25th

Symposium on Naval Hydrodynamics

St. John’s, Newfoundland and Labrador, CANADA, 8-13 August 2004

Parametric Optimization of SWAT-Hull Forms

by a Viscous-Inviscid Free Surface Method

Driven by a Differential Evolution Algorithm

Stefano Brizzolara

(University of Genova - Italy, EU)

ABSTRACT

An optimization method capable to automatically

search for the optimal underwater hull form of

SWATH type vessels, with regards to total resistance,

is described in the paper. The optimization

environment is based on an original parametric

geometry definition module, capable to define a series

of underwater hull forms characterized by a double

hump and an intermediate hollow shape, and a set of

robust and well proven CFD codes, e.g. a linear free

surface panel method for the prediction of wave

resistance coupled with a thin boundary layer integral

method for the prediction of friction and form

resistance. The validity and good accuracy of the CFD

methods employed in the numerical optimization

procedure is proven in the paper by comparison of the

numerical results obtained in the case of a typical

SWAT demi-hull for which resistance test results were

conducted in towing tank.

The optimization strategy is driven by a differential

evolution algorithm which demonstrated its good

capability in finding the optimum solution, even in a

difficult design space, such as the one adopted for the

application examples presented in the paper. These

application examples regard always the same design

vessel (with fixed main dimensions and displacement),

whose underwater hull forms are optimized with

respect to different objective functions: pure wave

resistance or total resistance, at different given design

speeds. The substantial difference in the shape of the

optimized hull forms, obtained considering different

design objectives, highlights the importance of shaping

the hull form of SWATH ships on the base of a given

operating profile. The order of magnitude of the

reduction of total resistance achievable with an

optimized hull shape should convince the reader about

the importance of using automated CFD optimization

methods in the design of modern and highly efficient

SWATH ships.

INTRODUCTION

Recently a revival of interest about SWATH

(Small Waterplane Twin Hull) ships has been

registered among naval architects, shipyards and

research centres. In fact, aside the usual and never

abandoned military applications, SWATH typologies

are being designed for passenger ships, fast ferries and

pleasure mega-yachts, rather globally. Also in Italy the

department of Naval Architecture and Marine

Engineering of the University of Genova has started an

independent research, which just recently turn out to be

of interest for a pre-competitive research project in

collaboration with an Italian shipyard, aimed to the

design of a foil assisted SWATH vessel. The qualities

which attract interest to a SWATH ship typology are

the potential lowest motions and acceleration in rough

seas in comparison with an equivalent mono hull or

catamaran and the potential low wash characteristics at

high speed. Nowadays both attributes are of primary

importance for the success of a passenger ship: the first

for assuring the best comfort to passengers and the

widest operability of the ship in rough seas; the second

to obtain the acceptance for the ship operation on a

route coming inside a closed bay or passing by sensible

littoral areas.

However, the scientific arguments that motivated

the author to initiate this independent research are

manifold and of different nature:

– to integrate an inviscid and viscous CFD method to

be able to predict the total resistance of a SWATH

ship, in a time compatible with the usual design and

to validate it over a typical case

– to experiment the possibility to set up an integrated

computational environment for the optimization of

hull forms, as automated as possible

– to experiment the use of computer aided parametric

automated definition of (simple) ship hull geometries

– to test the applicability of global optimization

algorithms to the design of efficient ship hull forms

– to find, from the results of the application, a series of

guidelines for the design of underwater SWAT-Hull

forms, which can qualitatively drive the designer in a

new project.

The concept of computer assisted parametric

optimization of hull forms has been introduced in a

relatively recent time to naval architects and has

stimulated an immediate interest and fascination in the

community. This probably because, by tradition naval

architects have always been used to design the hull

forms through the definition of a restricted set of global

and local form parameters, determined on the basis of

available test results of systematic series of hulls. So,

the dream to derive (in a reasonable time) a complete

resistance database of a personalized series of hulls,

naturally captures the attention of the naval architect.

This kind of studies are done already as a routine job in

ship design, but relying almost always on a trial and

error procedure, which still use a lot of human

resources for low level tasks, such as the generation of

the modified ship’s geometry, the analysis of results

and relative decision on the new geometry adaptation.

The new fascinating idea, used also in this study, is

related to the use of an integrated computer assisted

(possibly fully automated) optimisation environment

based on global optimization algorithms. The idea,

here, is to shift the low added value and time

consuming work (i.e. generate the geometry, prepare

the mesh, run calculation, analyse the results) onto the

computer program, leaving the strategic part of the

work (i.e. choice of free parameters, definition of the

design constraints, definition of the optimization goal)

to the designer.

Some example of applications of these concepts in

the naval field were presented on papers (various,

2003) of different authors all involved in a joint

European research project dedicated to the

development new hull optimization methodologies,

involving different parametric approaches for the

variation of the hull shape and different CFD codes

(mainly panel methods) for the prediction of wave

resistance and/or ship motions. Other examples regards

the approach of Peri et al. (2001), who use a more

direct geometrical modification method that acts

directly on the control points of Bezier surface patches

which represent the hull surface. This methodology

requires the imposition of a set of not intuitive

continuity and fairing constraints to control the

behaviour of the modifying surface as well as other

design constraints, as for instance a fixed displacement,

to be included in the objective function. In this respect,

the smart approach of parametric geometry generation

used by Harries (1998, 1999) appears to overcome

these problems and to be based on a set of parameters

at least more meaningful and familiar to naval

architects.

SWATH types of hull, due to their modular and

geometrically simple constitutive elements, are an ideal

test case for the first application of an automatic

optimization procedure. In fact, already in the past,

several studies were dedicated to this problem.

Papanikolaou et al. (1991), present a very

comprehensive approach to the total design of fast

SWATH vessels, in which for the local form

optimization, he estimated the wave resistance with

Strettensky’s integral method (developed for ships

moving in a canal) and integrated the flat plate local

skin friction coefficients along the hull panels and an

empirical form factor to estimate the viscous

resistance. A Lagrange multiplier technique was used

to drive the optimization, keeping as constant the given

displacement and longitudinal centre of buoyancy.

Another very comprehensive example is given by

Salvesen et al. (1985) who first present a well

structured SWATH design environment made up of a

series of modules, which would be serially used by an

operator to design an SWATH hull geometry and

estimate its resistance. The only optimization

procedure used by Salvesen is regarding the wave

resistance component only, estimated through a slender

body theory. A Lagrange multiplier method is used ans

it acts directly on the distribution of the transverse

section areas of the underwater hull forms as requested

in input by the adopted slender body theory.

The method presented in this paper, which extends

and completes the first work on SWAT underwater hull

optimisation, with regards to wave resistance only

(Brizzolara, 2003), uses a combined viscous-inviscid

method for the evaluation of total resistance of

SWATHs, and in this respect should be more general

and should give more accurate trends than the

previously cited works. Moreover, the automatic proce-

dure based on the parametric geometry generation

driven by the optimization algorithm, and the use of the

coupled CFD methods, made possible to investigate a

huge number of cases (of the order of 1000) for each

optimization exercise, in a reasonable time, and to

easily single out (hopefully) the best hull form.

The marked difference in the shape of the

optimized hull forms obtained using the total resistance

as a criteria for optimization instead of just the wave

resistance, at the different considered speeds confirms

the importance of the significance of using a viscous-

inviscid CFD like the one used in our study.

NUMERICAL OPTIMIZATION ENVIRONMENT

A computer program has been developed

according to the flow chart presented in figure 1. We

define it ‘optimization environment’ since the program

is complemented by a series of analysis and graphic

routines, which help the user to easily check the status

of the convergence, as well as to discuss the optimum

solution (against some other good candidates) and the

history of the free parameters and the objective

function, in the end. The core of the procedure is made

up by the parametric geometry generation module, the

CFD solvers and the optimization algorithm, but also

the routines that integrate various modules by

elaborating creating a valid input data for one from the

output of the other were required and time consuming.

With reference to the independent and automatic

character which the whole procedure must have, a very

important detail is also the recurrent check of error,

which must be done at all levels to prevent a break in

the optimization flow.

Figure 1: Flow chart of the optimization program, with main

details of the major component modules.

The optimization environment has been created in

SCILAB, a platform-independent public domain scienti-

fic computation system in continuous evolution. This

environment offers a very complete and powerful set of

scientific functions for calculus, data manipulation and

graphics. A SCILAB macro can easily manage Fortran

and C subroutines together with system commands and

compiled executables. So apart from the CFD codes

written in Fortran, all the rest of the optimization

procedure is written in SCILAB, including the

optimization algorithm and the geometry generation

module. The structure in figure 1, reflects the hierarchy

of the highest level routines. After the definition of the

main input, the top level program, that drives the rest

of the modules, is the optimization algorithm, which

according to its strategy generates each iteration a

different set of free variables, which identify the

geometry of a SWAT-Hull, and successively starts the

serial launch of CFD modules and post processor

routines. The control of the various module is reserved

to the top level routine written in SCILAB, which

manages unwelcome crashes of the different modules.

The theoretical and numerical topics of the different

modules are briefly described in the next paragraphs.

PARAMETRIC GEOMETRY DEFINITION

Parametric representation of the hull surface is

essential for the automated optimization procedure.

SWAT-Hull forms are easy to be represented

parametrically. In our case we can assume: the struts

having parabolic or circular arc sections, defined by a

maximum thickness and mean length, a taper ratio and

an inward or outward cant; the underwater hulls either

as body of revolution, defined by one generatrix

profile, or as body with elliptic cross sections, having

the two symmetry axes defined by two profiles: one in

the longitudinal and the other in the horizontal

symmetry planes.

Following a previous work (Brizzolara, 2003), the

underwater hulls were kept as bodies of revolution with

a particular parameterization of the generatrix curve.

The intention is to generate unconventional underwater

hull forms that can maximize the wave cancellation

effects between the generated wave trains along the

length. For this reason a profile typology with at least

three relative extremes (maxima or minima), at given

longitudinal positions (Xm1, Xm2, Xm3), has been

defined. Its length, L, maximum diameter, D, and

therefore displacement or CP, are also fixed. The

curvature radii, RL.E. and RT.E., at the leading and

trailing edge of the curve are given and the thickness of

the profile in correspondence of the second and third

relative extremes.

Figure 2 represents these parameters on a non-

dimensional coordinate system assumed for the rest of

the paper: the transversal coordinates are non-

dimensionalized with respect to the maximum diameter

D, while the longitudinal coordinates with respect to

the maximum length L.

Initial Range of Free Var.:

Min&Max{r0, r1, m1, m2, m3}

Design and Reference Data:

CP, L, Dmax. Ls, Bs

Fni, CWrif, CTrif, CVrif

INP

UT

DA

TA

INP

UT

DA

TA

MO

DU

LE

Determine the Generatrix Curve:solve the linear system of eq.s

3D Panel Mesh:

Body with elliptic or circular sections

Input file generation forFree Surface B.E. method

D.E. algorithm parameters:

CR, F, NP, VTR, iter_max

Free Variables Value:

r0, r1, m1, m2, m3

CF

D

MO

DU

LE

S

Free Surface B.E. Method:

Wave resistance, wave pattern and streamlines

Thin B.L. Integral Method:

Cf and Separation areas: Cvp

Convergence Criteria & Output Data:

Cw, Cf, Cvp, CT

Input

file

Inte

racti

on:

Tra

nsp

. V

el.

&

se

para

ted

regio

n

MO

DU

LE

Calculate Objective Function:

∆CT, DCW for one or more weighted speed

VTR ?

ITERmax ?

Historic

Data

GE

OM

ET

RY

Historic

Data

OP

TIM

IZA

TIO

N

Generate New Parameter VectorAccording to D.E. algorithm

STOPYES:

NO:Historic

Data

In the affine non-dimensional plane {y,x}={Y/D,X/L},

the generatrix curve has been analytically defined

through the following polynomial of 8th

degree:

( ) ∑=

⋅=8

1

2

k

k

k xaxy (1)

The polynomial representation was inspired by

that used by Gertler (1950) for DTMB Series 58 of

underwater streamlined body of revolution, as well as

the way he used for determine the coefficients ka . In

fact, these coefficient are found by imposing the

following physically meaningful constraints:

- the non-dimensional length of the curve 1=� ;

- the non-dimensional first maximum 5.0/11 == DYy

- the non-dimensional longitudinal position of the three

relative extremes: 321 ,, mmm

- the total displaced volume, imposed through the

longitudinal prismatic coefficient: 24 LDCP π∇=

- the non-dimensional value of the curvature radii at

the leading and trailing edge of the curve,

respectively: 2

..1

2

..0 ; DLRrDLRr ETEL ==

The above conditions expressed in mathematical form,

result in the following system of equations:

( )( )( )

( )

=

=

=

=

1

3

2

1

01

0'

0'

0'

y

my

my

my

&

( )

( )

( )[ ] ( )

( )[ ] ( )

=+

=+

=

=

∫

12

32

02

32

1

0

2

1

1''1'1

0''0'1

4

21

ryy

ryy

Cdxxy

my

P (2)

Substituting the polynomial representation of the curve

(1) into the system (2), one obtains the following

system of linear equations, with respect to the unknown

coefficients ka of the polynomial function (1):

=

=⋅⋅

=⋅⋅

=⋅⋅

∑

∑

∑

∑

=

=

−

=

−

=

−

0

0

0

0

8

1

8

1

1

1

8

1

1

2

8

1

1

1

k

k

k

k

k

k

k

k

k

k

k

a

mak

mak

mak

& ( )

=⋅

=

=+

=⋅

∑

∑

∑

=

=

=

1

8

1

01

8

1

8

1

1

41

21

rak

ra

Cka

ma

k

k

P

k

k

k

k

k

(3)

In the optimization examples presented next in the

paper, the length, maximum diameter and volume of

the hull were fixed, so each iteration a reduced set of

free variables, FV={m1,m2,m3,r0,r1} is generated by

the differential evolution algorithm, the corresponding

generatrix curve is found by solution of the system (3)

and finally the hull panel mesh is generated. The

surface of the underwater body is intersected, without

any fillet radius, with the surface of the strut, that for

this study has been assumed vertically cylindrical with

parabolic section of the form:

( ) 2

32

2

1X

L

t

L

txY

S

S

S

S −= (4)

in which tS is the maximum thickness of the strut and

LS is its length, and the reference point for local

coordinate are on the mid point of the strut length.

Figure 2: Type of generatrix curve of the underwater body of

revolution and relative parameters used to define it.

It is obvious that many other mathematical

representation of the generatrix curve are possible. For

instance, a convenient formulation would be a B-Spline

curve, defined by 7 vertexes, two fixed at the L.E. and

T.E., other two placed right vertically over the L.E. and

T.E. vertexes, to control the curvature radii, and the last

three free to move in any direction (within a certain

maximum range). This representation, though having

the advantage to depend on parameters defined over a

certainly continuous domain, looses the implicit

satisfaction of the required fixed volume, so it requires

an additional iterative optimization procedure to satisfy

this design constraint. On the contrary, the five free

variables FV employed to define the polynomial

generatrix curve of (1), result defined over a non-

continuous domain, which is made continuous, for the

sake of good convergence, returning an artificially high

object function value, where the system (3) has not

solution. This fact complicates the convergence toward

the minimum of the objective function. In fact, for an

objective function defined in such a complex domain

the usual analytical (deterministic) minimization

algorithms, would fail to find the absolute minimum.

Stochastic optimization algorithms, like genetic types

or differential evolution types, instead, have more

chance to explore the whole unconnected domain,

without getting stuck from a barrier of non-existent

solutions.

Examples of generated profiles and corresponding

demi-hull panel meshes are given in figure 3, which

compares the reference demi-hull form (with

underwater hull taken from model #40050165 of Series

58) with the four optimized hulls obtained at four

reference speeds. The corresponding generatrix curves

of the underwater body of revolution are represented in

figure 14. The body panel mesh is automatically

obtained from a net of cubic spline curves, defined

over several patches, which can have different panel

densities and longitudinal distributions to better define

the strut intersection with the hull body and especially

at the leading and trailing edges of both bodies.

Figure 3: Perspective view of the panel mesh automatically

generated by the parametric geometry module. From top to

bottom: reference SWATH demi-hull form, with underwater

hull of Series 58, and the four demi-hulls optimized at

Fn=0.30, 0.35, 0.41, 0.40.

VISCOUS-INVISCID FREE SURFACE METHOD

An incompressible irrotational potential flow is

assumed by enforcement of the Laplace equation to the

total velocity potential in the fluid domain bounded by

the hull surface SB and the free surface SF. We use a

indirect boundary element method, linearized with

respect to the double model flow as developed by

Bruzzone (1994) and further adapted and successfully

validated in the case of high speed mono- and multi-

hull vessels (Brizzolara et al., 1998) also with the

possibility of including dynamic attitude prediction and

flow behind dry transom sterns (Brizzolara &

Bruzzone, 2000).

A Cartesian right-handed reference frame {XYZ}

travelling with the ship at U∞ constant speed is centred

at an arbitrary point on the intersection of the

longitudinal symmetry plane with the undisturbed free

surface; the x axis oriented aft-wards, and the z axis

oriented upwards.

The total velocity potential φ+=Φ ∞ xU , and the

unknown perturbation potential φ, with respect to the

uniform incident flow, must both satisfy the Laplace

equation in the complete domain:

0,0 =∆=∆Φ φ (5)

together with the following boundary conditions:

0=Φ∇⋅n�

on the hulls (6)

0=∇⋅Φ∇ ζ on the free surface (7)

2

2

1

2

1∞=Φ∇⋅Φ∇+ Ugζ on the free surface (8)

0, →→Φ ∞ φxU for x → −∞ (9)

namely, the Neumann condition on hulls surfaces (6),

the kinematic and dynamic condition on the free

surface (8,9) and the radiation condition for the

disturbance upstream (10). ζ=ζ(x,y) represents the

unknown free surface equation.

In our method the free surface boundary

conditions (7) and (8) are linearized using a small

perturbation theory, by which the total velocity

potential Φ is considered as the sum of a main

contribution represented by the potential ΦD of the

flow around a double model symmetrical with respect

to the undisturbed free surface, considered as deeply

immersed in the fluid, and the contribution of the

perturbation potential 0Φ due to presence of the wavy

free surface. Congruently the free surface is thought as

composed by the Bernoulli wave Dζ calculated for the

double model, and a smaller order component oζ , i.e.:

oD ζζζ += (10)

gg

U DDD

22

2 Φ∇⋅Φ∇−= ∞ζ on z=0 (11)

Using this assumption, it is possible to combine

the free surface boundary conditions (7) and (8) in the

following analytical linear expression valid,

congruently with the linearization, on z=0:

S58

O30

O35

O41

O50

DyDxzxyDyDx

DyyyDxxxyx

bUag

ba

Φ+−Φ=+ΦΦ+

+Φ+Φ++

∞ 2)(22

22 22

φφ

φφφφ (12)

where:

DyyDyDxyDx

DxyDyDxxDx

b

a

ΦΦ+ΦΦ=

ΦΦ+ΦΦ= (13)

The double model and the linear free surface

potential flow problems are both solved by a boundary

element method which discretized the continuous

problem, approximating the hull surface (SH) and the

undisturbed free surface (SF), with a structured set of

quadrilateral planar panels each having constant

distribution of Rankine sources on it.

Defining the influence coefficient vector as the

velocity vector induced at the centroid of panel i by a

panel j having a uniform distribution of sources with

constant strength σj:

),,()1

( ijijijjjquad

ij

ZYXdSr

≡∇∫ (14)

we discretized the boundary conditions (6) and (8),

imposing them on each panel centroid taking into

account for the contribution of any panel on the hull

(NH in number) and on the free surface (NF in

number). As a result, the following linear system of

equations in the unknown sources intensities is

obtained:

Hzi

NN

j

ijyiijxiij NiUxnZnYnXFH

,11

==++∑+

=

(15)

NFiy

bUx

ag

x

Y

xxy

Y

y

x

X

xYbXa

D

i

D

ii

j

ijDDij

i

D

ij

i

D

ijiij

NN

j

i

FH

,12)(22

]))((2)(

)(22[

2

2

1

=∂

Φ∂+−

∂

Φ∂=+

+∂

∂

∂

Φ∂

∂

Φ∂+

∂

∂

∂

Φ∂+

+∂

∂

∂

Φ∂++

∞

+

=

∑

σπ

σ (16)

To compute the derivatives of the potential, a four

points differential operator is used, everywhere on the

free surface in both longitudinal and transversal

directions, except behind transom sterns. As known

this operator gives an implicit property of numerical

damping of the disturbance which otherwise should be

enforced by other means. The free surface waves are

found by substituting the total velocities calculated

over the free surface panels in (8).

The wave resistance is found, in this study, by the

classical integration of the dynamic pressure found at

the each panel centre, with Bernoulli equation over all

the hull panels. Other studies (Brizzolara et al., 1998)

used a (numerical) transverse cut method, to calculate

the wave resistance from the energy content of the

generated wave pattern. No attempt were made in this

study to compare the two methods, being the second

preferable, in general, in case of fast ship hulls with a

relevant dynamic trim and sinkage. The SWATH hull

attitude was kept fixed in the computation of this study.

The viscous effects are approximated by a bi-

dimensional thin boundary layer integral method

applied on the three dimensional inviscid streamlines

distributed to cover the largest portion of the body

surface. This approach is widely used in several

viscous-inviscid panel methods developed in the

aeronautical field, for instance by Maskew (1968). The

methods have been validated with success on many

different airships fuselage forms and aircraft (Dvorak

et. al., 1977) at Reynolds numbers typical of those

reached during ship’s hulls towing tank tests. Usually

they give sufficiently accurate results when the cross

flow is negligible and when separation is absent or it is

confined in a limited portion of the body. The classical

bi-dimensional integral boundary layer equation is

solved for the laminar flow using Thwaites’ (1949)

method as revised by Curle (1967):

( ) fCd

dU

UH

d

d

2

12 =++

η

θ

η

θ (17)

where θ is the b.l. momentum thickness at a point η on

the curvilinear abscissa of the considered inviscid

streamline which has on any point a potential flow

velocity U=U{x(η),y(η),z(η)} interpolated on it from

the known solution of the inviscid free surface problem

(16), or from that of the double model problem.

( )( )0

2

=∂∂= ξξν uUC fis the local friction coefficient

and H=δ*/θ is the shape factor; u(ξ) describes the

velocity profile across the b.l., being ξ the local

coordinate perpendicular to the body surface.

Defining a parameter K proportional to the

gradient of the external velocity U, found from the

solution of the potential flow over the body, and the

other two correlated parameters l and L:

ην

θ

∂

∂=

UK

2

,

0=∂

∂=

ξξ

θ u

Ul , ( )[ ]22 +−= HKlL (18)

the b.l. equation (17) can be transformed in a new

differential equation:

U

LUK =

∂

∂

∂

∂

ηη (19)

which can be solved using the empirical closing

relations between L and K found by Curle (1967):

( ) ( )µµ ,645.0, KgKKL +−= (20)

where µ obeys to Curle’s differential equation:

2

24

ην

θµ

d

UdU= (21)

Equation (19) is integrated over the streamline length,

obtaining the solution for the b.l. momentum thickness:

( )( )

( )[ ] ( ) ( ) ( )( )

6

2

0

5

2

2 00,22.21

45.0

++= ∫ η

θηηµη

νηθ

η

U

UdUKg

U

(22)

which is integrated assuming an initial value of the

momentum thickness at the stagnation point η=0 on the

streamline (see Maskew ,1981), with an iterative

procedure, since g(K,µ) is depending in turn by θ. The

first iteration g=0 may be assumed. Once (22) is solved

the other integral b.l. parameters are found by means of

(18) and using the Karman-Polhausen relations to

derive the b.l. thickness .

Transition is predicted by the Granville method,

which uses empirical analytical relations between the

local momentum thickness Reynolds number and the

local external velocity gradient parameter K, previously

defined. If laminar separation is predicted, an empirical

relation, based again on Rnθ, is used to check if the b.l.

will reattach as turbulent b.l. or will separate for the

rest of the streamline. After transition or reattachment,

the turbulent b.l. computation begins, assuming the

same momentum thickness found at the last calculated

laminar b.l. point and using an empirical relation for

the initial shape factor.

The turbulent form of the integral equation (17) is

solved using Nash and Hicks (1981) method, which is

based on the solution of three integro-differential

equations, obtained by integrating the continuity

equation along the boundary layer thickness, on every

point along the streamline. Assuming a Cole’s velocity

profile across the boundary layer:

( )

−+

+

=

δ

πξ

ν

ξξη

βττ cos12

ln,u

Cu

K

uu (23)

K=0.41 and C=2.05, uτ is the friction velocity and uβ , a

parameter with dimension of a velocity, a set of three

ordinary differential equation is obtained of the form:

αα

τα

βαα

ηδλ

ηδλ

ηδλ

η

δ

δλ

Λ+=

=++

d

dU

d

du

d

du

d

d

1

111

3,

3,2,1,

(24)

with the coefficients λα,i and Λα (α=1..3, i=1..4) that

are depend only on the following unknown parameters:

δ, uτ , uβ , and Cτ . This last parameter is found

introducing another equation:

[ ]τττ

δηCC

d

dC−=

15.0 (25)

where ∫=δ

τ ξτδρ

0

22

1d

UC

21

1025.0

−=

HCτ

A standard first order finite difference scheme is used

to integrate the three eq. (24) with (25), by refreshing

the values of the coefficients λα,i and Λα along the

streamline, at each iteration. The relations valid for the

Cole’s profiles are used at the end to calculate the

boundary layer momentum and displacement thickness

as a function of the standard thickness, found as part of

the solution.

At this level the two methods are simply related.

The real iterative coupling scheme between the two

flows may be performed (see Cebeci, 1998), for

instance, using the transpiration velocity concept of

Lighthill. Virtual normal velocity components on the

body (named transpiration velocities) are needed in the

potential flow problem to impose that the calculated

edge of the boundary layer is a streamline also of the

potential flow problem. The transpiration velocities,

depend, as from Lighthill, from the boundary layer

growth on the body. So, interpolating the transpiration

velocities found on each streamline back at each panel

centroid, it is possible to solve the potential flow

problem, with the BEM, assigning a finite normal

velocity instead of the usual Neumann condition in (7).

The new solution of the inviscid method, with

transpiration velocities, is in turn used to solve a new

b.l. problem and the procedure is iterated until the

convergence is found. For the optimization procedure

though, no iterative coupling was used between the tow

methods. Only a rough estimation of viscous pressure

resistance is made, by excluding the contribution of

those panels which belong to a region of separated

flow, from the integration of the dynamic pressure

which gives the total (dynamic) pressure resistance.

VALIDATION OF THE CFD METHOD

The viscous-inviscid method outlined in previous

section, has been applied in the case of a SWATH like

monohull which was tested in towing tank for the

validation study of CFD methods to be used in an

ongoing research project on similar hulls. The test case

is a mono-hull of the SWATH type having elliptic

cross sections and a symmetric strut having a circular

arc section. The horizontal and longitudinal profiles of

the hull are represented in figure 4, while the main

characteristics of the model are reported in table 1.

Table 1: Main Characteristics of the SWATH demi-hull

model tested in towing tank for validation of CFD methods

LOS [m] Bmax [m] Hmax

[m]

T [m] S [m2] ∇∇∇∇ [m3]

Underwater Hull

3.625 0.463 0.350 0.475 3.06 0.258

Strut

1.72 0.12 0.125 - 0.542 0.021

Free surface inviscid calculations were performed

in the complete range of Fn tested and the thin

boundary layer, based on the inviscid pressure

distribution, was calculated for the corresponding

Reynolds number in model scale. A number of about

700 panels was used with about 30 streamlines to

describe the (half of the) body. The free surface was

discretized with about 3000-4000 panels depending on

the Froude number, for an extension of about 3 hull

length by one hull length aside.

Figure 4: Longitudinal and horizontal profile of the SWATH

demi-hull model tested in towing tank for resistance

measurements.

Bare hull resistance tests were conducted without

the use of any turbulence stimulator, so also in the

numerical calculations the natural transition criteria of

Granville was used.

Figure 5 presents the comparison between the

numerically predicted and experimentally measured

total resistance. Evidently, the agreement is excellent in

the whole speed range, also near the peak due to wave

resistance; poorer, on the contrary, for Fn<0.28, where

probably the interactions between viscous-inviscid

flows are more pronounced and highly non linear

(large separated regions also in the laminar flow).

Probably in this regime, direct viscous-inviscid

interaction method, with a proper description of the

separated flow regions and those with laminar bubbles,

would lead to better correlations.

2

4

6

8

10

12

14

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

FnLos

CT*1

03

Numerical

Experimental

Figure 5: Comparison of predicted and measured total

resistrance for the test SWATH demihull

Anyhow, the thin boundary layer method used did

not predict any flow separation up to the body

truncated end, so frictional resistance were the only

component of viscous resistance. The numerical

frictional resistance coefficient is compared in figure 6

with the reference value obtained for the whole body

using the correlation curves of the turbulent flat plate

of Schoenherr and of the ITTC’57.

These reference curves are named ‘composed’

since the total friction resistance coefficient is obtained

by summation of the two partial coefficients of the strut

and of the hull, each at its characteristic (length)

Reynolds number, and weighted by the correspond-

ding wetted surface, i.e.:

( ) ( )TOT

HULL

LFTOT

STRUT

LF

TOT

FS

SRnC

S

SRnCC

HULLSTRUT+= (26)

2.5

2.7

2.9

3.1

3.3

3.5

3.7

3.9

4.1

4.3

4.5

2 4 6 8 10 12 14 16 18

RnLos*10-6

CF*1

03

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Fn

Los

Numerical

ITTC'57 composed

Schoennher composed

Rn-Fn

Figure 6: Comparison of numerical total frictional resistance

predicted for the test case at model scale Reynolds numbers,

with total frictional resistance calculated on the basis of

classical correlation curves.

For highest Reynolds number the numerical curve is

practically identical to the turbulent flat plate curve of

Schoenherr, while for 61012 ⋅<Rn , it results even

lower, in spite of any form factor. In fact at these low

Rn, a considerable portion of the hull and of the strut

are interested by laminar flow, according to the

transition criteria used. This large laminar portion of

flow, clearly visible from figure 8 which presents the

plot of the local friction coefficient at a typical model

scale Rn, is also due to the very fine entrance body of

the underwater hull. Laminar flow region on the

streamlines extend up to the magenta color.

When numerical calculations for the full scale

Reynolds number, [ ] 6101200:100 ⋅∈Rn , are compared

(figure 7), then a certain form factor re-appears in the

numerical calculations with respect to considered

friction lines. In this respect the results obtained for full

scale were judge realistic, keeping in mind that the

scope of the optimization, for our optimization scope,

has a comparative more than absolute meaning.

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0 200 400 600 800 1000 1200 1400

RnLos*10-6

CF*1

03

Numerical

ITTC'57 composed

Schoennher composed

Figurre 7: Same comparison as in figure 6, but at typical full

scale Reynolds numbers.

Figure 8: Plot of streamlines coloured in relation to the local

friction coefficient (colour scale on the right); one of the

lowest model scale numbers RnLos=4.84⋅106.

OPTIMIZATION PROBLEM DEFINITION

The problem of optimizing the shape of the underwater

hull in our case is easily identified and consists in a

minimization problem of the numerical resistance

experienced by the a parametric hull shape which is let

free to change in a certain design space bounded and

limited by a number of constraints. In general, also

other less obvious optimization problems can anyhow

be reduced to a minimization problem of a certain

function named objective function, which (also

indirectly) depends from the values of the free

parameters of the problem. In our case, in the most

general sense, the objective function can be assumed as

the weighted average of the resistance deviation with

respect to that of a reference solution, calculated at one

or more speeds NV, i.e.:

[ ]( ) ( )

( )( ) ( )

( ) ∑

∑

=

=

=−

=∆

∆⋅=ℑ

NV

i

i

iHullf

iHullfi

i

NV

i

iij

pVR

VRVRVR

VRpmmmrr

1.Re

.Re2

%

1

%32110

1,10

,,,, (27)

R is the resistance (or a component of it) evaluated for

the hull defined by the currently generated FVj vector;

the optimization procedure is naturally an iterative

procedure in which a number of successive tentative

solutions are evaluated and the convergence is

controlled by a deterministic or stochastic optimization

strategy.

The constraints used for the optimisation were,

first of all, the range limits of the free parameters. In

our case, allowing the larger variation:

∞<<

<<

10

321

,0

1,,0

rr

mmm (28)

Depending on the method adopted, these

constraints may explicitly be part of the formulation of

the numerical algorithm or can be implicitly included

inside the evaluation of the objective function, leaving

the parameter vector unbounded.

Often the objective function is non-linear, with

respect to the free variables and may not behave well in

some regions of the design space, i.e. can be singular or

non definite. Our case has all these characteristics,

since for instance there will be some combination of

the free parameter vector FV={m1,m2,m3,r0,r1}, which

lead to non real solution of system (3). There are other

solutions which lead to unfeasible curve shapes, in

which the underwater hull surface is slimmer than the

strut local breadth at some intersection point; or in

which the generatrix curve of the underwater hull has

more than three relative extremes over its length due to

the eight degree polynomial form (1) used for its

definition.

When any of the above mentioned situation

happens, the optimization procedure must not stop and

the optimization algorithm must always have in return

a real value of objective function. So one keeps these

unfeasible individuals alive, but practically discard

them from being possible solutions adding a high

penalty figure to the objective function value. In our

case in addition to the F.V. bounds of (11), we checked

these conditions:

( ) ( ) [ ].... ;,: ET

STRUT

EL

STRUTSTRUTHULL xxxxyxyx ∈∀>∃ (29)

( )i

iHULL

i xdx

xdyniniix ∀=>=∃ ,0:3&,1, (30)

( ) 0:8,1, ≠Ι=∃ kk amka (31)

and we assigned them the following penalty values:

Condition F.V.out of range (28) (29) (30) (31)

Penalty

Function

( ) 1,10 2 >jj mm

( ) 0,1102 <− jj mm

100

500

1000

OPTIMIZATION ALGORITHM

From all the above reasons, adding that the possibility

that the CFD solvers, though adapted, may crash or

give unrealistic results, it is clear that the minimization

problem is not an easy task to be solved. The high non-

linearity and bad behaviour of the objective functions,

which are analytically non-differentiable (or numeri-

cally differentiable but at a great cost in terms of

calculation time) and characterized by many possible

local extremes and peaks (a penalty value is assigned

for unfeasible solutions), convinced us to opt for a non-

deterministic stochastic type algorithm. Among them,

the “Differential Evolution” is a relatively recent and

efficient direct search method first developed by Storn

& Price (1995), which is having good resonance in the

scientific world, being among all inherently parallel.

This method uses a rather simple population based,

stochastic approach and its global convergence

properties have been proven over several test cases

with real valued multi-modal non-linear and non

differentiable objective functions. Its efficiency

secured it the third ranking in the . A short description

of the main principles of the adopted scheme is given

below but more details can be found in Price (1996)

and Storn (1996) for the practical applications.

The D.E. algorithm create successive generations

each one having NP population individuals, each one

characterized by a F.V. vector. The generation

individuals do not change in number during the

evolution minimization process. The first generation

has individuals uniformly (in probabilistic sense)

distributed within a certain guessed initial domain, or

around a known solution. Up to this point, like genetic

algorithms. The idea behind D.E. is a new scheme to

generate the trial free variables vectors: D.E. generates

new F.V. vectors by adding the weighted difference

vector between two population members to a third

member. If the resulting vector scores a lower objective

function value, then the new vector replaces the

individual with which it was compared. The

comparison vector can, but not need to be, a member of

the population. To keep track of the progress of the

minimization procedure, the best individual is saved

for each generation. Several variation of the algorithm

exist, depending on the way of calculating the random

deviations on the base of the vector difference of two

generation members.

The scheme DE1 uses the following relation:

( )321 xxFxxnew

����−⋅+= (32)

in which 321 ,, xxx���

are three vectors taken randomly

among the individuals of the same generation. F is a

constant factor which controls the amplification of the

differential variation. In order to include a certain

mutation among individuals of the same species, as in

genetic algorithms, a direct cross-over (direct substi-

tution of each vector component) is performed on the

newly generated vectors. The factors which control the

number of substituted vectors components is the cross-

over factor [ ]1;0∈CR , which is related to the

probability of that component to be substituted. The

new mutated vector will be part of the next generation

if its objective function will be lower than that obtained

with the original base vector.

The scheme DE2, differs from the DE1 basically

in the generation of the new trial vectors:

( ) ( )3211 xxFxxxx bestnew

������−⋅+−⋅+= λ (33)

The additional control variable λ regulates the

greediness of the scheme to converge towards a local

minimum by adding a vector deviation which depends

also by the distance to the best member of the current

generation bestx�

. Other variations may be introduced by

modifying the crossover function (exponential or

binomial) and the base vector 1x�

in for the new vector

differential generation formula (15) that instead of

being randomly selected it may be fixed to the best

vector of the current generation.

In any case no attempt of personalizing the

algorithm has been done for this study, and the

standard DE/rand/1/exp algorithm has been used with

the following constant factors: F=0.6, CR=0.8, NP=50.

Usually the minimum value is reached after 20-30

generations, after that no more improvement is found.

Being the minimum value not known, the value to

reach has been set to an extreme value of -100% so the

procedure always stops when the maximum number of

iteration is reached. In such a difficult and irregular

free variables design space, it is difficult to ascertain if

the absolute best vector is really the optimum. A weak

confirmation may come from the comparison of the

shape determined for the last three or four best cases: if

these forms are similar, and were found in the last

generations, hopefully they represent the optimum hull

forms. This is comparison is shown, in the examples of

figure 14.

OPTIMIZATION EXAMPLES

In Brizzolara (2003) a first work on the optimization of

the underwater hull form of SWATH was presented

and already showed promising potential possibilities

for an eventual automated ‘intelligent’ procedure. In

the previous study, in fact, the same parametric hull

forms, with the same given displacement, were used

and a systematical evaluation of wave resistance was

performed, exploring a certain range of variation of the

five free variables FV, without any optimization

strategy. The best hull shape was found, a posteriori,

by ordering the solutions with respect to the calculated

wave resistance. The calculation were done for two

different design speeds, Fn=0.30 and Fn=0.50, and

already some interesting and marked trend of the shape

of the best hull were found.

The first set of new optimization activities is

intended to complete and extend those first calcu-

lations, introducing, as a novelty, the optimization

algorithm and a couple of new design speeds, to better

resolve the effect of forward ship’s velocity.

The object function (27) has been calculated

assuming, as reference, the calculated resistance for a

conventional SWATH design, which features two

underwater bodies of revolution in similitude from

model #40050165 of Series 58 (Gertler, 1950) and a

parabolic strut, whose form was kept constant during

the parametric optimization. Main characteristics of the

reference hull are summarized in table 2, while the 3D

representation of panel meshes are illustrated in figure

3, together with those of the optimum hulls found.

Table 2: Main geometric characteristics of the reference

SWATH design with underwater hull from Series 58.

Underwater Hull Strut

LH [m] 80.0 LS/LH 0.70

DH/LH 0.075 BS/LS 0.07

CP 0.65 XLE/LH 0.15

Global

S/∇2/3 11.9 T/L 0.131

L/∇1/3 4.89 LCB% (+ aft) -4.6%

At each design speed, the optimization algorithm

created some 2000 set of parameters with an effective

total calculations of about 900 (only the feasible

geometries were calculated). The considered speeds

were Fn=0.30, 0.35, 0.41, 0.50, so ranging from

medium to high speed. The four best geometries found

at the end of the optimization are represented in figure

14, which shows the 2D plot of the hull generatrix

curves together with the strut profile and the reference

S58 profile. It may be noted that the shape of the four

best candidates (the ones which scored the 4 minimum

values of the wave resistance) are very close together,

although their FV vectors are distant in the domain

space. The corresponding generated 3D geometries are

represented in figure 3.

Figure 13 presents an example of correlation plots

between the objective function and the values assumed

by the geometrical parameters found during the

optimization, at Fn=0.41. It is clear the distinct increase

of wetted surface (S) needed to obtain a lower

resistance (the wavy form of the curve increase its

length with respect to the reference hull), as well as the

best position of the first maximum diameter (m1) is

more advanced or the best of LCB at this speed is

towards midship and the T.E. curvature radius is higher

than the reference hull.

In addition, from the analysis of the best generatrix

curves, it emerges that when the speed is increased, the

through position advance from the most rear

corresponding to Fn=0.30 to a more central one at the

highest speeds (Fn=0.41,0.40). This shift can be

correlated to wave interference effect. In table 3 which

summarizes the main characteristics of the inviscid

optimized hull forms, it has been reported also the

distance between two peaks (m3-m1) of the optimized

hull, and that between the two peaks and the

intermediate through (m3-m1 or m2-m3), which seems to

be correlated to the half or full (λ/L) fundamental wave

length. At Fn=0.35 and 0.41, for instance, the hull peak

to through distance is close to half the fundamental

wave number. In fact the scatter plots of figure 9, it

may be noted that the best four solutions are laying in

the area of the graph where the non-dimensional

coordinate of the first maxima is close to the leading

edge (m1≅0.1) and second relative maxima is close to

trailing edge (m2≅0.8).

Table 3: Main characteristics of the optimized hull forms

with inviscid method

Des

Fn λ /

(2L)

R0 r1 m2 -

m1

m3 -

m1

m2 -

m3

LCB

% S /

∇2/3

0.30 0.28 2.02 0.30 0.60 0.44 0.16 -6.1 12.1

0.35 0.38 1.74 1.19 0.68 0.40 0.28 -1.6 12.3

0.41 0.53 2.60 1.95 0.73 0.23 0.50 -0.8 12.4

0.50 0.78 3.22 2.18 0.77 0.28 0.49 -0.6 12.4

Of course, the positive interference effects are

valid around the design speed only: in general at other

speeds the optimized hull will, on the contrary,

experience negative interference effects. This fact is

clearly shown by the comparison of the wave

resistance coefficients of the four optimized hulls over

a wide speed range, represented in figure 11. It turns

out, in fact, that the hull which is optimized for

Fn=0.30 is worse at highest Fn, with respect to the hull

optimized for these highest speeds. The contrary holds

for hulls optimized at high speed, which have the

highest wave resistance at low Froude numbers. In the

same figure it is reported also the wave resistance

calculated for the reference hull, which turns to be the

worst in absolute in the whole high Fn numbers range.

The above correlation is confirmed by the

examination of the generated wave pattern at different

speeds. This comparison is shown in figure 16.

Evidently, each optimized hull creates the lowest

waves at its optimum speed, while in general the

conventional SWATH produced always the highest

waves at high speeds. The wave cancellation effects are

visible mainly in the transversal wave system induced

between the demi-hulls and in the wake, but also in the

divergent wave components, important at the highest

Froude numbers.

If the optimization criteria incorporates a weighted

average of the resistance at different speeds, as

formulated in (27), then of course, the optimized hull

forms will appear different, but in any case always with

a double peak shape. In this case the resistance curve

will also show a compromise trend with respect to the

ones shown in figure 11, depending on the weight

given at the wave resistance at each considered speed.

From the designer point of view, however, it is

more interesting the comparison of the results obtained

with the viscous-inviscid automatic optimization. The

calculations were made for a full scale length of 80m,

keeping the same maximum diameter, draft and

prismatic coefficient of the reference hull (table 2) as in

the previous optimization cases.

Table 4 present the characteristics of the optimum

hull forms found at the same target speed of the

inviscid design case. Figure 15 shows the generatrix

curves of the optimized hull forms obtained with

viscous-inviscid method (plain thick line) in

comparison with those obtained at the same speed with

the inviscid method (dashdot line).

Table 4: Main characteristics of the optimized hull forms

with viscous-inviscid method

Des

Fn λ /

(2L)

R0 r1 m2 -

m1

m3 -

m1

m2 -

m3

LCB

% S /

∇2/3

0.30 0.28 0.05 0.41 0.15 0.08 0.07 -4.4 11.8

0.35 0.38 2.87 1.15 0.64 0.58 0.06 -4.6 12.3

0.41 0.53 1.79 1.14 0.69 0.35 0.34 -2.0 12.3

0.50 0.78 1.62 1.75 0.63 0.25 0.38 -0.8 12.3

Being the total resistance used as objective function for

this new series of optimizations, the best hull form will

be the one which realizes the best compromise between

the (usually contrasting) optimum shapes to minimize

each individual component of the total resistance

(friction, viscous-pressure and wave resistance, in our

scheme). The compromise implicitly takes into account

the share which each resistance component has in the

total resistance at each considered design speed. For

instance, the hull optimized at Fn=0.30, where the

wave resistance is not the main part of total resistance,

shows, with respect to the inviscid case, a reduced

gradient of the generatrix curve at the L.E., and a lower

peak to through distance. This modifications tends to

reduce the resistance due to friction and b.l. separation,

which result to be the main components of the total

predicted resistance, at this speed.

Same type of trend in shape modification is noted

increasing the speed up to Fn=0.50, at which, on the

contrary, the optimum hull form is nearly the same as

for the inviscid case. At this speed, in fact, the wave

part is dominant with respect to the viscous part of the

total resistance. This concept of dominance of the

various components of the object function is well

rendered by the two Pareto plots of figure 10, which

represent the viscous resistance reduction vs. the wave

resistance reduction, for all the calculated cases during

the optimization at Fn=0.35 and Fn=0.41. The four best

cases are positioned all on the Pareto frontier which

defines the limit curve on which it is not possible to

decrease one component of the object function without

increasing also the other. From all the above reasons it

is natural that the optimum solution moves towards

higher viscous resistance (with respect to the reference

hull form) which are more than compensated by the

considerable decrease obtained on the wave resistance.

Interesting, at last, is the comparison between the

predicted total resistance for the optimized hulls. As

expected the optimized hull for Fn=0.30 (O-30) has the

lowest predicted total resistance in the lower speed

range, while already above 18 knots the other three

hulls (O-35, O-40, O-50) experiment the lower

resistance. Also in this case the conventional hull form

with S-68 hull, is the worst at the highest speeds, due to

its high wave resistance properties.

CONCLUSIONS AND FUTURE PROSPECTS

Without repeating main conclusions already drawn in

the previous sections, we summarize the followings.

The validity of our viscous-inviscid CFD method

for the prediction of the total resistance of SWATH

like hull shapes has been verified against experimental

results, with very satisfactory, as even unexpected,

correlations. The possibility to use an automated

optimization procedure involving parametric geometry

generation methods, a viscous-inviscid CFD method

and an efficient global optimization algorithm, is

proven to be effective and brought in general to

acceptable and feasible solutions. The importance in

considering the total (wave+friction+viscous_pressure)

resistance for the optimization of the SWATH hull

forms is also hopefully demonstrated in the paper and

obviously depends on the design speed. The

advantages in using a non-conventional underwater

hull forms, shaped in such a way to reduce total

resistance, is also demonstrated in the paper. The

maximum gains in terms of wave resistance, respect to

a (non optimized) conventional solution can be up to

70%, while in terms of total resistance the gain is more

limited but still considerable (order of 30%). These

gains, naturally, vary with the relative Froude number.

Some topics for future developments regard:

The thin boundary layer method used could be

improved and made more robust, especially in the

prediction of the transition location and separation

point. The parametric geometry generation method,

though very efficient and inherently respecting the

design volume constraint, demonstrated some

weakness in global application, giving from half to one

third of unfeasible solutions, which in general slower

the convergence of the optimization algorithm. Alter-

native formulations are being tested using splines.

More optimization cases will be run, considering

different design scenarios, including also the effect of

separation of the demi-hulls, the dimension and shape

of the struts, the number of underwater hulls, etc.

AKNOWLEDGEMENTS

Author wishes to thank Rodriquez Engineering for

according to publish towing tank test results of the

SWATH like demi-hull together with the validation

study made for the an ongoing research project with

them dedicated to the design of a hydrofoil hybrid

SWAMH. Special thanks to Claudia for her continuous

support.

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Figure 9: Correlation plots between the position of the two

relative extremes m1 and m2 (top) on the generatrix curve of

the underwater body of revolution, obtained from all the

cases calculated for the optimization at Fn=0.41. Bottom:

same correlation but for Fn=0.35.

Figure 11: Comparison of the predicted wave resistance

coefficients for the four optimized hulls (with respect to wave

resistance only) over a large speed range. The wave

resistance coefficient of the reference hull is also represented.

Figure 10: Pareto plot of the wave resistance gains vs.

viscous resistance gains during the optimization of the hull

forms at two different speeds: Fn=0.35 (top) and Fn=0.41

(bottom). Reference resistance calculated for the S58 hull.

0.0

0.5

1.0

1.5

2.0

2.5

10 15 20 25 30V [knt]

RT

[M

N]

0.25

0.3

0.35

0.4

0.45

0.5

Fn

Series-58

Optimum Hull for Fn=0.30

Optimum Hull for Fn=0.35

Optimum Hull for Fn=0.41Optimum Hull for Fn=0.50

Fn

Figure 12: Comparison of the predicted total resistance for

the four optimized hull (with respect to full scale total

resistance) over the complete speed range. Predicted

resistance of the reference hull S58 is also represented.

Figure 13: Correlation plots between geometry characteristics and calculated wave resistance for all the about 900

cases analysed during the automatic optimization procedure of SWATH hull, at Fn=0.41.

Fn=0.30

Fn=0.35

Fn=0.41

Fn=0.50

Figure 14: Best underwater hull generatrix curves found at the end of the optimization procedure for wave resistance alone, at four different design Froude numbers.

Fn=0.30

Fn=0.35

Fn=0.41

Fn=0.50

Figure 15: Comparison between hull generatrix curve optimized with the viscous-inviscid method (plain thick curve) and those optimized with the inviscid method

only (dash-dot curve), at four different design Froude numbers. The generatrix curve of the reference Series 58 hull (plain thin curve)is also plotted.

O-30,Fn=0.30

S-58,Fn=0.30

O-50,Fn=0.30

O-35,Fn=0.30

O-30,Fn=0.50

S-58,Fn=0.50

O-50,Fn=0.50

O-35,Fn=0.50

O-30,Fn=0.35

S-58,Fn=0.35

O-50,Fn=0.35

O-35,Fn=0.35

Figure 16: Contour plots (ζ/LH*100) of the wave patterns generated by the original hull (S58) and by the three

optimized hulls (O-30, O-35, O-50), each optimized at a different optimization speed (Fn=0.30, 0.35, 0.50).