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ELSEVIER

Applied Ocean Research 17 1995) 11-11

0 1995 Elsevier Science Limited

0141-1187(95)00007-0

Printed in Great Britain. All rights reserved

0141-I 187/95/ 09.50

New higher-order boundary element methods for

wave diffraction/radiation

B. Teng* R. Eatock Taylor-t

Depart ment of Engineeri ng Science, Uni versit y of Oxf ord, O xford OX I 3PJ, U

(Received 2 February 1995; accepted 19 April 1995)

This paper describes some higher-order boundary element methods and presents a

novel integral equation for the calculation of the wave diffraction and radiation

problem. A higher-order element discretisation of the resulting integral equation

is used. An examination of the convergence and CPU time is carried out, and the

results demonstrate the advantages of the new method.

INTRODUCTION

The analysis of wave diffraction and radiation by a body

at or near a free surface requires the solution of a

boundary value problem based on Laplaces equation.

Because the problem involves an unbounded domain,

the integral equation method brings certain advantages

to its solution. Based on this method, the constant panel

representation introduced by Hess and Smith is

widely used, for example by Faltinsen and Michelsen,

Garrison3 Inglis and Price,4 and Korsmeyer et al5

In this, the submerged body surface alone is discretised

by a set of quadrilateral or triangular flat panels.

Pulsating sources or sinks are placed at the centres of

all panels, the strengths of these sources being constant

in each panel. For a body with a curved surface,

however, such a representation allows leaks between

panels, with the result that large numbers of panels

are required to achieve computations with sulhcient

accuracy.

Recently, the use of higher-order element methods for

this problem has been investigated, for example by Liu

et CZI.,~*~ atock Taylor and Chat? and Eatock Taylor

and Teng. Higher-order element methods are believed,

in general, to give more accurate results than the

constant panel method for the same computational

effort, although there appear to be no direct compar-

isons for the water wave problem. For complex

structures and analysis of the non-linear diffraction

*Present address: Department of Civil Engineering, Dalian

University of Technology, Dalian 116024, Peoples Republic

of China.

In higher-order element methods, special attention

has to be paid to the specification of solid angles at

nodes on the body surface, and to some singular

integrals when field points are very close to a source

point. The integration of the singular values where the

slope of the surface is discontinuous only exists in the

Cauchy principal value (CPV) sense, so special techni-

ques have to be used. Most researchers use indirect

methods to avoid computing them directly. A method of

cancelling the solid angle and CPV integrals was devised

by Eatock Taylor and Chau. It combines the integral

equation outside the body with another inside the body,

obtained using the same pulsating source Green func-

tion as outside. In this paper, we have derived another

novel integral equation, by using a simple Green

function inside the body, which satisfies a rigid lid

condition on the free surface, and a weakly rigid

condition on the sea bed. This equation appears to be an

improvement over that of Eatock Taylor and Chau.*

tTo whom all correspondence should be addressed.

In the following, we describe the new formulation,

problem by Stokes expansion techniques, accuracy is a

vital factor in a successful calculation, together with

minimum computer storage and CPU time. In a higher-

order boundary element approach to this analysis,

the body surface is discretized by a set of curved

elements, and the velocity potentials at nodes on the

element sides and corners become the unknowns. The

velocity potential and its derivatives inside an element

are expressed in terms of the corresponding nodal values

and shape functions. Thus higher-order element

methods are convenient for the calculation of wave

run-up and second-order forces on structures, where the

potential at the water surface and its spatial derivatives

are needed.

71

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72

B. Teng R. Eatock Taylor

and give results from some numerical experiments.

These were carried out to investigate the efficiency of the

method, and to study convergence in relation to meshes

both on the body surface and the inner sea bed. The

latter mesh is a feature of the new method, but is only

required for structures reaching to the seabed.

2 THE BASIC INTEGRAL EQUATION

In order to illustrate the various approaches, we

consider the first-order wave diffraction and radiation

problem. It should be noted, however, that the novel

formulation introduced here has also been applied to

the calculation of second-order diffraction problems,

where it is particularly effective. We assume here that

the first-order incident waves have frequency w, and

the time factor of the complex potentials is taken as

e iwf.We use the oscillating source G(x, x0) as Greens

function, which satisfies the linear free-surface bound-

ary condition, the radiation condition at infinity and

the impermeable condition on the horizontal seabed at

depth d. Here x and x0 are the field and source points,

respectively. Use of Greens identity leads to the

following Fredholm integral equation over the body

surface Ss

C(XO)~(XO)Js.G(;;o)b(x)ds

=-

G(x, x01 x)ds (1)

SB

Here 4(x) is the unknown scattering potential associated

with a normal velocity V(x) prescribed on Ss. The

positive direction of the normal to the body surface is

defined as being out of the fluid. C(Q) has the value

unity if x0 lies strictly inside the fluid region, and zero if

x0 is outside the fluid. When x0 is on the body surface,

47rC is the solid angle over which the fluid is viewed

from x0 (2?rC at the water line where the body surface

intersects the free surface).

In the higher-order boundary element method, the

body surface can be discretized by Na isoparametric

elements. After introducing shape functions hk([, r]) in

each element, we can write the velocity potential and its

derivatives within an element in terms of nodal values in

the form

(2)

where K is the number of nodes in the element, 4k are

the nodal potentials, and (

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New higher-o rder boundary element methods

73

diagonal terms of the left-hand side matrix [A].) To

deal with these difficulties, a number of alternative

approaches have been developed and applied to the

wave diffraction problem.

Eatock Taylor and Chau applied Noblesses contin-

uous integral equation

to the higher-order boundary

element method, and obtained an indirect formulation,

For later comparison we call this method A. By

applying Greens theorem inside the body, they obtained

the following integral equation

(1- c>+o>

Is,

G;;

) (x0)ds

=U

J

,

G(X>%MXo)ds

where Sw is the inner water plane, and v = w*/g. Eatock

Taylor and Chau then combined eqn (5) with eqn (1)

to obtain the new integral equation

+

JJ

x, x0>

&

@(x0) 4(x))ds

SB

=--

J J

W, o>x>s 6)

SB

In this equation, the coefficient C has been eliminated

and the singular parts in the double-layer integration are

cancelled. Instead, attention has to be paid to the

calculation of an integral on the inner water plane.

For floating bodies, the following alternative integral

equation has been developed by Wu and Eatock

Taylor12

The auxiliary Green function Go, which corresponds to a

rigid free surface condition, is defined by

Go(x,

x0) =

-&-

where

Y =

[R2+

(2 -

zo)2]12,

l =

[R+

(2 +

zo)2]12,

and

R =

[(x - xo)2 (y - yo)2]12

(8)

in Cartesian axes with the z-axis measured vertically

upwards from the mean free surface. This equation

avoids the integral on the inner water plane, and the

singular kernel in the integral on the body surface is still

cancelled.

In the case of a body extending to the seabed, one can

use a Green function Gt, which corresponds to an

infinite sum of images of the foregoing sources and their

reflections about the horizontal seabed

where

r2,, = [R* + (z - z. - 2nd) ]

1/2

r3n = [R2+ (z + zo + 2nd)]j2

r4 = [R2 + (z - z. + 2nd) ]

l/2

r5,, = [R2 (z + z. - 2nd)2] 12

(11)

We can then obtain another integral equation as

follows

This equation avoids the integration on the inner water

plane, but the calculation of G1 is time consuming. Since

eqns (12) and (7) have the same form, we designate the

solution based on these equations as method B.

We can avoid the drawbacks in the above methods, by

using the simple Green function

G2 = -f

7r

f,;+ +

1

>

(13)

in the integral equation