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OCR for page 963
24th Symposium on Naval Hydrodynamics
FuLuoka, JAPAN, 8-13 July, 2002
A Panel-E`ree Method for Time-Domain Analysis
W. Qiu, H. Peng (Martec Limited, Canada)
C.C. Hsiung (Dalhousie University, Canada)
Abstract
A panel-free method (PFM) has been developed
to solve the radiation and the diffraction prob-
lems of floating bodies in the time domain. The
boundary integral equations in terms of source
strength distribution were desingularized by remov-
ing the singularity in the Rankine source of the
time-dependent Green function. The geometry of
a body surface was mathematically represented by
NURBS surfaces. The integral equation was then
globally discretized over the body surface by using
Gaussian quadratures. No assumption is needed, as
in the conventional panel methods, for the degree
of approximation of distributed source strength on
the body surface.
The accuracy of PFM was demonstrated by its
application to the radiation and diffraction prob-
lems for a Wigley hull in the time domain. The
computed force response functions, hydrodynamic
coefficients, and the wave exciting forces were com-
pared with published results.
INTRODUCTION
The predictions of wave-induced motions and load-
ings are essential elements of ship design. Over
the past few decades computational hydrodynam-
ics has been developed as a powerful tool for both
ocean engineers and naval architects. It allows eval-
uation of preliminary designs or ship performance
at a relatively low cost compared with experimental
tests. Strip theory was applied as the first analyt-
ical ship motion theory for computations and has
been used as a practical prediction method, but it
gives unsatisfactory predictions at low frequencies
and at high forward speeds, and it is not applica-
ble to ships of low length-beam ratios due to its
1
slender body assumptions. Also, the strip-theory
approach is not able to compute the hydrodynamic
pressure distribution over the hull surface except
on sections. Some of the deficiencies of strip the-
ory can be removed by the three-dimensional flow
theory. Hess and Smith (1964) pioneered the panel
method, in which a source distribution on a body
surface was employed for flow computation. The
body surface was subdivided into a number of flat
quadrilaterals over which the source strength was
assumed to be constant. Since the singularity so-
lution of integration can be obtained by using pla-
nar quadrilateral panels or triangles and constant
source strength over a panel, the constant-source-
flat-panel method has been used in a wide variety
of problems both in the frequency domain and the
time domain. It is often referred as a lower-order
panel method. Normally, a large number of panels
are required to achieve accurate results.
H~gher-order panel methods have been devel-
oped in various degrees to overcome the deficien-
cies of the constant-source-flat-panel method. Most
higher-order methods allow for linear or quadratic
panels and first- or second-degree polynomial dis-
tribution of source strength over a panel. It nor-
mally requires more computational effort than the
lower-order panel method. Maniar (1995) ap-
plied B-splines and developed a higher-order panel
method in which the potential and the geometry
of a body are allowed any degree of continuity.
Recently, Lee et al. (1998) and Danmeier (1999)
have presented a geometry-independent higher-
order method which separates the geometric and
hydrodynamic representations. The body was
represented by subdividing the body surface into
patches. B-splines were employed to represent the
geometry and the velocity potential on each patch.
Lee and Newman (2001) used B-splines only to rep-
resent the velocity potential. The accuracy of the
solution can be refined by controlling the degree
of B-Splines and/or subdividing patches. In the
OCR for page 964
higher-order panel methods, the singular 1/r term
can be evaluated numerically in a variety of ways.
For example, in the work of Danmeier (1999), an
adaptive subdivision and triangulation scheme was
used to evaluate the singularity of the Rankine
term.
Landweber and Macagno (1969) proposed a
desingularized procedure which removed the singu-
larity of 1/r before discretizing the integral equa-
tion and applied it to the problem of uniform flow
past an ellipsoid. The numerical solution then
could be applied to the exact boundary geometry,
and the integral equation could be discretized over
the body surface by Gaussian quadratures. Theo-
retically, this eliminates the errors due to both the
geometrical approximation and the assumed degree
of approximation of source strength distribution as
in the panel method. Kouh and Ho (1996) further
developed this method and applied it to solve prob-
lems of uniform flow past a sphere, an ellipsoid and
a Wigley hull in which geometries were represented
by analytical expressions.
Recently, Qiu and Hsiung (2001a,b,c) have de-
veloped a so-called panel-free method for the time-
domain analysis. In their work, the desingular-
ized integral equation in terms of source strength
distribution has been developed by removing the
singularity due to the Rankine term in the time-
dependent Green function. The Non-Uniform Ra-
tional B-Splines (NURBS) were adopted to de-
scribed the body geometry so that the desingular-
ization method can be applied to arbitrary bod-
ies. The integral equations were then globally dis-
cretized over the body surface by Gaussian quadra-
tures. They applied the panel-free method to the
radiation and the diffraction problems of a hemi-
sphere. Computed response functions, added mass
and damping coefficients showed excellent agree-
ment with published results by the analytical solu-
tion and the panel method.
The work presented in this paper is the contin-
uing development of the panel-free method in the
time domain. It has been extended to the radia-
tion and the diffraction problems for a Wigley hull
at zero speed.
MATHEMATICAL FORMU-
LATION
It is assumed that the fluid is incompressible, invis-
cid and free of surface tension and that the flow is
irrotational. Consider the radiation and the diffrac-
tion problems of a three-dimensional body at zero
speed in a semi-infinite fluid with a free surface.
The potential functions, ~k(P(x, y, z); t), for
k =1,2, ...,6 and 7, satisfies the following governing
equation, boundary conditions, far-field conditions
and initial conditions:
V2¢k = 0
d Vb + 9 0¢k = 0
Wok - V
dn — ok
V°k ~ O
V¢k ~ o
0k = 0, ~k = 0
where
Vnk = nk (k + mk (k,
vn~c —
arlk
in Q (1)
on z = 0
on Sb
as R1 ~ x,
on z = 0
as z ~ -x
as t = to
for k = 1,2, ,6 (2)
for k = 7 (3)
where 9 is the gravitional acceleration, (I denotes
the incident wave potential, Q is the computa-
tional domain, Sb is the mean wetted body surface,
R1 = ~, t is time, to = 0 for the radiation
problem and to =—x for the diffraction problem,
(k iS the amplitude of unsteady motion in six de-
grees of freedom, mk is m-terms and Ilk iS the unit
inner normal vector pointing into the body surface.
With the time-dependent Green function, the
potential function can be obtained from the integral
equation in terms of source distribution as follows:
°k (P; t) = ~ dr ~ G(P, Q; t—T)ak (Q; 7)dS
t0 Sb
(4)
where Q = Q(x', y', z') and (Jk iS the source
strength which can be solved from
2
0¢k ~ p; t) = - - ~k (P; t)
OCR for page 965
+I dial ~G(P,Q;t—I) (Q;~)dS
to sb Nap
where the time-dependent Green function
G(P, Q; t - r) for the infinite water depth
(Wehausen and Laiton, 1960) can be written as
G(P, Q; tar) = Go(P, Q)~(t-~)+H(t-~)F(P, Q; to-)
(6)
with
Go(P,Q) = -4 (- -—) (7)
5(t—or) is the Dirac delta function and H(t - r) is
the Heaviside unit step function, and
F(P,Q;t—7) =
~ coo
.
2 ~ J N/~ sin[~/~(t—7)]ek(Z+Z ) JO (kR)dk
with
r = `/(x—X')2 + (y _ y/)2 + (z _ Z')2 (9)
rat = >/(x—X,)2 + (y _ y')2 + (Z + Z')2 (10)
R = v/(x - X')2 + (y _ y')2 (11)
and JO is the Bessel function of the zeroth order.
Based on Gauss's flux theorem, Eq. (5) can be
desingularized as
And =—(I; t)
+ ~ Ark (Q; t) ~ —irk (P; t) ~ ~ lids
+2 ~ ark (Q; t) In dS
+J~ dri ( ~Q; )ak(Q;~)dS (12)
where Go (P. Q) and G2 (P. Q) are defined as
GAP, Q) = - 4 (- +—) (13)
G2(P, Q) = 4 — (14)
After the source strength is solved from Eq.
(12), the velocity potential then can be obtained
from the following non-singular integral equation:
(5) 0k(P;t) =
~ Gt (P. Q) Ark (Q; t) - )(Q) -~t py ] dS
+2 Js~ ark (Q; t)G2 (P. Q)dS + To ~ ~ ; t)
+ J dri ak(Q; r)F(P, Q; t—r)dS (15)
to Sb
dSb L
where y(P) is the source distribution on Sb which
makes the body surface an equipotential surface of
potential on and satisfies the homogeneous integral
equation
)(p) = -I )(Q) on ids (16)
(8) Equation (16) can be desingularized in a similar
way to Eq. (12), and y(P) can be solved by find-
ing the eigenfunction of ~K(P, A)/ p associated
with the eigenvalue equal to -1, where K(P, Q) =
2Gi(P,Q). The potential, ho' is constant in the
interior of the equipotential surface, and can be
computed at the origin by
No=- Jo )(Q)(~Q~ + ~Q,~)dS (17)
where Aid and If denote distances between Q and
the origin, and Q', the image of Q. and the origin,
respectively.
Since the initial boundary value problems for
radiation and diffraction has been linearized, the
response function can be used to describe the re-
sponse of the linear system. According to the work
of Liapis and Beck (1985), the radiation force can
be expressed as:
. . .
Fik (t) =—~ jk Ok (t)—~ jk (k (t)—Ok (k (t)
rt
J K'Rk(t—T)(k(~)dT, j, k = 1, 2, ,6
o
(18)
where Ok is the added-mass; Ok is the hydrody-
namic damping coefficient; Ok is the coefficient of
the hydrodynamic restoring force; and K~Rk(t) is
the force response function which can be solved
by a direct solution scheme (Cong, et al., 1998)
from Eq. (18) using a non-impulsive input velocity
(k(t) = ~/57iexp(-oet2~. Here or is an arbitrary
number.
3
OCR for page 966
The impulse response function KjD(t) for the
diffraction problem can be solved from the follow-
sing equation:
Ioo
KjD (t—r)r10 (~)dr =—gj7 (t)—hj7 (t) (19)
—00
where
gj7 (t) = p ~ o7 (t)r? jdS (20)
sb
hj7 (t) = - p l o7 (t)mjdS (21)
sb
The non-impulsive incident wave ~10 and the corre-
sponding derivatives of the incident wave potential
were given by King et al (1988). In this work,
mj's are all zero for zero speed case.
NUMERICAL IMPLEMEN-
TATION
While many mathematical representations have
been adopted to describe the body surface, non-
uniform rational B-Splines (NURBS) have be-
come the preferred method (Farin, 1991). The
widespread acceptance and popularity of NURBS
are because they provide a general and flexible de-
scription for a large class of free-form geometric
shape. Their intrinsic characteristics of local con-
trol, low memory requirement, coupled with a sta-
ble and efficient generating algorithm, make them
a powerful geometric tool for surface description,
especially for complicated body geometry. In the
panel-free method, NURBS were adopted to de-
scribe the body surface mathematically.
It is assumed that Np patches are used
to describe a body surface. Each patch can
be represented by a NURBS surface. Let
P(x~u,v),y~u,v),ztu,v)) be a point on a patch;
x, y and z denote the Cartesian coordinates; and
u and v are two parameters for the surface defini-
tion. On a NURBS surface, P(u, v) can be defined
as follows:
=0 j=o ij C.,j. i q (v)
P(u,v) = ~i~-o~j ow',jN` ~(u)Njqtv) (22)
Since Eq. (12) is singularity free, it can be dis-
cretized by directly applying the Gaussian quadra-
ture and the trapezoidal time integration scheme.
The Gaussian quadrature points are arranged in
the computational space, rs, then their correspond-
ing coordinates, normals and Jacobian in the phys-
ical space can be obtained based on Eq. (22~.
Therefore, Eq.~12) can be written as
0~a~p;; t) = -ok (Pi; t)
Up Nj Mj
+~ ,Wj fakfQ~S;t)VpG~(Pi, QUASI npi
j=1 r=1 s=1
—ark (Pi; t) VQG 1 (Pi, Q.:' S) · ngr~ ]
Np Nj Mj
~ ~ ~ Wj ak f Q; ; t) V pG2 (Pi Qrs) n
j=1 r=1 s=1
Np Nj Mj rs rs F(P, Qj; t)
+!5t[2 ~ OWE (JktQj ;to) one
~=1 r=1 s=1
kin—1 Np Nj Mj F p rs t—t
+ £ ~ `£ ~ Wjrs ~ i' Qj; k)
k=1 j=1 r=1 s=1
for i = 1,2,...,Np
where Wjrs = Wrw5J's' Nj and Mj are the num-
ber of Gaussian quadrature points in the u- and v-
directions on the jth patch. Pi = Pifu~,vm),n =
1,...Ni,m = 1,...Mi and Ads = Qj~ur~vs) are the
position vectors of Gaussian quadrature points on
the ith and jth patches in the physical space, re-
spectively; npi and nets are the corresponding unit
normals; wr and ws are the weighting coefficients
in the a- and v-directions; Jjrs is the Jacobian of
QjrS; t is the time; to is the lower limit of time; and
At is the time step, to = to+ki\t and t = to+k~\t,
where k and kit are the time constants at any in-
stant and for the total time, respectively. It can
be seen that the algorithm can be easily controlled
by changing the number and the arrangement of
Gaussian quadrature points.
NUMERICAL RESULTS
The panel-free method was applied to a Wigley hull
where wij are the weights; Ci,j form a network of at zero speed. The hull geometry is defined by:
control points; and Ni,ptu) and Nj~qtv) are the nor-
malized B-spline basis functions of degrees p and q 7' ~1 _ `2 y ~1 _ <2' t1 + 0 2~2) + `2 (1 - (~) (1 _ tE2~4
in the u- and v-directions, respectively. (24)
cJk chars; tk)]
(23)
4
OCR for page 967
where the nondimensional variables are given by
= 2x/L, ~1 = 2y/B, and ~ = z/T, where L is the
ship's length, B is the beam, and T is the draft.
The hull has L/B = 10, L/T = 16 and a block
coefficient of 0.5606. NURBS and analytical rep-
resentations were both used for the computation.
In the NURBS expression, the Wigley hull was de-
scribed by a 13 x 13 control net (Np=1) with degrees
3 in both u- and v-directions. Figure 1 shows the
distribution of 6 x21 Gaussian quadrature points
and the control net, where the Gaussian points are
denoted by "+" and the solid lines represents the
control net.
Yet
Figure 1: Control net and Gaussian point distribu-
tion (6x21) for a Wigley hull
The computations were carried out for the ra-
diation problem. Figure 2 shows the computed
heave and pitch response functions at a time step
dt = 0.25s using 6x21 and 6x25 Gaussian points
on the hull represented by NURBS surfaces. In
order to investigate the convergence of the compu-
tation, lOx30 Gaussian points were also applied
on the analytically represented surface. It was
found that the computation was not sensitive to
time step and the number of Gaussian points. In
these figures, the heave and pitch response func-
tions, K33 and K55, are nondimensionalized as
K33 / ~ pgV /L) /7[ and K55 / ~ pgV ~ x:, respec-
tively. The time t is nondimensionalized as t~/37~.
The heave and pitch added mass for the Wigley
hull at zero speed was computed from the response
functions. The heave and pitch added mass and
the frequency are nondimensionalized as A33/(pV),
A33/(pVL2) and w = w~/57~, respectively. The
computed heave added mass by PFM was com-
pared with those of TiMIT and WAMIT. TiMIT
and WAMIT are two panel-method codes from
MIT for time-domain and frequency-domain wave
analysis, respectively. Note that results of TiMIT
and WAMIT used here were taken from the work
of gingham (1994~. Figure 3 shows the compari-
son. The irregular frequencies are shown at ~ ~ 5.8
and w ~10 for both TiMIT and WAMIT in which
the half-hull was discretized by 144 panels. PFM
shows an oscillation around w = 5.8, but its behav-
ior is different from those of TiMIT and WAMIT.
The curve around the irregular frequencies tends
smoother as more Gaussian points are distributed.
The computed heave and pitch damping co-
efficients were compared with those by Bing-
ham (1994) in Figures 4. The heave and pitch
damping coefficients are nondimensionalized as
B33 / (pVcv) 47; and B55 / (pVLai) x:, respec-
tively.
The wave exciting forces were also determined
for the Wigley hull. The heave and pitch response
functions for the Froude-Krylov forces were com-
puted according to the work by King et al. (1988~.
Then the exciting forces were compared with re-
sults from King (1987), where 120 panels were used
to represent the half-hull. In the computation of
diffraction problem, the time step was chosen as
0.25s. The computed heave and pitch response
functions due to the diffracted waves in compari-
son with the results of King (1987) are presented
in Figure 5, where the heave and pitch response
functions, K37 and K57, are nondimensionalized as
K37 / (P9V /L) ~ and K57 / ~ pgV ~ /7i, respec-
tively. The oscillation of the curve shown in results
by the panel method is not presented in the results
by PFM.
Applying Fourier transform to the diffraction
and Froude-Krylov response functions, we were
able to obtain the frequency-domain wave exciting
forces. The forces and phases were compared with
those results from King (1987) and the strip the-
ory results taken from his work in Figures 6 and 7.
To be consistent with the presentation of King, the
frequency is nondimensionalized as ~ = kL, and
the nondimensional heave and pitch exciting forces
are given as F3 7 / ~ pgV / L) and F5 7 / ~ pgV ), respec-
tively. The results by PFM show a good agreement
with those by the panel method and by the strip
theory. With the number of Gaussian points in-
creased for NURBS surface, the computed results
5
OCR for page 968
by PFM approached to solutions that were com-
puted by the analytical hull expression and large
number of Gaussian points.
CONCLUSIONS
A panel-free method (PFM) has been developed to
solve the radiation and the diffraction problems in
the time domain. In PFM, the integral equation
in terms of source strength is desingularized be-
fore it is discretized. The singularity-free integral
equation allows for application of Gaussian quadra-
ture globally over the exact body geometry. The
body geometry can be either described in an ana-
lytical definition or by a parametric representation.
The complex body geometry can be accurately de-
scribed by NURBS surfaces which have been widely
used in the field of computer aided design.
In general, compared with the panel method,
PFM involves less numerical manipulation, since
panelization of a body surface is not needed. Pro-
gramming of the PFM is easier than the panel
method. It is more accurate, since the assumption
for the degree of approximation of source strength
distribution as in the panel method is no longer
needed, and Gaussian quadrature can be directly
and globally applied to the body surface with a
mathematical description. The accuracy of the so-
lution can be easily enhanced and controlled by
changing the number and distribution of Gaussian
quadrature points.
The robustness and accuracy of PFM has been
demonstrated by its applications to the radiation
and the diffraction problems of a Wigley hull at
zero speed in the time domain. The computed radi-
ation and diffraction response functions, hydrody-
namic coefficients and wave exciting forces for the
Wigley hull agree well with published results. The
oscillatory error of the memory function at large
time tends to be reduced by PFM in comparison
with the panel method.
The evaluation of the waterline integral by
PFM will be developed in the near future for
the steady forward speed case. The concept of
the panel-free method can also be applied to the
frequency-domain computation.
ACKNOWLEDGMENT
This work was supported by the Natural Sciences
and Engineering Research Council of Canada.
REFERENCES
gingham, H.B. (1994). Simulation Ship Motions
in the Time Domain. Ph.D. Thesis, Massachusetts
Institute of Technology, Massachusetts.
Cong, L.Z., Huang, Z.J., Ando, S. and Hsiung, C.C.
(1998~. Time-Domain Analysis of Ship Motions
and Hydrodynamic Pressures on a Ship Hull in
Waves. Proceedings 2nd International Conference
on Hydroelasticity in Marine Technology,
Fuknoka, Japan, pp. 485-495.
Danmeier, D.G. (1999). A Higher-Order Panel
Method for Large-Amplitude Simulation of Bodies
in Waves. Ph.D. Thesis, Massachusetts Institute
of Technology, Massachusetts.
Farin, G.E. (1991). CURBS for Curve and Surface
Design. SIAM Activity Group on Geometric De-
.
sign.
Hess, J.L. and Smith,
Calculation of Nonlifting
about Arbitrary Three-Dimensional
Journal of Ship Research, Vol. 8, No.
22-44.
A.M.O. (1964).
Potential Flow
Bodies.
3, pp.
King, B.K. (1987). Time-Domain Analysis of
Wave Exciting Forces on Ship and Bodies. Ph.D.
Thesis, The University of Michigan, Michigan.
King, B.K., Beck, R.F. and Magee, A.R. (1988).
Seakeeping Calculations With Forward Speed
Using Time Domain Analysis. Proceedings 17th
_ymposium on Naval Hydrodynamics, The Hague,
Netherlands, pp. 577-596.
Kouh, J.S. and HO, C.H. (1996). A High Order
Panel Method Based on Source Distribution and
Gaussian Quadrature. Schiffstechnik, Bd. 43, pp.
38-47.
Landweber, L. and Macagno, M. (1969). Irrota-
tional Flow about Ship Forms. IIHR Report, No.
123, The University of Iowa, Iowa City, Iowa.
6
OCR for page 969
Lee, C.H. and Newman, J.N. (2001). Solu-
tion of Radiation Problems with Exact Geometry.
Proceedings 16th International Workshop on Water
Waves and Floating Bodies, Hiroshima, Japan,
pp. 93-96.
Lee, C.H., Farina, L. and Newman, J.N. (1998~.
A Geometry-Independent Higher-Order Panel
Method and Its Application to Wave-Body Inter-
actions. Proceedings of Engineering Mathematics
and Applications Conference, Adelaide.
Liapis, S.J. and Beck, R.F. (1985~. Seakeeping
Computations Using Time Domain Analysis.
Proceedings 4th International Conference on
Numerical Ship Hydrodynamics Washington
, ,
D.C., pp. 34-54
Maniar, H.D. (1995~. A Three Dimensional Higher
Order Panel Method Based on B-Splines. Ph.D.
Thesis, Massachusetts Institute of Technology,
Massachusetts.
Qiu, W. and Hsiung, C.C. (2001a). A Panel-Free
Method for Time-Domain Analysis of Radiation
Problem. Proceedings 16th International
Workshop on Water Waves and Floating Bodies,
Hiroshima, Japan, pp. 129-132.
Qiu, W. and Hsiung, C.C. (2001b). A Panel-Free
Method for Time-Domain Analysis of Radiation
Problem. Ocean Engineering, accepted for publi-
cation.
Qiu, W. and Hsiung, C.C. (2001c). Time-
Domain Analysis of Diffraction Problem by a
Panel-Free Method. Proceedings 6th Canadian
Hydromechanics and Marine Structure Conference.
Vancouver, Canada, pp. 15-21.
Wehausen, J.V. and Laitone, E.V. (1960~. Sur-
face Waves. Handbuch der Physik (ea. S. Flugge),
Springer-Verlag, Vol. 9.
7
OCR for page 970
1
0.8
a:
~ 0.6
o
._
u:
._
o
z
0.4
0.2
o
-0.2
s
4
vet
us
~ 2
o
._
I;;
._ 1
a:
a
z
o
1.2
I \ PFM (NURBS' 6x25)
. \ I PFM (analy.,lOx30) ~ |
. ~ W_ ~
0 1 2
3 4 5 6
Nondimensional time
l
3~1~
I ~ PFM (NURBS' 6X25)
\ I PFM (analy.,lOx30)
1 \ ~ ==~
-2
-
0 1 2 3 4 5 6
Nondimensional time
Figure 2: Heave and pitch radiation force response functions for a Wigley hull
8
OCR for page 971
1.5
ce
o
._
cat
._
o
He
us
id
o
._
1
0.5
o
0.06
0.05
0.04
0.03
0.02
0.01
o
2
\ C ~
>~ Z4.5 5 5.5 6 6.5
.,~
PFM (NURBS, 6x21)
PFM (analy., lOx30) -
TiMIT
WAMIT 0
, , , 1
0 2 4 6 8
Nondimensional frequency
10 12 14 16
l l l l l l
| PFM (NURBS, 6x21)
I PFM (analy., lOx30) ~ |
~ \
1 ,
0 2 4 6 8 10 12 14 16
Nondimensional frequency
Figure 3: Heave and pitch added mass for a Wigley hull
9
OCR for page 972
0.08
us
m~ 0.06
-
o
A:
0.04
z
o
1
0.8
0.6
o
._
a:
cot
._
O 0 4
0.2
PFM (NURBS, 6x21)
;~ PFM (analy., lOx30)
Y \, TiMIT
WAMIT 0
1 ~
Z ~ ~ 0
51 ~ Ha 5 5.5 6 6.5 7 7.5
t ~~ .,, . .... l I
o
0.1
2 4 6 8 10 12 14 16
Nondimensional frequency
PFM (NURBS, 6x21)
PFM (analy., lOx30) ~
TiMIT .......
1 ~
0.02 / ~
J ~~ :. .
O :z 4 ~
A r TO 12
Nondimensional frequency
Figure 4: Heave and pitch damping coefficients for a Wigley hull
10
14 16
OCR for page 973
The impulse response function KjD(t) for the
diffraction problem can be solved from the follow-
sing equation:
Ioo
KjD (t—r)r10 (~)dr =—gj7 (t)—hj7 (t) (19)
—00
where
gj7 (t) = p ~ o7 (t)r? jdS (20)
sb
hj7 (t) = - p l o7 (t)mjdS (21)
sb
The non-impulsive incident wave ~10 and the corre-
sponding derivatives of the incident wave potential
were given by King et al (1988). In this work,
mj's are all zero for zero speed case.
NUMERICAL IMPLEMEN-
TATION
While many mathematical representations have
been adopted to describe the body surface, non-
uniform rational B-Splines (NURBS) have be-
come the preferred method (Farin, 1991). The
widespread acceptance and popularity of NURBS
are because they provide a general and flexible de-
scription for a large class of free-form geometric
shape. Their intrinsic characteristics of local con-
trol, low memory requirement, coupled with a sta-
ble and efficient generating algorithm, make them
a powerful geometric tool for surface description,
especially for complicated body geometry. In the
panel-free method, NURBS were adopted to de-
scribe the body surface mathematically.
It is assumed that Np patches are used
to describe a body surface. Each patch can
be represented by a NURBS surface. Let
P(x~u,v),y~u,v),ztu,v)) be a point on a patch;
x, y and z denote the Cartesian coordinates; and
u and v are two parameters for the surface defini-
tion. On a NURBS surface, P(u, v) can be defined
as follows:
=0 j=o ij C.,j. i q (v)
P(u,v) = ~i~-o~j ow',jN` ~(u)Njqtv) (22)
Since Eq. (12) is singularity free, it can be dis-
cretized by directly applying the Gaussian quadra-
ture and the trapezoidal time integration scheme.
The Gaussian quadrature points are arranged in
the computational space, rs, then their correspond-
ing coordinates, normals and Jacobian in the phys-
ical space can be obtained based on Eq. (22~.
Therefore, Eq.~12) can be written as
0~a~p;; t) = -ok (Pi; t)
Up Nj Mj
+~ ,Wj fakfQ~S;t)VpG~(Pi, QUASI npi
j=1 r=1 s=1
—ark (Pi; t) VQG 1 (Pi, Q.:' S) · ngr~ ]
Np Nj Mj
~ ~ ~ Wj ak f Q; ; t) V pG2 (Pi Qrs) n
j=1 r=1 s=1
Np Nj Mj rs rs F(P, Qj; t)
+!5t[2 ~ OWE (JktQj ;to) one
~=1 r=1 s=1
kin—1 Np Nj Mj F p rs t—t
+ £ ~ `£ ~ Wjrs ~ i' Qj; k)
k=1 j=1 r=1 s=1
for i = 1,2,...,Np
where Wjrs = Wrw5J's' Nj and Mj are the num-
ber of Gaussian quadrature points in the u- and v-
directions on the jth patch. Pi = Pifu~,vm),n =
1,...Ni,m = 1,...Mi and Ads = Qj~ur~vs) are the
position vectors of Gaussian quadrature points on
the ith and jth patches in the physical space, re-
spectively; npi and nets are the corresponding unit
normals; wr and ws are the weighting coefficients
in the a- and v-directions; Jjrs is the Jacobian of
QjrS; t is the time; to is the lower limit of time; and
At is the time step, to = to+ki\t and t = to+k~\t,
where k and kit are the time constants at any in-
stant and for the total time, respectively. It can
be seen that the algorithm can be easily controlled
by changing the number and the arrangement of
Gaussian quadrature points.
NUMERICAL RESULTS
The panel-free method was applied to a Wigley hull
where wij are the weights; Ci,j form a network of at zero speed. The hull geometry is defined by:
control points; and Ni,ptu) and Nj~qtv) are the nor-
malized B-spline basis functions of degrees p and q 7' ~1 _ `2 y ~1 _ <2' t1 + 0 2~2) + `2 (1 - (~) (1 _ tE2~4
in the u- and v-directions, respectively. (24)
cJk chars; tk)]
(23)
4
OCR for page 974
18
16
14
LO 12
cot
o
c: 10
c'
._
O 8
z
6
4
2
o
n 2 4
150~
100 t
50L
L)
is 0
I:
AL
-50~
lain
-150
l
_'\L
_
'I?
\^ 1 - - 1
PFM (NURBS, 6x21)
PFM (analy., lOx30)
King (1987)
Strip theory
A)
.,
PFM (NURBS, 6x21)
PFM (analy., lOx30)
King (1987)
Strip theory
i''/
,,
-200 ~ ~ ~ ~ _
0 2 4 6
kL
10
14
10 12 14
Figure 6: Heave wave exciting forces and phases for a Wigley hull
12
OCR for page 975
2.5
2
l l l
/~' \ PFM (NURBS, 6x21)
_ ,y \ PFM (analy., lOx30)
at ~ King (1987) ~
ILL 15 | ~ . ~ I Strip theory ~ l
E ¢: ~
to 1 _ ~ \
0.5 _/ \"
11
, . . . . .
o
150
100
50
cam
ct
us
l
4 6 8 10 12 14
kL
l
o
-50
-100
l l l
PFM (NURBS, 6x21)
PFM (analy, lOx30)
King (1987)
Strip theory ~
_ ; I
~ ~ . -I- ~ ~ ~
l l l
/:
f
, . . .
2 4 6 8
kL
10 12 14
Figure 7: Pitch wave exciting forces and phases for a Wigley hull
13
OCR for page 976
DISCUSSION
J. Tao
VBD-European Development of Centre for
Inland and Coastal Navigation, Germany
I would like to express my appreciation for the
author who presented a new method which
seems to have a large development potential for
many time domain problems. From your
preliminary results, I have the impression that
this panel-free method works well for radiation
problems. We can see the good agreement force
radiation force response functions as well as for
added mass and damping. However, there is a
large difference in predicting diffraction-force
response functions, particularly in Figure 5 for
the pitch mode. Could you explain this result? I
would expect that a validation for your results
would clarify the advantage of the new method.
AUTHORS' REPLY
There are some differences between the
computed pitch force response functions by the
panel-free method and by the conventional panel
method. We, unfortunately, were not able to
find the experimental results of wave exciting
forces of the Wigley hull at zero speed for
comparison.
DISCUSSION
L.J. Doctors
The University of New South Wales, Australia
In my own panel-code for ship motions, I use a
"lid" on the internal free surface. This is an
extremely simple and effective way of totally
eliminating the problem of the irregular
frequencies.
Can you implement a similar idea in your
method in order to deal with the matter of the
irregular frequencies?
AUTHORS' REPLY
The irregular frequencies shown in the computed
obtained by Fourier transform of the response
function computed in the time domain. They
were due to the oscillation of the response added
mass and damping coefficients in the
frequency domain were function in the time
domain, particularly at large time. In the panel-
free method, this could be improved by
increasing the number of Gaussian points and/or
changing the distribution of Gaussian points.
The "lid" may be useful for the frequency-
domain computation, but we wonder if it could
improve the time-domain computation.
Representative terms from entire chapter:
wigley hull
