tomoaki utsunomiya and eiichi watanabe- accelerated higher order boundary element method for wave...

8/3/2019 Tomoaki Utsunomiya and Eiichi Watanabe- Accelerated Higher Order Boundary Element Method for Wave Diffractio

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Accelerated Higher Order Boundary Element Method forWave Diffraction/Radiation Problems and Its Applications

Tomoaki Utsunomiya and Eiichi WatanabeDept. of Civil Engineering, Kyoto University

Kyoto, Japan

ABSTRACT

This paper presents an accelerated higher order boundary elementmethod for wave diffraction/radiation problems and its applications,especially for wave response analysis of VLFS (Very Large Floating

Structures). The Fast Multipole Method (FMM) has been implementedon the higher order boundary element code using 8-node quadrilateralelement. The method utilizes an iterative solver, multipole expansionof Greens function, and hierarchical algorithm using quadrant-tree.

For solving hydroelastic problem efficiently using iterative solver, anew algorithm, where the equations of motions representing platevibration are solved at each iterative step, has been introduced. Thenumerical benchmark calculations have shown the efficiency of themethod both in storage requirement of O(N) and computation time of

O(N log N), where N is the number of unknowns for the velocity

potential.

KEY WORDS: fast multipole method; higher order boundary elementmethod; diffraction; radiation; very large floating structure; VLFS;hydroelasticity.

INTRODUCTION

The Boundary Element Method (BEM) using the free surface Green

function has been used as a basic tool for solving diffraction/radiationproblems of a floating body. However, when the method is applied to aVery Large Floating Structure (VLFS) such as a Mega-Float, the largerequirement of computer resources with O(N2) for storage and with

either O(N2) or O(N3) for CPU time, which depends on the selection ofthe solver either as iterative type or LU-factorization type (N: the

number of unknowns), has made the computation of velocity potentialsby BEM impractical.

Therefore, the computation for a VLFS has been tackled by the finiteelement method (FEM) where band storage characteristics of thesystem matrix are utilized, by the semi-analytical approaches, or by theBEM utilizing higher-order panels such as B-spline functions (see e.g.Ohtsubo and Sumi, 2000). However, the higher-order panel methodbased on B-spline functions is hardly convergent for iterative solvers.

This paper employs an alternative approach using Fast Multipol

Method (FMM) (Rokhlin, 1985; Barnes and Hut, 1986; Greengard an

Rokhlin, 1987; Greengard, 1988; Fukui and Katsumoto, 1997, 1998The FMM is applied here to accelerate the calculation of the Highe

Order Boundary Element Method (HOBEM) using 2nd order 8-nodequadrilateral panels. The essential theoretical background of the FMMis based on Graf's addition theorem of Bessel functions which appear ithe free surface Green function, introduction of hierarchical algorithmto compute the system equations, and employment of the iterativ

solver. Although similar approach to accelerate BEM for wavdiffraction/radiation problems can be found using the pre-correcteFFT algorithm (Korsmeyer, et al., 1996, 1999; Kring, et al., 2000), it ilimited up to now to a low-order panel and a deep-water case.

Formulations are presented with the several benchmark calculation

including wave response analysis of VLFS. The storage requirement oO(N) and CPU time characteristics ofO(NlogN) have been confirme

by the benchmark calculations.

FORMULATIONS

The Boundary Value Problem

As shown in Fig. 1, consider a box-like VLFS (Very Large Floatin

Structure) of the length 2a (=L), the width 2b (=B), the draft d, anfloating in the open sea of variable depth (with constant depth h ainfinity). The variable depth sea-bottom surface (SB) is defined suc

that the boundary line (B) touches on the flat bottom base-surface oz=-h and the SB must locate higher than the base surface (z=-h). Th

coordinate system is defined such that the xy plane locates on thundisturbed free surface and the z axis points upwards. The center othe VLFS is on thez axis as shown in Fig. 1.

Consider the long-crested harmonic wave with small amplitude. Thamplitude of the incident wave is defined by A, the circular frequenc

by , and the angle of incidence by (=0 corresponds to the heawave from positive x direction and = /2 to the beam wave fropositivey direction).

305

Proceedings of The Twelfth (2002) International Offshore and Polar Engineering Conference

Kitakyushu, Japan, May 2631, 2002

Copyright 2002 by The International Society of Offshore and Polar Engineers

ISBN 1-880653-58-3 (Set); ISSN 1098-6189 (Set)


2/8

Assuming the water to be perfect fluid with no viscosity andincompressible, and the fluid motion to be irrotational, then the fluid

motion can be represented by a velocity potential . Also, we considerthe steady-state harmonic motions of the fluid and the structure, with

the circular frequency . Then, all of the time-dependent quantities canby represented similarly as follows:

The Boundary Integral Equation

]),,(Re[);,,( tiezyxtzyx = (1)

In this study, we employ the following integral equation as fundamental equation, which is applicable to a floating body witvariable-depth sea bottoms and/or breakwaters (Teng and Eatoc

Taylor, 1995; Utsunomiya, et al., 1998):

where i is the imaginary unit (i2=-1) and tis the time. In the following,all time-dependent variables are represented in the frequency domain

unless stated explicitly.

Incident wave

Assuming also that the fluid and structural motions are small so that thelinear potential theory can be applied for formulating the fluid-motionproblem. With these assumptions, the following boundary valueproblem may be formulated:

02 = in (2)

Kz

=

on S (3)F

0=

n

on S (4)

B

0=z

on B (5)

0

Fig. 1 Configuration of the analytical model.

)(4)(

),(

}),(

)(),(

)({

)()),(

4(

2

2

xdSn

xG

dSn

xGx

n

xG

xdSz

xG

ISSS

SSS

S

BHSHB

BHSHB

I

rr

rr

rrr

rrr

rrr

=

+

+

(11

),( yxwin

=

on S (6)HB

0=

n

on S (7)

HS

0))()(

(lim =

II

rik

rr

on (8)S

)sincos(

cosh

)(cosh

yxik

Ie

kh

hzkgAi

++= (9)Here, and . The symbol S designates th

inner plane ofz inside the boundary . It is noted that th

integral in Eq. (11) is performed for and the norma

derivative is made on the same surface. Using Eq. (11), the evaluationof solid angles and CPV integrals that may needed when usuaFredholm type integral equations are used, can be avoided. Th

functionr

represents the Green function representing wate

waves, and satisfying the boundary conditions on the free surface, othe flat sea bottom (z=-h), and the radiation condition at infinity. Th

series form can be represented as follows (Newman, 1985; Linton1999):

),,( zyxx =r

=

),( r

xG

),,( =r

I

)

hB

,( , =r

where is the incident wave potential. The symbols , , ,

and represent the fluid domain, the free surface, the variable depth

sea-bottom surface, and the flat-bottom base surface on z ,

respectively. The symbols S , , and represent the bottom

surface of the floating body, the wetted side surface of the floatingbody, and the artificial fluid boundary at infinity, respectively. The

symbol Krepresents the wave number at infinite depth sea ( ;

g is the gravitational acceleration), whereas k is the wave number at

constant depth h satisfying the following dispersion relation:

I

0

FS

=

2=

BS

h

g/

B

HB HSS S

)(cos)(cos)(2

),(0

0 hkhzkN

RkKxG

mmm

m

m ++=

=

rr(12

Kkhk =tanh (10)

)2

2sin1(

2 hk

hkhN

m

m

m+= (13

The symbol n represents unit normal vector (the positive direction

points out of the fluid domain). The complex-valued variable

represents the vertical deflection of the bottom surface of the floating

body. The variable rin Eq. (8) represents horizontal distance betweenthe origin and the referred point.

),( yxw

Khkkmm

=tan (14

where ( ) is a positive real number, and k . Th

following relationship is also used:

mk 1m ik=

0

306


3/8

)(2

1)( )2(

00kRiHikRK = (15) = HBS wdSyxpV ),( (23

Substituting Eq. (18), Eqs. (21)-(23) into Eq. (20), and calculating thvariations with respect to w, we haveHere, is the modified Bessel function of the second kind, and

is the Hankel function. The symbol R represents horizontal distance

between andr

. The functionr

G is given as (Teng and

Eatock Taylor, 1995):

0K

)2(

0H

xr

),(2

r

x

31211

2

1111),(

rrrr

xG +++=rr

(16)

2/122 ])([ += zRr

=

+

+

+

+

+

HB

HBHB

HB

S

SS

S

wdSdyxi

wdSwwdSwkdSyx

w

yx

w

x

w

y

w

y

w

x

w

y

w

y

w

x

w

x

wD

),,(

])1(2

[

2

22

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

(24

2/122

1])([ ++= zRr

2/122

21])2([ hzRr +=

Here, substituting Eq. (19) and the variation into Eq. (24), an

considering the arbitrarity of , we finally obtain the followin

equation of motion in the modal coordinates:

l

2/122

31])2([ hzRr +++=

=

=P

lS jljljl HB

dSfdyxiMK1

2 ),,()( (25

Equation of Motion for VLFS DeflectionsHere,

It is now widely accepted that VLFS response in terms of the vertical

deflection can be captured well by modeling the whole VLFS as anelastic plate. In this study, we employ the thin plate theory on elasticfoundations (which model the hydrostatic restoring forces):

+++

+

+

=

HB

HB

S jl

jljl

jljl

S

jl

lj

dSffkdSyx

fyxf

xf

yf

y

f

x

f

y

f

y

f

x

f

x

fDK

])1(2

[

22

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

(26

),(),(),(),( 24 yxpyxkwyxwyxwD =+ (17)

whereD is the plate rigidity, is the mass per unit area, k ( :

density of fluid), and is the dynamic pressure on the bottom

surface of the VLFS. The dynamic pressure relates to the

velocity potential on the bottom surface of the VLFS from thelinearized Bernoulli's equation:

)y

g= ,(xp

),( yxp

=HBS jllj

dSffM (27

),,(),( dyxiyxp = (18)

are the generalized stiffness matrix and the generalized mass matrix

respectively. The selection of the modal function has in fact somarbitrariness; e.g., dry-modes of the plate may be frequently used. Ithis study, we choose the modal function made as a tensor product o

modal functions of a vibrating free-free beam in thex andy direction(Utsunomiya, et al., 1998).

Our target is to solve the stationary solution which is governed by Eq.(17) and the free boundary conditions are specified along the edges of

the plate. In the following, we firstly represent the deflection

by series of the products of the modal functions and the

complex amplitudes , and then derive the equation of motion for the

modal amplitudes using Ritz-Galerkin's method:

),( yxw

),( yxfl

l

Formulation for Hydroelastic Analysis

Substituting Eqs. (4), (6) and (7) into Eq. (11), we have

)(4),(),(

}),(

)(),(

)({

)()),(

4(

2

2

xdSwxGi

dSn

xGx

n

xG

xdSz

xG

IS

SSS

S

HB

BHSHB

I

rrr

rrr

rrr

rrr

=

+

+

(28

=

=P

lll

yxfyxw1

),(),( (19)

The Hamilton's principle is used to derive the equation of motion.

0)( =+ VTU (20)Moreover, substitution of Eq. (19) into Eq. (28) yields

Here, U represents the strain energy, T the kinetic energy, and V thepotential energy due to external force.

)(4),(),(

}),()(),()({

)()),(

4(

1

2

2

xdSfxGi

dSnxGx

nxG

xdSz

xG

IS l

P

ll

SSS

S

HB

BHSHB

I

rrr

rr

r

rrr

rrr

=

+

+

=

(29

dSkwyx

w

y

w

x

w

y

w

x

wDU HBS

]}))(1(2

2)(){([2

1

22

2

2

2

2

2

2

2

2

2

2

2

+

+

+

+

=

(21)

dSwTHBS

=22

2

1 (22)

If we determine the unknown modal amplitudes ( l ) an

the unknown nodal potentials ( ) to be distributed on th

surface , and , the problem is solved. This can be done b

l P,,1 K=

i Ni ,,1 K=

HBS

HSS

BS

307


4/8

solving the simultaneous equations composed of (25) and (29).

The right-hand side of Eq. (25) may be rewritten as

),,1(}]{[][

],,[),,(

8

1

1

8

1

1

81

PjLL

dSfNNdSfdyx

j

e

NHB

eej

e

NHB

eS jS j eHB

KM

MK

==

=

=

=

=

(30)

In Eq. (30), the symbol S represents each element (or panel) on the

bottom surface of the floating body , and the number of elements is

represented by . That is

e

HBS

NHB

e

NHB

eHB

SS1=

= (31)

The symbols N represent shape functions to interpolate the

potential inside each element from the nodal values of

(when 8 noded panel is used). The complex valued vector { has

elements of the potentials at nodes ( ), and the row

vector [ (the size isN) is composed by a superposition of [ at

the corresponding nodes. Substituting Eq. (30) into Eq. (25), andexpressing it in the matrix form, we have

81 ,, NK

]j

L

81 ,, K

}

ejL ]

i Ni ,,1 K=

}]{[}]{[ 2 LiMK = (32)

Here, }{ is the complex valued vector of which elements are

( ), [ is the real matrix (the size is P ) of

which elements are ( l ), and [ is the real

matrix (the size is P ) of which row is composed by

( ).

l

P,,1 K

j 1=

l =

][j

L ,

]2MK

ljK

P,

P

]lj

M2

N

Pj ,,1, K= L

K

Eq. (29) may be represented in a matrix form as

}{4}]{[}]{[I

BiA = (33)

where [ is a complex valued matrix (the size is N ), is a

complex valued matrix (the size is N ), and { is a vector (of

the sizeN). From Eq. (32),

]A N}

I

][B

P

}]{[][}{ 12 LMKi= (34)

Substituting Eq. (34) into Eq. (33), we have

}{4}]){[]][[]([ 122I

LMKBA = (35)

In the above equation, we have unknowns only for ( i ).

Solving first Eq. (35) for ( i ) and then substituting them

into Eq. (34), we also have ( l ). Thus we have solved the

given problem.

i N,,1 K=i

l

N,,1 K=

P,,1 K=

In Eq. (35), we have to evaluate LU factorization of [ in

advance; however, we have to be aware that for most cases P

and also [ is a real banded matrix, thus the LU

decomposition would be trivial in the total computation time. When wemploy Eq. (35) for solving the velocity potential by using an iterativsolver, we can obtain the final solution with one iterative procedureThis feature is most important advantageous point over th

conventional modal method when an iterative solver is used. (Thcomputation time is reduced by a factor of 1/(number of modes)).

]2MK


5/8

Here, ( ) represent the polar coordinate values of the original

multipole expansion point O with respect to the newly definedmultipole expansion point O. Similarly, the following relationship issatisfied:

,

)()(

~ni

nmn

mnm ekIn

M

n

M

=

=

(42)

ddJNn

G

ddJxn

GN

n

G

xddJn

GddJ

z

G

e

NE

xeexe k

kk

e

NE

exe k

kk

e

NE

exe

NI

ee

||

||)}({

)()||||4(

)ofnear:()(1

1

1

8

1

)(1

21

1

8

1

)(1

1

1

2

1

1

1

2

=

=

=

=

=

=

+

+

+

rr

r

r

r

r

Implementation to Higher Order Boundary Element

Method

)(4||),(

)(cos)(

||),(

||

)(cos)(

1

11

0)fromfar:(

1

)ofnear:(1

1

11

1

1

8

1

0)fromfar:()(

1

xddJfM

hzkerkKi

ddJfGi

ddJNn

M

hzkerkK

Ie

P

lllmn

m n

NHB

xee

m

in

mn

e

NHB

xee

P

lll

ke

kkmn

m n

NE

xeexe

m

in

mn

r

r

r

rr

=

+

++

=

=

= =

=

=

=

=

=

=

(47

In Eq. (29), the distribution of the potential within each element (orpanel) may be expressed by using the 2nd order interpolation functions

and the potentials at the nodal points of the element as follows:

k

k

kN ),(),,(

8

1

=

= (43)

where ( , ) represents the coordinate values in the curved plane

coordinate system defined on the surface of each element. Substitution

of Eq. (43) into Eq. (29) yields

)(4||),(

||)}({

)()||4(

1 1

1

1

1

21

1

8

1

1

1

1

2

xddJGfi

ddJxn

GN

n

G

xddJz

G

Ie

P

l

NHB

ell

e

NE

e k

kk

NI

ee

r

r

r

=

+

+

= =

=

=

=

(44) Application of the Fast Multipole Method

where ),( ee

JJ = represents the Jacobian defined for eachelement e. The symbols NHB , , , and NI represent the

number of elements on the surfaces of S , , , and S ,

respectively, and

NHS NB

HB HSSS

B I

NBNHSNHBNE ++= (45)

When Eq. (47) is solved by an iterative method, { } is calculated fo

an assumed { using Eq. (34), and then the left-hand side of Eq. (47

is obtained for the assumed { . The values of{ are then update

to minimize the error between the left-hand side and the right-hand sidvalues of Eq. (47). In the calculation of the left-hand side values of Eq(47), the diagonal terms of the coefficient matrices are stored in th

core memory; on the other hand, the off-diagonal terms are calculatein each step of the iterative process. This enables the drastic reductioof the storage requirement to O(N). However, of course, we have taccelerate the computation concerning the off-diagonal terms of thcoefficient matrices.

}

} }

is satisfied. Eq. (44) may be rearranged as

ddJNn

G

ddJxn

GN

n

G

xddJn

GddJ

z

G

e

NE

exe k

kk

e

NE

exe k

kk

e

NE

exe

NI

ee

||

||)}({

)()||||4(

)(1

1

1

8

1

)(1

21

1

8

1

)(1

1

1

2

1

1

1

2

=

=

=

=

=

=

+

+

+

r

r

r

r

r

The calculation concerning the off-diagonal terms may be separate

into two parts: the first one is on the calculation for elements near x(3rd and 6th lines in Eq. (47)), and the second one is on the calculation

for elements far from (4th, 5th, 7th and 8th lines in Eq. (47)). In th

following, we explain the accelerated calculation method of the termin the 4th and 5th lines in Eq. (47) as an example for the second part.

r

xr

=

=

=

=

+=

1

1

8

1

0)fromfar:()(

1

||

)(cos)()(

ke

kkmn

m n

NE

xeexe

m

in

mn

ddJNn

M

hzkerkKxb

rr

r

(48)(4||),(1

1

11

xddJfGiIe

NHB

e

P

lll

r =

=

=(46)

where corresponds to the case when the collocation point x is

included in the element e, andr

to the case whenr

is outside of

the element e. Further substitution of Eqs. (37) and (39) into (46)yields

ex r r

ex xAs has already been defined, the values of ( ) in Eq. (48) definthe collocation point x measured from the multipole expansion poin

O. The multipole expansion point O can be located in an arbitrar

horizontal position under the restriction of ; in this study, th

multipole expansion point O is located on the projection of the origin o

the local coordinate system (

zr ,,

r


6/8

As is clear from Eq. (40), the integrand in Eq. (48) includes only the

values concerning ( ),, ; this fact means that the integral can be

evaluated beforehand regardless of the position ofx . By storing thevalues of the calculated integral in the core memory, we can obtain anefficient algorithm very simply; however, still this has an efficiency ofonly O(N

r

2).

Next, we introduce the concept of "cell" gathering the influences from

many far field elements into an influence from one large cell (Rokhlin,1985). For this purpose, we reform Eq. (48) as follows:

=

=

= =

+=

ke le

llmn

m n

Ncell

xkk

m

in

mn

ddJNn

M

hzkerkKxb

Cell

1

1

8

1

0)fromfar:(

1

||

)(cos)()(

r

r

(49)

In Eq. (49), the summation for elements is replaced by the summation

for cells. In Eq. (49), the cell k should be as large as possible (toinclude elements as many as possible) from the computationalefficiency. In Eq. (49), the multipole expansion point O is located on

the center of each cell, and ( , the coordinates of the collocation

point , is newly defined correspondingly. In general, as larger the

cell size, the number of cells, Ncell, becomes small, and thus leads toshorter computation time.

),, zr

xr

In this study, we employ the quadrant-tree to define the tree structure ofcells and the hierarchical computation algorithm (Greengard andRokhlin, 1987; Greengard, 1988; Fukui and Katsumoto, 1997, 1998).For simplicity, the shape of each cell is chosen to be a square. Thehierarchical cell structure is organized on the xy plane (on the

undisturbed free surface). Firstly, define the level-0 cell of square

shape, which must include all of the analyzed area ( , , and

). Then, divide the level-0 square cell into four even square cells,

and define them to be level-1 cells. Similarly, define level-2 cells bydividing each level-1 cell into four even square cells. This process isrepeated until the bottom level cells (leaf cells) include someappropriate number of elements. In the dividing process, the cell that

includes no elements is deleted from the definition list of cells.

HBS

HSS

BS

The method to define "far" cells and "near" cells is made similarly asGreengard and Rokhlin (1987). That is, the cell i that includes the

collocation point x inside and the cells that enclose the cell i adjacentto it are defined as "near" cells; otherwise the cells are categorized to be

"far" cells. Note that this categorization into "near" and "far" cellsshould be made on the same level of the tree structure. In thecalculation of Eq. (47), the computation for "near" cells is made foreach element (corresponding to the lines 3 and 6 in Eq. (47)).

r

In the actual computation, firstly calculate the integral of Eq. (49)around the center of each element as the multipole expansion point.Then using Eq. (42), the multipole expansion coefficients are convertedto the coefficients for the center of a leaf cell. By gathering the

coefficients of all elements included in the leaf cell, the multipoleexpansion coefficient for the leaf cell can be obtained. The calculation

of the multipole expansion coefficients for upper level cells can bemade similarly, by using Eq. (42) and gathering the coefficientsincluded in the upper level cell. With these procedures, all of the

necessary multipole coefficients can be set up, without specifying the

collocation pointr

.x

Next, we specify the collocation point x to coincide a nodal point on

either of , , or . At this stage, we had better calculate from

upper-level to lower-level, by selecting larger cells as far as possibleThis hierarchical algorithm is known to be O(N log N) in thcomputation time (Barnes and Hut, 1986).

r

HBS

/6[ h

2.1 ka

HSS

]R

15+

BS

ka

The truncation of the infinite summations in Eq. (49) is made a

(Newman, 1985), ( ), an

( ) where a is the maximum cell size to b

used in the actual computation.

maxm =

[max

=n

20max

=n 5


7/8

analyzed model becomes 67,098. The computation time was 38.4 hour

using 5 CPUs for the wave period of 18 seconds, the wave direction of=/4, and the residual tolerance =10-2.

Fig. 4 shows the deflection amplitudes in the variable depth sea and inthe constant depth sea (h=8m). We observe a considerable difference

of the response characteristics when the VLFS is located in the variabledepth sea from those in the constant depth sea, showing the importanceto include the effect of variable depth under the VLFS.

Fig. 5 shows a snapshot of the surface elevation around the floatingbody, where the diffraction and the radiation waves are both included.

As seen in Fig. 5, the refraction of waves, and the change of wave

length (about =157m inside the reef corresponding to 8m water depth)by dispersion are clearly observed.

CONCLUSIONS

The accelerated higher order boundary element method using fastmultiople algorithm has been presented. The analysis has been madefor a VLFS on the variable depth sea configuration as an example to

show the effectiveness of the method. The proposed method utilizesthe multipole expansion of the Greens function of the series form,introduction of the hierarchical algorithm based on the quadrant-tree,and a modern iterative solver such as GMRES. The theoreticalrequirements of the memory allocation of O(N) and the computationtime ofO(NlogN) have been demonstrated by the developed program

through numerical examples. The large scale hydroelastic problemswith about 300,000 unknowns have been demonstrated to be solvableby the proposed accelerated BEM program within about one day,

showing the advantage of the proposed method for large scale analysissuch as VLFS problems. This study was supported by the Program forPromoting Fundamental Transport Technology Research from theCorporation for Advanced Transport & Technology (CATT).

REFERENCES

Barnes, J, and Hut, P (1986). "A Hierarchical O(N log N) Force-

Calculation Algorithm,"Nature, Vol 324, pp 446-449.

Fast multipole method Direct method

CPU timeModel

Typicalpanel

size

Numberof

nodesNumber

of iter.

CPU time

per iter.CPU time

Memory

allocationLU

factorization

GMRES

solver

Memory

allocation

ABCD

E

25.0m12.5m6.25m

3.125m

1.5625m

1,6095,37719,39373,345

284,929

31323232

31

3 sec15 sec86 sec570 sec

1081 sec*

1.8 min10.4 min77.4 min775 min

1689 min*

27 MB89 MB315 MB1.15 GB

1.75 GB*

1.55 min28.5 min(907 min)(708 hr)

(1658 days)

3.28 min36.4 min(474 min)

(113 hr)

(71 days)

54 MB489 MB(6 GB)(85 GB)

(1.2 TB)

Number of iterations CPU timeModel L/

Waveperiod 310

= 410= 310= 410= C

CCDD

9.57

17.933.262.062.0

18 sec

10 sec6 sec4 sec4 sec

25

112252287287

32

148497

1.15hr

3.46hr8.23hr63.8hr

14.5hr*

1.29hr

4.40hr15.7hr

Table 2 Number of iterations of GMRES and CPU times for various L/.

* Parallel computation using 6 CPUs; number of modal function is 120.

Table 1 Performance of the fast multi ole method and the direct method.

* Parallel computation using 5 CPUs.

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311


8/8

Its Application to VLFS in Variable Water Depth and Topography,"

Proc 20th Int Conf on Offshore Mech & Arctic Eng, OMAE01-5202.

(b) =10-2

(a) =10-3

(b) in variable depth sea.

(a) in constant depth sea (h=8m).

Fig. 4 Deflection amplitude at T=18sec.Fig. 2 Deflection amplitudes of VLFS in constant depth sea formodel B (solid line: FMM, dash-dotted line: direct method).

Fig. 5 Snapshot of the surface elevation around the

floating body in the variable depth sea.Fig. 3 Contour plot of the variable depth configuration.

312

tomoaki utsunomiya and eiichi watanabe- accelerated higher order boundary element method for wave...

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