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7/25/2019 Higher Order BEM
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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 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
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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
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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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