Research Article Interaction of Submerged Breakwater by a

Research ArticleInteraction of Submerged Breakwater by a SolitaryWave Using WC-SPH Method

Afshin Mansouri1 and Babak Aminnejad2

1 Departmentof Civil Engineering The Islamic Azad University of Roudehen Branch Tehran Iran2Department of Civil Engineering Faculty of Civil Engineering The Islamic Azad University of Roudehen Branch Tehran Iran

Correspondence should be addressed to Afshin Mansouri afshinmansoorigmailcom

Received 3 January 2014 Revised 8 March 2014 Accepted 11 March 2014 Published 13 April 2014

Academic Editor Agostino Bruzzone

Copyright copy 2014 A Mansouri and B Aminnejad This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Interaction of a solitary wave and submerged breakwater is studied in a meshless Lagrangian approach For this purpose a two-dimensional smoothed particle hydrodynamics (SPH) code is developed Furthermore an extensive set of simulations is conductedIn the first step the generated solitary wave is validated Subsequently the interaction of solitary wave and submerged breakwateris investigated thoroughly Results of the interaction of solitary wave and a submerged breakwater are also shown to be in goodagreementwith published experimental studies Afterwards the effects of the inclination and length of breakwater aswell as distancebetween two breakwaters are evaluated on damping ratio of breakwater

1 Introduction

Due to development of ports and marine protection of theseashores against the sea waves must be considered as animportant issue in the field of marine structure In thiscontext breakwaters have been used as appropriate toolsin previous decades A wide range of breakwaters has beenintroduced so far Submerged breakwater is one of themwhich has been utilized in many countries Generally someadvantages of submerged breakwater can be mentioned asfollows

(1) simple structure

(2) cost effectiveness

(3) safety of marine transport vessels

(4) short construction period

(5) preserving beauty of the coast (touristy aspect)

(6) high strength of the structures against sea waves

(7) preventing coastal erosion area with scattering ofwave energy

In addition the performance of submerged breakwatersdepends on geometry material properties and environ-mental characteristics of surrounding breakwater [1] Inthe following paragraphs several studies dealing with thesubmerged breakwaters are reviewed

Effects of overtopping submerged obstacle on wavecharacteristics were investigated by Hara et al [2] Theyused a numerical model as well as a regression method tocalculate the rate of the wave breaking Hayakawa et al[3] also studied flow field around a trapezoidal submergedbreakwater numerically They compared the obtained 2Dand 3D numerical results with an experimental findingSolitary wave interaction with submerged multibodies wasconsidered by Wan Decheng and Guoxiong [4] theoreticallyAn experimental investigation of the wave transmission onsubmerged breakwaters was also considered by Armono andHall [5] An experimental comparison of wave field behinda submerged trapezoidal breakwater and double submergedtrapezoidal breakwater was performed by Yoshida et al[6] In their study the ratio between breakwaters and thewave height has been studied Interaction of a solitary waveand permeable rectangular submerged breakwater had beenevaluated by Huang et al [7] They measured the wave

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2014 Article ID 524824 9 pageshttpdxdoiorg1011552014524824

2 Modelling and Simulation in Engineering

transmission as well as the amount of dragon the break-water Stamos et al [8] performed an experimental studyand considered performance of absorption and reflectionof waves from hemicylindrical and rectangular submergedbreakwaters Johnson et al [9] conducted an experimentaland theoretical study of currents and waves around trape-zoidal submerged breakwater Muni-Reddy and Neelamani[10] presented a numerical and experimental investigationfor measurement of wave breaking and force on a protectedoffshore base with trapezoidal breakwater Chen et al [11]experimentally studied transformation of waves with variousprofiles between a trapezoidal submerged breakwater andsea wall Christou et al [12] presented numerical simulationof regular nonlinear waves interaction with rectangularsubmerged breakwaters for investigation of transmitted andreflected waves profile Zaman et al [13] investigated theamount of deformation of waves passing over the parabolicbreakwater and the extent of waves reflected area Losadaet al [14] investigated transmission reflection turbulenceand breakage of regular and irregular waves overtoppingof rubble mound breakwaters Cao et al [15] numericallystudied hydrodynamic properties of wave propagation whichsurrounded two impermeable trapezoidal submerged break-waters

In the area of breakwater computations smoothed par-ticle hydrodynamics (SPH) can also be implemented as anovel meshless numerical method that is being developedfor the study of near shore waves and navy needs In thiscontext several numerical models have been developed withSPH method for modeling free surface flows Dalrymple etal [16] Gotoh et al [17] and Shao [18] have presented somenumerical techniques in conjunction with SPH which havebeen widely used Furthermore Dalrymple and Rogers [19]showed that SPH method offers a variety of advantages forfluid modeling especially those with a free surface Yimet al [20] investigated waversquos interaction with porous rigidsemisubmerged rectangular breakwater with RANS and SPHmethods Cherfils et al [21] also considered flow surroundinga horizontal submerged plate as a breakwater near shoreswithSPHmethodDidier andNeves [22] studiedwave speed waveheight and breakage of wave over sloped sea wall for a regionof the Portuguese with SPH method

Furthermore to simulate water waves with SPH manydifferent approaches have been presented so far Shao [23]offered a coupled incompressible smoothed particle hydro-dynamics (ISPH) porous medium method to simulate freesurface profile Liu et al [24] optimized 3D SPH methodfor application of water wave modeling Generally interestedreader can refer to some articles [25 26] for finding reviewliterature of SPH development

Due to necessity of obtaining appropriate comparisonbetween the different factors affecting the performance ofsubmerged breakwater and lack of widespread study onsubmerged breakwaters with SPHmethod a relatively exten-sive parametric study on trapezoidal submerged breakwatersis presented For this purpose various geometrical aspectsof breakwater are considered and a comparative study isperformed Furthermore it is being tried to introducean appropriate combination of numerical techniques in SPH

for accurate simulation of solitary wave-breakwaters interac-tion

2 Governing Equations

In smooth particle hydrodynamics fluid flow can be rep-resented by 119873 discrete particles Each of these particlesmay be specified by an individual mass Moreover particlesmove according to the equations of Newtonian mechanicsincluding a pressure force For each of these particles theequations are based on Lagrangian form of the Navier-Stokesequations

119889120588

119889119905= minus120588nablaV

119889V119889119905= minus1

120588nabla119901 + 119892 + Π

(1)

where V 119901 and 120588 are the velocity pressure and densityrespectively 119892 is the gravitational acceleration and Π refersto the diffusion terms

3 SPH Formulation

Navier-Stokes equations solution in SPHmethod is by defin-ing a kernel function119882(119909 minus 1199091015840 ℎ) around every 1199091015840 locationof an SPH particle The maximum of this function occurs at119909 = 119909

1015840 The features of kernel function must be similar toDirac Delta function Therefore behavior of kernel functionis completely dependent on value of 119903 = 119909 minus 1199091015840 The valueℎ is the parameter which defines the ldquosizerdquo of influence areaof kernel function and is called the smoothing length Thekernel is normalized to unity in the following way

intΩ

119882(119909 minus 1199091015840 ℎ) 119889119909

1015840= 1 (2)

and we also havelimℎrarr0

119882(119909 minus 1199091015840 ℎ) = 120575 (119909 minus 119909

1015840) (3)

We should regard the SPH kernel as a defining region ofinfluence of the SPH particle It will turn out with only SPHparticleswhich liewithin each otherrsquos regions of influence andare able to interact with each other

Furthermore kernel function is implemented to approx-imate field variables at any point in a computational domainFor instance an estimate value of a function 119891(119909) at thelocation 119909 is given in a continuous form by an integral of theproduct of the function and a kernel function119882(119909 minus 1199091015840 ℎ)

⟨119891 (119909)⟩ = intΩ

119891 (1199091015840)119882(119909 minus 119909

1015840 ℎ) 119889119909

1015840 (4)

where the angle brackets ⟨⟩ denote a kernel approximationIn addition to the abovementioned characteristics kernelfunction must satisfy several other features such as positivitycondition [27] Moreover (4) can be also approximated by asummation as follows

⟨119891 (119909)⟩ cong sum

119895

119898119895

120588119895

119891 (119909119895)119882(

10038161003816100381610038161003816119909 minus 119909101584010038161003816100381610038161003816 ℎ) (5)

Modelling and Simulation in Engineering 3

where119898119895120588119895is the volume associated with the particle 119895 This

equation can be used if the function 119891(119909) is only known at119873discrete points

The kernel function description can be finalized bydefining a specific kernel function Numerous possibilitiesexist A large number of kernel function types are discussedin literature ranging from polynomial to Gaussian Themost common is the B-spline kernel that was proposed byMonaghan and Lattanzio [28]

119908 (119903 ℎ) = 120572119863

1 minus3

21199022+3

411990230 le 119902 le 1

1

4(2 minus 119902)

3

1 le 119902 le 2

0 119902 ge 2

(6)

which is considered in the present paper and 119902 = 119903ℎ and 120572119863

is equal to 10120587ℎ2 in two dimensional cases

4 SPH Implementation onGoverning Equations

Now based on the SPH formulation mentioned in previoussection governing equations will be discretized In this con-text the continuity equation for particle 119894 can be consideredin the SPH formulation as

119889120588119894

119889119905= minus

119873

sum

119895=1

119898119895(V119895minus V119895) sdot nabla119882

119894119895 (7)

where the sum extends over all neighboring particles and119882is the smoothing kernel evaluated at the distance betweenparticles 119894 and 119895

The SPH formulation of the momentum equations basedon artificial viscosity can also be obtained through theparticle approximation procedure as in

119889V119894

119889119905= minus

119873

sum

119895=1

119898119895(119875119894

1205882

119894

+

119875119895

1198752

119895

+ Π119894119895)nabla119882

119894119895+ 119892 (8)

where Π119894119895is the viscosity term To introduce viscous dissi-

pation an artificial viscosity concept which has been devel-oped using Monaghan [29] is implemented This term issymmetric conserves momentum and introduces a shearviscosity into the momentum equation The SPH form of themomentum equation is written as

Π119886119887=

minus120572119862119886119887120583119886119887

120588119886119887

119881119886119887sdot 119903119886119887lt 0

0 for other values(9)

where

120583119886119887=ℎV119886119887sdot 119903119886119887

1199032

119886119887+ 1205782

(10)

120588119886119887=1

2(120588119886+ 120588119887) 119862

119886119887=1

2(119862119886+ 119862119887)

1205782= 001ℎ

2

(11)

120572 is also a constant Monaghan [29] suggests that a value of120572 = 001 can be used for most computations

5 Numerical Details

51 XSPH Technique Tomodify the particlersquos velocity XSPHtechnique is used and particle 119894 is moved based on thistechnique In XSPH scheme it is formulated as follows

119889119903119894

119889119905= V119894+ 120576sum

119895

119898119895

120588119894119895

V119894119895119882119894119895 (12)

Velocity of particle 119894 is recalculated in such manner thatthe velocity of particle 119894 and the average velocity of the closestneighborhood particles interacting with particle 119894 are takeninto account

52 Time Stepping Procedure In the present paper predictor-corrector time stepping algorithm is implemented In thepredictor-corrector formulation variation of particles prop-erties (such as velocity) is estimated as [17] Consider

V119899+12119894

= V119899119894+Δ119905

2119865119899+12

119894 (13)

And by implementing forces at half step the values ofvelocity can be calculated at the end of any step which isformulated as follows [17]

V119899+1119894= 2V119899+12119894

minus V119899119894 (14)

53 Density Filter In weakly compressible SPH formulationlarge pressure oscillation has been observed These pressureinaccuracies can be modified using some procedures Inaddition to pressure modification the free surface profilewill be also improved Therefore MLS (moving least square)density filter which is a first order density correction is usedto reproduce the linear variation of the density field

120588119886= sum

119887

120588119887119882

MLS119886119887

119898119887

120588119887

= sum

119887

119898119887119882

MLS119886119887 (15)

The corrected kernel is also evaluated as follows

119882MLS119886119887

= 119882MLS119886119887

(119903119886) = 120573 (119903

119886) sdot (119903119886minus 119903119887)119882119886119887 (16)

More details can be found in many references in theliterature [30 31]

54 Boundary Conditions Two rows of stationary waterparticles at the boundary (Figure 1) remain fixed or moveaccording to an externally imposed function for say wavemaker boundaries When a fluid particle approaches theboundary the density and hence pressure exerted by theboundary particles increase This generates the necessaryrepulsive force on the water particles A major deficiency ofthis method is the generation of numerical boundary layersWhenwater particles get close to the boundary particles theytend to stick to the boundary thus they generate a boundarylayer which is not physically observed


dx dx

dx2 dz2

Figure 1 Interaction betweenfluid particles and boundary particles

6 Solitary Wave Generation

Solitary wave is generated by Prescribe Piston Wave Maker[5] Value of paddle displacement is applied in the followingequation [22]

119883119901(119905) =

119867

119896[tanh (119883 (119905)) + tanh(119896

119889120582)]

119883 (119905) =119896

119889(119888119905 minus 119883

119901(119905) minus 120582)

(17)

where119867 is the wave height 119896 = radic((31198674119889)) is wave numberand 119888 = radic(119892(119889 + 119867)) is speed Wave speed can also beobtained from the following equation [22]

119906119901(119905) =

119888119867

119889

1

cosh2119883 (119905) + (119867119889) (18)

61 Validation of Generated Solitary Wave According toMaiti and Sen [32] simulation which is a benchmark test thelength and flat length of the tank water depth and slope ofthe right hand side of the tank were respectively consideredto be 10m 97m 03m and 45 degree (Figure 2)Themotionof wave paddle is calculated for 119867119889 = 01 Therefore waveheight and 120582 are 003 and 6882

The result of solitary wave generation is illustrated inFigure 3 Simulation is conducted from 0 till 10 sec It isobserved that the wave height is equal to 003m and obtainedfree surface profile is relatively accurate for further studiesHowever some inconsistencies are visible at second 4 Inaddition average error of SPH simulation (point to point) isequal to 189 percent

7 Results and Discussion

In this section various aspects of solitary wave-breakwaterinteraction will be studied At first step free surface related toa specific breakwater under interaction with a solitary waveis computed using SPH to ensure that our numerical setupis suitable for investigation of more complicated problemsAfterward various geometrical characteristics of breakwatersare also discussed Formore detailed studies two breakwatersare considered and effects of distance between these breakwa-ters will be evaluated

71 Validation For further evaluation of SPH accuracy thefree surface profile generated in solitary wave-breakwater

H

d

R

L

120573

Figure 2 Computational domain

0

002

004

0 2 4 6 8 10 12Wav

e hei

ght

(m)

Maiti and Sen (1999)SPH

minus002X (m)

Figure 3 Comparison of generated solitary wave with Sagnitarsquosresults

Table 1 Wave flume solitary wave and breakwater features

Value Characteristics03m Wave height (119867)10m Water depth (119889)45m (119871

0)

70m (119871overal)08m Height of breakwater (ℎ)04m Breakwaterrsquos beam1 2 Slope of breakwaterrsquos wall

interaction is compared with experimental work of Grilliet al [33] To perform this comparison a computationaldomain is defined (Table 1) Furthermore the characteristicsof wave flume solitary wave and breakwater are presentedResults of this simulation are illustrated at Figure 4 Featuresof the presented simulation are also reported in Table 2 Freesurface profiles in Figure 4(a) through 4(d) are at 10 1181 13and 1392 sec respectively RMS of obtained solutions whichare calculated relative to experimental findings seem to bereasonable Therefore it can be declared that our numericalsetup can model solitary wave-breakwater interaction withsatisfying precision

72 Investigation of Breakwater Inclination A subject studiedhere is considering the effects of breakwaterrsquos slope on1198671198671015840ratio For this purpose computational domain as well assolitary wave is considered to be the same as previous sectionSlope of breakwater at aft part (120573

2) which is shown in

Figure 5 is varied systematically All considered angles are


0

05

0 1 2 3 4

Wav

e hei

ght

(m)

X (m)minus05

minus1minus2minus3minus4

Experimental dataSPH

(a)

0

05

0 1 2 3 4

Experimental data

Wav

e hei

ght

(m)

X (m)minus05


SPH

(b)

0

05

0 2 4

Wav

e hei

ght

(m)

X (m)minus2minus4


(c)


0

05

0 1 2 3 4

Wav

e hei

ght

(m)

X (m)minus05


(d)

Figure 4 Comparison of SPHrsquos free surface profile with experimen-tal data [33]

Loveral

L0 L1

L2

1205731

1205732

d

h


Table 2 Computational error

Time (sec) Figure RMS10 Figure 4(a) 00361181 Figure 4(b) 003413 Figure 4(c) 00491392 Figure 4(d) 0053

presented in Table 3 Furthermore geometrical characteris-tics of breakwater are also proposed in this table

Table 3 Breakwaterrsquos characteristics

1205732

1205731

ℎ 1198711

0 tanminus1(05) 04 215 tanminus1(05) 04 245 tanminus1(05) 04 260 tanminus1(05) 04 275 tanminus1(05) 04 290 tanminus1(05) 04 2

0010203

0 20 40 60 80 100X (m)

minus01minus02minus03W

ave h

eigh

t (m

)

Outcome wave (120573 = 0)

Incoming wave

Figure 6 Free surface profile for 1205732= 0 degree

Free surface profile related to each considered conditionis illustrated in Figures 6 7 8 9 10 and 11 In each caseobtained free surface profile is compared with generatedsolitary wave without considering breakwater effect Exactevaluation of breakwaterrsquos slope on the free surface profileis presented in Table 4 It is obvious that wherever 120573

2= 90

maximumwave reduction can be achieved Correspondinglywhen 120573

2decreased to zero minimum breakwater effect

occurs Generally Figure 12 also shows a representative frameof SPH solution

73 Investigation of Breakwater Length To perform furtherparametric studies the effects of breakwaterrsquos length onfree surface can also be evaluated In this context previouscomputational characteristics are kept and only the lengthratio L1L2 is changed systematically Considered situationsare presented in Table 5 Figures 13 and 16 show the obtainedresults at various length ratios In all cases breakwatergenerates a hollow in free surface at breakwater positionwhich leads to a reduction in wave crest Generally basedon Table 6 it is observed that L1L2 ratio decreased sobreakwater performance will be improved In addition itcan be concluded that smaller length ratio leads to morereduction of wave amplitude To complete this section a viewof SPH solution is also presented in Figure 17

74 Investigation of Distance between Two Submerged Break-waters In this part influences of distance variation betweentwo breakwaters are evaluated Distance between two break-waters (119883) is considered to be less than wavelength at thebeginning of simulations Consequently this distance willincrease until the gap becomes larger than the wavelengthIn addition wave height water depth and length of waveflume are considered to be 119867 = 03m 12m and 70msuccessively Breakwater is located 30 meters from the leftGeometrical characteristics of breakwaters are also presented


Table 4 Effect of breakwater slope on HH1015840 ratio

Values 1205732= 0 120573

2= 15 120573

2= 45 120573

2= 60 120573

2= 75 120573

2= 90

HH1015840 1078973 1185936 1297549 132081 1470258 1503094

Table 5 Geometrical characteristics

11987111198712

1205731

1205732

1198712

1198711

9 tanminus1 (05) tanminus1(05) 02 18633 tanminus1(05) tanminus1(05) 03 1942 tanminus1(05) tanminus1(05) 05 21367 tanminus1(05) tanminus1(05) 06 22

0010203

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave


in Table 7 Figure 18 shows the gap effects between twoconsidered breakwaters It is observed that when wavereaches the first breakwater the wave height is equal to025m However second breakwater decreases the waveheight in correspondence with considered distance betweentwo breakwaters In this context two parameters 119867 (waveheight above first breakwater) and 1198671015840 (wave height abovesecond breakwater) are introduced Breakwater influencecoefficient is also defined as 1198671198671015840 Precise values of definedparameter are presented in Table 8 It is obviously clear thatfor 119883 = 03m maximum wave reduction is achievedFurthermore it can be concluded that optimum gap betweenbreakwaters is affected by wave length In reality when gapbetween breakwaters is smaller than wave length effects ofsecond breakwater will be decreased Additionallymaximumeffectiveness of breakwater is observedwhenever the distanceis equal to wave length A representation of our SPH solutionis also depicted in Figure 19

8 Results

The present paper focuses on the breakwater performanceunder a solitary wave Furthermore we have tried to imple-ment meshless and Lagrangian approach of smoothed par-ticle hydrodynamics method to study considered physicsIn this context a two-dimensional SPH code is developedTo consider diffusion terms an artificial viscosity model isincluded Particles movement is also calculated using XSPHvariant and predictor-corrector scheme which is investigatedas time stepping algorithm To reinitialize density field MLSdensity filter is utilized

To start computations generated solitary wave is verifiedby comparing previous studies Afterwards a breakwater

001020304

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave


0010203

0 20 40 60 80 100


Incoming wave

Wav

e hei

ght

(m)

X (m)minus01minus02


00204

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus02

minus04


Incoming wave


0

05

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus05


Incoming wave


Figure 12 A representative frame of SPH solution


Table 6 Exact evaluation of length ratio effect on free surface reduction

Values 11987111198712= 9 119871

11198712= 633 119871

11198712= 42 119871

11198712= 367

HH1015840 1094116 1105263 1111112 1222223

0

02

04

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus02

Incoming waveOutcoming wave (120574 = 90)

Figure 13 Free surface profile with ratio 9 in length

0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01

Incoming waveOutcome wave (120574 = 633)

Figure 14 Free surface profile at length ratio 633

Table 7 Breakwater characteristics

(119883) Left sideslope

Right sideslope

Breakwaterheight

Lowerlength

02 tanminus1(05) tanminus1(05) 04 203 tanminus1(05) tanminus1(05) 04 204 tanminus1(05) tanminus1(05) 04 205 tanminus1(05) tanminus1(05) 04 2

Table 8 Effects of breakwaters distance

Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05

HH1015840 1247285 142718 13381 1337985

is also included in computational domain To validate ourSPH solutions obtained free surface profile is compared withexisting experimental results and it is observed that our SPHsolutions have a reasonable accuracy In the second phasevarious geometrical aspects of breakwater are also studied

Generally effects of implementing two breakwaters incli-nation of breakwater and breakwaterrsquos length on free surfacedepression are evaluated It is found that when gap betweenbreakwaters is smaller than wave length effects of secondbreakwater will be decreased Maximum effectiveness ofbreakwater also occurred whenever the distance is equal towave length In addition when the inclination of breakwaterincreased free surface deformation will be modified Fur-thermore it is observed that length ratio of breakwater caneffectively influence free surface profile (Figures 14 and 15)

0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01



Figure 17 A view of SPH solution

0

02

04

020 40 60 80 100W

ave h

eigh

t (m

)

X (m)minus02

Incoming waveOutcome wave (X = 02)

Outcome wave (X = 03)


Figure 18 Simulated free surface profiles at different distancesbetween breakwaters

Figure 19 Particle representation of SPH solution


To perform more precise computation extending SPHsolutions to three-dimensional case will be in attendanceVarious simple breakwater geometries will also be introducedin near future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Tokura and T Ida ldquoSimulation of wave dissipation mech-anism on submerged structure using fluid-structure couplingcapability in LS-DYNArdquo in Proceedings of the 5th European LS-DYNA Conference 2005

[2] M Hara T Yasuda and Y Sakakibara ldquoCharacteristics of asolitary wave breaking caused by a submerged obstaclerdquoCoastalEngineering vol 1 no 23 1992

[3] N Hayakawa T Hosoyamada S Yoshida and G TsujimotoldquoNumerical simulation of wave fields around the submergedbreakwater with SOLA-SURF methodrdquo in Proceedings of the26th International Conference on Coastal Engineering (ICCErsquo98) pp 843ndash852 June 1998

[4] W Decheng and W Guoxiong ldquoNumerical simulation of asolitary wave interaction with submerged multi-bodiesrdquo ActaMechanica Sinica vol 14 no 4 pp 304ndash305 1998

[5] H D Armono and K R Hall ldquoWave transmission on sub-merged breakwaters made of hollow hemispherical shape arti-ficial reefsrdquo in Canadian Coastal Conference pp 313ndash322 June2003

[6] A Yoshida S Yan M Yamashiro and I Irie ldquoWave FieldBehind a Double-Submerged Breakwaterrdquo in Proceedings of the12th International Offshore and Polar Engineering Conferencepp 785ndash790 May 2002

[7] C-J Huang H-H Chang and H-H Hwung ldquoStructuralpermeability effects on the interaction of a solitary wave anda submerged breakwaterrdquo Coastal Engineering vol 49 no 1-2pp 1ndash24 2003

[8] D G Stamos M R Hajj and D P Telionis ldquoPerformanceof hemi-cylindrical and rectangular submerged breakwatersrdquoOcean Engineering vol 30 no 6 pp 813ndash828 2003

[9] H K Johnson T V Karambas I Avgeris B Zanuttigh DGonzalez-Marco and I Caceres ldquoModelling of waves andcurrents around submerged breakwatersrdquo Coastal Engineeringvol 52 no 10-11 pp 949ndash969 2005

[10] M G Muni-Reddy and S Neelamani ldquoWave interaction withcaisson defenced by an offshore low-crested breakwaterrdquo Jour-nal of Coastal Research no 39 pp 1767ndash1770 2006

[11] H-B Chen C-P Tsai and C-C Jeng ldquoWave transformationbetween submerged breakwater and seawallrdquo Journal of CoastalResearch no 50 pp 1069ndash1074 2007

[12] M Christou C Swan and O T Gudmestad ldquoThe interactionof surface water waves with submerged breakwatersrdquo CoastalEngineering vol 55 no 12 pp 945ndash958 2008

[13] M H Zaman H Togashi and R E Baddour ldquoDeformationof monochromatic water wave trains propagating over a sub-merged obstacle in the presence of uniform currentsrdquo OceanEngineering vol 35 no 8-9 pp 823ndash833 2008

[14] I J Losada J L Lara R Guanche and J M Gonzalez-OndinaldquoNumerical analysis of wave overtopping of rubble moundbreakwatersrdquoCoastal Engineering vol 55 no 1 pp 47ndash62 2008

[15] Y-G Cao C-B Jiang and Y-C Bai ldquoNumerical study onflow structure near two impermeable trapezoid submergedbreakwaters on slop bottomsrdquo Journal of Hydrodynamics vol22 no 5 pp 190ndash196 2010

[16] R A Dalrymple O Knio D T Cox M Gesteira and S ZouldquoUsing a lagrangian particle method for deck overtoppingrdquo inProceedings of the 4th International SymposiumWaves pp 1082ndash1091 ASCE September 2001

[17] H Gotoh S Shao and T Memita ldquoSPH-LES model for numer-ical investigation of wave interaction with partially immersedbreakwaterrdquo Coastal Engineering Journal vol 46 no 1 pp 39ndash63 2004

[18] S Shao ldquoIncrompressible SPH simulation of wave breaking andovertopping with turbulence modellingrdquo International Journalfor Numerical Methods in Fluids vol 50 no 5 pp 597ndash6212006

[19] R A Dalrymple and B D Rogers ldquoNumerical modeling ofwater waves with the SPHmethodrdquoCoastal Engineering vol 53no 2-3 pp 141ndash147 2006

[20] S C Yim D Yuk A Panizzo M di Risio and P L-F LiuldquoNumerical simulations ofwave generation by a vertical plungerusing RANS and SPH modelsrdquo Journal of Waterway PortCoastal andOcean Engineering vol 134 no 3 pp 143ndash159 2008

[21] J M Cherfils L Blonce G Pinon and E Rivoalen ldquoSimulationof water wavemdashcoastal structure interactions by smoothedparticle hydrodynamicsrdquo in Proceedings of the 8th InternationalConference onHydrodynamics Nantes France September 2008

[22] E Didier and M G Neves ldquoWave over topping of a typicalcoastal structure of the portuguese coast using A SPH modelrdquoJournal of Coastal Research vol 56 pp 496ndash500 2009

[23] S Shao ldquoIncompressible SPH flow model for wave interactionswith porous mediardquo Coastal Engineering vol 57 no 3 pp 304ndash316 2010

[24] C Liu J Zhang and Y Sun ldquoThe optimization of SPHmethod and its application in simulation of water waverdquo inProceedings of the IEEE 7th International Conference on NaturalComputation (ICNC rsquo11) pp 2327ndash2331 July 2011

[25] PW ClearyM Prakash J Ha N Stokes and C Scott ldquoSmoothparticle hydrodynamics status and future potentialrdquo Progress inComputational Fluid Dynamics vol 7 no 2-4 pp 70ndash90 2007

[26] M B Liu and G R Liu ldquoSmoothed particle hydrodynamics(SPH) an overview and recent developmentsrdquo Archives ofComputational Methods in Engineering vol 17 no 1 pp 25ndash762010

[27] G R Liu and M B Liu Smoothed Particle HydrodynamicsmdashAMeshfree Particle Methods World Scientific Publishing 2007

[28] J JMonaghan and J C Lattanzio ldquoA refined particlemethod forastrophysical problemsrdquo Astronomy and Astrophysics vol 149pp 135ndash143 1985

[29] J J Monaghan ldquoSmoothed particle hydrodynamicsrdquo AnnualReview of Astronomy and Astrophysics vol 30 no 1 pp 543ndash574 1992

[30] A Colagrossi and M Landrini ldquoNumerical simulation ofinterfacial flows by smoothed particle hydrodynamicsrdquo Journalof Computational Physics vol 191 no 2 pp 448ndash475 2003

[31] A Crespo Application of smoothed particle hydrodynamicsmodel SPHysics to free surface hydrodynamics [Doctoral thesis]2008


[32] S Maiti and D Sen ldquoComputation of solitary waves duringpropagation and runup on a sloperdquo Ocean Engineering vol 26no 11 pp 1063ndash1083 1999

[33] S T Grilli M A Losada and F Martin ldquoCharacteristics ofsolitary wave breaking induced by breakwatersrdquo Journal ofWaterway Port Coastal ampOcean Engineering vol 120 no 1 pp74ndash92 1994

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



transmission as well as the amount of dragon the break-water Stamos et al [8] performed an experimental studyand considered performance of absorption and reflectionof waves from hemicylindrical and rectangular submergedbreakwaters Johnson et al [9] conducted an experimentaland theoretical study of currents and waves around trape-zoidal submerged breakwater Muni-Reddy and Neelamani[10] presented a numerical and experimental investigationfor measurement of wave breaking and force on a protectedoffshore base with trapezoidal breakwater Chen et al [11]experimentally studied transformation of waves with variousprofiles between a trapezoidal submerged breakwater andsea wall Christou et al [12] presented numerical simulationof regular nonlinear waves interaction with rectangularsubmerged breakwaters for investigation of transmitted andreflected waves profile Zaman et al [13] investigated theamount of deformation of waves passing over the parabolicbreakwater and the extent of waves reflected area Losadaet al [14] investigated transmission reflection turbulenceand breakage of regular and irregular waves overtoppingof rubble mound breakwaters Cao et al [15] numericallystudied hydrodynamic properties of wave propagation whichsurrounded two impermeable trapezoidal submerged break-waters

In the area of breakwater computations smoothed par-ticle hydrodynamics (SPH) can also be implemented as anovel meshless numerical method that is being developedfor the study of near shore waves and navy needs In thiscontext several numerical models have been developed withSPH method for modeling free surface flows Dalrymple etal [16] Gotoh et al [17] and Shao [18] have presented somenumerical techniques in conjunction with SPH which havebeen widely used Furthermore Dalrymple and Rogers [19]showed that SPH method offers a variety of advantages forfluid modeling especially those with a free surface Yimet al [20] investigated waversquos interaction with porous rigidsemisubmerged rectangular breakwater with RANS and SPHmethods Cherfils et al [21] also considered flow surroundinga horizontal submerged plate as a breakwater near shoreswithSPHmethodDidier andNeves [22] studiedwave speed waveheight and breakage of wave over sloped sea wall for a regionof the Portuguese with SPH method

Furthermore to simulate water waves with SPH manydifferent approaches have been presented so far Shao [23]offered a coupled incompressible smoothed particle hydro-dynamics (ISPH) porous medium method to simulate freesurface profile Liu et al [24] optimized 3D SPH methodfor application of water wave modeling Generally interestedreader can refer to some articles [25 26] for finding reviewliterature of SPH development

Due to necessity of obtaining appropriate comparisonbetween the different factors affecting the performance ofsubmerged breakwater and lack of widespread study onsubmerged breakwaters with SPHmethod a relatively exten-sive parametric study on trapezoidal submerged breakwatersis presented For this purpose various geometrical aspectsof breakwater are considered and a comparative study isperformed Furthermore it is being tried to introducean appropriate combination of numerical techniques in SPH

for accurate simulation of solitary wave-breakwaters interac-tion

2 Governing Equations

In smooth particle hydrodynamics fluid flow can be rep-resented by 119873 discrete particles Each of these particlesmay be specified by an individual mass Moreover particlesmove according to the equations of Newtonian mechanicsincluding a pressure force For each of these particles theequations are based on Lagrangian form of the Navier-Stokesequations

119889120588

119889119905= minus120588nablaV

119889V119889119905= minus1

120588nabla119901 + 119892 + Π

(1)

where V 119901 and 120588 are the velocity pressure and densityrespectively 119892 is the gravitational acceleration and Π refersto the diffusion terms

3 SPH Formulation

Navier-Stokes equations solution in SPHmethod is by defin-ing a kernel function119882(119909 minus 1199091015840 ℎ) around every 1199091015840 locationof an SPH particle The maximum of this function occurs at119909 = 119909

1015840 The features of kernel function must be similar toDirac Delta function Therefore behavior of kernel functionis completely dependent on value of 119903 = 119909 minus 1199091015840 The valueℎ is the parameter which defines the ldquosizerdquo of influence areaof kernel function and is called the smoothing length Thekernel is normalized to unity in the following way

intΩ

119882(119909 minus 1199091015840 ℎ) 119889119909

1015840= 1 (2)

and we also havelimℎrarr0

119882(119909 minus 1199091015840 ℎ) = 120575 (119909 minus 119909

1015840) (3)

We should regard the SPH kernel as a defining region ofinfluence of the SPH particle It will turn out with only SPHparticleswhich liewithin each otherrsquos regions of influence andare able to interact with each other

Furthermore kernel function is implemented to approx-imate field variables at any point in a computational domainFor instance an estimate value of a function 119891(119909) at thelocation 119909 is given in a continuous form by an integral of theproduct of the function and a kernel function119882(119909 minus 1199091015840 ℎ)

⟨119891 (119909)⟩ = intΩ

119891 (1199091015840)119882(119909 minus 119909

1015840 ℎ) 119889119909

1015840 (4)

where the angle brackets ⟨⟩ denote a kernel approximationIn addition to the abovementioned characteristics kernelfunction must satisfy several other features such as positivitycondition [27] Moreover (4) can be also approximated by asummation as follows

⟨119891 (119909)⟩ cong sum

119895

119898119895

120588119895

119891 (119909119895)119882(

10038161003816100381610038161003816119909 minus 119909101584010038161003816100381610038161003816 ℎ) (5)





119908 (119903 ℎ) = 120572119863

1 minus3

21199022+3

411990230 le 119902 le 1

1

4(2 minus 119902)

3

1 le 119902 le 2

0 119902 ge 2

(6)





119889120588119894

119889119905= minus

119873

sum

119895=1


119894119895 (7)



119889V119894

119889119905= minus

119873

sum

119895=1

119898119895(119875119894

1205882

119894

+

119875119895

1198752

119895

+ Π119894119895)nabla119882

119894119895+ 119892 (8)



Π119886119887=

minus120572119862119886119887120583119886119887

120588119886119887

119881119886119887sdot 119903119886119887lt 0


where

120583119886119887=ℎV119886119887sdot 119903119886119887

1199032

119886119887+ 1205782

(10)

120588119886119887=1

2(120588119886+ 120588119887) 119862

119886119887=1

2(119862119886+ 119862119887)

1205782= 001ℎ

2

(11)


5 Numerical Details


119889119903119894

119889119905= V119894+ 120576sum

119895

119898119895

120588119894119895

V119894119895119882119894119895 (12)



V119899+12119894

= V119899119894+Δ119905

2119865119899+12

119894 (13)


V119899+1119894= 2V119899+12119894

minus V119899119894 (14)


120588119886= sum

119887

120588119887119882

MLS119886119887

119898119887

120588119887

= sum

119887

119898119887119882

MLS119886119887 (15)


119882MLS119886119887

= 119882MLS119886119887

(119903119886) = 120573 (119903

119886) sdot (119903119886minus 119903119887)119882119886119887 (16)




dx dx

dx2 dz2




119883119901(119905) =

119867

119896[tanh (119883 (119905)) + tanh(119896

119889120582)]

119883 (119905) =119896

119889(119888119905 minus 119883

119901(119905) minus 120582)

(17)


119906119901(119905) =

119888119867

119889

1

cosh2119883 (119905) + (119867119889) (18)






H

d

R

L

120573


0

002

004

0 2 4 6 8 10 12Wav

e hei

ght

(m)


minus002X (m)




0)







0

05

0 1 2 3 4

Wav

e hei

ght

(m)

X (m)minus05



(a)

0

05

0 1 2 3 4

Experimental data

Wav

e hei

ght

(m)

X (m)minus05


SPH

(b)

0

05

0 2 4

Wav

e hei

ght

(m)

X (m)minus2minus4


(c)


0

05

0 1 2 3 4

Wav

e hei

ght

(m)

X (m)minus05


(d)


Loveral

L0 L1

L2

1205731

1205732

d

h






1205732

1205731

ℎ 1198711


0010203

0 20 40 60 80 100X (m)


ave h

eigh

t (m

)


Incoming wave



2= 90








Values 1205732= 0 120573

2= 15 120573

2= 45 120573

2= 60 120573

2= 75 120573

2= 90

HH1015840 1078973 1185936 1297549 132081 1470258 1503094


11987111198712

1205731

1205732

1198712

1198711


0010203

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave



8 Results



001020304

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave


0010203

0 20 40 60 80 100


Incoming wave

Wav

e hei

ght

(m)

X (m)minus01minus02


00204

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus02

minus04


Incoming wave


0

05

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus05


Incoming wave





Values 11987111198712= 9 119871

11198712= 633 119871

11198712= 42 119871

11198712= 367

HH1015840 1094116 1105263 1111112 1222223

0

02

04

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus02



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01





Right sideslope

Breakwaterheight

Lowerlength



Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05

HH1015840 1247285 142718 13381 1337985



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01




0

02

04

020 40 60 80 100W

ave h

eigh

t (m

)

X (m)minus02










References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119908 (119903 ℎ) = 120572119863

1 minus3

21199022+3

411990230 le 119902 le 1

1

4(2 minus 119902)

3

1 le 119902 le 2

0 119902 ge 2

(6)





119889120588119894

119889119905= minus

119873

sum

119895=1


119894119895 (7)



119889V119894

119889119905= minus

119873

sum

119895=1

119898119895(119875119894

1205882

119894

+

119875119895

1198752

119895

+ Π119894119895)nabla119882

119894119895+ 119892 (8)



Π119886119887=

minus120572119862119886119887120583119886119887

120588119886119887

119881119886119887sdot 119903119886119887lt 0


where

120583119886119887=ℎV119886119887sdot 119903119886119887

1199032

119886119887+ 1205782

(10)

120588119886119887=1

2(120588119886+ 120588119887) 119862

119886119887=1

2(119862119886+ 119862119887)

1205782= 001ℎ

2

(11)


5 Numerical Details


119889119903119894

119889119905= V119894+ 120576sum

119895

119898119895

120588119894119895

V119894119895119882119894119895 (12)



V119899+12119894

= V119899119894+Δ119905

2119865119899+12

119894 (13)


V119899+1119894= 2V119899+12119894

minus V119899119894 (14)


120588119886= sum

119887

120588119887119882

MLS119886119887

119898119887

120588119887

= sum

119887

119898119887119882

MLS119886119887 (15)


119882MLS119886119887

= 119882MLS119886119887

(119903119886) = 120573 (119903

119886) sdot (119903119886minus 119903119887)119882119886119887 (16)




dx dx

dx2 dz2




119883119901(119905) =

119867

119896[tanh (119883 (119905)) + tanh(119896

119889120582)]

119883 (119905) =119896

119889(119888119905 minus 119883

119901(119905) minus 120582)

(17)


119906119901(119905) =

119888119867

119889

1

cosh2119883 (119905) + (119867119889) (18)






H

d

R

L

120573


0

002

004

0 2 4 6 8 10 12Wav

e hei

ght

(m)


minus002X (m)




0)







0

05

0 1 2 3 4

Wav

e hei

ght

(m)

X (m)minus05



(a)

0

05

0 1 2 3 4

Experimental data

Wav

e hei

ght

(m)

X (m)minus05


SPH

(b)

0

05

0 2 4

Wav

e hei

ght

(m)

X (m)minus2minus4


(c)


0

05

0 1 2 3 4

Wav

e hei

ght

(m)

X (m)minus05


(d)


Loveral

L0 L1

L2

1205731

1205732

d

h






1205732

1205731

ℎ 1198711


0010203

0 20 40 60 80 100X (m)


ave h

eigh

t (m

)


Incoming wave



2= 90








Values 1205732= 0 120573

2= 15 120573

2= 45 120573

2= 60 120573

2= 75 120573

2= 90

HH1015840 1078973 1185936 1297549 132081 1470258 1503094


11987111198712

1205731

1205732

1198712

1198711


0010203

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave



8 Results



001020304

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave


0010203

0 20 40 60 80 100


Incoming wave

Wav

e hei

ght

(m)

X (m)minus01minus02


00204

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus02

minus04


Incoming wave


0

05

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus05


Incoming wave





Values 11987111198712= 9 119871

11198712= 633 119871

11198712= 42 119871

11198712= 367

HH1015840 1094116 1105263 1111112 1222223

0

02

04

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus02



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01





Right sideslope

Breakwaterheight

Lowerlength



Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05

HH1015840 1247285 142718 13381 1337985



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01




0

02

04

020 40 60 80 100W

ave h

eigh

t (m

)

X (m)minus02










References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










dx dx

dx2 dz2




119883119901(119905) =

119867

119896[tanh (119883 (119905)) + tanh(119896

119889120582)]

119883 (119905) =119896

119889(119888119905 minus 119883

119901(119905) minus 120582)

(17)


119906119901(119905) =

119888119867

119889

1

cosh2119883 (119905) + (119867119889) (18)






H

d

R

L

120573


0

002

004

0 2 4 6 8 10 12Wav

e hei

ght

(m)


minus002X (m)




0)







0

05

0 1 2 3 4

Wav

e hei

ght

(m)

X (m)minus05



(a)

0

05

0 1 2 3 4

Experimental data

Wav

e hei

ght

(m)

X (m)minus05


SPH

(b)

0

05

0 2 4

Wav

e hei

ght

(m)

X (m)minus2minus4


(c)


0

05

0 1 2 3 4

Wav

e hei

ght

(m)

X (m)minus05


(d)


Loveral

L0 L1

L2

1205731

1205732

d

h






1205732

1205731

ℎ 1198711


0010203

0 20 40 60 80 100X (m)


ave h

eigh

t (m

)


Incoming wave



2= 90








Values 1205732= 0 120573

2= 15 120573

2= 45 120573

2= 60 120573

2= 75 120573

2= 90

HH1015840 1078973 1185936 1297549 132081 1470258 1503094


11987111198712

1205731

1205732

1198712

1198711


0010203

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave



8 Results



001020304

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave


0010203

0 20 40 60 80 100


Incoming wave

Wav

e hei

ght

(m)

X (m)minus01minus02


00204

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus02

minus04


Incoming wave


0

05

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus05


Incoming wave





Values 11987111198712= 9 119871

11198712= 633 119871

11198712= 42 119871

11198712= 367

HH1015840 1094116 1105263 1111112 1222223

0

02

04

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus02



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01





Right sideslope

Breakwaterheight

Lowerlength



Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05

HH1015840 1247285 142718 13381 1337985



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01




0

02

04

020 40 60 80 100W

ave h

eigh

t (m

)

X (m)minus02










References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

05

0 1 2 3 4

Wav

e hei

ght

(m)

X (m)minus05



(a)

0

05

0 1 2 3 4

Experimental data

Wav

e hei

ght

(m)

X (m)minus05


SPH

(b)

0

05

0 2 4

Wav

e hei

ght

(m)

X (m)minus2minus4


(c)


0

05

0 1 2 3 4

Wav

e hei

ght

(m)

X (m)minus05


(d)


Loveral

L0 L1

L2

1205731

1205732

d

h






1205732

1205731

ℎ 1198711


0010203

0 20 40 60 80 100X (m)


ave h

eigh

t (m

)


Incoming wave



2= 90








Values 1205732= 0 120573

2= 15 120573

2= 45 120573

2= 60 120573

2= 75 120573

2= 90

HH1015840 1078973 1185936 1297549 132081 1470258 1503094


11987111198712

1205731

1205732

1198712

1198711


0010203

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave



8 Results



001020304

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave


0010203

0 20 40 60 80 100


Incoming wave

Wav

e hei

ght

(m)

X (m)minus01minus02


00204

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus02

minus04


Incoming wave


0

05

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus05


Incoming wave





Values 11987111198712= 9 119871

11198712= 633 119871

11198712= 42 119871

11198712= 367

HH1015840 1094116 1105263 1111112 1222223

0

02

04

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus02



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01





Right sideslope

Breakwaterheight

Lowerlength



Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05

HH1015840 1247285 142718 13381 1337985



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01




0

02

04

020 40 60 80 100W

ave h

eigh

t (m

)

X (m)minus02










References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Values 1205732= 0 120573

2= 15 120573

2= 45 120573

2= 60 120573

2= 75 120573

2= 90

HH1015840 1078973 1185936 1297549 132081 1470258 1503094


11987111198712

1205731

1205732

1198712

1198711


0010203

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave



8 Results



001020304

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01minus02


Incoming wave


0010203

0 20 40 60 80 100


Incoming wave

Wav

e hei

ght

(m)

X (m)minus01minus02


00204

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus02

minus04


Incoming wave


0

05

0 20 40 60 80 100

Wav

e hei

ght

(m)

X (m)minus05


Incoming wave





Values 11987111198712= 9 119871

11198712= 633 119871

11198712= 42 119871

11198712= 367

HH1015840 1094116 1105263 1111112 1222223

0

02

04

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus02



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01





Right sideslope

Breakwaterheight

Lowerlength



Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05

HH1015840 1247285 142718 13381 1337985



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01




0

02

04

020 40 60 80 100W

ave h

eigh

t (m

)

X (m)minus02










References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Values 11987111198712= 9 119871

11198712= 633 119871

11198712= 42 119871

11198712= 367

HH1015840 1094116 1105263 1111112 1222223

0

02

04

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus02



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01





Right sideslope

Breakwaterheight

Lowerlength



Damping ratio 119883 = 02 119883 = 03 119883 = 04 119883 = 05

HH1015840 1247285 142718 13381 1337985



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01



0010203

0 20 40 60 80 100Wav

e hei

ght

(m)

X (m)minus01




0

02

04

020 40 60 80 100W

ave h

eigh

t (m

)

X (m)minus02










References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

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