hydrodynamic design study on ship bow and stern hull form...

Research ArticleHydrodynamic Design Study on Ship Bow and Stern Hull FormSynchronous Optimization Covering Whole Speeds Range

Yu Lu1 Xin Chang 1 Xunbin Yin2 and Ziying Li1

1Naval Architecture and Ocean Engineering College Dalian Maritime University Dalian 116026 China2College of Shipbuilding Engineering Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Xin Chang changxin heuoutlookcom

Received 2 February 2019 Revised 31 May 2019 Accepted 13 June 2019 Published 8 August 2019

Academic Editor Giorgio Besagni

Copyright copy 2019 Yu Lu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The main objective of this article is to describe an innovative methodology of synchronous local optimization which considers thewhole ship speed range being presented for aKRISOContainer Ship (KCS) Parametric form approaches are adopted by employing afairingB-spline curve in order to generate variants of the bowand stern of formsusing formdesignparametersmodified resulting inan optimization system based on NSGA-IIThe total resistance is calculated by the Rankine source panel method and the empiricalformula which agrees well with the corresponding experimental data and further acquires validation with the overall error of20 Accordingly the ship forepart and stern form have been optimized under conditions of the single design speed and wholespeeds range based on the considerations of generally distributed and variable operational speeds for the operating characteristicsof modern container ships synchronously The optimized result presents well-balanced drag reduction benefits which averagelyremain above 40 of ship resistance decrease Compared to the traditional optimization process which is based on a specificdesign speed the newly developed method is more practical and effective in both automation and integration

1 Introduction

Energy saving and low-carbon environmental protection arethe two inevitable trends in shipbuilding industry Mean-while the aims of ship energy saving and emission reductionhave promoted the rapid development of the research in shipdesign and optimization In recent years driven by economicinterests and environmental protection missions domesticand international liner companies adopted new measures toslow down the shiprsquos service speed the benefits by doing soare even more remarkable to container ships

The operation speed of modern container ship has beenreduced year by year [1] At the same time the operatingspeed distribution is more decentralized and uniform Dueto the influence of the slow sailing strategy the resistanceperformance of the single speed and the high speed containerships has been paid less and less attention At the sametime the container ships designed for high speed have beensubjected to great impact because they cannot play a goodresistance performance in themiddle and low speed segment

The present work presents an appropriate resistanceoptimization method of the bow and stern which considerswhole speeds range combining the parametric hull formtransformation method a RSM hydrodynamic solver andmodern optimization techniques It should be noted that allthe codes are in-house developed

The methodology for hull form transformation based onthe parametric ship-hull model is presented in Section 2 In1950 Lackenby made an attempt to build the ship parametricmodel [2] He obtained hull parametric variants throughmodifying the prismatic coefficient the center of buoyancyand the extent and position of parallel mid-body Lackenbytransform is applied to optimization of ship design byHarries [3] who completed the ship-hull transformation bycontrolling the transverse section area curve containing thedesign parameters see eg Harries andAbt [4]Nowacki andKaklis [5] Harries and Nowacki [6] Lee [7] Maisonneuveet al [8] Saha Suzuki and Kai [9 10] Kim [11] Perez andSuarez [12ndash14] Abt and Harries [15] Harries [16] Zhang andZhu [17] Tahara Peri Campana and Stern [18] and AI

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 2356369 19 pageshttpsdoiorg10115520192356369

2 Mathematical Problems in Engineering

Ginnis and PD Kaklis et al [19ndash22] In this work the methodis concerned for the modification and reconstruction of thehull bow and stern form using a fairing B-Spline parametriccurve

Section 3 is devoted to the prediction method of thetotal resistance performance which has been defined asthe optimization objective function in this work The ITTC1957 model-ship correlation formula and the Rankine sourcepanel method are applied to calculate the frictional resis-tance and wave-making resistance respectively The Rankinesources method is based on the potential-flow withoutconsidering viscosity Nevertheless for a sophisticated ship-hull optimization system it is more simple and convenientand has been proved to be accurate reliable and efficient soit has been investigated in various studies (Abt and Harries etal [23] Lu Chang and Hu [24] Lowe and Steel [25] SuzukiKai and Kashiwabara [26] Zhang Kun and Ji [27] Zhang[28] Chi Huang and Noblesse [29] Choi Park and Choi[30] and Chi Huang and Kim [31])

Section 4 presents the Fast Elitist Nondominated SortingGenetic Algorithm II (NSGA-II) used to search for theminimum ship total resistance for the ship-hull optimizationNSGA-II is a kind of modified GAs with low computationalrequirements In 1975 Holland [32] a professor of MichiganState University firstly proposed Genetic Algorithm (GA)which is a computational model simulating evolutionary pro-cess in nature Deb et al improved GA and proposed NSGA-II [33 34] with Fast nondominated sorting method proposedto reduce the computational difficulty the evaluation andcomparison of congestion put forward to help to make quickjudgment for the same level of individual behavior and elitiststrategy introduced to expand the space sampling

Finally two optimization cases for KCS [35ndash37] are setup and presented in Section 5 The first case deals witha ship-hull optimization at design speed while the secondcase involves a total resistance minimization problem againstmultiple objective functions covering all speed range

2 Hull Form Transformation Method

21 Definition of Fairing B-Spline Parametric Curve In thisarticle the hull form transformation is realized by the fairingB-spline parametric curve transformation The deformationof B-spline parametric curves is realized by adjusting theinput characteristic parameters and then the deformationof the characteristic parameter curve is mapped to thegeometric deformation of the hull

The fairing B-spline parametric curve is formed byapplying the constraint conditions on a B-spline curve Thespatial spline curves parameterized by 119906 are expressed by119903(119906)

119903 (119906) = (119909 (119906) 119910 (119906) 119911 (119906)) (1)

The equation corresponding to the curve with mth fairnesscan be expressed as

119871119898 = int1

0(119863119898119903 (119906))2 119908ℎ119890119903119890 119863119898 = 119889119898

119889119906119898 (2)

Some control constraints should be satisfied in the fairingprocess

(1) Distance Constraint Condition The Euler formula is usedto express the distance between the control point 119875119894 andthe spline curve 119903(119905) The distance shall meet the followingcondition

119860 =119899

sum119894=0

[119908119894 (119903 (119906119894) minus 119875119894)]2 le 120576 (3)

where 119908119894 is distance weight and 120576 is a tolerance greater thanzero

(2) Curve Ends Constraint Condition The tangent slope andcurvature of the starting point or the end point of the splinecurve shall meet the following conditions

1198791 = 1198631119903 (119906119894) minus119876119894 = 0 (4)

1198792 = 1198632119903 (119906119894) minus119870119894 = 0 (5)

where 119894 = 0 or 119894 = N If 119894=0 it represents the starting point andif 119894=n it represents the end point

(3) Area Constraint Condition The area 119878 under the splinecurves should meet the condition of the fixed value that is

119878 = 1198780 (6)

where 1198780 is a given areaAt the same time if necessary other conditions can be

taken into account in the same waymentioned above to com-plete the corresponding constraints Finally the constrainedoptimization problem can be transformed into the extremevalue problem of the unconstrained objective function I bythe linear superposition of these constraints

119868 = 119871119898 + 1205821198601015840 + 12058311198791 + 12058321198792 + ]119865119908ℎ119890119903119890 1198601015840 = 119860 + 1198892

(7)

where 120582 1205831 1205832 and ] are all Lagrange operators and d2 is aslack variable

22 Hull Deformation Method by Mapping This sectionshows how to realize the deformation of the transversesection lines by the fairing B-spline parametric curves Thismethod is also applicable to other lines of the hull Thehull geometry model is stored in the form of offsets thatis the hull is discretized into a finite number of transversesection lines On the YZ plane the mapping formula of thecoordinate change from the point on the B-spline curve tothe point on the transverse section line is expressed as

119909119906 = 119909 + 119903120575119909119906 (119909) (8)

119911119906 = 119911 + 119903120575119911119906 (119911) (9)

where 120575119910119906 and 120575119911119906 are the maximum displacement variationsof fairing B-spline curve in the Y axis and Z axis respectively

Mathematical Problems in Engineering 3

(y z) and (119910119906 119911119906) are the coordinates of any observationpoint before and after the hull deformation respectively119903120575119910119906(119910) and 119903120575119911119908(119911) represent the displacement variationof the point on the fairing B-spline curve with the sameoriginal coordinates as the observation point when the inputparameters are 120575119910119906 and 120575z

Thus the displacement variations of all points on the B-spline curve can be mapped to the ship hull curve with thesame original coordinates For the B-spline parametric curveitself is highly fairing while beingmapped to the hull the hullgeometry deformation must also be fairing

In the process of hull form transformation one or moreof the fairing B-spline curves can be applied as deformationweights to avoid the geometric distortion Similarly assumingthat 120575119910119908 and 120575119911119908 are the maximum variations of the curvesin the Y direction and the Z direction since they are used asweight functions 120575119910119908 and 120575119911119908 should meet 0 lt 120575119910119908 le 1 and0 lt 120575119911119908 le 1 respectively Thus the coordinate displacementformula of the points on each transverse section line can beexpressed as

119909119906 = 119909 + 119903120575119909119906 (119909) sdot 119903120575119909119908 (119909) (10)

119911119906 = 119911 + 119903120575119911119906 (119911) sdot 119903120575119911119908 (119911) (11)

where 119903120575119910119908(119910) and 119903120575119911119908(119911) represent the weight value of thedisplacement variation of the point on the fairing B-splinecurve with the same original coordinates as the correspond-ing point on the hull surface when the input parameters are120575119910119908 and 120575119911119908

3 Resistance Prediction Method

31 Total Resistance Calculation Formula In this article thehull form transformation is realized by the fairing B-splineparametric curve transformation The deformation of B-spline parametric curves is realized by adjusting the inputcharacteristic parameters and then the deformation of thecharacteristic parameter curve is mapped to the geometricdeformation of the hull

The total resistance can be calculated by the empiricalformula as follows

119862119905 = 119862119908 + (1 + 119896) 1198621198910 (12)

where119862119905 is the total resistance coefficient and 119862119908 is the wave-making resistance coefficient which can be obtained by theRankine source method 119896 is form factor 1198621198910 is the frictionalresistance coefficient which can be calculated by the ITTC1957 model-ship correlation line formula

1198621198910 = 0075(lg119877119890 minus 2)2 (13)

where 119877119890 is the Reynolds number and 119877119890 = 119881119871] 119881 isthe ship speed 119871 is the length of the waterline and V is thecoefficient of the kinematic viscosity of the fluid

32 Rankine Source Method Taking the o-xyz as the Carte-sian coordinate system fixed on the ship the X-axis is in the

direction and the Z-axis is perpendicular to the surface ofthe water The Y-axis is determined by the right hand ruleAssuming that the ship speed is 119880 because of the motion ofthe hull the velocity potential Φ and perturbation potential120593 with the free surface effect constitute the total velocitypotential 120601 of flow field around the ship whose expressionis 120601 = Φ + 120593 In the potential flow the ideal fluid isnonviscous incompressible and nonrotating Then in thefluid computational domain the total velocity potential 120593subjects to the Laplace equation and the following boundaryconditions

nabla2 (Φ + 120593) = 0 (14)

(1) e Hull Boundary ConditionThis is

nabla (Φ + 120593) sdot 997888rarr119899 = 0 (15)

where997888rarr119899 is the unit outward normal vector of the hull surfaceand is represented by 997888rarr119899= (nx ny and nz)

(2) e Free Surface ConditionThis is

12 120601119909 (nabla120601 sdot nabla120601)119909 + 1

2 120601119910 (nabla120601 sdot nabla120601)119910 + 119892120601119911 = 0 (16)

where the subscripts denote the partial derivatives and 119892 isthe gravitational acceleration

(3) e Radiation ConditionThis is

nabla120593 997888rarr (119880 0 0)119886119904 1199092 + 1199102 + 1199112 997888rarr infin

(17)

Employ the Rankine source panel method by the iterationprocedure to solve the above boundary conditions Once thevelocity potential120601 is resolved the free surfacewave elevation120578 can be obtained In addition through the instrumentalityof the Bernoulli equation the pressure coefficient 119862119901 at eachpanel can be defined as

119862119901 = 1 minus nabla120601 sdot nabla1206011198802 minus 2 119911

1198651198992 (18)

where 119865119899 is the Froude number with the expression 119865119899 =119880radic119892119871119882119871 Hence the wave-making resistance coefficient 119862119908can be solved by integral calculus of the pressure throughoutthe wetted hull surface S and formulated as

119862119908 = minus 1119878 int119878

119862119901119899119909119889119904 (19)

When each resistance coefficient component is obtained thetotal calm water resistance Rt could then be determined by 119877119905= 05120588U2SCt where 120588 is the density of fluid

33 Validation of the Total Resistance Calculation Method forKCS KCS (KRISO Container Ship) is a modern containership designed byKorea Research Institute of Ships andOceanEngineering It has been selected as the standard form of


Figure 1 The body plan of the KCS

Figure 2 The side view of the KCS

Table 1 Principal characteristics of the KCS model

Principal dimensions Symbol Unit ValueScale 120582 [-] 315994Length betweenperpendiculars 119871119875119875 [m] 72786

Waterline length 119871119882119871 [m] 73568Beam 119861 [m] 10190Moulded depth 119863 [m] 05696Draught 119879 [m] 03418Displacement nabla [m3] 16493Wetted surface area S [m2] 95579Block coefficient 119862119861 [-] 06508Midship section coefficient 119862119872 [-] 09849Prismatic coefficient 119862119875 [-] 06608Design speed 119865119899 [-] 026

CFD results validation by Goteborg ship CFD symposiumand Tokyo ship CFD symposium with a wealth of open testdata and a large number of numerical results The principaldimensions are listed in Table 1 The body plan the side viewand the mesh generation are shown in Figures 1 2 and 3respectively There are 1132 hull surface meshes and 2485 freesurface meshes

According to the calculation method of the ship totalresistance mentioned above five speed points are selected forthe hydrodynamic resistance performance evaluation and thecorresponding calculation conditions are consistent with theshipmodel test data published in the ship hydrodynamicCFDsymposium Comparison of the total resistance coefficientbetween the calculated results and the open test results isshown in Figure 4 It can be seen from the figure that thenumerical results are in good agreement with the open testresults The overall error is controlled within 2 whichconfirms the validity of the calculation method

Comparisons of wave profile and longitudinal wave cutsprofiles between the calculation results and test results areshown in Figures 5 and 6 respectively which also showthat the calculated results are in good agreement with theexperimental results The wave pattern calculated is shownin Figure 7

The comparative analysis shows that the empirical for-mula combined with the Rankine source method based onthe potential flow theory to predict the resistance perfor-mance of ship has a high reliability which can meet therequirements of ship hull optimization well

4 Optimization Strategy

The optimization strategy is a critical part of the resistanceoptimization design methodology for bow and stern line Alarge number of studies related to the optimization strategieshave been explored and implemented for the ship designIn this study the NSGA-II (Nondominated Sorting GeneticAlgorithm II) is used as the resistance optimization algorithmfor searching for the optimal hull form with a minimal shiptotal resistance considering all speed range

41 Fast Elitist Nondominated Sorting Genetic Algorithm

(1) e Fast Nondominated Sorting Method The fast non-dominated sorting method is implemented Each individual119894 is set with two attributes 119878i and 119899i The former is the setof solution individuals dominated by the individual 119894 thelatter is the number of solution individuals that dominatethe individual i At first all individuals in the populationare found in terms of 119899i =0 and put into the current setF1 Then for each individual j in the current set F1 everyindividuals set 119878j dominated by individual j is examinedThe 119899k of each individual 119896 belonging to individuals set 119878j isdecremented by one If 119899kminus1 = 0 then the individual k is thenondominant individual within 119878j and is put into another setH Finally the same nondominated rank irank is assigned to


Figure 3 The mesh generation of the hull and free surface

20

40

60

014 018 022 026 03

ExpCal(Ori)

Ｎtimes10

3

Ｈ

Figure 4 Comparison of total resistance of the KCS between experiment and calculation

0

001

002

0 05 1xL

Fn=026

ExpCal

minus002

minus001

００

Figure 5 Comparison of wave profile of the KCS between experiment and calculation at Fn = 026

all the individuals within 1198651 and the same process continuesuntil all individuals are graded

(2) Density Estimation and the Crowded Comparison Oper-ator In order to ensure the diversity of the population inthe process of optimization calculation the technology ofdensity estimation is employed representing the densityof any individual 119894 as crowding distance 119894d that is thesmallest cube containing only the individual 119894 When 119894d issmaller it means that the feasible solutions are concentratedaround an individual 119894 For maintaining the diversity ofthe population an operator for density estimation and thecrowded comparison is needed to ensure that the algorithm

can converge to a uniform distribution of Pareto-optimalfront After sorting and crowded comparison any individual119894 in the population has two attributes nondomination rank119894rank and crowding distance 119894119889 For any two individuals i j ifthe conditions 119894rank lt 119895rank or 119894rank =119895rank but 119894119889 lt 119895119889 meetthen the individual 119894 is preferred(3) Algorithm Procedure In this study the step-by-stepprocedure shows that NSGA-II algorithm is simple andstraightforward A random parent population P0 is createdinitially Then a combined population 119877119905 = 119875119905 cup 119876119905 is formedin which the tth is the generation of the proposed algorithmtherefore the population 119877119905 will be of size 2NThe population


000

001

002

0 025 05 075 1 125

Wav

e Ele

vatio

nL

xL

yL=01024

ExpCal

minus002minus025

minus001

(a)

000

001

002

000 025 050 075 100 125

Wav

e Ele

vatio

nL

xL

yL=01509

ExpCal

minus002minus025

minus001

(b)

000

001

002

000 025 050 075 100 125

Wav

e Ele

vatio

nL

xL

yL=030

ExpCal

minus025minus002

minus001

(c)

000

001

002

0 025 05 075 1 125W

ave E

leva

tion

L

xL

yL=040

minus025minus002

minus001

ExpCal

(d)

Figure 6 Longitudinal wave cuts profiles of the Series 60 ship at different values of yL under the condition of Fn = 026 (a) yL = 01024 (b)yL = 01509 (c) yL = 03 (d) yL = 04

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

0

3

6

9

minus02

minus05

minus04

minus6

minus3

minus03

minus02

minus01

times10minus3

Figure 7 Wave pattern of the KCS at Fn = 026

119877119905 is sorted according to nondomination The new parentpopulation 119875119905+1 is formed by adding solutions from the firstfront till the size exceeds N Thereafter the solutions of thelast accepted front are sorted according to ⩾n and the first Npoints are picked This population of size N is now used forselection crossover and mutation to create a new population119876119905+1 of size N It is noted that we use a binary tournamentselection operator but the selection criterion is now based on

the niched comparison operator ⩾n The NSGA-II algorithmflowchart is shown in Figure 8

42 Optimization Design Flow of Hull Lines The implemen-tation of the hydrodynamic optimization system required theuse of three main components presented above namely thehull form transformation the resistance prediction by Rank-ine source method and the optimization strategy Through


Genetic operation(Selection Crossover and

Mutation)

isgenltmaxgen

Exit

Start

Fast nondominated sorting

Initialize all parent population(gen=0)


Mutation)

gen=gen+1Fast nondominated sorting

Combine parent and child population

Crowded comparison

Output new offspring population

Yes

No

Elitism strategy(Combine parent and child)

Figure 8 The NSGA-II algorithm flowchart

the analysis and establishment of hull form optimizationmodel proposed above the optimization design process ofhull lines is shown in Figure 9

5 KCS Ship Optimization

51 Optimization Based on Single Design Speed

(1) Definition of Optimization Problem for Single Design Speed

(a) Objective FunctionThe total resistance at the design speedis used as the optimization goal

119865 = 1198621199051198621199050 (20)

where1198621199050 is the total resistance coefficient of original hull and119862119905 is the total resistance coefficient of the feasible solution inthe optimization process

(b) Design Variables Because the KCS ship has a largerbulbous bow aswell as a complex shape of stern surface whenit is sailing the first shear wave and the tail wave will interfere

with each other after the ship The interference may bebeneficial or harmful to wave making resistance Thereforethe bow and the stern are selected as the optimization designarea as shown in Figure 10

A total of 7 variables are set for the bow area that is1198891199091 1198891199101 1198891199102 1198891199103 1198891199104 1198891199111 and 1198891199112 which are mainlyused to control the expansion changes of bulbous bow inthe direction of X axis the harmomegathus changes in thedirection of Y axis the translational changes of the bow inthe direction of Z axis and the precursor smooth transitionbetween bulbous bow and hull A total of 3 variables areset for the stern area which are 1198891199113 1198891199114 and 1198891199115 mainlyused to control the translational changes of the stern in thedirection of Z axis the sag at the upper end of the stern andthe uplift of the lower end of the stern The different typesof transformation for the KCS ship in offsets circumventedwith the above 7 variables are shown in Figures 11ndash17 whereoriginal hull lines are marked with red and the transformedhull lines are marked with blue

(c) Constraints Conditions The constraint of the change ofdisplacement is |nabla nabla0minus1| lt 1 the constraint of the changeof wetted surface is |SS0minus1| lt 1The constraint conditions


Parent ship hull

Hull transform module

Hydrodynamic performance prediction module

Optimization algorithm module

Hull form parameterized

Simulation (objective function)Design speed and whole speed

range

Optimization algorithm

Condition satisfied Optimized solution

No Yes

Constraint conditions objective functions

Design parameters (variants)

Hull form

Numerical modelResistance calculation

Design variants

Constraint conditions

Parametric transform

Figure 9 The process chart of hull lines optimization

Bow area Stern area

Figure 10 The transformation regions of the KCS forebody and aftbody respectively

on the 10 design variables that control the translation of thehull are listed as follows

minus20 le 1198891199091 le 20minus25 le 1198891199101 le 15160 le 1198891199102 ( ∘) le 180

minus02 le 1198891199103 le 12minus10 le 1198891199104 le 02minus15 le 1198891199111 le 15minus05 le 1198891199112 le 25minus03 le 1198891199113 le 06minus06 le 1198891199114 le 03minus03 le 1198891199115 le 08

(21)

(2) Optimization Results for Single Design SpeedAfter settingthe population size M=20 the evolution generations N=20

Table 2 The specific presentations of OptS01

1198891199091 16 1198891199111 08F 9247 1198891199101 -20 1198891199112 -041003816100381610038161003816nabla nabla0 minus 11003816100381610038161003816 074 1198891199102 174 1198891199113 01510038161003816100381610038161198781198780 minus 11003816100381610038161003816 093 1198891199103 001 1198891199114 -048

1198891199104 -07 1198891199115 047

the crossover probability Pc=08 and the mutation proba-bility Pm=03 the optimization based on the design speedis carried out The convergence courses for the objectivefunction and various design variables are shown in Figures18 and 19 respectively

As shown in Figures 18 and 19 the objective functionand design variables are convergent in the last iteration Thefinal convergence solution is marked by the red boundaryline denoted by OptS01The design variables and constraintsof the optimal one are listed in Table 2 Comparison of thetransverse section lines between the optimal hull and original


Figure 11 X-direction transformations of forebody offsets compared to original hull form

Figure 12 Y-direction transformations of forebody offsets com-pared to original hull form

Figure 13 Z-direction transformations of forebody offsets com-pared to original hull form

Figure 14 Z-direction transformations of aftbody offsets comparedto original hull form

Figure 15 Upper concave transformations of aftbody offsets com-pared to original hull form

Figure 16 Lower convex transformations of aftbody offsets com-pared to original hull form

hull is shown in Figure 20 and comparison of surface wavesat design speed of Fn=026 is shown in Figure 21

As shown in Figure 21 the first peak wave movesbackward in the vicinity of the bow and the tail peak wavemaximum amplitudes of the optimized hull are obviouslysmaller than those of the original hull

52 Optimization Based on Whole Speed Range

(1) Definition of Optimization Problem forWhole SpeedRangeIn the optimization based on the whole speed range thedesign area and the hull deformation are consistent with theprevious text Therefore the design variables and functionalconstraints (drainage volume and wet surface area) willremain unchanged The target condition is changed to 5


Figure 17 Fair transition transformations of forebody offsets compared to original hull form

9000

9500

10000

10500

11000

0 50 100 150 200 250 300 350 400

Obj

ectiv

e fun

ctio

n (F

)

Number of variants

Figure 18 The convergence course for the objective function

typical speed points covering the whole speed range of Fn =015 sim 030 which is the normal operation condition area ofthe container ship Therefore the total resistance coefficientsat the 5 speed points are all used as subobjective functionswhich are expressed as

1198651 = 119862119905111986211990501

1198652 = 119862119905211986211990502

1198653 = 119862119905311986211990503

1198654 = 119862119905411986211990504

1198655 = 119862119905511986211990505

(22)

where Ct01 Ct02 Ct03 Ct04 and Ct05 are the total resistancecoefficients of KCS at the speeds of Fn=0152 Fn=0195Fn=0227 Fn=026 and Fn=0282 respectively Ct1 Ct2Ct3 Ct4 and Ct5 are the total resistance coefficients of thecorresponding speed points of feasible scheme in the processof optimization F4 is the subobjective function for designspeed of Fn=026

(2) Optimization Results for Whole Speed RangeAfter settingthe population size M=20 the evolution generations N=20the crossover probability Pc=08 and the mutation probabil-ity Pm=03 the optimization based on the whole speed range

is carried out The solution set of each subobjective functionis shown in Figures 22ndash25

For the optimization objective function of KCS at thedesign speed point is the subobjective function F4 only thesolution sets of the other subobjective functions relative tothe design speed objective function F4 are showed in Figures22ndash25 Because in the absence of special requirements orrestrictions the optimal solution at the design speed pointis still the main object the optimal solution of the multi-objective optimization can be obtained from the ldquoexclusiverdquoPareto solutions of the subobjective functions relative tothe subobjective function F4 In Figures 22ndash25 the feasiblesolutions are shown together with the infeasible solutionsbeyond the constraints In order to select the optimal solutionfrom the solution set conveniently the feasible solutionswill be marked by the blue hollow ldquoIrdquo and the infeasiblesolutions will be marked by the purple hollow ldquordquo The twooptimal solutions Opt01 and Opt02 are obtained from thesolution set which are marked by red solid ldquo◼rdquo At the sametime the convergence courses for various design variables ofoptimization are shown in Figure 26

From the Pareto solution set in Figures 22ndash25 it can beseen that there are two final optimal solutions at the speedsof Fn=0152 Fn=0195 Fn=0227 Fn=026 and Fn=0282respectively denoted by Opt01 and Opt02 whose objectivefunction corresponding design variables and constraints arelisted in Table 3

Comparison of the transverse section lines between theoptimal schemes Opt01 and Opt02 and the original hullis shown in Figure 27 It can be seen that compared tothe original hull the bulbous bows of Opt01 and Opt02


000

100

200

300

0 100 200 300 400Number of variants

minus300

minus200

minus100

Des

ign

varia

bled

x1

(a)

050

150

250


minus350

minus250

minus150

minus050

Des

ign

varia

bled

y1

(b)

155

160

165

170

175

180

185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

200


minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 19 Continued


010

030

050


minus050

minus070

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)

Figure 19 The convergence course of single design speed optimization for various design variables (a) dx1 (b) dy1 (c) dy2 (d) dy3 (e) dy4(f) dz1 (g) dz2 (h) dz3 (i) dz4 and (j) dz5

Figure 20 Comparison of the optimized hull form (blue) and original hull form (red)

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

01

02

03

04

05

0

3

6

9Optimized Hull

Original Hullminus05 minus6

minus3

minus04

minus03

minus02

minus02

minus01

times10minus3

Figure 21 Comparison of wave patterns between the optimized hull and original hull at Fn = 026


9300

9500

9700

9900

10100

10300

10500

10700

9200 9400 9600 9800 10000 10200 10400O

bjec

tive f

unct

ionF4

Objective function F1

Figure 22 The solution set of objective functions F1 and F4

9300

9500

9700

9900

10100

10300

10500

10700

9200 9400 9600 9800 10000 10200 10400

Obj

ectiv

e fun

ctio

nF4



9300

9500

9700

9900

10100

10300

10500

10700

9400 9600 9800 10000 10200 10400 10600

Obj

ectiv

e fun

ctio

nF4


Figure 24The solution set of objective functions F3 and F4

9300

9500

9700

9900

10100

10300

10500

10700

9700 9900 10100 10300 10500 10700

Obj

ectiv

e fun

ctio

nF4




000

100

200

300


Des

ign

varia

bled

x1

minus300

minus200

minus100

(a)

050

150

250


minus350

minus250

minus150

minus050

Des

ign

varia

bled

y1

(b)

155

160

165

170

175

180

185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

200


minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 26 Continued


010

030

050


minus070

minus050

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)

Figure 26 The convergence course of whole speed range optimization for various design variables (a) dx1 (b) dy1 (c) dy2 (d) dy3 (e) dy4(f) dz1 (g) dz2 (h) dz3 (i) dz4 and (j) dz5

(a) (b)

Figure 27 Comparison of the hull forms (a) optimized hull form Opt01 (blue) and original hull form (red) (b) optimized hull form Opt02(blue) and original hull form (red)

stretch forward and fullness is reduced The upper end ofthe transition region of the hull is gradually recessed whilethe lower end connecting with bulbous bow is graduallyprotruding outwards At the same time the aft ends of Opt01and Opt02 translate along the Z axis positive and the overallshape becomes more slender The total resistance of theoptimized hulls based on different optimization algorithmsand original hull at various Froude numbers are calculatedby the method mentioned above and listed in Table 4

The total resistance of the optimized hulls based ondifferent optimization algorithms is all less than that oforiginal hull at various Froude numbers smaller than 026The resistance reduction effect of the optimal scheme OptS01based on design speed is better than that of the two optimalschemes Opt01and Opt02 based on all speed range But atthe nondesign speed points the resistance reduction effectsof OptS01 are obviously worse than those of Opt01 andOpt02and as the speed decreases the situation is more obvious

The wave contours are compared for the optimal schemeOpt01 and original hulls for multiple Froude numbers onthe left row in Figure 28 Correspondingly on the right

row in Figure 28 the wave contours are compared for theoptimal scheme Opt02 and original hulls for multiple Froudenumbers

In Figure 28 it can be seen that when the Froudenumber is small the first peak wave maximum amplitudesof the optimized hull are obviously smaller than those ofthe original hull With the increase of Froude number thecontrast difference is gradually reduced When Fn=026 andmore the first peak wave moves backward in the vicinity ofthe bow At the same time the tail wave amplitudes of theoptimized hull form are obviously smaller than those of theoriginal hull under multiple Froude numbers The optimalschemes Opt01 and Opt02 have similar law of comparisonand the difference is not significant

In summary despite the comparison of the total dragcoefficient the comparison of the surface wave shows thatthe resistance performance of the two optimization schemesis better than that of the original hull At the same time it isverified that the resistance optimization design methodologyfor bow and stern line considering whole speeds range isfeasible and reliable


(a1)

(b1) (b2)

(c1) (c2)

(d1) (d2)

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3

Original Hull

Opt01

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt01

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt01

(a2)

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt01

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3Opt02

Original Hull

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt02

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt02

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt02

minus02

minus02

minus02

minus02

minus02minus02

minus02 minus02

minus2

minus2

minus3minus3

minus3

minus6

minus3

minus6

minus1

minus2

minus2

minus1

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3

times10minus3times10minus3

times10minus3 times10minus3

Figure 28 Continued


(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01


Figure 28 Comparison of wave patterns at various Froude numbers (a1) the optimal scheme Opt01 and original hull at Fn = 0152 (a2) theoptimal schemeOpt02 and original hull at Fn = 0152 (b1) the optimal schemeOpt01 and original hull at Fn = 0195 (b2) the optimal schemeOpt02 and original hull at Fn = 0195 (c1) the optimal scheme Opt01 and original hull at Fn = 0227 (c2) the optimal scheme Opt02 andoriginal hull at Fn = 0227 (d1) the optimal scheme Opt01 and original hull at Fn = 026 (d2) the optimal scheme Opt02 and original hull atFn = 026 (e1) the optimal scheme Opt01 and original hull at Fn = 0282 (e2) the optimal scheme Opt02 and original hull at Fn = 0282

Table 3 The specific presentations of the optimums Opt01 (left) and Opt02 (right)

F1 9597 F1 9377F2 9493 dx1 13 dz1 02 F2 9511 dx1 17 dz1 03F3 9523 dy1 -18 dz2 02 F3 9514 dy1 -21 dz2 01F4 9553 dy2 171 dz3 -024 F4 9579 dy2 172 dz3 -03F5 9952 dy3 064 dz4 -027 F5 9957 dy3 0535 dz4 -0361003816100381610038161003816nabla nabla0 minus 11003816100381610038161003816 014 dy4 -049 dz5 0397 1003816100381610038161003816nabla nabla0 minus 11003816100381610038161003816 012 dy4 -058 dz5 03971003816100381610038161003816SS0 minus 11003816100381610038161003816 085 1003816100381610038161003816SS0 minus 11003816100381610038161003816 078

Table 4 Comparison of total resistance between the optimized hulls based on different optimization algorithms and original hull at variousFroude numbers

KCS Opt01 Opt02 OptS01Fn Ct0(times103) Ct(times103) Reduction Ct(times103) Reduction Ct(times103) Reduction0152 3697 3548 403 3466 623 3598 2660195 3549 3369 507 3376 489 3377 3850227 3487 3321 477 3318 486 3338 3280260 3653 3490 447 3499 421 3378 7530282 4489 4467 048 4469 043 4499 033

6 Conclusions

A resistance optimization design methodology for bow andstern line considering all range of speeds has been imple-mented and applied Firstly the partially parametric mappingdeformation method is presented via the features parametriccurve Accordingly the deformation theory formula has beenput forward for launching the three-dimensional partialparametric hull deformation and geometry smoothing tran-sition Secondly the method for evaluating the ship totalresistance is proposed by combining the ITTC 1957 model-ship correlation formula with the potential flow Rankinesource panel method By this method the total resistanceof KCS container ship is calculated which agrees well withthe corresponding experimental data and further acquiresvalidation with the overall error of 20

Accordingly the ship bow and stern of KCS have beenoptimized under conditions of the single design speed Theoptimization results show a decrease of 70 in the totalresistance at the design speed of Fn=026 but the resistancereduction benefit is weakened at other speed points Forexample at low speed of Fn=0152 the drag reduction benefitis reduced to about 2 and at high speed of Fn=0282 theresistance is slightly increased The reason of drag reductionis analyzed through the comparison of wave profiles for theoriginal and optimized hulls

Then the ship bow and stern of KCS have been optimizedunder conditions of whole speeds range and two optimalschemes are obtained The optimization results are compre-hensively analyzed by comparing the total resistance andthe wave profiles between the original and optimized hullsThrough the analysis it can be seen that when the Froude


number is in 0152-026 the drag reduction benefits of thetwo optimization schemes at each speed point all averagelyremain above 40 of ship resistance decrease some as highas 6 At high speed of Fn=0282 the drag reduction benefithas some drop

Overall the drag reduction benefits of the optimalscheme based on whole speed range are not as good as thosebased on the design speed but more balanced Especially formodern container ships its operating condition is often notfixed In order to save energy the measures of reducing speedare often adopted and the ships rarely sail at design speedTherefore while there is no special requirement or definitelimit the resistance optimization design methodology con-sidering whole speeds range present in this paper can providesome guidance in early stages of ship design

In conclusion the present study shows an effectiveapproach for the resistance optimization ofmodern containerships design which considers whole speed range and offersconstructive assistance for designers who are attempting toobtain superior resistance performance through optimizeddesigns

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

The authors thank the reviewers for their comments andsuggestions in improving the quality of the article Thiswork was funded by the National Natural Science Foun-dation of China (Grant No 51809029) the PhD ScientificResearch Fund of theNatural Science Foundation of LiaoningProvince (Acceptance No 1552440867180) the Polar Ship-ping and Safety Research Institute of China DMU (GrantNo 3132019306) and the Fundamental Research Funds forthe Central Universities under Grant Nos 3132019113 and3132018207

References

[1] C Banks O Turan A Incecik et al ldquoUnderstanding shipoperating profiles with an aim to improve energy efficientship operationsrdquo in Proceedings of the Low Carbon ShippingConference pp 9-10 London UK 2013

[2] H Lackenby ldquoOn the systematic geometrical variation of shipformrdquo Trans INA vol 92 pp 289ndash316 1950

[3] S Harries Parametric design and hydrodynamic optimizationof ship hull forms [PhDesis] Institut fur Schiffs-und Meer-estechnik Technische Universitat Berlin Germany 1998

[4] S Harries and C Abt ldquoParametric curve design applyingfairness criteriardquo in Proceedings of the International Workshopon Creating Fair and Shape-Preserving Curves and Surfaces1998

[5] HNowacki andEDHorst ldquoCreating fair and shape preservingcurves and surfacesrdquo in Proceedings of the International Work-shop Organized by the EU Network Fairshape Kleinmachnow1997

[6] S Harries and H Nowacki ldquoForm parameter approach tothe design of fair hull shapesrdquo in Proceedings of the 10thInternational Conference on Computer Applications ICCASMIT Cambridge MA USA 1999

[7] Y S Lee Trends validation of CFD predictions for ship designpurpose [PhD thesis] Institut fur Schiffs-und MeerestechnikTechnische Universitat Berlin Germany 2003

[8] J J Maisonneuve S Harries J Marzi H C Raven U Vivianiand H Piippo ldquoToward optimal design of ship hull shapesrdquo inProceedings of the 8th international marine design conferenceIMDC03 pp 31ndash42 Athens 2003

[9] G K Saha K Suzuki andH Kai ldquoHydrodynamic optimizationof ship hull forms in shallow waterrdquo Journal of Marine Scienceand Technology vol 9 no 2 pp 51ndash62 2004

[10] G K Saha K Suzuki andH Kai ldquoHydrodynamic optimizationof a catamaran hull with large bow and stern bulbs installed onthe center plane of the catamaranrdquo Journal of Marine Scienceand Technology vol 10 no 1 pp 32ndash40 2005

[11] H C Kim Parametric design of ship hull forms with a complexmultiple domain surface topology [PhD thesis] University ofBerlin Germany 2004

[12] F Perez and J A Clemente ldquoConstrained design of simple shiphulls with B-spline surfacesrdquo Computer-Aided Design vol 43no 12 pp 1829ndash1840 2011

[13] F Perez J A Suarez J A Clemente and A Souto ldquoGeometricmodelling of bulbous bows with the use of non-uniform ratio-nal B-spline surfacesrdquo Journal ofMarine Science andTechnologyvol 12 no 2 pp 83ndash94 2007

[14] F Perez-Arribas J A Suarez-Suarez and L Fernandez-Jambrina ldquoAutomatic surface modelling of a ship hullrdquoComputer-Aided Design vol 38 no 6 pp 584ndash594 2006

[15] C Abt and S Harries ldquoA New Approach to Integration of CADand CFD for Naval Architectsrdquo in Proceedings of the 6th Inter-national Conference on Computer Applications and InformationTechnology in Maritime Industries COMPIT Cortona Italy2007

[16] S Harries ldquoSerious play in ship design tradition and futureof ship design in Berlinrdquo Tech Rep Technical University ofBerlin 2008

[17] P Zhang D Zhu andW Leng ldquoParametric approach to designof hull formsrdquo Journal of Hydrodynamics vol 20 no 6 pp 804ndash810 2008

[18] Y Tahara D Peri E F Campana and F Stern ldquoComputationalfluid dynamics-based multiobjective optimization of a surfacecombatant using a global optimization methodrdquo Journal ofMarine Science and Technology vol 13 no 2 pp 95ndash116 2008

[19] A I Ginnis C Feurer K A Belibassakis et al ldquoA CATIAship-parametric model for isogeometric hull optimization withrespect to wave resistancerdquo in Proceedings of the InternationalConference on Computer Applications in Shipbuilding 2011 vol1 pp 9ndash20 Italy 2011

[20] A I Ginnis R Duvigneau C Politis et al ldquoA multi-objectiveoptimization environment for Ship-Hull Design based on aBEM-Isogeometric solverrdquo in Proceedings of the 5th Conferenceon Computational Methods in Marine Engineering SpringerHamburg Germany 2013


[21] A Ginnis K Kostas C Politis et al ldquoIsogeometric boundary-element analysis for the wave-resistance problem using T-splinesrdquoComputerMethods AppliedMechanics and Engineeringvol 279 pp 425ndash439 2014

[22] K Kostas A Ginnis C Politis and P Kaklis ldquoShip-hull shapeoptimization with a T-spline based BEMndashisogeometric solverrdquoComputerMethodsAppliedMechanics and Engineering vol 284pp 611ndash622 2015

[23] C Abt S Harries J Heimann and H Winter ldquoFrom redesignto optimal hull line by means of parametric modelingrdquo inProceedings of the 2nd International on Computer Applicationsand Information Technology in the Marine Industries pp 444ndash458 Hamburg Germany 2003

[24] Y Lu X Chang and A Hu ldquoA hydrodynamic optimizationdesign methodology for a ship bulbous bow under multipleoperating conditionsrdquo Engineering Applications of Computa-tional Fluid Mechanics vol 10 no 1 pp 330ndash345 2016

[25] T W Lowe and J Steel ldquoConceptual hull design using a geneticalgorithmrdquo Journal of Ship Research vol 47 no 3 pp 222ndash2362003

[26] K Suzuki H Kai and S Kashiwabara ldquoStudies on the optimiza-tion of stern hull form based on a potential flow solverrdquo Journalof Marine Science and Technology vol 10 no 2 pp 61ndash69 2005

[27] B Zhang K Ma and Z Ji ldquoThe optimization of the hull formwith the minimum wave making resistance based on rankinesource methodrdquo Journal of Hydrodynamics vol 21 no 2 pp277ndash284 2009

[28] B J Zhang ldquoResearch on optimization of hull linesfor mini-mum resistance based on Rankine source methodrdquo Journal ofMarine Science and Technology vol 20 no 1 pp 89ndash94 2012

[29] C Yang F Huang and F Noblesse ldquoPractical evaluation ofthe drag of a ship for design and optimizationrdquo Journal ofHydrodynamics vol 25 no 5 pp 645ndash654 2013

[30] H J Choi D W Park and M S Choi ldquoStudy on optimizedhull formof basic ships using optimization algorithmrdquo Journalof Marine Science and Technology - Taiwan vol 23 no 1 pp60ndash68 2015

[31] C Yang F Huang and H Kim ldquoHydrodynamic optimizationof a triswachrdquo Journal of Hydrodynamics vol 26 no 6 pp 856ndash864 2014

[32] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence MIT Press 1992

[33] K Deb S Agrawal A Pratap and T Meyarivan ldquoA fast elitistnon-dominated sorting genetic algorithm for multi-objectiveoptimization NSGA-IIrdquo in Proceedings of the InternationalConference on Parallel Problem Solving From Nature vol 1917pp 849ndash858 Springer

[34] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[35] L Larsson F Stern and V Bertram ldquoBenchmarking of com-putational fluid dynamics for ship flows The Gothenburg 2000workshoprdquo Journal of Ship Research vol 47 no 1 pp 63ndash812003

[36] T Hino Proceedings of CFD Workshop Tokyo 2005 NationalMaritime Research Institute Tokyo Japan 2005

[37] L Larsson F Stern and M Visonneau Numerical ShipHydrodynamics-An assessment of the Gothenburg 2010 Work-shop Springer Science amp Business Media Netherlands 2014

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Submit your manuscripts atwwwhindawicom


Ginnis and PD Kaklis et al [19ndash22] In this work the methodis concerned for the modification and reconstruction of thehull bow and stern form using a fairing B-Spline parametriccurve

Section 3 is devoted to the prediction method of thetotal resistance performance which has been defined asthe optimization objective function in this work The ITTC1957 model-ship correlation formula and the Rankine sourcepanel method are applied to calculate the frictional resis-tance and wave-making resistance respectively The Rankinesources method is based on the potential-flow withoutconsidering viscosity Nevertheless for a sophisticated ship-hull optimization system it is more simple and convenientand has been proved to be accurate reliable and efficient soit has been investigated in various studies (Abt and Harries etal [23] Lu Chang and Hu [24] Lowe and Steel [25] SuzukiKai and Kashiwabara [26] Zhang Kun and Ji [27] Zhang[28] Chi Huang and Noblesse [29] Choi Park and Choi[30] and Chi Huang and Kim [31])

Section 4 presents the Fast Elitist Nondominated SortingGenetic Algorithm II (NSGA-II) used to search for theminimum ship total resistance for the ship-hull optimizationNSGA-II is a kind of modified GAs with low computationalrequirements In 1975 Holland [32] a professor of MichiganState University firstly proposed Genetic Algorithm (GA)which is a computational model simulating evolutionary pro-cess in nature Deb et al improved GA and proposed NSGA-II [33 34] with Fast nondominated sorting method proposedto reduce the computational difficulty the evaluation andcomparison of congestion put forward to help to make quickjudgment for the same level of individual behavior and elitiststrategy introduced to expand the space sampling

Finally two optimization cases for KCS [35ndash37] are setup and presented in Section 5 The first case deals witha ship-hull optimization at design speed while the secondcase involves a total resistance minimization problem againstmultiple objective functions covering all speed range

2 Hull Form Transformation Method

21 Definition of Fairing B-Spline Parametric Curve In thisarticle the hull form transformation is realized by the fairingB-spline parametric curve transformation The deformationof B-spline parametric curves is realized by adjusting theinput characteristic parameters and then the deformationof the characteristic parameter curve is mapped to thegeometric deformation of the hull

The fairing B-spline parametric curve is formed byapplying the constraint conditions on a B-spline curve Thespatial spline curves parameterized by 119906 are expressed by119903(119906)

119903 (119906) = (119909 (119906) 119910 (119906) 119911 (119906)) (1)

The equation corresponding to the curve with mth fairnesscan be expressed as

119871119898 = int1

0(119863119898119903 (119906))2 119908ℎ119890119903119890 119863119898 = 119889119898

119889119906119898 (2)

Some control constraints should be satisfied in the fairingprocess

(1) Distance Constraint Condition The Euler formula is usedto express the distance between the control point 119875119894 andthe spline curve 119903(119905) The distance shall meet the followingcondition

119860 =119899

sum119894=0

[119908119894 (119903 (119906119894) minus 119875119894)]2 le 120576 (3)

where 119908119894 is distance weight and 120576 is a tolerance greater thanzero

(2) Curve Ends Constraint Condition The tangent slope andcurvature of the starting point or the end point of the splinecurve shall meet the following conditions

1198791 = 1198631119903 (119906119894) minus119876119894 = 0 (4)

1198792 = 1198632119903 (119906119894) minus119870119894 = 0 (5)

where 119894 = 0 or 119894 = N If 119894=0 it represents the starting point andif 119894=n it represents the end point

(3) Area Constraint Condition The area 119878 under the splinecurves should meet the condition of the fixed value that is

119878 = 1198780 (6)

where 1198780 is a given areaAt the same time if necessary other conditions can be

taken into account in the same waymentioned above to com-plete the corresponding constraints Finally the constrainedoptimization problem can be transformed into the extremevalue problem of the unconstrained objective function I bythe linear superposition of these constraints

119868 = 119871119898 + 1205821198601015840 + 12058311198791 + 12058321198792 + ]119865119908ℎ119890119903119890 1198601015840 = 119860 + 1198892

(7)

where 120582 1205831 1205832 and ] are all Lagrange operators and d2 is aslack variable

22 Hull Deformation Method by Mapping This sectionshows how to realize the deformation of the transversesection lines by the fairing B-spline parametric curves Thismethod is also applicable to other lines of the hull Thehull geometry model is stored in the form of offsets thatis the hull is discretized into a finite number of transversesection lines On the YZ plane the mapping formula of thecoordinate change from the point on the B-spline curve tothe point on the transverse section line is expressed as

119909119906 = 119909 + 119903120575119909119906 (119909) (8)

119911119906 = 119911 + 119903120575119911119906 (119911) (9)

where 120575119910119906 and 120575119911119906 are the maximum displacement variationsof fairing B-spline curve in the Y axis and Z axis respectively





119909119906 = 119909 + 119903120575119909119906 (119909) sdot 119903120575119909119908 (119909) (10)

119911119906 = 119911 + 119903120575119911119906 (119911) sdot 119903120575119911119908 (119911) (11)





119862119905 = 119862119908 + (1 + 119896) 1198621198910 (12)


1198621198910 = 0075(lg119877119890 minus 2)2 (13)




nabla2 (Φ + 120593) = 0 (14)


nabla (Φ + 120593) sdot 997888rarr119899 = 0 (15)



12 120601119909 (nabla120601 sdot nabla120601)119909 + 1

2 120601119910 (nabla120601 sdot nabla120601)119910 + 119892120601119911 = 0 (16)




(17)



1198651198992 (18)


119862119908 = minus 1119878 int119878

119862119901119899119909119889119904 (19)



















20

40

60

014 018 022 026 03

ExpCal(Ori)

Ｎtimes10

3

Ｈ


0

001

002

0 05 1xL

Fn=026

ExpCal

minus002

minus001

００






000

001

002

0 025 05 075 1 125

Wav

e Ele

vatio

nL

xL

yL=01024

ExpCal

minus002minus025

minus001

(a)

000

001

002

000 025 050 075 100 125

Wav

e Ele

vatio

nL

xL

yL=01509

ExpCal

minus002minus025

minus001

(b)

000

001

002

000 025 050 075 100 125

Wav

e Ele

vatio

nL

xL

yL=030

ExpCal

minus025minus002

minus001

(c)

000

001

002

0 025 05 075 1 125W

ave E

leva

tion

L

xL

yL=040

minus025minus002

minus001

ExpCal

(d)


Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

0

3

6

9

minus02

minus05

minus04

minus6

minus3

minus03

minus02

minus01

times10minus3







Mutation)

isgenltmaxgen

Exit

Start




Mutation)



Crowded comparison


Yes

No








119865 = 1198621199051198621199050 (20)







Parent ship hull






range



No Yes



Hull form


Design variants




Bow area Stern area





(21)




1198891199104 -07 1198891199115 047
















9000

9500

10000

10500

11000

0 50 100 150 200 250 300 350 400

Obj

ectiv

e fun

ctio

n (F

)

Number of variants



1198651 = 119862119905111986211990501

1198652 = 119862119905211986211990502

1198653 = 119862119905311986211990503

1198654 = 119862119905411986211990504

1198655 = 119862119905511986211990505

(22)








000

100

200

300


minus300

minus200

minus100

Des

ign

varia

bled

x1

(a)

050

150

250


minus350

minus250

minus150

minus050

Des

ign

varia

bled

y1

(b)

155

160

165

170

175

180

185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

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minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 19 Continued


010

030

050


minus050

minus070

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)



Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

01

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0

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6

9Optimized Hull


minus3

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9300

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Obj

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000

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x1

minus300

minus200

minus100

(a)

050

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Des

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y1

(b)

155

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Des

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y2

(c)

010

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Des

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y3

(d)

000

040


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minus080

minus040

Des

ign

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y4

(e)

000

100

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minus100Des

ign

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z1

(f)

000

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Des

ign

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z2

(g)

010

030

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minus010

Des

ign

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bled

z3

(h)

Figure 26 Continued


010

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050


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minus050

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minus010

Des

ign

varia

bled

z4

(i)

010

040

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100


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Des

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z5

(j)


(a) (b)









(a1)

(b1) (b2)

(c1) (c2)

(d1) (d2)

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3

Original Hull

Opt01

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt01

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt01

(a2)

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt01

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3Opt02

Original Hull

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt02

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt02

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt02

minus02

minus02

minus02

minus02

minus02minus02

minus02 minus02

minus2

minus2

minus3minus3

minus3

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minus3

minus6

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minus2

minus2

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minus04

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minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3



Figure 28 Continued


(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018












119909119906 = 119909 + 119903120575119909119906 (119909) sdot 119903120575119909119908 (119909) (10)

119911119906 = 119911 + 119903120575119911119906 (119911) sdot 119903120575119911119908 (119911) (11)





119862119905 = 119862119908 + (1 + 119896) 1198621198910 (12)


1198621198910 = 0075(lg119877119890 minus 2)2 (13)




nabla2 (Φ + 120593) = 0 (14)


nabla (Φ + 120593) sdot 997888rarr119899 = 0 (15)



12 120601119909 (nabla120601 sdot nabla120601)119909 + 1

2 120601119910 (nabla120601 sdot nabla120601)119910 + 119892120601119911 = 0 (16)




(17)



1198651198992 (18)


119862119908 = minus 1119878 int119878

119862119901119899119909119889119904 (19)



















20

40

60

014 018 022 026 03

ExpCal(Ori)

Ｎtimes10

3

Ｈ


0

001

002

0 05 1xL

Fn=026

ExpCal

minus002

minus001

００






000

001

002

0 025 05 075 1 125

Wav

e Ele

vatio

nL

xL

yL=01024

ExpCal

minus002minus025

minus001

(a)

000

001

002

000 025 050 075 100 125

Wav

e Ele

vatio

nL

xL

yL=01509

ExpCal

minus002minus025

minus001

(b)

000

001

002

000 025 050 075 100 125

Wav

e Ele

vatio

nL

xL

yL=030

ExpCal

minus025minus002

minus001

(c)

000

001

002

0 025 05 075 1 125W

ave E

leva

tion

L

xL

yL=040

minus025minus002

minus001

ExpCal

(d)


Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

0

3

6

9

minus02

minus05

minus04

minus6

minus3

minus03

minus02

minus01

times10minus3







Mutation)

isgenltmaxgen

Exit

Start




Mutation)



Crowded comparison


Yes

No








119865 = 1198621199051198621199050 (20)







Parent ship hull






range



No Yes



Hull form


Design variants




Bow area Stern area





(21)




1198891199104 -07 1198891199115 047
















9000

9500

10000

10500

11000

0 50 100 150 200 250 300 350 400

Obj

ectiv

e fun

ctio

n (F

)

Number of variants



1198651 = 119862119905111986211990501

1198652 = 119862119905211986211990502

1198653 = 119862119905311986211990503

1198654 = 119862119905411986211990504

1198655 = 119862119905511986211990505

(22)








000

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minus300

minus200

minus100

Des

ign

varia

bled

x1

(a)

050

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(b)

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y2

(c)

010

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y3

(d)

000

040


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minus040

Des

ign

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bled

y4

(e)

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minus100Des

ign

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z1

(f)

000

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Des

ign

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z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 19 Continued


010

030

050


minus050

minus070

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)



Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

01

02

03

04

05

0

3

6

9Optimized Hull


minus3

minus04

minus03

minus02

minus02

minus01

times10minus3



9300

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Obj

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Obj

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Obj

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e fun

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nF4




000

100

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300


Des

ign

varia

bled

x1

minus300

minus200

minus100

(a)

050

150

250


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minus250

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minus050

Des

ign

varia

bled

y1

(b)

155

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170

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185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

200


minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 26 Continued


010

030

050


minus070

minus050

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)


(a) (b)









(a1)

(b1) (b2)

(c1) (c2)

(d1) (d2)

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3

Original Hull

Opt01

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt01

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt01

(a2)

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt01

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3Opt02

Original Hull

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt02

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt02

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt02

minus02

minus02

minus02

minus02

minus02minus02

minus02 minus02

minus2

minus2

minus3minus3

minus3

minus6

minus3

minus6

minus1

minus2

minus2

minus1

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3



Figure 28 Continued


(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018
























20

40

60

014 018 022 026 03

ExpCal(Ori)

Ｎtimes10

3

Ｈ


0

001

002

0 05 1xL

Fn=026

ExpCal

minus002

minus001

００






000

001

002

0 025 05 075 1 125

Wav

e Ele

vatio

nL

xL

yL=01024

ExpCal

minus002minus025

minus001

(a)

000

001

002

000 025 050 075 100 125

Wav

e Ele

vatio

nL

xL

yL=01509

ExpCal

minus002minus025

minus001

(b)

000

001

002

000 025 050 075 100 125

Wav

e Ele

vatio

nL

xL

yL=030

ExpCal

minus025minus002

minus001

(c)

000

001

002

0 025 05 075 1 125W

ave E

leva

tion

L

xL

yL=040

minus025minus002

minus001

ExpCal

(d)


Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

0

3

6

9

minus02

minus05

minus04

minus6

minus3

minus03

minus02

minus01

times10minus3







Mutation)

isgenltmaxgen

Exit

Start




Mutation)



Crowded comparison


Yes

No








119865 = 1198621199051198621199050 (20)







Parent ship hull






range



No Yes



Hull form


Design variants




Bow area Stern area





(21)




1198891199104 -07 1198891199115 047
















9000

9500

10000

10500

11000

0 50 100 150 200 250 300 350 400

Obj

ectiv

e fun

ctio

n (F

)

Number of variants



1198651 = 119862119905111986211990501

1198652 = 119862119905211986211990502

1198653 = 119862119905311986211990503

1198654 = 119862119905411986211990504

1198655 = 119862119905511986211990505

(22)








000

100

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300


minus300

minus200

minus100

Des

ign

varia

bled

x1

(a)

050

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Des

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(b)

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Des

ign

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y2

(c)

010

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Des

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bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

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minus100Des

ign

varia

bled

z1

(f)

000

100

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minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 19 Continued


010

030

050


minus050

minus070

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)



Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

01

02

03

04

05

0

3

6

9Optimized Hull


minus3

minus04

minus03

minus02

minus02

minus01

times10minus3



9300

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10700

9200 9400 9600 9800 10000 10200 10400O

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9300

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Obj

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Obj

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Obj

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000

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ign

varia

bled

x1

minus300

minus200

minus100

(a)

050

150

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minus350

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minus050

Des

ign

varia

bled

y1

(b)

155

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170

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185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

200


minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 26 Continued


010

030

050


minus070

minus050

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)


(a) (b)









(a1)

(b1) (b2)

(c1) (c2)

(d1) (d2)

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3

Original Hull

Opt01

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt01

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt01

(a2)

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt01

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3Opt02

Original Hull

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt02

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt02

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt02

minus02

minus02

minus02

minus02

minus02minus02

minus02 minus02

minus2

minus2

minus3minus3

minus3

minus6

minus3

minus6

minus1

minus2

minus2

minus1

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3



Figure 28 Continued


(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









000

001

002

0 025 05 075 1 125

Wav

e Ele

vatio

nL

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yL=01024

ExpCal

minus002minus025

minus001

(a)

000

001

002

000 025 050 075 100 125

Wav

e Ele

vatio

nL

xL

yL=01509

ExpCal

minus002minus025

minus001

(b)

000

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Wav

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vatio

nL

xL

yL=030

ExpCal

minus025minus002

minus001

(c)

000

001

002

0 025 05 075 1 125W

ave E

leva

tion

L

xL

yL=040

minus025minus002

minus001

ExpCal

(d)


Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

0

3

6

9

minus02

minus05

minus04

minus6

minus3

minus03

minus02

minus01

times10minus3







Mutation)

isgenltmaxgen

Exit

Start




Mutation)



Crowded comparison


Yes

No








119865 = 1198621199051198621199050 (20)







Parent ship hull






range



No Yes



Hull form


Design variants




Bow area Stern area





(21)




1198891199104 -07 1198891199115 047
















9000

9500

10000

10500

11000

0 50 100 150 200 250 300 350 400

Obj

ectiv

e fun

ctio

n (F

)

Number of variants



1198651 = 119862119905111986211990501

1198652 = 119862119905211986211990502

1198653 = 119862119905311986211990503

1198654 = 119862119905411986211990504

1198655 = 119862119905511986211990505

(22)








000

100

200

300


minus300

minus200

minus100

Des

ign

varia

bled

x1

(a)

050

150

250


minus350

minus250

minus150

minus050

Des

ign

varia

bled

y1

(b)

155

160

165

170

175

180

185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

200


minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 19 Continued


010

030

050


minus050

minus070

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)



Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

01

02

03

04

05

0

3

6

9Optimized Hull


minus3

minus04

minus03

minus02

minus02

minus01

times10minus3



9300

9500

9700

9900

10100

10300

10500

10700

9200 9400 9600 9800 10000 10200 10400O

bjec

tive f

unct

ionF4



9300

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9700

9900

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10700

9200 9400 9600 9800 10000 10200 10400

Obj

ectiv

e fun

ctio

nF4



9300

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10700

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Obj

ectiv

e fun

ctio

nF4



9300

9500

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10100

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10500

10700

9700 9900 10100 10300 10500 10700

Obj

ectiv

e fun

ctio

nF4




000

100

200

300


Des

ign

varia

bled

x1

minus300

minus200

minus100

(a)

050

150

250


minus350

minus250

minus150

minus050

Des

ign

varia

bled

y1

(b)

155

160

165

170

175

180

185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

200


minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 26 Continued


010

030

050


minus070

minus050

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)


(a) (b)









(a1)

(b1) (b2)

(c1) (c2)

(d1) (d2)

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3

Original Hull

Opt01

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt01

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt01

(a2)

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt01

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3Opt02

Original Hull

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt02

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt02

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt02

minus02

minus02

minus02

minus02

minus02minus02

minus02 minus02

minus2

minus2

minus3minus3

minus3

minus6

minus3

minus6

minus1

minus2

minus2

minus1

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3



Figure 28 Continued


(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018










Mutation)

isgenltmaxgen

Exit

Start




Mutation)



Crowded comparison


Yes

No








119865 = 1198621199051198621199050 (20)







Parent ship hull






range



No Yes



Hull form


Design variants




Bow area Stern area





(21)




1198891199104 -07 1198891199115 047
















9000

9500

10000

10500

11000

0 50 100 150 200 250 300 350 400

Obj

ectiv

e fun

ctio

n (F

)

Number of variants



1198651 = 119862119905111986211990501

1198652 = 119862119905211986211990502

1198653 = 119862119905311986211990503

1198654 = 119862119905411986211990504

1198655 = 119862119905511986211990505

(22)








000

100

200

300


minus300

minus200

minus100

Des

ign

varia

bled

x1

(a)

050

150

250


minus350

minus250

minus150

minus050

Des

ign

varia

bled

y1

(b)

155

160

165

170

175

180

185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

200


minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 19 Continued


010

030

050


minus050

minus070

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)



Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

01

02

03

04

05

0

3

6

9Optimized Hull


minus3

minus04

minus03

minus02

minus02

minus01

times10minus3



9300

9500

9700

9900

10100

10300

10500

10700

9200 9400 9600 9800 10000 10200 10400O

bjec

tive f

unct

ionF4



9300

9500

9700

9900

10100

10300

10500

10700

9200 9400 9600 9800 10000 10200 10400

Obj

ectiv

e fun

ctio

nF4



9300

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10700

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Obj

ectiv

e fun

ctio

nF4



9300

9500

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10100

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10500

10700

9700 9900 10100 10300 10500 10700

Obj

ectiv

e fun

ctio

nF4




000

100

200

300


Des

ign

varia

bled

x1

minus300

minus200

minus100

(a)

050

150

250


minus350

minus250

minus150

minus050

Des

ign

varia

bled

y1

(b)

155

160

165

170

175

180

185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

200


minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 26 Continued


010

030

050


minus070

minus050

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)


(a) (b)









(a1)

(b1) (b2)

(c1) (c2)

(d1) (d2)

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3

Original Hull

Opt01

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt01

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt01

(a2)

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt01

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3Opt02

Original Hull

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt02

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt02

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt02

minus02

minus02

minus02

minus02

minus02minus02

minus02 minus02

minus2

minus2

minus3minus3

minus3

minus6

minus3

minus6

minus1

minus2

minus2

minus1

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3



Figure 28 Continued


(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









Parent ship hull






range



No Yes



Hull form


Design variants




Bow area Stern area





(21)




1198891199104 -07 1198891199115 047
















9000

9500

10000

10500

11000

0 50 100 150 200 250 300 350 400

Obj

ectiv

e fun

ctio

n (F

)

Number of variants



1198651 = 119862119905111986211990501

1198652 = 119862119905211986211990502

1198653 = 119862119905311986211990503

1198654 = 119862119905411986211990504

1198655 = 119862119905511986211990505

(22)








000

100

200

300


minus300

minus200

minus100

Des

ign

varia

bled

x1

(a)

050

150

250


minus350

minus250

minus150

minus050

Des

ign

varia

bled

y1

(b)

155

160

165

170

175

180

185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

200


minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 19 Continued


010

030

050


minus050

minus070

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)



Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

01

02

03

04

05

0

3

6

9Optimized Hull


minus3

minus04

minus03

minus02

minus02

minus01

times10minus3



9300

9500

9700

9900

10100

10300

10500

10700

9200 9400 9600 9800 10000 10200 10400O

bjec

tive f

unct

ionF4



9300

9500

9700

9900

10100

10300

10500

10700

9200 9400 9600 9800 10000 10200 10400

Obj

ectiv

e fun

ctio

nF4



9300

9500

9700

9900

10100

10300

10500

10700

9400 9600 9800 10000 10200 10400 10600

Obj

ectiv

e fun

ctio

nF4



9300

9500

9700

9900

10100

10300

10500

10700

9700 9900 10100 10300 10500 10700

Obj

ectiv

e fun

ctio

nF4




000

100

200

300


Des

ign

varia

bled

x1

minus300

minus200

minus100

(a)

050

150

250


minus350

minus250

minus150

minus050

Des

ign

varia

bled

y1

(b)

155

160

165

170

175

180

185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

200


minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 26 Continued


010

030

050


minus070

minus050

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)


(a) (b)









(a1)

(b1) (b2)

(c1) (c2)

(d1) (d2)

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3

Original Hull

Opt01

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt01

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt01

(a2)

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt01

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3Opt02

Original Hull

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt02

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt02

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt02

minus02

minus02

minus02

minus02

minus02minus02

minus02 minus02

minus2

minus2

minus3minus3

minus3

minus6

minus3

minus6

minus1

minus2

minus2

minus1

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3



Figure 28 Continued


(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018





















9000

9500

10000

10500

11000

0 50 100 150 200 250 300 350 400

Obj

ectiv

e fun

ctio

n (F

)

Number of variants



1198651 = 119862119905111986211990501

1198652 = 119862119905211986211990502

1198653 = 119862119905311986211990503

1198654 = 119862119905411986211990504

1198655 = 119862119905511986211990505

(22)








000

100

200

300


minus300

minus200

minus100

Des

ign

varia

bled

x1

(a)

050

150

250


minus350

minus250

minus150

minus050

Des

ign

varia

bled

y1

(b)

155

160

165

170

175

180

185


Des

ign

varia

bled

y2

(c)

010

050

090

130


minus030

Des

ign

varia

bled

y3

(d)

000

040


minus120

minus080

minus040

Des

ign

varia

bled

y4

(e)

000

100

200


minus200

minus100Des

ign

varia

bled

z1

(f)

000

100

200

300


minus100

Des

ign

varia

bled

z2

(g)

010

030

050

070


minus050

minus030

minus010

Des

ign

varia

bled

z3

(h)

Figure 19 Continued


010

030

050


minus050

minus070

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)



Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

0

01

02

03

04

05

0

3

6

9Optimized Hull


minus3

minus04

minus03

minus02

minus02

minus01

times10minus3



9300

9500

9700

9900

10100

10300

10500

10700

9200 9400 9600 9800 10000 10200 10400O

bjec

tive f

unct

ionF4



9300

9500

9700

9900

10100

10300

10500

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(h)

Figure 26 Continued


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z5

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(a) (b)









(a1)

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(d1) (d2)

Fn=0152

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p

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00102030405

0

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Original Hull

Opt01

Fn=0195

xLpp

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0

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4Opt01

Original Hull

Fn=0227

xLpp

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0

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Original Hull

Opt01

(a2)

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00102030405

0

3

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9

Original Hull

Opt01

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xLpp

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00102030405

0

1

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3Opt02

Original Hull

Fn=0195

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0

2

4Opt02

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0

3

6

Original Hull

Opt02

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xLpp

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00102030405

0

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6

9

Original Hull

Opt02

minus02

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minus05

minus04

minus03

minus02

minus01

times10minus3

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times10minus3

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Figure 28 Continued


(e1) (e2)

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p

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00102030405

0

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9Opt01

Original Hull

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xLpp

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p

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00102030405

0

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6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









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(g)

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(h)

Figure 19 Continued


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z4

(i)

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z5

(j)



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0

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(d)

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y4

(e)

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ign

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(f)

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(g)

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(h)

Figure 26 Continued


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z5

(j)


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(d1) (d2)

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0 02 04 06 08 1 12 14 16

00102030405

0

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Original Hull

Opt01

Fn=0195

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0

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Original Hull

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6

Original Hull

Opt01

(a2)

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0

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9

Original Hull

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0

1

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3Opt02

Original Hull

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0

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4Opt02

Original Hull

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0

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6

Original Hull

Opt02

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0

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6

9

Original Hull

Opt02

minus02

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minus01

minus05

minus04

minus03

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minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3



Figure 28 Continued


(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









010

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0

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Obj

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Des

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x1

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(d)

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(e)

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(f)

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(g)

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z3

(h)

Figure 26 Continued


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z5

(j)


(a) (b)









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(d1) (d2)

Fn=0152

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0 02 04 06 08 1 12 14 16

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0

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Original Hull

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Fn=0195

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0 02 04 06 08 1 12 14 16

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0

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Original Hull

Fn=0227

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Original Hull

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Original Hull

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0 02 04 06 08 1 12 14 16

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Original Hull

Fn=0195

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0

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Original Hull

Fn=0227

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0

3

6

Original Hull

Opt02

Fn=026

xLpp

yLp

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0 02 04 06 08 1 12 14 16

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0

3

6

9

Original Hull

Opt02

minus02

minus02

minus02

minus02

minus02minus02

minus02 minus02

minus2

minus2

minus3minus3

minus3

minus6

minus3

minus6

minus1

minus2

minus2

minus1

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3



Figure 28 Continued


(e1) (e2)

Fn=0282

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p

0 02 04 06 08 1 12 14 16

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0

3

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Original Hull

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p

0 02 04 06 08 1 12 14 16

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0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









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Obj

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ign

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x1

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(c)

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(d)

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Des

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(e)

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(f)

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(g)

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Des

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z3

(h)

Figure 26 Continued


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(i)

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Original Hull

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Original Hull

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minus02

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minus02minus02

minus02 minus02

minus2

minus2

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minus04

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minus01

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minus04

minus03

minus02

minus01

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times10minus3

times10minus3

times10minus3

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Figure 28 Continued


(e1) (e2)

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0 02 04 06 08 1 12 14 16

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0 02 04 06 08 1 12 14 16

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0

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Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









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Des

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y2

(c)

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Des

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y3

(d)

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Des

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y4

(e)

000

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z1

(f)

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Des

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z2

(g)

010

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Des

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z3

(h)

Figure 26 Continued


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z4

(i)

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Des

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z5

(j)


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(b1) (b2)

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Original Hull

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Original Hull

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Original Hull

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Original Hull

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Original Hull

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Original Hull

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minus2

minus2

minus3minus3

minus3

minus6

minus3

minus6

minus1

minus2

minus2

minus1

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

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minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3



Figure 28 Continued


(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









010

030

050


minus070

minus050

minus030

minus010

Des

ign

varia

bled

z4

(i)

010

040

070

100


minus020

minus050

Des

ign

varia

bled

z5

(j)


(a) (b)









(a1)

(b1) (b2)

(c1) (c2)

(d1) (d2)

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3

Original Hull

Opt01

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt01

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt01

(a2)

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt01

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3Opt02

Original Hull

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt02

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt02

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt02

minus02

minus02

minus02

minus02

minus02minus02

minus02 minus02

minus2

minus2

minus3minus3

minus3

minus6

minus3

minus6

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minus04

minus03

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minus01

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minus04

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minus01

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minus04

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minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3



Figure 28 Continued


(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









(a1)

(b1) (b2)

(c1) (c2)

(d1) (d2)

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3

Original Hull

Opt01

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt01

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt01

(a2)

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt01

Fn=0152

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

1

2

3Opt02

Original Hull

Fn=0195

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

2

4Opt02

Original Hull

Fn=0227

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

Original Hull

Opt02

Fn=026

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9

Original Hull

Opt02

minus02

minus02

minus02

minus02

minus02minus02

minus02 minus02

minus2

minus2

minus3minus3

minus3

minus6

minus3

minus6

minus1

minus2

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minus04

minus03

minus02

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minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01

times10minus3

times10minus3

times10minus3

times10minus3



Figure 28 Continued


(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









(e1) (e2)

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt01

Original Hull

Fn=0282

xLpp

yLp

p

0 02 04 06 08 1 12 14 16

00102030405

0

3

6

9Opt02

Original Hull

minus02 minus02

minus3

minus6

minus3

minus6

minus05

minus04

minus03

minus02

minus01

minus05

minus04

minus03

minus02

minus01







6 Conclusions








Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018












Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018

































Journal of












Journal of







Volume 2018






Volume 2018








hydrodynamic design study on ship bow and stern hull form...

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