiffness reduction factor

P Main optimization problem (3.1)

R Radial contribution (3.3)

s Scaling vector (5.12)

S MMA approximated subproblem (3.9)

t Boolean vector identifying common harmonic

identities

(??)

T Transformation matrix (3.27)

u Displacement

u Displacement vector (either global or elemental)^u Virtual displacement vector (either global or el-

emental)

(3.25)

^

U Virtual displacement matrix (concatenation of

different^u’s corresponding to different unit

loads

(5.4)

v Horizontal displacement component of u (3.29)

v Set of mode numbers used in virtual loadcases

w Vertical displacement component of u (3.29)

W Dual objective subproblem (3.16)

x Design variable or elemental density (3.20)

x Design variable vector (3.20)

14

Greek symbols

Symbol Description Defined

β Link angle (3.26)

Γ Feasible domain

∆ Difference

ε Strain

ζ Displacement amplitude vector (??)

η Number of DOFs in an element

θ Angle about the axis of axisymmetry

Θ Domain containing all elements with FE model figure 5.2

ι(n) Vector of size n with all elements equal to 1

κ Total number of iterations to reach convergence

κ Vector containing rearranged stiffness coeffi-

cients

λ Vector of Lagrangian multipliers

Π Domain containing all plane elements that

model the main bearing

figure 5.2

ρ DOF number

σ Stress

τ Constraint tolerance (3.1)

Υ Domain containing relative stress link-elements figure 5.2

Φ Domain containing all bearing affiliated ele-

ments, both link and plane elements

figure 5.2

χ Vector containing all density ranges of all design

variables

χ−

Lower density boundary

χ Upper density boundary

ω Boolean vector identifying common frequencies (??)

Ω Domain containing all design variables xj figure 5.2

15

Acronyms

BES Bluewater energy services

DAO Derivatives for arbitrary orientation

DOF Degree of freedom

FEA Finite element analysis

FEM Finite element method

FPSO Floating storage production and offloading

KKT Karush-Kuhn-Tucker (conditions)

MMA Method of moving asymptotes

OTC Offshore Technology Conference

SAO Sequential approximate optimization

SIMP Solid isotropic material with penalization

SLP Sequential linear programming

SPM Single point mooring

TOP Topology optimization

16

Chapter 1

Introduction

1.1 A brief history

The ‘peak oil theory’ is being disputed more and more frequent as the projected

maximum petroleum extraction in the year 2020 is probably not met. In late

2013 a KPMG publication [10] stated that a receding oil and gas market is till

date unfounded, backed by the fact that across the globe considerable invest-

ments are made off the coast of the US, Brazil and Northern Australia. The

Artic regions are expected to undergo similar developments in the no so distant

future, and advances in the continues struggle with ice-induced problems are

made. The recent decrease in oil prices is supply-triggered, rather than a de-

crease in demand. This increase in supply, without losing ourselves in too much

speculation, is in large part caused by recent political instabilities in, among oth-

ers, the Middle East. The current demand itself was anticipated by the Interna-

tional Energy Agency (IEA) which also states that the global energy market is

expected to rise by a third between 2011 and 2035. The development of sustain-

able energies are not yet able to cope with a demand of this magnitude, hence,

fossil fuels are the only candidate to fill in the gap. Offshore developments will

naturally play a substantial role -certainly considering the controversy related

to shale gas extraction- and consequently own of its key components, the FPSO.

An FPSO is a ship-shaped vessel that remains moored at sea for moderate peri-

ods of times while operating a pre-developed subsea oil or gas field. The abbre-

viation FPSO (Floating Production Storage and Offloading) has been steadily

gaining more and more recognition since its first application in 1977; the Shell

Castellon, an FPSO operating an oil field in 117 m of water in the Spanish

Mediterranean. It stems from the vessels ability to process (produce) hydro-

carbons and storing them for certain amounts of time before being offloaded

to a shuttle tanker. The term floating indicates that the vessel does not need

17

to be supported other than its own buoyancy. The need for deeper and ever

remote oil and gas field development has proved to catalyze extensive research

in the technology concerning these vessels, slowly pushing the concept of FPSO

towards offshore energy market dominance.

As far back as 1891, the first submerged oil wells were developed in fresh water

lakes in Ohio, USA, using small platforms supported by piles driven into the

lake bed. After the Second World War the first permanent offshore installations,

pioneered by Kerr-McGee Corporation, struck oil beyond the sight of land. In

both cases the dependency on land required infrastructure that connected both

offshore and onshore facilities; pipelines had to be built in order to transport

oil ashore for further processing and logistics. This remained the case up until

developments in both economy and engineering favored ventures that further

extended the borders known to the offshore sector. This consequently launched

the era of floating production, which began in 1975 with the Transworld 58

becoming operational, a converted semi-submersible drilling rig deployed in the

Argyll field off the coast of the UK. From here on the floodgates were opened,

spawning various types of FPS structures: compliant towers, spars and tension

leg platforms, added to the existing semi-submersibles and FPSO-like vessels.

The latter without doubt the industry’s favorite, whereas 63 percent of all FPS

installations are accounted for by FPSOs, a grand total of 186 worldwide based

on the 2013 statistics.

Among recent FPSO records are the Pioneer, which operates in record water

depth of 2.6 kilometers (8,520 ft.), and the Kiszomba with a storage capacity

of 2.2 million barrels (350,000 m3). The Royal Dutch Shell ordered the largest

vessel ever constructed: the Shell Prelude FLNG, build by Samsung Heavy In-

dustries. The 488 meter (1,601 ft.) vessel, expected to weigh approximately

600,000 tonnes when operational, is designed to extract liquefied natural gas

(LNG) from the Browse Basin, 200 kilometers off the coast of north Australia.

It weighs more than five aircraft carriers combined.

1.2 FPSO fundamentals

A key feature of conventional FPSOs is the turret, normally found in the bow

of the vessel. In contrary to spread moored vessels, which have multiple con-

nections to the seabed at the bow and stern, the turret is the only component

physically anchored. The FPSO can rotate (also known as weathervane) freely

around the turret, while subjected to the prevailing environmental conditions.

Since all anchor chains are attached to the submerged part of the turret, such

18

a system is called a single point mooring (SPM) system. The advantage of

weathervaning is that it reduces the forces exerted on the anchor chains and

connections by minimizing the roll and heave motions. Spread moored systems,

while maintaining their angular position regarding wind, waves and/or swell,

might be subjected to enormous forces when confronted with a certain angle,

considering their significant hull surface area. SPM turret systems allow the

vessel to orientate the bow of the FPSO facing oncoming weather, thereby re-

ducing the loads on the mooring system. This also proves an advantage during

offloading procedures: shuttle-tankers may connect at the stern of the FPSO

while the weather may safely attempt to push the vessels apart and reducing

the change of collision. As a consequence offloading procedures might continue

in harsher weather conditions.

Many FPSO hulls are conversions of surplus tankers; tankers that had a deck

structure suitable for carrying a process facility. Conversion meant that acquir-

ing an FPSO was relatively cheap and significantly faster than building from

scratch, which certainly added to the popularity in the early years of these off-

shore installations. There are, however, certain drawbacks to converting old

tankers, the most important of which is the restriction on the weather condi-

tions and water depth. For these conditions, the demands for integrating the

turret into the hull can become quite elaborate and is therefore usually not eco-

nomically feasible. As a result, initially, FPSOs were designed to produce small

to medium sized oil fields in remote locations, ranging from moderate to deep

waters, where pipelines and fixed infrastructure would prove inefficient. With

the development of turret mooring and new-build ship-shaped hulls the number

of FPSOs operating in very deep water and harsh weather conditions has grown

substantially. In short, new build FPSO are designed keeping in mind roughly

four requirements:

1. Installation of the turret (usually in the bow).

2. Oil storage capacity.

3. Space for process facilities and accommodation.

4. Displacement and ballast capacity as to reduce the effect of motions on

the mooring and riser systems.

Nowadays, the modular-based FPSO construction is very much standardized

and automated. It out-competes the application of large jackets, since modular

assembly at a shipyard reduces the need for heavy-lifting vessels at the instal-

lation site. Consequently, this reduces the lead-time to first oil.

19

Figure 1.1: The Rosebank FPSO as designed for Chevron; the

turret is highlighted. The Rosebank will serve as a case study

throughout this thesis.

FPSOs commonly owned by contractors (such as the Dutch founded Bluewater

Energy Services (BES) and SBM) and leased by oil companies if need arises.

This a sharp contrast with production platforms which are usually owned by

the oil companies themselves. The reason for this shift is the fact that oil com-

panies rather lease oil fields with a small or uncertain reservoirs than own them,

although a lack of operational experience within a certain region may also be a

factor.

20

Chapter 2

Thesis subject and goals

2.1 Problem description

While applying larger turret diameters (from about 20+ meter diameter), off-

shore engineers have found that the increased relative displacements between

the inner and outer rings of the main bearing causes unevenly distributed loads

and even direct damage on the rollers. This in turn leads to excessive wear and

down-time of the turret system and its weathervaning capability. Bluewater

Energy Services (BES) is interested in a structural solution to this problem.

The main goal in this thesis would therefore be to optimize the main bearing

support structure (as well within the turntable as the turret casing) in such a

way as to limit non-uniform load distributions on the bearing rollers. In the

next paragraphs we shall take a small step back and look at the complete prob-

lem before deciding on a structural approach.

Consider two typical turret designs: a stiff turret with large radius/height ratio

and a less stiff turret with smaller radius/height ratio. The first concept po-

tentially provides a lot of space for equipment within the turret, perhaps even

an inverted swivel stack system thereby drastically reducing the turret height.

It can also cope with more risers. By increasing the diameter we consequently

increase the stiffness of the turret, and this has a drawback, which is discussed

in the next paragraph. The mooring chains exert a force on the spider, the ver-

tical component of which will be transferred by the turret to the main bearing.

In absence of a lower guiding bearing, the horizontal component will have to be

passed up through the entire turret by shear force until it can be transferred to

the ship hull, where it will cause a massive moment. To prevent this, it would

be logical to add a lower bearing to transfer the horizontal load directly to the

hull. However, the increased stiffness of a larger diameter turret would limit

the amount of horizontal forces being transferred simply because the turret in-

21

trinsically does not want to bend, but rather deal with the loads using mostly

shear. Slender turrets that, for obvious reasons, have less bending stiffness do

not display this problem. They deform until they are restricted by the lower

bearing and from thereon transfer a significant proportion of the horizontal load

to the turret casing.

The upper and lower radii of the turret each have their own specific limitations.

We cannot make the upper radius too small taking into account the space needed

for the risers, umbilicals and such. The lower radius is limited by the spider

(the mooring chains system) as well as the risers. As a result of the above, the

upper radius being smaller than the lower radius, the turret has a cone shape.

A cone shaped turret means that, when being lowered into the ships hull during

installation, the radius of the casing in the ship has to be greater than the lower

radius of the turret. When installed, a gap has to be bridged between the main

bearing of the turret and the hull. The structure that closes this gap and adds

stiffness to the outer bearing ring is called the torsion box.

Earlier, within BES, a simplified FEM model was made concerning the SKARV

FPSO project which included a three-raceway roller bearing and a cone shaped

outer ring support structure. The inner and outer ring interaction was based

on contact and gap elements. The model is constraint at the lower part of

de cone which simulates a rigid hull connection. By introducing an axial and

horizontal force to the upper part of the turret we can see how the inner ring

behaves with respect to the outer ring; this without the influence of an inner

support structure. The point of this quick study was to establish the difference

between working with a shell and a solid model. However, we can use it simply

to illustrate the problem with extensive relative bearing deformation. In figure

2.1a and figure 2.1b we see both the creation of a gap and an angular difference

between the two rings. Note that these are visually exaggerated displacements;

the turret and the turret casing seem to actually touch or cross at certain points.

This, of course, does not happen in reality.

In internal correspondence (as far back as 2008) within BES it was noted that

the trend of turret design is towards larger main bearings and more riser space

because of design and installation simplicity; among other things, the risers can

be pulled up straight rather than at an angle. Also, main bearings that span the

entire casing diameter are not a serious financial setback with respect to smaller

bearings, considering the total projected costs are not uncommonly 100 million

plus. As a result the bearings got bigger in diameter over the years. However,

the cross section did not grow in the same fashion because of limitations to

the roller ton size. Because of this growing habit of up-scaling (and because of

22

(a) Bearing displacement (b) Bearing rotation

Figure 2.1: The SKARV main bearing displacement and rota-

tion shown on either side of the turret. Note that the displace-

ment and rotations are exaggerated. In reality the inner and

outer ring never come in contact.

the fact that the turntables inevitably became more massive) the forces on the

bearings could actually deform the rings in such a way that they could slide over

the roller tons axially, which causes damages to the ring. So, on the outer ring

directed we have the loads induced by hogging and sagging of the hull; on the

inner ring we have loads caused by the massive turntable inertia; all of which

are combined with the static axial force caused by the turret weight. As long as

the relative deformation of both rings is the same, there would be no problem

as the roller will be loaded in an acceptable manner. This has hitherto been

the case concerning smaller diameters, but the boundaries are currently being

pushed, explaining the industries interest in solutions, whether they are created

from a structural or mechanical perspective.

Other types of bearings have been considered to some extend. Basically, the

turret main bearing can be based on three different concepts, i.e.

1. Roller bearings (currently applied)

2. Container ring bearings

3. Bogey bearings

BES has shown to favor roller bearing application in SPM systems, including

CALM buoys, mooring towers and turrets. Both container ring and bogey

applications -having significantly larger diameters- suffer from pitting corrosion

and fatigue induced cracks, and this is explains (in part) BES’ penchant for

23

sticking with smaller diameter bearings. The known issues regarding roller

bearings include (and quote for internal correspondence)

1. ‘Maintenance issues; insufficient greasing or the usage of inferior quality

grease causing internal corrosion and ingress of water and debris.’

2. ‘The inaccuracy of supporting surfaces causing peak loads in certain areas

and edge loads on the rollers.’

3. ‘Incorrect design of support structures; deformation of structures leading

to the rotation between and opening of the bearing rings with respect to

each other, also causing non-uniform load distribution.’

Although all of the above potentially pose relevant and significant problems, the

latter will obviously be the one this thesis will address.

2.2 Problem approach

One is of course allowed to question the entire existence of the single point

moored FPSO and come op with an radical new concept. This thesis, however,

will work within the concept of the current FPSOs capable of weathervaning.

Having established this, we may fundamentally approach the problem in the

following ways:

1. Limit the loading on the main bearing; this is already partly done by

applying a lower bearing, though the horizontal forces induced by the

turntable inertia still remain.

2. Acquire more proportional bearing dimensions; this can mean two things:

either you increase the cross section of the bearing along with the radius

or you keep the bearing diameter small.

3. Different bearing design; an inner bearing ring that can impose pulling

forces on the outer ring is already applied (using a fourth raceway), but

there are various other options that could be considered.

4. Design the structures adjacent to the main bearing in such a way it limits

the relative deformations between the inner and outer bearing ring.

Approach 1 holds a fundamental question that we always have to ask ourselves

repeatedly during any design process. The inertia forces from the turntable can

only be reduced by decreasing the mass or radius of gyration. This requires

extensive redesigning of the turntable and would likely cause more problems

than it would solve. Approach 2 and 3 will need to be investigated by an expert

24

on large diameter bearings since it requires far going knowledge about tribol-

ogy 1 and contact mechanics. It is therefore wise to consider the bearing itself

as a black box and work with the dimensions and properties provided by the

manufacturer. Approach 4 lies within the realm of structural mechanics and we

should be able to analyze this particular problem using finite element method

(FEM).

Various attempts were made by other companies to solve this pending problem

via a structural approach. Two of them are discussed here; a few are referred to

in [25, 26, 27]. SBM filed a patent [24] in 1989 which described the use of a ‘rigid

ring’ in order to protect the outer main bearing ring from deformations. This

torsion box is by no means an innovative way of distributing loads but rather a

very stiff circular structure on which the main bearing is mounted; in essence it

is a extension of the bearing itself making the construction stiffer on the whole.

The torsion box rests on top of the extended turret casing and is free on all

sides, it is not embedded in the structure of the hull itself. The observation that

this construction does not strike one as an valid invention does not change the

fact that its patent held for 20 years after which it expired in 2009.

In 1997 a paper [12] (also in affiliation with SBM) was published in the OTC con-

cerning the so-called ‘forgiving tanker/turret interface’ stating problems caused

by relative deformation could be solved by applying a flexible coupling between

turret and casing using an elastomeric suspension. The system included several

elastomeric pads installed at and angle that could be hydraulically aligned in

situ. Still, it leaves one to question the fatigue performance of these types of

support (since the entire loads have to be transferred through a few pads) and

contingencies of this design have been found sporadic during further literature

study.

It might be worth pointing out a perhaps rather salient contradiction: On one

hand engineers have tried to tackle this problem by adding stiffness by intro-

ducing a rigid torsion box, on the other they have done the exact opposite by

installing flexible couplings. Granted, a difference might exist between inner

and outer ring stiffness in order to reach an optimum, but both papers made

opposite adaptions to the outer only. This peculiar fact might tell us something

useful about how the industry responds to problems of relative nature, rather

than the conventional absolute ones.

1The science concerned with interacting surfaces, friction, lubrication and wear.

25

2.3 A step toward topology optimization

2.3.1 Optimization potential

The conclusion of the previous section was that the design of the main bearing

support structure (i.e. the structures directly connected to the main bearing,

the hull and the turret turntable) would be a suitable candidate for further,

more extensive research. This means we will start searching for indications that

changes in topology of this structure could actually improve the stress distribu-

tions within the main bearing. It also means ruling out the possibility that the

solution might be a very straight forward and simple one; a scenario to avoid

at all cost would be one which, after months of research and modeling, yields

a fundamental, text book solution to the problem. This is not to be expected

though, considering the industry still struggles with the problem, but it is nev-

ertheless a desirable approach to any new problem.

Figure 2.2: The model geometry with the main bearing cross

section shown in red, the turntable and hull in gray.2

In figure 2.2 a section of the hull and the turntable are represented by two

discretized areas with appropriate dimensions. Connecting the hull and the

turntable is a simple cross sectional model of the main bearing in which two

links are placed for each of the four raceways (therefore 8 links in total). Each

pair of links is a simplified model for a roller, where the links represent either

edge of the roller. When subjected to an uniform load distribution or vertical

displacement, the stresses in these links will be equal. If these stresses are not

equal, we may conclude a certain degree of relative deformation between the

coupled substructures has occurred. This is only valid if we allow the links

26

to only respond to compression, not tension, since rollers can only exert pres-

sure on a surface. Thus, by evaluating the relative stress between the links

we evaluate the relative deformations between the structures. Since the rollers

are relatively small in comparison to the turntable and hull structure (where

we expect smooth, near zero curvatures), we can assume the stress distribution

between the links to be a linear function. The load distribution will therefore

be optimal when both stresses are equal. Upon performing a FEM analysis

one may derive the relative stress between the two members within a set. The

higher this relative stress, the less uniform the load distribution on a bearing

roller would be in reality (and vice versa).

The stiffness corresponding to a particular degree of freedom in a structure

is always dependent on the modulus of elasticity and the topology. If we as-

sume both are interchangeable, then we are allowed to simulate an arbitrary,

non-defined change in topology by varying the elasticity of both design spaces

independently, i.e. E ⊂ Ω. While going through different combinations of stiff-

ness we can monitor the effect it has on the relative stress between each set of

links. Note that we are trying to monitor the effect of topology in general, not

a specific topology.

The algorithm grades the load distributions within the link sets of each structure

by introducing a dependent performance variable φ , defined by

φ(Eh, Et) =1

n

n∑i=1

(σmax

σmin

)i

φ ∈ [1,∞〉 (2.1)

In figure 2.3a we can see the landscape described by the performance variable φ

as a function of the hull and turntable elasticity ratio (Eh and Et). The ratios

are defined by

Eh,t =E

210 GPaEh,t ∈ [0.5, 2]

As φ approaches unity (or perhaps becomes equal to 1) the better the load

distributions in all link sets become. Optimizing the performance variable for

changing E would therefore mean

min φ(Eh, Et) : Eh,t ∈ Γ (2.2)

with Γ representing the feasible domain. The feasible domain ranges from 0

to 1, since a ratio of 1 represents the full elasticity modulus of commonly used

steel (210 GPa). A ratio equal to 0 means we discarded the design space and

a ratio equal to 2 would mean we are using a very stiff but fictional topology

(since no change in topology can result in a higher stiffness than the fully closed

design space). Figure 2.3 shows that, in both feasible and non-feasible domain,

27

(a) Partly non-feasible domain (b) Feasible domain

Figure 2.3: Performance variable φ as function of turntable and

hull stiffness ratio (Et and Eh respectively) in both feasible and

non-feasible domain.

the optimal solution is not a simple matter of making both hull and turntable

as stiff or flexible as possible, nor is it a unique ratio between stiffness. The

numerical results of this analysis are shown in table 2.3.1. In the chosen design

space and bearing cross section geometry, a φ of 1.1056 is the best attainable

performance, resulting from a sem-stiff turntable (Et = 0.5) and a stiff hull

structure (Eh = 1.0).

min φ Eh Et

Feasible 1.1056 0.50 1.00

Non-feasible 1.1045 0.72 1.22

Based on this simple test we can conclude that there is potential for a structural

optimization approach, since the ratio between the hull and the turntable stiff-

ness is not a straight forward one. We have to realize, though, that this model

is two dimensional, and as such it does not take into account the full geometric

effects of the structure in 3D. It does, however, give us more confidence that a

3D model will support our conclusions based on 2D.

2.3.2 Why topology optimization?

Before the need for excessively large, segment-fabricated main bearings to sup-

port the colossal turrets we find in the new generation FPSOs, non-uniform

load distributions were not the main concern. Smaller diameter bearings have a

28

more proportionate cross section in contrast to their large diameter cousins and

as such they are less susceptible to relative inner and outer ring deformations,

or, more specific, to deformations in general. Also, as the bearing diameter

decreases, we expect the mass of the turret, the turret radius of gyration and

the loads induced by the risers to decrease accordingly (decreasing the diame-

ter obviously means decreasing the feasible amount of risers). Because of these

relations, the importance of load distributions have hitherto not been an active

constraint in the design process. They only emerged as a scaling effect of con-

tinuously pushing the main bearing dimensions to its limits.

While pushing these boundaries, initial designs have been based (in most cases)

on pre-existing but smaller turrets build for similar conditions; turrets that had

proven themselves in practice but were designed without an active constraint

concerning load distributions. These initial designs would be adapted so as to

tackle the problems at hand, resulting in a larger, slightly modified turret. This

manner of designing is very much like an evolutionary process, at the end of

which we now find ourselves approaching the boundaries imposed by excessive

wear and gapping.

Further modification of the turret design might well be possible, at least to a

certain extend. The Rosebank’s main bearing support structure has been suc-

cessfully adapted, although only in numerical models, to reduce the non-uniform

load distributions on the rollers to an acceptable limit. This also included the

introduction of a new type of bearing, one with an additional fourth (outer)

raceway to cope with the rings’ newly developed gapping behavior. The inter-

esting question remains: Can engineers maintain this approach of constantly

modifying designs for ever growing diameters and still regard the result as an

structural optimum? Since a suitable initial design is a prerequisite to finding

an optimum within a design domain, it is not realistic to positively answer that

question because the earlier structures were not designed to deal with future

problems concerning relative deformations of the main bearing. The structure

has demand added to it, demand that was not previously there.

We might even go further and ask ourselves: Do engineers (as they are ul-

timately human) have a reliable intuition when it comes to finding optimal

structures that have to satisfy one or multiple relative constraints? Also this is

not obvious. Engineers in general have developed an intuition that helps them

design and recognize stiff structures (structures that tend to resist deforma-

tion when subjected to a certain load) as well as avoiding stress concentrations.

When it comes to relative deformation, that intuition looses its grasp because a

structure no longer has to be either stiff or compliant. As we saw earlier, when

29

dealing with coupled structures, such as the main bearing and its support struc-

ture, matters get only more complex, especially when we increase the number of

such constraints. In many FPSOs the outer main bearing ring is reinforced with

an circular boxed construction, also referred to as the torsion box or ‘rigid ring’

(the term used in the SBM patent [24]). This construction was added to make

the outer ring more stiff in attempt to solve these relative constraint problems.

The point being, there is no easy, straight forward method to determine the

desired compliance or stiffness for each individual structure when dealing with

these type of constraints.

There is a overlapping field within mathematics and engineering which attempts

to approximate an optimum structure, given a design domain and a certain set

of boundary conditions, loads and constraints, called ‘structural optimization’.

We can divide structural optimization into three global categories:

1. Size optimization

2. Shape optimization

3. Topology optimization

wherein topology optimization (TOP) would be considered the most fundamen-

tal approach. An example of size optimization could be to optimize the cross

sectional area of a cantilever bar subjected to a horizontal load and horizontal

displacement constraint. Naturally, this concept can also be applied to multi-

variable problems, but the geometry of the model is predetermined. Shape op-

timization deals with solely the contour of structural boundaries. Basically, it

can manipulate the shape of a boundaries, but cannot create new boundaries or

change the connectivity within the structural domain. Topology optimization,

in general, is able to change boundaries and connectivity when given an appro-

priate design domain. Consider a model which is basically a boxed construction

with diagonal members in which the design variables are the cross sectional

areas. Let’s assume we want to minimize cross sectional area (i.e. minimize

mass or volume) but maintain a certain stiffness. Obviously, the optimization

should only decrease the area of those members that, in their given position

and orientation, do not contribute significantly to the structure’s performance

or could be sacrificed in order to add area to other, better positioned members.

TOP would be able to reduce the cross sectional area of a member thus far such

that the member is essentially removed from the structure.

30

(a) (b)

Figure 2.4: From a initial starting point (a), with the cross sec-

tion of the members as design variables, TOP is able to eliminate

those whose contribution is not significant (b).

By removing a member altogether we effectively change the boundaries of the

structure, but it could also create new or merge existing boundaries and change

the connectivity of the structure as a whole. For more examples on how TOP

can shape and sculpt structures, take a quick look at chapter 4 in which various

simple test models are examined.

By increasing the number of design variables, we increase the complexity and

possible detail within a structure. Doing so enables TOP to help us find a

structural optimum within a feasible, bounded domain. Given a space in which

to shape a structure, one will find that it has a (practically) infinite number of

possible structures to choose from. Most of these structures will preform horri-

bly or worse, nevertheless, if they are part of the domain, they are considered

a possibility. Imagine a design space composed of blocks that can be switched

‘on’ and ‘off’, i.e. either they are added to the structure or removed. If we are

looking for an optimal structure within a domain with n of such blocks, we are

looking at 2n possibilities. When we consider design spaces with 2000 blocks or

more, the number of possibilities is more than the staggering amount of atoms

in the observable universe. In this thesis a design space with over 12.000 design

variables is considered. How one navigates through this maze of possibilities is

explained in detail in chapter 3.

2.4 Goals and boundaries

To conclude this chapter, a list of goals and subgoals is formulated. Whether

or not these goals are achieved, and to what extend, will be evaluated in the

conclusion (chapter 7). The list differs somewhat from the goals in the thesis

proposal, which is due to the fact that during the actual modeling, unexpected

31

problems were encountered. The goals are as follow:

1. Investigate the potential of topology optimization for the pending main

bearing issues (has already been discussed, see subsection 2.3.1).

2. Write a general mathematical framework for optimization with relative

stress constraints.

3. Write an optimization program capable of handling

(a) multiple (relative) constraints.

(b) the intrinsic non-linear behavior of roller bearings.

(c) axisymmetric as well as non-axisymmetric loadcases.

(d) data exchange between optimization and external FEM-algorithms.

4. Compute an optimum solution which can be used as an initial design and

provides structural insights concerning relative constraints.

Since this thesis is mainly concerned with the potential for applying topology

optimization to problems of axisymmetric and relative nature, boundaries are

set on the various failure modes. Mostly likely, the resulting structure will fail

on one or multiple of these criteria, but, as discussed earlier in this chapter,

it should provide insights in what a theoretical optimum structure should look

like, from which point on practical engineering can take over. In the future, this

model may be expanded with a whole range of failure modes. All boundaries

set in this thesis are now listed:

1. The loadcases are assumed to be static. No dynamic effects are examined.

2. The main bearing is considered a black box (it will not be part of the

design domain Ω).

3. Effects of gravity on the structure are neglected.

4. No fatigue, buckling or yield critera are considered.

5. All material is assumed to behave elastic.

6. The structure will be axisymmetric.

7. The stress distribution on the rollers can be approximated in a linear

fashion (as long as contact is maintained).

32

2.5 Thesis structure

Some effort will now be made to describe the structure of this thesis; how it will

attempt to achieve these goals. The best way to structure this thesis turns out

the be rather chronologically.

Chapter 3 will discuss the basics of optimization and, in particular, the

methods used in this thesis. The mathematical description of the prob-

lem is formulated such that we can always return should we find ourselves

lost in detail. Of great importance throughout the other chapters are the

concepts of finding topological derivatives and their manipulation by us-

ing penalization (SIMP). The method of moving asymptotes, duality and

Newton’s method for finding multi-constraint optima, although important,

will be less prominent in later chapters.

Chapter 4 is there to confirm the actual theory by conducting a series

of simplified model tests. Not all performed tests are included in this

chapter; the ones that are were highlighted mainly because they reassured

the algorithm’s performance under special conditions, such as:

1. Link elements that change their angle during optimization.

2. The coupling of two separate design spaces and their susceptibility

to ill-conditioning.

3. The introduction of relative constraints, rather than conventional

absolute ones.

Also, the ways in which performance can be monitored and evaluated are

introduced. This, of course, might help in the interpretation of the data

produced in chapter 6.

Chapter 5 could be considered the actual in-depth modeling of the main

bearing support structure problem. It represents the application of the

math proposed in 3 (verified by chapter 4) to a set of Matlab and Ansys

programs controlled by a master of ‘governor’ program, all of which can

be found in the appendices ?? and ??. These programs together perform

the model building, finite element analyses, necessary data exchange and

optimization in an iterative fashion for both axisymmetric and bilateral

loadcases. Concerning the latter, some new mathematical concepts are

introduced which would be out of place in chapter 3, concepts that are

needed to determine topological derivatives for superimposed harmonic

functions (these type of functions cause tangial deformations which greatly

complicate matters).

33

Chapters 6 and 7 will introduce different sets of loadcases, their behavior,

results and the conclusions that can be drawn from them. The goals, as

described in 2.4, will be evaluated.

34

Chapter 3

Mathematical modeling

The focus of this chapter is to establish a mathematical framework on which we

can then proceed building the final bearing and support structure model (which

will be done in chapter 5: ‘Applications to main bearing support structure’).

It might occur that the connection between theory and application within the

FPSO is not explicitly mentioned in this chapter, e.g.: it does mathematically

formulate the relative constraint functions, but the actual modeling of the main

bearing rollers, in which these constraints are used, will be left to chapter 5.

Naturally, chapter 5 has references to the mathematics where needed.

Within this framework we will discuss all mathematical concern within the

topology optimization, among others, the

the formal definition of the main optimization problem denoted P

the definition of the objective and constraint functions

methods for approximating and subsequently solving P

The order in which we shall discuss these topics is based on the order in which

the algorithm deals with the optimization (which is rather similar). This could

well proof to be more convenient when evaluating the actual programming in

Matlab.

3.1 Formulation of main problem

Let us first define our main optimization problem and refer to it as P (shown in

(3.1)). In contrast with conventional topology optimization, we are not directly

concerned with either the stiffness or compliance of the structure. Of course

there are practical limitations to how flexible a structure can actually become,

but the real restrictions are the stress distributions on the roller bearings. If

35

we would set out to optimize these stress distributions we will find ourselves

facing a multi-objective optimization, considering the amount of roller bearings.

Intuitively, one would like to stay away from such mathematical formulations as

they could pose a problem far too complex for practical applications. Regarding

the stress distributions as constraints functions with corresponding constraint

limitation values (or tolerances) avoids this problem altogether. A logical choice

for the objective function would be the reduction of mass. In general, the

optimization problem could then be written as

P :

min f0 (x)

fi (x) 6 τi i = 1, . . . ,m

x(k)j ∈ χj j = 1, . . . , n ∧ ∀k

x(κ)j =

χj

χ−j

(3.1)

wherein f0 is the objective function, fi is one particular constraint function with

corresponding tolerance τi and x is the set of design variables or element densi-

ties. κ is the amount of iterations needed to reach convergence. The fourth and

last demand within P is a solid/void demand, i.e. the resulting structure should

mainly consist of either solid or void design variables. The tendency towards

these types of structures has to be mathematically built into the algorithm.

This is done using the so called SIMP method which is explained in detail in

subsection 3.3.2.

3.1.1 Objective function

The objective function in this thesis has two separate definitions: a plane stress

one and an axisymmetrical one. The plane stress definition is a rather simple,

linear function of x defined as

f0 (x) =

n∑j=1

xj (3.2)

Since densities are proportionate to the total mass of the structure, minimizing

(3.2) as demanded by P is equal to minimizing the mass of the structure. This

particular definition is only used in chapter 4 where the main purpose it to test

the eventual algorithms characteristics and performance. For axisymmetrical

models, such as the main bearing support structure model described in chapter

5 and onward, f0(xj) becomes a function of the distance that element j is located

from the symmetry axis. Consider the volume enclosed by a ring-shaped object

(as depicted in figure 3.1)

36

Figure 3.1: A design variable xj within an axisymmetric model

represents a solid ring of material (here cut in half) with area

A and inner and outer radii ri and ro.

V = 2

π∫0

ro∫ri

Ar dr dθ

in which ri and ro are its inner and outer radii respectively. A is the area of an

element, and, since the design spaces will be uniformly meshed, A will simply

be a constant. Its normalized mass can then be described by

m

πA= xj

(r2o − r2

i

)j

= xjRj (3.3)

From here its only a small step defining the axisymmetric objective function

f0 (x) =

n∑j=1

xjRj = xTR (3.4)

3.1.2 Relative stress constraints

The need arises for a mathematical notation in order to compare stresses within

each roller. Choosing a stress-ratio approach could cause problems in case of

severe uneven load distributions, causing the algorithm to deal with constraint

values approaching infinity or zero. In these circumstances, this approach would

fail to supply the optimization with useful constraint information and cause com-

putational errors. In reality these situations should not occur, provided that no

turret was designed that poorly, but when running models test we want the al-

gorithm to be able to deal with these types of hypothetical situations. We could

imagine a scenario in which one roller would be found rudimentary, then the op-

timization should be able to eject this roller by removing all element connecting

it without encountering instabilities; instabilities resulting from perturbations

within an almost unloaded roller that might well cause huge constraint value

37

responses. Therefore, defining a constraint function using the difference ∆σ

between two stress values would be a more suitable approach. The formulation

would then become

f∆σ (x) = |∆σ| − τ 6 0 (3.5)

But dealing with absolute values while calculating topological derivatives is

not preferable. The derivatives loose their indication whether they increase or

decrease the constraint value per change in density of an element. In order to

still use these constraint formulations, we have to assign two separate functions

for each constraint; one that deals with positive difference and one that deals

with negative difference, i.e.

f∆σ =

f∆σ = ∆σ − τ 6 0

f∗∆σ = −∆σ − τ 6 0(3.6)

in which

∆σ = σ1 − σ2

Both σ-terms represent the stress in a roller bearing. Rollers can only transfer

loads when being compressed, and therefore both σ1 6 0 and σ2 6 0. To ensure

this condition is met at each iteration, pre-tensioning of the main bearing is

modeled at a later stage (see subsection 5.1.3). Looking at (3.6), one can easily

verify that f∗∆σ is a horizontally mirrored image of f∆σ, the symmetry axis of

which is the line −τ . This is shown in figure 3.2. Consequently, only one of

these constraint can be active at a time, or both are dormant. Applying this

to each constraint function effectively means doubling the amount of relative

constraints. Since the eventual algorithm is well suited to deal with a large

number of constraints, this should not pose any computation problems.

3.1.3 Load preservation constraints

Running an optimization solely on relative stress constraints will yield only

trivial solutions wherein both f0 and fi approach 0. This is due to the fact

that the algorithm thinks discarding the loaded elements within the structure

as a valid solution (and it is right to do so since the problem is not properly

bounded). This leads to instabilities within the FEM analysis. To avoid these

trivial solutions the algorithm uses ‘load preservation’ constraints, which are

displacement constraints that apply at the loaded nodes. These constraints are

defined by

fi = ui − τi 6 0 (3.7)

38

Figure 3.2: A hypothetical development of a constraint function

and its mirror about the line g = −τ . Which one is which does

not matter.

It is inevitable that these constraints demand structural stiffness of some degree,

but, in reality, there are of course limitations on how compliant the turntable

and casing may become; e.g. a support structure made of Jell-O would not

be helpful. Even though they are chosen in a heuristic manner, displacement

constraint can be of use in finding an optimum by setting these restrictions.

3.2 Optimization

To solve for P (3.1), one of the core elements in optimization in this thesis is

the Method of Moving Asymptotes (or MMA) as first proposed by K. Svanberg

[20]. In his paper he devised a new method for dealing with a large number

of design variables and constraints as typically found in structural optimiza-

tion. Furthermore, it was thought out in such a way as to keep the method

flexible, handling various types of constraints and elements as well as element

sizes, shape variables and material orientation. MMA belongs to the class of

approximation methods, in which we also find, among others, the well known

Sequential Linear Programming (SPL) and Sequential Approximate Optimiza-

tion (SAO). As the exact functions for structural response to a change in design

variable xj are non-linear and usually too complicated to be obtained analyti-

cally, the general approach had always been to create approximation functions

on which an iterative-type optimization can be based. Commonly, this entailed

the application of the a first order Taylor-expansion, as in SPL. MMA is an

alternative method which produces asymptotic, convex approximate functions.

Here convexity indicates the property of a given set S such that each linear con-

nection between two arbitrary data points x1 and x2 represents a subset within

39

Figure 3.3: A convex (left) and non-convex (right) example of

set S

the original S, i.e. x ⊆ S. Schematic representation of a convex and non-convex

set S are shown in 3.3. With the approximate description of P (later denoted S)

a new domain can be defined (using Lagrangian duality) in which a minimum

has to be found which satisfies the KKT-conditions (also discussed later).

3.2.1 MMA approximation

First, let us go into detail concerning the convex asymptotic approximations.

Since the optimization is an iterative (evolutionary) process, we will have k

denote the current iteration number, such that k ∈ N. The approximation

function (which shall be symbolically distinguished from the exact function as

f) is then defined as

f(k)i (x) = r

(k)i +

n∑j=1

(p

(k)ij

U(k)j − xj

+q

(k)ij

xj − L(k)j

)(3.8)

This means that for every objective or constraint function f we create a domain

in Rn. Each independent design variable within the set x has its own unique

asymptote, the characteristics of which are defined by

p(k)ij =

(U

(k)j − x(k)

j

)2

∂f/∂xj if ∂f/∂xj > 0

0 if ∂f/∂xj 6 0

q(k)ij =

0 if ∂f/∂xj > 0

−(x

(k)j − L

(k)j

)2

∂f/∂xj if ∂f/∂xj < 0

in which U and L represent the upper and lower asymptote boundary respec-

tively. The partial derivatives are evaluated at x(k) and therefore simply repre-

sent a either positive or negative scalar value. The sign of the partial derivative

determines which of the two asymptotes is activated, the lower or upper. The

term ri is a variable, independent of any partial derivative, which controls the

40

vertical shift in the approximation in order to line it up with the exact function.

Therefore

r(k)i = fi

(x(k)

)−

n∑j=1

(p

(k)ij

U(k)j − x(k)

j

+q

(k)ij

x(k)j − L

(k)j

)

One can now easily verify that

f(k)i

(x(k)

)= fi

(x(k)

)∂

∂xjf

(k)i

(x(k)

)=

∂

∂xjfi(x(k)

)∀k ∈ N

which means that the original and approximation function, as well as their

derivatives, are equal when evaluated at the iteration point x(k). A good ob-

servation would be that the only remaining control regarding the shape of fi

is the positioning of the asymptotes using U and L. Svanberg [20] proposes

a somewhat heuristic approach to determine the boundaries, but, in any case,

it is clear that the iteration point should lay in between the boundaries, i.e.

L(k)j < x

(k)j < U

(k)j . By expanding or contracting the boundaries of each design

variable separately, MMA has dynamic control of the optimization; it is able to

slow down or speed up the alteration of each xj . When ∆xj/∆k is consistently

positive or negative its asymptotes are expanded, allowing the variable to in-

crease or decrease more rapidly. When elements have a clear purpose (or lack

of purpose) this is an advantage, since it will take less iterations to converge.

When elements are in a structural zone where their contribution is less obvious,

∆xj/∆k might alternate; an element may sway back and forth in density. Its

boundaries are then contracted in order to limit its behavior. At a later stage

in optimization the element’s purpose may become clear and as it does, the

increase or decrease in density will become consistent and the boundaries are

readily expanded. The exact conditions can be read in [20] and the particular

adaptions made in this thesis are discussed in 5.2.6.

With these approximated functions we can define an optimization subproblem

S based on P discussed earlier, i.e.

S(k) :

min f

(k)0 (x)

f(k)i (x)− τi 6 0 i = 1, . . . ,m

x(k)j ∈ χj j = 1, . . . , n

(3.9)

Solving S(k) with the approximation functions obtained using x(k) yields a new

set of design variable values which shall be used as the next iteration point

x(k+1). Thus, we sequentially solve a series of approximated subproblems, each

of which will hopefully bring us closer to the solution to main problem P defined

41

in (3.1) until some convergence criteria is met. Note that within S, the solid/void

demand has disappeared. This is due to the fact that intermediate solutions may

vary in density to whatever extend the algorithm pleases.

3.2.2 Lagrangian duality

S is solvable using the Lagrangian duality principle. From hereon, f simply de-

notes the approximated function (earlier denoted by f). We make a distinction

between the ‘primal’ variables and ‘dual’ variables, which consist of the de-

sign variables x and the so called ‘Lagrangian multipliers’ λ respectively. Both

primal and dual variables are used to incorporate the constraint function into

the objective function (a nested function if you will) to form the Lagrangian

L : Rn × Rm → R

L (x,λ) = f0 (x) +

m∑i=1

λifi (x) = f0 (x) + λT f(x) (3.10)

The dual variables λ will prove a useful tool for solving these types of opti-

mization problems. Because L is the sum of the objective function and weighed

summation of constraint functions, for any particular set λ, L is a hyper surface

with n dimensions. Since all functions within L are convex, this hyper surface

has to be convex as well when we demand λi ≥ 0,∀i. 1 Thus, for any par-

ticular λ, the minimum W = minx

L will have a certain value. As ‖λ‖ → ∞,

W = minx

L→ −∞, and, more importantly

lim‖λ‖→0

W (λ) = minx

(lim‖λ‖→0

L (x,λ)

)= min

xf0 (x) : fi 6 τi (3.11)

which means that as the norm of λ approaches 0, the minimum of the La-

grangian converges to a feasible optimum within the objective function. This

characteristic is formally defined by the ‘Lagrange dual function’ (or simply

‘dual function’) defined as

g (λ) = infx⊂χ

L (x,λ) = infx⊂χ

(f0 (x) +

m∑i=1

λifi (x)

)(3.12)

where g is the infimum, or ‘greatest lower bound’, of the Lagrangian L. There-

fore, if we find the solution to g, we find the solution to the original problem

S. The KKT-conditions set the requirements that a solutions needs to satisfy

1Subproblem S satisfies Slater’s condition for ‘strong’ duality. Slater’s condition is a reg-

ulatory condition (or constraint qualification) for strong duality within the KKT-conditions.

Only when the problem can be characterized as having strong duality, the difference between

the primal and the dual solution, better known as the ‘duality gap’, is equal to 0. It will not

be further discussed in this thesis.

42

in order for it to be an minimum. For inequality constraints (such as we have

defined in P) one needs to satisfy the following criteria

∂f0

∂x∂s + λT

∂f

∂x∂s = 0 , f = 0 (3.13)

in which ∂s represents a vector in any feasible direction.

(a) Objective (solid) and constraint func-

tion (dashed).

(b) Lagrangian L as function of λ (dot-

ted) and set of corresponding optima (thick

solid) denoted W (λ)

Figure 3.4: Visualization of using the Lagrangian duality prin-

ciple considering a simple 2 dimensional optimization problem

with 1 objective and 1 constraint function.

A simple duality optimization example is shown in figure 3.4 using one objective

and one constraint function within a 2 dimensional domain. Figure 3.4b shows

how the set of optima corresponding to a sequential set λ converges to the

minimum of the original objective function while remaining within the feasible

domain. In this particular example the optimum lies in between the upper and

lower bound defined by the constraint function, i.e. the constraint is not active.

Figure 3.5 shows the obvious effect of adapting the constraint function in such

a way it does becomes active. The optimum now converges to the boundary

imposed by the constraint function. Incorporating the MMA approximation

defined in (3.8) yields

L (x,λ) = r0 − λTb +

n∑j=1

(p0,j + λTpjUj − xj

+q0,j + λTqjxj − Lj

)(3.14)

in which r0 − λTb is the dual variable counterpart of ri. The set of optima as

a function of the dual variables λ is defined as W , such that for a particular

design variable xj we obtain

Wj (λ) = minxj

Lj (xj ,λ) (3.15)

43

Figure 3.5: Visualization of the same problem as in figure 3.4,

only with an active constraint. The infimum coincides with the

objective function evaluated at the lower bound of the feasible

domain, as is the case when considering strong duality.

As we determined earlier, the dual objective function Wj (λ) is a concave func-

tion, and as such finding the global optimum means finding the unique solution

to the equation

dLjdxj

= 0

Doing so allows us to define λ as a function of xj , hence Wj becomes an implicit

function of xj instead of the explicit formulation in (3.15). To finally find a

solution to the dual function g (3.12) (and consequently the problem S) we have

to solve for subproblem W, defined for iteration k as

W(k) : maxW (λ) (3.16)

This is relatively easy function that can be solved using an arbitrary search

algorithm, such as a conjugate gradient method, to find a solution that abides

the KKT-conditions (3.13). The algorithm in this thesis uses the primal-dual

Newton method, which is basically the well-known first order Newton method

adapted for primal-dual problems. This method is not discussed in this thesis,

the reader is referred to [16].

3.3 Topological derivatives

Since the method of moving asymptotes is a first order approximation tech-

nique, we will need to provide subproblem S with first derivative information. In

44

optimization problems such as these they represent the effect a change in topol-

ogy has on the objective and constraint functions, hence the name ‘topological

derivatives’ , also frequently referred to as elemental ‘sensitivities’. Whether

or not these derivatives are easily obtained depends on the complexity of the

elements and the constraints involved. Not in the sense of expanding the set of

degrees freedom within each element (this only increases computation time) but

in the sense of using more extraordinary, exotic elements like contact elements,

more on which later. In reality, the structure’s response is described by a rather

large set of differential equations which one would not want to derive analyti-

cally. Fortunately, only the first derivatives of various responses to structural

change need to be obtained.

3.3.1 Objective derivative

The plane stress objective function was defined by (3.2), a relatively easy func-

tion. Determining the derivative is also rather easy, i.e.

∂f0

∂xj= 1 ∀j (3.17)

The axisymmetrical variant, defined by (3.4), becomes

∂f0

∂xj= Rj ∀j (3.18)

Rj was defined earlier in (3.3). Note that ∂xj means a infinitesimal increase in

density, therefore the objective function would increase as well. Of course, we

set out to achieve the opposite: decreasing densities and decreasing objective

values. This might be a little counter-intuitive and the reader will be reminded

a few times throughout this thesis.

3.3.2 Suppressing intermediate densities using SIMP

Up to this point there is no mathematical tendency built into the algorithm for

creating a homogeneous structure. As far as the optimization is concerned, an

optimum solution x(κ) might well consist of various intermediate values. This, of

course, is completely impracticable from the engineers’ perspective. We might

in some circumstances allow structures involving different material types or

laminates, but these are considered outside the scope of this thesis. Gradually

decreasing and increasing densities throughout the structure are simply not

feasible. Therefore, the demand imposed on the optimization is

x(κ)j = χ

−j

∨ x(κ)j = χj (3.19)

45

wherein κ is the amount of iterations needed to satisfy the convergence criteria.

This was already included in P (3.1). The demand in (3.19) can be achieved by

imposing so called ‘Solid Isentropic Material with Penalization’ method (SIMP)

on the moduli of elasticity corresponding to each independent design variable,

such that

Ej = Egxpj p ∈ [1,∞) (3.20)

wherein Eg is the initial modulus for all design variables and p is a penalization

factor which can be freely chosen to control the severeness of the penalty ele-

ments receive for not being completely solid (usually p = 3 produces satisfactory

results). Thus, by using SIMP, the density does not influence the stiffness of an

element j in a direct manner, but by affecting the material property Ej . Before

each iteration the elasticity vector E has to be updated within the FEM model,

which shall be further discussed in subsection 5.2.7. If the stiffness matrix of

an arbitrary element j is given by kj = Ejdj , then SIMP has the following

influence on the global stiffness matrix:

K = Eg∑

xpjdj (3.21)

This relation is needed in subsection 3.3.3.

3.3.3 Constraint derivatives

S (3.9) deals with both stress-based and displacement-based (load preservation)

constraints as defined earlier in (3.6) and (3.7). Since stress ultimately de-

pendents on displacement, obtaining the topological displacement derivatives

should enables us to calculate both. How this is done exactly is described in

detail in chapter 5; for now, the focus is solely on the displacement derivative.

We will mainly be concerned with the so-called ‘adjoint method’ [9]. Our main

objective is to calculate the derivative for a particular degree of freedom, re-

ferred to as uρ. Eventually, ρ will be a set of DOFs, but for simplicity let’s

assume it consists of just one DOF. This DOF can also be written as

uρ = fTρ u (3.22)

in which u is the displacement vector and fρ is a dimensionless unit vector with

the ρ-th component unity and all other component equal to 0. At each iteration

we demand a static, determined solution, defined by

K− fu = 0 (3.23)

46

Combining both equation (3.22) and (3.23) we define the adjoint equation as

uρ = fTρ u + λ (K− fu) (3.24)

in which λ is a vector. The partial derivative with respect to xj of the adjoint

equation becomes

∂uρ∂xj

=∂ fTρ∂xj

u + fTρ∂u

∂xj+∂λ

∂xj(f −Ku) + λ

∂f

∂xj+ λ

(∂K

∂xju + K

∂u

∂xj

)When we assume fρ, f and λ are independent of x, then after eliminating some

terms we are left with

∂uρ∂xj

=(fTρ − λK

) ∂u

∂xj+ λ

∂K

∂xju

In this we recognize another equilibrium condition if we define λ =^uT

ρ , which

represents the displacement caused by a unit load. By defining λ in this way,

the derivative is reduced to

∂uρ∂xj

=^uT

ρ

∂K

∂xju

Using (3.21) the displacement derivative can now be formulated as

∂uρ∂xj

= pxp−1j

^uT

ρ kju (3.25)

It is important to note that the notation^u is deliberately chosen to make a

clear distinction regarding the conventional displacement u. This can be best

explained performing the following dimension analysis

^uT

ρ

[m

N

]= K−1

[m

N

]· fTρ [−]

∂uρ∂xj

[m

−

]= pxp−1

j [−] · ^uT

ρ

[m

N

]· kj

[N

m

]· u [m]

^uρ therefore represents a virtual displacement in DOF ρ per unit load in the

DOF of the element being evaluated. Throughout this thesis a clear distinction

is made between virtual and physical displacements, wherein virtual are merely

displacements that allow us to calculate topological derivatives and physical

displacement are the actual displacements found given a certain loadcase.

47

3.3.4 Analysis of penalization

It is perhaps valuable to evaluate the effect of penalization on the derivatives

and see if they behave as expected. The range evaluated is p ∈ [1,∞), wherein

p = 1 would mean the solid/void constraint is discarded from P and solutions

with intermediate densities are condoned. Assume the displacement uρ has a

certain value for xj = 1 and xj → 0 and some undefined intermediate values in

between. Take the extreme case where p → ∞, then for any particular xj the

derivatives would yield

limp→∞

∂uρ∂xj

∣∣∣∣x=1

= p · ^uT

ρ kju =∞ ∀j

This makes sense; any decrease in density would be infinitely penalized the

next iteration, resulting in elimination of element j. Its gradient then becomes

infinite, as the actual change in density ∆x approaches ∂x will result in the

expected ∆uρ between the solid and void state. A variable xj → 1 applied in

the same situation yields

limp→∞

(limxj→1

∂uρ∂xj

)=

^uT

ρ kju · limp→∞

(limxj→1

pxp−1j

)= 0

This is also logical since all density values within that range will be reduced to

0, meaning the element effectively no longer ‘exists’. The structure does not

respond to elements that do not exist, hence the derivative return is equal to

0; it does not matter what changes you make to that element, due to infinite

penalization its effect on structural response is infinitely small.

Besides forcing a homogeneous structure, a high factor p makes the algorithm

less conservative and speeds up the optimization, but it could also easily cause

instabilities. A penalization factor equal to 3 already creates enough tendency

to converge to a solid/void solution, so that sense we do not have to be con-

cerned extending the upper limit such that χj = 1. Theoretically, the lower

limit should be χ−j

= ε in which ε is a small number in order to prevent compu-

tational problems caused by 0. In practice, however, also any arbitrarily small ε

will cause numerical problems better known as ‘ill-conditioning’. More on this

problem in subsection 5.2.3. If we define the stiffness reduction Pj such that

Pj(p, x) = pxp−1j for any particular j, we can visualize the effect of p for x ∈ χ,

which is done in figure 3.6.

3.3.5 Derivatives for arbitrary orientation (DAO)

Ideally, we want to give the algorithm as much freedom as is practically possible

to steer the topology towards more stiffness or more compliance. Imagine the

48

Figure 3.6: The effect of penalization factor p on the stiffness re-

duction factor P (p, x) for an arbitrary element. The blue surface

represents P (p, x) = 1, and so the intersection of both surfaces

is the curve x (p) =1

p−1√p

which the density where penalization

has no influence.

optimum would demand a rather compliant structure. Increasing compliance

could cause the rollers to rotate in space, i.e. their local coordinate systems

are no longer parallel to the global coordinate system. We do not expect large

rotations, but considering the enormous loads in play a slight rotation can al-

ready have a significant effect on the distributions. We can easily transform the

displacement of the bearing nodes in the global system to their respective local

systems, but including this transformation within the displacement derivatives

needs some elaboration.

First, a quick recap on how to transform global displacements to the local dis-

placements of a link element. In the local coordinate system a link element has

only 2 DOFs, that of each node in the direction tan (∆u/∆v) (which is the ori-

entation of the local coordinate system). The values of these local displacements

can be obtained via transformation of the global displacements, i.e.

(u1′

u2′

)=

(sin (β) cos (β) 0 0

0 0 sin (β) cos (β)

)u1

v1

u2

v2

(3.26)

or

u′ = Tu (3.27)

49

Figure 3.7: A schematic representations of definitions and con-

ventions concerning the arbitrary orientation of a link element

(shown in red). The nodes are denoted 1 and 2.

This is an easy geometrical relation, visualized in figure 3.7. The stress in link

element i would therefore become

σi = Eε = EBui′ = EBTui

Since both T and u are a function of xj its derivative becomes

∂σ

∂xj= EB

∂ (Tu)

∂xj(3.28)

When we abbreviate c = cos(β) and s = sin(β) then the local displacement

derivative can be written as

∂ (Tu)

∂xj=

∂u1s

∂xj+∂v1c

∂xj∂u2s

∂xj+∂v2c

∂xj

(3.29)

Regarding the calculations, there is only a distinction between the horizontal

and vertical displacement derivatives since they are multiplied with a factor s

and c respectively. Only the calculations for the vertical displacement derivative

are shown, the horizontal counterpart is omitted. Using the product rule we can

further expand each individual term. The vertical terms become

∂vc

∂xj= c

∂v

∂xj+ v

∂c

∂xj

50

The term ∂v/∂xj we already obtained in (3.25). The second term can be ex-

panded even further

∂c

∂xj=dc

dh· ∂h∂xj

with h = ∆u/∆v and c = cos(arctan(h)). This term will account for the change

of angle due to a unit load. But, since this effect is small and is multiplied with

the displacement, which is an even smaller number, this term is insignificant

compared to the regular displacement derivative (i.e. c · ∂v/∂xj v · ∂c/∂xj).It is therefore neglected.

51

Chapter 4

Test model verification

To ensure the proper functioning of the algorithm, various test models were

programmed to monitor different effects. This chapter shall evaluate a few of

them. They start simple and gain more complexity as we add more constraints

en geometrical challenges, in preparation of the eventual FPSO main bearing

model. The details of each model are not discussed, merely the stability and

convergence are what actually matters at this point. The actual application of

the mathematical model is shown in detail in chapter 5. The following conditions

are modeled and tested for response and stability:

1. Derivatives for arbitrary orientation (DAO) of link elements, as discussed

in subsection 3.3.5.

2. Relative constraints, as formulated in subsection 3.1.2.

3. Coupled design spaces, as the eventual main bearing model will have to

deal with separate design spaces connected only by the main bearing.

The topology plots are shown in a gray-scale fashion, wherein black means full

density and white insignificant density. These two states should be the only

ones present at k = κ, as required in P (3.1) (or at least as close as possible).

In between k = 1 and k = κ we find the intermediate iterations which can

-and always will- contain design variables in a fuzzy (grey) state, making the

structure non-homogeneous. Regions which hold a substantial amount of these

fuzzy elements can be thought of as regions where the optimizer is still looking

for possible solutions. Fully white regions, where elements obviously contribute

little to none to a certain constraint, will most probably remain white unless

some form of perturbation is present within the system.

52

4.1 Arbitrary orientation with stress and dis-

placement constraint

The purpose of this test is to verify the optimizer obtains valid topological stress

derivative for a link which is allowed to rotate (mildly) during optimization.

Derivative information is based on subsection 3.3.5 (Derivatives for arbitrary

orientation). The objective is to reduce mass while not exceeding a certain stress

threshold in the link connecting the center and the top left node. Furthermore, a

horizontal displacement constraint is added at the top left corner. The structure

is subjected to a conservative horizontal force as shown in figure 4.1. Note that

the model represents a plane stress situation, not a axisymmetrical one.

Figure 4.1: Test for validating the optimization using derivatives

for arbitrary orientation. In this model a horizontal displace-

ment constrain is added at the top left corner. The diagonal

link is subjected to a stress constraint.

The results of this test are shown in figure 4.2 and 4.3. Since the link is consid-

ered weightless, the optimizer should use it to its full extend in order to further

the reduce mass of the surrounding structure. Indeed it does just that, as shown

in figure 4.2. However, it needs to maintain a link support structure to ensure

it does not exceed its stress limit. There are some checkerboarding patterns

involved -thereby introducing some artificial stiffness- but not to a worrying

degree. Checkerboarding itself is explained in section 8.6.

If we remove the stress constraint from the link, the optimizer simply iterates

towards a minimal triangular structure in which the link forms a connection

between the top left corner and a structure spanning the distance between the

bottom right corner and center, as would be expected.

4.2 Displacement and relative stress constraints

In order to evaluate the ability of the algorithm to deal with relative stress con-

straints (as formulated in (3.6)), the block model from the previous section is

53

Figure 4.2: Optimization of an arbitrary orientated link element

within a structure.

used once more. The design space is now subjected to a compressing vertical

force and held in place by two sets of links, 1 set on the left and 1 on the right

(shown in figure 4.4) The difference between the stress in a set may not exceed

a threshold value τ and an addition vertical displacement constraint is added at

the force location. Since the block is not uniformly loaded, the force will have

to be distributed on both side via shear stress. This means the 2 inner links

will start off with a higher initial stress than their neighbors, but τ is chosen

in such a way that x(1) ⊂ Γ. In essence the algorithm is forced toward a more

complex geometry than would be needed in order to satisfy the displacement

constraint alone, in which case is could simply form a pyramid structure using

the 2 inner links. The relative stress constraint prevents the outer links from

being decoupled from the structure entirely.

The results of this test are shown in figure 4.5. The outer links are kept at the

minimum amount of stress while still maintaining a feasible solution. The way

in which the optimizer maintains pressure on the outer links is rather interest-

ing, i.e. it recognizes the importance of the angle of the outer legs with respect

to the outer links. Only the vertical component of the force applied to the link

will result in an actual contribution to the relative stress. When the angle is

decreased, that contribution diminishes. However, the angle needed in this case

cannot be obtained by a direct connection, hence the slight bend supported by

an perpendicular member.

Figure 4.9a and b show the constraint values for both displacement and relative

stress. In reality there are 4 relative stress constraints, but only 1 is shown since

the others are either exactly the same -due to structural and load symmetry-

54

(a) Displacement constraint values.

(b) Link stress constraint values.

(c) SIMP values.

Figure 4.3: Results of the DAO test model. All data show con-

vergence: the displacement (a), stress (b) and the SIMP values

(c), which means the optimum is bounded by both the displace-

ment and stress constraint. SIMP convergence strengthens the

confidence in the penalization method.

or a mirrored function. This was discussed earlier in subsection 3.1.2 and an

example is shown in figure 3.6. It seems clear that the relative stress constraint

is active, whereas the displacement constraint is not.

55

Figure 4.4: Test for validating the optimization using relative

stress constraints.

Figure 4.5: Optimization of an block model with relative con-

straints. The topology is a direct effect of the imposed con-

straints.

56

(a) Displacement constraint values.

(b) Relative stress constraint values.

Figure 4.6: Results from the relative stress constraint model as

depicted in figure 4.1. Both constraint functions show intent to

remain feasible. One might notice the relative stress violates the

tolerance briefly, but corrects itself right away. The algorithm

is sufficiently resilient to deal with these violations.

57

Figure 4.7: A simple model to test the response of the optimizer

to coupled structures and possible ill-conditioning of the global

stiffness matrix. The relative constraints apply to the set of ver-

tical and the set of horizontal links. The dashed vertical line on

the left end of the model represents the axis of symmetry. Note

that this model is only 2D symmetrical, not axisymmetrical.

4.3 Coupled design spaces with relative constraints

Coupling structures is another important step within the optimization since we

are eventually planning on connecting both hull and turntable design spaces

using a simplified main bearing model. An anticipated problem -regarding FEA

and the optimization which is dependent upon it- is the possibility of an ill-

conditioned system of equations, which tends to arise when e.g. coupling a rigid

to a compliant structure. This phenomenon might well turn out to be pernicious

to obtaining the structural response and its derivatives. The problem can really

be reduced to the eigenvalue characteristics of global stiffness matrix K.

As mentioned earlier, not much is know about the effects of relative constraints

on topology optimization. If K(1) is ill-conditioned, the optimizer might only

exacerbate this initial state (more on ill-conditioning can be found in subsection

5.2.3). Or perhaps the algorithm might -after a few or so iterations- stumble

upon a structure that translates mathematically to ill-conditioning, causing the

optimization to become unstable. Although this test does not completely guar-

antee safeguarding from ill-conditioning, it might reinforce confidence regarding

the application in larger models. In order to test algorithm’s response to such

models, the model in figure 4.7 on page 58 is optimized using relative constraints

and a load preservation (displacement) constraint.

The results are not as straight forward as in the previous models. First of all, the

58

Figure 4.8: Objective and topology results from the model as

depicted in figure 4.7.

lower horizontal links are quickly rendered useless and therefore disconnected

from the structure as a whole. Disconnection inherently satisfies the relative

stress constraint since both stresses approach 0. The remaining relative stress

constraint does not seem to be active, but nevertheless it is satisfied. Instead the

displacement becomes and remains active from as early as k = 4. A possibility

might be the optimization is not bounded by the relative stress constraints.

In such a case, the bounding constraint would become the load preservation

constraint itself. As long as the relative constraints are met as well, the solution

remains within Γ.

59

(a) Displacement constraint values.

(b) Relative stress constraint values.

(c) SIMP values.

Figure 4.9: Results of the coupled design space model as de-

picted in figure 4.7.60

Chapter 5

Application to main

bearing support structure

With the mathematical framework and definitions in place, it is time to put it

to actual use. This chapter will discuss the precise details of how the support

structure, the main bearing cross section and the individual rollers and various

loadcases are modeled. It will do so in the following order:

1. FE modeling; Describes the general modeling of turntable, turret casing

and main bearing, but also the simplified modeling of the main bearing

rollers. Here the different domains within the FE-model are defined which

should help the reader to distinguish different sets of elements and their

relation with the optimization process.

2. General programming for all loadcases; Describes the application of

chapter 3 to the model presented in subsection 5.1.2. Some of these adap-

tations are of a programming nature, they describe how Matlab prepares

the input data for the optimizer. Others are mathematical adaptations to

cope with problems unique to the turntable/casing model. Furthermore,

an overview of the Ansys/Matlab-coupling is given. These adaptations are

general in the sense that they apply to both axi -and non-axisymmertical

loadcases. They are all the optimizer needs to handle axisymmetric load-

cases.

3. Additional programming for bilateral loadcases; Describes the addi-

tional adaptations unique to bilateral loadcases. These types of loadcases

are most interesting since they can describe the limit states to which the

turret and turntable are subjected, but they are also significantly more

complex.

61

To avoid confusion, keep in mind that structural axisymmetry is always main-

tained, but loadcases may be axisymmetric as well as bilateral. In the previous

chapter only plane stress models were evaluated. Of course, in reality, ax-

isymmetrical structures could never be accurately modeled with plane stress

assumptions; for one the stiffness would be severely underestimated and -even

worse- the horizontal translation of an element as a whole would not induce any

resistance. Also, one has to apply unrealistic boundary conditions in order to

avoid rigid body motions. On the brighter side, the only aspects that really

change are the elemental stiffness matrices and the switch from a Cartesian to

cylindrical coordinate system.

5.1 Finite element modeling

5.1.1 Element description

Three types of elements are used throughout this thesis (in both the turret

model and the test models discussed earlier in chapter 4). These elements are:

1. link180 for main bearing roller modeling

2. plane183 for axisymmetric loadcases

3. plane25 for bilateral loadcases

link180 is used in its most simple form, i.e. linear stress development and

compression as well as tension capability. Ideally, links should only be capable

of compression, not tension, since bearing rollers can only convey loads while

compressed. In order for link180 to be compression-only, Ansys needs to per-

form non-linear static analyses; these are analyses based on a Newton-Raphson

iterative solving method. However, non-linear analyses are incompatible with

plane25 and the axisymmetric option of plane183. This incompatibility is

avoided by issuing pre-tension in both the axial and radial rollers, making sure

the rollers always remain compressed to a certain extend. This, in itself, in-

cludes an assumption which shall be further discussed in subsection 5.1.3.

plane183 is an 8 node element with each node having 2 DOFs; radial and axial

displacement. The introduction of mid-side nodes makes this element better

at describing pure bending situations and avoid shear locking during iterations.

The increased number of DOFs does increase the computation time regarding

FEA and the calculation of topological derivatives, but does not affect the opti-

mization since the amount of elements within the design domain stays the same.

62

plane25 also has 8 nodes but with each having 3 DOFs; radial, axial and

tangial displacement. This ‘harmonic’ element is only used when applying bi-

lateral (non-axisymmetric) loadcases. It is, in essence, and expanded version

of plane183. The radial and axial displacement relations are similar, but

plane83 also has tangial displacements which only interact with other tan-

gial displacements within the element. These elements are further discussed in

section 5.3 in which bilateral loadcases are discussed.

5.1.2 Modeling and discretization

The Rosebank FPSO (which is being designed at the time of writing this thesis)

is used as a case study, and as such its dimensions are roughly used throughout

the modeling. The program only needs the global measurements, i.e. the sizes

of the bearing and turntable; the internal geometry is not of importance since

that will be the result of topology optimization. The size of the casing design

space (the design space that is actually part of the hull) is chosen freely, but

of sufficient size to make sure the optimization can utilize it in order to satisfy

structural constraints. It could well prove to be as important as the topology

within the turntable itself. Figure 5.2 and table 5.1 provide all dimension as

used in Ansys. All distances are dividable by 70, which is the element size that

provides sufficient resolution without overreaching the available computation

power. These dimensions are readily changeable, meaning the user is free to

chose different shapes as long as the chosen dimensions remain a multiplicity of

70.Table 5.1: Value of spatial dimensions as shown in figure 5.1.

Symbol Dimension Value [mm]

a Core radius 4200

b Outer radius 12460

c Turntable overhang 1120

d Turntable height 4900

e Turntable depth 700

f Inner radius 12460-700

g Casing width 2800

h Casing height 3500

The various areas are divided in different domains. In the previous chapters

we already defined (or hinted at) some domain specifications. The precise def-

initions of all specified domains is listed for convenience in table 5.2 and their

relation to each other can be found in figure 5.2.

63

Figure 5.1: The default model dimensions of Θ as used in FEA.

All dimensions are adaptable to a certain extend. The model

resembles the one used in figure 4.7 and the mirrored image of

figure 2.2, except that this model is axisymmetric about the axis

shown on the left. This means that this model has a cylindrical

gap in the center.

Table 5.2: Domain definitions

Symbol Domain Type

Θ Entire FEM domain. plane,link

Ω The set design variables x. plane

Φ All elements within main

bearing.

plane,link

Π Elements modeling main

bearing cross section area.

plane

B Link elements affiliated

with roller modeling.

link

Υ Link elements affiliated

with roller relative con-

straint functions.

link

The domain dimensions determine the eventual mesh resolution of Ω. The finer

Ω is discretized, the more detailed the topology can be described, but the more

demanding the algorithm will become. The element size of 70 mm is chosen in

an empirical fashion, i.e. based on experience obtained for chapter 4. Ansys

64

Figure 5.2: The domains represented schematically in relation

to each other. The size of the set is approximately indicated by

the size of its bubble. Throughout the thesis, a domain might

either represent the DOF, node or element numbers of which it

consists. Domain definitions are given by table 5.2. It can be

verified that, e.g. Υ ⊂ Φ ⊂ Θ.

will try to mesh with element size 70 where possible, but it will deviate when

the geometry is not dividable by 70. Therefore, the dimensions of the Rosebank

will be slightly adapted so as to ensure the mesh algorithm creates an uniform

plane mesh. This is only a necessity for elements within Ω. plane-elements

within Π can be meshed in any shape or form, as long as they abide the shape

criteria. They will be discussed in 5.1.3.

5.1.3 Main bearing modeling

The main bearing is the bridge between two design spaces; a substantial part

of the loads have to be conveyed by it. The manner in which these loads

are conveyed will determine the distribution on the bearings and hence their

performance. Figure 5.3 a through c shows different perspectives on the main

bearing model. In the following paragraphs the different aspects of main bearing

modeling will be explained.

Bearing rings

The model has two bearing rings: an inner ring connected to the turntable and

an outer ring connected to the casing. Both are unaffected by the optimiza-

tion (as Π 6⊂ Ω), they only serve as a container for the rollers and bridge the

discretization difference between the design spaces and the rollers (hence the

irregular mesh). In order to simplify the programming as much as possible,

65

clearances between the rings were neglected. In reality, gaps ranging from 0.15

to 0.75 mm are maintained between the rings to account for the space needed

for lubrication and deformation. The model does not include that space and

one might therefore misinterpret the mesh in figure 5.3c as if the two rings were

connected. However, the inner and outer elements share no nodes and contact

between the inner and outer ring is not modeled; they are only allowed to con-

vey forces through the rollers. In certain loadcases this could result in the rings

seemingly overlapping, this has no implications on the accuracy of the model

and can therefore be disregarded.

Rollers

There are various ways in which a roller bearing could be modeled. One of the

most accurate ways would be to introduce a special branch of elements, called

‘contact elements’, also commonly used by the industry. However, given the

non-linear behavior of these elements and the fact that their topological stress

derivatives will probably be rather difficult to obtain, these are outside of the

scope of this thesis. In light of the goal to just maintain uniform stress distri-

bution, the choice is made to introduce a simpler bearing model using only a

set of link elements. A major drawback of this method is the incompatibility of

non-linear analyses and the plane-axisymmetry option in Ansys. This problem

is solved using bearing pre-tension (see 5.1.3).

As shown in figure 5.3b, each bearing is modeled with ‘constraint’ links and

‘support’ links. The support links purpose is solely to provide a better force

transfer from the inner to the outer main bearing ring (a minimum of 5 links

in total is required). The constraint links are the elements which are actu-

ally monitored by the optimizer in order to compare stresses and establish the

relative stress function values f∆σ. They are allowed to rotate freely, thereby

ensuring the support structure is allowed to rotate. Of course, large rotations

are not expected (and actually constraint in some sense by the displacement

constraints) but since the main bearing has to convey such large forces it could

be that even small rotations might cause significant derivative errors if they are

based on links with fixed orientation. This effect is account for in ‘Derivatives

for arbitrary orientation’, subsection 3.3.5.

The properties of the rollers can be chosen freely by either adapting their

Young’s modulus or increase their ‘area’ real constant in Ansys. This area

has no physical meaning, but it can be set such that the links combined provide

a good approximation of the roller stiffness. Both types of links together make

up set B and Υ (see figure 5.2), meaning they are not part of the design domain

66

and their stiffness will remain unchanged during optimization. Although the

eventual stiffness within a roller will vary linearly to ensure initial feasibility,

i.e. x(1) ⊂ Γ, more on which later (see subsection 5.2.5).

Bearing pre-tension

In reality, the main bearing is hydraulically pre-tensionsed in the axial direc-

tion and this has a significant effect on the roller load distributions. Rothe

Erde applies this pre-tension in order to keep the upper and the lower raceway

rollers under constant compression. From a FEM modeling point of view, this

is perfect since linear static analyses will still be accurate even when describ-

ing intrinsic non-linear behavior of the rollers. However, the radial bearings

are not subjected to any pre-tension and, as a result, they may theoretically

become partially of fully unloaded. This is unacceptable since linear analyses

will cause the respective links to becomes tension loaded, thus rendering the

solution invalid; the stress in all links must remain negative (as discussed ear-

lier, see subsection 5.1.1). To circumvent this problem, radial, as well as axial

pre-tension, is included. This is an assumption of which the potential technical

implications are not studied in detail, but they do not strike one as unfeasible

when dealing with a 4-raceway bearing. It would not be possible to use this

approach with a 3-raceway bearing because in order to ‘pinch’ the rings in the

radial direction you do need two separate radial raceways. Both axial and radial

pre-tension are modeled with a single link and the inistate-command in Ansys.

5.2 General programming for all loadcases

Models describing 3 dimensional, axisymmetric structures can be reduced to 2

dimensional models when they abide certain criteria. When subjected to an

axisymmetric load, the deformation will also be axisymmetric. The stress dis-

tribution in the tangent direction (also referred to in the literature as ‘hoop’

or circumferential stress) is then assumed to be equal for all radial planes. In

any radial plane, all corresponding elements will only deform within that ra-

dial plane. Though the structure might be presented in 3D space, the problem

can be reduced to two cylindrical dimensions: radial and vertical DOFs. This

greatly decreases computation costs, therefore most FEM programs include ax-

isymmetric elements, including Ansys.

5.2.1 Topological derivatives

We can write the objective and constraint values and derivatives as they will be

determined by the optimization algorithm. Matlab is a matrix-based computa-

67

(a) Technical drawing of main bearing as produced by Rothe Erde. It shows both

rings, 4 roller bearings and the axial pre-tension bolt.

(b) A schematic cross section of the 4 raceway main bearing with left the inner ring

and right the outer ring. Each roller is modeled with 2 constraint links (solid red) and

3 support links (grey dashed), except the main (or axial) roller which is modeled with

5 support links sine it is significantly larger. The numbers show the sequence in which

the constraint links are ordered within the set Υ and in each subsequent dependent

matrix. The set B contains both the constraint and support links.

(c) The discretized main bearing as modeled in Ansys. The mesh is irregular, which

is caused by the size difference between the link spacing and design elements (the link

space is half the design element size). The irregular mesh does not complicate the

optimization since Π * Ω. Note that although the rings seem to be connected, their

elements do not share any nodes.

Figure 5.3: Various representations of the 4-raceway main bear-

ing system.

68

tion program, and as such it is preferable to perform matrix-based calculations

rather than a series of nested programming loops. Preparing the functions and

derivatives basically entails the application of various algebraic manipulations

(using various mapping matrices denoted M), but it will in any case proof use-

ful when understanding and/or checking the programming itself. During this

progress, references to chapter 3 will be made when necessary. The mapping

matrices used in the following equations are defined by

M1 =

1 0

0 1

−1 0

0 −1

, M2 =

1

0

1

0

T

, M3 =

0

1

0

1

T

, M4 =

(1 −1

−1 1

)

The first derivatives of all constraints are assembled in the Jacobean J(k) such

that all relative stress constraint functions f∆σ are found at the top and the

displacement constraints fu at the bottom, such that

J = ∇f =∂f

∂x=

(∂f∆σ∂x

∂fu∂x

)T(5.1)

From hereon, let’s denote the subset DOFs belonging to nodes of the constraint

links as uΥ ⊂ u. The set containing the horizontal and vertical distances

between these nodes then becomes

∆uΥ = uTΥ

(I(8) ⊗M1

)+ LT (5.2)

With ∆uΥ we can easily determine the set of link orientation angles β (as

defined in figure 3.7). These angles are used to determine the contribution of

the global displacements to the displacements in the local coordinate system.

This contribution factor is defined

c = sin (β)T(I(8) ⊗M2

)+ cos (β)

T(I(8) ⊗M3

)(5.3)

and is not dependent on j. gj represents the local displacement derivative values

for uB, i.e.

gj = c

(^

UT

ku

)j

(5.4)

wherein the first term accounts for the change in global displacements trans-

formed to local values. The contribution of the vertical and horizontal dis-

placements of each node have to be added in order to determine the total local

displacement of each node, which is defined by n. This is a simple summation

of the appropriate components within g. We then obtain

nT = gTj ·(I(16) ⊗ ι(2)

)(5.5)

69

Consequently, the stress in each constraint link becomes

∂σ

∂xj= pEgx

p−1j Bn (5.6)

By mapping (5.6) using M4 we attain the derivatives for the constraint functions

within the set Υ

∂f∆σ∂xj

=(I(4) ⊗M4

)· ∂σ∂xj

(5.7)

Therefore, the Jacobean as defined (5.1) becomes

J = ∇f =

∂f1∂x1

· · · · · · · · · ∂f1∂xn

− ∂f1∂x1

· · · · · · · · · − ∂f1∂xn

.... . .

...∂fm−1

∂x1· · · · · · · · · ∂fm−1

∂xn∂fm∂x1

· · · · · · · · · ∂fm∂xn

(5.8)

J is a [m× n]-sized matrix wherein n m. As mentioned earlier, the relative

stress constraint f∆σ are arranged at the top of the matrix, the load preservation

constraints fu at the bottom. The load preservation constraints have no mirrored

cousins, hence the absence of a minus sign at the bottom two (displacement)

constraint functions in (5.8). A last note on the constraint function derivatives:

It’s easy to confuse the appropriate sign of the derivatives. Note that each

derivative provides the response of the constraint function upon a ∂x increase

in density, not decrease. This might prove a little counter intuitive since the

objective is -of course- to decrease density and structural mass alike. Recall the

model shown in figure 4.4 (page 56) and its vertical displacement constraint at

the top center node

fu = −v − τ 6 0

∂fu∂xj

= − ∂v

∂xj

Now imagine a increase of a particular density in the model. As a result, the

displacement at the top will decrease. As a downward displacement is defined

by a negative sign, a decrease in displacement yield a positive displacement

derivative. The constraint function derivative will therefore have positive sign.

In short, when looking at derivative contour plots, ∂fi/∂xj 6 0 indicates regions

where removing elements would bring more ‘pressure’ onto the constraint func-

tion. ∂fi/∂xj > 0 indicates regions that will relieve pressure on the constraint.

70

5.2.2 Displacement decomposition

Testing axisymmetric structures on their response derivatives revealed that -

as mesh resolution was increased- the turntable regions with less deformation

returned faulty derivative values, visualized as noise in contour plots. This is

due to the fact that the turntable is only supported at the main bearing; upon

loading, a substantial part of the nodal displacements is due to vertical motion

of the turntable as a whole, here referred to as element translation (ET). The

elemental deformations (ED) are rather small with respect to the translations,

even more so when the element size is decreased, therefore numerical errors oc-

cur when calculating the derivatives. They are not cumulative as the derivative

information is not exchanged between iterations, but they do confuse the op-

timizer and cause instability. Regions with relatively large ED (e.g. the main

bearing where all forces are transferred to the hull) are less susceptible to these

errors. The casing itself has very limited ET since it contains the boundary

conditions. Two dimensional non-axisymmetric structures are less affected by

this type of error because they lack the increased stiffness due to increased stress

in the tangial direction. Their translation/deformation ratio is much more fa-

vorable than would be the case in axisymmetric structures.

The numerical errors are a pure result of computational precision. Matlab reads

FEA data from Ansys and stores it in double-precision floating point format.

A 32 bit operating system uses two 1 storage locations to store such a number,

allowing for a 15 digit precision for mathematical manipulations. Now, a nodal

displacement of the turntable can be split in both element translation and el-

ement deformation. In the axisymmetric case there is a significant difference

in magnitude between the two components. It is this difference that results in

numerical noise within the derivative contour since Matlabs finite precision in-

evitably causes loss of information when evaluating such numbers. E.g., consider

the following number

+0. 100000000000000︸ ︷︷ ︸Double precision

619150012︸ ︷︷ ︸Deformation

e - 08

which represents a nodal displacement of an arbitrary point in the turntable.

This number consists of the sum of a general element translation of 0.1e−8 and a

nodal contribution to element deformation. If Matlab were to store this number

in double-precision format, some deformation information would be lost since

the precision is bound to 15 digits. Thus, all mathematical procedures within

Matlab will be performed with the number

1Hence the name ‘double’.

71

+0.100000000000001e− 08

which includes a round-off error which results in a 62% overestimation of the

deformation displacement of that particular DOF. Errors of this size are capable

of creating substantial noise within the derivative contour, even rapid alterna-

tion of sign within a group of adjacent elements.

Because this effect is due to the difference in magnitude between ET and ED, it

means that it is a intrinsic property of the structure itself. It cannot be solved

by changing the unit convention. If one adapts the modulus of elasticity Eg the

magnitudes of k and u would lie closer together. However, is does not solve the

problem of the displacement having two components of different magnitude.

Fortunately, Matlab does store all information provided by Ansys, whose output

can range up to 25 digit precision. It is only upon performing calculations that

Matlab reduces the precision to a double format. If the displacements were to

be split in two separate terms of different magnitudes, one can retain numer-

ical precision in both cases. This method will be referred to as ‘displacement

decomposition’.

The numerical precision problems that arise with turntable-like structures can

be tackled using displacement decomposition, which means we describe the

derivative as a superposition of ET and ED components, such that

^uT

ρ ku =^uT

ρ,RkuR +^uT

ρ,εkuR +^uT

ρ,Rkuε +^uT

ρ,εkuε (5.9)

in which the index R (for ‘rigid’) refers to the ET component and ε refers to the

ED component. In each term, round-off errors are limited since only information

of the appropriate magnitude is used. The ET displacement itself is composed

of a horizontal and a vertical displacement vector, i.e.

uR = vR ⊗

1

0

+ wR ⊗

0

1

(5.10)

in which

vR = ι(n) min(|v|)

(1

n

∑n

sign (vn)

)α

wR = ι(n) min(|w|)

(1

n

∑n

sign (wn)

)α

72

For a plane element with mid-side nodes, n will be equal to 8. The ED displace-

ment is simply defined by

uε = u− uR (5.11)

If α is a positive odd number (i.e. α ∈ 2N − 1), then the following condition

holds

limα→∞

(1

n

∑n

sign (un)

)α=

0 for sign (u) 6= ±n1 for sign (u) = n

−1 for sign (u) = −n

Per default, α = 101 deems sufficient to reduce the effect on elements not

prone to ET. In both an axisymmetrical and plane stress structures a vertical

translation of an element does not result in any nodal forces (i.e. fR = 0),

it merely means a solid ring of material is being moved either up or down.

Therefore, these displacements do not contribute to the topological derivative

and we can neglect them without consequence. A simplified turntable/casing

structure model was constructed to pinpoint the source of this (then unknown)

numerical error. The effect of displacement decomposition can be seen in figure

5.4a and b. By retaining precision in both the ET and ED terms in (5.9) the

derivative contour shows smooth gradients in all regions, also those far removed

from boundary conditions.

(a) Without displacement decomposition. (b) With displacement decomposition.

Figure 5.4: The effect of rigid body motion correction on the

topological derivatives of a simplified turntable/hull structure

(both figures have the same color scale and range).

5.2.3 Coefficient ratios

Another computational/numerical error occurs when the stiffness matrix (which

Ansys generally refers to as ‘coefficient’ matrix since not all problems are of a

structural nature) consists of values having vastly different magnitudes, or (the

73

way that Ansys refers to the problem) the ratio between coefficients becomes

too small. This phenomenon is also known as ‘ill-conditioning’. In a sense, these

numerical errors are of the same ‘precision’-related nature as the ones described

in subsection 5.2.2, however, they do not necessarily lead to ill-conditioning

since this also depends on the specific composition of any particular matrix. A

telling example of this is given in [5] where two springs with greatly varying

stiffness are connected in-line. The manner in which forces and boundary con-

ditions are applied determines whether or not the system will be susceptible to

ill-conditioning.

As it turns out, the global stiffness matrix K(k) of the support structure model

is susceptible to ill-conditioning, depending on the chosen χ. A tell-tale sign

of this is the fact that results from the optimization shows violent perturba-

tions in some (or all) structural responses that, consequently, cause unstable

optimization. This means the algorithm still optimizes using correctly calcu-

lated derivatives, but Ansys returns faulty values based on a K that became

ill-conditioned somewhere during the iteration process (since K(1) most likely

is not ill-conditioned due to initial conditions x(1)j = χj , ∀j). Hence, one needs

to be cautious when choosing the lower bound of χ.

The choice of lower bound is a trade-off. On one hand the structural influence of

‘discarded’ elements must be insignificant (χ−→ 0), on the other, ill-conditioning

has to be avoided (χ−→ χ). Determined in an empirical fashion, χ

−= 0.1 proves

an appropriate value to ensure both. This means that, account for the effect of

penalization, the stiffness of an discarded element is a factor 10−3 of the original

stiffness.

5.2.4 Scaling

Another important adaptation to the objective and constraint functions and

derivatives is scaling. This is an adaptation regarding programming considera-

tions, and it also applies to the design variables themselves. The algorithm is

unable to deal with very large or very small values and derivatives which increase

the chance of instability (also cause by the same numerical errors discussed in

the previous subsections). To avoid this problem, we introduce a scaling vector

s and applied a element-wise product of the objective and constraint functions

as defined in S (3.9). This yields

f = (r− τ ) s 6 0

∇f = ∇r(ι(n),T ⊗ s

)6 0

: 1 6 τisi 6 100 ∀i (5.12)

wherein r is the set consisting of multiple types of response values (in this case

74

displacement, stress and relative stress).

5.2.5 Initial feasibility

The Rosebank is already designed such that the roller load distributions -in

theory- are within acceptable ranges, so FEM analysis of a detailed model should

yield a solution within Γ given the applied loads and their positions. But this is

an already completed structure. In order to give the optimizer the most flexible

set up possible, the initial design spaces are fully closed and solid, i.e. x(1) = χ.

This means that, if the same loadcase is applied, the initial constraint values

f(1)i could well prove to be non-feasible. Certain optimization algorithms do

not necessarily require a feasible start-off, the primal-dual Newton method can

only cope with these types of situations conditionally. It is therefore advisable

to adapt the structure to provide acceptable initial constraint values. These

adaptations can be made in the following ways:

1. Adaptation of the loadcase: Change the force magnitude and locations or

change the bearing pre-tensioning.

2. Change elemental properties within Φ.

3. Change elemental properties within Ω, thereby changing the initial topol-

ogy x(1).

Adaption of the loadcase is not obviously not preferable; one inevitably looses

the guarantee the resulting structure will perform satisfactory under different

loadcases. Remember that by taking a severe loadcase we do not automatically

satisfy less severe loadcases, due to the fact that there are relative constraints

involved. Changing the initial elemental properties within Ω means we are

basically already introducing a bias towards a certain structure, which actually

impedes the main advantage of topology optimization to certain extend and

therefore also not really preferable. What remains is the adaptation of the

rollers. To achieve initial feasibility, the elasticity within each roller is assumed

to be a linear function of its length. Varying elasticity in the manner might, for

example, simulate the angle or tapered geometry of an roller of each individual

raceway. However, this is an assumed condition and no further research is done

within this thesis. In future models, it could be considered converging the roller

elasticity to a uniform state during iterations, or perhaps find an approach for

changing the initial state x(1) in such a way that no definite structural bias is

introduced within the optimizer.

5.2.6 Asymptotal increase

As discussed in subsection 3.2.1, one of the main advantages of MMA is the fact

that the optimization can be sped up of slowed down by setting separate bound-

75

aries for all asymptotic approximations, hence the name ‘moving’ asymptotes.

As admitted by Svanberg [20], the factor with with these boundaries are allowed

to extend or contract are chosen in a heuristic manner. In a multi-constraint

problem with lots of conflicting interests, such as the current problem, it is

preferable to air on the safe side regarding boundary expansion. If we allow the

boundaries to expand too rapidly, especially in the first iterations, the algorithm

might not notice and skip a path towards a more optimal design. Once passed,

it is less likely, though not inconceivable, it will reroute and still converge on

this solution. Altogether, it is better to accept slightly increased computational

demands than running this risk.

5.2.7 Ansys-Matlab coupling

It would be possible to contain the entire algorithm within Matlab. This means

the FEM analyses would also have to been done internally. As the size and geo-

metrical complexity of the model grows, so does the programming in Matlab, to

the point it becomes rather strenuous. For this exact reason, the algorithm in

this thesis calls upon Ansys (a dedicated FEM program) to build models, assign

numbering and perform the needed structural analyses. To solve for W(k) each

iteration both Matlab and Ansys are used and they exchange data by writing it

to the harddisk. Matlab governs the entire program, calculates function values

and their derivatives and does the actual primal-dual optimization. The entire

algorithm is shown schematically in figure 5.5 in which it is split up in sev-

eral blocks representing the most important activities. Notice that the blocks

which are evaluated by Ansys are enclosed in the dashed rectangular section,

all others are accounted for by various (nested) Matlab scripts. Completing the

primal-dual Newton optimization, the algorithm can either restart the process

for the next iteration k+ 1 or it can reach convergence after which the solution

is presented.

Almost all steps are described at some point in this thesis, but a note has

to made concerning the central path in figure 5.5 regarding the stiffness matrix

export and conversion: In order to ensure Matlab and Ansys work with identical

stiffness matrices kj ∀j, they are exported forehand. They are exported only

once with a separate algorithm and need not be part of the optimization loop

since (3.20) adapts each kj through its modulus of elasticity. Furthermore,

elements with the same radial position will have equal k(1), thus the algorithm

will only have to export matrices for each radial position. The reason for using

Harwell-Boeing format is not of immediate importance, but will be elaborated

in the corresponding section(s) in appendix ??.

76

Figure 5.5: Schematic representation of the entire algorithm.

The core programming is done Matlab, but embedded is a func-

tion which calls upon Ansys to perform FEM analyses. This

Ansys module is shown within the red dashed rectangular sec-

tion. S refers to the subproblem defined in (3.9) and W refers

to the dual objective function defined in (3.16).

77

5.3 Additional programming for bilateral load-

cases

(This section has been omitted for legal reasons. Please contact E.vanVliet-

2@student.tudelft.nl for more information.)

78

Chapter 6

Results

Two separate loadcases are examined: an axisymmetric and a bilateral one.

As explained in chapter 5, axisymmetric loadcases use plane183 as their main

element, whereas bilateral uses the harmonic plane25. The axisymmetric load-

cases are only able to describe the gravitational loads on the turret, whereas the

bilateral loadcases can describe a wide range of different situations. For each of

the two loadcases the results shown in this thesis include (in this order)

1. Relative constraint values

2. Displacement constraint values

3. Objective function values

4. SIMP values

5. Computation time

6. Topology

all of which are displayed as a function of iteration number. The topology (or

density visualization) actually shows what parts of the structure are degraded

and subsequently removed, and which parts remain essential as the iterations

pass. The results shall be commented on in the corresponding caption of that

particular graph.

79

6.1 Axisymmetric loadcase

The gravitational loads include the turntable as well as the turret and riser/um-

bilical weights, but can be adapted as required. This loadcase only incorporates

the math discussed in section 5.2, not that of section 5.3. The results of the

loadcase shall be discussed in the caption of each graph (figures 6.2 through

6.7). The detailed set-up of the algorithm is shown in the table below.

Symbol Description Value

τ∆σ Relative stress tolerance 0.05 MN

τρ Displacement tolerance 10%

Initial boundary expansion 0.5

Explicit boundary expansion 1.1

Explicit boundary contraction 0.7

p Penalization factor 3

χ−

Lower density boundary 0.1

Turntable load 15 MN

Riser hang-off load 5 MN

Element types plane183, link180

Boundary conditions Casing edges clamped

Element size 70 mm

Figure 6.1: Axisymmetric loadcase.

80

Figure 6.2: Relative stress constraint values of the axisymmetrical load-

case. In reality there are 8 relative stress constraints (as explained in subsection

3.1.2), but only the 4 closest to the 0-boundary are relevant. The other, mir-

rored functions are left out for clarity. These results show that 3 of the 4 relative

constraint sets are active, i.e. they converge towards the feasible boundary. The

fourth, the lower axial roller, is not active but slightly swaying back and forth.

Since the objective is to minimize mass, the constraints do (theoretically) not

necessarily need to converge. However, as more elements are removed, they will

most likely be forced to. The lower axial constraint will also converge, when

given enough time. During this convergence the overall structure will not change

significantly, it is therefore decided to save the additional computation costs.

81

Figure 6.3: Displacement constraint values of axisymmetrical loadcase.

Both displacement constraint show convergence towards the feasible boundary.

This being the case, they do impose some structural stiffness on the solution,

which needed since relative stress constraints alone do not provide a sufficiently

bounded optimization problem (as discussed in chapter 4). During this test, the

displacement constraints were set at 10% of their initial displacement values, but

can always be adapted as required.

Figure 6.4: Objective function values of axisymmetrical loadcase. The

objective shows an overall decrease with steady speed, converging on a lower

bound that represents the minimal mass needed to support the applied loads

and satisfy all imposed constraints at the same time. The detailed view (right)

does show the algorithm’s decision to slightly back track and add mass, rather

than removing it (at iteration number 16 and 17). At that time, none of the

constraints are that close to the feasible limit, so it is hard to point out an clear

reason why that decision is made.

82

Figure 6.5: SIMP values of axisymmetrical loadcase. The SIMP values

also show a nice convergence towards 0. Since the initial design space is fully

enclosed, none of the variables are in an intermediate stage. In the beginning

iterations a peak value in intermediates can be observed, after which the amount

steadily declines as penalization filters them out. This behavior is logical since

the optimizer is stuck to the initial boundary expansion, it cannot simply remove

rudimentary elements in one iteration.

Figure 6.6: Optimization computation time. The time per iteration starts

off at about 45 minutes, but as the constraint functions reach the limits of

the feasible domain and start to show more profound contradictory demands,

the algorithm takes longer to find a minimum in W. A salient dip is found

between iteration 32 and 37. This is due to the fact that the lower axial relative

constraint finds some leeway and start to move away from its boundary.

83

Figure 6.7: Topology. Iteration 3, 7, 11, 18 and 40 of the axisymmetrical

loadcase. The intermediate iteration between 18 and 40 show only small adjust-

ments which are less important. Note that in each figure the left edge represents

the axis of symmetry. Between the turntable on the left and the casing on the

right we see a solid black square which represents the main bearing.

84

6.2 Bilateral loadcaseTo run the complete model, the bilateral loadcase representing the rolling motion

of an FPSO is used, as shown in figure 6.8 (as was already used in an example

in figure ??). As in the previous section, the results will be explained in the

caption of each result (figures 6.9 through 6.14). The detailed set-up used in

the bilateral loadcase is shown below.

Symbol Description Value

τ∆σ Relative stress tolerance 0.05 MN

τρ Displacement tolerance 10%

Initial boundary expansion 0.5

Explicit boundary expansion 1.1

Explicit boundary contraction 0.7

p Penalization factor 3

χ−

Lower density boundary 0.1

Turntable load 15 MN

Riser hang-off load 5 MN

Transverse load 1 MN

Overturning load 1 MN

Element types plane25, link180

Boundary conditions Casing edges clamped

Element size 70 mm

Figure 6.8: Bilateral loadcase.

85

Figure 6.9: Relative stress constraint values of the bilateral loadcase. In

reality there are 8 relative stress constraints (as explained in subsection 3.1.2),

but only the 4 closest to the 0-boundary are relevant. The other, mirrored func-

tions are left out for clarity, as was also done in the axisymmetric counterpart

in figure 6.2. All relative constraint functions seem to converge, although the

outer radial constraint is lagging behind just a bit.

Figure 6.10: Displacement constraint values of bilateral loadcase. In

comparison to its axisymmetric counterpart in figure 6.3, the bilateral loadcase

creates more volatile displacement constraint responses and although they seem

to converge, they do so in a less behaved fashion. This probably due to the fact

that with the inclusion of bilateral load distributions, the relative constraints

gain more prominence.

86

Figure 6.11: Objective function values of bilateral loadcase. The objec-

tive shows an overall decrease with steady speed, converging on a lower bound

that represents the minimal mass needed to support the applied loads and sat-

isfy all imposed constraints at the same time. The detailed view (right) does

show the algorithm’s decision to slightly back track and add mass, rather than

removing it (at iteration number 31 to 37). This is to counteract the sudden

increase in both displacement constraints (see figure 6.10). Although currently

the optimizer reaches a objective minimum at iteration 32, it will probably reach

a lower value when given enough time. But, since the overall structure will not

change significantly, this is considered unnecessary.

Figure 6.12: SIMP values of bilateral loadcase. As in the axisymmetric

counterpart (see figure 6.5, the SIMP values also show a nice convergence to-

wards 0. Since the initial design space is fully enclosed, none of the variables are

in an intermediate stage. In the beginning iterations a peak value in intermedi-

ates can be observed, after which the amount steadily declines as penalization

filters them out. This behavior is logical since the optimizer is stuck to the

initial boundary expansion, it cannot simply remove rudimentary elements in

one iteration.

87

Figure 6.13: Optimization computation time of bilateral loadcase. The

time per iteration starts off at about 45 minutes, but as the constraint func-

tions reach the limits of the feasible domain and start to show more profound

contradictory demands, the algorithm takes longer to find a minimum in W, as

was also the case in the axisymmetric loadcase.

88

Figure 6.14: Topology. Iteration 2, 5, 9, 15 and 40 of the bilateral loadcase.

The intermediate iteration between 18 and 40 show only small adjustments

which are less important. Note that in each figure the left edge represents the

axis of symmetry. Between the turntable on the left and the casing on the right

we see a solid black square which represents the main bearing.

89

Chapter 7

Conclusions

7.1 Program stability and convergence

Within the set of assumptions as shown in section 2.4, both the TOP-algorithms

(axi -and bilateral symmetric loadcases) show stable behavior and convergence,

which ensures the derivative calculations as proposed in chapter 5 are accu-

rate (at least to a satisfactory extend). Not only do the solutions converge

toward multiple or all constraints imposed on the structure, they also converge

on the demand for solid models using the SIMP-method. Usually, the displace-

ment constraint functions show a kind of dynamic behavior in which they first

overshoot their boundaries into the non-feasible domain. This, in itself, is no

problem; intermediate solutions are not required by MMA to stay within Γ and,

hypothetically, it could be the case that in order to reach an optimum, the

algorithm has to navigate through some non-feasible intermediates. Although,

there is no clear reason to believe that this is currently the case. A more likely

candidate is fact that the initial asymptotic boundary expansion (as discussed

in subsection 5.2.6) is set too high, allowing the optimizer to try and speed

up convergence. This initial increase factor determines the speed at the first

two iterations, after which the normal boundary expansion (which is an explicit

function, using the data from previous iterations) takes over. It was only the

latter that was decreased manually, the initial retained its original value.

The amount of design variables and constraint functions cause moderate com-

putation times, between 1 and 3 hours per iteration, with around 40 iterations

to reach full convergence. There are, however, certain demands which tend to

cause excessive computation time, indicating that the Newton-method has some

difficulty searching for minimums. These occurrences were rare and are prob-

ably linked to some ill-posed combination of constraint tolerances or an initial

start condition that is sufficiently outside the feasible domain Γ, but also a too

90

conservative limitation to the boundary increase of the moving asymptotes (as

discussed the previous paragraph) might prove to be a culprit.

7.2 Resulting structure

A detailed view of both solutions is shown in figure 7.1, wherein the focus is on

the structure directly connected to the main bearing. For both solutions, the

axisymmetric (7.1a) as well as the bilateral loadcase (7.1b), the optimizer has

a clear penchant for creating a tubular (or circular) structure incorporated in

the turntable. Furthermore, a slightly different structure emerges on the casing

side, supporting the outer main bearing ring. This is undoubtedly linked to the

deformations cause by the transverse, bilateral loads.

(a) Solution of axisymmetrical loadcase. (b) Solution of bilateral loadcase.

Figure 7.1: A close up view of the axisymmetric and bilateral

solution x(κ). Clearly, the solution shows to favor a tubular

structure around the inner main bearing ring. The solid square

in the center represents the main bearing itself (which remains

untouched by the algorithm).

The circular structure on the turntable side is actually more of a circle segment

as shown in figure 7.2a, it most likely emerges from particular characteristics

of such a geometrical shape. Then, what are these apparently favorable char-

acteristics? To answer that question the torsion behavior within two separate

models are examined, i.e.

1. A numerical axisymmetric model

2. A analytic cantilever model

91

In case of axisymmetric deformations (that is: axisymmetric loadcases) all radial

cross sections of a ring-shaped volume, described by a particular element, will

deform within that same radial plane, and they all deform exactly the same way.

This phenomenon is better know as ring torsion. Without any external bound-

ary conditions, a ring will resist rotation about its center line due to increasing

tangial stresses, sometimes referred to as ‘hoop’ stresses. A higher torsion re-

sistance will decrease the rotation of the inner bearing ring as well, which is

what the optimizer seems to prefer in almost all examined circumstances. Why

not a boxed construction instead of a circular one? To answer that question, a

separate numerical model was made comparing the two geometries as depicted

schematically in figure 7.2b and 7.2c (using the same plane183 elements as in

the TOP-programming). Since the algorithm is trying to minimize structural

mass, different geometries with equal cross section areas have to be examined.

Therefore the model takes radius r and thickness t as independent variables and

a as a dependent variable defined specifically to satisfy the equal surface areas,

i.e.

a (r, t) : AC = AB =1

2πr + t

(1− 1

4π

)(7.1)

(a) Schematic representation of

the circular structure adjacent to

the main bearing (Π) on the

turntable side (as proposed by the

optimizer).

(b) Circular structure with rota-

tion about the center line φC .

The red lines indicated constraint

equations coupling rotation and

displacements.

(c) Boxed structure with rotation

about the center line φB . The red

lines indicated constraint equa-

tions coupling rotation and dis-

placements.

Figure 7.2: Schematic models of torsion rings.

The resistance to torsion is approximated by applying a moment to each geom-

etry and determine the rotation about its center line, as shown in figure 7.2b

and 7.2c. Repeating the process while varying r and t produces the results in

figure 7.3a. Figure 7.3a is a torsional stiffness indicator of a circular ring, in

this case represented by 1/φC . The smaller the rotation φC about the center

92

line, the higher the torsional stiffness. As expected, we see an increase in this

particular stiffness when either increasing the cross section radius or thickness,

or both. The same effect can be found in an axisymmetric boxed cross section.

Upon examining the optimal shape (i.e. the shape with the least surface area),

figure 7.3b shows that, at least within a realistic range, the circular cross section

constantly has a higher resistance to torsion, especially profound in structures

with large radii and small thicknesses. When setting out to create a structure

that decreases the rotation of the inner main bearing ring, circular shapes prove

to be the most efficient.

(a) Indication of stiffness (1/φC) of a circular ring-

shaped structure as a function of r and t

(b) The ratio of rotation about the center line when

subjected to moment M (φC/φB).

Figure 7.3: Results of numerically evaluating ring-shaped struc-

tures with circular and boxed cross section subjected to torsion.

93

Also in non-axisymmetric loadcases, in which the cross section deformations

vary as a function θ, circular shapes are preferable. This can be shown in an

analytical fashion, by considering the torsional stiffness of a cantilever using

well-known relations as given by Roark [14] for cross sections as depicted in

figures 7.2b and 7.2c, i.e.

JC =1

2π(r4 − (r − t)4

)JB = t(a− 2)

2(a− t)

(7.2)

in which J denotes the polar inertia for a circular and boxed cross section

respectively. Now taking r and t as independent variables, t is defined as

t(r, t) : AC = AB =2πr − 4a

π − 4(7.3)

in which t ∈ [0,∞).

t equals 0 when the perimeter of both cross sections are equal, i.e. 2πr = 4a.

The surface plot of t(r, t) is given by figure 7.4a. The ratio of polar inertia, much

the same as the ratio in figure 7.3b, is examined in figure 7.4b. By looking at

figure 7.4b and 7.4a both, it is evident that circular shapes are favorable when

dealing with decreasing thickness. The polar inertia of a solid beam, however,

is higher that that of a circular one with equal cross section area, but solid

sections are of course avoided when trying to minimizing mass.

(a) Thickness t as a function of r and a. Note the un-

defined white areas in which t < 0 or t > r.

(b) The ratio of polar inertia of a cantilever model of a

boxed and circular cross section (JC/JB).

Figure 7.4: Results of analytically evaluating cantilever models

with circular and boxed cross section subjected to torsion.

94

Figure 7.5: The normalized bearing stress results from the bi-

lateral loadcase (shown in figure 6.8). The stress increase factor

shows how much the stress has increased from its original value

σ(1). We can see that the stresses pair up and stick together;

this is obviously due to the optimizer trying to satisfy the rel-

ative stress constraints. Note that all bearing stresses increase

due to pre-tension effects.

So, the optimizer sets out to increase the torsion stiffness of the inner bearing

ring, whereas the outer ring is made somewhat more compliant. Although this

solution emerges under a number of assumptions (such as neglecting the presence

of a lower bearing), it is has a sharp contrast with the ‘rigid ring’-concept [24]

which adds stiffness to the outer ring.

7.3 Increased main bearing loads

The constraint functions show us the relation between stresses on a main bearing

roller, they do not show the stresses themselves. Plotting the stress development

within the main bearing, as is done in figure 7.5, reveals an increasing load on

all rollers. This might seem odd, given the fact that the loads do not increase;

how can, for example, the loads on the axial orientated roller increase if we do

not increase the static vertical loads? The cause of this is directly linked to

removal of elements. During the first iteration all elements start off as solids

(x(1) = ι), and as such, the elements in the turntable adjacent to the main

bearing will be compressed due to the applied pre-tension; they are subjected

to ‘pinching’, if you will. As these elements are subsequently removed from

the structure, more and more of the pre-tension load is redirected to the main

bearing, hence the increase in roller loads. Pre-tension is there to compress

the rollers, and, in this algorithm, it is essential to keep them that way since

95

axisymmetrical FEA does not support non-linear analysis. Since the elements

which make up the main bearing (Π) are not part of the design domain, the

pre-tension cannot be ejected. There is no way for the algorithm to adjust or

reduce pre-tension when needed. This leads to the automatic assumption that

pre-tension (of a certain magnitude) needs to be incorporated, which might also

affect the solution. Incorporating pre-tension adjustment into the optimizer has

to be examined in future models, and since it has been discussed here, it will

only be mentioned shortly in the recommendations (chapter 8).

7.4 Evaluation of set goals

As a final conclusions, let’s recapitulate what the set of goals exactly was and

evaluate whether or not these were achieved. The list, as formulated in subsec-

tion 2.4, is once more given by:

1. Investigate the potential of topology optimization for the pending main

bearing issues.

2. Write a general mathematical framework for optimization with relative

stress constraints.

3. Write an optimization program capable of handling

(a) multiple (relative) constraints.

(b) the intrinsic non-linear behavior of roller bearings.

(c) axisymmetric as well as non-axisymmetric loadcases.

(d) data exchange between optimization and external FEM-algorithms.

4. Compute an optimum solution which can be used as an initial design and

provides structural insights concerning relative constraints.

The only goals that was not achieved (i.e. in the exact way it was meant) is goal

3b: the program’s capability of describing the intrinsic non-linear behavior of

the roller bearings. This is due to the fact that Ansys cannot perform non-linear

analyses while using axisymmetric option of plane elements, and to circum-

vent this problem, horizontal as well as vertical pre-tension were assumed. This

assumption, and the compatibility problem leading to it, were discussed in sub-

section 5.1.1 and 5.1.3.

Most of the assumptions in this thesis can be overcome by methods proposed

in the recommendations (chapter 8). Therefore, topology optimization can be

considered a serious candidate for solving problems of relative nature, and one

can find this relativity in other places than just the main bearing; also the

96

lower bearing and the swivelstack will probably be interesting subjects. Since

topology optimization is a rather unknown concept within the offshore industry,

a company such as BES might stand a lot to gain in expanding and utilizing

this type of knowledge.

97

Chapter 8

Recommendations and

contingencies

8.1 Constraints for failure modes

Up till this point, all failure modes, such as yield and fatigue, were ignored. If

the goal is to create a more accurate model, capable of designing not only an

initial design, but a readily usable turret/casing structure, one has to find a

way to incorporate such additional constraints. This is not an idea that is too

far-fetched, but, with defining such constraints, scrutiny is advised. The more

complex the demands become, the more complex its derivatives will be, as is the

case in e.g. subsection 8.5 on replacing link180 for contact elements. Research

has been done on this subject and papers were published for adaptations con-

cerning stress constrained topology optimization using so called cluster methods

[8]. Stress criteria should, realistically, apply throughout the entire structural

domain, potentially creating a vast number of constraints. To reduce this num-

ber to an acceptable level a so called clustering method is used. This method

might prove to be the first step towards incorporating fatigue constraints into

the optimization cycle. The effect of fatigue performance on the optimization

process and its solution is something BES regards as interesting since turntables

are rather susceptible to this phenomenon. A lot of monitoring and maintenance

is required to ensure the structural safety of the FPSO, which puts a strain on

the companies resources. Furthermore, some advances are made obtaining the

topological derivatives for buckling constraints, which grapples with a whole

different set of mathematical problems, that of structural non-linear responses

and instabilities.

98

8.2 Constraints in multiple radial planes

The constraints on the main bearing load distributions should ideally be imposed

on all radial planes, not just a particular one. The load distributions should be

within the set tolerance at any point on the circumference. For axisymmetric

loadcases this is not relevant since the structural response is also axisymmetric.

Hence, if one radial plane satisfies the constraint, that means they all do. In

non-axisymmetric loadcases this is no longer the case. Because there are an in-

finite amount of radial planes, this would mean the algorithm has to deal with

an infinite amount of constraint functions. For practical purposes, an appropri-

ate angle is chosen between constraint radial planes such that the number of

constraint functions is acceptable A reasonable assumption would then be that

if the constraint radial planes meet all requirements, the intermediate plane will

meet them as well.

However, this means that the derivatives for all constraint radial planes will

have to be calculated. More constraint functions would intuitively increase the

amount of unit load analyses even further, possibly to an unacceptable number.

Fortunately, the unit load analyses are valid for all radial planes when they are

merely rotated to the appropriate angle. This opens up the possibility for the

derivatives to be determined as a function of θ by means of convolution with

the structures physical response, i.e.

∂uρ (θ)

∂xj= pxp−1

j

(^uT

ρ (θ) ∗ ku (θ)

)or -more useful- in integral form

∂uρ (θ)

∂xj= pxp−1

j

2π∫0

^uT

ρ (τ) ku (θ − τ) dτ (8.1)

To avoid any unnecessary ambiguity, the choice was made to use a simplified

form of (??) shown on page ??; meaning, (8.1) needs to be adapted to include

the change of ka due to the mode of the applied load(s).

8.3 Incorporate pre-existing structure

Other departments pursue their own structural needs. The support structure is

also there to support equipment, guide risers and other piping. From these view

points, demands such as the need for horizontal layering of decks might follow.

However, the optimizer, up till this point, does not care about such demands.

Among other possibility, one might define a pre-existing, initial structure within

Φ such that the optimizer cannot simply discard it. It can, however, decouple

99

the structure if some type of connectivity to the main bearing is not ensured.

Defining this pre-existing structure should therefore be subject to engineering

scrutiny.

8.4 Lower bearing modeling

The effects of the presence of a lower bearing should, when correctly designed,

greatly affect the eventual loads on the main bearing; a full and accurate topol-

ogy optimization of the support structure cannot be considered complete with-

out it. In essence, it should not even prove too difficult or time consuming to

make a good approximation of this interaction. The turret extending down-

wards to the lower bearing can, for example, be represented by a simple beam

model, the end of which is connected to a partly constraint horizontal link el-

ement. The crux of this proposed model is that this link element has to be

non-linear in nature, something we have seen earlier is incompatible with ax-

isymmetric modeling. In the main bearing, this is solved using pre-tensioning.

The lower bearing, however, is not, and cannot be pre-tensioned. Disregarding

this would mean that the lower bearing can start to pull on the casing. Even

if this happens for a limited amount of iterations, and even if it happens in a

certain part of its circumference, it could well be enough to contaminate the

solution. Therefore, the presence of the lower bearing is neglected in this thesis.

Finding a solution to the incompatibility between non-linear static analysis and

axisymmetry would not only solve this problem, but would also eliminate the

direct need for proper pre-tensioning within the main bearing.

8.5 Contact elements

For a more accurate analysis of the interaction between the inner and outer main

bearing ring, a better way of modeling the roller is paramount. In this thesis

sets of link180 elements were used in order to retain simplicity of derivative

calculations; constraints were based on the stress difference between the two

most outer elements, and stress itself following from well defined displacement

derivatives. To include contact elements means determining whether or not

they are compatible with harmonic elements (such as plane25) and gaining a

deep understanding in how these elements behave mathematically. Non-linear

behavior might well entail that derivatives will depend on some type of iterative

calculation, greatly increasing computational demand.

100

8.6 Radial basis functions

The phenomenon ‘checkerboarding’ is well known within the field of topology

optimization. Its the pattern created by the algorithm such that direct neigh-

bors of a dense element have 0 density, while the diagonal neighbors have high

densities. This pattern resembles that of a checker -or chessboard, and it gives

the model a type of stiffness that is either too expensive or too difficult to man-

ufacture. Any sufficiently large region which displays this pattern can also be

regarded as mimicking an intermediate density, something SIMP is supposed

to prevent. Unfortunately, it is unable to do so since, technically, all demands

within P are satisfied. Huang [9] solves this problem by make the derivative value

of each element dependent on its neighboring elements, i.e. elements within a

certain radius r. Using all values within r, based on some averaging function

the new derivative values is calculated. Since derivatives are supposed to show

a smooth gradient within a derivative field this does should not affect the solu-

tion by much (one might observe a more persistent fuzzy state near structural

boundaries during intermediate iterations). In regions where the derivatives do

not show a nice gradient, this averaging will prevent them from turning into

checkerboard patterns. A more general way of manipulating the derivative field

would be to introduce a radial basis function. Radial basis functions are any

type of function that depend on the distance from the functions center, in this

case an arbitrary element j. The function describes in what way the derivative

value of j is influenced by its own value and that of its surroundings. Charac-

teristics of the elements, such as location or density, can also be incorporated.

Make the radial basis function depend on an elements location from another

point in space might be useful when only particular areas in the design space

are susceptible to checkerboarding. Caution is adviced, though, since we are

artificially changing the derivative field as calculated from real FEA data. If

not done correctly, the optimizer might become unstable.

8.7 Removal of obsolete elements

The optimizer has the objective to reduce mass while being subjected to pe-

nalization, therefore it will try to remove as many elements as possible without

violating constraints. In most cases, a lot of elements will be rendered obsolete

within, give or take, the first 10 iterations. The optimizer should, ideally, be

able to reintroduce some of these elements, and indeed it does so in particu-

lar situations. However, some elements turn out the be obviously rudimentary;

they drop down to χ−j

and flat-line. Still, these elements contribute to the com-

putation time of all processes: FEA, data exchange, displacement decoupling,

derivative calculations and eventual optimization. A possibility to discard these

101

elements fully (also within the finite element model) could potentially save a

lot of time (or, since the optimization time increases towards later iterations,

it might level off the time needed). The condition for eliminating elements

should be chosen carefully; only when an element, and those within its imme-

diate vicinity, show no penchant to contribute even in a slight manner, it can

be safely removed. Ansys provides this option in the ekill-command, but, as

in the compression-only state of link180, it can only be used when preforming

non-linear static analyses.

8.8 Pre-tension adjustment

Pre-tension adjustment refers to the optimizer’s ability to adapt the initial com-

pression of the main bearing rollers. This has been discussed in section 7.3 of

the conclusion, to which the reader is now referred.

102

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105

Index

asymptotes, see MA112

axisymmetric, 119

axisymmetry, 32, 33, 45, 61, 62, 67, 71,

104

bilateral symmetric, 123

bilateral symmetry, 33, 78, 86, 100, 107

checkerboarding, 53, 109

coefficient ratio, see ill-conditioning

computation time, 98

constraint

derivative, 46

load preservation, 38

relative stress, 38, 53

stress, 53

contact element, 66, 108

convergence, 98

convex approxiation, see MA112

convolution, 107

derivative

constraint, 46

convolution, 107

displacement, 47, 108

objective, 45

relative stress, 70

stress, 70, 108

topological, 45

design space, 31

design variables, 36

duality, see Lgrangian duality112

feasible, 98

Fourier-series, 79

guiding bearing, see lwer bearing112

harmonic

element, 63, 78

load, 78, 79, 82

Harwell-Boeing, 76, 117

hoop stress, see blateral symmetry112

ill-conditioning, 58, 73

jacobean, 69

KKT-conditions, 40

Lagrangian, 42

Lagrangian duality, 33, 42

lower bearing, 108

main bearing, 20, 21, 25, 26, 29, 32,

65, 99, 103

MMA, 33, 39, 75, 98

objective function, 36

optimization, see tpology optimization112

patents, 25

penalization, 46, 48

performance variable, 27

physical loads, 79

pre-existing structure, 107

pre-tension, 67, 103

primal-dual Newton method, 44

rimal-dual Newton method, 33

roller, 108

Rosebank, 63

Rothe Erde, 67, 68

106

sensitivity, 45

SIMP, 46, 98

Slater’s condition, 42

superposition, 33, 80–82

tangial deformation, see bilateral sym-

metry112

topology optimization, 30, 98

torsion, 99

upper bearing, see main bearing

virtual loads, 79

107