numerical hull series for calm water

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Numerical Hull Series for Calm Water and Sea-Keeping

Patrick Couser, Formation Design Systems Pty Ltd, Fremantle/Australia, [email protected]

Stefan Harries, Friendship Systems GmbH, Potsdam/Germany, [email protected]

Fabian Tillig, SSPA Sweden AB, Gteborg/Sweden, [email protected]

Abstract

Naval architects draw inspiration from previous designs, literature reviews, statistical regression

models and systematic series. In this paper, a complementary approach, using simulation-driven

design, is presented: exploration of the multi-dimensional design-space using first-principles

methods. The vessel is modelled parametrically with the free-variables that define the design-space.

The design-space is then populated by systematic variation of these variables. The key benefit of the

proposed method is that it allows the design team to quickly explore the design-space and build up a

knowledgebase ahead of an anticipated project. This then allows quick interrogation of the numerical

model series to substantiate design decisions during the bidding and tendering process.

Nomenclature

BWL Beam on waterline

CB Centre of Buoyancy

CG Centre of Gravity

DWL Design Waterline

GMt Transverse metacentre above CG

GZ Hydrostatic righting lever

LCB Longitudinal Centre of BuoyancyLCG Longitudinal Centre of Gravity

LPP Length between perpendicularsVCB Vertical Centre of Buoyancy

VCG Vertical Centre of Gravity

n

n-dimensional (design) space

Abbreviations

CFD Computational Fluid Dynamics

COM Component Object Model

CPU Central Processing Unit

FLOPS Floating-point Operations per Second

FFW FRIENDSHIP-Framework(Software)

GPU Graphics Processing Unit

HM Hydromax (Software)

MSI Motion Sickness Incidence,

RAO Response Amplitude OperatorSK Seakeeper(Software)

1. Introduction

The aim of this paper is to demonstrate, by means of an example application to a mega-yacht, how

numerical simulation can be used to explore the design-space early in the concept design stage of a

project and how this information may be used to gain deeper insight into the design compromises

which will have to be made.

Table I shows the principal particulars of the proposed vessel; these would typically be given by the

client: Design me a mega-yacht thats a bit faster, a bit bigger and a bit more luxurious than the one Ibought last year! As can be seen, the design requirements are quite vague, so it is of utmost

importance to gain an understanding of the design-space in which the solution will lie (or even to

ascertain the feasibility of the proposal).

1.1. Why?

Information is power! Prior knowledge of the relevant design-space for a ship-design project enables

the design team to achieve a sensible compromise that meets the customers requirements. This

knowledge can be gained in several ways. For example, an existing vessel may serve as a basis design

from which a new, improved vessel that better fulfils the customers requirements can be derived. Or,

if there is little prior knowledge or the project requires a completely novel vessel design, then it is

important for the designer to gain an understanding of the design-space by some other means. Tosummarise, the proposed approach might be used to:

Paper presented at the 10th Internat onal Conference on Computer and IT Appl cat ons n the Mar t me Industr es,

Berlin, 2-4 May 2011, Hamburg, Technische Universitt Hamburg-Harburg, 2011, ISBN 978-3-89220-649-1


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Gain an insight of the design-space early in the project; Enable rapid prototyping of ideas for novel design solutions; Provide data for decision support for possible design changes required to achieve desired

performance; and

Anticipate consequences of requested design changes.Table I: Principal particulars of the proposed mega-yacht

minimum maximum

Length between perpendiculars,LPP [m] 68.00 72.00

Beam onDWL [m] 14.00 14.25

Design Waterline,DWL [m] 3.9

Displacement in seawater atDWL [tonnes] approximately 2200

Cruise speed [kts] 16.0

Maximum speed [kts] 20.0

1.2. How?

The example presented serves to illustrate the concepts and methodology. However, there is noreason why different design aspects could not be examined or different numerical tools used. The key

is being able to automatically vary the proposed vessel in a manner so as to produce viable variants

and then be able to predict the variants performance characteristics pertinent to the design

requirements.

The design-space investigation thus comprises three main tasks:

1. Definition of a suitable parametric model which can be used to generate feasible designvariants from a small number of key parameters.

2. Numerical analysis of the vessel using simulation tools which can provide an assessment ofthe vessel performance characteristics of interest (in this case, hydrostatics, resistance and

sea-keeping). These tools need to be selected so that they can provide sufficiently reliable

data within available time and computational resource constraints.

3. Automation of vessel design variation, analysis, results gathering and post-processingtasks.

The FRIENDSHIP-Framework (FFW Friendship Systems, 2009) is used to firstly define the hull

geometry in a parametric manner which can then be systematically varied and secondly to

systematically vary the design, control the analyses and collate the results for all the design variants.

1.3. What is Important?

What is of interest and importance to the designer will depend on the individual project being

undertaken. In this example, static stability as well as resistance and also passenger comfort when thevessel is under the influence of waves are considered.

The vessels calm water resistance was estimated using SHIPFLOW (Flowtech 2004, 2009), whilst

sea-keeping characteristics and hydrostatic stability were predicted using Seakeeper (SK) and

Hydromax(HM Formation Design Systems, 2011).

1.4. Computer Hardware

It is interesting to look at the increase in computer performance over time; this is shown in Fig. 1 for

the last 30 years (SUPERCOMPUTER 2011; Thibault et al. 2009; Koomey et al. 2009). There

continues to be exponential growth in not only the performance of supercomputers but also that of

personal micro-computers. What is also interesting is the application of GPUs rather than CPUs tosolving CFD flows (Thibault et al. 2009). GPUs can be optimised for floating-point calculations and


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matrix inversion much more than CPUs (which are required to perform a much broader range of

operations). The rate of increase in performance of GPUs is greater than that of both CPUs and

Supercomputers.

The rapid development of computer hardware and the advent of computer clusters and clouds (e.g.

Amazon Elastic Compute Cloud EC2) and other distributed systems now mean that the hardware

resources necessary for the type of numerical investigations described in this paper are now

accessible to even the smallest design teams.

Fig. 1: Super- and Micro-computer performance with time

2. Methodology of the Investigation

In this section, we shall look in some detail at the numerical method used for the design-space

investigation. The key concept to take from this paper is the methodology; different analysis software

can be substituted and different performance measures will be appropriate for different projects.

2.1. Parametric Modelling

The general hull-form chosen for the example mega-yacht was a classical twin-screw design with

bulbous bow and skeg. Appendages were not included at this initial phase of the design. The bulb was

modelled in some detail, since it had a significant impact on the hull resistance. The bulb was blended

into the main hull over a region of transition aft of the forward perpendicular. The main hull itself was

split into fore- and aft-body regions joined at the section with maximum cross-sectional area. A full3D model of this geometry was realised in the FFW.

The model was parameterised so the geometry could be manipulated by a small number of key

features which the designer would wish to vary. These parameters are the free-variables of the n-

dimensional design-space to be investigated and were used to generate design variants within that

space. The parameters (or free-variables), with their range of variation are given in Table II and the

primary curves describing the model are shown in Fig. 2. The body plan, plan, profile and perspective

views of a representative instance of the parametric model are shown in Fig. 3. Full details of the

parametric model are described inHarries (2010).


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Table II: The free variables that are used to define the parametric model

minimum maximum

Length between perpendiculars,LPP [m] 68.00 72.00

Beam onDWL [m] 14.00 14.25

Midship area coefficient 0.82 0.89

Prismatic coefficient of fore part of hull 0.60 0.63DWL half angle of entrance [deg] 14.0 18.0

DWL fullness coefficient 0.58 0.62

Bulb area to midship area ratio 0.092 0.098

Bulb fullness coefficient 0.75 0.85

Longitudinal position of section with max. cross-sectional area [%LPP] 44.0 48.0

Fig. 2: Parametric model of round bilge mega-yacht

Fig. 3: Example of typical bare hull with bulbous bow generated from the fully parametric model


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2.2. Performance Prediction

The numerical tools used to calculate the vessel performance are described below. The tools

presented cover three of the main areas of interest during initial design: resistance, sea-keeping and

static stability. However it would be entirely feasible to include other tools to compute other

performance parameters, for example production cost, manoeuvring, etc. The scope of the

performance parameters to be considered depends on the time available to complete the study as well

as the tools available and the level of detail of the ship model required to produce meaningful results.

2.2.1. Flow Simulation and Resistance Prediction

When predicting calm-water resistance, there is generally a trade-off between accuracy and the

computational effort required. Since only the bare hull was modelled, it was considered appropriate to

employ potential flow theory to solve the non-linear wave resistance problem with free sinkage and

trim combined with a thin boundary layer theory calculation for the frictional resistance, further

details are given in Harries (2010). When fine-tuning appendages, such as brackets, later in the

design, a RANSE calculation should be undertaken to accurately capture the viscous phenomena, for

example Brenner (2008).

The flow simulations were computed on a standard dual core notebook and took about four to five

minutes per variant and speed. With a CFD license for both cores, around 200 designs could be

computed in one overnight job. A typical panel arrangement and results are shown in Fig. 4.

Fig. 4: Typical panel arrangement of free surface and hull

with wave-wake height contours and hull streamlines at FN = 0.393

2.2.2. Motions in Waves and Comfort Measures

The vessel motions due to waves were predicted using Seakeeper a linear strip theory method in the

vein ofSalvesen et al. (1970). Two scenarios were considered (details are given in Table III):

1. Vessel at anchor or in a marina in a very slight sea-state the so-called Party condition.(Note that mooring forces were not considered.)

2. Vessel underway at a cruising speed of 16kts in a higher sea-state, as might be encounteredwhen traveling between two such Party locations the Cruise condition.

The motion sickness incidence (MSI) after two hours exposure was computed at different longitudinal

positions along the length of the vessel (Fig. 5). That is the percentage of people who can be expected

to vomit after having been subjected to the motions for a period of two hours, as calculated by themethod proposed by OHanlon and McCauley (1974) andMcCauley et al. (1976). The performance


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measure extracted from the analysis was simply the minimum MSI along the length of the vessel for

each of the two scenarios considered; assuming that the vessel layout could be adjusted so that MSI-

critical systems (e.g. the bar) could be sited accordingly.

Table III: Two scenarios considered for the sea-keeping calculations

Party Cruise

Vessel speed [kts] 0.0 16.0

Characteristic wave height [m] 0.5 2.0

Modal period [s] 2.0 7.1

Wave heading Head seas

Wave spectrum type 1-Parameter Bretschneider

Fig. 5: Typical MSI distribution over the length of the vessel

The sea-keeping model used 41 equally spaced sections through the hull. Conformal mapping wasused to model the sections and compute the sectional added mass and damping in heave (five

mapping terms were used to give a good fit to the hull sections). The vessel heave and pitch response

amplitude operators (RAOs) was then calculated at 200 frequencies and these were used to calculate

the MSI. The calculations, for 200 variants, were computed on an average desktop computer using

SK, again in an overnight job controlled by the FFW.

2.2.3. Hydrostatic Stability Criteria

Virtually all vessels must comply with hydrostatic stability criteria specified by class. A small subset

of intact-vessel stability criteria, which are typically applied to this class of vessel, were selected from

theLarge Commercial Yacht Code (Maritime and Coastguard Agency 2007) intact stability standards

for monohull vessels, section 11.2.1.1. These criteria are summarized in Table IV.

Table IV: Stability criteria considered

Section Description Required value

11.2.1.1.1a Area under GZcurve from 0 to 30 deg. heel shall not be less than 0.055 m.rad

11.2.1.1.1b Area under GZcurve from 0 to 40 deg. heel shall not be less than 0.090 m.rad

11.2.1.1.2 Area under GZcurve 30 to 40 deg. heel shall not be less than 0.030 m.rad

11.2.1.1.3 Maximum GZat 30 deg. or greater heel shall not be less than 0.2 m

11.2.1.1.4 Angle at which maximum GZoccurs shall not be less than 25 deg.

11.2.1.1.5 Initial metacentric height (GMt) shall not be less than 0.15 m

In order to obtain a meaningful performance measure of stability, the maximum vertical centre ofgravity (VCG) at which all criteria were just passed was calculated for a range of displacements. A


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typical curve of maximum allowable VCG against displacement, for three representative design

variants, is shown in Fig. 6. The measure of performance used was the area under the maximum

allowable VCG curve integrated over the displacement range of 1800t to 2600t. This measure was

chosen because early in the design process, neither the VCG nor the displacement would be known

with certainty; the measure gives some indication of the scope of VCG change that can be

accommodated whist still passing the criteria.

Damage stability has not been considered at this stage because this would depend on the

compartmentation layout which would not be available early in the initial design. Once a design

variant has been selected for detailed design development, the internal layout and compartmentation

would have to be chosen so that damage stability requirements were met.

The analysis was performed inHMusing a range of heel angles at each displacement to calculate the

GZcurve for a given VCG. The vessel was free-to-trim ensuring a longitudinal balance ofCG and CB

(theLCG being derived from theLCB of the upright vessel). The VCG was then systematically varied

to determine the maximum value ofVCG at which all the stability criteria were still passed. Managed

by the FFW, the calculations, for 200 variants, were computed in a matter of several hours.

Fig. 6: Typical MSI distribution along the length of the vessel

2.3. Software Integration

The FFWand the simulation software are developed by different software vendors. However, in order

to automate the task of generating design variants and analyzing their performance, it is essential that

the software systems are able to communicate. Under Microsoft Windows there exists a paradigm for

inter-process communication. This is known as the Component Object Model (COM). For full detailsof COM, the interested reader is referred toBox (1998). COM allows access to suitably COM-enabled

applications via a common interface from a variety of programming languages: C#, VBA, etc. and

also the FFWs own macro language.

Suitable macros were developed in the FFW to export the hull geometry and then import this

geometry and run the analyses inHMand SK. The results of the analyses were then read back into the

FFWfor post-processing to calculate the final performance measures for each variant. Fig. 7 shows a

screenshot of the FFW, HMand SKin action.


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Fig. 7: Screenshot of FRIENDSHIP-Framework, Hydromax and Seakeeper in use

2.4. Design of Experiments

The design-space was investigated using a Design of Experiments approach to populate the domain

with variants. The principal particulars of the vessel are given in Table I and the nine free-variable

which were used to define the variants are given in Table II. These nine free-variable thus establish a

nine-dimensional space 9. A Sobol algorithm, Press et al (2007) was used to give a quasi-random,

yet uniform sampling of these variables over the desired range (see Table II). The performance was

calculated for 200 variants. A typical distribution of variants (for one free-variable) is shown in Fig.

8; as expected, the Sobol algorithm provides a uniform, quasi-random sampling over the design-space. The design of experiments approach covers the design-space more economically than a regular

grid approach a regular grid of just two parameter variations in 9 dimensions would require 512 (29)

variants.

Fig. 8: Typical distribution of a free-variable using the Sobol algorithm

2.5. Response surfaces

The design-space exploration generates a large quantity of data and represents a not insignificant

amount of computational effort (especially if sophisticated numerical simulation tools have been


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used). It is useful then, to reuse this data, potentially for automated optimisation or other similar

applications. There are several ways in which this data can be captured so as to be able to determine

the vessel performance measures for a set of specified values of the free-variables. These include:

statistical regression; artificial neural networks (e.g. Couser 2004) and response surfaces. All of these

methods effectively allow interpolation of the performance measures of the design given a set of

values of the design parameters (free-variables) without having to redo the numerical simulation, thus

saving a lot of computational effort.

Following the work ofHarries (2010) a response surface, meta-model method has been used. The n-

dimensional (where n is the number of free-variables) response surfaces for the performance

measures are fitted using a Kriging approach, see Tillig (2010). Once the response surface has been

generated, interpolation is more or less instantaneous (compared with a CFD or sea-keeping

calculation which might take a few minutes to several hours to perform). Continuous iso-parametric

curves and surfaces can then be generated from the response surface making it easier for the designer

to visualise the design space: the designer is able to see the effect of continuously varying one or two

free-variables rather than seeing discrete results for variants where all the free-variables have been

modified (which is the raw output from the design of experiments investigation of the design-space).

3. Results

This section presents some results for the mega-yacht example. One should not forget that these

findings are only meaningful in the context of the chosen parametric model (the established design-

space) and that they rely on the validity of the simulations. Even though these simulations are built on

first principles, there are notable simplifications, for instance the wave resistance and sea-keeping

analyses, as used in this example, ignore viscosity.

3.2. Correlations

Some samples of the raw results from the Sobol investigation of the design-space are presented by

means of correlation plots (as shown in Figs. 9 to 14). These correlation plots can highlight generaltrends in the data but it should be noted that the points represent discrete variants where all of the

free-variables have changed; thus these diagrams do not accurately represent the continuous variation

of a single variable. The band-width of the scatter of points about the mean line gives an appreciation

of the difference that can be achieved due to variation of all the other free-variables. It should be

noted that even when there is reasonably strong correlation between performance and a free-variable,

there is often a significant range of performance (which thus depends on the other free-variables). For

example, in Fig. 11, at a length of 70m the Cruise MSI can vary between 4% and 5%. This also

implies that there is always room for improvement even though one (or several) free-variables need to

be fixed at a certain point in the design process. The range of performance can be taken as an initial

indication of how much potential for optimisation is available.

3.2.1. Principal Hull Geometry

Fig. 9 presents the vessel displacement against the length between perpendiculars. A general trend

towards higher displacement for longer vessels can be seen. Nevertheless, as discussed above, there

are instances of vessels with higher and lower displacements (for a fixed LPP) that depend on the

values of the remaining free-variables.

3.2.2. Calm Water Resistance

Fig. 10 shows the correlation between vessel length and predicted wave resistance coefficient. As

might be expected the longer the vessel, the lower the resistance. The interested reader is referred to

Harries (2010) for further details and results of the resistance calculations performed.


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Fig. 9: Typical correlation of a performance measure with a free variable (Displacement and Length)

Fig. 10: Strong correlation between Wave resistance coefficient and LPP

Fig. 11: Strong correlation between MSI and LPP


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Fig. 12: Very weak correlation between MSI and Beam

Fig. 13: Strong correlation between Stability performance measure and LPP

Fig. 14: Un-correlated relationship between Stability performance measure and Bulb Fullness coef.


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3.2.3. Sea-Keeping

As might be expected, the MSI shows a reasonably strong inverse correlation to the vessel length: as

the vessel length increases, the motion sickness incidence decreases, Fig. 11. Sometimes it may be

found that there are surprising correlations (or lack thereof); for example Fig. 12 shows that the

correlation between MSI and beam is not very strong, contrary to what might be expected.

3.2.4. Hydrostatic Stability Criteria

A strong inverse correlation between vessel length and stability was found Fig. 13. This is probably

due to the fact that the displacement range for the stability calculations was fixed irrespective of

vessel length. Shorter vessels would be broader and/or deeper in the water generally resulting in

greater intact stability (up to the angles of heel investigated). Other parameters showed little or no

influence on stability (Fig. 14) indicating that they can be varied to improve other performance

measures without penalising the stability performance.

3.3. Response Surfaces

Once the n-dimensional response surfaces have been fitted to the discrete data obtained from the

design-space exploration, continuous iso-parametric curves and surfaces can be generated for

continuous variation of only one or two free-variables (the others remaining constant). In Figs. 15 to

17, all but two free-variables are kept constant resulting in iso-surfaces through the design space. In

each diagram the range of each free-variable has been normalised to 1.0.

In most cases the response surfaces follow what might be expected: Fig. 15 shows that the delivered

power requirement is reduced for longer and generally narrower vessels; and Fig. 16 shows that MSI

is reduced for longer vessels, with the optimum beam being about the middle of the range. However

in the case of the stability performance measure response surface, Fig. 17, the effects of length and

beam are more complex. It should be noted that since the entire design-space exploration is not

covered by the variants tested, care should be taken to ensure that the response surface is used forinterpolation, and not extrapolation. The sharply raised corners in Fig. 17 are due to extrapolation

with insufficient variants to adequately describe the response surface in these regions.

Fig. 15: Response surface for Power vs. Length and Beam


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Fig. 16: Response surface for Sea-keeping vs. Length and Beam

Fig. 17: Response surface for Stability vs. Length and Beam

4. Taking Things Further / Practical Application

In this paper we have presented an example using a mega-yacht initial design project. Relatively

simple numerical simulation tools have been used to investigate three aspects of the design process:

calm-water resistance (using potential flow and boundary layer theory), sea-keeping (using strip-

theory) and static stability. However, there is no reason why the same methodology cannot be applied

to different problems using different simulation tools.


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The FRIENDSHIP-Framework is very useful in that it facilitates a parametric model of the hull

geometry and allows this geometry to be systematically varied. It then manages sending the geometry

to and retrieving the results from the external simulation tools. The COM interface provides a

relatively simple mechanism for inter-process communication on the Microsoft Windows platform.

(The coupling between the FFWand the analysis software, SKandHM, was achieved using COM.)

Although not presented in the current work, using the resulting response surfaces to drive an

optimisation search is entirely possible and would be the logical next step of the design process (see

Harries (2010) for an example).

5. Conclusions

One of the most challenging tasks for the ship designer is to gain an insight into the non-linear

relationships between competing objectives, constraints, free- and dependent-variables so as to be

able to obtain a suitable final design that meets the customers requirements. The methodology

described in this paper shows how first-principles simulations, coupled with a parametric model of

the vessel can facilitate rapid exploration of the design-space. The methodology can be summarised

as follows:1. Creation of a suitable parametric model of the vessel. The parameters chosen to be free-variables entirely define the vessel and span the design-space of interest. They can be

regarded as the free-variables of an optimisation problem.

2. Performance measures and constraints such as hydrostatics and hydrodynamic performanceare identified and determined by means of numerical simulations based on the vessel

obtained from the parametric model.

3. The design-space is then systematically and automatically explored using formal methods.4. The results of the design-space exploration are captured by response surfaces that allow for

very rapid interpolation of the performance measures for any of set of values of the free-

variables.

5. Once the design-space is known and understood, the data can be used to answer what if?type questions as well enabling optimisation searches to be performed quickly.

The key things to take from this paper is the methodology. The details of the specific parametric

model and analysis tools used are of less importance because they can (and should) be adapted and

tailored to the specific needs of the individual project. However, what this paper aims to show is how

an in-depth knowledge of the design-space in which one finds oneself can be gained by more formal

and extended use of numerical simulation. Of course, as computational power continues to increase

and the accuracy of numerical simulation techniques continues to improve, it will be appropriate to

change the hardware and software used to perform the design-space exploration. It is believed by the

authors that the approach described in this paper will aid naval architects during their design tasks by

providing familiarity with novel design ideas more quickly and allowing them to make appropriate

design modifications to match evolving client requirements more easily.

References

BOX, D. (1998),Essential COM. Addison Wesley, ISBN-0-201-63446-5

BRENNER, M. (2008),Integration of CAD and CFD for the hydrodynamic design of appendages in

viscous flow, Diploma thesis, TU Berlin

COUSER, P.; MASON, A.; MASON, G.; SMITH, R.C.; von KONSKY, B.R. (2004), Artificial neu-

ral networks for hull resistance prediction. 3rd Int. Conf. on Computer Applications and Information

Technology in the Maritime Industries (COMPIT), Sigunza, pp.391-402

HARRIES, S. (2010),Investigating multi-dimensional design spaces using first principle methods, 7th

Int. Conf. on High-Performance Marine Vehicles (HIPER), Melbourne.


15/15

220

JULIEN, C.; THIBAULT, J.C.; SENOCAK, I. (2009), CUDA implementation of a Navier-Stokes

solver on multi-GPU desktop platforms for incompressible flows, AIAA 2009-758, 47th

AIAA

Aerospace Sciences Meeting.

FLOWTECH Int. AB (2004),XPAN / XBOUND Theory, Gothenburg

FLOWTECH Int. AB (2009), SHIPFLOW 4.3 Users Manual, Gothenburg, http://www.flowtech.se/

FORMATION DESIGN SYSTEMS Pty Ltd (2011), Seakeeper User Manual, Hydromax User

Manual, Fremantle, http://www.formsys.com

FRIENDSHIP SYSTEMS (2009), User Guide FRIENDSHIP-Framework, Potsdam,

http://www.friendship-systems.se/

KOOMEY, J.G.; BELADY, C.; PATTERSON, M.; SANTOS, A.; LANGE, K.D. (2009), Assessing

trends over time in performance, costs, and energy use for servers.

http://www.intel.com/assets/pdf/general/servertrendsreleasecomplete-v25.pdf

MARITIME AND COASTGUARD AGENCY (2007), The Large Commercial Yacht Code (LY2).

Merchant Shipping Notice MSN 1792 (M) Edition 2.

http://www.mcga.gov.uk/c4mca/msn_1792_edition_2.pdf

McCAULEY, M.E.; ROYAL, J.W.; WYLIE, C.D.; OHANLON, J.F.; MACKIE, R.R. (1976),

Motion sickness incidence: Exploratory studies of habituation, pitch and roll and the refinement of a

mathematical model. Human Factors Research Inc., Technical Report 1733-2

OHANLON, J.F.; McCAULEY, M.E. (1974), Motion sickness incidence as a function of the

frequency and acceleration of vertical sinusoidal motion. Aerospace Medicine, 45(4), pp.366-369

PRESS, W.; TEUKOLSKY, S.; VETTERLING, W.; FLANNERY, B. (2007), Numerical recipes

The art of scientific computing, Cambridge University Press

SALVESEN, N.; TUCK, O.E.; FALTINSEN, O. (1970), Ship motions and sea loads. Trans. SNAME

78, pp.250-287.

SUPERCOMPUTER (2011), http://en.wikipedia.org/wiki/Supercomputer

TILLIG, F. (2010), Parametric modeling and hydrodynamic analysis of twin-skeg vessels, Diploma

thesis, TU Berlin

numerical hull series for calm water

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