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).
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