Automated Fairing of Ship Hull Lines
Automated Fairing of Ship Hull Lines
Automated Fairing of Ship Hull Lines
Received 06.08.2003
Abstract
The problem of creating fair ship design curves is of major importance in Computer Aided Ship Design
environment. The fairness of these curves is generally considered a subjective notion depending on the
judgement of the designer (eg., visually pleasing, minimum variation of curvature, devoid of unnecessary
bumps or wiggles, satisfying certain continuity requirements). Thus an automated fairing process based
on objective criteria is clearly desirable. This paper presents an automated fairing algorithm for ship
curves to satisfy objective geometric constraints. This procedure is based on the use of optimisation tools
and cubic B-spline functions. The aim is to produce curves with a more gradual variation of curvature
without deteriorating initial shapes. The optimisation based fairing procedure is applied to a variety of
plane ship sections to demonstrate the capability and flexibility of the methodology. The resulting curves,
with their corresponding curvature plots indicate that, provided that the designer can specify his objectives
and constraints clearly, the procedure will generate fair ship definition curves within the constrained design
space.
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Fairing is a part of most Computer Aided Ship tional based on variation of curvature. Nowacki and
Design (CASD) packages commercially available to- Lü (1994) proposed procedures for developing fair
day. In the earlier cases the fairing procedures curves under constraints in which the fairness crite-
were based on interactive routines where the de- rion is based on the linear combination of the square
signer visually observes the design curves and in- of the second and the third derivative norm and the
teractively modifies it until satisfactory fairness is constraints apply to approximation conditions, end
achieved (Snaith and Parker, 1972). Alternatively, conditions and an integral condition pertaining to
the designer is presented with curvature plots which the area under the curve, while Pigounakis and Kak-
help to identify the regions of unfairness (Horsham, lis (1996a) developed a 2 stage automatic algorithm
1988). These procedures can be seen as the com- for fairing cubic parametric B-splines under convex-
puterised version of the traditional manual fairing ity, tolerance and end constraints. An iterative knot
method achieved by flexible battens and weights. removal and reinsertion technique is employed which
Hence, the procedure has no objective criteria and adopts the curvature-slope discontinuity as the fair-
suffers the same drawbacks of the manual fairing ness measure. Pigounakis et al. (1996b) proposed
method, i.e. the need for excessive time and suit- 3 algorithms for fairing spatial B-spline curves: lo-
ably trained and experienced personnel. cal fairing by knot removal and local/global fairing
based on energy minimisation. Hahmann (1998) pro-
One of the main goals of the fairing process is posed an automatic and local fairing algorithm for
to automate the process and hence minimise sub- bi-cubic B-spline surfaces. In the proposed method
jective human intervention, which can lead to many a local fairness criterion selects the knot where the
inconsistencies in the resulting hull form geometry. spline needs to be faired and the control net is mod-
The development of automated procedures in which ified by a constrained least squares approximation.
the fairness is defined in an objective manner and Poliakoff et al. (1999) presented an automated curve
achieved within the boundaries set for the design fairing algorithm for cubic B-spline curves based on
problem is clearly desirable. During the past decade an extension of Kjellander’s (1983) algorithm, which
there have been numerous attempts to produce au- is based on finding and correcting the offending data
tomated fairing procedures for curves and surfaces point. The point to be faired is chosen by calcu-
to be employed in both the CAD and CASD en- lating for each point the distance to be moved and
vironment. Pramila (1978) used a linearised fair- then choosing the one for which the distance is great-
ness functional which minimises strain energy for est. Kantorowitz et al (2000) described a method
ship hull surfaces. Maccallum and Zhang (1986) for fairing ship hull lines which determines the suit-
described an automatic smoothing algorithm based able number of control points to produce the required
on B-splines’ curvature behaviour property and ap- shape of the body sections. Yang and Wang (2001)
plied this to some curve forms used in ship design. presented a method for planar curve fairing by min-
Nowacki et al. (1989) described a surface approxi- imal energy arc splines where as a first step the op-
mation scheme based on minimisation of the sum of timal tangents for curve interpolation are computed
the strain energy of mesh lines and the potential en- and the point positions are adjusted by smoothing
ergy of springs attached to the data points. Rogers discrete curvatures.
and Fog (1989) applied their constrained B-spline
curve/surface fitting algorithm to ship hull forms The 3-dimensional ship hull fairing problem is
to generate defining polygons for curves and defin- generally reduced to the fairing of 2-dimensional ship
ing polygonal nets for surfaces. Sapidis and Farin hull design curves, namely sections, waterlines and
(1990) proposed an automatic fairing algorithm for buttocks. The main reason behind this is the com-
B-spline curves. The algorithm is based on removing plexity of the 3-dimensional fairing problem. In ad-
and reinserting knots of the spline. Liu et al. (1991) dition a robust theory for the fairness of curves would
introduced constrained smoothing B-spline curve fit- greatly contribute to surface fairing methodologies.
ting for mesh curves of ships by minimising an en- Indeed, most of the existing surface fairing method-
ergy functional as a fairness measure. Huanzong et ologies either borrow or adapt ideas from curve fair-
al. (1991) proposed a fairing method by minimising ing. Thus, the curve fairing problem often arises
the elastic strain energy of mesh curves of hull sur- in ship hull design and deserves attention. There-
faces. Moreton and Sequin (1992) applied non-linear fore, in this paper an automated fairing algorithm
optimisation techniques to minimise a fairness func- for generating fair ship lines is presented. This algo-
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rithm is based on a variational scheme in which all of points on the curve and whose lengths are propor-
the numerical details of the fairing process are hidden tional to the magnitude of curvature at the associ-
from the designer. The fairing problem is specified in ated point. The characteristics of a curve are evi-
terms of functional optimisation in which the desired denced by the undulations of its curvature plot. In-
form of the curve has a minimum measure of shape flection points occur when the curvature plot crosses
quality. The measure of shape quality is selected the curve (sign change), flat regions produce zero
to provide as uniform a distribution of curvature as curvature values, bulging tendencies produce locally
possible. The resulting ship hull form should not increased values, and flattening tendencies produce
deviate significantly from the original form in order locally reduced curvature values.
not to degrade hydrostatic and hydrodynamic per- The fairness of a curve is intimately related to
formance characteristics already obtained. There- the distribution of curvature over the form, favour-
fore, a closeness constraint is imposed to ensure that ing gradual transitions and avoiding abrupt changes.
the deviations between the original and faired lines Hence, a variety of definitions can be found in the
are not excessive. Hence, the main goal of this fairing literature for fairness criteria mostly associated with
procedure can simply be defined as the reduction in the curvature properties of curves. The widely ac-
curvature variation while retaining the overall curve cepted ones are the following:
characteristics.
The presentation of this paper begins with a de- • A curve is fair if its curvature plot consists of
scription of the curvature concept, which is used as relatively few monotone pieces (Farin and Sa-
a basic indicator of fairness of curves and surfaces. pidis, 1989)
Then the fairing problem is formulated as a non- • A curve is characterised as fair if its curvature
linear optimisation problem in which the objective
plot is continuous, has the appropriate sign (if
is to produce ship definition lines with reduced cur-
the convexity of the curve is prescribed), and
vature distribution. The fairing method is applied to it is as close as possible to a piecewise mono-
typical cubic B-spline ship curves and the results are tone function with as few monotone pieces as
presented in order to demonstrate the effectiveness possible (Sapidis and Farin, 1990)
of the numerical procedure.
• A C2 curve is considered fair if it minimises the
Curvature concept and definition of fairness integral of the squared curvature with respect
for ship hull lines to arc length (Roulier and Rando, 1994)
A mathematical fairing process would require an ob- • A fair curvature plot should be free of any un-
jective fairing criterion, which may be defined in necessary variation, i.e. the distribution of cur-
terms of the distribution of curvature or local radius vature on a fair curve must be as uniform as
of curvature (=evolute) along the curve. possible (Pigounakis et al., 1996b)
The curvature κ(t) of planar curves r(t) =
Thus, based on the above definitions it is required
[x(t), y(t)] has a positive or negative sign depending
of a fair ship line to have the following characteris-
on whether it curves to the left or right. Thus, this
tics:
signed curvature is highly desirable detecting inflec-
tion points as well as convex and concave regions of a • Devoid of unintended noise (erratically dis-
curve. Hence, the signed curvature can be expressed tributed high frequency, high amplitude undu-
as follows (Farin and Sapidis, 1989): lations)
• Devoid of unintended flat regions, and flatten-
ẍ(t)ẏ(t) − ÿ(t)ẋ(t) ing/bulging tendencies
κ(t) = 3/2
(1)
[ẋ(t)2 + ẏ(t)2 ]
• Continuous first and second derivatives
Curvature plots are typically displayed as curva- • Free of unnecessary variation, i.e. limited and
ture vs. arc length or directly along the curve as specified inflection points,
an offset curve with the distance proportional to the
curvature values. The curvature plot consists of seg- • Curvature as uniformly distributed as possible,
ments normal to the curve emerging from a number i.e. monotonically increasing or decreasing
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NARLI, SARIÖZ
• The deviation of offset points should be mini- namely the design variables, objective function and
mal, i.e. within the boundaries defined by the constraints.
designer Design variables: X(X1 , X2 , . . . .,Xn )
• Shape preservation Objective function: F(X) = f (X1 , X2 , . . . , Xn )
All curves satisfying the conditions stated above Constraints: gj (X) > 0, j = 1,2, . . . , n
will look fair or pleasing to the eye of an experi-
The task is to minimise an objective function de-
enced designer, and these shape requirements should
fined in terms of design variables subject to given
be translated into mathematical terms in order to
constraints. The design variables for the ship lines
implement them into a fairing algorithm.
fairing problem are the predefined 2-D offset points
for the curve in consideration. An initial curve,
Formulation of ship hull lines fairing as an op-
which satisfies the required performance character-
timisation problem
istics, is assumed to be available. The objective is
Ship definition curves are generally obtained by in- to eliminate undesirable shape features in order to
specting corresponding curvature plots in an inter- produce a curve fairer than the original one. While
active fairing process. Due to the aforementioned removing the flaws of the curve (e.g., unintended in-
drawbacks of this process, researchers are in a con- flections, noise), the general shape of the curve must
stant search for alternative and objective methods. be preserved in order not to degrade specific per-
The main goal is to develop automated fairing pro- formance characteristics. This is mainly achieved
cedures to generate ship curves that have uniformly by including the constraints into the problem. The
distributed (smooth) curvature plots. structure of the optimisation problem is illustrated
Formulation of the ship lines fairing process as in Figure 1. The following must be available in order
an optimisation problem can be set as a typical en- to formulate and solve the fairing problem, which are
gineering problem. It is formed of 3 basic elements, discussed in detail in the following sections:
GEOMETRIC
CONSTRAINTS
Optimisation Procedure
Objective function
Step size
Termination criteria
PERFORMANCE
CONSTRAINTS
NO Termination
criteria
satisfied?
YES
Final Curves
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NARLI, SARIÖZ
• A set of initial offset points obtain the quantity which characterises the desirabil-
ity of the product curve under that metric. Hence,
• A fairness metric described in terms of design the required shape will obviously be the one that
variables satisfies the predefined geometric constraints while
optimising this quantity.
• Definition of constraining boundaries
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NARLI, SARIÖZ
value of the penalty parameter is chosen to have a • It seems sensible to move from b∗i in the direc-
large value (approx. 1000) to add a penalty to the tion (b∗i − bi ), since a move in this direction
objective function as soon as one or more constraints has already led to a decrease in the value of
g(X) are violated. f(X1 , X2 , . . . , Xn ). Therefore, move from b∗i
The Hooke and Jeeves method is based on 2 types to (2b∗i − bi ) and continue with a new sequence
of step-by-step searches alternating in turn: first a lo- of exploratory moves about (2b∗i − bi ).
cal search, which is a unidirectional variation of each
design variable resulting in the direction of steepest • If the lowest function value obtained during the
descent, and a pattern move, and which represents a pattern and exploratory moves of (2b∗i − bi ) is
rotation of the search direction which accelerates the less than b∗i , then a new base point b∗∗ i has
search by the aid of increasing the step widths. The been reached. In this case, return to (2b∗i − bi )
search routine can only proceed in a feasible space with all suffices increased by unity. Otherwise
because outside that space the penalty functions are abandon the pattern move from b∗i and con-
not defined. tinue with a new sequence of exploratory moves
If we consider the problem of minimising about b∗i .
f(X1 , X2 , . . . , Xn ), the general procedure, can be
described as follows: Application and evaluation of the algorithm
for actual ship lines
• Start with an arbitrarily chosen initial
base point (b1 , b2 , . . . , bn ) and step lengths The automated fairing procedure is applied to 3 typi-
(h1 , h2 , . . . , hn ) for the respective variables cal ship definition curves. These examples represent
(X1 , X2 , . . . , Xn ). offsets taken at transverse sections along the ship
length, called body sections. The small dots dis-
• The method proceeds by a sequence of ex- played in the figures represent the control points of
ploratory and pattern moves. The proce- the curves.
dure for an exploratory move about the point The first 2-D curve is from a mathematical
(b1 , b2 , . . . , bn ) is as follows: hull form, the well-known Wigley form, selected to
• Evaluate f (bi + hi ). If the move from bi to demonstrate the performance of the fairing proce-
bi + hi is a success, replace the base point bi dure (Wigley, 1934). The offset values y(x,z) of the
by bi + hi . If it is a failure, evaluate f(bi − hi ). hull surface, which is symmetrical in both fore-aft
If this move is a success, replace bi by bi − hi . and port-starboard, is defined by the following equa-
If it is another failure, retain the original base tion:
point bi .
" 2 # z 2
• Repeat the above procedure for each variable B 2x
y(x, z) = 1− 1− (4)
in turn finally arriving at a new base point af- 2 L D
ter (2n+1) function evaluations at most.
where
• If the new variable value b∗i is equal to the orig-
inal base point value bi , halve each of the step x : distance from amidships, positive forward,
lengths hi and return to the first step. The z : distance from baseline, positive downwards,
calculations terminate when the step lengths
have been reduced to some prescribed level. If L, B, D : length, breadth and depth of the hull, re-
b∗i 6= bi , make a pattern move from b∗i . spectively.
The mathematical hull form has an intrinsically
• A pattern move attempts to speed up the fair surface and hence fair waterlines, buttocks and
search by using information already acquired body sections. The mid-section curve of this hull
about f(X1 , X2 , . . . , Xn ). It is invariably fol- form is selected as the first application of this fairing
lowed by a sequence of exploratory moves, with procedure and is displayed in Figure 2a. The offset
a view to finding an improved direction of points of this section are then randomly distorted by
search in which to make another move. The adding erratically distributed undulations into the
procedure for a pattern move from b∗i is as fol- data, which is shown in Figure 2b. The offset points
lows: of the disturbed curve serve as the initial points of
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NARLI, SARIÖZ
the optimisation. After approximately 1500 itera- imately 1000 iterations. It is clear from the figure
tions the faired curve is obtained, which is displayed that the variation of curvature is more gradual in
in Figure 2c. The curvature plot is drawn along the the resulting curve. The deviation between the ge-
curve to make all the undulations and unfair regions ometric characteristics of the original and resulting
visible. It is clear from the figures that the faired curves is within acceptable design limits. Table 1a
curve is radically improved in terms of fairness. The shows the offset points of original and faired curves
percentage improvement in terms of the objective together with corresponding percentage of deviation
function is 95%, and this reduction in the objective from the original offset values of the curve. Table 1b
function value is mathematical proof of the improve- displays the number of iterations, the value of the ob-
ment in the optimised curve. The curvature plot of jective function, and the percentage of improvement
the resulting curve has an evenly distributed smooth in terms of the objective function.
shape like the original Wigley section. It should be
noted that for a fairing algorithm to be considered
a useful tool, the deviation between the original and
the faired curves should be within the acceptable de-
sign limits. For this application the difference of the
sectional area of the resulting curve from the orig-
inal Wigley mid-section is 1.2% and therefore the
difference is within the tolerated limits. A simple
algorithm using divided differences principle (Renz,
1982; Narlı and Sarıöz, 1998) can eliminate high fre-
quency errors of the distorted curve but these errors z
are deliberately left to show the effectiveness of the y
optimisation algorithm.
A bulbous bow section, shown in Figure 3, is se- Figure 2. (a) Original, (b) distorted, (c) faired Wigley
lected as a second example. The initial value of the hull mid-section (1500 iterations, 95% of im-
objective function is reduced by 45% after approx- provement in the objective function).
Table 1a. Comparison of original and optimised offset values of the bulbous bow section.
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NARLI, SARIÖZ
Table 1b. Number of iterations, the value of the objective function, and the percentage of improvement in terms of the
objective function.
No. of Objective Improvement of
Iterations Function objective function (%)
Initial Curve 0 3.671 0
.. .. ..
. . .
250 3.049 16.9
.. .. ..
. . .
500 2.580 29.7
.. .. ..
. . .
750 2.276 38
.. .. ..
. . .
Optimised Curve 952 2.024 45
y
z
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NARLI, SARIÖZ
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