Method of designing optimum seakeeping hull form

CHINA SHIP SCIENTIFIC RESEARCH CENTER

Method of Designing Optimum

Seakeeping hull Form

Yl

Que

Wang Zhen

_{Dai Renyuan}

January

1986

(Presented at Third Symposium on Seakeeping Behavior

of Ships in China, Sept.

1983)

P. 0 . BOX 116, WUXI, JIANGSU

CHINA

CSSRC Report

METHOD OF DESIGNING OPTIMUM

SEAKEEPING HULL FORM

Yi _QILE

_{WANG ZBr}

DAi RENYtrAN

Chinc Ship Scientific Research Center

Abstract

A method of designing ship hull form with optimtm seakeeping performance is presented. The proposed methodology is based on

previ-c'usly developed ship performance _{prediction and evaluation programme}

of SRC. The problem to be _{solved is posed as a constrained} optimiza-tion problem, amenable to soluoptimiza-tion by non-linear programming technique, in which a set of hull geometry are taken as design variables and the weighted mean of maximwn waves acceptable to a specified criteria are taken as performance index. _{In this paper, the relationship between a} set of hull parameters and Lewis section _{representation is uniquely} defined. On the basis of this relationship, a good approximation of ship responses, and hence the performance index, can be obtained.

Opti-s _u

mlzation is executed by the method of Flexible penalty factor which offers the possibility to reduce the problem into an unconstrained one. Finally examples of optimal design are given with encouraging results. It was concluded that the methodology provides a basis for early-design synthesis of ship hull form with superior seakeeping performance.

I. _Introduction

In recent ship design practice, design efforts are being made more

ernphasizingly ori the synthesis of hull _{geometries in the earlier}

stages

of design which would lead to a better ship performance in realistic seaways rather than on solely looking after calm-water performances. The concept cf optimum seakeeping hull form design is highly esteemed and is becoming gradually a main goal in s!ip design practice.

Within the past two decades, advances and refinement in ship sea-keeping performance predication (1-5) have led to the development of technologies wiich make possible for the ship designer, in the early stages of design, to perform a comprehensive study and compare diffe-rent versions of design. However, in spite of the large amount of

c-putationa.l time, what was obtained, at best, was a relatively good hull

form among comparisons of limited design versions.

To solve this, attempts have been made by N.K. Bales (6) in his effort to relate seakeeping quality to the hull characteristics of

des-troyer types. Within specified hull parameter ranges, an optimum sea-keeping hull form can be found in the earlier stages of design process. It was noted, however, that the methodology proposed by N.K. Bales wa only available to the homogeneous class of destroyer type hull. The initiative of the authors, therefore, was to develope a new synthesis

technology covering a much broader range of application. The main fee-'

tures of the technology are:

Cl) Direct computational ship responses are used instead of relying on data base for quantification of model's seakeeping performance. This leads to a broadening of the range of application and relaxes the res-triction on a specific hull form.

The criteria are chosen according to the expected operating environments and predominant mission requirements; therefore, they are more concretely goal-oriented.

Additional constraints according to realistic condition are imposed where necessary during the process of optimization, thus making the result applicable in practice.

The approach includes three steps: the first is the selection of design parameters. The second is the quantification of the seakeeping performance index and the third is the optimization of the model. They

are described in the following sections.

Lt. Selection of Design Variable

conventional hull form parameters in practical _{application. The}

fundamental. premise of modeling effort was to define _{a relationship}

between design varia1les _{and seakeeping performance}

index. 'ro do so, it was adequate to _{start with}

_Lewis

section representation. _{First, a}

method of defining

_Lewis

section parameters by a carefully selected

set of hull form parameters was introduced

and subsequently, design

variables were chosen _{among them.} Defining the design waterline

Let us postulate design _{waterline, shown in Fig.}

1, is co1Tposed of four different

segments, which can be determined by the parameters,

such as ship length

(L), breadth (B), waterplane area fore and aft the

midship (A

, kA), half Lreadth at station O and 20 (y(0), y(20)), half entrance angle of waterline (e), aigLe _{of run}

°A and specified

number of station _{(Xei X1}

A total of eleven hull form

para-meters are involved. _{Segments of waterline f1 (X)}

and f2 (X) can be ex-pressed by a polynomial

of fourth order, the coefficietns of which

are determined by the end point values, the condition _{of continuety} of their first derivative _and

the area enclosed under _{the curve}

yf2(x) _y=B/2

t Cx)

ykx

b

_"-L

Fig. 1 _{Design waterline}

Defining the sectional _{area curve}

Initially, a preselected

sectional area curve is chosen as a

reference curve which

reflects, to some extent, the considerations and experience of the _{ship designer in trying}

to meet a specific

de-mand. We postulate that _{the required sectional}

area curve, as shown

in Fig. 2, may be _{determined by performing}

successive transformations

fran this reference curve. At first, _{linear transformation is applied} arid a new curve which satisfies

both end point conditions _and

specifi-ed location of maximtun cross section is cbtainspecifi-ed. The _{next step is to} modify the cuz-ve in such a way that the horizontal shift _{at each point}

of the curve obeys the parabolic law while keeping the end _points fix-ed. Thus the specified

area under the curve is attained. The parameters

in defining the sectional _{area curve involve: ship length (L), breadth}

(E), draft (T), displacement _{volume fore and aft of midship}

(V,VF), specified station numbers, _{area coefficient of xnaxirnum cross section} and ratios of sectional _{area at station C)}

and 20 to that of the

maxi-mum cross section.

y=t ( x) y=l _{y=f (x)}

A _A C20 ₂ 1 c0 20 Xml X _O L A.E. _FE.

Fig. 2 _{Sectional area curve}

3. Defini _{the underwater profile}

With the aid cf _{a similar reference cuz-ve, we difine the}

longi-tudinal central profile of the _{ship. The reference longitudinal}

cen-tral profile is modified by linear transformation only _{to adjust the} cut-up point of the keel at the stern.

When all of these have been completed a Lewis section representa-tion of the ship hull is determined. It may be noticed _{that quite a} large number of parameters _{have been involved in the above}

mentioned

approach. However most of them are taken to be constant in the

opti-inizatian process, except eight _{of them which are listed below as}

se-lected design variables:

Length of ship (L)

Beam of ship (B) Traft of ship (T)

Waterplane

area forward of amidships

Waterplane area aft of amidships _(kA) Displaced volume forward of amidships(VF)

Displaced volume aft of amidships _(VA)

III. _{Quantification of Seakeeping Performance}

Index

As mentioned before, limiting wave height is selected as the

seakeeping performance index in the process of optimization. The "limi-ting wave heightU is defined as the maximum wave height limited by the

criterion

set for predominant performance factors of a ship, navigat-ing at a specified mission speed and in specified ocean area. The steps in quantification of seakeeping performance index are described

as follows:

Specify the following items according to the specifications of a design, which include: Ship speed; relative wave heading angle

ß together with weighting factors Wm in accordance with relative

im-portance cf each heading ztm; time percentage S for execution of

mis-sion in each ocean area L; a set of criteria C for various

seakeep-ing performance factors K accordseakeep-ing to mission

demands,

say, magnitude of heave, pitch, roll, vertical displacement, velocity and accelera-tion at a point, subjective moaccelera-tion magnitude, probabilities of slanmm-irmg, deck wetness, propeller emergence and sonnai search, ease of sea sickness etc.

Represent the expected seaway by wave spectrum as proposed by

15th XTTC, in

which the significant wave height H and modal period

are parameters in defining the spectrum. The joint probability dis-tribution of an equivalent ocean environment can be expressed as the weighted mean of the joint probability distribution Pjt(Hi.Tj) in each ocean area L. Thus

p .(H.,T ) ₌

E s

ijL

(T1T)

(1)

1)

1

j

£

-5-(1), (7)

Modifiec str.p theory of C-erristma _{is used to predict}

the notion responses of the ship in five degrees of freedon at

specif!-ed spespecif!-ed arid relative heading. computations _{of two dimensional}

hydrodyriaDtic coefficients were performed beforehand as a data base for

a series of Lewis sections in the practical design range. These basic data are used to determine hydrodynamic coefficients of the design version by interpolation method. Within the extent of linear analysis, the ppS values of motion or frecuency of _{occurrence of related} pheno-mena i.e. the statistical description of various seakeeping performance

factors in long-crested irregular waves can be obtained. Further, a set of acceptable rnaxinìun wave heights }i. _{relevant to different} sea-keeping performance criteria CK can be defined for waves with dif fe-rent modal r)eriod T.

J.

Determine the least set of the acceptable rnaximwn wave heights which ijinits the execution of missions and find out the

correspon-ding limiting set of seakeeping performance factor X, thus

H,

minjH.

(2)

jm j jksnj

The limiting significant wave height for a definite wave

heading } is ¿etermined by weighting as follcs:

H1

'E

(W. H

/Ewj)

(3)

j

j

where W may be calculated according to (4) or (5).

-. P. (H.,, T4) ₍₄₎

i

j

-i

which is the frequency of occurance of waves with modal period T or

the

alternative expression

W. P. . (H. , T ) (5)

j ij jin

j

Finally, the seakeeping performance index i.e. the limiting wave height _{, taking different wave headings into account, can} be obtained by weighting _H as follcsz

H1j

= E

(W H

/ E

W)

(6)

mentioned before.

IV. _Optimization

1. Mathematical model

With the above discussion

_{in mind,}

_{it is understood that ship} performance index is

defined

as a

function

_{of a set of design}

vari-able s

H. =H.

_um

_um

_{(L,B,T,CIASA,VFSVA)}

Let vector X be expressed _{as the set of design variables}

l' ... X8)

= (L, B, T, C, _{, A, VF, VA)}

The problem to be solved here is _{then posed as the determination}

* * * * * * *

X

(L ,B ,T

,

C1

_Aw

IVFI

VA)

which maximizes the H , i.e.

* Lun H1. Cx ) max I H1 (X) h (x) = O 1.

g (x)O

J *

where X _{is the optimal solution of the problem.}

_Constraints,

either

expressed in equality as (h.(x) _{= O, i = 1, .., p)or in unequality as} (g(x) > o, j = i ... q) reflect specific design

_{requirements. The}

optimization so founded is subject to any consistent combination of the folling constraints

Constraint of displacement: The _{optimization is proceeded'} under the assuirtion of

_constant

_{displacement of the ship, i.e.}

w=W

o

Constraints of hull geometry parameters

(XJ1. X( (x1)2 or

_(X)0

= 1, 2, .. 8

Constraints of hull form coefficient

7

I =

_p

,

j

= 1, ... q _J (9)

(..WF)1E

_WF2

S&2

(T/L) T/L (T/L)2 (C/L)1 C/L( (c/L)2

(Cp)

(C,)

2 (r )

<C

<(C

VPA1

7P

VPA2

where C and C _{are respectively wate rplane coefficient fore and aft}

of atnidships, C and C _{are vertical prismatic coefficients fore}

and aft of

_amidships

Constraint of calm water resistance Res(X)

cr es. (13)

where Res, is the calm water resistance of a reference ship,

cr is the aìlzable increase of calm _{resistance in percentage.}

Constraint of longitudinal position of L.C.B.

(LOE) _i. L (X) ₍₁₄₎

Constraint of metacentric height

GM(x) G!' ₍₁₅₎

nu.n

Subscripts 2,1 in above expressions indicate the upper and lower limit, while subscript O is the assigned value. _{Data of L.C.B and metacentric} height are obtained from hydrostatic curves. Calm water resistance is determined using the method described in _{(lo) or estimated by an} alter-nate approach.

2. _{Optimization methodology}

The problem outlined by equation _{(9) is ascribed to an optimization} process subject to both equality and unequallty _{constraints. Generally} it can be solved by

_{seguenti al un}

_{constrained minimization te chnique}

(SUM?), but the task of calculation _{seems rather cumbersome. In view} of this, a direct method is proposed, which convert the problem into

an unconstrained optimization

process. At first, an objective function

is defined as follows:

F(X,M) (H_s - H

um

(X) ) + M.S(X) ₍₁₆₎

where H _{is the assigned maximwn}

of limiting significant _{wave height,}

say 12m _{, in our program for optimization,}

S (X) is the constraint function S(X)

= E

_h (X) _{+ E[Tttin(g. (X) 0) ]} (17) 1=1 ₁ 2 q 2

Hence the optimization process can be described as _{a minimization} of objective function

F(X,M). Search for optimum of _{F(X,M) is} proceed-ed by using simplex nethod describproceed-ed by Nelder-Mead Cli)

Let M

be the 'tFlexible penalty factor"?

its magnitude varies according to the following expression in the optimization _process:

n+1 M max { M , Entier [l/(a

-

E

-

i] }

(18)

j=l

where H _{is the initial value; X31s the apex of simp1ex}

X, the

cen-trcìjd and n is the dimension _{of vector Y. It is worth noting}

that the

magnitude of M and its

rate of increase are controlled by regulating

factcr Cz, _{and M increases steadily}

as the simplex itself contracts in

seaxching for optitnun of _{F(X,M). Then? M}

- as

n +1

+r

EXD_XHO

The constrained

_condition

is hence satisfied and optimum _{sointion is}

finally reached.

In view of avoiding iterative _{process which is a necessarystep} in solving the problem by SUMT, the unconstrained _minimization tech-nique of flexible penalty factor or simply called _{method of flexible} penalty factor reduces

greatly the computational task in searching for

optimum.

V.. _Examples

The effectiveness of proposed methodology depends on whether or not the optimization procedure just outlined can lead to _appreciable

improvement in e heavily constrained case. Two examples that foll

address that question. Equations (3) and (4) were used in defining the limiting wave heights in both examples. Responses of motion or related

criteria were all expressed in terms of significant single amplitudes.

1. _{Example i High speed round bilge craft}

Requirements specified in the disign:

Operating area: South China Sea

Speed of ship : 32 knots

a

o

Relative wave heading: Head sea (180 )

o

Seakeeping performance factors

and

criteria: Pitch (4.8 ); vertical acceleration at station 3 ((D.4g); probability of occurance of

botton slamming at station 3 (3%).

Principal dimensions of original version, bui]. form coefficients,

constraints and results of optimum design are listed in Table 1.

Com-parison of responses for original version and its optixnun alternative

are shn in Fig.3 and Pig.4 Fig.5 to Fig.7 represent designed

waterline, sectional area curve and underwater profile respectively. The result indicated that vertical acceleration at station 3 was the

limiting performance factor. The redesigned version could sustain

limiting wave height up to 2.Blm, instead of 2.71n for the original

version. The result also showed in case of seaway in South Sea with most probable encountering waves of modal period T 6.5 sec, the

redesigned version could sustain acceptable maximimt wave height up to l.68m, instead of l.59m for the original version. Should the later

navigate in a seaway of significant s'ave height l.68m, voluntary speed

reduction of 6.5 knots is necessary as indicated in rig. 8 In view

of the strict constraints imposed in the process of

optimization

modifications to hull form were rather limited, the result

thtained

by the optimization process was quite appreciable.

* _{It was understood that the roll motion of the craft may be} adequate-ly controlled by proper appendage design.

z/h 1 .0 0.5 1 .0 0.5

m

A

I4I

opt imum

version

-optimum version

-li-originel version

Fig.3 Heave response

PC h V = 32 kn = 1800 original version V 32 kn B = 180° o ₂ 3 4 ) /L

Fig..4 pitch response

5/s

_mx

0.5

y (ni)

Fig.6 Sectional area curve

5 10 15 20 station

optimum version

origin_io-_

T (m)

Fig. 7 Underwater profile

5 10 15 20 station

Fig.5 Design waterline

optimum version 4 original version 2

.

Optimum version original version

141

lU

Al

10 15 20 stet ion

Ta1e 1 0timurn hull form desian of hiqh speed craft

-2.

_{Exaple 2}

_{0ptLmum hull form design of}

_{a shuttle coaster}

This is an exanple of

_{optirnusi design in which}

_seakeeping

perfor-mance both in longitudinal and transverse planes has been take

into

consideration in optixnizaticn. Specific requirements of design are:

Cpez-atina ocean district;

_East

_{ina Sea}

13

-Parameters

_{original version}

_Constraints

_{Cptinum versic}

(ton)

587

587

₅₈₇

L tm)

62.6

_{60.72 - 65.73}

_63.77

Cm)

8.16

7.75 -

_8.56

_8.37

T (n)

_2.50

_{2.37 -}

_2.62

_2.38

X

12.50

10.50 - 13.00

_13.00

AWF (in2)

142.7

126.0

- 167.0

_155.58

\

Cmn)

_244.0

_216.0

_{- 270.0}

_251.94

v

Cm3)

240.1

_197.0

- 297.0

_243.93

VA

(rr)

332.58

277.0

- 397.0

_328.76

C

_0.559

_0.550

_{- 0.590}

_0.583

0.955

0.940

- 0.960

0.944

T/L

r.C399

0.035

- 0.047

0.0372

C/L

4

-0.625

0.525

- 0.650

0.650

CF

0.673

0.66

- 0.68

0.660

C

_0.545

_0.54

_{- 0.56}

_0.549

L

_-4.57

_-6.0

- -2.0

_-4.19

Pes/.es0

_{< i.03}

_1.028

GI (in)

> 1.0

_>1.0

_>1.0

Limiting wave

height (in)

2.70

2.81

service speed for optimization:

12 knots

relative headings

Head

(1800)

and bean

(900)

_seas

selected seakeeping performance factors and criteria: pitch (20),

roll

(100),

_{vertical acceleration at station 5}

_{(C.l5g) probability of}

occurance of bottom slansning at station 3 (3%), frequency of shipping

of green water at station 3

(O.5/mirz) frequency of emergence of

pro-peller (0.5/mm).

In order to reveal the effect of weighting regarding the relative

importance of wave headings, different ratios of W1800,W90o were

chosen as 3:1, 1:1 and 1:2 in optimization. Results are presented in

Table 2, where redesigned versions are designated by "Optimum i ",

10ptjmui 2" and "Optimum 3". Fig. 9 exhibits significant variation of

limiting wave heights. This fact reveals the importance of proper

selection of wieghting regarding the wave headings in the process of

optimization. Fig. 10 - 12 present the response curves of original

version and the redesigned version "Optimum 2". improvement in

re-sponse characteristics is evident, while difference of roll rere-sponses

in beam seas is mainly attributed to the variation of

resonance

per-iods.

H (m) J1 1 .50 1.70 1.60V 1 .50 23 25 T=6.5see 27 29 31 voluntery speed reduction 6.5 kn.

Fig.8 Maximum wive height

_{versus speed}

H11,3=1 .68ni

z/h

1 .0 0.5 (m)

3.0

2.5

0.

2,0

1 2 3

origindi version

optimum ve.sion

Fig.10 Heave respose

. 15 -Ho (e) 5.0 4., 4.0 WI 800

Wo

Fig.9 Limiting wave height

versus wave heading weight

V = 12 kn

v'a

$ 1800

1 .0 0.5 4 3 2 fr/K h

original version

original version

optimum version II

Fig. 1]. Pitch response

h V = 12 kn ß = V = 12 kn ß 90 optimum

version II

2 3 4 X /L

Fig.12 Roll response

* _{Station nuxnber at cut-up point of the keel}

** _{C is the distance f roui station O to cut-up point}

sults listed refer to different weightings regarding relatIve

wave headings Parameters

Original

-version Constraints tirnuzn i Optjimt 2 Optimwi 3

t (ton) _{3575 3575 3575} 3575 3575 L (in)

₉₆₀₀

_92-100 99.37 98.03 96.97 E _{15.20 15-17 15.02} 15.04 15.81 T (in) _{4.94 4.5-5} 4.89 4.90 4.94 X 19.50 19.2-19.8 19.46 19.45 19.44 481 448-640 557.04 537.46 503.68

(2

_{566.60 517-E80 569.05 541.78} 537.86 7F :1426.7 1383-1725 1483.6 1491.9 1519.3 VA (in3) _2061.1 L762.8-2104.8 2004.2 1995.9 1968.5 0.659 0.65-0.75 0.746 0.729 0.657 0.777 0.7 -0.79 0.762 0.735 0.701 0.0514 0.045-0.058 0.0492 0.0499 0.0509 ** CIL 0.975 0.96-0.99 0.973 0.973 0.973 0.600 0.25-0.8 0.544 0.566 0.611 Cv 0.736 0.7 -0.85 0.720 0.752 0.741 LOE _{-4.86 5.00-O - 3.89} -4.01 -3.20 Res/Res o 1.08 1.01 1.0]. 1.01 GM (in) _>1.0 >1.0 >1.0 >1.0 >1.0, Limiting wave height in 2.27

head seas (in)

1.76 2.63 2.30

Limiting wave

height in 4.71

bean seas (in)

4.39 4.67 4.85

weighted mea 2.88

of limited

349

wave heights_(xTi)1 3.90

3.66

4.00

VI. Conclusion

The presented methodology provides _{an effective tool to optimize} a design from the

point

_{of view of seakeeping performance of ships in} early design stage.

_Assumptions

_made

_in

_{the course of deriving the} general model relating hull form to seakeeping performance are not

subject to the restriction for the _{types of displacement ships. Various} constraints which reflect specified _{requirements of a design are}

im-posed arid the optimization is _{proceeded according to the mission}

as-signed to the ship, thus making the methodology more practical in application. However, detailed reservations _{still exist, such as}

synthesis of the above-water hull _{form in further detailing the design,} proper weighting for different conditions in executing _{optimization.} In these respects further investigation must be emphasized. Accumula-tion of experience in practical _{application of the proposed}

methodo-logy is also needed.

Re feren ces

Gerritsvia, J. and Beukeirnan, W., "7na1ysis _{of the nodified strip}

theory for the caictiation of ship motions and _{wave bending} moments", I.S.P. Vol. 14, 1967.

N Salvesen, E.0. Tuck, Odd Fa].tinsen, _{"Ship motions arid sea} loads", Trans. SN»IE. Vol. 78, 1970.

S.Olsen, "bn evaluation of the seakeeping _{qualitities of naval} combatants", Naval Engineers Journal, Vol. 90, 1978.

E,N.Ccimstock and R.C.Keane, "Seakeeping by design", Naval

En-gineers Journal, Vol. 92, 1980.

Yi Que, "A Comprehensive Approach to _Seakeeping Performance Evaluation", Second Symposium on Seakeeping Behavior of Ships in China, 1981.

N.K.Bales, "Optimizing the seakeeping performance of destroyer-type hulls", The 13th symposium on naval hydrodynamics, Tokyo Japan, Oct. 1980.

A.I.Raff, "Program scores-ship structural response in waves,

SSC-230, AD-752468.

0. Grim, "Die sthwingtgen von schwinimeden, zweidimensionalen

Korpern"., HSVA

report,Nc..1171, Sep. 1959.

F.Tasai, "HydrodynaiTtic force and moment produced by swaying and

rolling oscillaticn of cylinders on the free surface", Reports of Research Institute for Applied Mechanics,

Kyushu

University

Japan. Vol. IV, 1961.

Ir.J.Holtrcp, "A statistical analysis of performance test

res-ults", I.S.P., Vol. 24, 1977.

(il) L.C.W.Dixon, "Nonlinear optimization", The English Universities

Press

Limited, London,

1972.

(12) D.H.Himmelblan,

"Applied

nonlinear programming", McGraw-Hill Book Company, New York, 1972.