An Improved Solution of Body Boundary Integral Equationfor Offshore Structure in Time Domain Simulation

DUAN Wen-yang, HAN Xu-liang, ZHAO Bin-bin(College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China)

Abstract: Based on a three dimensional transient free surface Green function, an improved solutionof body boundary integral equation in time domain is presented, to numerically simulate hydrodynam鄄ic forces of offshore structure in waves. Transient Green function is adopted under the mean wettedbody boundary condition and linearized free surface boundary condition. In order to avoid the highoscillation normal derivatives of the transient Green function memory term, an improved efficient so鄄lution is proposed here in time domain. An internal solution inside the offshore structure, which sat鄄isfies the Laplace equation, can replace the normal derivatives of the transient Green function mem鄄ory term so as to only calculate the transient Green function memory term itself in body boundaryintegral equation. Using the newly proposed approach, various cases for 3D submerged body and float鄄ing body are simulated. The numerical results show that this new solution can effectively save thememory usage and space for computer, and make simple calculation for integral equation than the tra鄄ditional method. The proposed method was validated by comparing its numerical results with someavailable results in the public domain, and satisfactory agreements were achieved.Key words: transient Green function; memory term; internal solution;

integral equation; hydrodynamic forcesCLC number: U661.1 Document code: A doi: 10.3969/j.issn.1007-7294.2013.03.003

1 Introduction

With urgent need of oil and natural gas resource exploration in deepwater and complexocean environment, the moored floating production systems play a significant role in this field.Due to the hydrodynamic forces acting on offshore structures, there may occur drift motion indeepwater. Once the exciting hydrodynamic forces can be calculated accurately, the equationof motions can be solved easily. Therefore, with the purpose to keep mooring system platformwithin preset bounds, it is very necessary to accurately predict the hydrodynamic forces actingon offshore structures in deepwater.

Received date： 2012-10-28Foundation item: Supported by the National Nature Science Foundation of China (No. 51079032)Biography： DUAN Wen-yang(1967-), male, professor of Harbin Engineering University,

E-mail: duanwenyang@hrbeu.edu.cn;HAN Xu-liang(1985-), male, Ph.D. candidate of Harbin Engineering University,E-mail: hanxuliang112@yahoo.com;ZHAO Bin-bin(1984-), male, lecturer of Harbin Engineering University, E-mail: zhaobin2_1984@163.com.

Article ID： 1007-7294（2013）03-0226-13

第 17卷第 3期 船舶力学 Vol.17 No.32013年 3月 Journal of Ship Mechanics Mar. 2013

Initially the 2D strip theory method[1] is established to calculate the hydrodynamic prob鄄lems of floating body, and predict its motions. Today, it is still widely used for ship design.Also frequency domain approaches [2] have been widely adopted to solve hydrodynamic prob鄄lems. Some commercial software, such as Wamit [3] and Hydrostar[4], can evaluate in practicaloffshore engineering in frequency domain. But it has some limitations and helpless for nonlin 鄄ear or non-steady-state problems.

With the development and application of the high performance of computer, many re鄄searchers prefer to simulate transient and nonlinear hydrodynamic problems using the transientGreen function[5-6] and the Rankine source method[7-8] in time domain. Although the Rankinesource method is simple to evaluate, it requires the discretization of the body surface and freesurface, also an artificial damping zone[9-10] should be given to absorb the wave energy. Where鄄as transient Green function satisfies the linear free surface boundary condition and the far-field radiation condition. The transient Green function is complex and difficult to calculate, butonly the wetted body surface needs to be discretized by this approach. It is found that tran 鄄sient Green function is relatively suitable to simulate lots of hydrodynamic problems.

Based on the pioneering work of Finkelstein[11] and Cummins[12], earlier work on transienthydrodynamic problems were to deal with a body performing radiation[5] and diffraction[13] prob鄄lem by impulse response function. The source distribution method was adopted to obtain thesource strength, and then evaluate the potential on panel elements. The convincing results aregiven on submerged bodies undergoing large-amplitude motions by Beck and Magee [14] andFerrant[15]. While Lin and Yue[16], Lin et al[17], Duan[18], Shin et al[19], Liu[20] used the same ap鄄proach to study the large amplitude ship motions. The detailed work for comparing linear andnonlinear ship motion using time domain method was well done by Sen [21], Singh and Sen[22].Datta et al[23] modified the numerical scheme of transient Green function and increased its ro鄄bustness and overall efficiency to study the motions of fishing vessels. Also the mixed sourcesand dipoles distribution method was employed to obtain the potential on panel elements di 鄄rectly. Zhu et al [24] showed the applicability of the transient Green function to investigate ofthe gap influence on the hydrodynamics for multiple floating structures. Regardless of the indi鄄rect source distribution method or direct mixed sources and dipoles distribution method, itshould be noticed that large amounts of data (transient Green function memory term, partialderivatives and its normal derivative) need to be stored in the body boundary integral equationfor time domain simulation. If the body boundary integral equation is evaluated directly as tra鄄ditional, computational burden and the memory demand will eventually be increasing with theprocess of computation for long time simulation. Liu et al [25] used the product of the transientGreen function and the velocity potential Fourier transformations to replace the convolution oftransient Green function to perform a long-time simulation with constant over time.

In the present study, based on the three dimensional potential transient free surface Greenfunction, the improved solution has been proposed to solve body boundary integral equation foroffshore structure hydrodynamic problem in time domain. The objective of this paper is to use

第 3期 DUAN Wen-yang et al: An Improved Solution of … 227

Fig.1 The scheme of wave interaction witha mooring offshore structure

an internal solution inside the offshore structure, which satisfies the Laplace equation, so as toonly calculate transient Green function memory term itself, and all of its normal derivatives arereplaced by internal solution. So there is only one quantity, transient Green function memoryterm itself, needed to be stored for long time simulation procedure. It can effectively save thememory usage and space for computer, and make simpler calculation for integral equation thanthe traditional method. The numerical results are compared with some available results in thepublic domain to validate the correctness and effectiveness of the present method. Satisfactoryagreements are achieved.

2 Mathematical formulations

Fig.1 shows the scheme of wave interaction with a mooring offshore structure. The body isfloating on the free surface. The earth-fixed Cartesian coordinate system O-XYZ is adopted with the X-axis pointswave direction, the Z-axis is verti鄄cally upward. The XOY plane restsparallel to the undisturbed free sur鄄face. The coordinate system O′-X′Y′Z′ is fixed on body. Origin point O′locates at the gravity center of the off鄄shore structures.2.1 Assumptions and body boundary conditions

The fluid is assumed to be ideal fluid which is incompressible, inviscid and irrotational.The water depth is infinite. The mooring offshore structure is floating on the surface.

In the fluid field 赘, velocity potential can be regarded as椎T p,� �t =椎I p,� �t +椎 p,� �t (1)

where 椎=椎T -椎I is the disturbed flow potential, and p x, y,� �z is a point in the fluid field 赘.

The incident wave potential 椎I in infinite water depth can be expressed as follows:

椎I x, y, z,� �t = g灼a棕 ekzsin k xcos茁+ysin� �茁 -棕� �t (2)

where 灼a is the amplitude of wave; 茁 is the wave heading; the dispersion relationship is k=棕2/g

and k is the wave number.The fluid filed is bounded by the free surface SF , the body surface SB, the bottom surface

SD and the infinite boundary surface S∞. The velocity potential satisfies boundary conditions and

initial conditions.

荦2椎 p,� �t =0 (in the fluid field 赘 ) (3)

228 船舶力学 第 17 卷第 3 期

坠2椎坠t

2 +g 坠椎坠z =0 (on SF ) (4)

坠椎坠n =Vn-

坠椎I

坠n (on SB ) (5)

荦椎→0 (on SD ) (6)

椎＝ 坠椎坠t =0 (on SF , t=0) (7)

椎, 坠椎坠t , 荦椎→0 (on S∞ ) (8)

In the above formula, the normal vector →n on body surface SB points out of the fluid field, andVn is the velocity of the body in the normal direction. g is the gravity acceleration. For the moor鄄ing offshore structure is often at zero speed of advance, the disturbed flow potential can be re鄄garded as the diffraction potential.2.2 Outside field integral equation

The transient free surface Green function, which includes the Rankine item 1rpq

- 1rpq′

and

the free surface memory item G軒 p, t; q,軒 軒子 , satisfies the linearized free surface condition.

G p, t; q,軒 軒子 = 1rpq

- 1rpq′軒 軒+G軒 p, t; q,軒 軒子 (9)

G軒 p, t; q,軒 軒子 =2∞

0乙 gk姨 ek z+軒 軒灼

J0 k軒 軒R sin gk姨 t-軒 軒子軒 軒dk (10)

where R= x-軒 軒孜2+ y-軒 軒浊

2姨 ; rpq= x-軒 軒孜2+ y-軒 軒浊

2+ z-軒 軒灼

2姨 ; rpq′= x-軒 軒孜2+ y-軒 軒浊

2+ z+軒 軒灼

2姨 .And p z, y,軒 軒z and q 孜, 浊,軒 軒灼 are the field and source points, respectively. J0 is the zero or鄄der Bessel function. k is the wave number. The table interpolation method[6] for infinite waterdepth transient Green function is applied here.

By applying Green identity to 椎 q,軒 軒子 and G軒 p, t; q,軒 軒子 , and integrating both sides with子 from 0 to t, we can obtain the boundary integral equation for velocity potential 椎 on the bodysurface SB as below[26].

2仔椎 p,軒 軒t +SB

蓦椎 q,軒 軒t 坠坠nq

1rpq

- 1rpq′軒 軒- 1

rpq- 1rpq′軒 軒坠椎 q,軒 軒t

坠nq蓦 蓦dsq=

t

0乙d子SB

蓦G軒 坠椎 q,軒 軒子坠nq

-椎 q,軒 軒子 坠G軒坠nq

軒 軒dsq蓦蓦

蓦蓦

蓦

蓦蓦

蓦蓦

蓦where the normal derivatives of the transient Green function memory term is expressed as

坠G軒坠n = 坠G

軒坠x nx+

坠G軒坠y ny+

坠G軒坠z nz (12)

2.3 Internal field integral equationFor the sake of avoiding the high oscillation normal derivatives of the transient Green

(11)

第 3期 DUAN Wen-yang et al: An Improved Solution of … 229

Fig.2 Schematic of the internal field model

function memory term in body boundary in鄄tegral equation in time domain, we assumethat there is an internal closed fluid field赘I inside the floating body, as Fig.2 shows.The internal field integral domain includesbody surface SB and an internal imaginarycontrol surface SC . The internal potential of

the closed fluid field is 追 pI ,� �t , whichsatisfies the Laplace equation 驻追 pI ,� �t =0.

The classical Green second identity is employed for 追 pI ,� �t and 1/rpI qIin the internal

closed fluid field 赘I . And pI x, y,� �z and qI 孜, 浊,� �灼 are the field and source points in internal

closed fluid field 赘I, respectively. The location of control points are the same on body surface

SB, what we need to do is to get the points on internal imaginary control surface SC . Then the

integral equation for internal close fluid field is obtained by applying Green identity as below.

SB +SC

蓦 1rpI qI

坠追 qI ,� �t坠nqI

-追 qI ,� �t 坠坠nqI

1rpI qI

� �蓦 蓦dSqI=-4仔追 pI ,� �t (13)

where 追 pI ,� �t represents the potential of internal field at time t. Let pI tend to the surface

of internal flow field along qI normal direction. Then the internal field integral equation is

given here.

SB +SC

蓦 1rpI qI

坠追 qI ,� �t坠nqI

-追 qI ,� �t 坠坠nqI

1rpI qI

� �蓦 蓦dSqI=-2仔追 pI ,� �t (14)

Then we assume that the relationship of potential between the internal closed fluid 赘 I

and outside field 赘 is equivalent on body surface SB, 追 pI ,� �t =椎 p,� �t . So we can obtain

SB

蓦椎 q,� �t 坠G軒 p, t; q,� �子坠nq

dSq=SB

蓦追 qI ,� �t 坠G軒 pI , t; qI ,� �子坠nqI

dSqI(15)

Obviously it is much harder to obtain the normal derivatives of the transient Green func鄄tion memory term on the internal imaginary control surface SC . So let the potential equal to ze鄄

ro 追 pI ,� �t =0 on SC . The relationship between the internal closed fluid and outside field is de鄄

scribed as

SB

蓦椎 q,� �t 坠G軒 p, t; q,� �子坠nq

dSq=SB+SC

蓦追 qI ,� �t 坠G軒 pI , t; qI ,� �子坠nqI

dSqI(16)

Also the classical Green second identity is employed for 追 pI ,� �t and G軒 pI , t; qI ,� �子 inthe internal closed fluid field. Here the integral equation can be expressed as

230 船舶力学 第 17 卷第 3 期

SB+SC

蓦 追 qI ,蓦 蓦t 坠G軒 pI , t; qI ,軒 軒子坠nqI

-G軒 pI , t; qI ,軒 軒子 坠追 qI ,軒 軒t坠nqI

軒軒軒軒

軒軒軒軒dSqI

=0 (17)

Substituting Eq.(17) to Eq.(16), the internal solution 坠追 qI ,蓦 軒t /坠n can replace the nor鄄mal derivatives of the transient Green function memory term so as to only calculate transient

Green function memory term itself G軒 pI , t; qI ,蓦 軒子 here.

SB

蓦椎 q,蓦 軒t 坠G軒 p, t; q,蓦 軒子坠nq

dSq=SB+SC

蓦G軒 pI , t; qI ,蓦 軒子 坠追 qI ,蓦 軒t坠nqI

dSqI(18)

Finally, combining Eqs.(11) and (18) the body boundary integral equation for disturbedpotential 椎 can be expressed by the new solution as below.

2仔椎 p,蓦 軒t +SB

蓦椎 q,蓦 軒t 坠坠nq

1rpq

- 1rpq′蓦 軒- 1

rpq- 1rpq′蓦 軒坠椎 q,蓦 軒t

坠nq軒 軒dsq

=t

0乙d子SB

蓦G軒 p, t; q,軒 蓦子 坠椎 q,軒 蓦子坠nq

dsq-t

0乙d子SB+SC

蓦G軒 pI , t; qI ,軒 蓦子 坠追 qI ,軒 蓦子坠nqI

dsqI

here 坠追/坠n is the internal solution of the internal close fluid field by Eq.(14).2.4 Hydrodynamic forces

According to Bernoulli equation, the unknown velocity potential 椎 on body surface can beobtained from Eq.(19), integrating the hydrodynamic pressure on the mean wetted body surfaceSB, the hydrodynamic force is given here.

F軋=-籽SB

蓦坠 椎+椎I軒 蓦坠t ·n軋dS (20)

Once the hydrodynamic forces are obtained, the harmonic analysis method is adopted here.

F軋ij軒蓦t =∞

n=0移 Aij

軒蓦n cosn棕t+Bij軒蓦n sinn棕軒 軒t (21)

where Aij軒蓦1 and Bij

軒蓦1 are the hydrodynamic coefficient for the added mass and damping co鄄

efficient, respectively.

3 Numerical implementation of present method

The Boundary Element Method (BEM) is adopted to solve the new boundary integral Eq.(19). The wetted floating body surface SB is divided into NB panel elements and the internal imag鄄

inary control surface SC is divided into NC panel elements. The velocity potential on each panelelement is assumed constant.

The discretized form of Eq.(14) and new boundary integral Eq.(19) can be expressed asfollows:

(19)

第 3期 DUAN Wen-yang et al: An Improved Solution of … 231

NB +NC

j=1移Rij追

M

nj =NB +NC

j=1移Rij

n追

M

j i=1, 2, …, NB+NC (22)

NB

j=1移 Aij椎

M

j +Bij椎M

nj移 移=Ci i=1, 2, …, NB (23)

where the influence coefficient matrix is

Rij =驻Sj

蓦1rijdSj ; Rij

n=

驻Sj

蓦 坠坠nj

1rij蓦 蓦dSj (24)

Aij =

2仔-驻Sj

蓦 坠坠nj

1rij′蓦 蓦dSj i=j

驻Sj

蓦 坠坠nj

1rij- 1rij′蓦 蓦dSj i≠

≠≠≠≠≠≠≠≠≠

j

; Bij =-驻Sj

蓦 1rij- 1rij′蓦 蓦dSj (25)

Ci=驻tM-1

m=0移着m

NB

j=1移 椎

m

nj驻Sj

蓦G軒 m

j d≠≠

≠≠

≠

軒≠

軒≠

軒S -驻t

M-1

m=0移着m

NB +NC

j=1移 追

m

nj驻Sj

蓦G軒 m

j d≠≠

≠≠

≠

軒≠

軒≠

軒S ,

着0=0.5, 着m=1.0 m＞蓦 蓦0here the symbol m represents for time t=M驻t and 子. The symbols i and j represent the panel el鄄ement of the field point p and source point q , respectively.

In the traditional method, it needs to be stored 4mNB

2groups of data (i.e. transient Green

function memory term G軒 , partial derivatives 坠G軒/坠x, 坠G軒/坠y, 坠G軒/坠z) at the mth time step. In theabove equations, it can be used to avoid storing large amounts of data. Now there is just tran鄄

sient Green function memory term G軒 needed to be stored, so only mNBNB+NC蓦 蓦groups of data

needs to be stored at the mth time step. If for the submerged body, mNB

2is only one quarter of

the traditional method. Therefore, the present method can effectively save the amount of memo鄄ry usage and space for long time simulation, also make simple calculation for integral equa 鄄tion. What is more, the new Eq.(19) does not reduce the accuracy of formula. Thus, this formulaprovides an efficient way to perform long simulation of hydrodynamic problems in time domain.

4 Numerical results and discussion

4.1 Heave motion of a submerged sphereThe heave motion of a submerged sphere, with the depth of submergence equal to the di 鄄

ameter z0 /R=2.0. A forced heave motion z=z0+Acos棕t is imposed at t=0. In order to verify thepresent method, the linear and nonlinear heave force for the submerged sphere is calculated, andthe added mass and damping coefficients are obtained by harmonic analysis method. The re 鄄

sults are divided by 4仔籽R3/3 for A33 and 2仔籽R

3/3 for B33 for non-dimensionalization. In Fig.3

and Fig.4, both of linear and nonlinear results of the added mass and damping coefficients give

(26)

232 船舶力学 第 17 卷第 3 期

Fig.3 Added mass for heave motion Fig.4 Damping coefficient for heave motion

a good correlation with Ferrant (1991). Figs.5 and 6 give the comparison of CPU (Inter I7, 3GBRAM) time and RAM usage amount of old method (transient Green function method) and pre鄄sent method with the same mesh panel number on submerged body. Result by the presentmethod calculated three periods and 20 time steps per period. It clearly shows the presentmethod costs less time than old method. In additional, the RAM usage amount is reduced bynearly three quarters.

4.2 Radiation computations of a hemisphereHere the mean wetted body is adopted to study the heave motion of a sphere. The sphere

is initially semi-submerged with its center on the water plane, and a forced heaving motion z!"t=-Asin 棕! "t is imposed on sphere at t=0. Here we set R=1.0, R is the radius of the sphere andA=0.1R is the amplitude of the sphere. Fig.7 shows that sketch of elements on the total sur鄄face can be regarded as the superposition of the body surface mesh and the internal imaginarycontrol surface mesh.

When the wetted body surface mesh sizes are invariant, to investigate the effect of meshsizes on the internal imaginary control surface. Fig.8 shows that the time history of verticalhydrodynamic force acting on the hemisphere obtained by different internal control point for

heave motion conditions CC=棕2R/g, non-dimensionalization as 籽gR

3. It can be seen that the

difference between the results obtained by the mesh sizes of case 1 and case 3 is negligible,but the result obtained by the mesh size of case 2 and case 4 is little different from the othertwo. To some extent, this implies that in the internal surface SC , the circle number Nl and axial

Fig.5 CPU time for different panel number Fig.6 RAM usage amount for different panel number

第 3期 DUAN Wen-yang et al: An Improved Solution of … 233

Fig.7 Sketch of elements on the hemisphere surface (left), internal surface (middle)and total surface (right)

number Nr of segments should match as much as possible to body wetted surface. Therefore,here we choose case 3 to meet the requirements of the calculation accuracy and the computa鄄tional efficiency. As it can be seen from Fig.9, satisfactory agreements are achieved about theadded mass and damping coefficient result for heave motion, when they compare with the se鄄mi-analytical method by Humle[27].

Figs.10 and 11 give the comparison of CPU (Inter I7, 3GB RAM) time and RAM usageamount of old and present method on the same mesh panel number on body surface. Result bythe present method calculated three periods and 30 time steps per period. The present methodjust costs a little more time than old method due to adding internal surface, but it saves morethan half of the RAM usage amount.

Fig.8 Heave force for different internal control Fig.9 Added mass and damping coefficientpoint (CC=2.0) for heave motion

Fig.10 CPU time for different panel number Fig.11 RAM usage amount for different panel number

234 船舶力学 第 17 卷第 3 期

4.3 Diffraction problem of barge hull at Fn=0.0The diffraction computation of wave body interaction is carried out by a barge hull in reg鄄

ular waves with unit amplitude, as shown in Fig.12. The principal particulars of barge hull aregiven in Tab.1. The water depth is infinite.

The hydrodynamic forces Fx/籽gAL2, Fz/籽gAL

2

and My/籽gAL3of barge hull for the different wave鄄

length ship length ratio in regular heading wavesare plotted in Figs.13 to 15, respectively. Where籽, g and A are the fluid density, acceleration ofgravity and wave amplitude, respectively; L is thelength of barge hull. It could be understood thatthe study of these results indicates that the hydro鄄dynamic forces from new approach are observed tomutually agree with not only frequency results byHydrostar software, but also results by transient Green function method (Old method). As aresult, the new approach proposed in this paper is correct and effective.

With the purpose of investigating the effect of mesh sizes on internal imaginary controlsurface, we choose two cases. Case 1 is the uniform distribution mesh, that is, the segment oninternal control boundary matching body wetted surface. Let mesh refinement, for case 2, be-

Fig.12 The scheme of barge hull diffraction in waves

Tab.1 The principal particularsof barge hull

Length between perpendiculars, Lpp

Breadth, B

Draught, T

Displacement, 驻

100.00 m

20.00 m

10.00 m

20 500.00 t

Fig.13 Hydrodynamic force Fx Fig.14 Hydrodynamic force Fz

(Fn=0.0, 茁=180°) (Fn=0.0, 茁=180°)

Fig.15 Hydrodynamic force My (Fn=0.0, 茁=180°)

第 3期 DUAN Wen-yang et al: An Improved Solution of … 235

twice as much as case 1. In Figs.16 and 17, it is observed that the results of time history of hy鄄drodynamic force Fx appear to be the same for 姿/L=2.0 of the two cases. However, for case 1,the short wave 姿/L=0.5 has obviously different from the result of transient Green function(TGF). Whereas it is the closest to the result of transient Green function (TGF) for case 2. Theseimply that the long wave is relatively simple flow field, so the internal imaginary control sur 鄄face mesh can be uniformly divided or less divided. While the short wave is relatively complexflow field, it is noted that the internal mesh should be appropriate refinement so as to obtainthe excellent results. Of course, the numerical results also show good convergence.

5 Conclusions

In this paper, based on the three dimensional potential transient free surface Green func 鄄tion, the new solution has been proposed to solve body boundary integral equation about lin 鄄ear problem between 3D floating bodies and waves interaction in time domain. An internal so 鄄lution inside the offshore structure, which satisfies the Laplace equation, can replace the nor 鄄mal derivatives of the transient Green function memory term so as to only calculate the tran 鄄sient Green function memory term itself for long time simulation procedure. It can effectivelysave the memory usage and space for computer, also make simple calculation for integral e 鄄quation with the same accurate as before. The proposed solution has been validated by com 鄄paring its numerical results with some available results in the public domain. Good agreementhas been achieved. The new solution is further extended to nonlinear problems with large am鄄plitude motions. The work in this paper demonstrates that the present approach can also be usedto evaluate wave loads and other dynamic responses in offshore structures design.

AcknowledgementThis work was financially support by the project of Natural Nature Science Foundation of

Fig.16 Hydrodynamic force Fx Fig.17 Hydrodynamic force Fx(Fn=0.0, 茁=180°) (Fn=0.0, 茁=180°)

236 船舶力学 第 17 卷第 3 期

China (No. 51079032).

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海洋结构物时域边界积分方程的改进方法

段文洋， 韩旭亮， 赵彬彬（哈尔滨工程大学 船舶工程学院， 哈尔滨 150001）

摘要: 文章基于三维时域格林函数理论，提出了一种时域边界积分方程的改进计算方法，用于预报海洋结构物在波浪中

运动受到的时域波浪力。 该方法通过利用物体内部构造的满足拉普拉斯方程的内部解，在时域边界积分方程中只需要

计及时域格林函数记忆项本身，从而有效地避免了求解具有高震荡积分特性的时域格林函数记忆项的法向导数。 对潜

体和浮体作了计算，结果表明，这种方法十分有效，可以有效节省内存使用量和计算机空间。 通过与相关已发表文献的

结果进行比较，计算结果令人满意。

关键词: 时域格林函数； 记忆项； 内部解； 边界积分方程； 水动力

中图分类号: U661.1 文献标识码: A作者简介： 段文洋（1967－），男，哈尔滨工程大学船舶工程学院教授，博士生导师；

韩旭亮（1985-），男，哈尔滨工程大学船舶工程学院博士研究生；

赵彬彬（1984-），男，哈尔滨工程大学船舶工程学院讲师。

238 船舶力学 第 17 卷第 3 期

海洋结构物时II边界积分方程的改进方法作者： 段文洋， 韩旭亮， 赵彬彬

作者单位： 哈尔滨工程大学 船舶工程学院, 哈尔滨 150001

刊名：船舶力学

英文刊名： Journal of Ship Mechanics

年，卷(期)： 2013(3)

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