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Proceedings

of

he FlIs t PaCljiIAs/G OjJshOle MeLilal l s Symposium

Seoul, Korea, 24 28 June, 1990

CopYllght 1990 by The IntelllatlOnal ouety ofOjJshOle and Palm Engllleeis

THE

ANALYSIS OF WAVE INDUCED DYNAMICS OF OCEAN PLATFORMS

BY HYBRID INTEGRAL EQUATION

METHO

T MatsuI

:- agol'a l'niwjsit.l

:- "gOYd. J P N

K

Kato

Tm ol

a C'oll('gt'

of T( clllloiogy

TOl Dta.

JAP.4S

BSTR CT

A relIable and economical hybrid integral-equation method

is

pro

posed for predicting wave-mduced dynamic responses of ocean

platforms of complex geometry ThIs is based on combming a

dIrect boundary element solution of the fluid region close to the

body wIth an eIgenfunction representation of the far field be

haviour To achieve accuracy and economy, the boundary sur

faces are discretised mto quadratic isoparametnc elements The

validity and accuracy of the proposed method are demonstrated

through several numerical examples, including the results for the

ITT senn-submersible. t is shown that the use of quadratIc

Isoparametric elements leads to significant improvement of accu

racy and efficiency of the hybrid integral-equation method, com

pared WIth classical boundary integral approaches based on con

stant element idealisations

KEY WORDS

Hybnd integral-equatIon method, Wave dIffrac

tIon,

Wave

radiation, Hydrodynamic load, Wave-mduced mohon,

Ocean platform

INTRODUCTION

The design of huge ocean platforms depends critically upon the

analysis of wave induced loads and motions. Thus, over the last

fifteen years, substantial efforts have been devoted to the devel

opment of reliable and economIcal methods for predicting these

effects, Computer programs based on three-dimensional diffrac

tion theory are now available for evaluating the hydrodynamic

coeffiCIents, loads and motIons for structures of practIcal form

For large bodIes with relatively simple geometry such

as

a ver

tical cylinder or sphere, these programs generally provide accu

rate predictions which compare well with analytical solutions and

model test results. Recent surveys for the

ITT

(Takagi et al.,

271

1985) and the ISSC (Eatock Taylor and Jefferys, 1986), however,

suggested that this was not the case for more complicated struc

tures such as a semi-submersible and tensIOn-leg platform (TLP)

The

ITT

Ocean Engmeering Committee has performed a

com-

parative

study

of motions of a senn-submersIble, whIch has been

published by Takagi

et al

(1985) PredIchons from 34 computer

programs were compared among themselves and WIth model test

results from three orgamsations \Vhile most computer programs

provided reasonable predictions of measured surge and

sway,

the

computed results for the other motIOns showed substantIal scatter

about the experiment.al data.

The

three-dImensional dIffractIOn

programs seemed

to

overestimate the heave added mass, lead

ing to longer natural periods

than

those predicted by Monson's

formula or experiment The survey for the ISSC, repubhshed

by Eatock Taylor and Jefferys (1986), also demonstrated sigmfi

cant and disturbing variability between the hydrodynamIC loads

and motions predicted for an example

TLP

by

17

dIffraction pro

grams

Most of the partIcipating organisatIOns employed a source-smk

dIstnbution or direct, boundary Integral formulation, with an as-

sumed constant distnbution of the governmg vanable (source

strength or potential) over each st;rface panel

t

would therefore

not be surprising if a large number of panels were required to

represent the complex geometry of the semi-submersible or TLP,

and the compleXIty of associated wave flows accurately. Indeed,

Korsmeyer et al (1988) adopted up to 12,608 constant source

panels to analyse another TLP with six columns Jefferys (1987)

studIed the same TLP configuration

WIth

different mesh arrange

ments, and cautioned that the constant source panel method did

not necessarily lead to correct answer, as the number of panels

was increased WIthout limit.

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The present work

IS

motivated by

the

need to investigate more

relIable and economical numerical methods for predlctmg wave

mduced dynamic responses of ocean platforms of complex geom

etry One such method proposed herein IS

the

hybrId mtegral

equation method involving the use of quadratic Isoparametnc

surface elements. To

deSCrIbe the

complexity of body geome

try and associated wave flows accurately, It appears adequate to

adopt higher-order elements (e g lInear or quadratic) To the

best of our knowledge, higher-order panels have never been used

m conjunction with the classical boundary integral formulatIon,

due to the difficulty in integrating Green s functIon and its graru

ents multiplied by the higher-order shape functions over each ele

ment The rufficulty may be overcome by introducing a free-space

Green s functIOn (Le. a Rankine source) instead of the complete

Green s function This hybrid integral-equation method is based

on combming a

duect

boundary element solutIOn of

the

fluid re

gIOn

close to the body wIth an eigenfunction representation of the

far field behaviour. The formulation based on a constant panel

IdealIsation has been gIven by Matsui, Kato and Shuai (1987),

and It

IS

extended herein to adopt quadratic isoparametric ele

ments. (In our expenence, the use of linear elements does not nec

essanly lead

to

the improvement of accuracy) Because of the

SIm

plICIty of the new Green s functIOn, no addItional rufficulty arises

m employmg hIgher-order elements. Following a brief review of

the mathematical formlliation, the validity ,and accuracy of the

proposed method are ,demons irated through several numerical

examples, mcluding

tbe

results for the ITTC semI-submersible

C0mpuison is made between our computed r,esults and existing

numerical solutions based on a direct boundary i ll.tegral (Garri

son, 1978) and hybrid finite element method (Yue, Chen and Mci.,

1978)

REVIEW

OF

HYBRID

INTEGRAL-

EQUATION METHOD

Governing Equations

ConSider the problem of diffraction and radiation of water waves

by a three-dimensional floating body. The body is oscillating si

nusoidally about a mean position in response to excitation by a

regular incident wave of frequency wand amplitude A. The Carte

sian coordinate system

oxyz

is defined as shown m Figure

1

with

the oxy-plane on the mean free surface and the oz-axis positive

, vertically upwards

With the assumption of ideal and irrotational flow, the wave field

at

time t may be defined by a velocity potential

(1)

The body motion may be represented as a linear superposition of

rigid-body modes, with complex amplitudes Xq q = 1 to 6). The

indices

q

= 1

to

6 correspond respectively to surge, sway, heave,

roll, pitch and yaw.

z

., -

/

I

r

( I

SF

X

I,

V

1 -.-._ ... I

1 - I - - - - IS . / I

IS

( )

--

- -V - - ii -S- - _

1

COSK:mh

-

9)

In the above expressions, H ~ I ) is the Hankel function of the first

kind of order

n, Kn

is

the

modified Bessel function of the second

kind of order

n,

and

m

are

the

positive real roots of the equat ion

10)

By applying Green s second identity to the inner fluid region V,

and making use of equations (3b-d) and the continuity require

ments

11)

on SR, the boundary-value problem for q is reduced to the inte

gral equation

Cpq P)

+

II

q Q)-aa

R

dS Q)

JJsus

B

nQ PQ

+ ffs

q Q)

[ a ~ (R:Q)

-

II(R:

Q

] dS Q)

+ ffs

R

; Q) (R:Q)

-

~ ~ ; Q ) R : J ] dS Q)

= is hq Q)

(R:

Q

) dS Q) 12)

where

P

and

Q

designate the reference and integration points on

the boundary surfaces av enclosing V aV = SU

SB

USF U

SR),

273

R

pQ

denotes the distance between the two points, dS Q is a dif

ferential area on av, C

p

is the solid angle enclosed by av

at

the

point P, and SB, SF are the portions of

bottom

and free surface

bounded by SR. Subscript Q on n designates that the normal

denvative is to be evaluated

at the

point

Q.

iscretisation

Scheme

Using Quadratic

Isoparametric

Elements

To

solve the integral equation (12) numerIcally, the boundary sur

faces

S,

SB and SF are discretised respectively into NJ, N2 and N3

quadrilateral isoparametric elements shown

in

Figure 2. The

el-

ement 60S, has nodes p, /)

(J

= 1 to 8) at its corner and mid.

side points, whose coordinates are x ~ / ) ,

y; ),

z;/).

All the

nodes

on the whole boundaries are numbered in sequence, such

that

P,(a = 1 to N

J

) is the a-th node with the coordinates (x y,-,z,),

where NJ is the total number of nodes. The potential

q

at the

arbitrary point

Q

within the element

60S,

can be approximated

by

8

q{Q)

=

LM, Q)q{P,CI)

(13)

=1

where M, are the shape functions interpolating q quadratically

between the nodal values. The boundary surfaces may also be

approximated by piecewisely quadratic curved surfaces, such

that

8

XQ

=

M , ( Q ) x ~ )

=1

8

YQ =

L

M, Q)y;/)

=1

8

ZQ

=L

M, Q)z;/)

3=1

where (xQ, YQ, zQ) denote the coordinates of the point Q.

(14)

By

s u b s t i t ~ t i n g

equations (13) and (14) into equation (12) which

is satisfied at the node

P

and truncating

the

double series of

eigenfunctions in equation (7)

at

n

=

Nand m

= M, the

integral

equation (12) is replaced by NJ algebraic equations of the form

FIgure 2 A quadrilateral isoparametric element with 8 nodes

7/25/2019 ISOPE-P-90-092

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N M

L L

F.mnCY

mn

+ G.

mn/3mn)

n Om O

N

8

=

'

B(I) h p l)

L...J tJ ,

1=1;=1

(15)

where =

Nl+N2

and =

N ~ N 3 .

The influence coefficients

A ~ ~ ,

B ; ~ ) ,

F.mn and G.

mn

in equation (15) are given by

(16)

(17)

( ) =

f s J a ~ Q R;,Q)

-bmn R;,Q)] Zm(zQ) ( ) ndS

(18)

where

1

Rp,Q

=

[(x.

-

XQ)2

+

Y.

-

YQ)2

+(Z.

-

ZQ?]

e . = t a n - l ~ : ) ,

= t a n - l ~ : )

(19)

1

k H ~ I ) I k r o )

m

=

0

H ~ I ) k r o ) ,

b

mn

=

K : m K ~ K : m r o ) ,

m> 1

Kn K:mro) -

(20)

and

C. is

the solid angle enclosed by the boundary surface at the

node p,. For a smooth boundary

C. IS

27l', otherwise it can be

a,pq .

h

evaluated from the requirement

that

- = 0 when t e poten

an

tJals distrIbute uniformly over the whole boundaries.

Unknown variables involved in equation (15) are N

J

potentials

and NR = M + 1) 2N +

1)

coefficients CY

mn

, /3mn. In order

that

the number of equations becomes equal to the number of un

knowns, N

R

additional control points must be placed on S

R

to

satisfy

8 N; 8

L L

A ~ ~ ) , p q ~ I ) +

L

L A ~ ~ ) - / l B ~ ) , p q p ; I )

1=1

;=1

I = N ~ 1 J=1

N M

L L

F.mnCY

mn

+G.

mn/3mn)

n Om O

N,

8

=

'

B(I)h

p l)

L . L J J

1=1

J=1

where

(22)

The integrals in equations (16) and (17) may be evaluated by

numerical quadrature

(e

g a four point Gaussian quadrature)

274

when p. is not on the element

/),.S,.

However, when P, coincides

with one ofthe nodes on /) S say ~ I ) , numerical quadrature may

not be used b.ecause of the singular behaviour of the integrands

as

Rp,Q

-->

0,

and analytical treatment is reqUired

to

evaluate these

singular integrals. I t can be shown

that, for

flat elements,

A ~ ~ )

IS

zero due to the orthogonality of

Rp,Q

and ii For sufficiently

small elements where the curvature effect is negligible, one may

therefore assume

that

(23)

To evaluate

B ~ ) , it is

convenient to rewrite equation (17) in the

form

(24)

where

(25)

For sufficiently small elements, the mtegral of equation (25) may

be evaluated explicitly, yielding

B ~ ) =

t l,

sin

/3

log (

t } l .)

=1

tan 2 tan 2

(26)

where

l

/3 are as defined in Figure

3,

and K takes the value

2 and 3 for the corner and mid-side nodes respectively. Since the

integral involved in equation (24) is regular everywhere, this may

be evaluated by numerical quadrature.

The evaluation of F.mn and G.

mn

requires more careful consid

eration because of the oscillatory behaviour of the integrands,

part icularly when m is large. For these evaluations,

it

is

conve

nient to use the Fourier series expansion of the Rankme source.

By substituting this mto equation (18) and integrating explic

itly with respect to the azimuthal angle e, the surface integral

over SR in equation (18) is reduced to a line integral, which may

be then evaluated numerically. For full details, see Matsui, Kato

and Shirai (1987). Solution of equations (15) and (21) leads to

the numerical approximations

to

the diffraction and radiation po

tentials.

P,

(a)

(b)

Figure 3 Definitions of

l /3

and

{ for

(a) corner nodes,

(b) mid-side nodes

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NUMERICAL EXAMPLES

Several numerical examples have been studied t o illustra te the va-

hdity and accuracy of our hybrid integral-equatIOn method Both

constant and quadratic isoparametric element idealisations were

adopted and compared with each other as well as with existing

numerical solutions based on a direct boundary integral (Gar

nson, 1978) and hybrid finite element formulations (Yue, Chen

and Mei, 1978). In the remainder of this section, the abbrevia

hons H.LE.M., D.B.I

M.

and H.F.E.M are used respectively to

stand for the hybrid integral-equation method, direct boundary

Integral method and hybrid finite element method. A constant

and quadratic isoparametric element are referred to

as

C.E and

Q.E. respectively.

ircular Dock

The first example studied is a circular dock in shallow water,

for which exact analytical solutions have been obtained by Gar

ret (1971). The dock analysed here has a radius a and a draft

D =

0 5a in water of depth

h =

0

75a

Due to the double sym

metry of the geometry, only one quadrant of the fluid region

bounded by the fictitious cylinder of radius

1 25a

has been anal

ysed with boundary element idealisations shown in Figure 4 and

Table

1,

and the far field eigenfunction representation truncated

at N

=

5 and M

=

4, which was sufficient to ensure the conver

gence of the solution. Our computed results for the wave exciting

moment are compared with the analytical solutions in Figure 5

Also included

In

this comparison are results obtained by Yuen

and Chau (1987), using H.LE.M. (C E.)

I t is

clearly seen

that

H.I E

M. Q.E)

provides much closer results to the analytical

predictions than H.LE.M. (C.E.), confirming the high accuracy

of the H.LE.M. formulation based on Q E. iqealisations. Fur

thermore, our H.I.E

M.

(C E ) results are seen

i

a.

-

):

L :

a:

' -

~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0

N

"'

D. B. . M. -c. E.-

-[ ] -

- )-

-* -

H.1.E.M.-C.E.

H.1.E.M.-Q.E.

H.F.E.M.-Q.E.

EXPERIMENT

04- - - - - . - - - - . - - - - , - - - - - r - - - - . - - - - , - - - - - , - - - - i

C().8

1 6

2 4

3 2

4 0

T

( s e c )

Figure 8 Added mass in heave for the semi-submersible

normalised with displacement

P V)

~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

o

'

D.B.1.M.-C.E.-

-[ ] - H.1.E.M.-C.E.-

- )-

H. .E.M.-Q.E.-

-*- H.F.E.M.-Q.E.- _--

XP RIM NT / ,,/'

,,/

L

, , / , , /

0 4 - - - ~ r - - - - r - - - - . - - - _ , ~ - - _ r - - - - , - - - _ , - - - - ~

C().8 1 6

2 4

3 2

T ( s e c )

Figure 9 Surge amplitude of the semi-submersible in

head

waves

normalised with wave amplitude

A

4 0

277

CPU times were obtained on the FACOM M780/20 and VP-

200 at

Nagoya UniversIty Computation Center.

It

is evident

that

H.LE.M. offers a considerable saving of computer time over

D.B I.M. and H.F.E.M., especially when a wide range of frequen

cies must be studied.

a:

' -

x

a:

' -

x

o

N

Table 4 Comparison of CPU times

Method To(s)

TF(S)

DBIM CE)* 0.05 72.2

HIEM CE)* 36.4

19.1

HIEM QE)* 32.8 14.6

HFEM QE)** 0.15 56.7

-[ ]-

- ) -

-* -

O.B. .M.-C.E.

H. .E.M.-C.E.

H. .E.M.-Q.E.

H.F. E. M.

-Q.E.

EXPERIMENT

0 4 - - S ~ r - - - - r - - - - . - - - _ , - - - - _ r - - ~ , - - - _ , - - - - ~

C().8

o

N

1 6

2 4

3 2

T ( s e c )

a) Potential

flow

theory

D. B. .

M.

-C. E.-

-[ ] -

H. .E.M.-C.E.-

- )-

H. .E.M.-Q.E.-

l I

H.F.E.M.-Q.E.-

EXPERIMENT

r

,

I \

I

I \

I

I

I

I

I

I

I

I

I

I

I

4 0

0 4 - - m ~ . - - - - r - - - - . - - - ~ r - - - _ r - - - - - - - _ - - - - ~

C().8

1 6

2 4

3 2

T

( s e c )

b) With viscous correction A=0.023 m)

Figure

10

Heave ampli tude of the semi-submersible

in bow quartering waves normalised with

A

4 0

7/25/2019 ISOPE-P-90-092

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a:

-'

...

x

O.B.

I .H.-C.E.-

-[ ] -

H. I .E.H.-C.E.-

- ) -

H.I .E.M. Q.E.

H.F.E.M. Q.E.

0

EXPERIMENT

'

a

o

91.8

1.6

2.4

3 .2

4.0

T [sec]

Figure

11

Roll amplitude of the semi-submersible in beam

waves (normahsed with dimensionless wave number

kA

D.S I .M. C.E.

[CJ H. I .E.M. C.E.

- )-

H.I .E.M. O.E.

- -

H.F.E.M. O.E.

EXPERIMENT

x

a

1.6 2.4 3 .2

4.0

T

[sec]

Figure 12 Yaw amplitude of the sew-submersible m bow

quartering waves (normalised

With

kA

CONCLUSION

ThiS

work

was

motivated by the need to mvestIgate reliable and

economical numerical methods for predicting wave-induced dy

namic responses of ocean platforms of complex geometry One

such method proposed herem

S

the hybrid mtegral-equatlOn

method involving the use of quadrattc Isoparametric surface ele

ments The validity and accuracy of the proposed method were

confirmed by companng computed results With the analytical

so-

lution for the circular dock and the model test results for the

ITTC serm-submerslble Comparison of our results With eXlstmg

numencal solutions mdIcated that the use of quadratiC isopara

metrIC elements led to sigmficant Improvement of the accuracy

and effiCiency of the hybrId mtegral-equatlOn method, compared

278

with classical boundary integral approaches based on constant

element Ideahsations.

CKNOWLEDGEMENT

The

authors are grateful to Mr. Kimitoshl Sano of Nagoya

Uni-

versity for his careful typing ,of this manuscript.

REFERENCES

DnV (1981), Rules Classification of Mobile Offshore Units.

Eatock Taylor, R. and Jefferys, E.R. (1986), Variabili ty of Hy-

drodynamic Load Predictions for a Tension

Leg

Platform,

Ocean

Engineering

Vol

13, No.5, pp.449-490.

Garret,

C.J

R.

(1971), Wave Forces on a Circular Dock, Jour

nal of

Fluid Mechanics

Vol

46, Part 1

pp.129-139.

Garrison, C J. (1978), Hydrodynamic Loading of Large Off-

shore Structures Three-Dimensional Source Distribution

Methods, in

Numerical Methods in Offshore Engineering

Zlenkiewlcz, 0 C. et

al

eds), Wiley, Chichester, Ch.3,

pp.87-140.

Jefferys, J R (1987), Numerical Problems of First Order

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and

Floating Bodles

Report

No.

AM-87-06, Um-

verslty of Bristol

Korsmeyer, FT., Lee, C

-H,

Newman, J Nand Sclavounos,

P D (1988), The Analysis of Wave Effects on Tension

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of

the Seventh International

Conference

on

Offshore Mechanics

and

Arctic Engineering

ASME,

Vol

2, Houston, Texas, pp 1-14.

MatsUl, T., Kato,

K.

and Shirai, T (1987), A Hybrid Integral

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ComputatJOnal

Me-

chanlCs

Vol.2, pp.119-135.

Takagi, M , Arai, S., Takezawa, S , Tanaka, K and Takarada, N.

(1985), A Comparison of Methods for CalculatIng the

Mo-

hon

of a Semi-SubmefSlble,

Ocean EngineerIng

Vol

12,

No 1, pp 45-97.

Yue,

D K P, Chen, H.S. and Mel, C C (1978), A Hybrid El

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Numerical

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In

EngineerIng

Vol

12,

pp 245-266.

Yuen, M M F and Chau, F P (1987),

A

HybrId Integral Equa

tIon Method for Wave Forces on Three-DimenSIOnal Off-

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Journal

of

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and

Arc-

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TransactJOns of

the

ASME Vol 109,

No

3,

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