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OCR for page 977
24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Multi Objective Optimization of Ship Hull Form Design by
Response Surface Methodology
Yoshimasa Minami, Munehiko Hinatsu
(National Maritime Research Institute, Japan)
ABSTRACT
Recently multi objective design has been paid attention
for the development of high performance ship in actual
seas. So far the ship hull form has been mainly designed
from the viewpoint of the propulsive performance.
However it is important to evaluate the total performance
of ship including the seakeeping performance and the
propulsive performance in actual seas. In this study, we
use CFD and strip theory to estimate the propulsive
performance and seakeepin~ performance, respectively.
INTRODUCTION
r ~7 ~~ ¢ ~ ~
In optimization of ship hull form, we often optimize in
such a way that the wave resistance takes the minimum
value, because the wave resistance theory based on
potential theory doesn't require much computational cost.
However with great increase of performance of computers,
CFD have been used to minimize viscous resistance or to
optimize the nominal wake distribution by using nonlinear
programming method [1,23. But there are few examples to
apply multi objective design method using large-scale
Recently since CFD techniques have been developed computation tool, because it still costs a lot.
strikingly, CFD becomes one of most promising tools to This paper concerns the efficient multi-objective hull
catch the difference of hydrodynamic property when a ship form design which multi object functions are propulsive
hull form is slight changed. This is very important in the performance and seakeeping performance. The main
optimization procedure of hull form. However, today CFD interest is development of high performance ship in actual
still requires large amounts of computational cost. Hence sea. We improve the efficiency of multi objective design
the iteration number of optimization has to be reduced in by applying the approximation method in optimization
order to design the ship hull form efficiently. process. The RSM is used in the research of
The gradient-based numerical method that is used configuration design for aerospace engineering. Giunta
widely in optimization problem may search local optima
for the numerous peaks and may require large
computational cost to attain the optimum solution. Thus,
we use a response surface methodology (RSM) as the
optimization tool to search the global optimum for
multi-objective functions. The RSM approximates the
response surface by use of a polynomial function of design
parameters, and the RSM can be evaluated without
significant computational expense in the course of
optimization process. ~ --a-- ---- =--------- ---- ------------ -- ----I
In this study, Wigley model and Series 60 ship model form, it is important to choose suitable design parameters
are used as the initial ship hulls and optimize their hull and how to express the ship hull form mathematically by
form for multi-objective function. In the work, we choose use of the design parameters. We choose a cubic sine
ship resistance, added wave resistance, and amplitude of function that varied under constant displacement condition.
strip motion es the elements ofthe multi-objective function. As the design parameters are changed in optimization
Then we show that RSM is an effective method for the process, the grids used to solve Navier-Stokes (NS)
multi-objective design of ship hull form. equations have to be rearranged. This procedure is worked
_ _ _
applied the RSM to design of the configuration of High
Speed Civil Transport in order to estimate aerodynamics
performance and total gross weight and he successfully
showed the practical effectiveness of the RSM technique
A. In this case, the design parameters are independent
variables as principal dimension and wing configuration.
In this paper, we apply the RSM to design of ship hull
form using RSM and show the effectiveness for
multi-objective design. Since the ship hull form design is
carried out through the gradual modification of shin hull
OCR for page 978
out in design system. Furthermore in order to estimate the
seakeeping performance, we construct the interface
module that make the input data for the strip method by
use of the grid for NS computation. The multi-objective
optimizer is adopted the Genetic Algorithm. GA is used to
search global optimum from many response surfaces
simultaneously and effectively. The flow of our design
system is shown in Fig.1. Multi-objective design of ship
hull is done along this flow.
RESPONCE SURFACE METHODLOGY
The application of the RSM to design optimization is
aimed at reducing the cost of expensive analysis method
like CFD. The RSM make approximate Unction of
response y predicted by variable xi .
y = f(X}'X7 .~. An ) + £ (1)
In the expression of RS, the quadratic polynomials
model I often used because high order polynomials have
many coefficients and requires large computational cost to
approximate the polynomials. A quadratic model in m
variables has the form
y = cO + I< ~ C jX j + I< j irk< C jkX jXk (2)
Where y is the response,x is the design variable and
c is the tuning parameters. The quadratic polynomial
model has (m + b(m + 2) /2 coefficients in m variables.
Estimating the unknown coefficient requires nS analyses. nS
should be more than the number of total coefficients. This
polynomial model may be written in matrix notation as
Ty=c X
Where
y = [yell y~2~)~ ... yens']
x= :
(I)
xl
( no )
. .
xl (XnV )
c = LCo Cl
Cn~ ]
When c is substituted for c into Equation (3), values of
the response can be predicted at any location.
In most application of the RSM, the range of parameters
is defined from the lower and upper bounds on the design
variables. If n design parameters have 2n vertices, it
becomes impractical to evaluate the designs at all of the
vertices on the high order. Therefore we use D-optimal
design that can minimize the error of predicted value by
using the RSM on the lower order t43. The D-optimal
design states that the best set of points in calculation data
is selected to maximize the determinant |XTX| of
equation (4~. So "D" stands for the determinant of the
(XTX)-~ matrix associated with the model. The
D-optimal design leads to minimize maximum variance of
the predicted response surface of this model.
OPTIMIZATION SCHEME FOR MULTI
OBJECTIVE FUNCTION
The optimizer in multi-objective design must search
some candidate regions around the optimum point on each
response surface widely. The genetic algorithm is
commonly used to search a global optimum in wide region.
GA leads to reduce calculation time and increase search
efficiency of searching process for multi objective
functions. GA usually became worse convergence as the
population size becomes large. So we use Micro genetic
algorithm (,u GA) to optimize multi-objective functions.
,u GA has recently been used for faster convergence, since
this algorithm is small population size and less time
requirement in evaluating fitness [53. In recent researches,
it has been reported that,u GA is useful and prospective for
the optimal design A.
Multi-objective design needs generally trade-off
function to satisfy each performance for the designer or
owner's demand. In this study trade-off function is
normalize by the minimum desired level as follows,
0j(X) = Wi x | fi( t) ~' | : Minimize
Where O is normalized optimum value vector, fi(x) is
approximate function of each response surface, wi is the
weight function determined by designer and f is vector
of standard level which designer desire for each objection
functions. The optimizer for Multi-objective function
minimizes the norm of normalized optimum vector .
n, = (m + D(m + 2) nV : number of design variables
2
From a statistical point view, the unique least squares MODIFICATION FUNCTION OFHULLFORM AND
solution c to Equation (3) is denoted as; CALCULATION GRID
c = (XTX) 1XTy (4)
In the optimization processes, it is important how to
OCR for page 979
express the ship hull form with suitable parameters.
Generally, the constant displacement condition is
commonly used as the constraint condition in the ship hull
form optimization. In order to improve the total
performance, a modification function should not only
changes ship hull form locally but also changes globally.
We newly devise the modification function under the
constant displacement condition. This function is based on
a cubic sine function. The cubic sine function has keep
continuity in 1st and 2nd longitudinal and transverse
derivatives at the boundaries of modification of hull form.
In the ship hull form design, 5 parameters are used in
this study. Taking the Cartesian coordinate system such
that the origin is on the cross point of the water plane and
symmetry plane and midship section. X-axes is positive
toward ship stern, where x=-0.5 and 0.5 correspond to AP
and FP. y-axis is positive toward starboard. z-axis positive
vertically downward. The modification function is defined
as;
y(x, y) = yO (x, z) W(x, y) (6)
Where, Adz) express the original hull surface. w(X y) is
the modification function to change hull form in
transverse-direction. We consider the modification method
of hull form in two stages.
First stage, the prismatic curve Cp is deformed. The
modification functions in Cp curve are defined at front part
and apt part. They are:
:ACpf (x) = Cf · Am sins (xf ) ,~.5 < x < of
l /iCPa (x) = Ca Am sin (xa ) xa < x < o.s
Where x' = x x' =_ x
f (O.S+xr) a (0.5-xa)
tCp (X) = Cp (X)orignal + ~Cpf (X) _ 0.5 S x < Of
l Cp (X) = Cp (x)Orjgnal + pupa (X) xa < x < 0.5
Of and xa are the fore and aft boundaries of hull
modification area in Cp curve. Am=dmxB is midship
area. In order to keep displacement of ship constant, the
coefficient of modification function on Cp curve is define
as;
(0.5 -Xa)
=- cS
(0.5+ Xf )
In Cp curve, the fore part of modification area becomes
equal to the aft part of modification area by using this
equation.
Second Stage, the hull form is modified in breadthwise
direction at yz-plane. In order to keep smoothness at the
boundary of modification range, the modification function
in breadthwise direction is expressed with the cubic sine
function.
Corresponding to the fore part of modification in Cp curve,
' _z
z' = z ~ dcr
dm +df ~
Z'=_ z2dcf
ad 3
W(x,z) = ACpf (x) sin (fizz')
4tr
(10)
Corresponding to the aft part of modification in Cp curve,
ad 3
W(X, Z) = ACPa (X) sin (fizz )
4;r
_z
— z Idea
d +d
m ca (11)
z =— z2dCa
ca
Where dmis design draft, dcf ~ dCa is point of maximum
in above cubic sine function. The constant 3 d/4 fez in this
function is used to correspond the integral of this
function to the modified cross section area by equation (7).
This modification system of hull form is illustrated in
Fig.2. Hence, the design parameters become five that is
Lf ~La'Cf ,dcf ,dca -
For optimization of ship hull, it is important to rearrange
grid system in every optimization step. In this study, we
applied flexible grid method 173. The Initial grid is
generated by the use of a grid generation program. As the
design parameters change, the hull form (hull surface grid)
is changed through the above modification method. The
outer grid is rearranged in proportion to value which
normal vector of modification on hull surface is multiplied
into the inverse ratio of the length between the grid point
and outer boundary, if the grids at outer boundary are fixed.
These modification systems of grid are illustrated in Fig.3.
K is defined the number of grid in the normal direction.
(8) K is determined l at hull surface and kmaX at outer
boundaries.
XkneW = Xk°ld + wk (X1 ew _ X1 id ) (12)
Where Xk are the grids in yz-plane,
dk
w, =. __
~dk=~mE31Xm Arm 11 if k>2 (1
dkmax O if k=2
(9) dk is the length from grid point (K=k) to outer boundary
(K=kmax).
In order to estimate the seakeeping performance along
with the propulsive performance, the data for the strip
method has to be obtained automatically from the gird for
NS computation and we devise an interface module in our
OCR for page 980
design system for this purpose. Firstly the interface
module extracts the strip data from the grid data
corresponding to the strip section point in lengthwise
direction. This module provides the data for the strip
method at the following 25 sections.
As = (FP,—,—,—,1,1—, - - ,8—,9,9—,9—,9—, AP)
4 2 4 2 2 4 2 4
This module naturally provides the data for strip methods
as the ship hull is modified in the course of optimization.
CALCULATION OF RESISTANCE
We use CFD code to estimate the resistance of ship. The
CFD code in this study is NEPTUNE code developed in
our institute t83. The governing equations are the
Navier-Stokes equations with the artificial compressibility
assumption. The coordinate system in this study is shown where
in Fig.4. They can be given in conservative form as: 1
R(Q) = V ~(H-HV) = 0 (20)
'ilk Faces
A cell-centered finite volume approach for spatial
discretization is adopted. q and v, are placed at the
center of each grid cell and the grid cell is treated as a
control volume. Integration of the governing equations
(14) over a cell, yields
|,~V + ~ {(e + ev)dSx + ( f + fV)dSv + (g + gv)dSz }
At v
(18)
Where the integration of flux is conducted by using the
Gauss integration theorem. His the volume of the cell
and TV is its boundary. Sx, Sy and Sz are the area vector
components of cell boundaries.
The governing equations (18) are discretized as follows:
Balk +R(Q) 0 (19)
id +~(e+eV)+~(f+fv)+~(g+gv) =0 (14)
at do by Liz
The dependent variables q, the inviscid e, f ,g and the
viscous fluxes ev, fV, gv are written as
[ev fv gV]=-(R +V!
e
U2+p TV
vU v2+p
we WV
- o
2uX Ox + Vx
uy + vx 2vy
uz + wX vz + wy
O O
(16)
In Eq. (2), u, v, and w are the Cartesian components of the
fluid velocity. p is the modified pressure defined by
* z
P=P + 2
F.
n
Where p is the original pressure and Fn is the
Froude number based on the ship length and Re is the
Reynolds number and v, the non-dimensional kinematic
eddy viscosity determined from a turbulence model. ,6
is a positive constant of artificial compressibility.
Turbulence model in this study is used Baldwin-Lomax
model.
In the above equation, note i, j and k are the cell
numbering. H and H' are inviscid flux and viscous flux
on the cell's boundary. The change of the volume is
expressed as the sum of the six boundary faces of cell. The
inviscid flux H is defined using contravariant velocity
U as
(15) H - SxE + SyF + SzG =
,l]U
uU +PSx
vU + PSy
wU + pSz
(21)
U = Sxu+Syv+Szw (22)
The inviscid flux ~ is evaluated by MUSCL method
based on the flux differencing splitting scheme. The viscid
flux Hv cab be also expressed as;
Hv -SXEV +SyFV +SzGv (23)
Where the gradient of velocity on the boundary faces of
(17) cell is conducted using the Gauss integration theorem
Time integration is used by backward Euler formula as
follows;
Q(n+~)
ink + R(n+~) = 0 (24)
At
Here, the subscript /\t is time step. Equation (24) is solves
by an approximate Newton relaxation method as
OCR for page 981
The surge, heave and pitch motion can be obtained by
solution of ship motion equation. Ship motion equation is
the complex linear system as follows;
j=~3 5 [ (dip + Aid) + i~Bij + City pj = Ei (~27)
Where
* (n+l),m
I + { bR ~ ~Q(n+l),m
~ Q'(~- ) - Q~,k 1~ R(n+~),m ~ (25)
`~Q(n+l),m = Q(n+l),m+1 _ Q(n+l),m (26)
In the evaluation of R , the inviscid flux in R is
approximated by using the first order upwind differencing
for the inviscid flux in R . la is 4X4 identify matrix
except the first diagonal element which is zero in order to
satisfy the incompressible continuity equation. The
subscript m denotes iteration number on each time step.
Equation (24) starts from On+ = On at each time step and
after reaching the convergence, Qn+~ is set to Qn+i m+ . At
each step of the Newton iteration, a large sparse linear
system of equations has to be solved. A symmetric
Gauss-Seidel relaxation approach is used.
In order to accelerate the convergence of the equation, a
multigrid scheme (V-cycle) and a local time stepping
method are employed. The full multigrid strategy (FMG)
is used. In the FMG, the solution procedure starts with the
coarsest grid and after some cycles in the coarsest grid, the
solution is interpolated into the next finer grid. This time,
the solution is obtained with three-level full multigrid
method. The procedure repeats until the solution of the
finest grid is obtained.
CALCULATION OF SEAKEEPING PERFOMANCE
The seakeeping performance of the ship is evaluated by
use of a strip theory (Salvesen-Tuck-Faltinsen method) in
this design system t93. The Strip theory is usually used to
evaluate seekping performance in primarily design for
shipbuilders. This theory is practical design tool for
seakeeping performance.
Consider a ship advancing with constant speed U and
in the incident wave with small-amplitude motions of
angular frequency ~ in deep water. The longitudinal
coordinate system is shown in Fig.4. The x-axis is positive
toward bow, and z-axis is positive downward. X is the
angle between the direction of ship course and the
direction of incident waves. The wave condition in this
study is only one case. The direction of wave is in head sea,
the ratio of ship length and wave height is 1/50 and the
ratio of wave length and ship length is 1.0.
for i = 1,3,5
Where Mjj denotes the generalized mass matrix end A,j,
Bij and Cij are the added mass, damping and restoring
coefficients. These are evaluated independent of the
hydrodynamic analysis. The wave excitation forces and
moments are determined in accordance with the STF
method, and nonzero term among there are
ME ~ = M33 = pV M55 = It l
C33 = Maw C35 = C53 = -,~AWXW C = ,~VBML J
Here V is the displacement, IN the moment of inertia
about the y-axis, Aw the water plane area, xwthe center
of the water plane area, and BM, the longitudinal
metacentric height.
The added resistance is calculated from Mauro's
formula t103:
Ma to [ ~0 ~ ~ :iC(k)| + |S(k)| ~
x vie) {k - ko cos X) do (~28)
jv (k) - k
Where
v(k) = (~+kU2)1g = k+2~+k2 /Ko
k i=- 20 (I+2T+~). k i= 20 (1-2r+~)
K=~/, r=Uw/g, Ko=~2, a is wave height
Here C(k) and S(k) denotes the symmetric and
antisymmetric parts of the Kochin function respectively,
and they can bee written as:
C(k) = C7 (k) ~ Cj (k)
S(k) = S7 (k) -—Ad,—S j (k) |
(29)
where j denotes the mode of ship motion. The details of
the solution method for these velocity potentials aren't
described in this paper.
OCR for page 982
RESULTS OF DESIGN USING RESPONSE
SURFACE METHODLOGY
In the ship hull form design, we use 5 parameters
as Of ,Xa,Cf ,dcf Edna . The Constraint conditions in this study
are fixed displacement and keep space at local position
(e.g. engine room) and fixed principal dimensions
~ L, B. dm ). The object functions are resistance, added wave
resistance, amplitude of ship motion.
a) Wigley model (case of two design parameters)
The case of two design parameters is carried out and the
result can be visualized in order to recognize the effect of
design parameters easily. Wigley model is used for
simplicity to handle the hull form. The principal
dimensions of this model are Length 5.0 m, Breadth 0.5m
and draft 0.3125m. The two design parameter are Cf and
xa . The other parameter is fixed in optimization process.
The range of parameters is shown in table 1. The ranges of
design parameters are determined to consider geometry
constraint conditions roughly. Computational conditions
are as follows, Rn = 5.57 x 106 end Fn = 0.25 in the present
work. The only one of incident wave condition is studied.
The wave condition is in head sea and wavelength ratio is
1.0 and wave height ratio is 1/50. The numbers of grid
system for Wigley model is 105X33X33 and H-O grid
system is used. Although the present grid is relatively
coarse, it is sufficient to grasp the relation between the
solutions and hull deformation. The hull data used in Strip
theory is generated through interface grid module from
this grid.
First of all, the benchmark test of an optimization
problem is done to check the optimization efficiency of the
RSM. We have optimized a single objective function
putting weighting factor on multi objective functions of
resistance, added wave resistance and ship motion for
Wigley model. The simulation condition is above. Fig.5
shows the convergences of Optimization process using the
RSM. The results of Successive Quadratic Programming
and Genetic Algorithm have been shown in Fig.5. Thus,
the RSM has be able to search the optimum value by 12
calculations. On the other hand, the SQP has searched
optimum value almost by almost 50 calculations and the
GA have not converged by 150 calculations yet. We have
understood the optimization efficiency of the RSM from
these results
Next, the approximated response surface using RSM
compare with response surface using calculation data for
design parameters to investigate effectiveness of the RSM.
In order to express actual response surface is used the
results of calculation for design parameters (20 level in
each design parameter). Thus the number of calculations is
400 (20 X 20~. The components of multi objective
functions in this study are the resistance coefficient c' as
the propulsive performance, the added wave resistance
coefficient Craw, the nondimensional amplitude of heave
motion z' and the nondimensional amplitude of pitch
motion B' as the seakeeping performance. The 400
calculation conditions are eliminated from consideration
any infeasible design which exceeded any of the geometric
constraint. At least 12 calculations of estimation are
needed to fit a quadratic polynomial in two variables. Note
that preliminary research indicates that the number of
calculations in RSM process needs at least 1.5 to 2.5 times
than the number of unknown coefficients [11~. So we
choose 24 calculation conditions twice as many as the
number of coefficients in RSM model. The 24 parameters
are selected in order to minimize (XTX)~' by D-optimal
design method. The 24 configuration of ship hull modified
by using these parameters are evaluated using CFD tool
and strip theory. The RSM can construct response surface
by 24 results of calculation. The response surface made in
the RSM calls ARS (approximated Response Surface by
using the RSM. Also, in order to understand actual effect
of the difference of hull form, we constructed the response
surface of numerical experiment (RSNE) by interpolating
the results of calculations for all levels in each design
parameters. The correlation coefficient of the ARS and the
RSNE is almost 1.0. As one example, the correlation of
resistance is shown in Fig.6. So approximate values of
RSM agree with calculation results well. The ARS and
RSNE in each objective function are shown in Fig.7. We
have understood that the results of the ARS agree well
with the feature of response surface configuration and the
position of minimum from the view of response surface.
For cat and 0' in Fig.7, we understand that the RSNE
and the ARS are almost quadric surface for the change of
design parameters. On the other hand, for era,,, and din
Fig.7 the minimum of added wave resistance and the
amplitude of heave motion change linearly as the design
parameters change. The design parameter Cf seems to
influence on added wave resistance and amplitude of
heave motion. A linear term of RS model is predominant
for values of this estimation. From the response surface,
we recognize that enlarging the front area of the ship
minimize these estimation values.
To minimize multi-object functions by using ARS,
Genetic Algorism is applied as one of optimization
methods. The weights of objective functions are equal to
each estimation value. The constraint condition isn't
needed to decide in this optimization process, because the
upper bound and the lower bound of the design parameters
have already been decided under the geometrical
constraints conditions. The optimized solutions can be
obtained in 200 calculation cycle by using GA. This
optimization calculation takes 1 minute on Personal
Computer. The results of optimization are shown in % as
OCR for page 983
compared to that of original hull form in Fig.8. The design parameters become (-0.1, -0.01, 0.1, -0.04, -0.01~.
resistance is reduced in about 5%. The coefficient of The optimized solutions can be obtained in 500 calculation
Added wave resistance is reduced in about 1%. Other the cycle by using GA. This optimization calculation takes a
amplitude of ship motion is reduced a little. We understand few minutes on Personal Computer. We evaluate the
that this design system is successful and useful for multi performances of two designed hulls. The multi objective
objective optimization. Fig.9 shows the comparison of functions of two design hull forms are shown in % as
optimized hull form and original hull. The optimized compared to that of original hull form in Fin. 11.
hull form is to reduce the fore area for original hull form
and become V type at fore part. The stern of designed ship
expands slightly.
b) Series 60 ship model (case of five design
parameters)
Next, the five design parameters of Of if ~xa~dcf and
dCa are used in order to apply for practical problem.
Using five design parameters become more flexible. The
ranges of parameters are shown in table 2. The bounds of
design parameters are determined to consider geometry
constraint. Multi objective functions are the resistance
coefficient c`, the added wave resistance coeff~cientcr=,,,
the nondimesional amplitude of heave motion z' and
nondimensional amplitude of pitch motion B' . The
principal dimensions of Series 60 model in this study are
Length 7.0 m, Breadth 0.933m and draft 0.373m. The
computational conditions are Rn = 1.49 x 107 and Fn = 0.25 .
The only one of incident wave condition is studied in this
case. The wave condition is in head sea and wavelength
ratio is 1.0 and wave height ratio is 1/50. The numbers of
grid system for Series 60 model is 105X57X33 and H-O
grid system is used.
Firstly we choose 3125 design parameters (5 level in
, I, c7
Firstly in case of optimization of total performance, the
resistance (in calm water) is reduced in about 2.0%. The
coefficient of Added wave resistance decrease slightly.
Also the amplitude of pitch motion is reduced in about
1.0%. The amplitude of heave motion decreases slightly.
The resistance is slightly improved. Since Series 60 model
is thin hull form, the little modification of hull don't
influence greatly on the resistance. Fig.12 shows the body
plan of original and designed ship. The aft part of
modified hull form change near U type. The aft part is
smooth compared with original hull form.
For optimization of seakeeping performances, the
amplitude of pitch motion is reduced in around about 5%.
The amplitude of heave motion increases slightly. The
resistance and the coefficient of Added wave resistance
and increase slightly. For the increase of the added wave
resistance, we think that the sensitivities of weight
functions for the added wave resistance and amplitude of
ship motion aren't the same order. In this case, we have
found that the amplitude of heave motion is related to the
added wave resistance compared with the amplitude of
pitch motion. From the body plan of designed ship in
Fig.12, we understand that the fore and aft part of hull
form become near V type to enlarge damping force.
Thus, above simulation results show that the designer
each design parameter). The quadratic model in five analyzes the trade-off of multi-object functions easily
variables needs 21 coefficients. So we select 42 design ~~
parameters twice as many as number of coefficient by
using D-optimal design. These 42 configurations are
evaluated using by CFD tool and strip theory. The
response surfaces are constructed from 42 results of
calculation. The each correlation coefficient of ARS and
RSNE in multi objective functions is almost 1.0
respectively. Fig.10 show the correlation of the ARS and
the RSNE in c,. So we conclude that ARS can presume
an actual response for design parameters in this problem
enough. We minimize multi-object functions of ARS by
using optimization method GA. The constraint conditions
aren't considered in optimization process, since the bounds
of design parameters are determined under geometry
constraint. The weight of multi objective function is used
two cases. In one case, the weights are set to (1,1,1,1,1) for
c,, craw, z' and 0' to minimize totally objective
functions. As the results of optimization, design
parameters become (-0.35, -0.01, 0,35, -0.025, -0.040~. In
another case, the weights are (1,10,10,10,10) to minimize
the seakeeping performance. As the results of optimization,
without large computational cost.
CONCLUSION
This paper concerns ship hull design for multi-objective
function. The main focus is placed on the development of
optimization scheme.
The effectiveness of the RSM can be shown in comparison
with the RSNE and the ARS for two design parameters using
Langley model.
In this study, He quadratic model have sufficient accuracy
to express response surfaces of multi-objective functions for
hull form design.
The RSM model with 5 variables can be applied to design
Series 60 ship model for multi-objective functions. As
weight functions of multi objective function are changed, we
can design 2 ship configurations for op~ni~ion of total
estimation of performance and seakping performance. We
show that the desired ship hull can be estimated easily by
changing weight coefficient of a multi-objective function
OCR for page 984
without using large-scale calculation tools.
As next step, we'll try to add propeller effect and short-term
prediction to design system multi objective function propeller
effect estimate sea-margin (needed engine power) in actual seas.
On the other hand, the easy and practicable modification
method is used in this study. We need to devise much more
suitable modification method of ship hull form for
multi-objective design in the future. In additions as the coefficient
of RSM model increases to express complex configuration, we
have to develop a design technique that approximate the response
surface with quick response for change of design parameters.
ACKNOWLEDGEMENT
Optimal Design of lifting Bodies", Journal of the Kansai
Society of Naval Architect, No. 235, pp.1-8, 2001
to] Burgreen, GW. and Baysal, O.: "Three-Dimensional
Aerodynamics Shape Optimization of Wings Using
Sensitivity Analysis" ,ALAA Paper 94-0094, 1994.
t8] Hirata7 N. and Hino, T.:"An Efficient Algorithm for
Simulating Free-Surface Tublent Flows around an
Advancing Ship", Journal of the society of Naval Architects
of Japan7 Vol.185, pp.1-8, 1999
[9] Salvesen7 N., E.O. Tuck and O. Faltinsen: "Ship motion and
sea load ", TSNAME, Vol. 78, 1970
We would like to thank Dr. Todoroki of Tokyo Institute of t10] Mango, H.:'~Wave resistance of a ship in Regular Head
Technology for his Response Surface Methodology Software. Sea", Bulletin ofthe Faculty of Eng., Yokohama National
(http://floridames.titech.ac jp/todo-j.htrnl).
REFERENCES
[1] Hinn T. Kodama Y and Hirers N. . "nv`1r~1vnamin
~ ma, ^., ^~7 ^., FAX ~ ~ ~7 , ~ ~ Eva ~ `~ -
Shape Optimization of Ship Hull Forms Using CFD,7
Prceedings 3rd Osaka Colloquium on Advanced CFD
Applications to Ship Flow and Hull Form Design Osaka
Japan May 25-27, 1998, pp. 533-541.
[2] graham Y., Himeno, Y., and Tsu~har~ T.,: An
Application of Computational Fluid Dynamics to Tanker
Hull Form Optimization Problem,77 Proceedings 3 rd Osaka
Colloquium on Advanced CFD Applications to Ship
Flow and Hull Form Design, Osaka, Japan, May 25-27,
1998, pp.515-531.
[3] A. A. Giunta, V. Balabanov, S. Burgee, B. Geossman,
R.T.Haftka, W.H. Mason, and L. T. Watoaon,
"Multidisciplinary optimization of a super sonic
transport using design of experiments theory and
response surface modeling", Aeronautical J., vol.101
(1008), pp.357-356, 1997.
[4] Myera,R.H. and Montgomery,D.C. "Response
Surface Methodology, Process and Product
Optimization Using Design Experimets", pp.1-141,
279-401,462-480,John Willey & Sons, New York,
NY (1995)
[5] Kriah~rnar, K.: "Micro-Genetic Algorithms for
Stationary and Non-Stationa~y Function Optimization",
SPA: Intelligent Control `and Adaptive System, Vol. 1196,
Philadelphia, PA, 1989.
[6] Md. Mashud KARlM, M. Ikehat`a, K Suzuki `and H. Kai:
"Application of Micro-Genetic Argorithm ~ ,u GA) to the
Univ., Vol.9, (1960), pp.73-91
t11] Guinta, A.A., Dudley, J.M., Narducci, R. Grossman, B.,
Havana, R. T., Mason, W. H., and Watson, L. T.: "Noisy
Aerodynamic Response and Smooth Approximations in
HSCT Design" , Proceeding of He 5~
A AA/USAF/NASA/ISSMO Symposium on
Multidisciplinary Analysis and Optimization, pp.
1117-1128,Panama City Branch, FL, AIAA Paper
944376, 1994
| Initial ship configuration ~
I Desineen parameter set
I Modification ofhullform ~ DisplacementisfL'ced
Sinusoidal function
t~ )
Generate grid system |
~ . ~
~ .
Calculation grid /
I Interface of grid system 1l
~ StriD data at each section
| Calculation of Resistance | [ Calculation of Seakeeping ~
~ ,
CFD \
REM modeling
Search
| Multi objective optimizer
|optimum ship hull form |
,/ Strip Theory
Fig.1 The flow of design system for multi objective problem
OCR for page 985
I — O ri~n al H u 11 fo rm
/0.8
0.6
~ 0.4
_' 0.2
O
~ . _ .
~ ~ m o difie d h u 11 fo rm
. - . N_
5.2
5.0
~0 5 Xa O Xf 0.5 `' 4.8
X
0 0.1 0.2 0.3
o
-0.05
-0.1
N
-0.1 5
-0.2
-0.25
-0.3
-0.35
1
dcf' dca
............. ............... 7~.
l
Fig. 2 Modification system of ship hull form
I
Hull Surface/ K
~.
Outer Boundary
|K- kmaX
Fig. 3 Rearrangement of calculation and corresponding
to ~e modification of hull fo~m
~r
O1
~ Z
~_
~ X1
X
Fig.4 Coordinate system
~ _.~OP 1
~GA
· RSM
p~ ~_1-
~U
4.6
U~U 1
4.4
0 50 100 150 200
iteration no.
Fig. 5 Companson of convergence histo~y of Objective f
unction
8.0E-04
cn
c, 4.0E-04
,<
O.OE+OO /
O.OE+OO 4.0E-04 8.0E-04
Ct RSNE
Fig.6 Correlation of ~e RSNE and ~eARS
Table 1 Design parame~rs in ~lgley model case
Design Variables Range or Value
Xf 0.3 (fixed)
—0.40 < xa < 0.10
Cf -0.10 < CS < 0.10
dcf -0.03125 (Fixed)
dca -0.03125 (Fixed)
OCR for page 986
7
z
o.
o.~ -
O.` .
o.2 --
0.3 -
xa 0.4
RSNE
z
x Y
z
_ in, - u.
'O. 1 If
z
O.
ARS
7
z
Fig.7 Comparison of response surface of multi objective functions between forge RSNE and the ARS
OCR for page 987
OCR for page 989
OCR for page 990
Representative terms from entire chapter:
design parameters
Orignal
Ct
Craw
z'
v~
I I :: :
_ 1 ~ _ 1 Cl . .
Yll~ I ~ _ ~ R ~ ;
~ _
_~ _
=—
- _ . . .
~ __
=
l
l
l
-
-
_~
~_
0.9 0.92 0.94 0.96 0.98 1
1.02
Fig.8 Objective function for optimizes results as compared
to the value of original hull form (Wigley model case)
i
rllllrltmllllr
-0.04 -0.02 0 0.02 0.04
y
a) original
-
D!
-0.04 -0.02 0 0.02 0.04
y
b) modified
Fig.9 Comparison of original hull form and
optimized hull form
5.9E-04
~'
1 ~
k'! ! 1
5.7E-04
5.7E-04 5.9E-04 6.1 E-04 6.3E-04
Ct RSNE
Fig. 10 Correlation ofthe RSNE and the ARS
Table2 Design parameters in Series60 model case
Design Variables
Xf
xn
I Cf I
L dCf I
dca l
| Range or Value
1 o.lo
~;~
0.05
. . . . _
-0.05 0
y
a) Original
_ ~~\\\A.P. ~ p
I_ ~ 1\\\\\ 11 1
~ _ ~ ' ~ ~ ,
., 1,, , . _ ~ ., , ~
05 0 0.05
y
0.025
o
-0.025
-0.05
0.025
o
-0.025
-0.05
N
N
by Optimized hull form for minimization of resistance
F.P
n non
_~: ~
. . ., ., ~ 1-
0 05 ~ ~ 0.1 35
y
O
-0.025
-0.05
N
c) Op~ni~d hull form for minimization of seakeeping
Fig. 12 Comparison of body plan between for Me original
hull forth and optimized hull form
DISCUSSION
L.J. Doctors
University of New South Wales, Australia
The reviewer would like to congratulate the two
authors on a most practical and interesting piece
of research. Thus, skipping momentarily to the
results, one can see in Figure 7 and Figure 10,
for example, how the performance of the ship
has been improved by means of the described
optimization process, with respect to its total
resistance, added resistance in waves, and
motions in waves.
Furthermore, it is noteworthy to observe the
optimized hull form in Figure 11 displaying a
bulbous bow, when seakeeping is considered, as
one might expect.
My first question relates to the fact that the
resistance of the ship is computed on the basis of
a CFD code, in which one presumably solves the
Navier-Stokes (viscous-flow) equations with
nonlinear free-surface conditions. The only
essential approximation here would be the
particular turbulence model employed by the
computer program. On the other hand, the
authors have used the Salvesen, Tuck, and
Faltinsen (1970) theory for computing the ship
motions. This method is very practical and well
behaved but, nevertheless, assumes that the
motions are small and that the ship is slender.
The Maruo (1960) formula in Equation (31) is
similar in that linearized free-surface conditions
and a thin ship is assumed. To what degree do
the authors believe this inconsistency in the
different theories to be a concern? That is,
perhaps, one could utilize a simpler, thin-ship,
theory for the ship resistance, as was done by
Day and Doctors (1997a and 1 997b).
Secondly, I would appreciate the authors
indicating a typical number of required ship-
performance evaluations during the optimization
process. Thus, in the work of Day and Doctors,
in which the efficient Genetic Algorithm was
utilized, the vessel performance had to be
evaluated thousands of times. Only by using the
various features of the abovementioned linear
theories, could the computation be kept
reasonable. It would therefore seem, on the other
hand, that the optimization process using a CFD
solver would be extremely time-consuming.
Thirdly, could Equation (30) be clarified? The
standard Salvesen, Tuck, and Salvesen theory
predicts ship motions in five degrees of freedom
(indexes 2, 3, 4, 5, and 6~. Alternatively, for
simple motions in head or stern seas, only two
degrees of freedom are considered (indexes 3
and 5~. Have the authors here (indexes 1, 3, and
5) decided to include the surge degree of
freedom?
Once again, I would like to express my
appreciation to the authors for a most
informative paper.
AUTHORS' REPLY
Thank you for your interest in our work and your
questions.
For first question, we think that the inconsistency
of these theories is a difference where designer
gives priority in design process. Each evaluation
function of performances is calculated parallel
and independently in optimization process of the
RSM. The propulsive performance is influenced
by viscous flow around hull. Moreover, this
performance influences greatly on economy of
ship. So we have wanted to estimate this
performance strictly by using CFD. CFD tool
can simulate a viscous flow in high accuracy
and get information of flow field around hull for
hull design or propeller design etc. On the other
hand, ship motion can be estimated practically in
normal wave condition by strip method
excluding rough sea condition. We have checked
the relative improvement of seakeeping
performance for the original using by strip
method.
For second question, we have wanted to examine
whether the Genetic algorithm (GA) is effective
to the multi-objective optimization problem in
hull design. In this study, a multi-objective
function has been converted into a single object
function putting the weighting factor. The GA
can search the optimum of a single response
surface in several minutes. But the searching
time of the GA isn't shorter compared with that
of other optimization method when searching the
optimum of the response surface. One of reasons
is that the multi-peak of response surface doesn't
appear by using simple Response Surface model
and simplification of multi-objective function.
We understand that general optimization method
is enough for this optimization case. In the
future, we have wanted to investigate the
optimization method for searching the optimum
of high-order response surface model.
For third question, the equation of general
longitudinal motion is described in this thesis.
But I'm sorry, I have not been describing clearly
in this thesis that the influence of the surge
motion is not considered in optimization process
in order to simplify the calculation of ship
motion.
