Ship Hydrostatics and Stability 2010 Part 13 pot

One procedure for generating a surface can begin by defining a fam-ily of plane curves, for example ship stations, with the help of Bezier curves, non-rational or rational B -splines, or

Computer methods 305

Figure 13.10 shows that a surface can be described by a net of isoparametriccurves One procedure for generating a surface can begin by defining a fam-ily of plane curves, for example ship stations, with the help of Bezier curves,

non-rational or rational B -splines, or NURBS, with the parameter u Taking then the points u = 0 on all curves, we can fit them a spline of the same kind

as that used for the first curves Proceeding in the same manner for the points

u = 0.1, , u = 1, we obtain a net of curves Plane curves can be

prop-erly described by breaking them into spline segments and imposing continuity

conditions at the junction points Similarly, surfaces can be broken into patches

with continuity conditions at their borders The expressions that define thepatches can be direct extensions of plane curves equations such as those described

in the preceding sections For example, a tensor product Bezier patch is

13.2.7 Ruled surfaces

A particular case is that in which corresponding points on two space curves arejoined by straight-line segments For example, in Figure 13.11 we consider three

of the constant-it; curves shown in Figure 13.10 Then, we draw a straight line

from a u = i point on the curve w = 0.6 to the u = i point on the curve

w = 0.7, for i — 0, 0.1, , 1 The surface patch bounded by the w = 0.6 and

the w = 0.7 curves is a ruled surface A second ruled-surface patch is shown

between the curves w — 0.7 and w = 0.8 Ruled surfaces are characterized by

the fact that it is possible to lay on them straight-line segments

13.2.8 Surface curvatures

In Figure 13.12, let N be the normal vector to the surface at the point P, and V,

one of the tangent vectors of the surface at the same point P The two vectors, N

and V, define a plane, TTI, normal to the surface The intersection of the plane TTI

with the given surface is a planar curve, say C The curvature of C at the point

P is the normal curvature of C at the point P in the direction of V We

note it by k n A theorem due to Euler states that there is a direction, defined by

the tangent vector V i , for which the normal curvature, k - , is minimal, and

306 Ship Hydrostatics and Stability

u=0

"W=0.8

w=0.7

w=0.6

Figure 13.11 Two ruled surfaces

another direction, defined by the tangent vector Vmax, for which the normalcurvature, /cmax, is maximal Moreover, the directions Vmin and Vmax are

perpendicular The curvatures k min and /cmax are called principal curvatures.

For example, in Figure 13.12 the planes TTI and 7T2 are perpendicular one to theother and their intersections with the ellipsoidal surface yields curves that havethe principal curvatures at the point from which starts the normal vector N Thetwo curves are shown in Figure 13.13

Figure 13.12 Normal curvatures

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Figure 13.13 Principal curvatures

The product of the principal curvatures is known as Gaussian curvature:

•**• ~ "'min ' "'max \ij.l

and the mean of the principal curvatures is known as mean curvature:

~r

(13.20)

In Naval Architecture, curvatures are used for checking the fairness of surfaces

A surface with zero Gaussian curvature is developable By this term we stand a surface that can be unrolled on a plane surface without stretching Inpractical terms, if a patch of the hull surface is developable, that patch can

under-be manufactured by rolling a plate without stretching it Thus, a developablesurface is produced by a simpler and cheaper process than a non-developablesurface that requires pressing or forging A necessary condition for a surface

to be developable is for it to be a ruled surface Cylindrical surfaces are opable and so are cone surfaces The sphere is not developable and this causesproblems in mapping the earth surface Readers interested in a rigoroustheory of surface curvatures can refer to Davies and Samuels (1996) and Marsh(1999) The literature on splines and surface modelling is very rich To the booksalready cited we would like to add Rogers and Adams (1990), Piegl (1991),Hoschek and Lasser (1993), Farm (1999), Mortenson (1997) and Piegl and Tiller(1997)

devel-308 Ship Hydrostatics and Stability

13.3 Hull modelling

13.3.1 Mathematical ship lines

De Heere and Bakker ( 1970) cite Chapman (FredrikHenrikaf Chapman, SwedishVice- Admiral and Naval Architect, 1721-1808) as having described ship lines

as early as 1760 by parabolae of the form

y = 1 - x n

and sections by

In 1915, David Watson Taylor (American Rear Admiral, 1864-1940) published awork in which he used 5-th degree polynomials to describe ship forms Names oflater pioneers are Weinblum, Benson and Kerwin More details on the history ofmathematical ship lines can be found in De Heere and Bakker (1970), Saunders

(1972, Chapter 49) and Nowacki et al (1995) Kuo (1971) describes the state

of the art at the beginning of the 70s Present-day Naval Architectural computerprogrammes use mainly B-splines and NURBS

13.3.2 Fairing

In Subsection 1.4.3, we defined the problem of fairing A major object of thedevelopers of mathematical ship lines was to obtain fair curves Digital comput-ers enabled a practical approach Some early methods are briefly described inKuo (1971), Section 9.3 A programme used for many years by the Danish Ship

Research Institute is due to Kantorowitz (1967a,b) Calkins et al (1989) use

one of the first techniques proposed for fairing, namely differences Their idea

is to plot the 1st and the 2nd differences of offsets In addition, their softwareallows for the rotation of views and thus greatly facilitates the detection of unfairsegments

As mentioned in Subsections 13.2.2 and 13.2.8, plots of the curvature of shiplines can help fairing Surface-modelling programmes like MultiSurf and Sur-faceWorks (see next section) allow to do this in an interactive way More about cur-vature and fairing can be read in Wagner, Luo and Stelson ( 1 995), Tuohy, Latorreand Munchmeyer (1996), Pigounakis, Sapidis and Kaklis (1996) and Farouki(1998) Rabien (1996) gives some features of the Euklid fairing programme

13.3.3 Modelling with MultiSurf and SurfaceWorks

In this section, we are going to describe a few steps of the hull-modelling processperformed with the help of MultiSurf and SurfaceWorks, two products of Aero-Hydro We like these surface modellers for their excellent visual interface, the

by John Letcher; he called it relational geometry (see Letcher, Shook and

Shep-herd, 1995 and Mortenson, 1997, Chapter 12) The idea is to establish a hierarchy

of dependencies between the elements that are successively created when ing a surface or a hull surface composed of several surfaces To model a surface

defin-one has to define a set of control, or supporting curves To define a supporting

curve, the user has to enter a number of supporting points; they are the trol points of the various kinds of curves Points can be entered giving their

con-absolute coordinates, or the coordinate-differences from given, con-absolute points.

Moreover, it is possible to define points constrained to stay on given curves or

surfaces When the position of a supporting point or curve is changed, any dent points, curves or surfaces are automatically updated Relational geometry

depen-considerably simplifies the problems of intersections between surfaces and themodification of lines

Both MultiSurf and SurfaceWorks use a system of coordinates with the origin

in the forward perpendicular, the x-axis positive towards aft, the y-axis positivetowards starboard, and the z-axis positive upwards When opening a new modelfile, a dialogue box allows the user to define an axis or plane of symmetry, and

the units For a ship the plane of symmetry is y = 0.

We begin by 'creating' a set of points that define a desired curve, for example

a station Thus, in MultiSurf, a first point, pOl, is created with the help of thedialogue box shown in Figure 13.14 The last line is highlighted; it contains

310 Ship Hydrostatics and Stability

Figure 13.15 MultiSurf, points that define a control curve, in this case

a transverse section

the coordinates of the point, x = 17.250, y = 0.000, z = 3.000 There is a

quick way of defining a set of points, such as shown in Figure 13.15 In thisexample all the points are situated along a station; they have in common the

value x = 17.250 m.

To 'create' the curve defined by the points in Figure 13.15 the user has toselect the points and specify the curve kind A Bcurve (this is the MultiSurf

terminology for B-splines) uses the support points as a control polygon (see

Subsection 13.2.4), while a Ccurve (MultiSurf terminology for cubic splines)passes through all support points Figure 13.16 shows the Bcurve defined by

Y Z

X Figure 13.16 MultiSurf, a curve that defines a transverse section

Several curves, such as the one shown in Figure 13.16, can be used as support

of a surface To 'create' a surface the user selects a set of curves and then,through pull-down menus, the user choses the surface kind An example ofsurface is shown in Figure 13.17 Any point on this surface is defined by the two

parameters u and v The display shows the origin of the parameters, the direction

in which the parameter values increase, and a normal vector

To exemplify a few additional features, we use this time screens of the faceWorks package In Figure 13.18 we see a set of four points along a station.The window in the lower, left corner of Figure 13.18 contains a list of thesepoints Figure 13.19 shows the B-spline that uses the points in Figure 13.18 ascontrol points At full scale it is possible to see that the curve passes only throughthe first and the last point, but very close to the others The display shows againthe origin and the positive sense of the curve parameter

Sur-Figure 13.19 is an axonometric view of the curve Sur-Figure 13.20 is an graphic view normal to the x-axis In Figure 13.21, we see the same stationand below it a plot of its curvature In this case we have a simple third-degreeB-spline; the plot of its curvature is smooth In other cases the curve we are

ortho-interested in can be a polyline composed of several curves Then, the

curva-ture plot can help in fairing the composed curve Usually, it is not possible

to define a single surface that fits the whole hull of a ship Then, it is sary to define several surfaces that can be joined together along common edges

neces-A surface is defined by a set of supporting curves, for example, the bow profile,some transverse curves, etc

314 Ship Hydrostatics and Stability

Figure 13.22 The wireframe view of a powerboat

Figure 13.22 shows a wireframe view of a powerboat The hull surface iscomposed of the following surfaces: bow round, bulwark, bulwark round, hull,keel forward, keel aft, and transom

The software enables the user to view the hull from any angle, for example

as in Figure 13.23 Other views can be used to check the appearance and the

fairing of the hull The rendered view may be very helpful; we do not show an

example because it is not interesting in black and white

Three plots of surface curvature are possible: normal, mean or Gaussian

We have chosen the plot of normal curvature shown in Figure 13.24 The

•z

Figure 13.23 Rotating the wireframe view of a powerboat

316 Ship Hydrostatics and Stability

® Ship Lines: powerboat-3:3

Figure 13.26 The lines of a powerboat

13.4 Calculations without and with the computer

Before the era of computers, the Naval Architect prepared a documentation thatwas later used for calculating the data of possible loading cases The documen-tation included:

• hydrostatic curves;

• cross-curves of stability;

• capacity tables that contained the filled volumes and centres of gravity ofholds and tanks, and the moments of inertia of the free surfaces of tanks.For a given load case, the Naval Architect, or the ship Master, performed theweight calculations that yielded the displacement and the coordinates of thecentre of gravity The data for holds and tanks were based on the tables of

capacity The next step was to find the draught, the trim and the height KM by

interpolating over the hydrostatic curves Finally, the curve of static stability wascalculated and drawn after interpolating over the cross-curves of stability It is in

this way that stability booklets were prepared; they contained the calculations

and the curves of stability for several pre-planned loadings The same methodwas employed by the ship Master for checking if it is possible to transport someunusual cargo

The above procedure is still followed in many cases, with the difference thatthe basic documentation is calculated and plotted with the help of digital com-

puters, and the weight and GZ calculations are carried out with the aid of hand

calculators, possibly with the help of an electronic spreadsheet However, sincethe introduction of personal computers and the development of Naval Architec-tural software for such computers, it is possible to proceed in a more efficientway Thus, it is sufficient to store in the computer a description of the hull and

Computer methods 317

of its subdivision into holds and tanks The model can be completed with adescription of the sail area necessary for calculating the wind arm Then, theuser can define a loading case by entering for each hold or tank a measure ofits filling, for example the filling height, and the specific gravity of the cargo.The computer programme calculates almost instantly the parameters of the float-ing conditions and the characteristics of stability, and it does so without roughapproximations and interpolations For example, in a manual, straightforward

trim calculation one has to use the moment to change trim, MCT, read from

the hydrostatic curves Hydrostatic curves are usually calculated for the ship on

even keel; therefore, using the MCT value read in them means to assume that

this value remains constant within the trim range Computer calculations, onthe other hand, do not need this assumption The floating condition is found bysuccessive iterations that stop when the conditions of equilibrium are met with

a given tolerance

The ship data stored in the computer constitute a ship model; it can be nized as a data base In this sense, Biran and Kantorowitz (1986) and Biran,

orga-Kantorowitz and Yanai (1987) describe the use of relational data bases

John-son, Glinos, Anderson et al (1990), Carnduf and Gray (1992) and Reich (1994)

discuss more types of data bases Many modern ships are provided with boardcomputers that contain the data of the ship and a dedicated computer programme.Moreover, the computer can be connected to sensors that supply on line the tankand hold filling heights

13.4.1 Hydrostatic calculations

Some hydrostatic calculations are straightforward in the sense that we can form them in a single iteration For example, if we want to calculate hydrostatic

per-curves we must perform integrations for a draught TO, then for a draught T\, and

so on Chapter 4 shows how to carry out such calculations Other calculationscan be carried out only by iterations For example, let us assume that we want

to calculate the righting arm of a given ship, for a given displacement volume,

VQ, and the heel angle fa We do not know the draught, TO, corresponding to the

given parameters We must start with an initial guess, Tinit, draw the waterline,

WQ LQ , corresponding to this draught and the heel angle 0^, and calculate the

actual displacement volume If the guess Tjnit was not based on previous

calcu-lations, almost certainly we shall find a displacement volume Vi ^ VQ If the

deviation is larger than an acceptable value, e, we must try another waterplane,

WiLi, parallel to the initial guess waterline, WQ£O- This time we proceed in a

more 'educated' manner Readers familiar with the Newton-Raphson proceduremay readily understand why we use the derivative of the displacement volume

with respect to the draught, that is the waterplane area, Ayy We calculate a

draught correction