An Improved General Dual Reciprocity Boundary Element...

An Improved General Dual Reciprocity Boundary Element Model for WaveScattering by a Truncated Shoal

INTRODUCTION

The mild-slope equation (MSE) was derived by(1972) where he demonstrated that, if the bottom slope is 'mild',a three-dimensional wave problem could be well approximatedby a two-dimensional single equation. Since then, the MSE hasproved a useful tool for a wide range of water wave problems.

Several numerical approaches have been developed to solvethe MSE. For example, and (1977) and

(1981) developed the so-called hybrid method for theMSE, in which an infinite computational domain is divided intotwo: an outer region where infinite elements or eigen-functionscan be adopted and an inner region where finite element ordifference techniques can be used to obtain solutions. Most ofthe subsequent work differs only on the treatment of the outerdomain.

However, as the MSE is of the elliptical type, the amount ofdata and hence the size of the coefficient matrix become verylarge when finite element and hybrid element methods are used,and therefore the computation becomes very expensive forlarge domain simulation as shown by (1981).

The boundary element method (BEM) only requires adiscretization on the boundary of a computational domain and itis popular in solving wave propagation problems with constantwater depth ( and , 1983). However, when the waterdepth becomes a variable, the conventional BEM seems to bepowerless because that a domain integral arises and the domainhas to be discretized, which destroys the computationaladvantage of the conventional BEM.

Employing the dual reciprocity boundary element method(DRBEM), which was first proposed by and(1982), (1993a) presented a DRBEM wave model to solvethe MSE. It is shown that the DRBEM model possesses a greatadvantage in numerical efficiency over hybrid elementsmethod, in terms of both computational time and computermemory required. For example, for the case of Homma's island( , 1950) with incident wave of period =120 sec,

(1981) had to carry out his calculation with 10,560elements on the half of a symmetrical domain. For a generaldomain without symmetry to be utilized, he would have to use

21,120 elements which would lead to a linear system with21,600 real equations. However, using only 60 quadraticboundary elements and 192 internal collocation points for allthe calculations covering the full domain, (1993a) hasobtained an accurate result comparing with the results presentedby . (1976) and (1981).

Following the idea of (1988), where varyingwater depth is approximated with a perturbed constant depth,

(1997) proposed another DRBEM model to solve theMSE. This new model leads to a significant reduction of thenumber of unknowns in the resulting system of equations.However, since the governing equation solved by(1997) is only an approximate form of the MSE, the accuracy ofsolutions, as shown in the verification of the model forHomma's island, is satisfactory only for seabed geometry with aratio of the water depth in the deep ocean to that along thecoastal vertical wall less than 3. For Homma's original islandwith the depth ratio of 9, the accuracy is acceptable only for verylong waves. Hence, this perturbation DRBEM model is not asaccurate as the DRBEM model proposed by (1993a)

It is clear that, for a truncated shoal, the bed slope along the

BERKHOFF

BETTESS ZIENKIEWICZ

HOUSTON

HOUSTON

AU BREBBIA

NARDINI BREBBIA

ZHU

HOMMA

HOUSTON

ZHU

JONSSON HOUSTON

RANGOGNI

POULIN

POULIN

.

et al

ZHU

Then, considering that there is a restriction that the waterdepth is always nonzero in the DRBEM model ( , 1993a),which narrows the range of application of the DRBEM model incomparison with its counterparts of hybrid finite elementsmethod, . (2000) (see , 2001 also) developed anextension of Zhu's DRBEM model to a general DRBEM model(GDRBEM) for cases where zero-water-depth coastlines areallowed.

In this paper, we consider to modify the GDRBEM model forwave scattering by a submerged truncated shoal. For such kindof truncated shoals, numerical solutions from both hybrid finiteelements and DRBEM/GDRBEM solution have not yet beenreported. Although the similar problem was once solved by

and (1999) by using a standard error-checking Runge-Kutta method, their truncated shoal isrestricted the axi-symmetrical topography that permits theutilization of cylindrical coordinates and the technique ofvariable separation.

ZHU

ZHU LIU

CHAMBERLAIN PORTER

et al

boundary of the shoal top is discontinuous. Some attention has

Journal of Coastal Research SI 39 1463 - 1467 ICS 2004 (Proceedings) Brazil ISSN 0749-0208

Huan-Wen Liu†‡ and Peng-Zhi Lin‡§

† Dept. of Mathematics andComputer Sciences,Guangxi University forNationalities, Nanning, Guangxi530006, P.R. [email protected]

LIU, H.-W. and LIN, P.-Z., 2006.An improved general dual reciprocity boundary element model for wave scatteringby a truncated shoal. Journal of Coastal Research, SI 39 (Proccendigs of the 8th International CoastalSymposium),1463 -1467 . Itajaí, SC, Brazil, ISSN 0749-0208.

There are many numerical methods to model wave scattering by an island or a submerged shoal. However, for asubmerged truncated shoal, numerical solution has rarely been reported before. In this paper, the wave model basedon the general dual reciprocity boundary element method (GDRBEM) is extended to solve the mild-slope equationfor a truncated shoal. The computational domain, which is generally divided into two sub-domains in a conventionalGDRBEM model, is divided into three sub-domains in this study to separate the inner region with constant depth asan individual sub-domain. Using the current GDRBEM model, wave amplifications by a truncated conical shoal forvarious incident waves are calculated and compared with both the approximate analytic solutions and the previousGDRBEM solutions. It is found that the current GDRBEM model provides a much better comparison with theapproximate analytic solutions than its predecessor. This improved GDRBEM model is then used to investigate thedepth effects on the maximum wave amplification. The maximal wave amplification decreases when thesubmergence increases.

,ADDITIONAL INDEX WORDS: Water waves, the mild-slope equation wave amplification.

ABSTRACT

‡ Dept. of Civil Engineering,National University of Singapore,Singapore 119260, Republic [email protected]

Journal of Coastal Research Special Issue 39, 2006,

§ State Key Laboratory of Hydraulicsand Mountain River Engineering,Sichuan University, ChinaDepartment of Civil Engineering,National University of Singapore,Singapore

been paid to this discontinuity. According to and(1995) and and (1999), to

ensure the continuity of mass flow for waves crossing an oceanbottom with a discontinuity of the bed slope, the MSE should bereplaced by the extended MSE or the modified MSE togetherwith an interfacial jump condition being imposed there.

The aim of this paper is not to study the influence of thisdiscontinuity on the governing equation. Rather, we shall returnto the conventional MSE and study the influence of thediscontinuity on the numerical accuracy of the GDRBEMmodel. As we have known, in finite element and finitedifference method, the general way to resolve the influence ofthe topographic discontinuity is to refine the local meshes.However, we shall show in this paper that, this kind ofrefinement technique is not valid in the GDRBEM,

The differential equation (2) is defined on an infinitecomputational domain. As shown in Figure 1, such acomputational domain can be divided into three sub-domains

, and with denoting the finite inner region withconstant water depth denoting the finite middle region withvariable water depth and denoting the infinite outer regionwith constant water depth Since the governing equations inthese regions will be transformed into different form, theyshould be discussed separately.

where is a geometric parameter with respect to for, for and for

respectively, with being the internal angle of the boundaryat point . In the middle region , for any fixed source point ,

where is a geometric parameter with respect to :for , for and for

, respectively, with being the internal angle ofthe boundary at point .

In the outer region , , equation (4) is equivalent tothe Helmholtz equation

(10)

PORTER

STAZIKER CHAMBERLAIN PORTER

and in fact agood alternative is to divide the computation domain into sub-domains along the curve of bed-slope discontinuity.

Under a Cartesian coordinate system in which and denotea pair of orthogonal horizontal coordinates and denotes thevertical coordinate measured positively upwards from theundisturbed free surface, the incident wave potential of a trainof monochromatic waves propagating along the positive -axisover a submerged truncated shoal with variable water depth

can be expressed

(1)

where is the gravitational acceleration, the incident waveamplitude, the angular frequency and the wave number withrespect to the constant water depth outside the shoal.

According to (1972), the total wave potential,, by which the refraction of these waves due to the

topographic change of seabed and the existence of the shoal aredescribed, satisfies the MSE

(2)

in which and

(3)

Where is the local wave number determined by the dispersionrelation.

.

Generally, in the DRBEM ( and , 1982), tominimize the interpolation error, it is preferable that the right-hand side of the governing equation be kept as simple aspossible. On the other hand, the simplicity of the maindifferential operator should be taken into account in order toobtain the corresponding particular solution analytically.Particularly, for the problem of wave refraction and diffraction,in order to convert the original governing equation defined inthe whole infinite region into an equivalent integral equation ina finite region, we generally need utilize the far-field radiationcondition (S , 1949) to eliminate the integral alongthe infinite circle (see , 2001, p.33). Therefore it isconvenient to choose the Helmholtz operator as the maindifferential operator, see . (2000) and (2001). Forthe same reason here, we deliberately rewrite the governingequation (2) as

(4)

where

(5)

Let be the Hankel function of the first kind ofzero order with being the distance between a source point

and a field point It is well-known that is thefundamental solution of the Helmholtz equation

(6)

In the inner region , since that , equation (4) isequivalent to

(7)

Where with being the local wave number in .Multiplying both sides of equation (7) by and usingGreen's second identity, one can rewrite equation (7) as

(8)

(9)

THE GOVERNING EQUATION

x yz

xh(x,

y)

g A

k

et al

BERKHOFF

NARDINI BREBBIA

OMMERFELD

LIU

ZHU LIU

THE EQUIVALENT INTEGRALEQUATIONS

Figure 1. A definition sketch of a submerged truncated conicalshoal.

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Liu and Lin1464

where is the scattered wave. Multiplying both sides of

equation (10) by and using the far-field radiationcondition ( , 1949), one can rewrite (10) as

(11)

were

substituting equation (12) into equation (8) yields

Substituting equation (14) into equation (11) yields

By putting equations (13) and (15) into equation (9), wehave

,

,

Considering that the main differential operators in governingequation is the Helmholtz operator, we shall follow .(2000) to choose the interpolation functions to be the followingradial basis functions:

Thus, the right-hand side term in equation (4) is expanded as

(18)

where , and are the coefficients to be determined

with the collocation method by demanding the satisfaction ofequations

, (19)

System (19) can also be expressed in matrix form:

(20)thus we have

(21)

It is noted that the existence and the recursion formulae of theparticular solution to the following equation

(22)

for a given have been given by (1993b). The first domain

integral in the right-hand side of equation (16) thereforebecomes

(23)

which involves boundary integrals only. Similarly, the seconddomain integral in the right-hand side of equation (16) becomes

(24)

for + + + , where , that is,

for , for and forEquation (25) involves boundary integrals only and afterappropriate discretization, a linear system of algebraicequations involving the unknown function Ø on points can beestablished. The details of a similar system for constantboundary elements can be found in theAppendixAin .(2000) or in (2001).

It is worth indicating that if we still adopt the conventionalGDRBEM model ( ., 2000 and , 2001) to model thewave scattering by a submerged truncated shoal, that is, weignore the discontinuity of the bed slope along the top boundaryand divide the computational domain into two sub-domainsonly along the footline , then we shall have the final integralequation as follows

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SOMMERFELD

ZHU

ZHU

ZHU

LIU

ZHU LIU

is the outward normal unit vector of the outer domain

and is also a geometric parameter with respect to

for for , and

respectively, with being the internal angle of

the boundary at pointIt is noted that, along the boundary between and , we

have

(12)

(13)

And the continuity of the wave potential and flux across thecommon boundary demands

, .(14)

(15).

(16)

where , , and is used.Up to this stage, the original governing equation (2) defined

on the infinite domain has been

transformed into the above equivalent integral equation (16)defined on a finite domain . That is, all the source

points and field points are now restricted to the inner regionand the middle region as well as their boundaries and .

This has greatly simplified the solution procedure incomparison with the hybrid finite element approaches( , 1981). All but two terms in equation (16) requireinformation only on the boundaries and To eliminate the

two domain integrals, which involve the unknown functionand its gradient the GDRBEM will be employed in the

next section.

(17)

at interior collocation points within boundary points on

interior points within and boundary points on

substituting equations (23) and (24) into equation (16) yields

(25)

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HOUSTON

THE DUAL RECIPROCITY BOUNDARYELEMENT

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An Improved General Dual Reciprocity Boundary Element Wave Model 1465

(26)

In this section, using the current GDRBEM model, we shallpresent some numerical results of wave amplification for asubmerged truncated conical shoal. In all calculations related tothe current GDRBEM model, 40 quadratic boundary elements(with 80 boundary nodes) on both and , 300 internal

collocation points in the middle domain and 450 internal

collocation points in the inner domain

. (2004) presented an approximate analyticsolution of the MSE for wave scattering by a special submergedtruncated conical shoal with the vertex being situated at theundisturbed free surface, where =10km, =30km,

=1.333km, =4km and

The wave amplifications along the x-axis for two cases=480s and =240s are first calculated by using the

approximate analytic model ( ., 2004), the currentimproved GDRBEM model based on equation (25) and theGDRBEM model ( ., 2000) based on equation (26),respectively. In which, the total number and locations of allcollocation points used in the GDRBEM model and in thecurrent improved GDRBEM model are same. These three setsof results are presented in Figure 3 for =480s and in Figure 4for =240s. As we can see, agreements between approximateanalytic solutions and the present GDRBEM solutions for bothcases are excellent, but significant discrepancies between theconventional GDRBEM solutions and approximate analyticsolutions are observed, especially in the case =240s. It is worthindicating that not much improvement on the accuracy of theGDRBEM model is obtained after we increase both boundarynodes and internal collocation points.

With the validation of the current GDRBEM model, it is thenused to investigate the influence of shoal submergence on waveamplification. In . (2004), to obtain an approximate

The fundamental difference between equation (25) andequation (26) is that there is one more integral term along the topboundary in equation (25). It is clear that the errors caused by

the lack of this term cannot be patched up through the meshrefining near

As it will be shown in the next section, with this extra termbeing included in equation (25), the accuracy of our currentGDRBEM model will be greatly improved.

, are always used (see

Figure 2).

=0km, see Figure 1. Their results

are ideal for testing our improved GDRBEM model.

Ã

Ã

i

i .

NUMERICAL EXAMPLES ANDDISCUSSION

Ã Ã

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Ù

i o

m

i

i o

LIU

LIU

ZHU

LIU

et al

a bh h h’

T Tet al

et al

TT

T

et al

Figure 4. Wave amplifications along the x-axis for =240scalculated by the approximate analytic model ( ., 2004),the GDRBEM model ( ., 2000) and the presentGDRBEM model.

LIU

ZHU

et alet al

Figure 5. Wave amplifications along the x-axis for varioussubmergences in the case of =480s.

Figure 2. The distribution of boundary element nodes andinternal collocation points used in the current GDRBEM.

Figure 3. Wave amplifications along the x-axis for =480scalculated by the approximate analytic model ( ., 2004),the GDRBEM model ( ., 2000) and the presentGDRBEM model.

LIU

ZHU

et alet al

� ��

��o

III dG qqGc o**

)( )()( ��

� �� o

dqqc jjj

N

j

mi

j])ˆ*ˆ()(ˆ[ *

1

),( ��


Liu and Lin1466

analytic solution of the MSE, the vertex of the conical shoal hasto be located at the undisturbed free water surface. Thisrestriction does not apply to the GDRBEM model. In this study,we let represent the distance of the vertex of the shoal fromthe still water surface. In two cases of =480s and =240s forthe same truncated shoal studied above, wave amplificationsalong the x-axis are calculated for =-1.0km, -0.5km, 0km,0.5km and 1.0km, and the results are presented in Figures 5-6,respectively. It is observed the maximal wave amplificationdecrease when the water depth increases as the impact from theshoal on the wave becomes weaker and weaker. On the otherhand, it is interesting to observe that, for =480s, the focal pointmoves upstream as the increase of water depth, however, for

=240s, the focal point moves downstream as the increase ofwater depth. The wave reflection becomes more evident forshorter wave (e.g. =240s), which forms partially standingwave in front of the shoal when the local water depth is shallow.

The first author would like to acknowledge the support fromthe Natural Science Foundat ion of P.R. China(NSFC10162001), the Excellent Young Teachers Program ofMinistry of Education P.R. China (EYTP1784), Guangxi

Natural Science Foundation (GNSF0135001) and Guangxi Shi-Bai-Qian Scholars Program (No. 2001224). Both authors like toacknowledge the support from the research grant from theNational University of Singapore (R-264-000-121-112) andresearch grant from the Meteorological Services of Singapore(R-264-000-121-291).

J.C.W., 1972. Computation of combined refraction-diffraction.

(Vancouver, Canada), pp.471-490.

P. and O.C., 1977. Diffraction andrefraction of surface waves using finite and infiniteelements.

, 2, 1271.P.G. and D., 1999. Scattering and

near-trapping of water waves by axisymmetric topography., 388, 335-354.

S., 1950. On the behaviour of seismic sea wavesaround circular island. , 21, 199-208.

J.R., 1981. Combined refraction and diffraction ofshort waves using the finite element method.

, 3, 163-170.I.G.; O. and O., 1976.

Diffraction and refraction calculations for waves

h’T T

h’

T

T

T

Proceedings of the 13th InternationalConference Coastal Engineering

International Journal for Numerical Methods inEngineering

Journal of Fluid Mechanics

Geophysical Magazine

Applied OceanResearch

CONCLUSIONS

ACKNOWLEDGEMENTS

LITERATURE CITED

In this paper, wave scattering by a submerged truncated shoalis modeled numerically. Considering that the gradient of thewater depth is discontinuous along the boundary of the shoaltop, we extended the conventional GDRBEM wave model( ., 2000 and , 2001), in which the computationaldomain was divided into two sub-domains along the footline, toan improved GDRBEM model with the computational domainbeing divided into three sub-domains where the inner regionwith constant depth is an individual sub-domain. This treatmentleads to a new integral term appearing in the final boundaryintegral equation in the improved GDRBEM model. It is shownthat, with this new term being included, the present GDRBEMmodel provides much more accurate results than theconventional GDRBEM model does.

M.C. and C.A., 1983. Diffraction of water wavesfor vertical cylinders using boundary elements.

, 7, 106-114.

incident onan island. , 34, 469-496.

H.-W., 2001. Numerical modeling of the propagation ofocean waves. Wollongong, Australia: University ofWollongong, Ph.D. thesis, 132p.

H.-W.; Lin, P.-Z. and N.J., 2004. An analyticsolution of the mild-slope equation for scattering by atruncated conical shoal. To appear in

(Makuhari, Japan).D. and C.A., 1982. A new approach to free

vibration analysis using boundary elements. : Brebbia,C.A. (ed.), .NewYork: Springer-Verlag, pp. 312-326.

D. and D.J., 1995. Extensions of the mild-slope equation. , 300, 367-382.

S. A., 1997.. Denmark:

Technical University of Denmark, Ph.D. thesis.R., 1988. A simple procedure to solve the mild-

slope equation using BEM and perturbation technique.

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S.-P., 1993b. Particular solutions associated with theHelmholtz operator used in DRBEM.

, 4, 231-233.S.-P.; H.-W. and K., 2000. Ageneral DRBEM

model for wave refraction and diffraction., 24, 377-390.

ZHU LIU

AU, BREBBIA,

LIU,

LIU, SHANKAR,

NARDINI, BREBBIA,

PORTER, STAZIKER,

POULIN,

RANGOGNI,

SOMMERFELD,

ZHU,

ZHU,

ZHU, LIU, CHEN,

et al

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BERKHOFF,

BETTESS, ZIENKIEWICZ,

CHAMBERLAIN, PORTER,

HOMMA,

HOUSTON,

JONSSON, SKOVGAARD, BRINK-KJAER,

Figure 6. Wave amplifications along the x-axis for varioussubmergences in the case of =240s.T


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