research article development of an efficient hull form...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 838354 12 pageshttpdxdoiorg1011552013838354

Research ArticleDevelopment of an Efficient Hull Form DesignExploration Framework

Shinkyu Jeong1 and Hyunyul Kim2

1 Department of Mechanical Engineering Kyunghee University Yongin 446-701 Republic of Korea2Department of Computational and Data Science George Mason University Fairfax VA 22030 USA

Correspondence should be addressed to Shinkyu Jeong icaruskhuackr

Received 16 April 2013 Accepted 24 June 2013

Academic Editor Ming Li

Copyright copy 2013 S Jeong and H KimThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A high-efficiency design exploration framework for hull form has been developed The framework consists of multiobjectiveshape optimization and design knowledge extraction In multiobjective shape optimization a multiobjective genetic algorithm(MOGA) using the response surface methodology is introduced to achieve efficient design space exploration As a response surfacemethodology the Kriging model which was developed in the field of spatial statistics and geostatistics is applied A new surfacemodification method using shifting method and radial basis function interpolation is also adopted here to represent various hullforms This method enables both global and local modifications of hull form with fewer design variables In design knowledgeextraction two data mining techniquesmdashfunctional analysis of variance (ANOVA) and self-organizing map (SOM)mdashare appliedto acquire useful design knowledge about a hull formThe present framework has been applied to hull form optimization exploringthe minimum wave drag configuration under a wide range of speeds The results show that the present method markedly reducedthe design period From the results of data mining it is possible to identify the design variables controlling wave drag performancesat different speed regions and their corresponding geometric features

1 Introduction

In order to find the combination of design variables that gen-erates a hull form with optimal hydrodynamic performance(eg drag and seakeeping) a tremendous number of perfor-mance evaluations are required However it is obvious thatan increase in the number of evaluations causes an exces-sive increase in both the cost and time of the design pro-cess Recently computational-fluid-dynamics-(CFD-) basedoptimization techniques have become promising alternativetools in ship hydrodynamic design To be competitive inthe shipbuilding market a practical and efficient CFD-basedship hydrodynamic optimization tool is essential in thepreliminary and early design stages to aid in prompt andaccurate design

CFD-based hull form hydrodynamic optimization con-sists of a CFD solver that computes the flow field and eval-uates the performance of hull form (objective functions) hullform modeling with design variables and an optimization

technique that explores the combination of the design param-eters generating the hull formwith the optimumperformanceunder the given constraints

Recently optimization techniques that mimic the courseof biological evolution and themovement of living organismswhich is known as evolutionary algorithms (EAs) have beenwidely used in the field of ship design The typical examplesof methodology are genetic algorithms (GAs) [1 2] and par-ticle swarm optimization [3] Lowe and Steel [4] applied agenetic algorithm to search for a hull form satisfying pre-scribed primary and secondary form parameters within thedesign space Tahara et al [5ndash7] used a combination of suc-cessive quadratic programming and multiobjective geneticalgorithm to optimize tanker stern Delta Catamaran and soon Gammon [8] studied hull form optimization of fishingvessels using multiobjective genetic algorithm Three perfor-mance indices for resistance seakeeping and stability wereconsidered in this multiobjective optimization In particularautomatic selection of a few Pareto optimal solutions was

2 Mathematical Problems in Engineering

developed so that the designer can easily choose the morefavorable candidates More recently Knight et al [9] appliedparticle swarm optimization to incorporate uncertainty inparameters into the optimization of a planning craft

However the number of CFD runs required in thesemethods is too large Furthermore if these methods are com-bined with a time-consuming high fidelity CFD solver it isdifficult to complete the design process in reasonable timeTherefore it is very important to either have a highly efficientCFD tool or to use an accurate approximation model (egresponse surface) to partially replace the CFD runs for theCFD-based optimization tool to be useful in practical shipdesign

The response surface method is a well-established ap-proach for creating approximation models based on sampledata that can be obtained from physical experiments andcomputer simulations [10] A major advantage of using theresponse surface method in CFD-based design optimizationis to reduce the computational costs associated with the CFDruns A given optimization problem can be approximated bythe response surface method with smooth functions that canimprove computational efficiency in the optimization pro-cess The response surface method has been applied to manyCFD-based aerodynamics optimization problems includinga centrifugal fan and transonic airfoil and this approach wasshown to be quite efficient in exploring the design space[11 12]

In the present study the Kriging response surface model[13ndash15] is implemented in the CFD-based ship hydrodynam-ics optimization tool to provide the objective function valuesassociated with the hydrodynamics performance efficientlyThe Kriging model has a sufficient flexibility to representthe nonlinear and multimodal functions which are oftenappeared in hydrodynamic performances Another merit ofKriging model is that it predicts not only function value butalso the uncertainty of the estimated function value Basedon both the function value and its uncertainty it is possibleto improve the accuracy of Kriging model during the designprocess

In this study the Kriging model based multiobjectiveshape optimization method is applied to the optimal hullform design to minimize wave drag over a wide range ofdesign speeds For the modeling of hull form a surface mod-ificationmethod using shiftingmethod and radial basis func-tion interpolation is introducedThe shiftingmethodmakes itpossible to realize a smooth junction of the modified portionwith the original design The radial basis function is usedfor the local modifications of hull formThus these methodsenable both global and local modification of hull form withfewer design variables In design knowledge extraction twodata mining techniquesmdashfunctional analysis of variance(ANOVA) [16] and self-organizing map (SOM) [17]mdashareapplied to acquire useful design knowledge about a hull formFunctional analysis of variance (ANOVA) shows effect ofeach design variable on objective functions Self-organizingmap (SOM) identifies the design variables controlling thetrade-off between high-speed wave drag and low-speed wavedrag Numerical results show that the design explorationframework not only can explore superior hull forms within

a short computational time but also supplies the insight ofthe effect of each design variable on the drag performance atdifferent design speeds

The remainder of this paper is organized as follows Inthe next two sections design problem definition and designexploration used in this studywill be introduced respectivelyThis will be followed by results of shape optimization anddesign knowledge extraction

2 Design Problem Definition

21 Hull Form Definition and Design Variables An accurateand effective hull surface representation and modificationtechnique plays an important role in theCFD-basedhull formhydrodynamic optimization In the present study the Series-60 (block coefficient 119862119887 = 06) [18 19] hull model is adoptedas the initial hull form and modified to represent a new hullform using the shifting method [20ndash22] and the radial basisfunction (RBF) interpolation [23] during shape optimizationThese methods make it possible to represent both global andlocalmodifications of hull formwith smaller design variables

First the sectional area curve of the initial hull form ismodified using the shiftingmethod whichmoves the stationsof the initial hull form along the longitudinal direction Theamount of movement is determined by comparing themodified sectional area curve with the original curve Inthis study the sectional area curve of the initial hull form ismodified using (1a) and (1b)

119891119899(119909) = 119891

0(119909) + 119892 (119909 1205721 1205722) (1a)

119892 (119909 1205721 1205722)

=

1205721radic05 (1 minus cos(2120587119909 minus 1199091

1205722 minus 1199091

)) 1199091 le 119909 le 1205722

minus1205721radic05 (1 minus cos(2120587119909 minus 1205722

1205722 minus 1199092

)) 1205722 le 119909 le 1199092

0 elsewhere(1b)

where 119891119899(119909) and 1198910(119909) denote the sectional area curve ofthe new and initial hull forms respectively [21 22] Thus119892(119909 1205721 1205722) indicates the amount of shape modification atposition x and 1199091 and 1199092 are fixed points

As shown in Figure 1 the slope of the sectional areacurve is defined by 1205721 and the location of the fixed sta-tion is controlled using 1205722 By changing the parameters 1205721and 1205722 various sectional area curves can be obtained Incomparison with the spline polynomial representation thisformulation can prevent the generation of unrealistic hullforms associated with oscillation of the new sectional areacurve In this study the sectional area curves of both foreand aft bodies are modified simultaneously using four designvariables entrance angle (1205721119891) run angle (1205721119886) fore-bodyvariation (1205722119891) and aft-body variation (1205722119886)

Once the modified sectional area curve is obtainedlocal modification of the hull form is performed usingRBF interpolation In this study the displacement of point

Mathematical Problems in Engineering 3

Table 1 Summary of optimization applications

Design variable Definition Modification RangeDV1 Fore-body variation 120572

2119891030 plusmn 02

DV2 Entrance angle 1205721119891 plusmn05minus2

DV3 Aft-body variation 1205722119886 minus030 plusmn 02DV4 Run angle 1205721119886 plusmn05minus2

DV5ndashDV12 Local body variation 119909-coordinate of RBF 1ndash8 0050 plusmn 0005DV13 Stem line variation y-coordinate of RBF 9 0493ndash0500DV14 Stem line variation y-coordinate of RBF 10 0495ndash0501DV15 Stem line variation y-coordinate of RBF 11 0499ndash0507DV16 Stem line variation y-coordinate of RBF 12 0505ndash0520

1

05

00 05 1

f(x)

x

1205721

0

minus1205721

g(x)

x1 1205722 x2

g(x)

f0(x)

fn(x)

Figure 1 Section area curves and shape function

x = (119909 119910 119911) on the hull form is described by the followingRBF interpolation

119904 (x) =119873

sum

119895=1

120582119895120593 (10038171003817100381710038171003817x minus x119895

10038171003817100381710038171003817) + 119901 (x) (2)

where x119895 = (119909119895 119910119895 119911119895) is the center of the RBF N is thenumber of control nodes (centers) and 120593 is a given RBF withrespect to the Euclidean distance x In the present study theRBF 120593 in (2) is defined in terms ofWendlandrsquos C2 function asfollows

120593 (x) = (1 minus x)4 (4 x + 1) (3)

and the polynomial p in (2) is defined in terms of the linearpolynomial to recover the rotation and translation as follows

119901 (x) = 1198881 + 1198882119909 + 1198883119910 + 1198884119911 (4)

The coefficients 120582119895 in (2) and 119888119895 in (4) are determined by theinterpolation conditions

119904 (x119895) = 119891119895 119895 = 1 119873 (5)

91

475

6 8311

1012 2

(a)

(b)

Figure 2 RBF control nodes

with 119891119895 containing the discrete known values of the displace-ment on the boundary (control nodes) and the additionalrequirements

119873

sum

119895=1

120582119895119901 (x119895) = 0 119895 = 1 119873 (6)

The values of the coefficients 120582119895 and 119888119895 can be obtained bysolving the linear system

(119891

0) = (

119872 119875

1198751198790)(120582

119888) (7)

where 120582 = [1205821 1205822 120582119873] 119888 = [1198881 1198882 1198883 1198884] and 119891 =[1198911 1198912 119891119873]

119879 The elements of the matrices M and P aredefined as

119872119894119895 = 120593 (10038161003816100381610038161003816x119894 minus x119895

10038161003816100381610038161003816) 119894 119895 = 1 119873 (8a)

119875119894119895 = 119901119895 (x119894) 119894 = 1 119873 119895 = 1 4 (8b)

In this study 119909-coordinate of eight control nodes RBF 1ndash8 and 119910-coordinate of four control nodes RBF 9ndash12 aredefined as design variables for modification of the stationprofile and the stemprofile respectively as shown in Figure 2Table 1 summarizes the design variables and their range usedto define the new hull form All values in this table arenormalized by length of hull form which is unit


22 Objective Functions The present study is focused on thewave drag performance of hull form over a wide speed rangeTherefore the three objective functions (wave drag perform-ance at three different speeds) are defined as follows

1198911

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 022 (9a)

1198912

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 027 (9b)

1198913

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 0305 (9c)

where F119873 is Froude number 119862119894119882

and 119862119889119882

denote the wavedrag coefficient of the initial hull form and the hull formobtained during the optimization process respectively TheCFD tool for evaluating the wave drag (objective function)is the steady ship flow (SSF) which is based on Neumann-Mitchell (NM) theory [24] Even though NM theory is a lowfidelity method the wave dragtotal drag from this methodshows a fairly good agreement with experimental measure-ments [25]

3 Design Exploration Framework

Design exploration framework suggested in this study con-sists of multiobjective shape optimization and design knowl-edge extraction In themultiobjective shape optimization de-sign space is explored using multiobjective genetic algorithmand response surface model to find the optimum solutionsIn the design knowledge extraction data mining techniquesare applied to obtain the useful design information which ishelpful for understanding of design problem Figure 3 showsthe overall procedure the design exploration framework inthis study The details of each method are explained in thefollowing subsections

31 Multiobjective Shape Optimization

311 Initial Sample Selection The initial sample points areselected by Latin hypercube sampling (LHS) [26] to constructthe response surface model for each objective function It isdesirable to distribute sample points uniformly in the designspace The sample points distributed by LHS satisfied theorthogonal condition which means that points do not over-lap for all design variables Thus it is possible to distributepoints evenly in the design space with a small number of sam-ple points In this study 60 points are selected as initial samplepoints and wave drag performances of these hull forms areevaluated using SSF code

312 Response Surface Model Construction With the initialsample points selected in the previous step the responsesurface model is constructed for each objective functionTheresponse surface model adopted in this study is the Krigingmodel [11 13] which was developed in the field of spatial

statistics and geostatisticsThe Kriging predictor is expressedas follows

119910 (x) = 120573 + r119879Rminus1 (y minus 1120573) (10)

where x = 1199091 1199092 119909119898 is an m-dimensional vector ofthe design variable y is an n-dimensional column vector ofsampled response data 1 is an n-dimensional unit columnvector and R is the correlation matrix whose (119894 119895) elementis given by

R (x119894 x119895) = exp[minus119898

sum

119896=1

120579119896

10038161003816100381610038161003816119909119894

119896minus 119909119895

119896

10038161003816100381610038161003816

2

] 119894 119895 = 1 119899 (11)

The correlation vector between x and the n sampled data isexpressed as

r119879 (x) = [R (x x1) R (x x2) R (x x119899)] (12)

The 120573 can be calculated using the following equation

120573 =1119879Rminus1y1119879Rminus11

(13)

The unknown parameter 120579 for the Kriging model can beestimated by maximizing the following likelihood function

ln (120573 2 120579) = minus1198992ln (2) minus 1

2ln (|R|) (14)

where 1205902 can be calculated as follows

2=

(y minus 1120573)119879

R (y minus 1120573)119899

(15)

Maximization of the likelihood function is anm-dimensionalunconstrained nonlinear optimization problem A simplegenetic algorithm is used to determine 120579

The accuracy of the predicted value largely depends onthe distance from sample points Intuitively speaking thecloser point x is to the sample point the more accurate is theprediction 119910(x) This intuition is expressed as

1199042(x) = 2 [

[

1 minus r1015840Rminus1r +(1 minus 1Rminus1r)

2

11015840Rminus11]

]

(16)

where 1199042(x) is the mean squared error of the predictor and itindicates the uncertainty of the predicted value

Once the Kriging model is constructed cross-validationis performed to validate the accuracy of the model as shownin Figure 4 The objective function values estimated by theKrigingmodel show good agreement with those estimated bySSF solver

313 Design Space Exploration and Additional Sample PointsSelection In the constructed Kriging model the minimumwave drag hull form (nondominated solutions and Paretosolutions) has been explored using a multiobjective genetic


Initial sample selection

Response surface model construction

Design space exploration

using MOGA (Pareto solutions search)

Additional sample point selection

using clustering analysis

Design converged

Additional sample evaluation

End of multiobjective

shape optimization

Evaluation by response surface modelEvaluation by SSF code

No

Yes

Design knowledge extraction

ANOVA SOMOther data mining tools


Multiobjectiveshape optimization

Figure 3 Overall procedure of design exploration framework

algorithm (MOGA) The number of populations and gener-ations of MOGA were 512 and 100 respectively Figure 5(a)shows the nondominated solutions obtained after MOGAoperation in Figure 3 To investigate the performance of thesenondominated solutions and select additional sample pointsthe 119870-means clustering method [27 28] is applied to thenondominated solution obtainedThe119870-meansmethod sub-groups the nondominated solutions based on the similarityof three objective function values of solutions Figure 5(b)shows the nondominated solution after clustering analysisThe solutions with the same color have similar objectivefunction valuesThe center of each cluster is selected as repre-sentative of cluster and the wave drag of these points is eval-uated by SSF to check the performance Then these points

are used as additional sample points for the update of Krignigmodel This process is iterated until the Pareto front of thenondominated solutions is converged

32 Design Knowledge Extraction Two data mining tech-niques [16 29] analysis of variance (ANOVA) and self-organ-izing map (SOM) are applied to the results of multiobjectiveshape optimization to extract knowledge of hull designDetails of these will be introduced in following sections

321 Analysis of Variance (ANOVA) ANOVA is a commonlyused statistical analysis method which uses the variancemodel to show the effects of each variable on the functionThe

6 Mathematical Problems in EngineeringEv

alua

ted

by K

rigin

g m

odel

Evaluated by SSF

060504030201

0minus01

minus02

minus03

minus040605040302010minus01minus02minus03minus04

Obj1

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

060504030201

0minus01

minus02

minus03

minus0405 06040302010minus01minus02minus03minus04

Obj2

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

04

03

02

01

0

minus01

minus02

minus03


Obj3

Figure 4 Cross validation

ANOVA employed in this study decomposed the total vari-ance of the model into the variance due to each design vari-able on the Kriging response surfacemodel used in themulti-objective shape optimization

The total mean (120583total) and the variance (2total) of themodel are as follows

120583total equiv int sdot sdot sdot int 119910 (1199091 119909119898) 1198891199091 sdot sdot sdot 119889119909119898 (17)

2

total equiv int int [119910 (1199091 119909119898) minus 120583total]21198891199091 sdot sdot sdot 119889119909119898 (18)

0

minus01

minus02

minus03

minus04

minus05

minus08 minus06 minus04 minus02 0 02 04

020

minus02minus04

f2 ob

j

f1obj

f3 ob

j

(a) Nondominated solutions

0

minus01

minus02

minus03

minus04

minus05


f3 ob

j

f1obj

020

minus02minus04

f2 ob

j

(b) Nondominated solutions after clustering

Figure 5 Nondominated solutions obtained fromMOGA

The main effect (120583119894(119909119894)) and variance (2119894(119909119894)) of variable 119909119894

can be given as

120583119894 (119909119894)

equiv int sdot sdot sdot int 119910 (1199091 119909119898) 1198891199091 sdot sdot sdot 119889119909119894minus1119889119909119894+1 sdot sdot sdot 119889119909119898 minus 120583total

(19)

2

119894(119909119894) = int [120583119894 (119909119894)]

2119889119909119894 (20)

The proportion of the variance due to design variable x119894 tototal variance of the model can be calculated by dividing (18)by (20)This value indicated the effect of design variable x119894 onthe objective function

2

119894

2

total=

int [120583119894(119909119894)]2

119889119909119894

int sdot sdot sdot int [119910 (1199091 119909119898) minus 120583total]21198891199091 sdot sdot sdot 119889119909119898

(21)

Thus if this value is large x119894 has a large effect on the objectivefunction If not x119894 can be considered as less important designvariable for the objective function

322 Self-Organizing Map (SOM) SOM is an unsupervisedneural network technique that classifies organizes and visu-alizes large data sets SOM is a nonlinear projection algorithm


2D map

Input layer

Input vector

Weight vector[w1 w2 wm]

[f1 f2 fm]

Figure 6 Self-organizing map (SOM)

fromhigh- to low-dimensional spaceThis projection is basedon self-organization of a low-dimensional array of neuronsIn the projection algorithm the weights between the inputvector (input data) and the array of neurons are adjusted torepresent features of the high-dimensional original data onthe low-dimensional map The closer two-input data are inthe original space the closer the response of two neighboringneurons is in the low-dimensional spaceThus SOM reducesthe dimension of input data while preserving their features

A neuron used in SOM is associated with weight vectorw119894 = [1199081198941 1199081198942 119908119894119898] (119894 = 1 119873) where 119898 is equalto the dimension of the input vector and N is the number ofneurons Each neuron is connected to adjacent neurons by aneighborhood relation and usually forms a two-dimensionalhexagonal topology as shown in Figure 6

The learning algorithmof SOM[17] is startedwith findingthe best-matching unit (w119888) which is the closest to the inputvector f = [1198911 119891119898] as follows

1003817100381710038171003817f minus w1198881003817100381710038171003817 = min 1003817100381710038171003817f minus w119896

1003817100381710038171003817 (119896 = 1 119873) (22)

Then the weight vector of the best-matching unit and itsneighbors is adjusted to make its location much closer to theinput vector Repeating this learning algorithm the weightvectors become smooth not only locally but also globally Asa result the sequence of close vectors in the original spaceresults in a sequence of corresponding neighboring neuronsin the two-dimensional map

4 Results

41 Results of Shape Optimization In this study the mul-tiobjective shape optimization process shown in Figure 3 isiterated three times Figure 7 shows the initial sample pointsused to construct the Kriging model and the nondominatedsolutions obtained after three iterations of the multiobjectiveshape optimization It can be confirmed that the obj1-obj2 andobj1ndashobj3 show a strong trade-off relation while obj2-obj3

Case 1

Case 2

Case 3

Initial sampleNondominated solutions

06

08

04

02

0

minus02

minus04

minus06

minus08minus1 minus05 0 105

Obj1

Obj

2

(a) Obj1 versus Obj2

Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj2

Obj

3

(b) Obj2 versus Obj3

Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj1

Obj

3

(c) Obj1 versus Obj3

Figure 7 Results of shape optimization

shows a linear relation This implies that the performance interms of wave drag at the low-speed design speed (F119873 = 022)is opposed with those at both intermediate (F119873 = 027) andhigh design speeds (F119873 = 0305) and that the performance interms of wave drag at intermediate design speed is consistentwith that at high design speed The total computational time


taken to obtain the nondominated solutions was 20min ona PC (266GHz Intel Xeon 4 Gbytes RAM Mac OS) If thisshape optimization had been performed without the Krigingmodel even though the computational time of SSF for onecase is 10 seconds the total computational time taken toobtain the solutions would be about 18 days (10 seconds times 512populations times 100 generations times 3 iterations)

Among the nondominated solutions three typical opti-mum solutionsmdashCase 1 Case 2 and Case 3mdashare selected toinvestigate wave drag performance over a wide speed rangeas shown in Figure 7 Case 1 is the solution with low wavedrag performance at low design speed Case 2 is the solutionwith low wave drag performance at high design speedCase 3 is a compromise solution between Case 1 and Case2 Figure 8 compares the wave and total drag coefficient (119862119882and 119862119879) between the initial and each of these three solutionsCase 1 shows the best drag performance in the low-speedregime while Case 2 shows the best drag performance in thehigh-speed regime As expected Case 3 shows intermediatedrag performance at all speed ranges Table 2 shows thepercentages of wave and total drag coefficient variation fromthose of baseline configurationThe friction drag is calculatedaccording to the International Towing Tank Conference(ITTC) model-ship correlation line [30]

119862119865 =0075

(log10Reminus2)2

(23)

where Re is Reynolds number for the respective temperatureTable 3 displays variations in geometric properties Even

though each optimum hull exhibits very small changes inthe wetted surface area (119878wet the amount of area of hull thatis under water) and displacement (nabla the volume of fluiddisplaced by hull) the wave performance of these hull formsis improved very much

42 Results of Design Knowledge Mining

421 Analysis of Variance (ANOVA) To investigate the ef-fects of design variables on each objective function ANOVAis applied to the results of shape optimization According toresults DV10 and DV11 have a large effect on 1198911obj while DV8and DV12 have large effects on 1198912obj and 119891

3

obj as shown inFigure 9 1198911obj is the drag performance at relatively low speedand 1198912obj and 119891

3

obj are those at high speedFigure 10 compares the profile plan of the initial hull

form with that of Case 1 (low-speed hull form) and Case 2which has good drag performance at low and high speedsrespectively Compared with initial hull form Case 1 has aslightly wider section shape of the lower part (near DV11)and a narrower waterline in the bow part (near DV10)while Case 2 has a narrower waterline near DV12 and DV8According to the general knowledge about ship design hullform with a wide section shape shows the good wave dragperformance at the low speedwhile thatwith a narrow sectionshape shows the good wave drag performance at the highspeed Thus the information (design knowledge) obtainedthrough ANOVA corresponds to the general knowledgeabout ship design

02 025 03

4

3

2

1

CW

andCT

FN

CW (initial)CW (Case 1)CW (Case 2)CW (Case 3)

CT (initial)CT (Case 1)CT (Case 2)CT (Case 3)

times10minus03

Figure 8 Comparison of the wave and total drag coefficient for theinitial and three typical solutions

Table 2 Percentage of wave and total drag coefficient variation

Δ1198621

119882Δ1198622

119882Δ1198623

119882Δ1198621

119879Δ1198622

119879Δ1198623

119879

Wave drag () Total drag ()Case 1 minus7090 minus065 minus1462 minus1714 minus032 minus887

Case 2 2320 minus5660 minus5660 560 minus2747 minus3414

Case 3 minus4317 minus3218 minus3218 minus1044 minus1570 minus2261

Table 3 Variation of wetted surface area (119878wet) and displacement(nabla)

119878wet () nabla ()Case 1 082 minus004

Case 2 060 minus027

Case 3 016 minus019

422 Self-Organizing Map (SOM) To identify the param-eter controlling the trade-off between low- and high-speedgeometries SOM is applied to the nondominated solutionsobtained from shape optimization The number of solutionsused for SOM is 1024 Figure 11 shows the SOMobtained afterclustering based on the objective functions As described inSection 322 SOM consists of a large number of neurons thathave a hexagonal topology and the same dimensional vectorvalue as the input data Thus SOM can be colored by eachobjective function value and design variable value Figure 12shows the SOMs colored by each objective function value

The solutions on the left-hand side have a preferable 1198911objand unfavorable 1198912obj and 119891

3

obj while the solutions on the


22

22

7

8

4

40

151

DV10

DV11

DV8

DV7

DV11DV4

(a) 1198911obj

14

5

6

13

24

3

13

22

DV12

DV11

DV10DV8

DV5

DV4DV2

DV7

(b) 1198912obj

71

23

15

36

1

9

6

20

DV12

DV11

DV10

DV8

DV7

DV5DV4 DV2

(c) 1198913obj

Figure 9 Results of ANOVA

right-hand side have an unfavorable 1198911obj and preferable 1198912objand 1198913obj Thus it can be seen that there is a severe trade-offbetween low-speed wave drag (1198911obj) and high-speed wavedrag (1198912obj and 119891

3

obj) performances This trade-off was alsoconfirmed in Figure 7 To identify the relation between objec-tive functions and design variables the SOM is also coloredby each design variable value

Figure 13 shows the SOMs colored by DV4 DV7 DV8DV10 DV11 and DV12 which have relatively large effectson objective functions according to the ANOVA results inFigure 9

5

6

7

8

9

10

11

12

FrontBack

10

11

0025

0

0

minus0025

minus005

minus005 005

zH

yB

(a) Case 1 (red line) and initial hull form (black line)

5

6

7

8

9

10

11

12

FrontBack

812

0025

0

0

minus0025

minus005

minus005 005

zH

yB

(b) Case 2 (red line) and initial hull form (black line)

Figure 10 Comparison of the profile plan

Figure 11 SOM obtained after clustering based on the similarity ofobjective functions

The solutions on the left-hand side have relatively largevalues of DV4 DV7 DV8 and DV11 while those of theright-hand side have relatively small values Thus the low-speed geometry should have a wider section shape while thehigh-speed geometry should have a narrower section shapeThis corresponds to general knowledge of ship hull designAlthough DV10 and DV12 have large effects on objectivefunctions according to ANOVA the values of both DV10and DV12 are always small regardless of low- or high-speedgeometry It seems that small DV10 and DV12 (narrower


Obj1

minus085 minus072 minus058 minus045 minus032 minus018 minus005 008 022 035

(a) 1198911obj

Obj2

minus06 minus051 minus043 minus035 minus026 minus018 minus009 minus001 007 016

(b) 1198912obj

Obj3

minus06 minus054 minus047 minus041 minus035 minus028 minus022 minus016 minus009 minus003

(c) 1198913obj

Figure 12 SOMs colored by objective functions

waterline in the bow region) are necessary conditions for anondominated solution

Thus the design knowledge obtained through the presentdata mining can be summarized as follows (Figure 14)

(1) A narrower waterline in the bow region (small DV10and DV11) is a necessary condition for a nondomi-nated solution

(2) A wider section shape (large DV4 DV7 DV8 andDV12) is preferable for low-speed performance whilea narrower section shape (small DV4 DV7 DV8 andDV12) is necessary for high-speed performance

5 Conclusion and Future Works

In this study a high-efficiency design exploration frameworkconsisting of multiobjective shape optimization and designknowledge extraction has been suggested for hull form de-sign In the multiobjective shape optimization a multiobjec-tive genetic algorithm (MOGA) using the response surfacemethodology was adopted The present framework has beenapplied to hull form optimization exploring the minimumwave drag configuration over a wide range of speeds Theresults show that the present method markedly reduces thedesign period

In design knowledge extraction two data mining tech-niques functional analysis of variance (ANOVA) and self-organizing map (SOM) are applied to acquire useful designknowledge regarding hull form The results of data miningrevealed the design variables controlling wave drag perfor-mance and its corresponding geometric features

minus0

005

minus0

0039

minus0

0028

minus0

0017

minus0

0005

000

06

000

17

000

28

000

39

000

5

(a) DV4

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(b) DV7

004

5

004

6

004

7

004

8

004

9

005

005

1

005

2

005

3

005

4

(c) DV8

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

(d) DV10

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(e) DV11

004

5

004

5

004

5

004

5

004

5

004

6

004

6

004

6

004

6

004

6

(f) DV12

Figure 13 SOMs colored by design variables


5

6

7

8

9

1012

7

812 10

Low-speed hull form

Low-speed performance

Nondominated solutions

DV10 and DV12 are small

DV4 DV7 and DV12 are large and dv8 is

middle

DV4 DV7 DV8 and DV12 are small

5

6

7

8

9

1012

11 7

8

11

12 10

High-speed hull form

Front

Front

Back

Back

Hig

h-sp

eed

perfo

rman

ce

1111

0025

0

0

minus0025

minus005

minus005 005yB

zH

0025

0

0

minus0025

minus005

minus005 005yB

zH

Figure 14 Design knowledge obtained from data mining

As a futurework the presentmethodwill be applied to thenonlinear dynamics of ocean wave local sea level fluctuation[31 32] and seakeeping problems for the further validation

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Kluwer Academic Publishers Boston MassUSA 1989

[2] K Deb Multi-objective optimization using evolutionary algo-rithms John Wiley amp Sons Chichester UK 2001

[3] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works Part 1 (of 6) pp 1942ndash1948 December 1995

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

[5] Y Tahara S Tohyama and T Katsui ldquoCFD-based multi-objec-tive optimizationmethod for ship designrdquo International Journalfor Numerical Methods in Fluids vol 52 no 5 pp 499ndash5272006

[6] 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

[7] Y Tahara T Hino M Kandasamy W He and F Stern ldquoCFD-based multiobjective optimization of waterjet propelled highspeed shipsrdquo in Proceedings of the 11th International Conferenceon Fast Sea Transportation (FAST rsquo11) Honolulu Hawaii 2011

[8] M A Gammon ldquoOptimization of fishing vessels using a multi-objective genetic algorithmrdquo Ocean Engineering vol 38 no 10pp 1054ndash1064 2011

[9] J T Knight F T Zahradka D J Singer and M D ColletteldquoMulti-objective particle swarm optimization of a planing craft

with uncertanityrdquo in Proceedings of the 11th International Con-ference on Fast Sea Transportation (FAST rsquo11) Honolulu Hawaii2011

[10] G E P Box W G Hunter and J S Hunter Statistics for Exper-imenters John Wiley amp Sons New York NY USA 1978

[11] S Jeong M Murayama and K Yamamoto ldquoEfficient optimiza-tion design method using kriging modelrdquo Journal of Aircraftvol 42 no 2 pp 413ndash420 2005

[12] K Sugimura S Obayashi and S Jeong ldquoMulti-objective opti-mization and design rule mining for an aerodynamically effi-cient and stable centrifugal impellerwith a vaned diffuserrdquoEngi-neering Optimization vol 42 no 3 pp 271ndash293 2010

[13] D R JonesM Schonlau andW JWelch ldquoEfficient global opti-mization of expensive black-box functionsrdquo Journal of GlobalOptimization vol 13 no 4 pp 455ndash492 1998

[14] O Schabenberger and C A Cotway Statistical Methods ForSpatial Data Analysis New York NY USA Champmand andHallCRC 1st edition 2004

[15] T C Bailey and W J Krzanowski ldquoAn overview of approachesto the analysis andmodelling ofmultivariate geostatistical datardquoMathematical Geosciences vol 44 no 4 pp 381ndash383 2012

[16] S Jeong K Chiba and S Obayashi ldquoData mining for aerody-namic design spacerdquo Journal of Aerospace Computing Informa-tion and Communication vol 2 no 11 pp 452ndash469 2005

[17] T Kohonen Self-Organizing Maps vol 30 Springer BerlinGermany 1995

[18] F H Todd and F X ForestA Proposed New Basis For the Designof Single New ScrewMerchAnt Ship Forms and a Standard Seriesof Lines Transactions of SNAME 1951

[19] F H Todd Some Further Experiments on Single ScrewMerchantShip Forms-Series 60 Transactions of SNAME 1953

[20] H Lackenby On the Systematic Geometrical Variation of ShipForms vol 92 Transaction of Royal Institute of Naval Archi-tects 1950


[21] H Y Kim C Yang and F Noblesse ldquoHull form optimizationof reduced resistance and improved seakeeping via practicaldesign-oriented CFD toolrdquo in Proceedings of the Grand Chal-lenges in Modeling and Simulation Conference pp 375ndash3852010

[22] H Y Kim C Yang and H H Chun ldquoA combined local andglobal hull formmodification for hydrodynamic optimizationrdquoin Proceedings of the 28th Symposium on Naval HydrodynamicsPasadena CA USA 2010

[23] A de BoerM S van der Schoot andH Bijl ldquoMesh deformationbased on radial basis function interpolationrdquo Computers andStructures vol 85 no 11ndash14 pp 784ndash795 2007

[24] C Yang H Y Kim G Delhommeau and F NoblesseldquoThe Neumann-Kelvin and Neumann-Michell linear modelsof steady flow about a shiprdquo in Proceedings of the 12th Inter-national Congress of the International Maritime Association ofthe Mediterranean (IMAM rsquo07) pp 129ndash136 Varna BulgariaSeptember 2007

[25] C Yang H Y Kim and F Noblesse ldquoA practical method forevaluating steady flow about a shiprdquo in Proceedings of the 7thInternational Conference on Fast Sea Transportation 2007

[26] M D McKay R J Beckman andW J Conover ldquoA comparisonof three methods for selecting values of input variables in theanalysis of output from a computer coderdquo Technometrics vol21 no 2 pp 239ndash245 1979

[27] A K Jain M N Murty and P J Flynn ldquoData clustering areviewrdquo ACM Computing Surveys vol 31 no 3 pp 316ndash3231999

[28] P J Rousseeuw ldquoSilhouettes a graphical aid to the interpreta-tion and validation of cluster analysisrdquo Journal of Computationaland Applied Mathematics vol 20 pp 53ndash65 1987

[29] S Jeong and K Shimoyama ldquoReview of data mining for multi-disciplinary design optimizationrdquo Proceedings of the Institutionof Mechanical Engineers G vol 225 no 5 pp 469ndash479 2011

[30] ITTC ldquo1978 performance prediction methodrdquo ITTC Recom-mended Procedures and Guideline Procedure 7 5-02-01-03Revision 00 2002

[31] N Su Z-G Yu V Anh andK Bajracharya ldquoFractal tidal wavesin coastal aquifers induced both anthropogenically and natu-rallyrdquo Environmental Modelling and Software vol 19 no 12 pp1125ndash1130 2004

[32] M Li C Cattani and S-Y Chen ldquoViewing sea level by a one-dimensional random function with long memoryrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 654284 13pages 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


developed so that the designer can easily choose the morefavorable candidates More recently Knight et al [9] appliedparticle swarm optimization to incorporate uncertainty inparameters into the optimization of a planning craft

However the number of CFD runs required in thesemethods is too large Furthermore if these methods are com-bined with a time-consuming high fidelity CFD solver it isdifficult to complete the design process in reasonable timeTherefore it is very important to either have a highly efficientCFD tool or to use an accurate approximation model (egresponse surface) to partially replace the CFD runs for theCFD-based optimization tool to be useful in practical shipdesign

The response surface method is a well-established ap-proach for creating approximation models based on sampledata that can be obtained from physical experiments andcomputer simulations [10] A major advantage of using theresponse surface method in CFD-based design optimizationis to reduce the computational costs associated with the CFDruns A given optimization problem can be approximated bythe response surface method with smooth functions that canimprove computational efficiency in the optimization pro-cess The response surface method has been applied to manyCFD-based aerodynamics optimization problems includinga centrifugal fan and transonic airfoil and this approach wasshown to be quite efficient in exploring the design space[11 12]

In the present study the Kriging response surface model[13ndash15] is implemented in the CFD-based ship hydrodynam-ics optimization tool to provide the objective function valuesassociated with the hydrodynamics performance efficientlyThe Kriging model has a sufficient flexibility to representthe nonlinear and multimodal functions which are oftenappeared in hydrodynamic performances Another merit ofKriging model is that it predicts not only function value butalso the uncertainty of the estimated function value Basedon both the function value and its uncertainty it is possibleto improve the accuracy of Kriging model during the designprocess

In this study the Kriging model based multiobjectiveshape optimization method is applied to the optimal hullform design to minimize wave drag over a wide range ofdesign speeds For the modeling of hull form a surface mod-ificationmethod using shiftingmethod and radial basis func-tion interpolation is introducedThe shiftingmethodmakes itpossible to realize a smooth junction of the modified portionwith the original design The radial basis function is usedfor the local modifications of hull formThus these methodsenable both global and local modification of hull form withfewer design variables In design knowledge extraction twodata mining techniquesmdashfunctional analysis of variance(ANOVA) [16] and self-organizing map (SOM) [17]mdashareapplied to acquire useful design knowledge about a hull formFunctional analysis of variance (ANOVA) shows effect ofeach design variable on objective functions Self-organizingmap (SOM) identifies the design variables controlling thetrade-off between high-speed wave drag and low-speed wavedrag Numerical results show that the design explorationframework not only can explore superior hull forms within

a short computational time but also supplies the insight ofthe effect of each design variable on the drag performance atdifferent design speeds

The remainder of this paper is organized as follows Inthe next two sections design problem definition and designexploration used in this studywill be introduced respectivelyThis will be followed by results of shape optimization anddesign knowledge extraction

2 Design Problem Definition

21 Hull Form Definition and Design Variables An accurateand effective hull surface representation and modificationtechnique plays an important role in theCFD-basedhull formhydrodynamic optimization In the present study the Series-60 (block coefficient 119862119887 = 06) [18 19] hull model is adoptedas the initial hull form and modified to represent a new hullform using the shifting method [20ndash22] and the radial basisfunction (RBF) interpolation [23] during shape optimizationThese methods make it possible to represent both global andlocalmodifications of hull formwith smaller design variables

First the sectional area curve of the initial hull form ismodified using the shiftingmethod whichmoves the stationsof the initial hull form along the longitudinal direction Theamount of movement is determined by comparing themodified sectional area curve with the original curve Inthis study the sectional area curve of the initial hull form ismodified using (1a) and (1b)

119891119899(119909) = 119891

0(119909) + 119892 (119909 1205721 1205722) (1a)

119892 (119909 1205721 1205722)

=

1205721radic05 (1 minus cos(2120587119909 minus 1199091

1205722 minus 1199091

)) 1199091 le 119909 le 1205722

minus1205721radic05 (1 minus cos(2120587119909 minus 1205722

1205722 minus 1199092

)) 1205722 le 119909 le 1199092

0 elsewhere(1b)

where 119891119899(119909) and 1198910(119909) denote the sectional area curve ofthe new and initial hull forms respectively [21 22] Thus119892(119909 1205721 1205722) indicates the amount of shape modification atposition x and 1199091 and 1199092 are fixed points

As shown in Figure 1 the slope of the sectional areacurve is defined by 1205721 and the location of the fixed sta-tion is controlled using 1205722 By changing the parameters 1205721and 1205722 various sectional area curves can be obtained Incomparison with the spline polynomial representation thisformulation can prevent the generation of unrealistic hullforms associated with oscillation of the new sectional areacurve In this study the sectional area curves of both foreand aft bodies are modified simultaneously using four designvariables entrance angle (1205721119891) run angle (1205721119886) fore-bodyvariation (1205722119891) and aft-body variation (1205722119886)

Once the modified sectional area curve is obtainedlocal modification of the hull form is performed usingRBF interpolation In this study the displacement of point




2119891030 plusmn 02




1

05

00 05 1

f(x)

x

1205721

0

minus1205721

g(x)

x1 1205722 x2

g(x)

f0(x)

fn(x)



119904 (x) =119873

sum

119895=1

120582119895120593 (10038171003817100381710038171003817x minus x119895

10038171003817100381710038171003817) + 119901 (x) (2)


120593 (x) = (1 minus x)4 (4 x + 1) (3)


119901 (x) = 1198881 + 1198882119909 + 1198883119910 + 1198884119911 (4)


119904 (x119895) = 119891119895 119895 = 1 119873 (5)

91

475

6 8311

1012 2

(a)

(b)



119873

sum

119895=1

120582119895119901 (x119895) = 0 119895 = 1 119873 (6)


(119891

0) = (

119872 119875

1198751198790)(120582

119888) (7)

where 120582 = [1205821 1205822 120582119873] 119888 = [1198881 1198882 1198883 1198884] and 119891 =[1198911 1198912 119891119873]


119872119894119895 = 120593 (10038161003816100381610038161003816x119894 minus x119895

10038161003816100381610038161003816) 119894 119895 = 1 119873 (8a)

119875119894119895 = 119901119895 (x119894) 119894 = 1 119873 119895 = 1 4 (8b)




1198911

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 022 (9a)

1198912

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 027 (9b)

1198913

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 0305 (9c)


and 119862119889119882








119910 (x) = 120573 + r119879Rminus1 (y minus 1120573) (10)


R (x119894 x119895) = exp[minus119898

sum

119896=1

120579119896

10038161003816100381610038161003816119909119894

119896minus 119909119895

119896

10038161003816100381610038161003816

2

] 119894 119895 = 1 119899 (11)


r119879 (x) = [R (x x1) R (x x2) R (x x119899)] (12)


120573 =1119879Rminus1y1119879Rminus11

(13)


ln (120573 2 120579) = minus1198992ln (2) minus 1

2ln (|R|) (14)


2=

(y minus 1120573)119879

R (y minus 1120573)119899

(15)



1199042(x) = 2 [

[


2

11015840Rminus11]

]

(16)











Design converged



shape optimization


No

Yes











alua

ted

by K

rigin

g m

odel

Evaluated by SSF

060504030201

0minus01

minus02

minus03


Obj1

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

060504030201

0minus01

minus02

minus03


Obj2

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

04

03

02

01

0

minus01

minus02

minus03


Obj3





2


0

minus01

minus02

minus03

minus04

minus05


020

minus02minus04

f2 ob

j

f1obj

f3 ob

j


0

minus01

minus02

minus03

minus04

minus05


f3 ob

j

f1obj

020

minus02minus04

f2 ob

j




can be given as

120583119894 (119909119894)


(19)

2

119894(119909119894) = int [120583119894 (119909119894)]

2119889119909119894 (20)


2

119894

2

total=

int [120583119894(119909119894)]2

119889119909119894


(21)




2D map

Input layer

Input vector


[f1 f2 fm]






1003817100381710038171003817 (119896 = 1 119873) (22)


4 Results


Case 1

Case 2

Case 3


06

08

04

02

0

minus02

minus04

minus06


Obj1

Obj

2


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj2

Obj

3


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj1

Obj

3







119862119865 =0075

(log10Reminus2)2

(23)





3


3



02 025 03

4

3

2

1

CW

andCT

FN



times10minus03



Δ1198621

119882Δ1198622

119882Δ1198623

119882Δ1198621

119879Δ1198622

119879Δ1198623

119879






Case 2 060 minus027

Case 3 016 minus019



3



22

22

7

8

4

40

151

DV10

DV11

DV8

DV7

DV11DV4

(a) 1198911obj

14

5

6

13

24

3

13

22

DV12

DV11

DV10DV8

DV5

DV4DV2

DV7

(b) 1198912obj

71

23

15

36

1

9

6

20

DV12

DV11

DV10

DV8

DV7

DV5DV4 DV2

(c) 1198913obj



3



5

6

7

8

9

10

11

12

FrontBack

10

11

0025

0

0

minus0025

minus005

minus005 005

zH

yB


5

6

7

8

9

10

11

12

FrontBack

812

0025

0

0

minus0025

minus005

minus005 005

zH

yB






Obj1


(a) 1198911obj

Obj2


(b) 1198912obj

Obj3


(c) 1198913obj









minus0

005

minus0

0039

minus0

0028

minus0

0017

minus0

0005

000

06

000

17

000

28

000

39

000

5

(a) DV4

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(b) DV7

004

5

004

6

004

7

004

8

004

9

005

005

1

005

2

005

3

005

4

(c) DV8

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

(d) DV10

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(e) DV11

004

5

004

5

004

5

004

5

004

5

004

6

004

6

004

6

004

6

004

6

(f) DV12



5

6

7

8

9

1012

7

812 10

Low-speed hull form





middle


5

6

7

8

9

1012

11 7

8

11

12 10


Front

Front

Back

Back

Hig

h-sp

eed

perfo

rman

ce

1111

0025

0

0

minus0025

minus005

minus005 005yB

zH

0025

0

0

minus0025

minus005

minus005 005yB

zH



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2119891030 plusmn 02




1

05

00 05 1

f(x)

x

1205721

0

minus1205721

g(x)

x1 1205722 x2

g(x)

f0(x)

fn(x)



119904 (x) =119873

sum

119895=1

120582119895120593 (10038171003817100381710038171003817x minus x119895

10038171003817100381710038171003817) + 119901 (x) (2)


120593 (x) = (1 minus x)4 (4 x + 1) (3)


119901 (x) = 1198881 + 1198882119909 + 1198883119910 + 1198884119911 (4)


119904 (x119895) = 119891119895 119895 = 1 119873 (5)

91

475

6 8311

1012 2

(a)

(b)



119873

sum

119895=1

120582119895119901 (x119895) = 0 119895 = 1 119873 (6)


(119891

0) = (

119872 119875

1198751198790)(120582

119888) (7)

where 120582 = [1205821 1205822 120582119873] 119888 = [1198881 1198882 1198883 1198884] and 119891 =[1198911 1198912 119891119873]


119872119894119895 = 120593 (10038161003816100381610038161003816x119894 minus x119895

10038161003816100381610038161003816) 119894 119895 = 1 119873 (8a)

119875119894119895 = 119901119895 (x119894) 119894 = 1 119873 119895 = 1 4 (8b)




1198911

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 022 (9a)

1198912

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 027 (9b)

1198913

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 0305 (9c)


and 119862119889119882








119910 (x) = 120573 + r119879Rminus1 (y minus 1120573) (10)


R (x119894 x119895) = exp[minus119898

sum

119896=1

120579119896

10038161003816100381610038161003816119909119894

119896minus 119909119895

119896

10038161003816100381610038161003816

2

] 119894 119895 = 1 119899 (11)


r119879 (x) = [R (x x1) R (x x2) R (x x119899)] (12)


120573 =1119879Rminus1y1119879Rminus11

(13)


ln (120573 2 120579) = minus1198992ln (2) minus 1

2ln (|R|) (14)


2=

(y minus 1120573)119879

R (y minus 1120573)119899

(15)



1199042(x) = 2 [

[


2

11015840Rminus11]

]

(16)











Design converged



shape optimization


No

Yes











alua

ted

by K

rigin

g m

odel

Evaluated by SSF

060504030201

0minus01

minus02

minus03


Obj1

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

060504030201

0minus01

minus02

minus03


Obj2

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

04

03

02

01

0

minus01

minus02

minus03


Obj3





2


0

minus01

minus02

minus03

minus04

minus05


020

minus02minus04

f2 ob

j

f1obj

f3 ob

j


0

minus01

minus02

minus03

minus04

minus05


f3 ob

j

f1obj

020

minus02minus04

f2 ob

j




can be given as

120583119894 (119909119894)


(19)

2

119894(119909119894) = int [120583119894 (119909119894)]

2119889119909119894 (20)


2

119894

2

total=

int [120583119894(119909119894)]2

119889119909119894


(21)




2D map

Input layer

Input vector


[f1 f2 fm]






1003817100381710038171003817 (119896 = 1 119873) (22)


4 Results


Case 1

Case 2

Case 3


06

08

04

02

0

minus02

minus04

minus06


Obj1

Obj

2


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj2

Obj

3


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj1

Obj

3







119862119865 =0075

(log10Reminus2)2

(23)





3


3



02 025 03

4

3

2

1

CW

andCT

FN



times10minus03



Δ1198621

119882Δ1198622

119882Δ1198623

119882Δ1198621

119879Δ1198622

119879Δ1198623

119879






Case 2 060 minus027

Case 3 016 minus019



3



22

22

7

8

4

40

151

DV10

DV11

DV8

DV7

DV11DV4

(a) 1198911obj

14

5

6

13

24

3

13

22

DV12

DV11

DV10DV8

DV5

DV4DV2

DV7

(b) 1198912obj

71

23

15

36

1

9

6

20

DV12

DV11

DV10

DV8

DV7

DV5DV4 DV2

(c) 1198913obj



3



5

6

7

8

9

10

11

12

FrontBack

10

11

0025

0

0

minus0025

minus005

minus005 005

zH

yB


5

6

7

8

9

10

11

12

FrontBack

812

0025

0

0

minus0025

minus005

minus005 005

zH

yB






Obj1


(a) 1198911obj

Obj2


(b) 1198912obj

Obj3


(c) 1198913obj









minus0

005

minus0

0039

minus0

0028

minus0

0017

minus0

0005

000

06

000

17

000

28

000

39

000

5

(a) DV4

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(b) DV7

004

5

004

6

004

7

004

8

004

9

005

005

1

005

2

005

3

005

4

(c) DV8

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

(d) DV10

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(e) DV11

004

5

004

5

004

5

004

5

004

5

004

6

004

6

004

6

004

6

004

6

(f) DV12



5

6

7

8

9

1012

7

812 10

Low-speed hull form





middle


5

6

7

8

9

1012

11 7

8

11

12 10


Front

Front

Back

Back

Hig

h-sp

eed

perfo

rman

ce

1111

0025

0

0

minus0025

minus005

minus005 005yB

zH

0025

0

0

minus0025

minus005

minus005 005yB

zH



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198911

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 022 (9a)

1198912

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 027 (9b)

1198913

obj =(119862119889

119882minus 119862119894

119882)

119862119894

119882

at 119865119873 = 0305 (9c)


and 119862119889119882








119910 (x) = 120573 + r119879Rminus1 (y minus 1120573) (10)


R (x119894 x119895) = exp[minus119898

sum

119896=1

120579119896

10038161003816100381610038161003816119909119894

119896minus 119909119895

119896

10038161003816100381610038161003816

2

] 119894 119895 = 1 119899 (11)


r119879 (x) = [R (x x1) R (x x2) R (x x119899)] (12)


120573 =1119879Rminus1y1119879Rminus11

(13)


ln (120573 2 120579) = minus1198992ln (2) minus 1

2ln (|R|) (14)


2=

(y minus 1120573)119879

R (y minus 1120573)119899

(15)



1199042(x) = 2 [

[


2

11015840Rminus11]

]

(16)











Design converged



shape optimization


No

Yes











alua

ted

by K

rigin

g m

odel

Evaluated by SSF

060504030201

0minus01

minus02

minus03


Obj1

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

060504030201

0minus01

minus02

minus03


Obj2

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

04

03

02

01

0

minus01

minus02

minus03


Obj3





2


0

minus01

minus02

minus03

minus04

minus05


020

minus02minus04

f2 ob

j

f1obj

f3 ob

j


0

minus01

minus02

minus03

minus04

minus05


f3 ob

j

f1obj

020

minus02minus04

f2 ob

j




can be given as

120583119894 (119909119894)


(19)

2

119894(119909119894) = int [120583119894 (119909119894)]

2119889119909119894 (20)


2

119894

2

total=

int [120583119894(119909119894)]2

119889119909119894


(21)




2D map

Input layer

Input vector


[f1 f2 fm]






1003817100381710038171003817 (119896 = 1 119873) (22)


4 Results


Case 1

Case 2

Case 3


06

08

04

02

0

minus02

minus04

minus06


Obj1

Obj

2


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj2

Obj

3


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj1

Obj

3







119862119865 =0075

(log10Reminus2)2

(23)





3


3



02 025 03

4

3

2

1

CW

andCT

FN



times10minus03



Δ1198621

119882Δ1198622

119882Δ1198623

119882Δ1198621

119879Δ1198622

119879Δ1198623

119879






Case 2 060 minus027

Case 3 016 minus019



3



22

22

7

8

4

40

151

DV10

DV11

DV8

DV7

DV11DV4

(a) 1198911obj

14

5

6

13

24

3

13

22

DV12

DV11

DV10DV8

DV5

DV4DV2

DV7

(b) 1198912obj

71

23

15

36

1

9

6

20

DV12

DV11

DV10

DV8

DV7

DV5DV4 DV2

(c) 1198913obj



3



5

6

7

8

9

10

11

12

FrontBack

10

11

0025

0

0

minus0025

minus005

minus005 005

zH

yB


5

6

7

8

9

10

11

12

FrontBack

812

0025

0

0

minus0025

minus005

minus005 005

zH

yB






Obj1


(a) 1198911obj

Obj2


(b) 1198912obj

Obj3


(c) 1198913obj









minus0

005

minus0

0039

minus0

0028

minus0

0017

minus0

0005

000

06

000

17

000

28

000

39

000

5

(a) DV4

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(b) DV7

004

5

004

6

004

7

004

8

004

9

005

005

1

005

2

005

3

005

4

(c) DV8

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

(d) DV10

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(e) DV11

004

5

004

5

004

5

004

5

004

5

004

6

004

6

004

6

004

6

004

6

(f) DV12



5

6

7

8

9

1012

7

812 10

Low-speed hull form





middle


5

6

7

8

9

1012

11 7

8

11

12 10


Front

Front

Back

Back

Hig

h-sp

eed

perfo

rman

ce

1111

0025

0

0

minus0025

minus005

minus005 005yB

zH

0025

0

0

minus0025

minus005

minus005 005yB

zH



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Design converged



shape optimization


No

Yes











alua

ted

by K

rigin

g m

odel

Evaluated by SSF

060504030201

0minus01

minus02

minus03


Obj1

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

060504030201

0minus01

minus02

minus03


Obj2

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

04

03

02

01

0

minus01

minus02

minus03


Obj3





2


0

minus01

minus02

minus03

minus04

minus05


020

minus02minus04

f2 ob

j

f1obj

f3 ob

j


0

minus01

minus02

minus03

minus04

minus05


f3 ob

j

f1obj

020

minus02minus04

f2 ob

j




can be given as

120583119894 (119909119894)


(19)

2

119894(119909119894) = int [120583119894 (119909119894)]

2119889119909119894 (20)


2

119894

2

total=

int [120583119894(119909119894)]2

119889119909119894


(21)




2D map

Input layer

Input vector


[f1 f2 fm]






1003817100381710038171003817 (119896 = 1 119873) (22)


4 Results


Case 1

Case 2

Case 3


06

08

04

02

0

minus02

minus04

minus06


Obj1

Obj

2


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj2

Obj

3


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj1

Obj

3







119862119865 =0075

(log10Reminus2)2

(23)





3


3



02 025 03

4

3

2

1

CW

andCT

FN



times10minus03



Δ1198621

119882Δ1198622

119882Δ1198623

119882Δ1198621

119879Δ1198622

119879Δ1198623

119879






Case 2 060 minus027

Case 3 016 minus019



3



22

22

7

8

4

40

151

DV10

DV11

DV8

DV7

DV11DV4

(a) 1198911obj

14

5

6

13

24

3

13

22

DV12

DV11

DV10DV8

DV5

DV4DV2

DV7

(b) 1198912obj

71

23

15

36

1

9

6

20

DV12

DV11

DV10

DV8

DV7

DV5DV4 DV2

(c) 1198913obj



3



5

6

7

8

9

10

11

12

FrontBack

10

11

0025

0

0

minus0025

minus005

minus005 005

zH

yB


5

6

7

8

9

10

11

12

FrontBack

812

0025

0

0

minus0025

minus005

minus005 005

zH

yB






Obj1


(a) 1198911obj

Obj2


(b) 1198912obj

Obj3


(c) 1198913obj









minus0

005

minus0

0039

minus0

0028

minus0

0017

minus0

0005

000

06

000

17

000

28

000

39

000

5

(a) DV4

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(b) DV7

004

5

004

6

004

7

004

8

004

9

005

005

1

005

2

005

3

005

4

(c) DV8

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

(d) DV10

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(e) DV11

004

5

004

5

004

5

004

5

004

5

004

6

004

6

004

6

004

6

004

6

(f) DV12



5

6

7

8

9

1012

7

812 10

Low-speed hull form





middle


5

6

7

8

9

1012

11 7

8

11

12 10


Front

Front

Back

Back

Hig

h-sp

eed

perfo

rman

ce

1111

0025

0

0

minus0025

minus005

minus005 005yB

zH

0025

0

0

minus0025

minus005

minus005 005yB

zH



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










alua

ted

by K

rigin

g m

odel

Evaluated by SSF

060504030201

0minus01

minus02

minus03


Obj1

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

060504030201

0minus01

minus02

minus03


Obj2

Eval

uate

d by

Krig

ing

mod

el

Evaluated by SSF

04

03

02

01

0

minus01

minus02

minus03


Obj3





2


0

minus01

minus02

minus03

minus04

minus05


020

minus02minus04

f2 ob

j

f1obj

f3 ob

j


0

minus01

minus02

minus03

minus04

minus05


f3 ob

j

f1obj

020

minus02minus04

f2 ob

j




can be given as

120583119894 (119909119894)


(19)

2

119894(119909119894) = int [120583119894 (119909119894)]

2119889119909119894 (20)


2

119894

2

total=

int [120583119894(119909119894)]2

119889119909119894


(21)




2D map

Input layer

Input vector


[f1 f2 fm]






1003817100381710038171003817 (119896 = 1 119873) (22)


4 Results


Case 1

Case 2

Case 3


06

08

04

02

0

minus02

minus04

minus06


Obj1

Obj

2


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj2

Obj

3


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj1

Obj

3







119862119865 =0075

(log10Reminus2)2

(23)





3


3



02 025 03

4

3

2

1

CW

andCT

FN



times10minus03



Δ1198621

119882Δ1198622

119882Δ1198623

119882Δ1198621

119879Δ1198622

119879Δ1198623

119879






Case 2 060 minus027

Case 3 016 minus019



3



22

22

7

8

4

40

151

DV10

DV11

DV8

DV7

DV11DV4

(a) 1198911obj

14

5

6

13

24

3

13

22

DV12

DV11

DV10DV8

DV5

DV4DV2

DV7

(b) 1198912obj

71

23

15

36

1

9

6

20

DV12

DV11

DV10

DV8

DV7

DV5DV4 DV2

(c) 1198913obj



3



5

6

7

8

9

10

11

12

FrontBack

10

11

0025

0

0

minus0025

minus005

minus005 005

zH

yB


5

6

7

8

9

10

11

12

FrontBack

812

0025

0

0

minus0025

minus005

minus005 005

zH

yB






Obj1


(a) 1198911obj

Obj2


(b) 1198912obj

Obj3


(c) 1198913obj









minus0

005

minus0

0039

minus0

0028

minus0

0017

minus0

0005

000

06

000

17

000

28

000

39

000

5

(a) DV4

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(b) DV7

004

5

004

6

004

7

004

8

004

9

005

005

1

005

2

005

3

005

4

(c) DV8

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

(d) DV10

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(e) DV11

004

5

004

5

004

5

004

5

004

5

004

6

004

6

004

6

004

6

004

6

(f) DV12



5

6

7

8

9

1012

7

812 10

Low-speed hull form





middle


5

6

7

8

9

1012

11 7

8

11

12 10


Front

Front

Back

Back

Hig

h-sp

eed

perfo

rman

ce

1111

0025

0

0

minus0025

minus005

minus005 005yB

zH

0025

0

0

minus0025

minus005

minus005 005yB

zH



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2D map

Input layer

Input vector


[f1 f2 fm]






1003817100381710038171003817 (119896 = 1 119873) (22)


4 Results


Case 1

Case 2

Case 3


06

08

04

02

0

minus02

minus04

minus06


Obj1

Obj

2


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj2

Obj

3


Case 1

Case 2

Case 3


08

06

04

02

0

minus02

minus04

minus06


Obj1

Obj

3







119862119865 =0075

(log10Reminus2)2

(23)





3


3



02 025 03

4

3

2

1

CW

andCT

FN



times10minus03



Δ1198621

119882Δ1198622

119882Δ1198623

119882Δ1198621

119879Δ1198622

119879Δ1198623

119879






Case 2 060 minus027

Case 3 016 minus019



3



22

22

7

8

4

40

151

DV10

DV11

DV8

DV7

DV11DV4

(a) 1198911obj

14

5

6

13

24

3

13

22

DV12

DV11

DV10DV8

DV5

DV4DV2

DV7

(b) 1198912obj

71

23

15

36

1

9

6

20

DV12

DV11

DV10

DV8

DV7

DV5DV4 DV2

(c) 1198913obj



3



5

6

7

8

9

10

11

12

FrontBack

10

11

0025

0

0

minus0025

minus005

minus005 005

zH

yB


5

6

7

8

9

10

11

12

FrontBack

812

0025

0

0

minus0025

minus005

minus005 005

zH

yB






Obj1


(a) 1198911obj

Obj2


(b) 1198912obj

Obj3


(c) 1198913obj









minus0

005

minus0

0039

minus0

0028

minus0

0017

minus0

0005

000

06

000

17

000

28

000

39

000

5

(a) DV4

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(b) DV7

004

5

004

6

004

7

004

8

004

9

005

005

1

005

2

005

3

005

4

(c) DV8

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

(d) DV10

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(e) DV11

004

5

004

5

004

5

004

5

004

5

004

6

004

6

004

6

004

6

004

6

(f) DV12



5

6

7

8

9

1012

7

812 10

Low-speed hull form





middle


5

6

7

8

9

1012

11 7

8

11

12 10


Front

Front

Back

Back

Hig

h-sp

eed

perfo

rman

ce

1111

0025

0

0

minus0025

minus005

minus005 005yB

zH

0025

0

0

minus0025

minus005

minus005 005yB

zH



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119862119865 =0075

(log10Reminus2)2

(23)





3


3



02 025 03

4

3

2

1

CW

andCT

FN



times10minus03



Δ1198621

119882Δ1198622

119882Δ1198623

119882Δ1198621

119879Δ1198622

119879Δ1198623

119879






Case 2 060 minus027

Case 3 016 minus019



3



22

22

7

8

4

40

151

DV10

DV11

DV8

DV7

DV11DV4

(a) 1198911obj

14

5

6

13

24

3

13

22

DV12

DV11

DV10DV8

DV5

DV4DV2

DV7

(b) 1198912obj

71

23

15

36

1

9

6

20

DV12

DV11

DV10

DV8

DV7

DV5DV4 DV2

(c) 1198913obj



3



5

6

7

8

9

10

11

12

FrontBack

10

11

0025

0

0

minus0025

minus005

minus005 005

zH

yB


5

6

7

8

9

10

11

12

FrontBack

812

0025

0

0

minus0025

minus005

minus005 005

zH

yB






Obj1


(a) 1198911obj

Obj2


(b) 1198912obj

Obj3


(c) 1198913obj









minus0

005

minus0

0039

minus0

0028

minus0

0017

minus0

0005

000

06

000

17

000

28

000

39

000

5

(a) DV4

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(b) DV7

004

5

004

6

004

7

004

8

004

9

005

005

1

005

2

005

3

005

4

(c) DV8

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

(d) DV10

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(e) DV11

004

5

004

5

004

5

004

5

004

5

004

6

004

6

004

6

004

6

004

6

(f) DV12



5

6

7

8

9

1012

7

812 10

Low-speed hull form





middle


5

6

7

8

9

1012

11 7

8

11

12 10


Front

Front

Back

Back

Hig

h-sp

eed

perfo

rman

ce

1111

0025

0

0

minus0025

minus005

minus005 005yB

zH

0025

0

0

minus0025

minus005

minus005 005yB

zH



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










22

22

7

8

4

40

151

DV10

DV11

DV8

DV7

DV11DV4

(a) 1198911obj

14

5

6

13

24

3

13

22

DV12

DV11

DV10DV8

DV5

DV4DV2

DV7

(b) 1198912obj

71

23

15

36

1

9

6

20

DV12

DV11

DV10

DV8

DV7

DV5DV4 DV2

(c) 1198913obj



3



5

6

7

8

9

10

11

12

FrontBack

10

11

0025

0

0

minus0025

minus005

minus005 005

zH

yB


5

6

7

8

9

10

11

12

FrontBack

812

0025

0

0

minus0025

minus005

minus005 005

zH

yB






Obj1


(a) 1198911obj

Obj2


(b) 1198912obj

Obj3


(c) 1198913obj









minus0

005

minus0

0039

minus0

0028

minus0

0017

minus0

0005

000

06

000

17

000

28

000

39

000

5

(a) DV4

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(b) DV7

004

5

004

6

004

7

004

8

004

9

005

005

1

005

2

005

3

005

4

(c) DV8

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

(d) DV10

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(e) DV11

004

5

004

5

004

5

004

5

004

5

004

6

004

6

004

6

004

6

004

6

(f) DV12



5

6

7

8

9

1012

7

812 10

Low-speed hull form





middle


5

6

7

8

9

1012

11 7

8

11

12 10


Front

Front

Back

Back

Hig

h-sp

eed

perfo

rman

ce

1111

0025

0

0

minus0025

minus005

minus005 005yB

zH

0025

0

0

minus0025

minus005

minus005 005yB

zH



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Obj1


(a) 1198911obj

Obj2


(b) 1198912obj

Obj3


(c) 1198913obj









minus0

005

minus0

0039

minus0

0028

minus0

0017

minus0

0005

000

06

000

17

000

28

000

39

000

5

(a) DV4

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(b) DV7

004

5

004

6

004

7

004

8

004

9

005

005

1

005

2

005

3

005

4

(c) DV8

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

004

5

(d) DV10

004

5

004

6

004

7

004

8

004

9

005

1

005

2

005

3

005

4

005

5

(e) DV11

004

5

004

5

004

5

004

5

004

5

004

6

004

6

004

6

004

6

004

6

(f) DV12



5

6

7

8

9

1012

7

812 10

Low-speed hull form





middle


5

6

7

8

9

1012

11 7

8

11

12 10


Front

Front

Back

Back

Hig

h-sp

eed

perfo

rman

ce

1111

0025

0

0

minus0025

minus005

minus005 005yB

zH

0025

0

0

minus0025

minus005

minus005 005yB

zH



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5

6

7

8

9

1012

7

812 10

Low-speed hull form





middle


5

6

7

8

9

1012

11 7

8

11

12 10


Front

Front

Back

Back

Hig

h-sp

eed

perfo

rman

ce

1111

0025

0

0

minus0025

minus005

minus005 005yB

zH

0025

0

0

minus0025

minus005

minus005 005yB

zH



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article development of an efficient hull form...

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