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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


2/7

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 inside the body. We thereby

obtain the new integral equation

( JJ

+

s 2 dx dr

)

+(x0)

I

+

~(xo)~-~w~)

ds = -

JJsBGvds

(14)

where St is the area of the structure resting on the seabed

(analogous to Sw, the area of the structure piercing the

water-plane). Since this Green function contains a term

corresponding to reflection about the seabed of the

source l/r, the integral on St is never singular: it can be

evaluated by direct numerical methods. Furthermore,

since the vertical derivative of the Green function is very

small at the sea bed, coarse meshes can be used for that

calculation. And for a floating body, St vanishes and the

calculation can be further simplified. In fact, eqn 14)

reduces to eqn (7) in that case, since there are no

singularities on S, due to r2r and r31. Hence

We describe this last approach as method C.

(15)


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74

B. Teng, R. Eatock Taylor

Table 1. Convergence of new method method C) with difiereot meshes on the bode

hIa

Mesh

2.0

1Cl

20 2c2

2.0 4c4

2.0 Analytical

1.0 1Cl

1.0 2c2

1.0 4c4

1.0 Analytical

0.5 lC1

0.5 2c2

0.5 4c4

0.5 Analytical

Case

1

2

3

4

5

6

7

8

9

ka=0.5

ka=

1.0

ka=

1.5

ka=2.0

4.71794 4.20143 2.79375 2.15075

4.79228 4.15299 2.39285 1.75941

4.79817 4.15394 2.63219 1.76064

4.79871 4.15405 2.63227 1.76072

2.84447 3.26350 2.43544 1.81624

2.90674 3.27865 2.39285 1.69756

2.91140 3.28154 2.39433 1.69847

2.91175 3.28175 2.39443 1.69853

1.50231 1.96959 1.68940 1.38274

1.54077 1.98940 1.67907 1.34109

1.54285 1.99097 1.68003 1.34185

1.54321 1.99129 1.68019 1.34186

Table 2. Convergence of new method method C) with different meshes on the inner sea bed

h/a

Mesh Case ka=0.5 ka= 1.0 ka= 1.5

ka=2.0

2.0

4Cl

2.0

4C2

2.0 4c4

2.0

Analytical

1.0 4Cl

1.0

4C2

1.0

4c4

1.0 Analytical

0.5

4Cl

0.5

4C2

0.5 4c4

0.5

Analytical

1

2

3

4

5

6

7

8

9

4.79630 4.15323 2.63208 1.76066

4.79805

4.15386 2.63218

1.76065

4.79817

4.15394 2.63219 1.76064

4.79871 4.15405 2.63227

1.76072

2.90859

3.28012

2.39405

1.69847

2.91122 3.28146 2.39431

1.69847

2.91140

3.28154

2.39433

1.69847

2.91175

3.28175

2.39443

1.69853

1.54158

1.99007

1.67945

1.34113

1.54263 1.99084 1.68001 1.34186

1.54285 1.99097 1.68003

1.34185

1.54321 1.99129 1.68019 1.34186

4

NUMERICAL RESULTS AND DISCUSSION

Computer programs have been developed based on the

methods outlined in the preceding sections. The

programs allow the simplification of two planes of

geometric symmetry to be used where relevant. To

investigate the efficiency of these methods, we have

carried out some benchmark tests. Quadratic isopara-

metric elements were used (six-noded triangles and

eight-noded quadrilaterals). Thus for these tests, the

shape functions I?((, n) in eqn (2) were taken to be

parabolic, and the geometry was represented by a

parabolic variation between the nodes. Typical results

are given in the accompanying tables, where a factor

pgAa* is used for non-dimensionalisation. Here A is the

amplitude of the incident waves and

a

is a characteristic

length of the structure.

To demonstrate the good convergence of the new

method (method C) based on eqn (14), the first-order

surge forces on a uniform cylinder of radius a, stretching

from the seabed to the free surface in water depths of 2a,

a and

0.5a

re presented in Table 1. Four dimensionless

Table 3. Convergence of method A with different meshes on the inner sea bed

h/a

Mesh

Case

ka=0.5 ka= 1.0

ka=

1.5

ka=

2.0

2.0 4Al

2.0 4A2

2.0 4A4

2.0 Analytical

1.0

4Al

1.0

4A2

1.0

4A4

1.0

Analytical

0.5

4Al

0.5

4A2

0.5 4A4

0.5

Analvtical

1

2

3

4

5

6

7

8

9

4.80287 4.14502

2.63127 1.76840

4.80222 4.15764

2.63238 1.75980

4.79817 4.15398

2.63220 1.76063

4.79871

4.15405

2.63227 1.76072

2.91312 3.27392

2.39340 1.70610

2.91233

3.28262 2.39451

1.69815

2.91134

3.28148

2.39432 1.69848

2.91175 3.28175

2.39443

1.69853

1.54349 1.98590

1.67947 1.34871

1.54316 1.99128

1.68563 1.34171

1.54276 1.99080

1.67997 1.34188

1.54321 1.99129

1.68019 1.34186


5/7

h/a

New higher- order boundary element methods

Table 4. Convergence of method B with different tolerances for truncation of series

Tolerance

Case

ka=OS ka= 1.0

ka= 1.5

75

ka=2.0

2.0 T20

2.0 T5

2.0

T2

2.0

Analytical

1.0

T20

1.0

T5

1.0

T2

1.0 Analytical

0.5 T20

0.5

T5

0.5 T2

0.5 Analytical

1

2

3

4

5

6

7

8

9

4.78880

4.15025

2.63165 1.76078

4.79724

4.15355

2.63213 1.76066

4.79800

4.15384

2.63218 1.76065

4.79871 4.15405

2.63227 1.76072

2.90012

3.27586 2.39337 1.69869

2.91041

3.28104

2.39424

1.69849

2.91121

3.28144 2.39431 1.69847

2.91175

3.28175 2.39443 1.69853

1.53371

1.98572 1.67901 1.34210

1.54219

1.99061 1.67998 1.34186

1.54278 1.99095 1.68004 1.34184

1.54321 1.99129 1.68019 1.34186

Table 5. Comparison of relative CPU times of the various methods

hIa

Case New method (method C) Method A Method B

2.0

1

0.97 0.93 1.32

2.0 2 0.98 0.95 1.61

2.0 3 1 oo 1.07 2.20

1.0 4 0.95 0.92 1.35

1.0 5 0.96 0.93 1.78

1.0 6 1 oo 1.04 2.64

0.5 7 0.98 0.97 1.12

0.5 8 0.98 1 oo 1.62

0.5 9 1 oo 1.12 2.56

frequencies ka were considered, k being the wave

number. Results from the analytical solution given by

McCamy and Fuchs13 are compared with the boundary

element calculations. Three meshes were used for these

calculations. Mesh 1Cl is a mesh with one element on a

quadrant of the body surface. Mesh 2C2 has four

elements (2 depthwise x 2 circumferentially), and mesh

4C4 has sixteen elements (4 x 4). The numbers of

elements on the inner sea bed (surface S1) are the same as

those on the body surface. It can be seen that the

convergence of the method is very fast. The results from

mesh 4C4 have an accuracy to four significant figures for

all frequencies and water depths investigated. Tables 2

shows how the accuracy of the new method depends on

the mesh on the inner seabed St. Mesh 4Cl is a mesh

with one element on a quadrant of Si, mesh 4C2 has 4

elements (2 x 2), and mesh 4C4 has 16 elements (4 x 4)

on the inner sea bed. The body mesh has 16 elements on

a quadrant of the cylinder.

Table 3 shows how the convergence of method A,

based on eqn (6), depends on the mesh on the inner

water plane Sw. The meshes in this case are designated

mAn, where for method A the number n now indicates

the fineness of the mesh on the inner waterplane SW.

It can be seen that from results based on meshes 4C4

and 4A4, the new method and method A yield similar

values for the forces, and have almost the same accuracy

when compared with the analytical solution. With

meshes 4C1 and 4C2, the new method (method C)

seems to give slightly higher accuracy than the

equivalent results from method A (i.e. meshes 4Al and

4A2), since the Green function G2 and its derivative in

the z-direction are very small. This is more obvious at

the higher frequencies.

Table 6. Surge exciting force on a truncated cylinder

ka

Method

0.5 1.0 1.5 2.0

Method A, mesh 4Al 5.3814 4.2084 2.6361 1.7683

Method A, mesh 4A2 5.3799 4.2219 2.6374 1.7604

Method A, mesh 4A4 5.3753 4.2197 2.6375 1.7611

Method B 5.3753 4.2197 2.6374 1.7611

Method C 5.3753 4.2197 2.6374 1.7611

Analytical 5.3714 4.2202 2.6376 1.7612


6/7

76

B. Teng, R. Eatock Taylor

Table 7.

Heave exciting force on a truncated cylinder

Method

0.5

1.0

ka

1.5

2.0

Method A, mesh 4Al 0.29181

2.796x lo-O2 2.926x lo-O3

3.530x lo-O4

Method A, mesh 4A2

0.29177

2.793x lo-O2 2.895x lo-O3

3.406x10-O4

Method A, mesh 4A4 0.29178 2.796x 10m02 2.892x 10m03 3.19ox1o-o4

Method B 0.29178 2.796x lo-O2 2.892x lo-O3

3.190x lo-O4

Method C

0.29178

2.796x 10-O* 2.892x lo-O3

3.190x lo-O4

Analytical 0.29036 2.768x 10m02 2.845x lo-O3

3.079x lo-O4

Table 4 shows results from method B, based on eqns

(7) and (12). In the calculation of the Green function G1,

tolerances of 0.2, 0.05 and 0.02 were specified for the

truncation of the infinite series (corresponding to cases

T20, T5 and T2 in Table 4). On the body surface, the

mesh of 16 elements per quadrant of the cylinder was

adopted. It can be seen that with these tolerances, less

accurate results are obtained compared with the

corresponding results in Tables 2 and 3.

It is evident that these tolerances are somewhat

coarse, but they were selected on the basis of attempting

to achieve similar computational effort with the different

methods. Table 5 shows the comparison of CPU times

for the three approaches used for the assembly of the

left-hand side matrix [A] in eqn (4). The numbers given

in the table are the ratios of CPU times relative to those

achieved with the new method (method C) and mesh

4C4. The nine cases listed correspond to the nine cases

in Tables 2-4, associated with different meshes on the

waterplane and the inner sea bed, and different

tolerances in the summations. For the cases indicated

in these tables, the relative CPU times are insensitive to

wave frequency. It can be seen from case 1 that with

mesh 4A1, method A requires less CPU time, since

method C has to spend more time on the body integral.

But with an increase in the number of elements on the

auxiliary horizontal plane (the inner sea bed for method

C and the water plane for method A), our new method

will use less CPU time. It can also be seen that method B

uses more time than methods A and C, since the

calculation of the Green function G, is time consuming,

even with the large tolerances specified.

Tables 6 and 7, respectively, compare the heave and

surge wave exciting forces on a truncated cylinder,

calculated by the three methods. Results are also given

for a semi-analytical solution based on the approach

described by Yeung.r4 The cylinder has a radius a and a

draft h =4a, and is in water of depth d= 10a. In the

calculations, a mesh of 40 elements (4 circumferentially

x 8 depthwise x 2 radially) was used to discretise one

quadrant of the body surface for all three numerical

methods. In the case of method C, the inner sea bed

vanishes, so no auxiliary surface domain is needed in the

discretised integral equation. For method A, three

meshes were used on the water plane, identified as in

the case of the vertical cylinder. It can be seen from these

tables that the results from methods B and C are exactly

the same as expected from the discussion after eqn (14).

With method A, the results from mesh 4A4 employing

the finest mesh on the water plane have the same

accuracy as methods B and C, but the results from

meshes 4Al and 4A2 differ from those based on

methods B and C.

5

SUMMARY AND CONCLUSIONS

We have proposed here a novel boundary element

method, which is derived by applying a simple Green

function in an integral equation written for points inside

the body. The method replaces the integration on the

water plane in Eatock Taylor and Chaus method* by an

integration on the inner sea bed. The latter vanishes in

the case of a body which does not touch the sea bed.

Because the proposed Green function can be very easily

calculated, and its value is in any case small at large

depths, the method can use coarse meshes on the inner

sea bed without losing much accuracy, especially at high

wave frequencies. When very accurate calculations are

required, using a fine mesh on an auxiliary plane, the

new method (method C) uses less CPU time than the

other methods (A and B). In the specific case of floating

bodies, the new method is equivalent to method B.

ACKNOWLEDGEMENTS

This work was supported by the National Natural

Science Foundation of China, and the UK Behaviour

of Fixed and Compliant Offshore Structures research

programme sponsored by SERC through MTD Ltd and

jointly funded with the Admiralty Research Establish-

ment; Aker Engineering, A. S.; Amoco Production

Company; Brown and Root; BP Exploration; HSE; Elf

UK; and Statoil.

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New higher-or der boundary element methods

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Higher Order BEM