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 [email protected] 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 2.1 Problem description . . . . . . . . . 2.2 Problem approach . . . . . . . . . . 2.3 A step toward topology optimization 2.3.1 Optimization potential . . . . 2.3.2 Why topology optimization? 2.4 Goals and boundaries . . . . . . . . 2.5 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Mathematical modeling 3.1 Formulation of main problem . . . . . . . . . . 3.1.1 Objective function . . . . . . . . . . . . 3.1.2 Relative stress constraints . . . . . . . . 3.1.3 Load preservation constraints . . . . . . 3.2 Optimization . . . . . . . . . . . . . . . . . . . 3.2.1 MMA approximation . . . . . . . . . . . 3.2.2 Lagrangian duality . . . . . . . . . . . . 3.3 Topological derivatives . . . . . . . . . . . . . . 3.3.1 Objective derivative . . . . . . . . . . . 3.3.2 Suppressing intermediate densities using 3.3.3 Constraint derivatives . . . . . . . . . . 3.3.4 Analysis of penalization . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 21 24 26 26 28 31 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SIMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 35 36 37 38 39 40 42 44 45 45 46 48 . . . . . . . . . . . . . . . . . . . . . 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 5.1 Finite element modeling . . . . . . . . . . . . . . 5.1.1 Element description . . . . . . . . . . . . 5.1.2 Modeling and discretization . . . . . . . . 5.1.3 Main bearing modeling . . . . . . . . . . 5.2 General programming for all loadcases . . . . . . 5.2.1 Topological derivatives . . . . . . . . . . . 5.2.2 Displacement decomposition . . . . . . . . 5.2.3 Coefficient ratios . . . . . . . . . . . . . . 5.2.4 Scaling . . . . . . . . . . . . . . . . . . . 5.2.5 Initial feasibility . . . . . . . . . . . . . . 5.2.6 Asymptotal increase . . . . . . . . . . . . 5.2.7 Ansys-Matlab coupling . . . . . . . . . . . 5.3 Additional programming for bilateral loadcases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 62 62 63 65 67 67 71 73 74 75 75 76 78 6 Results 79 6.1 Axisymmetric loadcase . . . . . . . . . . . . . . . . . . . . . . . . 80 6.2 Bilateral loadcase . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7 Conclusions 7.1 Program stability and convergence 7.2 Resulting structure . . . . . . . . . 7.3 Increased main bearing loads . . . 7.4 Evaluation of set goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 90 91 95 96 8 Recommendations and contingencies 8.1 Constraints for failure modes . . . . 8.2 Constraints in multiple radial planes 8.3 Incorporate pre-existing structure . . 8.4 Lower bearing modeling . . . . . . . 8.5 Contact elements . . . . . . . . . . . 8.6 Radial basis functions . . . . . . . . 8.7 Removal of obsolete elements . . . . 8.8 Pre-tension adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 98 99 99 100 100 101 101 102 Index 106 4 (This page is left blank intentionally) 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 bearing 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 solution 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 manufacturer, starting at some chosen initial design. Since this initial design itself was not designed for -or possibly already had problems with- roller load distribution, one cannot be guaranteed a solution, let alone an optimal design. This thesis proposes a new perspective on this problem using topology optimization. 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 bilateral 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 derivatives 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 discards 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 response. 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 solutions 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 loadcases. The product of this thesis is an optimization program that is capable of handling multiple relative constraints in an axisymmetrical structure subjected to non-axisymmetrical loadcases. Further research has to be done in order to provide 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 bearing 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 thesis. 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 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, 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. Between 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 halfcommitted 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. 1 Whose 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. 2 The organ commonly known as the ‘voice box’, considering its link to sound production 3 In 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, especially 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 mathematics 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 differences. 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 complex of demands. There has been substantial development in this field, and the 4 Derived from the Greek word ‘topo’ meaning place. In this context in means where to put material, or how to distribute it throughout ‘Euclidian’ space. 5 In 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 a b B B c e ^ e E E Eg f0 fm f̂ρ g g h H i I(n) j k 6 unless Description Mode number Mode number Set containing all link180 elements modeling main bearing rollers Strain-displacement matrix Contribution factor; transforming global to local displacements Amplitude vector for physical harmonic displacements Amplitude vector for virtual harmonic displacements Elasticity vector; containing Young’s modulus all elements within Θ Elasticity matrix; describing the deformation of an element under load General elasticity (210 GPa) Objective function Constraint functions with m ∈ N and m 6= 0 Dimensionless unit load in DOF number ρ Lagrangian dual fuction Local displacement vector Double harmonic vector Harmonic primitives vector Constraint number6 Identity matrix with size n Element number Iteration number otherwise specified 13 Defined figure 5.2 (5.6), (3.28) (5.3) (??) (??) (3.20) (??) (3.2) (3.6), (3.7) (3.22) (3.12) (5.4) (??) (??) k K L L m n p p P P R s S t T u u ^ u Elemental stiffness matrix Global stiffness matrix Roller length vector Lagrangian function Total number of constraint functions Total number of design variables (number of elements in Ω) Penalization factor Set of mode numbers used in physical loadcase Stiffness reduction factor Main optimization problem Radial contribution Scaling vector MMA approximated subproblem Boolean vector identifying common harmonic identities Transformation matrix Displacement Displacement vector (either global or elemental) (5.2) (3.10) (3.20) (3.1) (3.3) (5.12) (3.9) (??) (3.27) Virtual displacement vector (either global or elemental) (3.25) Virtual displacement matrix (concatenation of ^ different u’s corresponding to different unit loads Horizontal displacement component of u Set of mode numbers used in virtual loadcases Vertical displacement component of u Dual objective subproblem Design variable or elemental density Design variable vector (5.4) ^ U v v w W x x 14 (3.29) (3.29) (3.16) (3.20) (3.20) Greek symbols Symbol β Γ ∆ ε ζ η θ Θ ι(n) κ κ χ − Description Link angle Feasible domain Difference Strain Displacement amplitude vector Number of DOFs in an element Angle about the axis of axisymmetry Domain containing all elements with FE model Vector of size n with all elements equal to 1 Total number of iterations to reach convergence Vector containing rearranged stiffness coefficients Vector of Lagrangian multipliers Domain containing all plane elements that model the main bearing DOF number Stress Constraint tolerance Domain containing relative stress link-elements Domain containing all bearing affiliated elements, both link and plane elements Vector containing all density ranges of all design variables Lower density boundary χ̄ ω Ω Upper density boundary Boolean vector identifying common frequencies Domain containing all design variables xj λ Π ρ σ τ Υ Φ χ 15 Defined (3.26) (??) figure 5.2 figure 5.2 (3.1) figure 5.2 figure 5.2 (??) figure 5.2 Acronyms BES DAO DOF FEA FEM FPSO KKT MMA OTC SAO SIMP SLP SPM TOP Bluewater energy services Derivatives for arbitrary orientation Degree of freedom Finite element analysis Finite element method Floating storage production and offloading Karush-Kuhn-Tucker (conditions) Method of moving asymptotes Offshore Technology Conference Sequential approximate optimization Solid isotropic material with penalization Sequential linear programming Single point mooring 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 investments 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 decrease in demand. This increase in supply, without losing ourselves in too much speculation, is in large part caused by recent political instabilities in, among others, the Middle East. The current demand itself was anticipated by the International 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 sustainable 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 periods of times while operating a pre-developed subsea oil or gas field. The abbreviation 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) hydrocarbons 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 Industries. 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 connections 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 reducing 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 acquiring an FPSO was relatively cheap and significantly faster than building from scratch, which certainly added to the popularity in the early years of these offshore installations. There are, however, certain drawbacks to converting old tankers, the most important of which is the restriction on the weather conditions and water depth. For these conditions, the demands for integrating the turret into the hull can become quite elaborate and is therefore usually not economically 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 installation 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 companies 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), offshore 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 problem 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 potentially 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 vertical 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 rotation shown on either side of the turret. Note that the displacement 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 tribology 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 concerning 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 introducing 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. 1 The science concerned with interacting surfaces, friction, lubrication and wear. 25 2.3 2.3.1 A step toward topology optimization 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 distributions 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 nevertheless 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 pressure 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 assume 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 stiffness 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 n 1X φ(Eh , Et ) = n i=1 σmax σmin φ ∈ [1, ∞i (2.1) i 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 GPa Eh,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). Feasible Non-feasible min φ 1.1056 1.1045 Eh 0.50 0.72 Et 1.00 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 stiffness 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 support 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 diameter 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 continuously 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 successfully 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 interesting 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 ultimately 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 deformation 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 structure, 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 fundamental 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 multivariable problems, but the geometry of the model is predetermined. Shape optimization 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 appropriate 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 section 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 horribly 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 problem 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 using 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 min f0 (x) f (x) 6 τi i(k) xj ∈ χj P: χ̄j (κ) x = j χ − i = 1, . . . , m j = 1, . . . , n ∧ ∀k (3.1) j 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 densities. 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 X xj (3.2) j=1 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 . Zπ Zro V =2 Ar dr dθ 0 ri 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 = xj ro2 − ri2 j = xj Rj (3.3) πA From here its only a small step defining the axisymmetric objective function f0 (x) = n X xj Rj = xT R (3.4) j=1 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 computational errors. In reality these situations should not occur, provided that no turret was designed that poorly, but when running models test we want the algorithm 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 optimization 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 38 (3.7) 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 optimization. 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 Optimization (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 analytically, 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 connection 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 (k) (k) f˜i (x) = ri + (k) n X pij (k) Uj j=1 − xj (k) + ! qij (k) xj − Lj (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 (k) pij (k) = qij = (k) Uj (k) − xj 2 0 0 − ∂f /∂xj if ∂f /∂xj > 0 if ∂f /∂xj 6 0 if ∂f /∂xj > 0 (k) xj − (k) Lj 2 ∂f /∂xj if ∂f /∂xj < 0 in which U and L represent the upper and lower asymptote boundary respectively. The partial derivatives are evaluated at x(k) and therefore simply represent 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 (k) ri = fi x(k) − (k) n X (k) j=1 (k) pij Uj (k) − xj + ! qij (k) xj (k) − Lj One can now easily verify that (k) f˜i x(k) = fi x(k) ∀k ∈ N ∂ ˜(k) (k) ∂ f fi x(k) x = ∂xj i ∂xj 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 observation would be that the only remaining control regarding the shape of f˜i 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. (k) (k) (k) Lj < xj < Uj . 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 increase 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) ˜(k) min f0 (x) (k) : f˜i (x) − τi 6 0 (k) xj ∈ χj i = 1, . . . , m 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 denotes the approximated function (earlier denoted by f˜). We make a distinction between the ‘primal’ variables and ‘dual’ variables, which consist of the design 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 : R n × Rm → R L (x, λ) = f0 (x) + m X λi fi (x) = f0 (x) + λT f (x) (3.10) i=1 The dual variables λ will prove a useful tool for solving these types of optimization 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 particular λ, the minimum W = min L will have a certain value. As kλk → ∞, x W = min L → −∞, and, more importantly x lim W (λ) = min kλk→0 x lim L (x, λ) kλk→0 = min f0 (x) : fi 6 τi x (3.11) which means that as the norm of λ approaches 0, the minimum of the Lagrangian 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) = inf L (x, λ) = inf x⊂χ x⊂χ f0 (x) + m X ! λi fi (x) (3.12) i=1 where g is the infimum, or ‘greatest lower bound’, of the Lagrangian L. Therefore, 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 1 Subproblem S satisfies Slater’s condition for ‘strong’ duality. Slater’s condition is a regulatory 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 ∂f ∂s + λT ∂s = 0 , f = 0 ∂x ∂x in which ∂s represents a vector in any feasible direction. (3.13) (b) Lagrangian L as function of λ (dotted) and set of corresponding optima (thick solid) denoted W (λ) (a) Objective (solid) and constraint function (dashed). Figure 3.4: Visualization of using the Lagrangian duality principle 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 T L (x, λ) = r0 − λ b + n X j=1 p0,j + λT pj q0,j + λT qj + Uj − xj xj − Lj ! (3.14) in which r0 − λT b 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) = min Lj (xj , λ) xj 43 (3.15) 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 function, and as such finding the global optimum means finding the unique solution to the equation dLj =0 dxj 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) : max W (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 technique, 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 topology 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 analytically. 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 function. Determining the derivative is also rather easy, i.e. ∂f0 = 1 ∀j ∂xj (3.17) The axisymmetrical variant, defined by (3.4), becomes ∂f0 = Rj ∀j ∂xj (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 (κ) xj (κ) =χ ∨ xj − j 45 = χ̄j (3.19) 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 = Eg xpj {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 elements 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 = Ej dj , then SIMP has the following influence on the global stiffness matrix: K = Eg X xpj dj (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 dependents 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, referred 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 = f̂ρT u (3.22) in which u is the displacement vector and f̂ρ 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 46 (3.23) Combining both equation (3.22) and (3.23) we define the adjoint equation as uρ = f̂ρT u + λ (K − fu) (3.24) in which λ is a vector. The partial derivative with respect to xj of the adjoint equation becomes ∂ f̂ρT ∂uρ ∂u ∂λ ∂f = u + f̂ρT + (f − Ku) + λ +λ ∂xj ∂xj ∂xj ∂xj ∂xj ∂K ∂u u+K ∂xj ∂xj When we assume f̂ρ , f and λ are independent of x, then after eliminating some terms we are left with ∂u ∂K ∂uρ = f̂ρT − λK +λ u ∂xj ∂xj ∂xj T ^ In this we recognize another equilibrium condition if we define λ = uρ , which represents the displacement caused by a unit load. By defining λ in this way, the derivative is reduced to T ∂K ∂uρ ^ = uρ u ∂xj ∂xj Using (3.21) the displacement derivative can now be formulated as T ∂uρ ^ = pxp−1 uρ kj u j ∂xj (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 T hmi = K−1 · fρT [−] N N T hmi ∂uρ m N ^ p−1 = pxj [−] · uρ · kj · u [m] ∂xj − N m ^ uρ hmi ^ 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 T ∂uρ ^ lim = p · u ρ kj u = ∞ p→∞ ∂xj x=1 {∀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 lim p→∞ ∂uρ xj →1 ∂xj lim T ^ = uρ kj u · lim p→∞ lim pxjp−1 xj →1 =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 concerned extending the upper limit such that χ̄j = 1. Theoretically, the lower limit should be χ = in which is a small number in order to prevent compu− j tational problems caused by 0. In practice, however, also any arbitrarily small 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−1 for any particular j, we can visualize the effect of p for x ∈ χ, j 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 reduction factor P (p, x) for an arbitrary element. The blue surface represents P (p, x) = 1, and so the intersection of both surfaces 1 is the curve x (p) = p−1 √ which the density where penalization p 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 already 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 displacements 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 orientation of the local coordinate system). The values of these local displacements can be obtained via transformation of the global displacements, i.e. u1 0 u2 0 ! = sin (β) 0 cos (β) 0 0 sin (β) 0 cos (β) ! u1 v1 u2 v2 (3.26) or u0 = Tu 49 (3.27) Figure 3.7: A schematic representations of definitions and conventions 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 0 = EBTui Since both T and u are a function of xj its derivative becomes ∂σ ∂ (Tu) = EB ∂xj ∂xj (3.28) When we abbreviate c = cos(β) and s = sin(β) then the local displacement derivative can be written as ∂u1 s ∂v1 c + ∂ (Tu) ∂xj ∂xj = ∂u2 s ∂v2 c ∂xj + ∂xj ∂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 ∂v ∂c =c +v ∂xj ∂xj ∂xj 50 The term ∂v/∂xj we already obtained in (3.25). The second term can be expanded even further ∂c dc ∂h = · ∂xj dh ∂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 displacement 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 displacement 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 considered 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 constraints (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 interesting, 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 symmetry54 (a) Displacement constraint values. (b) Link stress constraint values. (c) SIMP values. Figure 4.3: Results of the DAO test model. All data show convergence: the displacement (a), stress (b) and the SIMP values (c), which means the optimum is bounded by both the displacement 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 constraints. The topology is a direct effect of the imposed constraints. 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 vertical 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 illconditioned 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 guarantee 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 depicted 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 adaptations 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 loadcases. 3. Additional programming for bilateral loadcases; Describes the additional 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 maintained, but loadcases may be axisymmetric as well as bilateral. In the previous chapter only plane stress models were evaluated. Of course, in reality, axisymmetrical 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 5.1.1 Finite element modeling 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 perform 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, includes 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 optimization 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 bilateral (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 tangial 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 a b c d e f g h Dimension Core radius Outer radius Turntable overhang Turntable height Turntable depth Inner radius Casing width Casing height Value [mm] 4200 12460 1120 4900 700 12460-700 2800 3500 The various areas are divided in different domains. In the previous chapters we already defined (or hinted at) some domain specifications. The precise definitions 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 Θ Ω Φ Π B Υ Domain Entire FEM domain. The set design variables x. All elements within main bearing. Elements modeling main bearing cross section area. Link elements affiliated with roller modeling. Link elements affiliated with roller relative constraint functions. Type plane,link plane plane,link plane link 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 optimization (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 convey 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 distribution, 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 actually 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 direction 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 describing 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 earlier, 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 distribution 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 radial 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 axisymmetric 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 bearing 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 useful 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 −1 0 0 1 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 T ∂f∆σ ∂fu (5.1) ∂x ∂x 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 J = ∇f = ∂f = ∂x ∆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 T I(8) ⊗ M2 + cos (β) I(8) ⊗ M3 (5.3) and is not dependent on j. gj represents the local displacement derivative values for uB , i.e. gj = c ^ T ! U ku (5.4) j wherein the first term accounts for the change in global displacements transformed to local values. The contribution of the vertical and horizontal displacements 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 = gjT · I(16) ⊗ ι(2) 69 (5.5) Consequently, the stress in each constraint link becomes ∂σ Bn = pEg xp−1 j ∂xj (5.6) By mapping (5.6) using M4 we attain the derivatives for the constraint functions within the set Υ ∂σ ∂f∆σ = I(4) ⊗ M4 · ∂xj ∂xj (5.7) Therefore, the Jacobean as defined (5.1) becomes ∂f1 ∂x1 ∂f1 − ∂x1 .. . J = ∇f = ∂f m−1 ∂x1 ∂fm ∂x1 ········· ········· .. . ········· ········· ∂f1 ∂xn ∂f1 − ∂xn .. . ∂fm−1 ∂xn ∂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 ∂v =− ∂xj ∂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 function. ∂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 occur when calculating the derivatives. They are not cumulative as the derivative information is not exchanged between iterations, but they do confuse the optimizer 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 favorable 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 element 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 inevitably causes loss of information when evaluating such numbers. E.g., consider the following number +0. 100000000000000 | {z } 619150012 | {z } e - 08 Double precision Deformation 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 1 Hence 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 alternation 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 numerical 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 T ^ T T ^ T ^ T ^ ^ uρ ku = uρ,R kuR + uρ,ε kuR + uρ,R kuε + uρ,ε kuε (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 ) in which !α 1X vR = ι min(|v|) sign (vn ) n n !α 1X (n) wR = ι min(|w|) sign (wn ) n n (n) 72 (5.10) For a plane element with mid-side nodes, n will be equal to 8. The ED displacement 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 lim α→∞ 1X sign (un ) n n !α 0 = 1 −1 for sign (u) 6= ±n for sign (u) = n 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 conditions 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 perturbations in some (or all) structural responses that, consequently, cause unstable optimization. This means the algorithm still optimizes using correctly calculated derivatives, but Ansys returns faulty values based on a K that became ill-conditioned somewhere during the iteration process (since K(1) most likely (1) is not ill-conditioned due to initial conditions xj = χ̄j , ∀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 → χ̄). 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 considerations, 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 τi si 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) = χ̄. This means that, if the same loadcase is applied, the initial constraint values (1) fi 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 topology 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 bound75 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 geometrical 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 several 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 function which calls upon Ansys to perform FEM analyses. This Ansys module is shown within the red dashed rectangular section. 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 loadcases (This section has been omitted for legal reasons. Please contact [email protected] 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 loadcases 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/umbilical 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 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 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 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. 84 6.2 Bilateral loadcase To 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 functions 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 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 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 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. 87 Figure 6.13: Optimization computation time of bilateral loadcase. 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, 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 accurate (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 displacement 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 computation 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 probably 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 characteristics? 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 boundary conditions, a ring will resist rotation about its center line due to increasing tangial stresses, sometimes referred to as ‘hoop’ stresses. A higher torsion resistance 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. 1 1 {a (r, t) : AC = AB } = πr + t 1 − π 2 4 (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 rotation about the center line φC . The red lines indicated constraint equations coupling rotation and displacements. (7.1) (c) Boxed structure with rotation about the center line φB . The red lines indicated constraint equations coupling rotation and displacements. Figure 7.2: Schematic models of torsion rings. The resistance to torsion is approximated by applying a moment to each geometry 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 ringshaped 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 structures 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. 1 4 4 π r − (r − t) 2 2 JB = t(a − 2) (a − t) JC = (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 undefined 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 bilateral 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 relative 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 subsection 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 circumvent this problem, horizontal as well as vertical pre-tension were assumed. This assumption, and the compatibility problem leading to it, were discussed in subsection 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 concerning 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 number 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 infinite amount of radial planes, this would mean the algorithm has to deal with an infinite amount of constraint functions. For practical purposes, an appropriate 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. T ∂uρ (θ) p−1 ^ = pxj uρ (θ) ∗ ku (θ) ∂xj or -more useful- in integral form ∂uρ (θ) = pxjp−1 ∂xj Z2π T ^ uρ (τ ) ku (θ − τ ) dτ (8.1) 0 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 topology optimization of the support structure cannot be considered complete without it. In essence, it should not even prove too difficult or time consuming to make a good approximation of this interaction. The turret extending downwards 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 element. 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 axisymmetric 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 neighbors 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 manufacture. 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 solution 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. Characteristics 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 penalization, 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 particular situations. However, some elements turn out the be obviously rudimentary; they drop down to χ and flat-line. Still, these elements contribute to the com− j 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 immediate 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 compression of the main bearing rollers. This has been discussed in section 7.3 of the conclusion, to which the reader is now referred. 102 (This page is left blank intentionally) Bibliography [1] J.R. Barber Elasticity. Department of Mechanical Engineering and Applied Mechanics, University of Michigan, 3rd edition, Springer Science, 2010. [2] S.P. Boyd Convex Optimization. Department of Electrical Engineering, Stanford University, USA, 7th edition, Cambridge University Press, 2009. [3] Y.K. Cheung Finite Strip Method, CRC Press, Boca Ratom, Florida, 1998. [4] P.W. Christensen An Introduction to Structural Optimization. Division of Mechanics, Linköping University, Sweden, Springer Science, 2009. [5] R.D. Cook, Concepts and Applications of Finite Element Analysis. University of Wisconsin, Madison, 4th edition, 2002, John Wiley and sons inc. [6] T. Handreck Analysis of Large-diameter Antifriction Bearings in Conjuction with Customer-specified Companion Structures [7] D. Henery Prospects and Challenges for the FPSO. Offshore Technology Conference, 1995. [8] E. Holmberg Stress Constraint Topology Optimization Devision of Mechanics, Institute of Technology, Linköping University, Sweden, Struct Multidisc Optim, 48:33-47, 2013. [9] X. Huang, Evolutionary Topology Optimization of Continuum Structures: Methods and Applications. RMIT University, Australia, 1st edition, 2010, John Wiley and sons inc. [10] KPMG Being relevant; Suppliers to the Oil and Gas Industry Can Control Their Own Future. 2013. [11] S. Nishiwaki Optimal structural design considering flexibility. Computer methods in applied mechanics and engineering, Elsevier, 190 (2001) 44574504. [12] J. Pollack Latest Breakthrough in Turret Moorings for FPSO Systems: The forgiving Tanker/Turret Interface. Offshore Technology Conference, 1997. 104 [13] A. Rietz A first laboratory exercise in topology optimization using Matlab. Department of Mathematics, Linköping University, Sweden. [14] W. C. Young Roark’s formulas for stress and strain. 7th edition, McGrawHill, 2002. [15] B.F. Ronalds FPSO Trends. Society of Petroleum Engineers Inc., 1999. [16] C. Roos Interior Point Methods for Linear Optimization. Delft University of Technology, the Netherlands, 2nd edition, Springer Science, 2005. [17] O. Sigmund A 99 line topology optimization code written in Matlab. Struct Multidisk Optim 21, 120-127, Springer-Verlag, 2001. [18] O. Sigmund Topology Optimization - Theory, Methods and Applications. Spirnger Verlag, Berlin, 2003. [19] M. Sliwinksi Design Considerations in the Development of a Modular FPSO. Vencor Technologies Limited, Ottowa, Canada, 2004. [20] K. Svanberg, The Method of Moving Asymptotes - A New Method for Structural Optimization. International journal for numerical methods in engineering, 24 359-373, 1987. [21] K. Svanberg, A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations. SIAM Journal of Optimization, 12 555-573, 2002. [22] K. Svanberg, MMA and GCMMA, versions September 2007. KTH, Stockholm, Sweden, 2007. [23] H. De Sterek, Introduction to Computational Mathematics. Department of Applied Mathematics, University of Waterloo, 2006. [24] European Patent Specification 0 338 605 B1 Ship with Mooring Means. Single Buoy Mooring Inc, 1989. [25] United States Patent 6 263 822 B1 Radial Elastomeric Spring Arrangement to Compensate for Hull Deflections at the Main Bearing of a Mooring Turret. FMC Corporation, 2001. [26] United States Patent 7 063 032 B2 Upper Bearing Support Assembly for Internal Turret. FMC Technologies Inc., 2006 [27] United States Patent 5 860 382 Turret Bearing Structure for Vessels. M.A. Hobdy, 1999. 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 guiding bearing, see lwer bearing112 harmonic element, 63, 78 load, 78, 79, 82 Harwell-Boeing, 76, 117 hoop stress, see blateral symmetry112 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 ill-conditioning, 58, 73 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 objective function, 36 optimization, see tpology optimization112 feasible, 98 Fourier-series, 79 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 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 symmetry112 topology optimization, 30, 98 torsion, 99 upper bearing, see main bearing virtual loads, 79 107

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