49
1 INTRODUCTION
Prediction of Ship performances in calm and rough
waterisoneofthemostimportantconcernsfornaval
architects, already at the earliest design stage. From
this point of view seakeeping performanceisoneof
the most important performances in the ship hull
form optimization. It is possible to achieve
considerable improvements in terms of habit
ability,
operabilityandsurvivabilitybymeansofchangesin
hull form even when displacement and main
dimensionshavebeenfixed.
It is worth noting that for a comprehensive and
detailedshiphydrodynamicoptimizationallobjective
functionssuchasresistance,stability,seakeepingetc.
must be considered, because it is clear that
consideration of an objective function without the
otheronesgivesunrealisticandimpracticalresults.
Some researchers have considered two or three
objectivefunct
ionsforoptimizinghullformandsome
others only one objective functions. For example
Gammon (2011) uses three objective functions in his
study, Biliotti et al. (2011) and Grigoropoulos and
Chalkias(2010)utilizetwoobjectivefunctionsintheir
work and many researcher use only one objective
function(Hanetal.,2012,Zakerdoostetal.,2013,A.
Scam
ardellaandV.Piscopo,2014).
Zhang(2009and2012),Kimetal.(2009and2008)
andSahaetal.(2004)employeddifferenttypesofthe
Nonlinearlinearprogramming(NLP)asoptimization
techniques. Evolutionary Algorithm (EA) and
Artificial Neural Networks (ANN or NN) offer
effective method for conducting op
timization and
data analysis. EA techniques may be separated into
Genetic Algorithm (GAs), Evolutionary Strategies
(ESs)andEvolutionaryProgramming(EP).However
at present Genetic Algorithm (GA) and evolution
strategies (ESs) are most widely used in hull shape
modification. In this work, the term GA is used to
solving op
timization problem. Day and Doctors
(2001) studied hull form optimization using a GA
technique in which the objective was to minimize
Optimizing the Seakeeping Performance of Ship Hull
Forms Using Genetic Algorithm
H.Bagheri,H.Ghassemi&A.Dehghanian
DepartmentofOceanEngineering,AmirKabirUniversityofTechnology,Tehran,Iran
ABSTRACT:Hullformop
timizationfromahydrodynamicperformancepointofviewisanimportantaspectof
shipdesign.Thisstudypresentsacomputationalmethodtoestimatetheshipseakeepinginregularheadwave.
Intheoptimizationprocessthe Genetic Algorithm(GA)is linkedtothecomputationalmethodtoobtain an
optimumhullformbyta
kingintoaccountthedisplacementasdesignconstraint.Newhullformsareobtained
fromthewellknownS60hullandtheclassicalWigleyhulltakenasinitialhullsintheoptimizationprocessat
twoFroudenumbers(Fn=0.2andFn=0.3).Theoptimizationvariablesareacombinationofshiphulloffset
sand
maindimensions.Theobjectivefunctionoftheoptimizationprocedureisthepeakvaluesforverticalabsolute
motionatapoint0.15LBPbehindtheforwardperpendicular,inregularheadwaves.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 8
Number 1
March 2014
DOI:10.12716/1001.08.01.06
50
resistance. Jun and Kuniharu (2004) presented a
singleobjective optimization algorithm based on
genetic algorithm to improve hull form of a
catamaran.
Duetotheimportanceofseakeepingperformance,
seakeeping optimization has become a popular
researchtopicforthelastthreedecades.
Bales(1980)optimizedadestroyertypehullform,
in head seas and at various speed, on the basis of
analyticalpredictions,subsequentlyderivingbysome
regression formulas correlating relevant
performancestoformparameters,theoptimumhull.
Griogoropoulos and Loukakis (1988) developed a
numericalmethod,basedonanonlineardirectsearch
algorithm to minimize RAO peak values in head
regularwaves.Similarstudieshavebeenalsocarried
outbyHearnetal.(1991),whodevelopedaninverse
design procedure, based on the optimum hull
nonlinear direct search process. Kukner and Sariöz 
(1995) optimized the seakeeping qualities of a high
speed vessel, generating by the Lackenby method
(Lackenby, 1950), several derived
hulls having
differentformparametersasregardstheparentones.
Peacock et al.(1997) defined a mathematical model
based on a multiobjective research algorithm for
displacement monohulls. Sariöz and Sariöz (2006)
proposedanewoptimization procedure,basedona
nonlinearproblemsolvedbydirectsearchtechniques.
Campana et al.
(2009) proposed a new optimization
techniquefortheheavemotionoftheS175container
ship,adoptedbytheITTCSeakeepingCommitteeasa
benchmark test, considering two different
optimization procedures, namely a filled function
based algorithm anda Particle SwarmOptimization
method. Diez and Peri (2010) presented a new
approachfor
therobustoptimizationofabulkcarrier
conceptual design, subjected to uncertain operating
and environmental conditions, so extending the
standard deterministic formulation for design
optimization to take into account the uncertainty
relatedtodesignvariables,operatingconditionsand
computational results of the simulations. Finally
Özüm et al. (2011) investigated the
seakeeping
qualities of fast ships, systematically varying both
maindimensionsandhullformparameters.Anyway,
in almost all cases, optimization procedures were
based on the assumption that the optimum hull is
foundwhenthe vertical plane motionsandabsolute
vertical acceleration in regular head waves due to
combinedpitch,heavemotions,
isminimized.
In this study, after problem formulation and
especially the explanation of strip theory and a
particularformoftheoptimizationalgorithm(genetic
algorithm),resultsofapplicationofthismethodology
usingtwo differentcasesof theWigleyhull andthe
S60 hull are presented, and in both ones allowing
principal parameters of length, beam and draft to
changesimultaneouslywiththeoffsetofhullsurface.
Itshouldbenotedthatthecurrentdesignprocedure
is restricted to the minimization of vertical plane
motions and roll is not included for the following
reasons:
The sensitivity of roll motion to weight
distributioncharacteristicswhicharegenerallynot
availableattheearlystagesofdesign.
Difficultiesinpredictingnonlinearrolldamping.
Thefactthatexcessiverollcanalwaysbereduced
bychangingheadingtoheadorbowseas.
2 OPTIMIZATIONPROBLEMANDGA
The general mathematical form of a numerical
constrained optimization problem has been
represented here. Design variables and constraint
conditionsareusedtocharacterizetheproblem.The
role of
design variables in hydrodynamic
optimizationproblemsiscontrollingthegeometryof
the hull during optimization procedure. Constraints
are the values by which the design variables are
restrictedandmaybeseparatedintwotypes,equality
and inequality constraints. A function being
maximized or minimized by users is known as the
objective
function andthevalueofthisfunctionisa
criterion to determine the efficiency of design
optimization methodology. If in an optimization
problem only one objective function is used, the
optimizationisknown assingle objectiveand iftwo
ormoreobjectivefunctionsareused,theoptimization
isknownas
multiobjective.Thestandardformulation
of an optimization problem mathematically is as
follows:
Optimize
12
n
( ) ( ), ( ),..., ( )
m
Fx f x f x f x x


(1)
Subjectto
( ) 0 i 1,...,q
g ( ) 0 i 1,...,
i
i
hx
x
p


(2)
where
i
f
x
is the objective function, m is the
number of objective function,
q is the number of
equality constraints,
p
is the number of inequality
constraints and
1
, . . .,
n
x
xx S
 is a
solutionor individual. The set
n
S defines the
search space and the set
S defines a feasible
searchspace.Thesearchspace
S
isdefinedasann
dimensional rectangle in
n
(domains of variables
definedbytheirlowerandupperbounds):
1
i
li x ui i n

The constraints define the feasible area. This
means that if the design variables vector
x be in
agreement with all constraints

i
hx (equality
constraint) and
i
gx (inequality constraint), it
belongstothefeasiblearea.
In this study design varia bles vector include the
main parameters (length, beam, draft) and the hull
offset which are limited by the lower and upper
bounds. The ship hull displacement also is an
inequalityconstraint.
Among the class of evolutionary algorithms,
genetic
algorithm(GA)isthemostpopularalgorithm
forsolvingcontinuousoptimizationproblems,i.e.for
optimizing realvalued function
f
defined on a
subset of
n
for some dimension n . Genetic
algorithm can be applied to combinatorial problems
aswell.Geneticalgorithmisinspiredbytheevolution
51
theory(DarwinianTheoryofbiologicalevolution)by
meansofaprocessthatisknownasnaturalselection
andtheʺsurvivalofthefittestʺprinciple.Thecommon
idea behind this technique is similar to other
evolutionary algorithms: consider a population of
individuals; the environmental pressure causes
natural selection which leads to
an increase in the
fitness of the population. It is easy to see such a
process as optimization. Consider an evaluation
functiontobeminimized.Asetofcandidatesolutions
canberandomlygeneratedandtheobjectivefunction
can be used as a measure of how individuals have
performedin
theproblemdomain(anabstractfitness
measure) the lower the better. According to this
fitness, some of the better solutions are selected to
seed the next generation by applying recombination
and/or mutation operators to them. The
recombination(alsocalledcrossover)operatorisused
togeneratenewcandidatesolutions(offspring)from
existing
ones, they take two or more selected
candidates (parents) from the population pool and
exchange some parts of them to form one or more
offspring.Mutationoperatoris usedtogenerateone
offspringfromoneparentbychangingsomepartsof
the candidate solution.Applying recombination and
mutation operators causes
a set of new candidates
(theoffspring)competingbasedontheirfitnesswith
theoldcandidates(theparents)foraplaceinthenext
generation.
This procedure can be iterated until a solution
with sufficient quality (fitness) is found or a
previouslysetcomputationaltimelimitisreached.In
otherwords,
theendconditionsmustbesatisfied.The
composed application of selection and variation
operators (recombination and mutation) improves
fitness values in consecutive population. A general
flowchartofgeneticalgorithmisshowninFigure1.
Figure1.Generalflowchartofgeneticalgorithm
Genetic algorithm variables are divided into two
categories: object and genetic variables. Variables in
genetic algorithm commonly are as realvalued
vectors because this algorithm is usually used for
continuous parameters. A form of an individual in
GAisasfollows:
1
,...,
n
x
x
where
i
x
is the object variable. In object variables
mutation, each gene (biologic name of a vector)
changed whit mutation rate (genetic variable) in
rangeoftheirlowerandupperbounds.Themutation
methodologyfor
i 1 , . . . , n isasfollows:
 
11
,..., ,...,
,,
nn
ii
x
xxx
where
xx li ui


(3)
Scatter recombination is one of main type of
recombination (crossover) used in GA. This type of
crossover creates a random binary vector. So, the
genes are selected from the first parent where the
vector is a 1, and from the second one where the
vectorisa0.The
μ , λ
survivorselectionscheme
has advantages over its competitor, the
μ λ
selection scheme but the

μ λ selection scheme
is an elitist mechanism that can maintain the best
solutiontoeachgeneration(EibenandSmith,2003).
3 SEAKEEPINGCALCULATION
Thedeterminationofhydrodynamicforcesactingon
ashipcanbeformulatedasalinearboundaryvalue
problem in potential theory. Under the assumption
that motion responses are
linear, or at least can be
linearizedandareharmonic,theequationsofmotion
fortheadvancingshipinwavesmaybewritteninthe
followinggeneralform:
kj j k
L H, U η F , k,j 1,2, ,6
 (4)
where
H
represents the hull geometry, ω is the
wavefrequencyandUistheforwardspeed.Typically
theoperator
k
j
L
isoftheform
2
kj kj kj kj
MAωiBωC
kj
L
 (5)
where
M
is the generalized mass matrix, A and
B
represent the added mass and fluid damping
matricesassociatedwithforces/momentsinducedin
the
k th mode, as a consequence of motion in the
j
th mode and C is the hydrostatic restoration
matrix. The degrees of freedom,
j
, correspond to
surge, sway, heave, roll, pitch and yaw as
j
assumesthevalue1 6,respectively. Thedependence
ofthehydrodynamiccoefficientsandthehydrostatic
restoration upon the hull form shape may be
expressedas:


,,
,,
kj kj
kj kj
kj kj
AAH U
B
BH U
CCH
(6)
Thewaveexcitation
k
F isalsoafunctionofwave
heading
,, ,
kk
FFH U
(7)
52
Figure2.ComparisonofheaveandpitchRAOcoefficientformodelsoftheWigleyhull
The added mass, damping, restoring force and
waveexcitingforcetermscanbecalculatedbyusing
well established numerical procedures. In order to
reduce the computing time a linear strip theory
approach is adopted as described by Salvesen et al.
(1970). The sectional added mass and damping
coefficients are calculated by
using the wellknown
Frank CloseFit method (1967). The seakeeping
responses in head sea are generally the most
important responses for monohulls. Thus, all
calculationswerecarriedoutforverticalmotionsand
related kinematics certainly. The computed ship
responsesincludeverticalmotionandaccelerationat
bow region (at
a point 0.15LBP behind the forward
perpendicular). All the results are given for regular
headwaves.
ThecomparisonofDelftUniversityofTechnology
(DUT) Report experiment(1992) with a 3 m Wigley
hullswithlengthtobeamratioL/B=10andlengthto
draftratio L/T = 16inheadregular
wavewhit4 cm
wave height, heave and pitch RAO respectively are
shown in Figure 2. Using the numerical method
described above for computing Ship vertical motion
leads to good agreement and errors between
predictions and the experiments (white respect to
lineartheorywasemployed)liewithinabout%10for
the design Froude number (Fn=0.3). It should be
noted that according to the figure 2, the heave and
pitchRAOat
/L1.2thatthepeakvalueoccurs,have
a 180 degree phase difference. The vertical bow
motion(objectivefunction) isafunctionofthemain
dimension (length, beam and draft) and the hull
offsetsoftheshipintheoptimizationprocesswhich
mustbeminimized.
4 PROCEDUREOFTHEHULLFORM
OPTIMIZATION
Theprocedureofoptimizingashiphullforminorder
to find a hull shape with minimum bow vertical
motion is as follows. The optimization of hull form
can be performed by evaluating the hull forms that
are generated by variation operators and then
selecting the best forms of
lower vertical motion at
bowreignineachgeneration.
TheWigleyandS60hullformsareconsideredas
initialhullforms.Eachchromosome(biologicnameof
a solution) in the optimization algorithm consists of
ship offsets, length (waterline length), beam (in
waterline) and draft. Because of large number of
variables, the
genetic algorithm is a successful
technique for the hull form optimization problems
from a seakeeping point of view. The design
constraintsthatwereusedforthisstudyarethatthe
optimizer allowedno change in the total
displacement of the ship. In addition, sinkage and
trim effects are not considered
as a hydrodynamic
designconstraint.Somelimitshavebeenimposedon
theprincipaldimensionsandthehulloffsets.Inorder
to restrict the search space and to keep the optimal
hullneartheoriginaloneforcomparison,thelength,
beamanddraftarelimitedto±10percentvariationin
the principal
dimensions and the offsets points are
limitedto±3percentoftheinitialhulloffsets.Table1
representsvariationpercent of variablesusedintest
cases.
Table1.Variationpercentofvariablesusedintestcases
_______________________________________________
VariablesHulloffsetsL  B T
_______________________________________________
Variationpercent ±3±10 ±10 ±10
_______________________________________________
The Wigley model is a popular and wellknown
model in ship hydrodynamics experiments. Many
experimental and numerical results can be found in
theliteratureforthismodel.
We employed this model to compare numerical
results. The standard Wigley hull is a mathematical
displacement hull form, the geometric surface
of
whichcanbedefinedas: