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Applied ocean
Research
Applied Ocean Reseai-ch 28 (2006) 223-233
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A pFFT-FE coupling for hydroelastic analysis of floating flexible structures
in waves
Ken Takagi^'*, Jun Noguchi^
'^Deparlment of Naval Architecture and Ocean Engineering, Osaka Uuiversiry, 2-1 Yamadaoka, Suita, Osaka 565-0871,
Kawasaki Shipbuilding Co., 1-1 Kawasakicyou, Akashi, Hyogo 673-8666, Japan
Japan
Received 30 June 2006; received in revised form 4 November 2006; accepted 6 November 2006
Available online 22 December 2006
Abstract
A pFFT-FE coupling method, which can calculate the hydroelastic behavior of floating flexible structures, has been developed. The method
can handle a very large number of constant hydrodynamic panels in a reasonable CPU time. The scheme uses a consistent way of the data passing
in which the energy is conserved between the generalized modal damping and the radiation waves i f the hydrodynamic analysis is accurate
enough. In addition, the scheme satisfies the generalized Haskind-Newman relation between the modal diffraction force and the Kochin function.
These properties are important to ensure the numeiical accuracy. The numerical convergence and the accuracy of the method are demonstrated in
various ways including the comparison with experimental data. Finally an application to the sailing type offshore wind-power plant is shown to
demonstrate the applicability of this method to the challenging problem.
© 2006 Elsevier Ltd. A l l rights reserved.
Keywords: Very lai-ge offshore structure; Floating flexible structure; Hydroelastic behavior; Panel method; PrecoiTected FFT method
1. Introduction
example, Kashiwagi [4] proposed the hierarchical interaction
theoiy
Recently concepts of very large floating structures such as
floating airports (Suzuki [ 1 ] ) , floating wind-power plants [2]
or mobile offshore bases (Palo [3]) have been presented. They
require structural and hydrodynamic methods o f analysis that
differ f r o m those used f o r conventional marine structures. For
example, because o f exceptional geometry and size, very large
floating structures may require hydroelastic analysis to account
properly f o r the interaction between the structure and the
fluid. The flexibility o f these structures may result i n natural
frequencies that can be excited by w i n d waves.
The panel method based on the potential theory w i t h
the linear free surface condition is usually used f o r the
hydrodynamic analysis of these structures. Since the structures
are very large, a large number o f panels are requned
f o r accurate computation, and various methods have been
developed to reduce the computational time and memory. For
f o r accelerating
supported
floating
the
computation f o r the
column
airport. Utsunomiya et al. [5] developed
the fast multipole method to accelerate the computation o f
the pontoon type, very large floating structure i n a shallow
water region. Newman and Lee [6] showed the application o f
the precoiTected Fast Fourier Transform (pFFT) method f o r a
column supported pontoon and a mobile offshore base. These
methods reduces the computational time and memory f o r the
panel method below the order o f A''^, where the submerged
suiface o f the body is represented b y N panels. Newman and
Lee [6] also showed the higher-order panel method, w h i c h
dramatically reduces the number of panels w i t h the same
accuracy. However, the lower-order method may be more robust
i f the structure has sharp corners where local quantities such as
the fluid velocity and second order pressure are important.
The
structural analysis
is usually
carried
out w i t h
a
commercial Finite Element (FE) code, w h i c h provides the
eigenmodes o f the structure. Using commercial FE codes
* Con-esponding author. Tel.: -f81 6 6879 7571; fax: -f81 6 6879 7594.
E-mail address: takagi@naoe.eng.osaka-u.ac.jp (K. Takagi).
0141-1187/$ - see front matter © 2006 Elsevier Ltd. A l l rights reserved.
doi:10.1016/j.apor.2006.11.002
is very useful f o r engineers not simply because o f the
trustworthy result but because o f r i c h pre-post applications.
224
K. Takagi, J. Nogitchi /Applied
The hydroelastic analysis is achieved to couple the F E analysis
and the panel method. This is done b y the linear modal
analysis i n w h i c h the dry modes calculated by the F E code are
used as the generalized modes o f the motion. I n theoi7, this
procedure has been established. For example, W A M I T , w h i c h
is one o f the famous hydrodynamic analysis codes, includes the
capability to analyze the generalized modes o f the body motion
(Newman [7]). However, i n practice, the coupUng between the
panel method and the commercial F E code is not so simple.
Fujikubo [8] reviewed the standard procedure o f the
structural analysis o f a very large floating stnrcture i n w h i c h the
analysis has two stages. I n the first stage, the global response o f
the structure is estimated w i t h hydroelastic analysis. Using the
global response, the stress analysis o f the detailed structure is
performed i n the second stage. I n the global response analysis,
the size o f finite elements is relatively large, since the detailed
stiTictural design has not been decided yet i n this stage. I n
addition, creating a detailed finite element model costs a lot.
Thus, the incompatibility between the finite element and the
hydrodynamic panel often occurs.
Using bar elements is very efficient i n the global response
analysis i f the strucUare is composed of slender components.
This is the case of floating wind-power plants or the column
supported floating aiiport. Even f o r the pontoon type stmcture,
Fujikubo [8] showed a capability o f using the bar elements.
W h e n w e use the bar element, suiface panels f o r the
hydrodynamic analysis, w M c h require smaller panels than the
bar element, should be created. I n this case, we should carefully
treat the data passing between the FE code and the panel
method. Suzuki et al. [9] has developed a computer code named
V O D A C , w h i c h uses the bar element model and the hierarchical
interaction theoiy, f o r analyzing the hydroelastic behavior of
the column supported floating auport. However, they d i d n ' t
show about the energy conservation or other hydrodynamic
properties, w h i c h are often used f o r the validation o f the
hydrodynamic calculations.
I n this paper, we present a consistent way of data passing i n
which the energy is conserved between the generalized modal
damping and the radiation waves i f the hydrodynamic analysis
is accurate enough. I n addition, the Haskind-Newman relation
is satisfled between the modal diffraction force and the Kochin
f u n c t i o n f o r the generalized modal motion. These properties
are very important to ensure the accuracy o f the hydrodynamic
analysis.
I n the numerical calculation, we use the p F F T method
to accelerate the low-order panel method, since i t has no
restriction on the geometry o f the structure. M S C - N A S T R A N ,
which is one o f the populai' commercial F E codes i n the
shipbuilding industry, is used as a typical commercial FE
code. Using the p F F T - F E coupling code, the consistency i n
the energy conservation and the Haskind-Newman relation
o f the numerical results is examined. Numerical results
o f hydroelastic responses and strains are validated w i t h
experimental results. Finally we investigate the viscous drag
force, w h i c h has a considerable effect on the hydrodynamic
response o f the slender structures.
Ocean Research 28 (2006)
223-233
2. Numerical method
2.1. Precorrected
FFT
The p F F T method has been developed by Phillips and White
[10] to accelerate the electrostatic analysis o f complicated 3¬
D stiTJcUires. Korsmeyer et al. [11] has extended this method
to the periodic free surface flow. We apply this method to the
hydroelastic problem of a flexible floating structure. Although
the detail o f the method is f o u n d i n reference [11], w e show i t
briefly f o r convenience.
We solve the problem o f the linearized radiation/diffraction
problem i n the frequency domain. I t is assumed i n this problem
that the fluid is ideal, the flow is iiTotational and the wave
slope is small. Under the above assumptions, the fluid velocity
has a scalar potential §{x,oS),
for: x € IH-' and a; is the
angular frequency o f the incident waves. Since the m o t i o n o f
the structure and the fluid is harmonic i n time, all functions
of time can be represented as H{t) = ^ [ / t e ' " ' ] . We use the
complex number h instead o f H{t) hereafter.
The Cartesian coordinate system is defined so that the z
axis is vertically upward and the x-y plane coincides w i t h the
calm water suiface. Green's theorem and the linearized free
suiface Green function yield the f o l l o w i n g w e l l laiown integral
equation to be solved f o r the velocity potential
-2n$(x,o^)+(
Js,
=
$i;',co)'-^M^dS(x')
dvix')
/" ^^^^jJ^G(x,x',a))dSix')
JSb
dv(x')
for xeSb
where Sb is the portion o f the suiface of the
under analysis f o r which z
(1)
structure
< 0 when at rest, and v(x)
is
its unit suiface normal directed into the fluid domain. The
Green function G (x, x', alj, w h i c h satisfies the linearized free
suiface condition is defined by
G (x, X', c o ) ^ - + -
+ —
-^MkR)dk
(2)
where
r = ^ i x - x ' f + i y - y ' f + (z-z')\
n - ^ J i x - x ' f + i y - y f + {z + z ' f ,
R = J { x - x ' f + ( y - y ' ) \
JQ(X) is the zeroth-order Bessel function. K is the wave number
of the incident waves. I n the case o f deep water, K = oj^/g,
where g is the gravitational acceleration. The Green function
can be calculated quickly and accurately by a F O R T R A N
subroutine o f which the source code is found i n a book by
Kashiwagi et al. [12]
A straightforward method f o r solving Eq. (1) is to discretize
the suiface Sh w i t h Np planar hydrodynamic panels upon w h i c h
the potential and its normal derivative are taken to be constant
and to enforce the discrete equation at Np collocation points on
K. Takagi, J. Nogiichi /Applied
Ocean Research 28 (2006)
225
223-233
Sb, taken to be the panel centroids. Eventually, we arrive at the
linear system o f equations, of which coefficients ai-e represented
by a dense matrix. I f the system o f equations is sufficiently w e l l
conditioned, it may be solved by an iterative method w i t h orderNp cost. I n contrast, the p F F T method uses equally spaced g t i d
points on which the potential and its normal derivative on the
hydrodynamic panels are projected, and E q . (1) is computed by
a F F T convolution technique w i t h order-A^g In Ng cost, where
Ng is the number of grid points.
2.2. Equation
of
motion
In the commercial FE code, the displacement o f the structure
is represented by a vector [g]
Fig. 1. Definition of the element coordinate (.Vg, ye, Ze) and the earth fixed
coordinate (x, y, z).
of the nodal displacement
w i t h six degrees of freedom at both ends o f the bar element,
where the frame o f reference is the Cartesian
coordinate
system defined i n the previous section. The deformation o f
the bar element is based on Euler's beam theoiy and St.
Venant's torsion problem. The vector {g] is represented
{$•} = {sn, Si2, gi3,
Sl6 , S2I' 522, ••• 5N,5,
SU,
we obtain
(cof,-a?)q„-Y:^q,^,[g^"^]^[f^'"^
Hi
as
SN^G}
=
|/^^}
(7)
where the subscript j o f gjk is the nodal number, lc represents
translational and rotational displacements at the node and Ne is
where $-„ is the ampUtude o f incident waves, { / ^ ' " ^ } is the
the total number of the nodes i n the entire stmcture.
modal radiation force due to the mXh modal motion and {ƒ'^^^
The F E code gives the « t h eigenvalue a>„ and the eigenmode
{g''"^}
o f the structure. Since an eigenmode is orthogonal to
is the modal diffraction force. Eq. (7) is the equation o f
motion f o r the amplitude q,, o f die n t h eigenmode. I t is noted
other eigenmodes, the nodal displacements are represented by
thatS
a summation o f the eigenmodes as
coefficient.
(3)
k i
Using nodal displacements, the equation o f motion f o r the
stmcture moving with a cncular frequency co is represented by
{s"^"^}^ { / ^ " ' ^ }
2.3. Hydrodynamic
Z'" coincides w i t h the modal damping
forces
W h e n the bar element is used f o r the stmctural modeling,
the eigenmodes are given i n discrete data o f the displacement
at the nodes o f the element. Thus, i t is necessai-y to reconstruct
( - « 2 [M] + [TT]) {g} = {ƒ},
(4)
a continuous data o f the displacement f o r the computation
of the hydrodynamic forces. The present process is similar
where [ M ] is the mass matrix, [K] is the stiffness matrix, and
to Yoshida's idea [13] that the V O D A C employs. However,
{ ƒ } is the external force vector acting on the nodes. Since the
his idea should be modified to suit the data stmcture o f the
eigenmodes by the F E code are normalized, the mass matrix
commercial F E code and to ensure the hydrodynamic properties
described later. I n order to reconstruct the continuous data o f
has the f o l l o w i n g relation w i t h the stiffness matrix
the displacement f r o m the output data o f the commercial FE
code, the f o l l o w i n g assumptions are used.
-(") I
(a) Displacement between two adjacent nodes is linear.
m —n
(b) Distortion due to the rotational displacement at the node is
(5)
m ^ 71.
negligible. Only the r i g i d rotation around the element axis is
taken into account.
Substituting Eq. (3) into Eq. (4) and u ü U z i n g E q . (5), we get
iT
{co^,-0?)qn
= [g^'^^]
W h e n the 7 th element is defined by the origin node lc and the
end node I as shown i n F i g . 1, the velocity ti^"^ at a point x on
{ƒ}.
(6)
Eq. (6) suggests that the mass matrix and the stiffness matrix
the element is given by
S(")=i«{|(")+ö7)x(x-A\)
,
(8)
are not necessary f o r the equation o f motion. E q . (6) is very
convenient f o r coupling between the commercial FE code and
where
the hydrodynamic code.
I n order to compute the interaction w i t h the fluid motion,
J ~ 1 '^•i ' '••2 ' ^-^'^ J '
the external force is divided into the radiation force and the
d i f f r a c t i o n force. I f the radiation force is moved to the l e f t side,
(9)
226
Takagi, J. Nogiichi/Applied
Ocean Research 28 (2006)
223-233
ƒ 0')/
Jk
1
_
^
—-j{xi-Xk)»{kl
2Lj
/^(»)
^('0^
]ixi-Xk)
and
\
\
(10)
M
-Aj)
x
f^"^
and
denotes the translational displacement at the node k
1^"^
denotes the rotational displacement. The first term
o f 9j"^ denotes the r i g i d rotation due to the translational
displacement o f both nodes and the second term represent the
r i g i d rotation around the element axis. I f we define the radiation
velocity potential f o r the ;ith mode as (p*^") = iux/)^"^ the
boundary condition f o r i t is
V • S^"'
on the body surface.
dv
Fig. 2. Definition of forces acting on an element.
According to the definitions shown i n F i g . 2, the equations
of equilibrium f o r each element are given by
(11)
(17)
Therefore, the boundary condition f o r the radiation problem
is given by
p { x - Xk) XV AS-
M i ^ ' = {XI - Xk) X //^'^
(18)
Si
(12)
where v is the normal vector inward to the fluid. The diffraction
problem f o r the hydroelastic motion is the same as that f o r the
r i g i d structure. The velocity potential of the incident wave is
defined as
= e
Kz~iK(x
cosx+y
sinx)
(13)
by
(<^o + M
(14)
= 0,
where cpd is the velocity potential f o r the diffraction problem.
Summarizing the radiation problem and the diffraction
problem, the overall velocity potential 'P is represented as
$
=
(-ƒ.
g^.
p {x - Xk) xvAS»
e.v
(19)
e.v,
{xi — Xk) I \x\ —Xk\ is an unit vector representing the
axial direction o f the element, Sj denotes the surface o f the j t h
element, f'^^^ and f'^^^ are nodal force acting on the origin node
where x is the wave direction. The boundary condition is given
dv
where
- i g a ^ (00 + 4>d) + i c o T qn4>^"^ •
CO
^
The linearized pressure p
potential is given as
(15)
obtained f r o m this velocity
(16)
where 4>D = i^o +
and Z'"^ is the vertical component o f
the nth eigenmode. Once we get the velocity potential w i t h
the p F F T method, the pressure o f the fluid is easily obtained.
The hydrodynamic forces are computed by integrating the
pressure. I n order to compute the nodal force i n Eq. (7), the
hydrodynamic forces and moment acting on an element should
be decomposed into the nodal forces and moments. Thus, the
f o l l o w i n g assumptions are used.
(a) Nodal moments are zero.
(b) A s an exception o f assumption (a), the twisting moinent is
taken into account. However, the simplest assumption that the
twisting moment is evenly distributed at both end nodes is used.
k and the end node / o f the ; t h element respectively and m[^^
is a moment i n axial direction, i.e. the twisting moment, acting
on the 7th element. Note fliat { a n d { ƒ ^ ^ ^ } are obtained
as the sum o f /^f^' or f^^', since more than two elements are
connected at a node.
Combining Eq. (17) w i t h Eq. (18) and solving i t , we can
get the nodal forces. However, axial forces at both end nodes
cannot be determined uniquely f r o m Eqs. (17) and (18). I f
we assume that the fluid force i n the axial direction is acting
only on the origin node /c, the energy conservation between
the modal damping force and the energy radiation at the far
field due to the radiation wave is ensured. I n addition, the
generalized Haskind-Newman relation is satisfied. Proofs are
shown i n Appendix A and Appendix B respectively. I f we
use, f o r example, the more natural assumption that the axial
force is evenly distributed at both end nodes, neither the energy
conservation nor the generalized Hasldnd-Newman relation is
not satisfied.
2.4.
Viscous
Since,
force-
the
structure
is
composed
of
thin
structural
components, the viscous force plays an important role at the
resonant frequency f o r the elastic modes. I t is assumed that the
viscous force is represented as a drag force and its direction is
parallel to the relative flow velocity
f^^\t) =
\pCdSj V{t)
V{t)
(20)
K. Takagi, J. Nogiichi /Applied
Ocean Research 28 (2006)
223-233
227
0.5m
0.5m
L=4.16m
4.5m
Fig. 3. Sketch of the experimental model.
where f^j-'^ ( f ) is the drag force acting on the centroid o f the jth
element,
is the drag coefficient Sj is the projected area of
coefficient Q is determined f r o m an experimental result o f
which the detail is described in Appendix C.
the element and V (t) is the relative flow velocity at the centroid
3. Numerical results
o f the element, which is represented as
V(t)
(21)
where Ve denotes the velocity o f the elastic motion at the
centroid o f the element. Since (20) is not sinusoidal i n time,
the Fourier averaged value is used f o r the frequency domain
analysis, w h i c h is given by
-,0)
fd
=
4
^P'^d^J
(22)
I n addition, an iterative method is used to solve the equation
o f m o t i o n f o r taldng the nonlinearity o f the drag force into
account. The viscous force is decomposed into the nodal forces
w i t h the same manner as presented i n Section 2.3. The drag
We discuss the accuracy o f the solution f r o m the
energy conservation, the Haskind-Newman relation and the
comparison with an experiment. Then we w i l l show a more
challenging application.
3.].
Numerical
model and
medwd
Takagi et al. [14] carried out an experimental study on the
hydroelastic motion and the d r i f t force o f very large mobile
offshore structure i n waves. Our first target is to make a
comparison w i t h this experiment. The sizes o f the model are
4.16 m length and 4.5 m width. A sketch o f the expeiimental
model is shown i n Fig. 3. The model has 5 transverse beams,
4 lower-hulls and 36 struts. Transverse beams are made o f
K. Takagi, J. Nogiichi /Applied
228
(a) 12,960 panels.
aluminum. Lower-hulls are made o f f o a m i n g urethane and
have an aluminum backbone. I n order to guarantee the smooth
bending motion o f the backbone, there is a small clearance
between the backbone and the foaming urethane parts. Thus
the suiface o f the lower h u l l is not smooth. This may affect the
viscous force, w h i c h is discussed i n Appendix C. Struts have an
aluminum core, which ensures the elastic connection between
the transverse beam and the lower-huU. Further details o f the
experiment are f o u n d i n the original paper.
The stiTicture is modeled by 192 bar elements to obtain
the diy modes. Since this model is very simple, the C P U
time f o r calculating the dry modes takes only a f e w seconds,
although the data input takes o f the order o f a day. On the
other hand, the panel meshing f o r the hydrodynamic analysis
can be done automatically by using F E data, since the sectional
shape o f the stracture is simple. Fig. 4 shows panel-meshing o f
the numerical model f o r the hydrodynamic analysis. Since the
entire model is very large, only a close-up view around a stmt
is shown. We w i l l show the numerical results f o r the coarse
panel (12,960 panels) and the fine panel (28,080 panels). I n the
case o f coarse panel, the average C P U time by X e o n 2.8 G H z
is about 0.2 h f o r one mode, while i t is about 1.0 h f o r the fine
panel. The memory requirement o f the code is approximately
given as the number o f grid points x 1.5 kb, i f the number o f
panels is very large.
The p F F T uses the cell concept (see Phillips and W l i i t e [10]).
The size o f the cell is decided so that the side length o f the cell
is slighdy longer than the longest side among all panels, and
each cell contains equally spaced 27 grids. We used G M R E S as
an iterative solver f o r solving Eq. (1). The eiTor tolerance 10~^
is used f o r a l l computations. We did not use a preconditioner,
however the iteration scheme had no problem to converge. The
number o f iteration is usually less than 20 except f o r the very
short wave case.
conseiyation
223-233
(b) 28,080 panels.
Fig. 4. Panel-meshing of the numeii
3.2. Energy
Ocean Research 28 (2006)
and die Haskind-Newman
relation
Table 1 shows the relative en'or of the energy conservation
between the modal damping coefficient and the radiated wave
energy. Since the number of modes is 30 i n this computation,
we get 900 modal damping coefficients. We randomly selected
4 modal damping coefficients. However, we do not include very
small coefficients, since the relative error o f them is apparently
very big compared to that o f normal coefficients.
Table 1 shows that the eiTor o f the fine panel is much smaller
than that o f the coarse panel and the fine panel has enough
model for the hydrodynamic analysis.
Table 1
Relative error of the energy conservation between the modal damping
coefficient and the radiated wave energy
Mode number
12,960 panels
(%)
28,080 panels
(%)
5.47
2.14
4.20
1.39
0.67
0.39
0.35
1.26
1.13
2.68
1.56
1.12
0.77
0.12
0.98
3.86
X/L = 0.2
5,5
12,12
15,12
25,14
X/L = 0.7
5,5
12,12
15,12
25,14
The eiTor is defined as : |The modal damping coefficient — the radiated wave
energyl/The radiated wave energy x 100 (%).
Table 2
Comparison of the relative eiTor of the energy conservation between a
conventional panel code and the pFFT for a half immersed spheroid
A/L
Conventional code
(%)
pFFT
(%)
1.0
1.5
0.25
0.03
0.61
0.35
The ratio of minor axis to major axis is 2.4 where the draft coincides with the
half minor axis and the body length coincides with the major axis. The number
of panels is 2000.
accuracy except f o r the higher order modal damping coefficient.
I t is considered f o r the higher order mode that even the fine
panel is not sufficient to obtain the exact solution since the
modal shape o f the higher order mode is veiy complicated.
Table 2 shows a comparison o f the relative eiTor o f the
energy conservation between a conventional panel code and the
pFFT f o r a half immersed spheroid, where the conventional
code and the pFFT use the same panel meshing and the
subroutine f o r the evaluation o f the Green function. I n this
case, since the body shape is very simple, both the conventional
code and the pFFT give veiy small eiTors. However, the error
o f pFFT is larger than that o f the conventional code. This
result implies that numeiical results by the pFFT always have
a certain small error because o f the approximation o f the far
field contribution and the grid projection o f the Green function.
K. Takagi, J. Nogiichi/Applied
Ocean Reseaivh 28 (2006)
223-233
229
Table 3
The Haskind-Newman relation of the modal wave exciting force
Mode number
12,960 panels
Diffraction
H - N Relation
28,080 panels
Diffraction
H - N Relation
0.01919
0.00024
0.00145
0.01920
0.00025
0.00145
0.01828
0.00019
0.00134
0.01819
0.00021
0.00135
0.05825
0.00028
0.00015
0.05900
0.00037
0.00014
0.05730
0.00024
0.00018
0.05725
0.00036
0.00018
A / L = 0.2
5
15
25
A / L = 0.7
5
15
25
2.50
Z/Ca
RIS2 12%0pand
P\
1 1
1 1
1
ft
1
1
•
1 \ 1
\
Jl '
1
RIS2 2S0S0panel
ft
o
f\ 1
1
RIS2 20 modes
n R2S2 20 modes
4\
\
\
0
0.2
R2S2 30 modes
1
« R2S2 2S0SOpanel
1
\
0
R1S2 30 modes
1 \
1 1
1 1
R2S2 12960pand
0.4
0.6
0.8
X/L
0.2
0.4
0.6
0.8
XTL
1
1
Fig. 5. Comparison between the 12,960 panels and the 28,080 panels. Lines
and marks present the vertical displacement in the beam sea. R1S2 denotes the
intersection between the second lower-hull from the right and the second strut
from the how. R2S2 denotes the intersection between the first lower-hull from
the right and the second stmt from the bow. Both positions are described in
reference [14].
Fig. 6. Comparison between the 30 modes and the 20 modes. Lines and
marks represent the vertical displacement in the beam sea. R1S2 denotes the
intersection between the second lower-huU from the right and the second strut
from the bow. R2S2 denotes the intersection between the first lower-hull from
the right and the second strut from the bow. Both positions ai-e described in
reference [14].
the longitudinal direction and the empty space is very small.
According to our experience, this e n w f o r more complicated
I n the fine panel calculation, the number o f grid points A^^ is
structures could be 1 % even i f the size o f the panel is very small.
524,288 w h i l e A'^^ f o r the t w i n - h u l l structure is only 262,144.
Table 3 shows the Haskind-Newman relation o f the modal
Thus we decided to use the results by the 12,960 panels f o r the
exciting force. I n this table, the nondimensional modal exciting
comparison w i t h the experimental result i n the next section.
force is presented. I t seems the modal exciting force is not
perfectiy converged i f we use the 12,960 panels, since a l l the
Newman and Lee [6] estimated that A^^ =
0{N'^I'^)
and
the computational cost is O {n'^I^ l o g A^) f o r the general body
values are shghtly different f r o m that by the 28,080 panels.
shape. However, i f the structure is slender i n one dimension, the
However, the Haskind-Newman relation is satisfied very w e l l
efficiency o f the p F F T w o u l d be better since Ng approaches to
at a l l modes.
0{N).
Finally we compare the displacement obtained by
the
12,960 panels w i t h the results by the 28,080 panels i n F i g . 5.
3.3.
Comparison
witli experimental
result
The agreement between them is veiy good. I n the case o f
conventional floating stmcture, it is w e l l k n o w n that the
convergence
of motions is faster than the convergence o f
hydrodynamic coefficients.
Before comparing w i t h the experimental result, we examine
the convergence o f the modal expansion and the accuracy o f
the viscous force. F i g . 6 shows the convergence o f the modal
F r o m these results, i t is said that the fine panel gives
expansion. The sotid Hne and the broken line show the vertical
accurate results. However, the numerical calculation by the fine
displacements calculated w i t h 30 modes. The diamond and the
panel takes a longer time than a very large t w i n - h u l l structure
rectangulai- marks show the results calculated w i t h 20 modes.
problem shown i n Section 3.2 which is a time consuming
F i g . 7 shows the modal amphtude at t w o different wavelengths.
challenging problem w i t h 84,592 hydrodynamic panels. The
These figure show that the numeiical results against the number
reason is that the p F F T method requests grid points at the empty
o f modes are s u f f i c i e n ü y converged. Note that, i n practice, the
space between adjacent lower hulls f o r the F F T convolution
convergence o f the modal decomposition is not so important
integral while the t w i n hull stmcture has a slender shape i n
f o r the first stage design, because the higher order modes are
K. Takagi. J. Nogiichi /Applied
230
Ocean Research 28 (2006)
223-233
2.5
Fig. 7. Modal amplitude in tlie beam sea.
3.50
i
Z/^a
n
3.00
—
Ris:cd=i.o
R2S2Cd^l.0
O
A
2.50
Fig. 9. Comparison between the experimental results and the calculated results
of vertical displacement i n the beam sea. R1S2 denotes the intersection between
the second lower-hull from the right and the second strut from the bow. R2S2
denotes the intersection between the first lower-hull from the right and the
second strut from the bow. Both positions are described in reference [14].
0
RIS2Time-domoin
•
R2S2 Time-domain
OA
2.00
li
O
1.50
1.00
w
k
1
1
1
M
A
RIS2Cd^
O
R2S2 C d ^
O
\
t\
0.50
0.00
0.2
0,4
0.6
0.8
X/L
1
Fig. 8. Comparison of tlie vertical displacement in tire beam sea obtained by
different treatments of the viscous drag force. R1S2 denotes the intersection
between the second lower-liull from the right and the second strut from the bow.
R2S2 denotes the intersection between the first lower-hull from the right and the
second strut from the bow. Both positions are described i n reference [14].
influenced by the local structural design w h f l e the local design
has not decided yet i n detail at the first stage.
Fig. 8 shows the influence of the viscous force. This example
shows that the influence o f the viscous force plays an important
role to reduce the peak value at the resonant frequency. This
figure also shows that the difference between the Fourier
averaged value and the exact one. The exact value is obtained
by the time domain simulation of the equation o f m o t i o n i n
w h i c h Eq. (20) is direcdy used as a nonUnear external force.
The Fourier averaged value has a slight difference w i t h the
exact time-domain simulation; however the difference may be
negligible f o r the engineering purpose.
F i g . 9 shows a comparison between the numerical results
and the experimental results o f the vertical displacement. I t
seems the agreement between the numerical results and the
experimental results are good except f o r some resonant peaks.
The main reason of disagreement is supposed to be that the
viscous force is not correct, because the drag coefficient is
obtained f r o m a simple forced oscillation test whfle the m o t i o n
of the structure is very complicated and the interaction between
the strut and the lower-hull is expected. Another reason may be
the structural damping, especially the connection between the
foaming urethane and the backbone, makes frictional damping.
These influences are very difflcult to estimate.
1
Xlh
1-2
Fig. 10. Comparison between the experimental resuhs and the calculated
results of stram on the transverse beam in the beam sea. B3-1 denotes the left
side of the third beam. B3-2 denotes the shghtly left fi-om center of the tlmd
beam. Exact position is shown i n reference [14].
Figs. 10 and 11 show comparisons o f the strain between the
numerical results and the experimental results. I t seems that the
agreement between the numerical results and the experimental
results are better compared to that o f the displacement. Since
the displacements are measured near a corner o f the structure
while the strains are measured at the mid-part o f the beam,
the relative displacement at the measurement point f o r the
vertical displacement is larger than that f o r the strain. Thus, i t is
supposed that noiflinear effects on the measured displacement
are larger than that on the strain and i t may spoil the accuracy
of the numerical results. Another possibility is that the modal
analysis f o r the experimental model by the bar element is
not correct because many separated floats are attached on its
backbone and these floats may affect not only on the damping
coefficient but also on the elasticity. M o r e detailed experimental
data are necessary to reveal these influences.
It is mentioned that Takagi et al. [14] has presented
numeiical results by the p F F T - F E coupling method. However,
their results are obtained by an early p F F T - F E code w h i c h does
not satisfy the energy conservation or the H a s k i n d - N e w m a n
relation.
K. Takagi, J. Nogiichi/Applied
D
Ocean Research 28 (2006)
O
\
L2-2EXP
\
\
i
1
L2-I CAL
\
\
•
L2-2 CAL
\
\
'
/
\
'
' '
\
\
•
\
s
\
0
0
0
\
0
O
O,
O
0.2
0.4
0.6
0.8
O
1
X/L
231
previous section, the average error is almost the same level
as the error o f the 12,960 panels' case shown i n the previous
section. Average CPU time by X e o n 2.8 G H z is about 0.5 h f o r
one mode.
O L2- 1 EXP
r
1
1
223-233
1-2
Fig. I I . Comparison between the experimental results and the calculated
results of strain on lower-hull in the head sea. L2-1 denotes the fore of the
left lower-hull. L2-2 denotes the center of the left lower-hull. Exact position is
shown i n reference [14].
Fig. 13(a) and (b) show examples o f the vertical
displacement o f the lower-hull at mid-ship and that of the
transverse beam at the bow respectively. The drag coefficient
f o r the viscous force is Cd = 1.4, which is obtained by the
forced heaving test o f the lower-hull section, and the same
value is used f o r the struts. The calculated points are indicated
with marks; however some points are eliminated because the
iteration scheme f o r the viscous force has not converged at
those frequencies. I t seems that the numerical instability occurs
for the iteration o f the higher order mode. I f we l i m i t the number
of modes to 30, instability doesn't occur and we can get the
result. However, this problem should be solved i n the future to
make the present method more robust.
4. Conclusions
Fig. 12. The finite element model of the floating wind-power plant.
3.4. More challenging
example
A numerical result o f a more challenging example is shown
i n this section. The target is a sailing type offshore wind-power
plant planned by the National Institute for Environment Studies,
Japan [ 2 ] . The finite element model o f this structure is shown
i n Fig. 12. The structure is composed o f two lower-hulls and
78 struts. Lower-hulls are connected w i t h 15 transverse beams.
The structure supports 11 w i n d turbines and 4 sails w h i c h
are modeled as vertical bar elements. The lower-hull has a
rectangular section and the strut has a w i n g section. Details o f
the structure are f o u n d i n reference [2].
The straicture is modeled w i t h 472 bar elements. The wetted
suiface o f the stmcture is divided into 84,592 panels, and the
number o f grids is 262,144. 40 modes are used f o r presenting
the elastic motion. Although the numerical eiror o f the energy
conservation depends on the modal shape as indicated i n the
A p F F T - F E coupling method, w h i c h can calculate the
hydroelastic behavior o f floating flexible structures, has been
developed. The method can handle a very large number o f
constant hydrodynamic panels i n a reasonable C P U time.
I n order to give an indication o f numerical eiTor, we have
presented a consistent way of data passing i n w h i c h the energy
is conserved between the generalized modal damping and
the radiation waves i f the hydrodynamic analysis is accurate
enough. I n addition, the scheme satisfies the generalized
Haskind-Newman relation between the modal d i f f r a c t i o n force
and the Kochin function.
Using these properties, we have demonstrated the accuracy
of the numerical results f o r comparison w i t h experimental data.
The numerical result o f the hydroelastic m o t i o n including the
nonlinear influence by the viscous force has been validated
through the comparison w i t h the experimental data. F r o m these
results, we have concluded that the present method is useful
for the estimate o f hydroelastic behavior of a floating flexible
stmcture especially f o r the first stage o f its design process.
Finally we have
i.e. the sailing type
used 84,592 panels.
applicability o f the
shown a more challenging apphcation,
offshore wind-power plant, i n w h i c h we
The numerical results demonstrated the
present scheme to very large floating
2.00
2.00
Z/Ca
Wave direclion - 90 de^
Wave direction = 0 deg.
0
0.1
(a) Lower-hull at mid-slup.
0.2
X/L
0.3
(b) Transverse beam at the bow.
Fig. 13. The vertical displacement of the sailing type offshore wind-power plant.
K. Takagi, J. Nogiichi /Applied
232
Structures w i t h complicated geometry; however the iteration
scheme f o r the estimation of the viscous force has a slight
Ocean Research 28 (2006)
Companng (A.7) w i t h (19) and (A.5), the f o l l o w i n g equation
is obtained
problem.
Ne
A(«)
(/»)
Acknowledgements
223-233
JSb
j=i'-
^
1
This w o r k is supported by Grant i n aid f o r scientitic research
No. 16360436. The authors are grateful to M r . S. Yamazaki f o r
his cooperation i n the forced oscillation test.
+
2
ƒ
(HI)
(A.8)
Appendix A . Energy conservation
where the last change o f Eq. (A.8) is obtained f r o m the
Consider the mth radiation problem induced by the ; M t h
modal motion. The pressure p'^'"^ of this problem is defined as
definition of the nodal displacement and the nodal force.
Eqs. (A.3) and (A.8) suggest that the nodal radiation force is
symmetrical.
(A.1)
(A.2)
d.S = 0.
Lb (
dv
I f we apply Green's second identity to the radiation potential
and its complex conjugate, we obtain
Green's second identity gives
Thus w e obtain the f o l l o w i n g relation
=f
9^
Sb
(A.3)
•dS.
p^"y.
9^
Jsb^
Jsi,
dv
dv
where the overbar denotes the complex conjugate and 5,
denotes the infinite boundary. Since
The cross product f o r Eq. (18) and e.v yields
(A.9)
dS,
L
("0
dS
dv
dv
dv
is real at Sb and the
radiation potential satisfies Eq. (A.2), the l e f t side o f Eq. ( A . 9 )
-3-
ƒ p[v
can be altered as
x i x - Xk)} X e,. d 5
hi'")
(A.4)
The inner product f o r Eq. (A.4) and (^i
-
(m)
2i
gives
- ^ / ^ ^ P v . [ ? . x ( | , - | . ) j
(A. 10)
•d^
4
//^•^(l;-|.) =
dS
dv
dv
Sh
9^
On the other hand, since the asymptotic f o r m of the radiation
potential is represented by the K o c h i n function, the right side o f
X ( X - Xk) dS - (e, . f,^^'>) \e, . (|, - |^)) .
(A.5)
Since we assume that the fluid force i n axial direction is
acting only on the origin node k, the last term o f Eq. ( A . 5 )
vanishes. O n the other hand, using Eq. (12), we obtain the
f o l l o w i n g equation.
Eq. (A.9) can be altered as
•L
- [
p('"K.v,\ll"^
JSj
X (x - Xk)
Hni{K,e)H,AK,e)
dS
(A. 11)
dö.
Jo
where
^
dS.
dv
dv
2n
d.S =
dv
^9<*('")'
0(")--L
1^ r 27t
hi")
•fsb
, sdèi")
hi"')l-L.
(A.6)
Hn (K, Ö) =
^
^Kz+iK(x
hi").
cosO+y AnB)
dv
dv
Using Eqs. (10) and (17), the right side of Eq. ( A . 6 ) can be
(A. 12)
altered as
('").
JSb
Eqs. ( A . 8 ) - ( A . 1 1 ) suggest the f o l l o w i n g relation
h('0
T
dv
(«)
— ƒ P^"'^v • le,
Lj Jsj
I
lils
^^'"^
X (I, - Ik)} X (X - Xk) dS
^
^I
- ^ k ) ^ ^ d S , eA e, . ( | , +
A
("0
-
po?K
An
X C"" d\{H,n{K,6)Hn{K,6)]d6.
Jo
•
(A.1)
(A.13)
Eq. (A.13) shows the energy conservation between the
modal damping force and the radiated wave energy at infinity.
K. Takagi, J. Nogiichi /Applied
Ocean Research 28 (2006)
223-233
233
Appendix B . Generalized Hasldnd-Newman relation
I f we apply Green's second identity to the radiation potential
and the diffraction potential, we obtain
A
X
V
A X
A
H b
X
A
• • • • A
JSb
OV
Js^
y
dv
dv
J
• Exnct
f
/
JSb
i
A
A
dé'-"^
( / i ^ d 5 .
3
(B.2)
dv
a=0.01ni
A Circular a=0.02m
X Exact
(K, 6) = pg^a
X
A
• Circular .-1=0.01 111
Eq. (B. 1) can be altered by using the Kochin function defined
by Eq. (A. 12) as
Pg^aHn
y
.i=0.02ni
4
5
6
7
8
9(y[rad/5]10
Fig. C.1. Drag coefficient obtained from the forced oscillation test.
Since the diffraction pressure is given as p^^^
=
~pg^a<pD,
the right side o f Eq. (B.2) can be transformed further
the Keulegan-Caipenter number and the Reynolds number, thus
the drag coefficient varies w i t h the amplitude and the frequency
pg^aH,,
{K, 9) = -
f
d5.
(B.3)
JSb
o f the motion. However, i t is not convenient to use a variable
drag coefficient f o r estimate o f the motion. Thus, we
Eq, (A.8) suggests that the right side of Eq. (B.3) coincides
w i t h the modal diffraction force. Thus we obtain
use
Cd = 1.0 i n the numeiical calculation, which coiresponds to
the experimental value of the exact section w i t h the oscillation
amplitude a = 0.01 m at low frequency range.
pgf„//„(/Ce) =
(B.4)
References
Eq. (B.4) is the generalized Haskind-Newman relation.
Appendix C . Experiment of the viscous force
The experiment shown i n Section 3 has been carried out
i n the regular waves o f 0.02 m wave height. The R A O o f the
model shows that the relative displacement o f the lower h u l l is
approximately 0-0.06 m and the frequency range is 4-8 rad/s.
I f w e use the average o f these values, the Keulegaii-Caipenter
number K ^
3.4, the Reynolds number Re = 8500 and the
frequency parameter p =
2500. Saipkaya and Isaacson [15]
give various experimental data o f the drag coefficient Q .
It
seems the drag coefficient o f our case is 1.0-1.4 f r o m their
data. However, our case is slighdy out o f their data range and
the lower h u l l o f the experimental model has a small clearance
between the backbone and the float. Thus we have cairied out
the forced oscillation test w i t h sectional models at the wave tank
of Osaka University.
The size o f the tank is 14 m length, 0.5 m depth
and
0.3 m w i d t h . One o f the models has the same section as the
experimental model shown i n Section 3, i.e. the section has a
clearance between the backbone and the float. The other model
has a circular section o f 0.56 m diameter, w h i c h is the same
as that o f the model w i t h clearances. A l t h o u g h the lower h u l l
moves not only i n the vertical dhection but also i n the horizontal
direction, the forced oscillation has been canied out only i n
the vertical direction. The drag coefficient is obtained by the
Fourier analysis, and the linear damping component due to the
wave making is subtracted.
Fig. C.1 shows the experimental results. I t seems that the
difference between the exact section and the circular section is
neghgible. I t is w e l l k n o w n that the drag force is a function o f
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