TIME-DOMAIN COMPUTATION METHODS FOR A FLOATING PLATFORM OFCOMPLICATED GEOMETRY WITH MULTIPLE WAVE INTERACTIONS

Yuichi Ashida, Takeshi Hara, Takuya Taniguchi and Masashi KashiwagiDepartment of Naval Architecture & Ocean Engineering, Osaka University

2-1 Yamada-oka, Suita, Osaka, Japanhara_takeshi@naoe.eng.osaka-u.ac.jp, kashi@naoe.eng.osaka-u.ac.jp

ABSTRACT: Linear, time-domain analysis is used to solve the radiation problem for theforced motion of a floating platform of complicated geometry at both zero and low constantspeed of the current. The velocity potential due to sinusoidal forced oscillation is obtained froman integral equation which is derived from Green’s second identity. The integral equation issolved numerically using higher order boundary element method (HOBEM). However, facingseveral difficulties, constant panel method (CPM) is used for the final computation. Possiblereasons of numerical difficulties in the time-domain HOBEM are discussed. Obtained time his-tories of the results are Fourier analyzed, from which the added-mass and damping coefficientsare computed and those results are compared with corresponding results computed indepen-dently in the frequency domain. Agreement is excellent in the zero-speed case.

1. INTRODUCTION

Recently, there are growing concerns over environmental problems, and much interest and ex-pectation are being placed on renewable energies like solar power, wind power, wave power,and so on. The wind power generation in offshore regions is one of those potentials. In fact, thegeneration of wind power is already popular in Europe, but most of the wind-power plants inthe offshore region in Europe are bottom-mounted type. In Japan, however, because there arefew offshore regions with shallow water depth, a floating type should be used.

Various types of floating platform, such as the spar-buoy type and semi-submersible type,are already proposed and studied for installing large-scale wind turbines. In order to install notonly wind turbines but also solar panels and other equipment for marine energy utilization, thesize of floating platforms tends to become large and the geometry of the submerged part tendsto be complicated.

In this paper, we consider a triangle-shaped floating platform as Fig. 1. The figure showsonly the submerged portion of the platform, which is laterally symmetric and consists of verticalcircular columns placed at apexes of the equilateral triangular plan view, fully submerged lowerhulls connecting vertical columns, and a large number of oblique braces connecting the lowerhulls with an upper deck. In order to compute hydrodynamic forces on this floating platform andresulting motions in waves, the frequency-domain computation methods have been applied [1].However, time-domain computations are also required to consider transient phenomena and toinclude nonlinear forces such as mooring and wind forces in addition to linear components. Wealso need to investigate effects of the current on hydrodynamic forces. In the linear theory, thetime-domain and frequency-domain results are related through the Fourier transform. Solutionsin the time domain can be obtained indirectly with the memory-effect function which can becomputed by the Fourier analysis using frequency-domain solutions [2]. However, when thecurrent (equivalently the forward speed of the body) exists, it is known that the frequency-domain Green function becomes complicated and solutions are not reliable, whereas the time-domain Green function method [3] [4] [5] [6] needs relatively a little extra work over the zero

LBy

z

x

0

β

Figure 1 - Coordinate system and overview of a floating platform of complex geometry

speed case. Thus, in this paper, the time-domain Green function method is applied in order tosee effects of the current on hydrodynamic forces.

The numerical computations have been performed first by the higher-order boundary ele-ment method (HOBEM), since theoretically this method is advantageous in both accuracy andcomputational time over the constant panel method (CPM). However, through several efforts,we found that the results by HOBEM tend to be unstable as the time stepping proceeds, whereasthe results by CPM are stable. Possible reasons for this unexpected result are discussed in thispaper. Consequently computed results in this paper are obtained by CPM. For confirmation ofthe results, comparisons are made with the corresponding results in the frequency domain [1].The results for the body without braces are also presented to see the degree of contribution fromthe braces to the hydrodynamic interactions.

2. FORMULATION OF PROBLEM

The shape of a floating body under consideration for marine energy utilization is shown in Fig. 1only for the submerged portion, which is laterally symmetric and consists of vertical circularcolumns, fully submerged lower hulls with hexagonal section shape, and a number of obliquebraces with circular cylinder. Wind turbines are supposed to be installed on the deck of verticalcolumns placed at apexes of the equilateral triangular plan view. Fig. 1 also shows the Cartesiancoordinate system, which is fixed in a current of speedU . The origin is taken at the geometriccenter of the plan view and on the undisturbed free surface,xy-plane. Thez-axis is positivedownwards and the water depth is assumed deep enough in this paper.

The linear potential-flow theory is adopted, and hence the velocity potential is introducedand expressed as

Φ(x, t) =U{−x+φS(x)

}+ΦU(x, t) (1)

ΦU(x, t) = φ0(x, t)+φ7(x, t)+6

∑j=1

φ j(x, t) (2)

whereφS represents the steady disturbance in a current;φ0 denotes the incident-wave potential;φ7 the scattered potential;φ j the radiation potential due to thej-th mode of body motion; andx= (x,y,z) is the position vector. In this paper, attention is focused on the radiation problem inthe time domain and on validation of the code developed.

The velocity potential will be sought to satisfy the Laplace equation and the boundary con-ditions on the free surface (z= 0) and body surface (SH) written as(

∂∂t

−U∂∂x

)2

φ j −g∂φ j

∂z= 0 onz= 0 (3)

∂φ j

∂n= Xj(t)n j +UXj(t)mj onSH (4)

whereXj(t) is the time-varying amplitude in thej-th mode of motion;n j is the j-th componentof the normal vectorn (defined positive when pointing into the fluid from the body surface),with n j for j = 4∼ 6 defined asn j = (x×n) j−3; andmj is the so-calledm-term defined as

(m1,m2,m3) =−(n ·∇)V , (m4,m5,m6) =−(n ·∇)(x×V ) (5)

whereV ≡ ∇{− x+ φS(x)

}. Furthermore, the solution must satisfy the initial condition of

φ j = 0 andφ j = 0 att = 0, and the deep-water condition of∇φ j → 0 asz→ ∞.The problem formulated above may be solved by using the time-domain Green function,

which has been well studied [3] [4] and can be expressed in the form

G(P,Q, t − τ) = G(0)(P,Q)δ(t − τ)+G(W)(P,Q, t − τ)u(t − τ) (6)

whereG(0)(P,Q) =− 1

4π

(1r− 1

r1

)(7)

G(W)(P,Q, t − τ) =− 12π

∫ ∞

0

√gksin

{√gk(t − τ)

}e−k(z+z′) J0(kR)dk (8)

and R=

√{x−x′+U(t − τ)

}2+(y−y′)2 (9)

For brevity, we have used notations of P= (x,y,z) for the field point, Q= (x′,y′,z′) for thesource point,r for the distance between P and Q, andr1 for the distance to P from the mirror-image point aboutz= 0. δ(t) andu(t) in (6) are Dirac’s delta function and the unit step function,respectively, andJ0(kR) denotes the first-kind of Bessel function of zero-th order.

Applying Green’s theorem to a combination of this Green function and the velocity potentialφ j(Q, t) to be sought and integrating the result with respect toτ from−∞ to+∞, we can obtainthe boundary integral equation for the velocity potential on the body surface in the form

C(P)φ(P, t)+∫ ∫

SH

φ(Q, t)∂

∂nQG(0))P,Q)dS=

∫ ∫SH

∂φ(Q, t)∂nQ

G(0)(P,Q)dS

+∫ t

0dτ

∫ ∫SH

{∂φ(Q,τ)

∂nQ−φ(Q,τ)

∂∂nQ

}G(W)(P,Q, t − τ)dS

−U2

g

∫ t

0dτ

∮CH

{∂φ∂x′

−φ( ∂

∂x′− 2

U∂∂τ

)}G(W)(P,Q, t − τ)

∣∣∣∣z′=0

dy′ (10)

where, for simplicity, the mode indexj in the velocity potential is omitted,C(P) is the solidangle at P on the body surface, which can be determined only from the body geometry andnumerically by using the equi-potential relation in the higher-order boundary element method.

The first term on the right-hand side of (10) can be computed from the body boundarycondition (4), and the remaining terms represent memory effects and can be computed withonly the values in the past. The last term in (10) is the so-called line-integral term along theintersection (CH) between the body and free surfaces, which vanishes for the zero-speed case.

3. NUMERICAL SOLUTION METHOD

For solving (10), it is customary to discretize the submerged portion of a floating body into anumber of plane panels and to assume the unknowns to be constant on each panel. Then theintegral equation can be transformed into a linear system of simultaneous equations for the un-knowns. This solution method is known as the constant panel method (CPM). In this method,we need to integrate the Rankine source term represented byG(0)(P,Q) and its normal deriva-tive over a flat panel, which can be precisely performed with analytical method developed byNewman [9]. Numerical calculation methods for the time-dependent part of the Green functionG(W)(P,Q, t − τ) were well studied and described in detail by Liapis [3] and King [4], whichare also used in the present paper. However, to keep sufficient numerical accuracy by usingthis CPM for a complicated geometry especially with sharp corners, we need a large number ofpanels and hence large computation time.

In order to surmount these deficiencies, we adopt the higher-order boundary element method(HOBEM), which can attain generally higher numerical accuracy only with less number ofpanels. In this method, the submerged surface of a body is discretized into a number of curvedpanels, and each panel will be transformed into a flat panel of triangle or quadrangle in termsof a quadratic shape functionNk(ξ,η). In the iso-parametric scheme, both of the coordinatesand the velocity potential to be determined on each panel are expressed with the same shapefunction as follows:{

x,y,z}T

=M

∑k=1

Nk(ξ,η){

xk,yk,zk}T

, φ(x,y,z) =M

∑k=1

Nk(ξ,η)φk (11)

wherek is the node number in a panel, and the total number of nodes on one panel isM = 7 fortriangular panels andM = 9 for quadrangular panels. Here(xk,yk,zk) andφk denote the valuesof coordinates and unknown velocity potential at thek-th nodal point; superscriptT means thetranspose; andξ andη are normalized coordinates varying over the range−1≤ ξ,η ≤ 1.

The normal vectorn, directing into the fluid domain from the body surface, can be computedwith (11) as follows:

n=a×b

|a×b |, a=

{∂x∂ξ

,∂y∂ξ

,∂z∂ξ

}, b=

{∂x∂η

,∂y∂η

,∂z∂η

}(12)

We note that|a×b | is equal to Jacobian∣∣J ∣∣ in the variables transformation.

Substituting these equations into (10), we can write (10) in the discretized form and eventu-ally in the form of simultaneous equations for the velocity potentials at nodal points. The resultcan be expressed as

Cmφm(t)NT

∑ℓ=1

Dmℓφm(t) =NP

∑n=1

Smn(t)

+∫ t

0dτ{ NP

∑n=1

Rmn(t − τ)−NT

∑ℓ=1

Tmℓ(t − τ)φℓ(τ)}

−U2

g

∫ t

0dτ{ NF

∑n=1

Wmn(t − τ)}

for m= 1∼ NT (13)

wherem is the consecutive number of nodal points (and hence field points) and to be taken up toNT (whereNT is the total number of nodal points) to obtain the same number of equations as thenumber of unknowns. Some coefficients to be evaluated on each discretized panel, appearing in

(13), are defined as follows:

Dmℓ =14π

∫ 1

−1

∫ 1

−1Nk(ξ,η)

(r

r3 −r1

r31

)·n

∣∣J ∣∣dξdη (14)

Smn(t) =− 14π

∫ 1

−1

∫ 1

−1

∂φ(t)∂n

(1r− 1

r1

)∣∣J ∣∣dξdη (15)

Rmn(t − τ) =∫ 1

−1

∫ 1

−1

∂φ(τ)∂n

G(W)(Pm,Qn, t − τ)∣∣J ∣∣dξdη (16)

Tmℓ(t − τ) =∫ 1

−1

∫ 1

−1Nk(ξ,η)

∂∂nQ

G(W)(Pm,Qn, t − τ)∣∣J ∣∣dξdη (17)

Wmn(t − τ) =∫ 1

−1

[{∂φ(τ)∂x′

−φ(τ)( ∂

∂x′− 2

U∂∂τ

)}G(W)(Pm,Qn, t − τ)

]z′=0

∂y′

∂ξdξ (18)

Here the global consecutive node number is denoted asℓ (whereℓ= 1∼ NT), which should becomputed with the local node numberk (= 1∼ M) on then-th panel (n= 1∼ NP); that is, theconnection vectorℓ(k,n) must be computed and stored in advance.

Regarding the line-integral term, the total number of segments along the intersectionCH

on z= 0 is denoted asNF. The segment and unknown velocity potential are expressed withone-dimensional shape function in the form{

x,y,φ}T

=3

∑k=1

Nk(ξ){

xk,yk,φk}T

N1 =12

ξ(ξ−1), N2 =12

ξ(ξ+1), N3 = 1−ξ2

(19)

whereξ is the normalized coordinate varying over the range−1≤ ξ ≤ 1. With this transforma-tion, the derivative of the velocity potential can be computed from

∂φ∂x

=1

|M |

(ny

∂φ∂ξ

− ∂y∂ξ

∂φ∂n

), M = ny

∂x∂ξ

−nx∂y∂ξ

(20)

With regard to the singular integrals in (14) and (15) related to the Rankine source whenthe filed point exists on the panel to be integrated, numerical integrations have been adoptedwith Gaussian quadrature after applying a coordinate transformation in terms of the local polarcoordinates; for more details, readers are referred to Kashiwagi [7].

4. HYDRODYNAMIC FORCES

The hydrodynamic pressure in the linear theory can be computed from the linear term ofBernoulli’s pressure equation. Thus the hydrodynamic force acting in thei-th direction in theradiation problem can be computed from the following expression:

Fi(t) =6

∑j=1

ρ∫ ∫

SH

(∂∂t

−U∂∂x

)φ j(x, t)ni dS≡

6

∑j=1

fi j (t) (21)

In this paper, to validate the computer code, we consider the forced oscillation in thej-th modewith circular frequencyω, and computed results for the case ofU = 0 are compared with cor-responding results in the frequency domain computed by using the well-established Green-function method.

In the time domain, the motion amplitudeXj(t) and resulting hydrodynamic forcefi j (t)are time-dependent and thus the time history can be approximated using the Fourier series. Inthe linear theory, we may consider only the fundamental frequency component in the obtainedFourier series, which are written as

Xj(t)≃ Re{(

a1− ib1)

eiωt}= Re

{∣∣Xj∣∣e−iδ eiωt

}(22)

fi j (t)≃ Re{(

c1− id1)

eiωt}= Re

{∣∣ fi j∣∣e−iε eiωt

}(23)

wherea1− ib1 =

2T

∫ T/2

−T/2Xj(t)e−iωt dt, c1− id1 =

2T

∫ T/2

−T/2fi j (t)e−iωt dt (24)

andT denotes the period given byT = 2π/ω. The forcefi j (t) can be rewritten as follows:

fi j (t) = Re

{∣∣ fi j∣∣e−i(ε−δ)∣∣Xj

∣∣ ∣∣Xj∣∣e−iδ eiωt

}

=−∣∣ fi j

∣∣cos(ε−δ)ω2

∣∣Xj∣∣ Re

{(iω)2

∣∣Xj∣∣ei(ωt−δ)

}+

∣∣ fi j∣∣sin(ε−δ)ω∣∣Xj

∣∣ Im{

ω∣∣Xj

∣∣ei(ωt−δ)}

(25)

From (22), we can find the following relation:

Xj(t) = Re{(iω)2

∣∣Xj∣∣ei(ωt−δ)

}Xj(t) = Re

{iω

∣∣Xj∣∣ei(ωt−δ)

}=−Im

{ω∣∣Xj

∣∣ei(ωt−δ)}

(26)

Thereforefi j (t) can be expressed with the added-mass (Ai j ) and damping coefficient (Bi j ) inthe form

fi j (t) =−Ai j Xj(t)−Bi j Xj(t) (27)

whereAi j =

∣∣ fi j∣∣cos(ε−δ)ω2

∣∣Xj∣∣ , Bi j =

∣∣ fi j∣∣sin(ε−δ)ω∣∣Xj

∣∣ (28)

0 5 10 15 20

-0.15

-0.05

0

-0.10

-0.20

0.05

00.1

0.15

00.2

f33/ρg X32L

g/Lt

0 5 10 15 20

-0.15

-0.05

0

-0.10

-0.20

0.05

00.1

0.15

00.2

f11/ρg X1

2L

g/Lt

Figure 2 - Time history of hydrodynamic forces in surge (left) and heave (right) on floatingplatform computed by HOBEM (KL= 13)

5. RESULTS AND DISCUSSIONS

5.1 Computed Results by HOBEM

First we have implemented numerical computations using HOBEM for the case ofU = 0, andfound that sometimes computed results were unstable especially for a forced oscillation in thehorizontal direction. Fig. 2 shows typical time histories of normalized hydrodynamic forcesacting in the surge and heave directions, exerted by the forced surge and heave motions, i.e.f11

and f33. We can see in this example (atKL= 13.0) that f11 becomes unstable aftert√

g/L= 10,while f33 remains stable. This unstable behavior can be attributed to extremely large variationof G(W)(P,Q, t) near the free surface (z+ z′ → 0) and as a result numerical inaccuracy in thecomputation of memory effects on the right-hand side of (13). Fig. 3 shows the absolute valueof G(W)(P,Q, t), plotted againstX = x− x′, Y = y− y′, Z = z+ z′ (whereX = Y is assumed).We can see that the value becomes extremely large whenX =Y → 0 andZ → 0; that is, whenP and Q are in the same panel which is nearest the free surface, especially P is right on thefree surface. In the surge motion, the normal velocity∂φ/∂n is relatively large near the freesurface and thus inaccuracy in the integration of largely varying function over a panel near thefree surface tends to be amplified and accumulated as the time proceeds, resulting in unstableresults and eventually breakdown.

0

50000

100000

150000

|GW|

0

0.005

0.01 X= Y

0

0.005

0.01

Z

T= 0.0500T= 0.1000T= 0.2000

Figure 3 - Time-domain Green functionGW forzero speed case

In particular, in HOBEM, the fieldpoint P is placed at nodal points, whichcan be onz= 0 and the numerical in-tegration over the panel is made byGaussian quadrature. Thus this unsta-ble phenomenon becomes prominent inHOBEM. On the other hand, in CPM,the field point P is placed at the cen-ter of panels and thusz = 0. Especiallywhen the size of the panel nearest thefree surface is taken as relatively large,the value ofG(W) is not large as seenin Fig. 3, and thus stable and seeminglyreasonable results can be obtained. Thismay imply that the contribution from aregion very close to the free surface issmall, in spite of extremely large varia-tion of G(W). If this could be proven tobe true, we may dispense with integra-tion on a thin region of the panel near-est the free surface or we may apply asemi-analytical approximate method forthe integration over that region.

5.2 Computed Results by CPM

As explained, the results by HOBEM are sometimes unstable; especially for the case ofU = 0,the breakdown tends to happen at much earlier stage in the time integration. On the otherhand, the results by CPM are comparatively stable and thus we will investigate some featuresin hydrodynamic forces with computed results by CPM.

Table 1 - Principal dimensions of a floating platform with and without braces

Item with without

LengthL [m] 103.92 103.92

BreadthB [m] 120.00 120.00

Draft d [m] 10.00 10.00

Displacement volumeV [m3] 4918.00 4318.30

Area of water planeAw [m2] 157.09 58.91

Metacentric heightGM [m] 43.32 23.70

Since we consider the equilateral triangle-shaped platform and the origin of the coordinateslocated at the center of the figure, the relationship ofA11 = A22, A44 = A55 exists in the addedmass andB11= B22, B44= B55 exists in the damping coefficient for the zero-speed case. Thus,hydrodynamic forces will be shown only for the diagonal components in the longitudinal modein this paper. The principal dimensions of the platform are shown in Table 1 for both cases ofwith and without oblique braces.

Figure 4 shows computed results of the added mass and damping coefficients obtained bythe time-domain CPM, with and without current. As the current speed,Fn = 0.01 (U = 0.3m/s) is chosen by considering a realistic situation. To see the degree of contribution of braces,computations for the body excluding braces have been also conducted. The number of panelsis 1992 for the whole body with braces and 840 for the body without braces. This means that alarger part of the panels is spent for representing the braces.

The results are compared with corresponding results computed by the frequency-domainGreen-function method for the zero speed case in terms of CPM. They are in good agreement.In the nonzero current case, we can observe that the peaks of the damping coefficient becomelower in the region of high frequencies. This can be explained as the current effect, because thewaves generated by the body may flow downstream, and hydrodynamic interactions becomeweaker [8].

When the results of with and without braces are compared, although the values themselvesare of course different, it is found that the main variation tendency is almost the same. Thus, itcan be said that the contributions from braces to hydrodynamic interactions are relatively small.It is therefore practical to compute firstly the hydrodynamic interaction forces for the bodywithout braces and secondly the inertia force and nonlinear viscous damping force on obliquebraces separately and to sum both results with assumption of no interactions.

5.3 Conclusions

Numerical computations have been performed by using the time-domain Green-functionmethod for radiation forces on a floating body with complex geometry which is to be usedas a floating platform for installing wind turbines. Both of the higher-order boundary-elementmethod (HOBEM) and the constant panel method (CPM) were applied. It was found that com-puted results by HOBEM were unstable especially for a forced oscillation in the horizontaldirection. This instability originates from inaccurate integration of extremely large variation ofthe time-dependent part of the Green function on a panel nearest the free surface, because thereis a case in HOBEM that the field point P is placed right onz= 0 and thus a problem may occurwhen P is on the same panel to be integrated over the region very close to the free surface. Onthe other hand, in CPM, this kind of problem was not observed, because the field point P is

0 2 4 6 8 100

0.1

0.2

0 2 4 6 8 100

0.5

1

1.5

Freq.-domain (Fn=0)

Time-domain (Fn=0)

Time-domain (Fn=0.01)

Freq.-domain (Fn=0, without braces)

Time-domain (Fn=0, without braces)

Time-domain (Fn=0.01, without braces)

0 2 4 6 8 10

0.5

1. 0

Freq.-domain (Fn=0)

Time-domain (Fn=0)

Time-domain (Fn=0.01)

Freq.-domain (Fn=0, without braces)

Time-domain (Fn=0, without braces)

Time-domain (Fn=0.01, without braces)

0 2 4 6 8 100

0.1

0.2

0 2 4 6 8 100

0.1

Freq.-domain (Fn=0)

Time-domain (Fn=0)

Time-domain (Fn=0.01)

Freq.-domain (Fn=0, without braces)

Time-domain (Fn=0, without braces)

Time-domain (Fn=0.01, without braces)

0 2 4 6 8 100

0.01

0.02

A11 ρ

∆

/

A33 ρ

∆

/

A L55

2

L2ρ

∆

/

B11 ρ

∆

/ !

B33 ρ

∆

/ !

B55 ρ

∆

/ !

KL/2 KL/2

KL/2 KL/2

KL/2 KL/2

Figure 4 - Added mass and damping coefficients: Comparison of frequency-domain andtime-domain results, with and without braces and current

placed at the center of panels.The added-mass and damping coefficients were obtained by the Fourier analysis from the

time histories of the radiation forces. Obtained results were compared with corresponding re-

sults computed independently as a frequency-domain problem, and very good agreement wasconfirmed for the zero-speed case. In the forward-speed case, weaker wave interactions due tocurrent effects were observed in a high-frequency region. Although there are no reliable resultsto compare with, it can be said that computed results look reasonable.

In the computation for the floating platform studied in this paper, more than half of thepanels are used for representing the braces. However, we found that the contributions fromthe braces to hydrodynamic interactions are negligible, and thus it may be practical to computehydrodynamic interactions for a body without braces first and then to evaluate and add theinertia and damping forces on braces by using the Morison formula [10].

6. REFERENCES

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2. Kashiwagi, M. (2000) A time-domain mode-expansion method for calculating transient elas-tic responses of a pontoon-type VLFS,Journal of Marine Science and Technology, Vol. 5,No. 2, pp. 89-100

3. Liapis, S. J. (1986) Time-domain analysis of ship motions, Dissertation, The Department ofNaval Architecture and Marine Engineering, The University of Michigan

4. King. B. K. (1987) Time-domain analysis of wave exciting forces on ships and bodies, Dis-sertation, The Department of Naval Architecture and Marine Engineering, The University ofMichigan

5. Beck, R. F. and Liapis, S. J. (1987) Transient motions of floating bodies at zero forwardspeed,Journal of Ship Research, Vol. 31, No. 3, pp. 164-176

6. Bingham, H. B., Korsmeyer, E. T., Newman, J. N. and Osborne, G. E. (1993) The simulationof ship motions,Proc. of the 6th International Conference on Numerical Ship Hydrodynam-ics

7. Kashiwagi, M. (1995) A calculation method for steady drift force and moment on multiplebodies (in Japanese),Bulletin of Research Institute for Applied Mechanics, Kyushu Univer-sity, Vol. 78, pp. 83-98

8. Kashiwagi, M. (1993) Interaction forces between twin hulls of a catamaran advancing inwaves (Part 1: Radiation problem),Journal of the Society of Naval Architects of Japan,Vol. 173, pp. 119-131

9. Newman, J. N. (1986) Distributions of sources and normal dipoles over a quadrilateral panel,Journal of Engineering Mathematics, Vol. 20, pp. 113-126

10. Leblanc, L., Petitjean, F., Le Roy, F. and Chen, X. B. (1993) A mixed-panel-stick hydro-dynamic model applied to fatigue life assessment of semi-submersibles,Proc. of the 12thInternational Conference on Ocean, Offshore and Arctic Engineering(Glasgow, Scotland),Vol. 1, Offshore Technology