a hybrid element method for calculating hybrid element method for calculating three - dimensional...
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A HYBRID ELEMENT METHOD FOR CALCULATINGTHREE - DIMENSIONAL WATER WAVE SCATTERING
BY
DICK K. P. YUE
H. S. CHEN
AND
CHIANG C. MEI
Report No. MITSG 76-10Index No. 76-310-Cbs
ABSTRACT
A HYBRID ELEMENT METHOD FOR CALCULATING
THREE-DIMENSIONAL SCATTERING OF WATER WAVES
DICK K. P. YVZ, H. S. CHEN and C. C. MEI
Under the assumption of a harmonic, small-amplitude incident wave,an effective hybrid finite element method for calculating wave scatteringby and rest.lting forces on a general three-di.mensional body is presented.
The method is applied to four different geometries. Analytic orindependently obtained soI.utions are available for two of the geometries andcomparison indicates perfect agreement. No other solutions are known forthe other two geometric
All numerical results are checked using Optical Theorem, snd convergencetests are performed. A user's manual for the computer program as well assuggested extensions to the program are also included.
While only the scattering probI em is formulated and solved in thepresent work, extension to the solution cf the mathematically similar radia-tion problem is quite straightforward.
The three-dimension-l hybrid element method is a practical one forengineering applicati.or.s in water wave problems, and offers a good alternateto the Green function method, especially for complex geometries.
AC KNCiWLEDGENENT
This is the final repcrt of the work sponsored jointly by the Public
Service Electric and Gas Company, Newark, New Jersey, as part of a study of
a proposed of fshore floating nuclear power facility, and by the National
Oceanic and Atmo pheric Administration as part of the M.I.T. Sea Grant
Program.
The entire work was carried out at the Ralph N. Parsons Laboratory for
Water Resources and Hydrodynamics of the Department. of Civil Engineering at
N.I.T. The material in thi.s report is submitted by Nr. Yue in partial
fulfillment of the recuirements for the degree of Naster of Scierce in Civil
Engineering at N.I.T.
The authors are grateful to Ns. Louise Ketchen and Nary Maison for the
excellent typing.
TABLE CF CONTENTS
~Pa e
TITLE PAGE
ABSTRACT
ACKNOWLEL3GEFENT
TABLE OF CONTENTS
1. INTRODUCTION
1.1 ~Sco s of the Present ~Stud 10
1.2 Mathematical Statement of the. Problem 12
1.3 ~Hbrid Finite Element FormuIation � The Variational 14
Pr iIBc ical
2. THE FINITE ELKMF.NT FORl'fULATION
2.1 The Volume~lute ral I
18
20
262.2 The -urface I~etc ral 12
282. 3 ~Inte rale on S, I ~to I
282. 3.1
2. 3. 2 lot~eral I
32
332.3.4 Integral I<
352.4 Quadrature
2.5 Sv32m3a and th» Total Global Natrix37
403. NUMERIC'. E3 AMPLES
413.1 Vertical Circular C linder in Conata
3.2 Circular~CIIndrical Platform in Constant Wat~er De th
Con..tant ~De th3. 3 Rectan~ul853.4 ~Elli tic Island with a Circular Base
~Pa e
1064. ERROR ESTIMATES AND CONVERGENCE TESTS
1064.1 Th
1074.2 ~Conver ence of the. Finite Element A roximation
4.3 Conv~er ence of Series Terse in tahe Ansi t~ic Pe resentstion 114
1175. COMP1.TATXONAL ASPECTS AN1! THE PROGRAM USAGE
1175.1 Fr~pram ~lm lementati~on ~Esca and Time Re nirements
1195.2 ~The H hrid Finite E1ement P~ro ram
1195.2.1 ~Pro ra~mLo ic
129
52.30t tF 129
1315.2.4 Illost t. E le: ~Pxo ram List~in
5. 2. 4.1 Variable h'.arne Table
5.2.4.2 Sample Input5.2.4.3 Program Listing5e2.4.4 Sample Output
131
134
134
180
1976. CONCLUDING REMARKS
6. 1 SLI~es ted I~mrovements 197
2026.2 Recommendations and Cost Estimates for the Atlantic
Generatin Station Problem
6. 3 Concl usi one 203
5.2.1.1
5.2.1.2
5.2.1.3
5,2.1.4
5.2.1.5
5.2.1..6
5.2.1.7
5.2.1.8
5.2.1.9
5.2.1.10
5.2.1.11
5.2.1.12
5.2.1..13
MAIN
INPUT
ME SHO, ME SH1KIGVALHANKEI,
ASEMKV and EI.KV
FMAP2
SURFA S
GROS SK, PLOAD, CLOAD, DIACKDENSF. and COEFFSOLV and BKSUB'1'
FORCE
OUTPUT
123
123
123
124
124127
1.2 7
127127
128
129129
129
L IS T OF S KF ERENCES
LIST OF FIGURES
LIST OF TABLES
LIST OF SYl%GLS
APPENDIX A: A PROOF OF THE OPTICAI, THFORE1f
~Pa e
205
207
212
213
2I8
1. INTRODUCTION
Along with the technological advances in the design and construction
of large ships and off � shore structures, the need for an efficient and
versatile method for calculating wave forces and other hydrodynamical effects
to supplement laboratory testing is now widely recognized. The present
work is also motivated by such needs in an imaginative project of. ocean
engineering, i.e., the Atlantic Generating Station project. To meet the
increasing demand of electric power and to minimize environmental hazards,
the Public Service Electric and Gas Company of New Jersey proposes to
construct a nuclear power plant at a site three miles offshore from Atlantic
City. Two nuclear power plants of ll50 MeW capacity will rest on two
square platforms floating in a basin protected by two giant breakwaters. To
ensure sufficient supply of cooling water at a low intake flow rate, two
openings of 130 ft. width will be made in the breakwater system which is
roughly 2000 ft. in diameter. For the proper design of mooring systems,
prediction of possible basin and platform oscillation, and the wave forces,
due to incoming sea is needed. This work deals with a modern numerical
method which is potentially useful for the Atlantic Generating station as
well as for other large ocean structures.
In reality, the waves in a severe storm are highly complicated, being
large in amplitude and of a random nature. For theoretical work, it is
often necessary to assume the incident wave amplitude to be small so that
one can work with a linearized approximation. In this limited framework.
useful general relationships among the hydrodynamical quantities can be
deduced; see for example Haskind, 1957, Wehausen, 1971 and Newman, 1976.
For a few very specialized geometries, analytic solutions are also possible
for example, MacGamy and Fuchs, 1952! . For general geometries, however,
a largely numerical method is necessary, the semi-analytic Green function
method being the most popular. By using a Green function of some sort, this
method transforms the boundary value problem into an. integral equation along
a line for two-dimensional, or over a surface for three-dimensional problems,
which usually includes the boundary of the solid structure� By dividing
the intervaI into smalI segments and making piece-vise approximation, the
integral equation is discretized to give an approximate set of algebraic
equations. Solving the algebraic system, the unknown in the interval is
found and the flow quantities in the entire fieId can be subsequently
computed. TIris method is conceptually simple but the Green function or the
kernel of the integral equation is singular, and numerical evaluation of
its integral or derivative is laborious and complicated. The resulting
algebraic equation has a fully popul.ated coefficient matrix with very involved
elements. Furthermore, the computation of flow quantities after the solution
of the integral eouation is found can a]so be quite time consurrring.
The method is consequently expensive, both computationally and in terms of
the analytic preparation required. Nevertheless, this method has been
successfully used by several authors; see for ex rmple Garrison, 3974.
There are also situations where the Green functions are either too
complicated or cannot be easily constructed. Alternative, more efficient
methods are desirable.
We present here a more direct method involving finite elements, The
general. idea is to use discrete finite elements in the region near
complicated bodies and/or bottom surfaces, while some formal analytical
representation is assumed in a region away from them. Similar ideas
have been used long ago in conj unction with finite difference schemes
where there is a singularity in the region of computatio». It is based on
a plausible argument that an analytical solution of reasonably elementary
form which satisfies the differential equation and some of the boundary
conditions, In particular the singularity condition, must provide a more
efficient representation than a simple interpolation function which
satisfies few or none of these conditions. Near solid surfaces of arbitrary
geometry, analytic representation is of ten too cumbersome or impossible;
it is here, however, that the. power of finite elements can be most fully
utilized.
In this hybrid approach, the boundary value problem is recast as a
variational principle which incorporates the matching of the solutions in
the finite element interior region and the analytic exterior region as
natural boundary conditions so that ail the unknown parameters associated
with both regions can be solved simultaneously after discretization. This
is the most important feature of the present hybrid method. Under this
scheme, the finite element region can be greatly confined; the radiation
condition is satisfied exactly; local and global matchi»g are implicit, and
no iteration is required. The total number of unknowns is greatly reduced,
and the coefficient. matrix involved is narroW-banded with quite simple
components.
For two � dimensional problems, such as plane waves incident on an
infinitely long cylinder, a body with rotational symmetry about the vertical
axis, or long waves in shallow water where the vertical dependence isomitted, this method has been applied with success by Chen and Mei, 1974,and Bai and Yeung, 1974, where computational experiments show that the
hybrid element scheme is an attractive alternative to the integral equation
method, especial ly for complex geometries.
For strictly three � dimensional bodies, the only numerical method
that has been successful is the Green function method, and it is valuableto extend the hybrid element method to general three-dirrensional problemsto study its feasibil.ity and ef ficiency. This is the task undertaken inthe present study. Because of the significant increase in computation timeand storage associated with an additional space dimension Black, 1975estimates a typical increase in computation time of the order 100 times forsame body dimensions and wavelength!, it is not clear a ~riori whetherthis method offers a viable alternative ro the Green function method.
'1.1 Scop«of the Present Study
The linerrrized boundary value problem for general three-dimensional
scattering of water waves is stated. By constructing an artificial surfaceto enclose all complex geometries, an equivalent variational statement,
which states that a functional made up of certain integrals involvirrg theunknown flow field is stationary, is given and proved. Based on this
variational principle, and using a vertical circular cylindrical surface. ahybrid element method is formulated, The fluid region inside is discretizedinto a number of hexahedral six-sided block ! isoparametric quadratic
finite elements, while a truncated eigenfunction series solution with
� 10
unknown coefficients is used in the exterior which is assumed to be of
constant depth. Imposing the stationarity of the discretized functional,
a system of linear algebraic equations is obtained, which includes as
unknowns the finite element nodal parameters and the coefficients of the
exterior analytic solution. This resultant system is solved by direct
Gauss elimination.
For illustration, the three-dimensional wave scattering problem is
solved numerically. We point out that the radiation problem~ X e- waves
due to oscillating bodies, is mathematical so close to the scattering problem
that the present method can be adapted with trivial modification. The
following geometries are considered:
�! a vertical circular cylinder in constant water depth,
�! a circular cylindrical platform in water of constant depth,
�! a square platform in constant depth, and
�! an elliptical island with a circular toe.
Results for a range of' wave periods and incidence angles for the cases are
presented in Chapter 3. While all the four geometries involve one or
more vertical planes of symmetry and therefore simpler methods of computation
are available, we have taken no advantage of symmetry and each example is
treated as a general three � dimensional problem. In particular, the first
two geometries possess axisymmetry and can be solved in a two � dimensional
manner, and analytic MacCamy and Fuchs, 1952 for the vertical cylinder! and
semi-analytic Black, Mei and Bray, 197l and Garrett, 1971 for the circular
platform! solutions are available for comparison. To our knowledge,
no other solution exists for the latLer two examples.
1.2 Mathematical Statement of the Problem
The general three-dimensional scattering of water waves in an ocean
is formulated under the following assumptions:
a! incompressible, inviscid, irrotational flow;
b! small wave amplitude;
c! simple harmonic incident waves;
d! impermeable, perfectly reflective solid boundaries.� i'd%
If we define the velocity potential as the real part of 4> = $ e
where a is the angular frequency of the oscill.ations, then $ must satisfy
see Fig. 1.1!:
�.F 1!in V
3Q dBz g
�.2.2!on z = 0 in F!
on all solid boundaries in B! �.2.3!
The scattered wave defined by $ = $ � $ where $ is the incident wave,s
must satisfy the radiation boundary condition:
� 12�
The mathematical background � the boundary value problem and the
variational principle is given in the remainder of this chapter while
Chapter 2 summarizes the hybrid element formulation. Convergence tests
and error estimates are presented in Chapter 4, computational aspects and
program usage are discussed in Chapter 5 where sample computer listingsare also included. Conclusions and suggestions for improving and extending
the program are given in the last chapter.
~=on
, Dqbdn
I ~ I ' -- � v I1
I/ l~ 8e'
Figure 1.2 Introduction of the artificial circular cylindrical boundary,
13�
Figure 1.1 The boundary value pxoblem
yg2$ p !gm
II
I
!
lim Px' � � ik ! I! = 0a
k~ Br o s0
�.2.4!on S
wherek is the wave number associated with t!!. The solid boundary B includes0
all floating or submerged bodies and the sea bottom.
1.3 H brid Finite Element Formulation � the Variational Princi leAssuming that all body and depth irregularities are reasonably cogfj n-
ed, say, within a radius r , we introduce an artificial vertical circular8
cylindrical surface S of that radius so Chat all solid bodies and depthvariations are just enclosed see Fig. 1.2! .
We now show the equivalence of the boundary value problem and thestationarity of the following functional Chen and Mei, 1974!:
2 ~s 3 h' s ~E ] dSJ t!! = I � v ! ' dv � � $' dF + [ � � $ !2 2g 2 s 3r 2 8rU S �.3.1.!
� 14-
For clar ity, we use the ! ' notation to denote quantities outside of Swhile the non-primed variables denote now only quantities associated withthe fluid region inside of S. The first integral is hence a volume integralover the fini.te element fluid volume, V; while the integrals over F and Sare surface integrals, the former only over the free surface in the interiorof S. The total wave potential outside of S, t!' and the scattered potential
are assumed to satisfy Eqts. �.2.1! � �.2. 3! and Eqt. �.2.4!I
respectively. Clearly, if $ also satisfies Eqs. �.2.1! � �.2.3! in V, andif Q and t!' are matched on S such that
on S
aycIr Br
l. 3. 3!
then $ and $' form the complete solution in V and V'
By assuming constant depth in V', it is easily shown that the scattered
potential g ' can be represented by a double series of eigeniunctions:S
Ix cos n0 + p sin n0! cosh k z + h!H k r!�!on on 0 n 0
n=Q
CO CO+ ! <x cos n6 + 8 sin n0! cos K z + h!K K' r! �,3.4!
mn IBn m n III
k tanh k h =�0 0 g
VJtanl<h=
m Ill g�. 3.5!2, 3.
The proof of equivalence goes as follows. Taking the. first variation,
we have
W2hJ g! = V$ 76$!dV � � !ALII dF
g
~s I+ ]ds
2 s dr 2 s cjr ~ 2 3r
15
where h is the constant depth in V', and k, tc, m = 1, 2, 3,
respectively the positive real roots of the equivalent dispersion relations
where use is made of the fact that $ is prescribed i.e., 6$ = 0! . Now
by Green's theorem:
t V'064 dF + ~ 6y dB+ ~ 6g dF + -~ 6y dsJ Bn F S Sr
where n is the unit normal pointing outward from the fluid, so
6J !! = � 7 $6$ dV + � � $�$ dF + 6$ dBRd> 3Bz g Bn
+ I >,� > �g- g- y'! > ]asay', aea'
S
+ -- f64' � �' � 4 ! � �' � 4 ! ]dS1, 3 36 b '2 Br I I 3r
S
The last integral
a6y[6$ ' � � t! ' ]dS
2 s Br 's 3rS
r g26y t 6~ r y2 y t !dye
S U-16-
vanishes by virtue of Eqs. �.2.1! � �.2.4!, whence
5J�! = � V'g 5$ dV + � � y�$ dF + � 6g dB3d> d 3k
Qz g 3n
on account of Eqs. l.2.1! � �.2.3! and the matching conditions �.3.2!,
l. 3. 3! on S. Thus 3 $! is s tationary if and only if $ is a solution to the
boundary value problem in V, $' a solution to the boundary value problem in
V', and the coef ficients of $ ' in Eqt. l. 3.4! are such that f, $' ares
matched on S according to Eqts. �.3.2! and �.3.3! .
It should be noted that only $ must be prescribed and a11 other match�I
ing and boundary conditions are satisfied as natural boundary conditions.
Before we conclude this chapter, a few comments are due. First, it
should be noted that while we have chosen a circular cylindrical surface for
S for simplicity of evaluating the integrals over it, S can in practice take
on quite general say elliptical, spherical etc.! shapes. Secondly, later
investigation see Sections 4.3! indicates that it is almost always more
efficient to choose S as small as possible at the cost of having to use more
series terms in the analytic expression for the solution in V'.
must still enclose all floating or submerged bodies and bottom changes, and
for numerical accuracy, S must allow finite element aspect ratio not to
deviate significantly from unity.
Ihirdly, the analytic expression in V' Eqt. �.3.4!! may in general be
chosen in a number of other ways, say, using a Green function representation.
The latter choice, however, would lead to an integral equation on S, and hence
possesses much of the undesirability of the direct integral equation approach,
� 17
2. THE FINITE ELEMENT FORMJLATION
We rewrite the functional �.3.1! as
> y! = 2 ~4! dV
Q 2-I � p ds'2g
F
I2
+ � <I> � $ asl 2 s 3r s
ds8 fBr s
+ g � $' asI Br s
3r II6
Here the incident wave $ can also be expressed as partial waves
iga ik r cos e � e ! cosh k z + h!0 0 I 0
e cosh k h0
�.1. a!
iga cosh k z + h! Z a i J k r!cos n e � el! �.1.1!cosh k h n n o
o n=O
n 0K ~ 1n
n = l, 2, 3,
where a and 9 are respectively the amplitude and incidence angle of theo I
incident wave, and the Jacobi symbol
is introduced. The scattered wave in the outer region V' is as before
[ x cos n6 + 5 sin n6] cosh k z + h!H k r!�!on on 0 n 0
$ts
n=0
+ Z Z cI cos n6 + 5 sin n6! cos K z + h!K � r! �.2.4!mzl mn Ill n m
m 1 n=0
where k , K , m = 1, 2, ... are defined as in Eq. �.2.5!. For simpli-0 m
city we define
�. 2!m = 1, 2, 3,k m Ill
and k are now the roots of
k tanh k h - � = 0m m g
m=0, l, 2, 3, ~ ~ ~
that for m = 1, 2, 3,
cosh k z + h! = cos Ic z + h!m m
H l! k ! K g ! Z +~!n m n m
l! ' k !,, ! 2 --Pi II+1!n m n m II
Ve can now rewrite, by redefining cx andmn mn
Z Z a cos n6 + 8 sin n6! cosh k z + h!H k r!�!mxl IBIl ll1 n
� 3!
The evaluation of each of the integrals denoted as T., I , ... I6!
are presented in the following sections.
which are real for m = 0 and pure imaginary for m = l, 2, 3, .... Note
2.1 The Volume Inte ral I
I = � V$! dV1 2
If we subdivide the domain V into M finite elements and choose NV
nodal points x in V, i = 1, 2, ...,N» such that the potential in eachelement can be approximated as a linear function of the potential values
eon nodes within or on that element, using superscript ! to denote
e
quantities associated with element e, we have, for an element with Mnodes
�. 1.1. a!
or if we let
1 + M
M
1xM
then
�.1. 1. b!
eHere the interpolation functions N. are in general functions of coordi~i
nates, whether global or local to the element e.
For properly chosen elements and associated interpolations, and. for
sufficiently large M , we can write
20�
"vI 1 Z 2 Vf ! dV
e=l e
e<V
vZ � VtN'! <4 !!' dV'
e=l e
ecV
Z 2 f]I] ! f VfN ! V N ! dV ] $ !e=l e
ecV
z � , {e'! [K'�] f0'!e=l
eeV
1 y!T [K,j f0! �. 1. 2!
where
[K ] = [ J V N ] ]' M ] dV ]V
H x M
aNi aN'. aN. SN'. aNi aN'.v ij
'~+ '~+ � i � 1! dx' dy dz'ax ax ay ay az az
e
is the union of all the j,]t }'s which in general are not mutually exclu-
sive.
21�
ei, j = 1toM �.13!
and in the last equality of Eq. �.1.2! an assemblage is implicit where
e eClearly each [K j is symmetric and real N are real!, and hence
the globally assembled result [K ] is also real and symmetric. Further-
more, with shrewd node-numbering, [K ] can usually be made banded.
In this study, we have chosen isoparametric hexahedral elements of
the "Serendipity" family containing 20 exterior nodes and quadratic
interpolation functions, N , i = 1 to 20 Zienkiewicz et al, 1969! see
Fig. 2. 1!:
i lto8
,, q., . = +1 corner nodes!i' i' i
� E ! � + qq ! � + !i 4 i
i 9 to 12
E = 0- q = +1i i
N = � � + g ! � � 'q ! � + g4.!4 i i
i = 13 to 16
q = 0; F., 4. = +1 midside nodesi i
i = 17 to 20
0; ., q. =+1i ' i' i
wa get
22
�.1.4!
where , rl, form the local coordinates and ,, g., g = 0, +1 are thei i
nodal coordinate values.
By choosing the same functions N to map a local point from the4 space into the global x, y, z space via the nodal coordinates,
20
l2
Figure 2.1 The 'i-node hexahedral element with exterior nodes only in
local coordinates!
Figure 2.2 The caughter arbitrarily oriented, auadratic � sided hexahedral
element in global coordinates after isoparametric mapping
� 23-
ex
[X ! N�.1.5. a!
where
1 2 ' 20
yl y2 ''' y20
1 2 20
lx ! =�.1.5,b!
is th» 3 x 20 matrix of the gLobal coordinates of the nodes and
[N! F -I «! ' N20 Z 0 0! ] �.1. 5. c!
Arbitrarily shaped and oriented hexahedral daughter elements with quadratic
sides can be adopted in region V see Fig. 2.2! . Note that aside from
aspec.t ratio considerations for good numerical convergence, the sides ofthe daughter element need not be of equal length nor the "midside" nodes
be at mid-distance, under this isoparametric mapping.
The choice of the element is made for several reasons. The block shape
allows for convenience of element arrangement and of node numbering. The
element is quadratic so that a much smaller number of them would be requiredto achieve the same accuracy; furthermore, isoparametric mapping results in
much better approximation of boundary surfaces, for example, for quadratic
surfaces, the representation is exact.
Returning to Eq. �.1, 3!, we note that since Eq. �.1.4! cannot be
readily inverted, the integration must be performed in local coordinates.
BN BNi e e e
! dx dy dzB. B,'aN'i BN'. BN'. BN'.
K 1V ij Bx Bx By By
V
�.1.e!
i, j = 1 to 20
wr"l ting
1 1 1
dx dy dz = ~J[ d dn dCeV -1 -1 � l
a
ac
= [J]
-1where I J~ is the determinant and [J] the inverse of the Jacobean matrix
~3an 3F
3z
an az[x, y, z][J] =
a
ar.
a+an ar.
ax
az
Substituting from Eq. �.1.5!
3
an[N![X][J] =
ewhich can be evaluated in terms of E, n, I. and [x ] since N. is expliciti
in E�rr, g as given in Fq. �.1.4! . Since the integrand in Eq. �.1.6! isin general a function of the nodal coordinates of the clement, numerical
quadrature is necessary in the integration.
2.2 The Surface Inte ral 12
I = � - � $ ! dF1 hl
2 2 g
Substituting from Eq. �. l.l.b!, we obtain
~2z~=~ !' - � !I <!!!�!! d!'eE;F
Fe
2Z ~ ! [< � � " > N'! N ! dF ]� !
eEFg
F
z ~! z ]� !eCF
�.2.1!
e eponding only to the nodes on the free surface, i.e., $ c F
For simplicity, we require elements on F to have exactly one face
on F, and N !, <f> ! are 8 x 1 so that
2K ! : � - � ! N ! N !T dF ]g
8x8
F
2K, ~ � � NN dF
ijg ij1 to 8 �. 2. 2!
� 26�
e e e ewhere N !, $ !are the subvectors of N ?, $ ! containing entries corres-
eE F are involved. Note also that as in I , [K ] and [K ] are real and
e esymmetric, and since I.g ! C k$ ], [K ] always has a bandwidth less thanF
or equal to that of [K ].V
eTo evaluate I and in particular K , we perform the integration
l ij
in local coordinates
l 12
F ij g NN. ~J I! dEdq
i j p =1�. 2. 3!
-I -1
where the Jacobian is given by
which is readily evaluated in terms of local coordinates and [X ] by using
Eqs. �.1.5!.
The resulting integral is again evaluated by quadrature.
27
Again assemblage is implied in the last equality of Eq. �.2.1! where only
2.3 Inta rale on 3 I ~eo I
For computation the series �. leb! and �.3! are truncated as
follows:
Niga cosh k z + h! o
Z c i J k r! cos n B � 6 ! �.3.1!I 0! cosh k h n n o E0 n=0
and
M Nm l!Z Z u cos nG + 8 sin nB! cosh k z+ h! H k r!s mn mn m n m
m=0 n=0�. 3. 2!
where N,, i = 0, l, ..., M, and M are chosen depending on wave period andi
geometry.
2.3.l
T = � $ � $ dS1, 82 s 3r s
S0 2m
dz dB r
0
M Nm
K Z �, cos n6+ P sinmn mn
m=0 n=0
M N
Z Z � cos nB + 8 sinmn mn
m=O n=0
nB!cosh k z + h!H ' k r !k ! l!,
28�
r
K2 R=O
M N
Z Z � x u +kn mn
m On=0 n
nB! cosh k z + h!H k r !! l!m n m S
0
r cosh k< z + h! cosh k z + h! dz� h
Mr m 27rk sin.h 2k h
2 E mn mn n m S n m S 4kZ Z G +8 !H kr!H ' kr! + !
2m=0 n=O n m
lf we define a coefficient vector
= I:00' 01' 01' 02' 02' ON ' ON
0 0
IO' ll' l1' 12' 12' 1N , 1N1.' 1
MO' Ml' Ml' M2' M2' ' ' ' ' MN ' MNM
1 " N say! �.3.3..1!
= 1 Iej~ [V ]<V! �.3.1.2!
where [K ] N ~ N is diagonal and
K = 2~ H k r !H ' k rS! sinh 2k h+ 2k h!�! �!,n mS n mS m m
i n
i= 1 toN
where the i-th element of �! is < or 8 . Clearly by our notationmn mn
in Eq. �.3!, K are complex for < ~ 2N � 1, and real otherwise.
29
noting that all the terms associated with B. , i = 0 to M are zero and soi0'
are excluded, then we can writeiO
2.3.2
Nm
N ! {f ! Z Z Q cos n6 + 5 sin n6!cosh k z+ h!eES m=0 n=0
S
� >, ...��.n mS' m
where we have substituted from Zqs. �.l.l.b! and �.3.2! and N !, $ !are subvectors of N !, ff ! where only entries corresponding to $ E Se e e
are included.
Again in this formulation, we require, for simplicity, elements on Se etG have exactly 1 face on S and {N !, $ ! are vectors of dimension 8.
Me now write, using the def inition �. 3. l. 1!
ehere [K ] is defined as the 8 < N matrix withC
K .. = � N. cosh k z + h!H ' k r !k ! dSij i m n m S msinn6S
To perform the integration in local coordinates, we write
30
where the j-th element of
is indicated depending on
gf s
g! is ~ !, and a choice of cos n6 or sin n6j, since ip! is defined to include both 0's and
2.3. 3
Ij' dS5 I br s
0 2m iga cosh k z + h! oN
d6 -! hkh Zc i J k Sc«n BI d cosh k h n n o So n=0� h
M
cos n6 + p sin nB!cosh k z + h!H ' k rs!k ! I! IIILn m n m S m Z
m=O
iga0 2 tt cosh k z+h! o
N
- � !r dz i d60 0 .nS J coshkh On n o S
Z c i J k r ! cos n6 cos neI
-h 0
+ sin nB sin n6 !! ~
M N
. g g ~ cos n6 + g sin n6!cosh k z + h!H ' k rS!k ! I!,mn mn m n m S m
m=O n=O
21' Niga 0 !r p F i J k rS! cos nB cos nBI + sin n6 sin nBI!!
s n n o S0
n=O
N
~ u cos n6 + g sin n6!cosh k z + h!H ' k rS!! I!,on on 0 n o S
n=0
sinh 2k h + 2k h!0 0
4 cosh k h0
iga
!
sinh 2k h + 2k h!0 0
2 coshkh0
32�
0
N
K umn
N0
Z a coson
n=O
nB + 8 sin nBI!i J k r !H ' k rs!vrn I!,I on I n o S n o S
Niga o
- � ! w r ! Z Q cos n8 + 8 sin n8 ! d S on on I
n=0
i J k rS H ' k rS,n �!,
sinh 2k h. + 2k h!2 cosh k h 0 0
0
l> = � jq ! >!; �.3. 3. l!
here Q ! is defined as the �N � l! by l vector with0
iga. sinh 2k h + 2k h! -" 'S! " ' '" 'S'"'k 'S' 2
i 0
cas n8
s~ ne ' �. 3. 3. 2!
0onwhere the i-th element of p! is ~ !-on
2. 3.4
3
6 ar
Substituting $ from Eq. �. l.a!,
iga ik rS cos {8 � 8 !j $ ! - ! ik cos 8 � 8 !e
6 e e m o IecS
S
cosh k z + h!ds
cosh k h0
33�
where the orthogonality properties in z and 8 are invoked. Now we write
K {q ! <> jeCS
�. 3. 4. 1!
where we have defined the 8 x 1 vector {q! ao thate
e e 0ik r cos� � 9 !
N. - !ik cos� � 8 !e0 I
i ld o Ii
S
cosh k z + h!0
ds
cosh k h0
ik x cos 6 + y sin. 6 !e cosh k z + h! IJSI! dg d4 �.3.4 2!
0 S q=l
which after expressing x, y, z in terms of local coordinates can be
integrated numerically. Assembling {Q !'s for all e E S, we obtain
6 = � {q j {4! �. 3. 4. 3!
where now {Q 3 is an N >< 1 complex vector.y S
34�
cosh k h0
iga - �!ik
1 1
N � cos 8 +~sin 6 !i r I r I
-1 � 1 S
2.4
Three different Gauss-Legendre quadrature formulas Kopal, 1955;
Abramowitz and Stegun, 1964! are used in the evaluation of the integrals
Il' I2' 14 and I6
�! for I a 21 point formula is used for the cubic volume:
f C, V, C!d dV dC
[-496f +128Z f +BZf +5Z f ]+0 R!m r f v
�.4. 1!
where f = f�, 0, 0!
Z f = sum of values of f at the 6 points midway from the center tor
the 6 faces.
Z f = sum of values of f at the 6 centers of the faces.
Z f = sum of values of f at the 8 vertices.v
r 1 1 3 3
f E, q!dg dq = Z Z .. f g.,q!i=1 j=l-1 -1
�.4.2!
where
W.i
.8888889
.5555556+ 0. 77459671,3
35
�! for I and I , the evaluation over the square surface is approximated
at 9 integration points:
�! for I, we note the presence of rapidly varying functions sin nB and
cos na in the. integrand and adopted a more accurate 16 point formula
for the square surface area:
r 1 1 4 4
f g!dF dr! = 2 Z w w. f E, ~ !i=1 g 1
-1 -1
�.4. 3!
where
E,qi
1,4
2 $3
-36-
+. 3478548
+.6521452
~ 7745967
.3399810
2. 5 Summar and the Total Global Matrix
becomes
J {�!, P!! = 2 �] [K ]�]
+ � �] [K ] y!
+ � u] [K ] p]
�!
�!
+ � p! [K ] �]- + � �] [Kc] 'p! �!
{5!
�. 5. 1!
For J to be stationary, we require
i 1 I ~ ~ N
and
�.5.2.a,b!i ly ~ ~ ~ ! NT
which gives
[K ] C! + [K,]�! + [K,] W] = < ! �.5 3.a!
and
[K j p! + [K<j�! = <Q<> �. 5. 3. b!
37�
Summarizing and treating P! as unknown also, the functional {2.0!
H we define
y}' = [ ~}', >}'] lx N +N!V T
�.5.4.a!
of g} to all g for nodes on S, i.e.
y.~s} , p} ] �.5.4.b!
and assign the last positions
4]' = «e, ly.~s}', g.l
then iEqs. �.4.3.a,b! combine to give a system of linear equations
[K] q} = Q}
whexe
Q = [ Qp , Q< lx N +N! �.5. 5!
and the total global matrix [K], N + N ! x N + N ! has the following
Nstructure:
N
N
�. 5. 6!
We note that [K ] + [K ] is real symmetric and banded and [K ] is diagonal
and complex while [K ] is full and complex; they are positioned as shown in
~2.5. 6!. [K] is consequently complex and symmetric which is partly a
xesult of the rearrangement in Eq. �.3.2.1!.
Having obtained [K], Q} can then be solved by inversion, or formally�} = [K] q} � 5 >!
38�
From Eq. �.5.3.b!
{V! = [K,] { , � [ ,] {e! �.5.8!
which when subs t i tuted into Eq. �.5. 3.a! gives
[q] + [KF] � [KG] [KD] [KC ~ {~P! [KC] [KD!
or
[K ] {V! = {q > �. 5.9 .a, b!
where
[K'] : [ ! + [~ ] � [K ] [K ] l G]
and
{a'! = {0,! � [KG] [K ] {q
Clearly, [K' ] is still symmetric and banded, and complex, and only the
symmetric half band of [K'] is stored.
A direct method Gauss elimination! is used for solution.
-39-
and the nodal potentials in V as well as the coefficients of the expansion
in V' are solved. For efficiency the total global matrix is condensed to
eliminate the unknowns {p!, which can be recovered by substitution after
is solved.
3. NUMERICAL EXAMPLES
Three-dimensional diffraction by four different geometries are treat-
ed and presented in the following sections. For the first case, the analytic
solution is known, while for the second, accurate numerical results are
available by other techniques. They are therefore useful for testing the
pre'sent me tbc~d. The 1 as t twc~ examples have no other known so 1 u t iona, and
are chosen to illustrate the versatility and efficiency of our method,
while for simplicity of grid generation, they are still chosen to have
relatively elementary geometries.
A3 I thc finite. «3 c ment grid systems c>r«gc.nera ted by computer with
automa L ic ally min imiac. d bandwid ths-
Figure 3. I Wave incident on a bottom-seated circular cylinder of radius
� 40�
3.1 Vertical l'ircvlar Cylinder in Constant Water De th~ Fi . 3.1!
The cylinder has radius a and is centered along the z-axis. The water
depth is everywhere constant h. For the case of h = a, we use a fini te
element grid consisting of 36 elements � two layers of ]8 elements each
stacki d vertically forming a ring surrounding the cylinder. See Fig. 3.2! .
There are a total of 342 nodes, and the average element dimension, L, ise'
about .48a. The incidence angle is taken as B = 'I .I
In the. analytical theory for this two-dimensional geometry, only the
coefficients corresponding to the first eigenvalue k are involved and all
other coefficients vanish. For generality and for an indication of the
accuracy however, 5 eigenvalues are chosen N = 4; N = 12; N = N = Ns o ' 1 2 3
N = 10!, which corresponds to a total of 99 terms in the exterior analy-
tic representation Eq. �.3.2!! .The following quantities are presented'.
�! coefficients of the series solution in the far field:
�! normalized free surface elevation on the cylinder r = a!: p/a 0
where t! a,0! = �.� $ a,0,0!; and,i 4!
g
�! horizltntal force coefficient,
We define dimcnsionless force and moment coefficients by
F.
pgv a Ha,
and �.l.l.a,b!3 = K y Z
� Z=Q
CIZ=�
2
3G� Z =-4
Z
r =1.5o IOpe
Figure 3,2 Finite Element Structure for a Uniform Cylinder
36 elements, 342 nodal points.42
F = Re[ F.! e ] , 8 = Re[ M ! ei
For example, in this case, we have
0 2zsos 0 d9 dz
vh igao
FC x
x Ogler aha�.1.2!
� h 0 - � !4>
which is evaluated by numerical integration of the so1 ution
Comparisons are made to known analytic results see, for example,
MacCamy and Fuchs, 1952!
J' k a! cos ne.n n o-c i , - . ! for m = 0n H l!' k cosh k h sin no
k a! 0n o �.1. 3!
form40
3' k a! !i cos n U � 0 ! [J k r! � � � � �,� � H k r! jI n o l!' k n o
k a!n o
tIa n
o n=0
�. l. 4!
and
4 cos 9 tanb khC I o
F �!'H » k a!1 o
�.1. 5!
43�
where H, a are the typical draft and horizontal dimension of the body
respectively. F. refers to the force component in the i direction, and M.i
the moment component about the i-axis, i. e.
N
NCc0
O
O
N QNO
CO O
O cClO O
00 cA
ChVl
PlO
CCCV
O
00N Ng0Q CCl
O
O
CG Pl
00~N CP
0c
NCV 00NI/l EA
NI I
O N
N'G0
80
IJ
0 0
accl
0ch
I!
A U
EO
II II
CIcd
I I I
O
O j ON
00
Q4J4JVV!
O cd4J
cdQI
dc6
dJ
'cdC cd
V cdC C dcdI
dICCI
'4 cdc0
cd
C C OJJcd'0dI
dcV dI
dI
C dI
V 0 CLI4 0
� 44�
IO IO O O'K
0r
X
«ir
ICr IOrri0O
0 E
0 0 «I~ I0
CVCr
PlIO IO
Cr
E 0'riC
IOIO cr
rrJ0SSC0JJrg0
I Cl
0 O '4 0W~ J
4I
0C'«4«0
«J0«
0IOI CI I Cr
4 0r«J
Cl«0«' i S
0
IO I ClIa«0IO «S«J
0 4 00rO
O I SlJJ0zrCl
45
plI I w 0a a o
I Io aX ««r« IIO Pl«0
lI IO CI
IJrJ
IIil0 0
0 IS
0 0 4 rS«l.E 0«J
0 0~ J
I 0 IrSr0
For 9 = m s the comparison for a range of wave1.engths k a for C and0 X
are presented in Table 3.1. It is evident that the results are quiten
0
satisfactory <1%! for up to k a ~ 4- This implies a maximum k L ! of0 oe max
about 1.9, A . > 3.3 L ! . In Table 3,2, a sample of the coefficientsItin e
calculated for the case k a = l is listed for comparison, Note that their0
contribution to f' decreases rapidly though the magnitude of the coeff'is
cients themselves are not necessarily small! due to the magnitude of the
Besse1 f unc t iona.
See also Section 4.2 for summary of present and other results.
3. 2 Circular C~tinctric >1 Platters in Constant Water~De thWe consider a fixed dock in the torm of a circular cylinder with its
axis vertical. The radius of the cylinder is designated by a and thedraf t by H. The water depth h is constant. Specifically we have chosenH = . 5a and h = . 75a see Fig. 3. 3!,
The f inite element grid system now has a total of 56 elements tworings of 16 elements each stacked vertically outside the dock and 24 underit! and 435 nodes see 1'ig. 3.4!, Note in particular the use of the hexa-hedral elements in the center core, and the choice of greater node densitynear sharp edges. The average element dimension for this example is L 4.36a.e
Again, 5 eigenvalues and a total of 99 terms N = 4; N = 12, N = N = N0
10! are used in the outer series solution Rq. �.3.2!!, and we chooseThe vertical and horizontal force coefficients are now respectively
I
Figure 3. 3 Nave incident on a floating cylindrical dock of radius
a e = ~; l = .75 a!
z=0
� z =-,30
z =-,5Q
� z = .625a
� z =-75a
r/a =0 .25 .5,8 1 1.15 1.3
36 eler:.ents,i;ure .4 Finite element structure for a floating dock:435 nodal points
a 27t ~ ! dO dr
0 0
FC, z
z pgvraHa0
�.2 l!
and
0 2m
! cos 0 d6 dzmH iga
0� H 0
FC x
x jglTaHa0
� 2.2!
A table of these coefficients for a range of k a and their plot in0
comparison to those obtained by Garrett, l971, is presented in Table 3.3
and Fig. 3.5.a,b. Tlute results are clearly satisfactory and discrepancies
are imperceptib1 e on th» graphs for the entire range we considered. A
complet» 1 ist of nodal potential and series coefficient solutions for k a = 10
is also given as a part of the sample computer output in Section 5. 2.4.4.
Tabl» 3.3 Computed horizontal and vertical force coefficients
for a semi-immersed circular cylinder
-- = .7S, � = .S!h Ha ' a
1.0
0.8
0.
O.
3
koa
Figure 3.5 a! Horizontal force coefficient for a semi-immersed cylinderh, H � =,75; � = .5 B = ml.a ' ' a I
+, amplitude; X, phase: f inite element solution
amplitude: � � � phase. Garrett, 1971 results
50
2.0
1,6
14
1.2
O.
0,
Fig~re 3. 5 b! Vertical force coefficient for a semi-iauoereed cylinder See description in Figure 3.5 a!!
51
3. 3 Rect~an nla~rBar e in Constan~t. De th
We now consider di ffraction by a fixed semi-immersed draft H! rectan--
gular box in constant water depth h. Unlik» the previous two examples, this
problem must be treated zs a fully three-dimensional one. We choose a
special case. of a square barge, length 2a, H = .5a and h = a, and consider
two dif ferent incidence angles 6 = vr and 5m/4,I
The finite clement grid employed is similar to the one we use for the
circular dock and has 56 elements and 435 nodes. The previouts choice of
the number of series terms is still kept.
Tfte force and mooient coefficients for both angles of incidencc over a
range of wave number k a are plotted in Figs. 3.6.a,b,c and Figs. 3.7.a,b,c.
In general, far a given grid, the firiitc element approximation becomes
more accurate as the w-velength relative to it increases k a decreases! .0
On acccunt of the comparisons in the last section for a cylindrical dock
where a similar grid system i.s used, we expect the present results to be
rea.onably reliable at least for k a up to ~ 6, and certainly far beyond the0
regions of peak values around k a = l.0
In Pigs. 3.8.a,b,c,d and Figs. 3.9.a,b,c,d, the run-up, q/a, on the0
sides of this sqtare barge for k a = l,2,3,4 and both incidence angles are0
presented. Note that this local value becomes quite highIy varying ss the
wavelength decreases to about the body length k a = 4!, and the interpola-
tion between the nodal values in those instances are somewhat unreliable.
Furthermore, die to the finite element smoothing, both the value of ~rI/a0
which is proportional to the potential! and its derivative are finite at
the sharp edges, while according to two-dimensional potential flow around
52
-iso present.
For a measzre of the argular intensity cf scattered wave energy, we
define the differential scattering cross-section IA H!I , where A 8! is th»
angular variation of the faz-field scattered wave, i.e.
� iga ikor� e cosh k z+h! A 8!
s 4Q v 'lT ovk r
0
�.3.1! k r»l!0
and if we substitute in this asymptotic form of $ into Eq. �.3.4!, wes
have
CO igaA 8! = 2 -i! u cos n8 + P sin n8! / � !
n=0 on on�-3 2!
The differential scattering cross-section, ~A 8! ~, as a function of 8 for
k a = 1,2,3,4 and the two inciderce angles are given in Figs. 3.10.a,b,c,d0
ard Figs. 3.11.a,b,c,d; ar d in polar coordinates in Figs. 3.12.a,b,c,d, i!,
d. ii! and in Figs. 3.13.a,b,c,d for a better physical picture.
53
-1/3a 90 wedge, the gradient of the. potential exhibits a r type singularity.
This local discrepency can in principle be corrected see Chen and Hei, 1974!
by constructing special analytic elements similar to S around all sharp
edges and corners, Such a procedure is however, unnece sary if one is only
interested in averaged g1oba] ! results such as forces and moments, or in
local values in the major region not close to singularities. This is
evidenced by comparisons for the circular dock where a bottom sharp edge is
IIH
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a.ka=l0
76�
Figure 3.12 b! Polar Plot of Differential Scattering Cross-section,
~A 8! !, for a Square Barge. Normal Incideace: 6b. ka=2
o
77
Figure 3.12 c! Polar Plot of Differential Scattering Cross- ection,iA 8! !, for a Square Barge. Normal Incidence: 8
c.ka=30
� 78�
Figure 3. I2 d! . i! Polar Plot of Di f ferential Scattering Cross-section,
!A�! ~, for a Square Barge. Normal Incidence: 8Id.i k a = 4
0
79
Figure 3.12 d! . ii! Polar Plot of Differential Scattering Cross-section, A 8! !, for a Square Barge. Normal Incidence: 8
d.ii ka= 40
7'; 3Tr 5m 71TDetails for � ' < 8 < � and � < 8 < �,4 4 4 4
Figure 3.13 a! Polar Plot of Differential Scattering Cross-section,IA 8! I, for a Square Barge. Oblique Incidence: 8 = 5'/4.
a. ka=lo
Pigure ..13 b! Polar Plot of Differential Scattering Cross-section,~4�! I, for a Square Barge. Oblique Incidence: 6 = 5w/4.
b.%a=20
82
Figure 3.l3 c! Polar Plot of Differential Scattering Cross-section,
IA 9! I, for a Square Barge. Oblique Ivciden-.e: 9 = Sm/4.c, k a 3
0
Figure 3.13 d! Polar Plot of Differential Scattering Cross-section,~A 9!I , for a Square Barge. Oblique Incidence: 8 = 5x/4.
d. ka=40
84�
3.4 Elliptic Island with a Circular Base
The scattering of a plare wave by an island has been of theoretical
and practic;1 interest in recent years. Lonpuet-Higgins, 1967 showed the
existence oF trapped wave . around circular islands, while Vastano ard Reid,
1966, 1967, Lautenbacher, 1970, Smith and Sprinks, 1975 and 3orsscn,
Skovgaard, Kjaer �975! studied the tsunami run-up os various islands of
somewhat specialized geometries. All existing treatments require either
an assumption allowing linear shallow water long wave! theory vhich ignores
vertical variations, or that the ocean bottom i s slowly v"rying mild sj ope
theory! . It is of interest to pcint out that owing to both bottom refrac-
tion and posribi e resonance of the. trapped medea, the maximum run-up on an
island can often be very 1 arge.
In this last example, we ccnsider the diffraction by ar: island th-t
extends from the tree surface in the form cf a vertical elliptic cy1inder
up to a depth H, then slope out "ccnically" to form a circular toe on an
ocean bottom of constant depth h . The geom try and finite element grid
�6 elements � two rirjgs of 18 each, and 342 nodes! are shown in Fig. 3.14.
For an «pper e}liptic cylinder of m-.jor anc'. minor axes A, B anc' length
stbmergenc e! H, i.e. from z = 0 to � 8!, the lover "conical" portion is
generated ly the parameters ! and 8:
x = t XA + � � A!R] cos 8
�.4.l.a,b,c!
y = f XB + � � A!R] sin 0
and
z+ h = A h � H!
-S5-
where R is the radius of the. circular toe at the sea bc ttom z = � h! . Cut
planes normal to the z-axis - h < z < � H! yield elliptic cut sections
given by
2 2
[X A-R! + R]z [A B-R! + P.]'�.4. 2!
with eccentricity. e, varying fram J B e to 0:s I v-ries from iA
z= � H! to 0 z= � h!:
A- B!!A+ L-X!R
�.4. 3!
As a particular example, we choose A = a, B = .5a, H = .5a, h = a and
R = 1,5a as in Fig. 3.14! . The "conical" beach now varies from a width c f
a wi th a 1 on 2 slope at 8 = m/2, 3Ti/2 tn a 1 on 1 beach of width G.5a at
8 = 0,1!. Three incider t wave-numbers k a = l,2,3! at two angles cf
incidence 8 = vr and 3m/2! are considered.
The run-up, q/a, on the island z = C. x = A ccs8, y = B sin 8! are
comparisons
-86-
plotted versus azimuth 8! in Figs. 3.15.a,b,c and Figs. 3.16.a,b,c.
Due to the rather shcrrt and steep beach, the run-up dces not exceed 2 for
8 = m'or ~2 5 for 8 = 3z/2.I I
Again, plots of !A 8! ~ as a function of 8 are given in Figs.
3.li.a,b,c anc' Figs. 3.18.a,b,c, and in polar coordinates in Figs. 3.19.a,b,c
and Figs. 3.20,a,b,c,
Unfortunately, our preser t geometry does nc t resemble suf ficiently
any of tho.e studied by other investigators to justify quantitative
Figure 3.14 Geometry an2 Finite-element Structure for an Elliptical
Island with a Circular Base. Major and N.nor axes: A = a, B = .5a;
"draft" of elliptic cylinder: H = .5a, radius of circular toe:
R = 1.5a; water depth: h = a.
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� 99�
Figure 3.19 a! Polar Plot of Differential Scattering Cross-section,IA�! ~, for an Elliptic Island with a Circular Base. WaveIncident from z ~ + ~. e = T!.I
a. ka=10
� 100�
Figure 3.19 b! Polar Plot of Differential Scattering Cross-section,
~A 9! !, for sn Elliptic Island with a Circular Base. Wave2
Incident from x ~ + ~. 9 = vr! .
b.ka=20
101
Figure 3.19 c! Polar Plot of Differential Scattering Cross-section,IA�! I, for an Elliptic Island with a Circular Base. Wave Incidentfrom x %+'-. � = 7I!.
Ic. ka=3
o
102-
Figure 3.2G a! Polar Plot of Differential Scattering Cross � section,
~A�! , for an Eliiptic Island vith a Circu].ar Base. Wave Incidentfrom y ~ + ~. B = 3'/2! .
I
a. ka=l0
103�
Figure 3.20 b! Polar Plot of Differential Scattering Cross-section,~A O!~ , for an Flliptic Island with a Circular Base. Wave Incidentfrom y s + ~. � = 3'/2! .
!b.ka=2
o
104
Figure 3.20 c! Polar Plot of Differential Scattering Cross � section,
~A B! ~, for an Elliptic Island with a Circular Base. Nave Incidentfrom y '4 + ~. � = 3vr/2! .
Ic.ka=3
o
105
4. ERROR ESTIMATES AND CONVERGENCE TESTS
4. 1 Th e ~0 tie a! Theo rem
Whi le a number of general relationships governing the scattering andradiat. ion of ocean structures Haskind, 19 57; Mehausen, 1971; Newman, 19 76!are availabl e, there is just one relationship that relates the far-field
1t is a statement of global conservation of wavescattered wave only.
energy «nd is well known in quantum physics as the Optical Theorem. Forscattering of. a plane wave by a two-dimensional body, it is the iamiliar
�, 1. 1!
For thr~ e-dimensional scattering, it can be stated, as
277 ! A�! f d0 = � � � � -- RetA� ! ]2TI cosh k h
�. 1. 2!
0
where h is the cons tan t depth at inf inity and A 0! is the angular variationof the far-f ie. ld s< at tered wave, as give n in Eqs. �. 3. 1! and �. 3. 2! ~
106
where R and T are respectively the reflection and transmission coeffi ients.
The integral on the left-h;.nd � side is proportiona3 to what is common]y
known as the total scattering cross-sectior in physics, while th~ expression
on the. right of the equality is a measure of the wave energy in the forward
direction, i.e. the- direction of the incident wave. There are several
slightly different proofs cf Eq. �.3..2! all based or~ Green's theorem, a
rather direct. one using the method of statiorary phase is given in Appendix
We define a norma]ized error measure based on this theorem;
L.H.S. � R.H.S
OPT L.H ~ S ~
where L.H.S. and F.H.S. refer to the left and right � hand side respectively
of Eq. �.1.2! .
-6 -21'or all our numerical examples, E ranged from ~ 10 tc ~ 10
depending on wave-number, fini.te element grid fineness ard number oi serie,
terms used in the "nalytic expression in the exterior. As an example,
in Fig. 4.1, we plot EOPT over a range of k a for the particular case of0
the square dock in Section 3.3 under obliqueiy incident. waves � = 5%/4! .
While E > is useful as a global error estimate, its expression involves
only the propagating modes directly, and it is possible for one to make
some small error in an evanescent mode without causing an equal discrepancy
in E . Hence, it is also necessary and useful to study some localOPT '
quantities sr ch as integrated forces on a body or a local pressure to
estimate the z c teal error.
4. 2 Conver ence of the Finite Element A rozimation
Once a geometry is given, an appropriate choice of finite element grid
size and nlunber of series terms must be made. Iv general, elements should
107
IO
Figure 4.1
Obliquely Incident Waves 9> = 5'll'/4!.
108
IO
8 6
IO
8 6
IO'86
EopT 4
IO
8 64
0-586
4
Relative Error in Optical Theorem for the Square Dock under
-210 8 6
IQ
IFI io 0koa
Figure 4.2 Relative Error in the Magnitude of the HorizontalForce on a Uniform Cylinder.
110�
IO 86
IO8
EIqI84
IO 8 6 Eigure 4. 3 Relative Error in the liagnitude of the Run � up on aUniform Cylinder at; + : 0 = 0; X.: I3 = -/2; o: 9 = Tt. � = Tt!
k I ~ 2.9 A. < 2.2 L ! for an integrated quantityo 'e min ' e
ma xdimension L
emax
such as forces and k L < 1.9 X , % 3.3 L ! for a local quantity sucho e ' min ' e
as pressure or wave height.
In Tab3e 4.1, we tabuIate the results for a given k a = 1 for several0
diff erert grid systems note that p x q means a finite element structure of
p vertica3. layers of elements arid q radial elements per layer! . Here we
define relative error in the total scattering cross-section by
2Tt 27r
I<«! I' «!,��,0 0
S
0
k a = 1! even though for the present geometry, only the first verticaln
cosh k z + h!mode is present g x, y, z! = 4 x, y, o! 0
cosh k h0
using a sufficient. number of elements in the vertical dimension should not
The importance of
be overlooked for a general prob3.em,
112
The results are quite satisfactory even when the number of radial elements
are reduced by about haI f � x 10!, but a few percent error is introduced
when arly 1 layer of elements is used � x 18! . This is somewhat surprising
since. the average element dimension in the radial direction is about .8a for
� x 30! grid whi.ch is not much smaller than a vertic" 1 dimension of a for
the � x 18! grid. The implication is that the vertical length scale of the
velocity is smaller or at best equal to the horizontal scale for this c.ase
O
JIQl
Q
n Q
O, O
Y QQ'Q QQ Q C
I I I IC C I C C
Xr rvO a
I
C C; C
j
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8 4 Q
Q
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4.3 of Series Terms in t esentation
� !
!o
and
where ! refers to a quantity computed with the original choice of numberD
of series terms and ! refers to that computed with the present set.P
114
For numerical computations, it is necessary to truncate the doubly
infinite series in Eq. �. 3.4! to the finite sum in Eq. �.3.2! . Unlike
many other "hybrid" finite element methods, the present formulation reducesmatching conditions to ratural boundary conditions and an arbitrary numberof vertical eigenfunctions M! as well as the angular eigenfunctions N !for each vertical mode is permitted. The number of such terms that is
required for a giver. accuracy depends on the incident wavelength, the: bodyand bottom geometry and depth, and more particularly on the size of theartificial boundary S and the finite element grid structure inside S. To
provide some guidelines on the number of series terms that is necessary fora given problem, we solve the particular example of the square barge inSection 3.3 length 2a, draft .5a, depth a! under a normally incident
9 = 'tt! wave. of unit. waienimLber k a = 1! for severaI sets of series terms.I 0
We start from the origin 1 choice M = 4 N =- 12 N = N = N = N = 10,o ' 1 2 3 4
total 99 terms! and reduce, first M,keeping N fixed, then reduce N. keepingi
8 fixed. The results cf a set of such convergence tests are presented in
Table 4.2. Note that
~ ItlIII
~ IIt4tt
C
It40CI
QIltI0
C
IIII
II:I
U:II
ll.'I4I IICI tJ 115
While the present results are not unrelated to the incident wave, the
particular geometry and element grid, they do indicate that the original
set using 99 series terms is extremely conservative with error in both local
and integrated quantities less than % lQ ' . Et is quite amazing to ncte for5
example that the solution converged to within much less than IX even when
only a total of l2 ternis Y. = 1, N = 4, N = 3! are used.
This confirms the earlier observation that it is almost always
numerically advantageous to use a smaller S, and hence reduce the number
of fini te element nodes, even at the cost of having to include more
unkncwns in the exterior series representation, since the total number of
series terms required is much smaller than the semi-bandwidth of the finite
element. grid for most problems.
It should be noted that a complete study of ccnvergence of the series
terms would recuire a systematic set of tests similar to the one performed,
for a range of incident. wave lengths, depth., ance body dimensions or
geometry!, as well as the size of S and of the fini.te elements. While
such extensive tests are beyond the scope of this work, this and especially
the last section on convergence of the fivite elements should already
provide the basic guidelines to the choice of number of series terms, and
the element size to use for a given problem.
5. COMPUTATIONAL ASPECTS AND THE PROGRAM USAGE
S.i ~Pro ram T~mlementatt'o~n, S ate and Time Se nirementa
The program is impEemented on an IBM 370/168 computer running in an
OS/MVT environment at the M.I. T. Information Processing Center IPC!-
The program is written in IBM FORTRAN Gl language extensive testing also
done using a WATFIV compi]er! . Three routines BJS, BYS, BKS! to calculate
Bessel and related functions are called from the IBM SL � MATH library and an
assembly language program to set consecutive array elements to zero, ERASE
not WATFIV compatible! supported and provided for unrestricted use by M.I.T.
IPC! is also called.
The FORTRAN program including comment statements! is 1308 cards long
and requires about 78 K � K = 1024 s torage locations bytes! of 8 bits each!
ob jec t code.
The total primary core storage requirements object code and arrays!
and the CPU processing time required for each of the examples in Chapter 3
and 4 are summari zed in Tab E e 5. l. Single precision accuracy � bytes,
equivalent to approximately 7.2 decimal digits accuracy! is adopted through-
out, and no s< condary stv>rage is used. The timing, which includes mesh
generation, but not input/output operations or program compilation, is
calculated from cperating system cost summaries and not from dirert timing
routines. It is hence onEy an approximate estimate, and in light of the
findings in Chapter 4, likely to bc on the conservative side since an
extravagent numher of coefficients and con..ervative finite element grids
are always used.
EE7
N
W 0
'Cd0 0
O 0Vl
ll8
4JCd0 4J
OJdd
Cd
C 0 F-CdCJ
5.2 The Hybrid Fini te Element P~ro ram
5 . 2 . 1 P r a~ram L otic
Thr total program consists of a main program and l9 subroutines.
The program is general with regard to geomet ry, < ttoice of grids or
co« fficients, or in ident wave period or angle, with the exception of
the fol lowing routines
l! MAIN, the main program which must be slightly modified to reflect
corre~ t array sizes;
�! 1NPU'1', the input p~ogram whi «h mus t be changed for any changes in
geometry or grid st.rue ture;
:3! FORC'E, this routine to calculate force and moment. coef ficients
is dependent on the partic.ular body geometry; and
minor modi fications for oth~ r geometries are veryexamp1es Lr~ ated.
advisablu.
The overal1 program logic is best illustrated in Fig. 5.1 hy the
flow of < on tro1 among the subprograms
119
�! HESHO, HESH1. While t hese mesh generation routines are quite genera1,
they are particularly suitable for cylindrical geometric as in the
initialize constants and I/O device numbers
set up and print geometrical and grid
data
loop over each frequency point
input and print physical data
assemble [K�] corresponding to the volume integral
assemble into [K ] the matrix [K ] resulting from
the integral over the free surface
calculate all required eigenvalues k
output summary of eigenvalues etc.
calculate all required noramlized Eankel and re-
lated functions
output summary of Hankel functions, normalization
factors, etc.
120�
assemble matrices / vectors due to
artificial boundary S
static condensation to eliminate coefficient
unknowns {p!
Gauss elimination to reduce equation matrix to
upper triangular
back substitution for solution {g!
print dimensionless nodal potential solution
substitute to recover
unnormalize and output all dimensionless
coefficient solutions
<,i I «<1;< I. e and output upi i < «] th«:.>r<. z results
calculate and output farce
coefficients
end of loop for one wavelength
Figure 5.1 Flow chart for the progra~
The following is a description of each of the programs used:
5. 2. 1, 1 MAIN
This main program is responsible for the transfer of control indi-
cated in Fig. 5.1. It also initiates universal constants,
input/output device numbers and performs basic conversions, input and
output, and error calculation Optical Thenrem! .
5.2.1.2 INPUT
This subroutine sets up and prints all geometric and grid data.
It must be rmxitten for any ohanqe in geo+et>~~. For all the
examples studied, no data is input and the grids are generated via
calls to mesh generation programs: MESHO and MESHl.
5. 2. l. 3 MESHO, MESHl
These routines generate vertical cylindrical rings of 20-node
hexahedral elements: the node coordinates and the element node connecti-
vities. The number of vertical, angular and radial layers of elements
and the positions of the nodal points, as well as the initial values for
node and element numbering can be arbitrarily specified. Node numbering
is radially, from top to bottom and sprials out. By using a suitable
set of calling parameters, the resulting bandwidth of the system can be
minimized. This is achieved in all the examples treated,
The element node cones <-t ivi ties aze set-up in the uzdez assumed
5.2.1.4 ETGVAL
This routine obtains the specified number of eigenvalues k of
Eq. �. 3! by using fixed-point iteration.
5. 2. 1. 5 HANKEL
The required functions J k r !, H k r !, K k r !, H k r !�! �! 'noS'no S�'nmS'noS
and K' k r ! are calculated and normalized in the following manner:n m S
�!
"' !!H x!
n normalized P n,x! H x!!
where
2 2 -1/4 2 2 r x � n ! for x � n > vx
-1/3x
2 n
for Ex � n~~vx2 2
2 -1/4 2 2 1/2 -1 nx ! exp I- n � x ! + n cosh � ! j2 2 7
for n � x > vx
P n,x!
124
in all the other subprograms.
For grid geometries that can be described in cylindrical coordinates
by constant coordinate surfaces, MESHO and MESH1 can be used directly to
generate the complete grid system. When the geometry deviates only
slightly from the former shapes, only those particular regions need to
be input see Pig. 5.2!. For a problem where grid systems of other
analytic such as rectangular, elliptic, etc.! shapes are more efficient,
NESHO, MESHl must be modified. The general program structure, however,
is still valid and required changes can be quite minor.
Figure 5.2 An example using both automatic generation in region V! and
direct input in region V! to set up finite element grids
125
andK x}
K x!!n normal ized Q n, x!
where
Q n x! = n + x ! e2 2-1/4 � ~n+x n+v'n +x n
It follows from recursive formulae that
H ! � � 8 !n-l x n l!
n normal iz ed P n x!
and
� K x! � � K x! l! n l!
<' x! !n-I x
n normal ized Q n, x!
J k r ! occurs only in the right-hand side vector and is notn o S
normalized.
The above normalization is clearly not unique but is adequate to
make the special functions involved to be of order unity for the
126�
entire range of m and n. These normalized values are used throughout the
programs. Although the final system of linear equations is well condi-
tioned, this normalization may be important especially when round-off
accumulation may be significant.
5.2.1. 6 ASKHKV and ELKV
ASkZKV assembles the global matrix [K ] from the element matricesV
e[K ] which are calculated in ELKV using numerical integration. The
mathematical procedure is as described in Section 2.1, which involves
the isoparametric mapping �.1.5!.
5,2.l.7 FMAP2
This mapping routine is called by four other programs SURFAS, CROSSK,
PLOAD and FORCE. It performs the numerical isoparametric mapping from
one face of the parent element in local coordinates onto a face of the
daughter element on the free surface F or the cylindrical surface S in
global x,y,z coordinates. The Jacobian and/or the corresponding global
coordinate values are returned as required for numerical quadrature
see Eqs. �. l. 5!, �. 2.4!, �. 3.2 . 0!! .
5.2.1.8 SURFAS
The matrix [K ] due to the integral over the free surface, I , is
assembled directly into [K ] from each element on the free surface.
Numerical integration i.s performed in local coordinates and mapped into
global coordinate space see Section 2.2!.
5.2.l.9 CROSSK, PLOAD, CLOAD, DIAGK
These four routines construct respectively [K ], {Q ]', [Q ], IK ],C P C D
which are the vectors and matrices resulti.ng from integrals on the
artificial boundary S I -I !. CROSSK and PLOAD require quadrature and
mapping, while Q ! and fK ! are evaluated directly. The procedures
follow quite directly from Section 2.3.
The load vectors Q !, fQ ! are nondimensionalized by the factorC
iga � !, i.e.,
d
{Q !
P dimensionless iga - � ! 0
and
Q ! Q !> .C dimensionless iga
- �!
The resulting nodal potential and series coefficient solutions are hence
also dimensionless:
!! f4!dimensionless iga
- . '!
and
~.!! .dimensionless iga - � '!
128
5. 2. l. 10 DENSE and COEFF
For computational efficiency in Gauss elimination, DENSE performs
static condensation to eliminate the unknowns p! see Eq. �. 5. 9. a!!. No
matrix inversion is required here as [K j is diagonal. Subroutine COEFF
recovers p! by substitution Eq. �.5.8!! of the solution $!.
5.2.1.11 SDLV and BKSUBT
We have used two general routines to perform Gauss elimination and
back � substitution respectively for a system of linear algebraic complex
equations. The coefficient matrix of the equations is syrrmretric and
banded, and is therefore stored in a symmetric packed form.
No pivoting partial or complete! is performed or is necessary!.
5. 2. 1. 12 FORCE
This program calculates force and moment coeffi-
cients Eqs. �.1,1 .a,b! from $!. Numerical integration is perFormed
using Eq. �.4.2! .
5.2.1.13 OUTPUT
This routine outputs the real and imaginary parts and the magnitude
and phase of each entry of a complex vector.
5.2.2 ln ut. Re uirements
In addition to the geometric input and the finite element grid as
described in Section 5.2.1.2, we must prescribe physical data regarding
the incident wave, the number of eigenvalues and of the coefficient un-
knowns for each frequency point to be solved. These are described here.
We remark that if the boundary geometry is compLicated so that additional
input cards are needed, they must preceed those described in the present
section. The following input is required:
]29
CARD NO. DESCRIPTION OF INPUT FORMAT
I51 NCASE: a right-justified integer giving the
number of frequency points to be analyzed in
the current run
I5, 2Fl5. 7NWK, THETAI, WAVET
NWK = right-justified integer representing no.
of eigenvalues k , kl, ... ~ 1 to be used for
first frequency point
THETAI = floating-point number for the wave
incident angle in radians
WAVZT = floating-point no. for the normalized wave
aperiod: WAVET T/v � where T is the wave periodg
20I40' 1' '''' NWK-1
if NWK ! 20, continue on as many cards as necessary!
right � justified integer indicating number of coeffi-
cient terms chosen for eigenvalues k0, k , ..., k.
e.g., if N0 = 3, then cos 00, cos 6, sin 0, cos 20, sin 26,
are included.
Repeat cards 2 and 3 or more cards! for as many points as indiciated in
card 1.
5.2.3
The INPUT program prints a summary of geometry and physical data:
a! far-field depth, radius of cylindrical surface S and other
geometrical data,
130
b! total number of nodes and elements,
c! elements on S, on F, on B, etc.,
d! a node-coordinate. table,
e! an element.-connectivity table,
f! information on serai-bandwidth of resulting matrix.
For each frequency, the following results are printed:
a! wave period and angle of incidence,
b! table of the eigenvalues and the number of coefficients chosen for
each,
c! table of Bessel functions required,
d! table of normalized modified! Hankel functions and their derivative
and the normalization factors P and Q,
e! complete list of all dimensionless nodal potentials solutions
I0!! .dimensionless'
f! complete list of unnormalized dimensionless coefficient solutions,
g! optical theorem results,
h! force coefCIcients.
5.2.4 Illustrative Errample: program Listing, Sample Input and Gutput
The program listing together with sample input and output are pre-
sented here for the case of the semi � immersed cylinder using
eigenvalues as described in Section 3.2.
5.2.4.1 Variable name table
The following is a list of variable names used in the main program,
these names are preserved as far as possible in the subroutines. The
Pro ram Variables Ex lanation/Notation in Text
Variables in common:
total number of elements/ M
total number of nodes/ N
number of elements on F
INTEGER*4NELE
NNOD
NB number of elements on S
number of nodes on S/N
number of equations in final
condensed system, equals NNOD
semi-bandwidth of final condensed
NODR
NEQT
system
N + l0
NTOTL
total number of coefficients used/ NT
number of eigenvalues chosen in
g'/M + ls
number of coefficient terms for
NTOTL2
M = 0/NMAX*2 � I
number of elements on circumference
of cylinder of radius a
132
radius of the cylinder a and the incident amplitude a are used as normali�0
zation factors throughout. They are assumed to be unity and do not appear
in the programs.
INTERGER*4NFZ
REAL*4
WAVFT
HH
important variables not in comm<>n ,'
NCASE
REA1,*4PAT
OPR
OP I.
COHPLEX*SCJ
Arrays
NFI.F.FX NFX!
NELEFZ NF Z!
WK NWK!
WK H NWK!
REAL*4
133�
I READ
IWRITE
THETAI
N .ON 3, NE LE!
NEl,ES NS!
NELEH NB!
NGOF NWK!
number of elements on bottom of.
cylinder of radius a
card input device number
printer output device number
wave incidence angle/01
Tnormalized wave period/-
/a/gradius of S/r
as ymp to t i c dep th /h
depth of immersion/H
number of frequency po i»ts t > solve INTEt,'ER~4
right hand side of Optical Theorem
lef t hand s ide of Optical Theorem
matrix of clem«nt node connectivities INTEGER+4
vector of numbt'rs ol elements on F
vector of numbers of elements on S
vector of number of coe f ficients chosen
for ea~ h eigenvalue/ N. + 1!1
vector of numbers of «1 ements on
cyl inder circumference
vector of number of «1ement s on
cy] inder base
vector ~>f eigenvalues km
vector of k h valuesm
REAL*4
FNORN NWK, NMAX! matrix of normalization factors for
Hankel functions, etc./P n,x!, Q n,x!
SYSK NEQT, NBAND! [K ] COMPL EX*8
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SYSKC HODR, HTOTL2! IKCJ
[K J diagonal!SY SKD NTOTL2 !
SXSQM NTOTL2!
XH NWK, HOAX!
DH NWK, HOAX!
Ks ~ r !! valuesn m S normalized
5.2.4. 2 Sample Input
For one wavelength solved, only three cards are used:
CARD COLUMN 12345678901234567890123456789012345678901234567890...
Card 1:
5 7.8839178
12 10 10 10 10
3.14159262
3
5. 2. 4. 3 Progr am 1 is t ing
334
WKR NWK!
xz max!
SN NNAX!
CS NMAX!
WORK NNAX!
XXZ �, NNOD!
vector of k r valuesm S
vector of J k r ! valuesn o S
workspace required by routine CROSSK
workspace required by routine CROSSK
workspace required by routine HANKEL
matrix of node coordinates
matrix of H k r !! . and�!n o S normalized
n m S normalized
matrix of H k rS l' d and�! 'n o S normalized
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6. CONCLE DING REMAMS
6.1 ~Su eet.ed
In general, especially when larger or more complex geometries, or
incident waves of higher frequency are to be studied, the following modifica-
tions and extensions to our present program are recomn~ended and may be
necessary!:
A. Second'ary storage: The large core requirements we experienced strongly
suggest the use of additional storage on secondary devices, which
enables many core � aving alternatives:
l! Data storage: When many different wavelengths or modes are to be
analyzed using the same grid structure, a large portion of th»
data such as geometrical information, the global matrix [K ] due
tc the volume integral! can be saved in secondary storage and
need be initiated arly once.
2! Program overlay: The present flow of control among the programs
require only a few programs to be in core at one time while all.
the oth»r subroutines may be stored externally. Such a scheme
results in only a small primary storage saving, however, as the
total object code is only ~ 80K long.
3! Partitioning: This is the most. useful and often necessary extention
to a program using Gauss elimination for solution. tdthi le partition-
ing a Gauss elimination solver is straight forward, efficient
assembling of the equation matrices in partitioned form require
special m-nipulations, Experience with ,imilar procedures in
other problems such as finite element methods in structural
197
mechanics!, however, can be adapted quite directly.
B. Conversion to double precision: In this type of problem, a sizeable
number of complex unknowns together with a large bandwidth are in-
volved. The great number of operations required to solve such a
system necessitates a careful consideration of possible round-off
accumulation. Our experience in this and other problems indicates
that in some of the examples presented � 500 unknowns, - 200 band-
width, � 10 complex operations!, we are already close to a regime7
where the accuracy af single precision becomes inadequate. By con-
verting to double-precision, round-off accumulation error is essenti-
ally eliminated, While in principle, the conversion is trivial in our
present program, the increased storage need makes other modificationsinvolving use of secondary storage such as partitioning! necessary.
The increase in computational time associated with double precision
is, however, not significant.
C. Substructuring: By treating the overall finite element domain as an
assembly of several smaller regions, the overall problem can be
analyzed separately in terms of each of these substructures. Such amodification is particularly effective when the geometry or expected
physical behavior suggests ready subdivisions of the element domain.In an attempt to extend the original Atlantic Generating Station two-
dimensional finite-element problem Chen and Mei, 1974! to higher
frequencies, a new program using secondary storage capabilities which
involved sub � structuring see Fig. 6.1! but not partitioning! was
198�
Figure 6.1 Substructuring the AGS into four regions
199
previously developed by the authors but was not completed for lack
of time. Our experience indicates that it is always important to
include all the elements on the artificial boundary in one substructure
to avoid unnecessary increase in bandwidth.
D. Use of symmetry: When the geometry under consideration possesses one
ar more vertical planes of symmetry, the program can be modified so
that only ore symmetric segment need be studied. The procedure is
illustrated schematically in Fig. 6.2 for the. case with - single
plane of symmetry. WhiIe two separate one symmetric, one antisymmetric!
problems must now be solved, each of them only involves half the
original number of nodal unknowns and bandwidth. The new computational
time required would only be one-fourth and the storage one half of
the non-symmetric problem. Again, a two-dimensional finite element
program using symmetry was previously developed and tested. The
treatment of the symmetric half! problem with a semi-circular outer
boundary and Neumann boundary conditions on the plane of symmetry is
similar to Chen and Mei for their rectangular and semi-circular
harbors. For the antisymmetric half! problem, the Dirichlet boundary
condi tions on the symmetry plane are imposed as essential boundary
conditions lo-dings! af ter the global matrix is assembled.
200
201
6.2 Reeommendatfons and Cost Estimates for the At1an tie Cene~ratinStation Problem
For the specific case of the offshore harbor problem for the Atlantic
Generating Station the following extensions to the present program are
recommended:
l! Change all storage variables from single- to double-precision
accuracy;
2! full partitioning in assemb1age and Gauss elimination solution by
using secondary storage; and
3! if an assumption of symmetry is reasonably valid i.e., if the ocean
bottom contours run approximately parallel to the length of the
generating station!, modifications be made to utilize symmetry to
solve only a symmetric! half of the problem.
Assuming a mean radius, a, of the circular breakwater of about 900
feet and a mean storm far-field depth of m 70 feet, we estimate, using
our earlier criterion Section 4.2!, that a total of w 6500 nodes with a
bandwidth of .f 550 is required for a w 8.5 second or higher ka 17!
incident wave. About 6 hours of computational time would be required for
each wavelength or mode!, and if symmetry is present, about 1. 8 hours
per point. The above estimate is arrived at hy a direct extrapolation
of the timing of the present examples which use no secondary storage.
Clearly, an appreciable cost associated with the necessary input/output
operations between primary and secondary devices, which is very much
machine dependent, must also be added.
Finally, the problem of radiation of waves due to forced osci]lations
of one or two platforms in the AGS harbor can be treated by the wave force
202
program with very little modification required. This is due to the math-
ematical similarities in the two problems. Since there are six degrees
of freedom for each platform, the number of added mass and damping coeffi-
cients involved is of course large and the total computer time needed
depends on the thoroughness of the investigation intended.
6.3 Conclusions
The hybrid element method presented in this work is found to be a
powerful one for three � dimensional water wave scattering problems. While
onIy relatively simpIe geometries have been studied, it is clear that by
using secondary storage and increasing precision of machine representation
to reduce round-off accumulation error, the extension to treating more
complicated structures incI uding depth variations is straightforward� . The
method is hence a practical one For engineering applications.
In our numerical experiments using simple geometries, the hybrid
cI pment method is aire ady quite competitive with known Green function
schemes. I',xpcrience indicates that our computing cost is essent ially
governed by the number of f inite element nodes. For finite element
method, the number of nodes increases with the fluid volume, whi! e for
Green function method the numher of area elements required increases with
the total wetted solid surface. Thus as the number and complexi ty of sol id
components increase in a given fluid region, wc expect the hybrid element
method to be less costly while the opposite is true for Green function
method In light of this, and of the virtual lack of any tedious analytic
preparation required, the hybrid elemene method, is, in our opinion a
203�
powerful alternative to the Green function method for three � dimensional
water wave problems,
Me emphasize once more that the problem of radiation by an oscillating
body is mathematically very similar to the scattering problem we have
t'reated. The linearized boundary value problem for a radiation problem is
obtained by setting the incident potential f to be zero and replacing
by $ = g radiated potential! in Fqs. 1 .2 .1! � � .2 .4! . and requirings r
further that
= Vd f!Bn n
on the oscillating body S where V is prescribed for a given oscillatingn
mode. The proper functional is now Bai and Yeung, 1974!:
2J f!! = � 'Vg! dV � � $ dF � V <f!dS
2 2g nS
+ � � f!! ~ dSyl g I2 Br
S
The modification of the present program to solve the radiation problem can
be trivially achieved; the forcing comes now from the integral over S and
not S, The total coefficient stiffness! matrix is completely unchanged,
while the forcing integrals 1 , I see Fq. �.0!! are no longer present5
and all the forcing terms come from the integral
V PdS
which can be directly evaluated by quadrature after V is given or computed.
204
LIST OF REFERENCES
Abramowitz, M. and Stegun, I. A. ed!, Handbook of MathematicalFunctions with Formulas, Graphs, and Mathematical Tables,Applied Math. series 55, National Bureau of Standards, U, S. Dept.of Commerce, June 1964, p, 895,
2. Bai, K. 3. and Yeung, R., "Numerical. Solution of Free-Surface PlowProblems", Proceedi~ns of the Tenth S mposium on Naval H~drod~namics,June 1974 in press! .
3. Black, 3, L. s "Wave Forces on Vertical Axisymmetric Bodies", Journa Iof Fluid Mechanics, Vol. 67, Part 2, 1975, pp. 369-376.
Black, J. L., Mei, C. C. and Bray, M. C. C., "Radiation and Scatteringof Water Waves by Rigid Bodies", Journal of Fluid Mechanics,Vol. 46, Part 1, 1971, pp. 15 I � 164.
4.
5. Chen, H. S. and Mei, C. C,, "Osci11ations and Wave Forces in a Man-
Naval ~g~drod nsmi<:e, June 1974 fn press! .
also
Chen, H. S. and Mei, C. C., "Oscillations and Wave Forces in an OffshoreHarbor", Technical. Report No. 190, Ralph M. Parsons Laboratory for'Water Resources and Hydrodynamics, Dept. of Civi] Engineering, M. l. T.,Cambridge, Mass. August 1974.
6. Garrett, C. 3. R., "Wave Forces on a Circular Dock", Journal of FluidMechanics, Vol. 46, Part 1, 1971, pp. 129-]39.
7. Garrison, C. J., "Hydrodynamics of Large Objects in the Sea; Part IHydrodynamic Analysis", Journal of H~dronautics, Vol. 8, No. 1,January I9 74, pp. 5 � 12.
Gottfried, K., Ijuantum I eel.anics, Volume. I: Fundamentals, W. A,Benjamin, Inc.. New York, kyar<*k 1966, pp. 106-109.
Haskind, M.Ds 9 "The Exci ting Forces and Wet t ing of Ships in Waves", in Russ ian! I zv. Akad. Nauk SSSR, Otd. Tekh. Nauk, 'l 957, Vol. 7,pp. 66-79. English trsnslatinn: David T~alryr Nodal Basin Translation,NA. 307.
10. Jonsson, I. G., Skovgaard, 0. and Brink � Kjaer, Oe 9 "Diffraction andRefraction Calculations for Waves Attacking an Island", submitted tn3ournal of Marine Research, Manuscript--November 1975.
11. Kopal, Z., Nume< ica~lAna 1 ais, Second Fdi t i m, .Iohn Wilef & Sons, InNew Vork, ]961, p. 562.
205
12. Lautenbacher, C. Ca a "Gravity Wave Refraction by Islands", Journalof Fluid Mechanics, Vol. 41, Part 3, 1970, pp. 655-672.
13. Longuet-Higgins, M. S., "On the Trapping of Wave Energy Round Islands",Journal of Fluid Mechanics, Vol. 29, Part 4, 1967, pp. 781-821.
14. MacCamy, R. C. and Fuchs, R. Aam "Wave Forces on Piles: a DiffractionTheory", Institute of Engineering Research, Waves Investigation I.ab.,Series 3, Issue 334, Berkeley, California, February 1952.
15. Newman, J. N., "The Interaction of Stationary Vessels with RegularWaves", proceed~in s oy the Eleventhde osinm on Naval Hyd~rod namics,April 1976 in press! .
16. Smith, R. and Sprinks, T., "Scattering of Surface Waves by a ConicalIsland", Journal of Fluid Mechanics, Vol. 72, Part 2, 1975, pp.37 3- 384.
I /. Vastano, A. C. and Reid, R. 0., "A Numerical Study of the TsunamiResponse at an Island", Dept. of Oceanography, Texas A 6 M University,A 6 M Project 471, Ref . 66-26T, 1966.
18. Vastano, A. C. and Reid, R. Oa a "Tsunami Response for Islands:Verification of a Numerical Procedure", Journal of Marine Research,Vol. 25, Part 2, 1967, pp. 129-139.
19. Wchausen, J. V., "The Motion of Floating Bodies", Annual Review ofFluid Mechanics, Vol. 3, 1971, pp. 237-268.
20. Zienkiewicz, 0. C., et al., "Iso-Parameter and Associated ElementFamilies for Two � and Three � Dimensional Analysis", Finite Elements
Press, Trondbeim, Norway, 1969, pp. 383-432.
206
LIST OF FIGURES ~Pa e131.1 The Boundary Val.ue Problem
1.2 Introduction cf the Artificial Circular CylindricalBoundary, S
232.1 The 20-node Hexahedral Element with Exterior Nodes Only in I.ocal Coordinates!
232.2 The Daughter Arbitrarily Oriented, Quadratic � sidedHexahedral Element in Global Coordinates AfterIsoparametric Mapping
403.1 Wave Incident on a Bottom-seated Circular Cylinder ofRadius a � = Tr; h = a!
423.2 Finite Element Structure for a Uniform Cylinder:36 Elements, 342 Nodal Points.
473.3 Wave incident on a Floating Cylindrical Dock of Radiusa �1 = vr; h = .75a!
3.4 Finite Element Structure for a Floating Dock:56 Elements, 435 Nodal Points
3.5 a! Horizontal Force. Coefficiert for a Semi-immersedh HCylinder � = .75; � = .5; 6 = m!.a'a'' I
+, ampli tude; x, phase: f inite element solutionampli.tude; � ---, phase: Garrett, 1971 results
51 b! Vertical Force Coef ficient for a Semi-immersedCyli nder See Description in Figure 3. 5 a! !
3. 6 Forces on a square dock: + magnitude;x � phase. Normal incider.ce: 8 = 7T.'7 I
a! Horizonta] Force Coef f icier.t
b! Vertical Force Coefficient
c! Momert Coefficient About y � axis
207�
~Pa e
Forces on a Square Dock: +, magni tude� x �, phase. Oblique Incidence: 8 = 5'/4.
3.7
a! Horizontal Force Coefficierit
b! Vertical Force Coefficient
c! Moment Coefficient About Horizontal Axis
57
Run-up, q/a, on the Sides of a Square. Barge: +! �: nodal! nagninnda; a! � �: noda! ! phase.Normal incidence: 0I
3.8
s! k�a = l
b! ka= 2
c! k a = 30
d! ka=4
60
61
62
63
3.9
a! ka=l0
b! ka=20
c! k a = 3o
d! ka=4
66
Differentia1 Scattering Cross-section, ~A e!~z, for aSquare Barge. Normal Incidence: 0 = Tr.
3. 1.0
a! k a 1
b! ka=2
c! k a = 3
d! k a = 40
68
69
70
71
� 208�
Run-up, q/ao, on the Sides o f a Square Barge: +! : nodal magnitude; x! -- � : nodal! phase.Oblique Incidence: 9 = 5T/4.I
~Fi ure ~Pa e
Differential Scattering Cross-section,~ 4 �!~ , fora Square Barge. Oblique Incidence: 6 = 5v/4.
3. 11
72 a! k a = 1
0
b! ka=2
c! k" = 30
d! ka=-40
I 3
74
75
Polar Plot of Differential Scattering Cross � section,~A 8! ~, fcr a Square Barge. Normal Incidence: 8
3 12
76 -! ka= 1
b! ka=2o
c! ka= 30
77
d!. i! k a = 40
7r 3m 5' 77r d! . ii! k a = 4. Details for � < 8 < � and � < 8
0 4
79
80
3.13
81 a! k a = 1
0
82 b! k a = 2
0
c! k a = 3C
d! k a = 40
84
Geometry and Finite � element Structure for an EllipticalIsland with a Circular Base. major and Minor axes:A = a, B = .5a; "draft" of elliptic cylinder: H = .5a,radius of circu]ar toe: R = 1.5a; water depth: h = a
3.14
Run � up, rr/a, on an Elliptic Island with a Circular Base: +!: ncdal! magni tude; x! � � �: nodal! phase.Wave Incident from x ~ + ~. 9 = rr! .
I
3.15
a! k a = 1
b! ka.= 20
c! k a = 3
89
90
209
Polar Plot of Differential Scattering Cross � section,~A e!~ , for a Square Barge. Ob1ique Incidence: 9 = 5l/4.
~Fi ute~Pa e
a! k a = 10
91
b' ka=20
c'; ka=30
92
93
3.17 Dif ferential Scattering Cross-section, ~A Q! ~, for anElliptic Island with a Circular Base.Wave Incident from x ~ + ~. � = v! .I
a! ka = 10
b! ka= 2o
c' k a = 30
94
95
3. 18 Differential Scattering Cross-section, ~ A�! !, for anE]liptic Island with a Circular Base,Wave Incident f rom y ~ + ~. � = 3' j 2! .I
a! k a = 10
b! ka=2
c} k a = 30
97
99
3.]9 Polar Plot of Differential Sc-ttering Cross-section, A�!I , for an Elliptic Island with a Circular Base.Wave Incident from x ~ + ~. � = m!.
]00 a! k a = 1
n1.01
b; ka= 2
102 c~ k a 3
0
3. ZG Polar Plot of Differential Scattering Cross-sec ion,~A�! ! ~, for an Elliptic Island with a Circular Base.Wave Incident from y ~ + ~. � = 3>/2! ~I
103 a! k a = 1
0
b! ka=20
c!ka=3
104
].05
2 io
3.16 Run � up, q/a, on an Elliptic Island with a Circular Base: +I �: nodal! magnitude; tx! � --: nodal! phase.Wave Incident from y ~ + gd. � = 3m/2! .
I
~Pa e
4.1 Relative Error in Optical Theore~ for the Square Dockunder Obliquely Incident Vaves e = 5~/4!.
I
108
110Relative Error in the Yagnitude c f tb e Hc rizontal Forceon a Uni form Cylinder.
4.2
4.3
120 � 1225.1 Flow Chart for the Program
An Example Using Both Automatic Generation in Region V!and Direct Input in Region V! to set up Finite ElementGrids.
1255.2
1996.1 SuLstructuring the AGS into Four Regions
Spli tting a problem with one vertical! Plane o f Symmetry 201into Two Half Problems.
6.2
211
Relative Error in the Magni.tude of the Run-up on a Uniform 111Cylinder at: + : 9 = 0; X: 9 = 7r/2; o: 8 = v. 8 = vr!.
LIST OF TABLES
TABLE~Pa e
443.1 Comparison Between Analytic and Finite Element Results
for Sc"ttering Off a Uniform Cylinder e = >!:IHorizontal Force Coefficient. CF and Free Surface
Elevation q/a at r = a.0
45Comparison Betweer Analytic and Finite Element Resu1tsfor Scattering off a Uniform Cylinder e = v!:Coefficien.ts u, 5 of the Exterior Solution. k a = l.
mn' mn 0
3.2
49Computed Horizontal and Vertical Force Coef ficients fora Semi-imziersed Circular Cylinder
H � = .75, � = .5!a ' ' a
3.3
113Comparison of Results for Diffraction ly a UniformCylinder, k a = 1, e = tr, Using a Variety of Finite Element
0Grid Structures: pxq p Rings Stacked Vertically of qFinite Elements Each.
4.1
115
1185.1 Summary o f Computational Requirements.
212
4.2 Comparison c f Results for Diffraction by a Square Barge,k a = 1, e = m, Using a Range. of Numbers of CoefficientTermse
LIST OP SYMBOLS
Typical horizontal dimension cf a body radius fora cylinder, half-length for a square!
Tncident wave amplitude
A�! Angular variation of the far � field scattered potential
All solid surfaces including bottom
Complex cor.jugale of preceding termC.C.
Relative error in the total horizontal force ona uniform cylinder
Relative error defined by the. Optical Theorem
Relative error in the total scattering cross-sectionE
Relative error in the run-up on the body at 6
Total free surfaceFinite element region free surface inside S
Total force vector
Component of total force in i-direction
Acceleration due to gravi.ty
Constant far-field! water depth
Typical vertical dimension of a body water depth fora uniform cylinder, draft for floating docks!
Hankel function of the first kind, order n argument. x
Imaginary unit; = ~l
C ,C ,C ,C ,C ,C Force and moment coefficients respectively iny z x y z x, y, z directions
Integrals in J $!Il, I2' .. I6
The stationary functional
Bessel function, order n argument x
rea1! wave-number associated with the incident wave
J x!
k0
K x! Modified Bessel function of the second kind, order nargument x
Average dimension of finite elements in a given gridLe
Number of vertical eigenfunctions iv the exterioranalytic representation minus one
Total moment vector
Component of total moment about i-axisM.1
Number of finite elements in V
M Number of ncdes in a finite element
Unit ncrmal vector pointing out of fluid region undercon sidera tion
Number of radial eigenfunction for each vertical modein the exterior analytic representation
N , Nl, N2 ''., N
e
iInterpolation functions for an element
Number of nodes on S
Total number of urknown coefficients in exterioran-lytic representation
Number of finite element nodes in V
Radial coordinate; = ~x2 + y
214
m = l, 2, 3 ... imaginary part of the imaginary rootsof the dispersion relation
Radius of the artificial cylinder SS
Complex! reflection coefficient in two-dimensionalscattering
Vertical circular cylindrical artificial surfaceseparating finite element and analytic regions
Control surface at infini.ty
Time
Complex! transmission coefficient in two � dimensionalscattering
Total fluid volumeFinite element fluid volume inside S
V
V Finite element region with grids generated by hand! input
Weighting function for quadrature point i
Rectilinear coordinates
Global! nodal coordinate values of a finite element
x, y, z
X. qy,qz
G mn' mnCoe f f icients of the ex terior eigenf unc tion expression
Absolute error in
Relative error in
A small number ; Belongs to
Jacobi symbol; = 1 for n = 0; = 2 for n = 1, 2, 3
Imaginary roots of the dispersion relation
Local coordinates of an element
Quadrature points in local. coordinates
Finite element regio~ with grids generated automatically
Free surface elevation
-l yPolar angle azimuth!; = tan !
e,
Incident velocity potential
= Re[ t>e ]
Imaginary part of
Real part of
Im !
Re !
d 3Gradient operator; = �, �, � !3x' 3y' 3z
3Bn
= n ~ 7
Absolute value of a real number!Nagnitude of a complex number!Determinant of a matrix
A subset of
Union of
A vector
!* Complex conjugate of
!' Of the analytic region exterior to SDerivative with respect to argument
!' Of an element
Transpose of
Inverse of a matrix!
!'
Incidence angle
Density of fluid
Radian frequency of the incident wave
Total velocity potential
Scattered velocity potential
of S
Of the free surface
Jacobian matrix of coordinate transformation forvolume, free surface and S
Total global! eruation matrix
Total global! matrix after static! condensation
Pull! matrix due to an integral ia J $!
Diagonal! matrix due to an integral in J $!
Matrix due to integral over free surface
Matrix due to the volume integral
Vector of interpolation func tions
Total load right-hand-side! vector af ter static!conden.,ation
Vectors due to integrals in J $!
Matrix of node coordinate values for an element
Vector of coeffic ient unknowns
Vector containing both FKN and coefficient unknowns
217
APPENDIX A
A PROOF OF THE OPTICAL THEOREM
Consider an incident wave given as in Eq. �.1.a!
iga cosh k z+h!0 O ik r cos B-01!ocosh k h
0
A. 1 .a!b f ikor cos B 01!o
where we have defined
v2 cosh k z+h!
f z!h+ 4 sinh k h
A. 1.b!
and
iga A + z sinh k ho ld 0
b = - ---!v2 cosh k h
0
A.1. c!a cons tan t
and h is the far-field constant depth.
At infinity, the scattered wave Fq. �.3.4!! can be written aa
ikorc A $! f z!
S A ro
A.2.a! k r -+ ~!0
*The proof here is quite well-known in classical physics and is notdifferent: in essence from that given in Gottfried, 1966.
218�
The Optical Theorem for three-dimensional scattering of water waves
Eq. 4.!.2! can be proved by app1ying Green's Theorem and using the method
of stationary phase.+
where we define
igaA 8! = � ! Z -i! a, cos nB + g sin n8!
on onn=0
A.2.b!
and
2 -i~/4c=b v' � e cosh k h7r 0
A.2.c!
0 = $V g � /~7'$!dV = g ~> � g* ~~!dS A. 3!
by virtue of Eq. �.2.1! in V, and Eq. �.2.2!, �.2.3! on F and 8
respectively.
It follows that
Im [g > ] dS = 0r Q+* A. 4!
and upon substituting in Eqs. A.l!, A.2!, expanding and performing depth
integration, we have
219
Applying Green's Theorem to 0 = $ + $ and $" over the entire fluid region,I s
V, we have
2m
0 = d9r Im f-ik b cos 8 � 8 ! ]k ~ 0 I
0
+lim e
k r~0
r 27r ikor [1-cos 8 � 9I! ]d9r Im f-ik bc* 4" 8!
0
0r
ikor [l-cos 8 � 8Z! ]b*c 4 9! ' cos 9 � 9 !]
0 A rI
0
8 A«> I ' A.s!
-it'2where order k r! terms vanish in the limpet.
0
8 � 8 ! ~ 0 k r! = c say!I 0
E I!
and
9 � 8 � m! ~ > 0 k r!I 0
� 220�
Now, the first integrand is periodic in 9 and the integral vanishes
identically. The second integral, I say!, can be evaluated using a
stationary phase argument since k r >! l. In this limit, contribution to0
the integral is neglegible except near stationary phase points 9 � 8 = 0I
and s, in fact, only for
Expanding cos� � 6 ! in Taylor series in turn about zero and 7r, keeping
only leading contribution in both exponent and coefficient, and evaluating
A at the stationary points, we have
I~Im [-i Arbc*A+�! e-ik «� � 0 !'�
0 Id� - 0 } � C.C.]
I
+ Im [ -i ~r bc* A*� + vr! e o d 8 � 0 � Tr! + C.C.]0 I I
k r ~ ~ ,c « 1!
where C.C. denotes the complex conjugate of the preceding term. Now the
second term vanishes identically upon taking the imaginary part, and the
first term can be. rewritten so that
2 2I~ 2 Im [ -i ~abc*A*� ! e o I d� � 6 !]o I I
kr~~, a<<1!0
Set
ik r � - 6I! '12, i e. 0 - 6I = / k2
0
then on taking the limit
221
PiI = 2 Im [ � iV'-bc* A+ 9 ! e ds]
i-Pi
I = � ~~> > Re[A R ! ]0
Vhence, Kq. A.5} becomes
27r20 = � ..".» �R.rA<e,» � I* J I<c~! I' «
0
or
21r
A e! I' de =�Re[A� ! ]
cosh k h0
as stated in Eq. �.1.2!.
222
The Fresrel integral is simply vm, and substituting in Eq. A.2.c! for b, we
finally have
