Hydrodynamic Forces and Motions in the Time Domain for an ...accuracy of ship motion predictions in...

Defence R&D Canada

DEFENCE DÉFENSE&

Hydrodynamic Forces and Motions in the

Time Domain for an Unappended Ship Hull

Kevin McTaggart

Technical Memorandum

DRDC Atlantic TM 2003-104

May 2003

Copy No.________

Defence Research andDevelopment Canada

Recherche et développementpour la défense Canada

This page intentionally left blank.

Hydrodynamic Forces and Motions in theTime Domain for an Unappended Ship Hull

Kevin McTaggart

Defence R&D Canada – Atlantic



May 2003

Abstract

This report describes numerical predictions of hydrodynamic forces and motions inthe time domain for an unappended ship hull in waves. The hydrodynamic forces inthe time domain are evaluated using previously computed values in the frequencydomain, which are based on the three-dimensional Green function for a body withzero forward speed. Forward speed effects are included in both the frequency andtime domains using suitable approximations. Due to the underlying importance ofnumerical predictions of hydrodynamic forces in the frequency domain, attentionhas been given to their accuracy, robustness, and efficiency. Comparisons betweenmotion predictions in the frequency and time domains show excellent agreement,suggesting correct theoretical and numerical implementations. Comparisons of nu-merical heave and pitch predictions with model test data for the frigate HALIFAXshow very good agreement. Time domain simulations of ship motion execute sig-nificantly faster than real time, indicating that the approach will be practical formodelling and simulation applications.

Resum e

Dans ce rapport nous decrivons les predictions numeriques calculees dans le do-maine temporel des forces hydrodynamiques et des mouvements d’une coque sansappendices dans les vagues. Nous avonsevalue les forces hydrodynamiques dansle domaine temporel en utilisant les valeurs calculees anterieurement dans le do-maine frequentiel avec une fonction de Green tridimensionnelle pour un corps devitesse avant nulle. Les effets de la vitesse avant ontete ajoutes dans les domainesfrequentiel et temporel en utilisant des approximations adaptees.Etant donne l’im-portance intrinseque des predictions numeriques des forces hydrodynamiques dansle domaine frequentiel, nous avons apporte une attention particulierea la precision,a la robustesse eta l’efficacite des calculs. L’accord entre les predictions des mou-vements dans les domaines temporel et frequentiel est bon, ce qui suggere que latheorie et son implementation logicielle soient correctes. L’accord entre les predic-tions numeriques du soulevement et du tangage pour la fregate HALIFAX avec lesdonnees sur maquette est tres bon. Le calcul des simulations des mouvements dunavire dans le domaine temporel est beaucoup plus rapide que le calcul en tempsreel, ce qui indique que l’on pourra utiliser notre methode pour la modelisation etles simulations.

DRDC Atlantic TM 2003-104 i


ii DRDC Atlantic TM 2003-104

Executive summary

Introduction

DRDC Atlantic is developing a new object-oriented library for simulation of shipmotions in waves. A previous report presented predictions of ship hydrodynamicforces and motions in the frequency domain using a panel method. This reportpresents a method for extending hydrodynamic force and motion predictions to thetime domain using previously computed values for the frequency domain.

Principal Results

Improvements have been made to accuracy, robustness, and computational speedof hydrodynamic force predictions in the frequency domain. New hydrodynamicforce predictions in the time domain produce motions which are in agreement withpredictions in the frequency domain, suggesting correct theoretical and numericalimplementations. Comparisons of heave and pitch predictions with experimentalresults show very good agreement.

Significance of Results

New object-oriented software is now available for simulating motions in wavesof an unappended ship. The new software uses a panel method, ensuring fidelityfor non-slender hull forms such as the Canadian Navy’s KINGSTON class. Theinclusion of nonlinear forces from buoyancy and incident waves can contribute toaccuracy of ship motion predictions in higher sea states. Both linear and nonlinearpredictions run significantly faster than real time, which is important for simulationapplications.

Future Plans

In the near term, forces from rudders, bilge keels, and other appendages will beincorporated into numerical predictions of ship motions in waves. In the longerterm, the motions of a freely maneuvering ship in waves will be predicted.

Kevin McTaggart; 2003; Hydrodynamic Forces and Motions in theTime Domain for an Unappended Ship Hull; DRDC Atlantic TM2003-104; Defence R&D Canada – Atlantic.

DRDC Atlantic TM 2003-104 iii

Sommaire

Introduction

RDDC assemble une nouvelle bibliotheque de logiciels orientes objet qui seront utilisespour simuler le mouvement des navires dans les vagues. Dans un rapport precedent, nousavons presente les predictions des forces hydrodynamiques exercees sur le navire et de sesmouvements, calculees dans le domaine des frequences avec la methode des panneaux.Dans ce rapport, nous presentons une methode permettant d’etendre les predictions desforces hydrodynamiques et des mouvements, au domaine temporel,a partir des valeursdeja calculees dans le domaine frequentiel.

Resultats principaux

Nous avons ameliore la precision, la robustesse et la vitesse du calcul des predictions desforces hydrodynamiques dans le domaine des frequences. Les nouvelles predictions pour laforce dans le domaine temporel se traduisent par des mouvements qui concordent avec lespredictions dans le domaine frequentiel, ce qui suggere que la theorie et son implementationlogicielle sont correctes. L’accord entre les predictions pour le soulevement et le tangage etles resultats experimentaux est bon.

Importance des r esultats

Un nouveau logiciel oriente objet permettant de simuler les mouvements d’un navire munid’aucun appendice dans des vagues est maintenant disponible. Les predictions de ce nou-veau logiciel sont effectueesa partir de la methode des panneaux, ce qui garantit l’exacti-tude des calculs pour les coques larges comme celles des navires de la classe Kingston dela Marine canadienne. L’ajout des forces non lineaires de la flottabilite et des vagues inci-dentes pourra contribuera la precision des predictions des mouvements des navires dans lesmers tres agitees. Le calcul lineaire et non lineaire des predictions est beaucoup plus rapideque le calcul en temps reel, unelement important pour les simulations.

Travaux ult erieurs pr evus

Nous incorporerons prochainement les forces exercees par les gouvernails, les quilles deroulis et autres appendices aux predictions numeriques des mouvements des navires dansles vagues.A plus long terme, on pourra predire des mouvements de navires manœuvrantlibrement dans les vagues.

Kevin McTaggart; 2003; Hydrodynamic Forces and Motions in theTime Domain for an Unappended Ship Hull; DRDC Atlantic TM2003-104; Defence R&D Canada – Atlantic.

iv DRDC Atlantic TM 2003-104

Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Sommaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Review of Hydrodynamic Coefficients in the Frequency Domain . . . 2

3 Hydrodynamic Forces for Transient Motions . . . . . . . . . . . . . . 6

3.1 Retardation Functions . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Hydrodynamic Damping and Stiffness . . . . . . . . . . . . . 8

3.3 Wave Excitation Forces . . . . . . . . . . . . . . . . . . . . . 9

4 CPF Hydroelastic Model Hull for Sample Computations ofHydrodynamic Coefficients . . . . . . . . . . . . . . . . . . . . . . . 10

5 Improvements to Computations of Frequency Domain Coefficients . . 13

5.1 Maintenance of Points on Waterline During PanelTransformation . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.2 Sign Correction to Speed Dependent Added Mass . . . . . . . 13

5.3 Improved Accuracy forx Derivatives of Velocity Potential . . 13

5.4 Gauss Quadrature for Integration from Source Panel . . . . . . 15

5.5 Galerkin Method for Evaluation of Velocity PotentialInfluence Coefficients . . . . . . . . . . . . . . . . . . . . . . 15

5.6 Additional C Code for Faster Computations . . . . . . . . . . 15

DRDC Atlantic TM 2003-104 v

5.7 Removal of Irregular Frequencies from Database . . . . . . . 17

5.8 Direct Evaluation of Incident Wave Forces in Database ofRadiation and Wave Excitation Forces . . . . . . . . . . . . . 25

5.9 Improved Efficiency for Evaluating Diffracted Wave Potentials 25

5.10 Interpolation of Inverse of Normal Velocity Influence Matrixwhen Evaluating Diffraction Potentials . . . . . . . . . . . . . 27

5.11 New Procedure for Computation of Diffraction Forces froma Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.12 Faster Versions of LAPACK Routines for Solution ofSystems of Linear Equations . . . . . . . . . . . . . . . . . . 29

6 Hydrodynamic Forces in the Frequency Domain for HALIFAX . . . . 31

6.1 Damping andx Derivative of Added Mass . . . . . . . . . . . 33

6.2 Wave Excitation Forces on HALIFAX . . . . . . . . . . . . . 40

6.3 HALIFAX Motions in the Frequency Domain Using NewComputational Methods . . . . . . . . . . . . . . . . . . . . . 43

7 Hydrodynamic Coefficients in the Time Domain for HALIFAX . . . . 46

8 Time Domain Simulation of Ship Motions . . . . . . . . . . . . . . . 53

9 Comparison of Motions in the Time and Frequency Domains . . . . . 54

10 Extension to Nonlinear Forces from Incident Waves and Buoyancy . . 57

11 Comparison of Numerical Predictions and Experimental Results forHeave and Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Document Control Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

vi DRDC Atlantic TM 2003-104

List of tables

1 Main Particulars for HALIFAX Class Frigate, CPF HydroelasticModel Deep Departure Condition . . . . . . . . . . . . . . . . . . . . 10

2 Hydrostatic Properties for Different Panel Meshes . . . . . . . . . . . 12

3 Typical CPU Times (seconds) Per Encounter Frequency forPrediction of Hull Added Mass and Damping Coefficients inFrequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Parameters for Databases of Frequency Domain Radiation andDiffraction Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Required CPU Time for Generating Database of Radiation andDiffraction Computations . . . . . . . . . . . . . . . . . . . . . . . . 31

DRDC Atlantic TM 2003-104 vii

List of figures

1 Coordinate System for Solution of Hydrodynamic Forces . . . . . . . 3

2 Sea Direction for Solution of Hydrodynamic Forces . . . . . . . . . . 3

3 Coarse Mesh for CPF Hydroelastic Model . . . . . . . . . . . . . . . 11

4 Medium Mesh for CPF Hydroelastic Model . . . . . . . . . . . . . . 11

5 Fine Mesh for CPF Hydroelastic Model . . . . . . . . . . . . . . . . 12

6 Dimensionless Forces from Zero Speed Heave Damping andxDerivative for HALIFAX at 20 Knots, Medium Mesh, with Centroidand Four Point Approximations . . . . . . . . . . . . . . . . . . . . . 16

7 Normalized Added Mass and Damping, and Normal VelocityCondition Number for HALIFAX Coarse Mesh, Longitudinal Modes . 19

8 Normalized Added Mass and Damping, and Normal VelocityCondition Number for HALIFAX Coarse Mesh, Lateral Modes . . . . 20

9 Normalized Added Mass and Damping, and Normal VelocityCondition Number for HALIFAX Medium Mesh, Longitudinal Modes 21

10 Normalized Added Mass and Damping, and Normal VelocityCondition Number for HALIFAX Medium Mesh, Lateral Modes . . . 22

11 Normalized Added Mass and Damping, and Normal VelocityCondition Number for HALIFAX Fine Mesh, Longitudinal Modes . . 23

12 Normalized Added Mass and Damping, and Normal VelocityCondition Number for HALIFAX Fine Mesh, Lateral Modes . . . . . 24

13 HALIFAX Limiting Condition Numbers for Longitudinal ModesSurge, Heave, and Pitch . . . . . . . . . . . . . . . . . . . . . . . . . 32

14 HALIFAX Limiting Condition Numbers for Lateral Modes Sway,Roll, and Yaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

15 Dimensionless Forces from Zero Speed Surge Damping andxDerivative of Added Mass for HALIFAX at 20 Knots . . . . . . . . . 34

16 Dimensionless Forces from Zero Speed Sway Damping andxDerivative of Added Mass for HALIFAX at 20 Knots . . . . . . . . . 35

viii DRDC Atlantic TM 2003-104

17 Dimensionless Forces from Zero Speed Heave Damping andxDerivative of Added Mass for HALIFAX at 20 Knots . . . . . . . . . 36

18 Dimensionless Forces from Zero Speed Roll Damping andxDerivative of Added Mass for HALIFAX at 20 Knots . . . . . . . . . 37

19 Dimensionless Forces from Zero Speed Pitch Damping andxDerivative of Added Mass for HALIFAX at 20 Knots . . . . . . . . . 38

20 Dimensionless Forces from Zero Speed Yaw Damping andxDerivative of Added Mass for HALIFAX at 20 Knots . . . . . . . . . 39

21 Wave Excitation Forces for HALIFAX, Medium Mesh, 20 knots,Stern Quartering Seas at 30 degrees . . . . . . . . . . . . . . . . . . 41

22 Wave Excitation Forces for HALIFAX, Medium Mesh, 20 knots,Bow Quartering Seas at 150 degrees . . . . . . . . . . . . . . . . . . 42

23 Frequency Domain Motions for HALIFAX from Direct andDatabase Computations, Medium Mesh, 20 knots, Stern QuarteringSeas at 30 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

24 Frequency Domain Motions for HALIFAX from Direct andDatabase Computations, Medium Mesh, 20 knots, Bow QuarteringSeas at 150 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

25 HALIFAX Surge Retardation ContributionsK011 andUKU

11 for U =20 knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

26 HALIFAX Sway Retardation ContributionsK022 andUKU

22 for U =20 knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

27 HALIFAX Heave Retardation ContributionsK033 andUKU

33 for U =20 knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

28 HALIFAX Roll Retardation ContributionsK044 andUKU

44 for U = 20knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

29 HALIFAX Pitch Retardation ContributionsK055, UKU

55, andU2KUU55

for U = 20 knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

30 HALIFAX Yaw Retardation ContributionsK066, UKU

66, andU2KUU66

for U = 20 knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

DRDC Atlantic TM 2003-104 ix

31 Motions for HALIFAX from Frequency Domain and Time DomainComputations, Medium Mesh, 20 knots, Stern Quartering Seas at 30degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

32 Motions for HALIFAX from Frequency Domain and Time DomainComputations, Medium Mesh, 20 knots, Bow Quartering Seas at150 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

33 Ship Referenced Coordinate System for Large Angular Motions . . . 57

34 Heave Buoyancy Forces for HALIFAX, Medium Mesh . . . . . . . . 60

35 Roll Buoyancy Moments for HALIFAX, Medium Mesh . . . . . . . . 60

36 Pitch Buoyancy Forces for HALIFAX, Medium Mesh . . . . . . . . . 61

37 Heave and Pitch for CPF Hydroelastic Model in Head Seas,Fn = 0.12 63

38 Heave and Pitch for CPF Hydroelastic Model in Head Seas,Fn = 0.20 64

x DRDC Atlantic TM 2003-104

1 Introduction

Simulations of ship motions in waves are required for numerous naval applications.When evaluating performance of crew and ship systems, the influence of ship mo-tions is often significant. The safety of a ship will be highly dependent on its abilityto resist capsize and wave-induced structural loads.

To facilitate simulation efforts, DRDC Atlantic is developing a new object-orientedlibrary for predicting ship motions in waves. During development of this library,attention is being given to balancing requirements for fidelity, efficiency, and ro-bustness. It was decided that these requirements could be satisfied by computingship motions in the time domain using hull hydrodynamic forces derived from com-putations in the frequency domain. Reference 1 describes computation of hull hy-drodynamic forces in the frequency domain using a panel method. The presentreport describes the evaluation of hydrodynamic forces in the time domain usingpreviously computed results in the frequency domain. Section 2 gives a review ofhydrodynamic coefficients in the frequency domain based on Reference 1. The the-ory for obtaining time domain forces from frequency domain values is presentedin Section 3. Section 4 gives particulars for the ship HALIFAX, which is used forsample computations throughout the remainder of the report. Improvements to fre-quency domain computations are given in Section 5, followed by sample frequencydomain coefficients in Section 6. Section 7 gives sample time domain coefficientsbased upon previously computed frequency domain values. Section 8 discussescomputation of ship motions in the time domain using time domain hydrodynamiccoefficients. To verify the new time domain approach, Section 9 compares mo-tions computed in the time and frequency domains. The extension of time domainpredictions to include nonlinear contributions from buoyancy and incident waves isdescribed in Section 10, followed by validation of computations using experimentaldata in Section 11. Section 12 gives concluding remarks.

DRDC Atlantic TM 2003-104 1

2 Review of Hydrodynamic Coefficients inthe Frequency Domain

As mentioned previously, this report describes prediction of hull hydrodynamicforces in the time domain using previously computed frequency domain values.For completeness, a review of frequency domain coefficients from Reference 1 ispresented here.

Figure 1 shows the coordinate system used for ship motion computations. The shipis assumed to move at steady forward speed and heading, and the coordinate systemis a translating earth system moving with the steady forward speed. The origin ofthe translating earth system is located at the ship centre of gravity when the ship isin calm water. The location of the translating earth system is not affected by wave-induced ship motions. Note that the present coordinate system is slightly differentfrom that used for solution of hydrodynamic forces in Reference 1, for which thevertical z axis passed through the centre of gravity but had its origin at the calmwater surface. Reference 1 used a different origin because many computationalterms were dependent upon elevation relative to the calm water surface. In thepresent axis system, elevation relative to the calm water surface is given by:

zwl = z + zCGwl (1)

wherezCGwl is the elevation of the ship centre of gravity relative to the calm water

surface when the ship is at rest. Figure 2 gives the sea direction convention, with aheadingβ of 180 degrees representing head seas.

Oscillatory ship motions in the time domain are related to ship motions in the fre-quency domain as follows:

ηj(t) = Real{ηj exp(i ωe t)} for j = 1− 6 (2)

whereωe is wave encounter frequency given by:

ωe = |ωI − kI U cos β| (3)

whereωI is incident wave frequency,kI is incident wavenumber,U is forward shipspeed, andβ is incident wave direction. Water depth is assumed to be large (i.e.,greater than half the incident wavelength), and the following dispersion relationapplies:

kI =ω2

I

g(4)

2 DRDC Atlantic TM 2003-104

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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.................

........

........

........

........

........

........

........

........

........

........

.........................

...................

zη3

......................................................................................................... ................... xCG•

........................................................................................................................................................

.....................

...................η5

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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........

........

........

........

........

.........................

...................

yη2

......................................................................................................... ...................x

η1•

..............................................................................

..........................................................................

.........................................

η6

.................

.................

.................

.................

.................

.................

........................................................................................................................................................................................................................................................................................................................................................

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............................................................................................................................y ........

........

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........

........

........

.........................

...................

z

CG•..................................................................

......................................................................................

.........................................η4

...........................................................................................................................................................................

Figure 1: Coordinate System for Solution of Hydrodynamic Forces

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........................................................................................................................................................................................................................................................................................................................................................................................................................................ ...............................Ship

.................................................................................................................................................................................................................................................................................................................................

..............................

.Sea

•..................................................................

................................................

......................................... β

Figure 2: Sea Direction for Solution of Hydrodynamic Forces


whereg is gravitational acceleration. The equations of ship motions in the fre-quency domain are given by:{

−ω2e ([M ] + [A]) + iωe[B] + [C]

}{η} =

{F I + FD

}(5)

where[M ] is the ship mass matrix,[A] is the added mass matrix,[B] is the dampingmatrix, [C] is the stiffness matrix,

{F I}

is the incident wave excitation vector, and{FD}

is the diffracted wave excitation vector. The added mass and damping matrixterms are given by:

Ajk = − ρ

ωe

∫Sb

[Imag(φk) +

U

ωe

Real

(∂φk

∂x

)]nj dS (6)

Bjk = −ρ

∫Sb

[Real(φk) −

U

ωe

Imag

(∂φk

∂x

)]nj dS (7)

whereρ is water density,Sb denotes the wetted surface of the ship,φk is the complexvelocity potential due to oscillatory motions for modek, andnj is the outwardnormal component for modej. Note that Equation (6) has a correction to a signerror for the speed term in Reference 1. The above equations can be rewritten toshow speed dependent terms as follows:

Ajk = − ρ

ωe

∫Sb

Imag(φk)nj dS − ρ

ωe

U

ωe

∫Sb

Real

(∂φk

∂x

)nj dS (8)

Bjk = −ρ

∫Sb

Real(φk) nj dS + ρU

ωe

∫Sb

Imag

(∂φk

∂x

)nj dS (9)

At zero and infinite frequency limits, the real part of the velocity potential goesto zero, and the velocity potential can be represented in terms of normalized realquantities as follows:

φ(ωe) = i φ′(0) ωe for ωe → 0 (10)

φ(ωe) = i φ′(∞) ωe for ωe →∞ (11)

At these limits the added mass and damping can be evaluated as:

Ajk = −ρ

∫Sb

φ′knj dS (12)

Bjk = ρ U

∫Sb

∂φ′k∂x

nj dS (13)

For pitch and yaw, the velocity potentials at forward speed include contributionsfrom heave and sway respectively as follows:

φ5(U, ωe) = φ5(0, ωe) +U

i ωe

φ3(0, ωe) (14)

φ6(U, ωe) = φ6(0, ωe) −U

i ωe

φ2(0, ωe) (15)


Added mass and damping coefficients for pitch can be expressed as:

Aj5 = − ρ

ωe

∫Sb

Imag(φ5(0, ωe)) nj dS +ρ

ωe

U

ωe

∫Sb

Real(φ3) nj dS

− ρ

ωe

U

ωe

∫Sb

Real

(∂φ5(0, ωe)

∂x

)nj dS

− ρ

ωe

(U

ωe

)2 ∫Sb

Imag

(∂φ3

∂x

)nj dS (16)

Bj5 = −ρ

∫Sb

Real(φ5(0, ωe)) nj dS − ρU

ωe

∫Sb

Imag(φ3) nj dS

+ ρU

ωe

∫Sb

Imag

(∂φ5(0, ωe)

∂x

)nj dS

− ρ

(U

ωe

)2 ∫Sb

Real

(∂φ3

∂x

)nj dS (17)

At the zero and upper frequency limits, the above equations can be re-written as:

Aj5 = −ρ

∫Sb

(φ′5)nj dS − ρ

(U

ωe

)2 ∫Sb

∂φ′3∂x

nj dS (18)

Bj5 = −ρ U

∫Sb

(φ′3)nj dS + ρ U

∫Sb

∂φ′5∂x

nj dS (19)

For yaw motion, added mass and damping terms are:

Aj6 = − ρ

ωe

∫Sb

Imag(φ6(0, ωe)) nj dS − ρ

ωe

U

ωe

∫Sb

Real(φ2) nj dS

− ρ

ωe

U

ωe

∫Sb

Real

(∂φ6(0, ωe)

∂x

)nj dS

+ρ

ωe

(U

ωe

)2 ∫Sb

Imag

(∂φ2

∂x

)nj dS (20)

Bj6 = −ρ

∫Sb

Real(φ6(0, ωe)) nj dS + ρU

ωe

∫Sb

Imag(φ2) nj dS

+ ρU

ωe

∫Sb

Imag

(∂φ6(0, ωe)

∂x

)nj dS

+ ρ

(U

ωe

)2 ∫Sb

Real

(∂φ2

∂x

)nj dS (21)

Yaw motion terms at the zero and upper frequency limits are:

Aj6 = −ρ

∫Sb

(φ′6)nj dS + ρ

(U

ωe

)2 ∫Sb

∂φ′2∂x

nj dS (22)

Bj6 = ρ U

∫Sb

(φ′2)nj dS + ρ U

∫Sb

∂φ′6∂x

nj dS (23)


For computational purposes, it is useful to be able to express ship hydrodynamiccoefficients at forward speed as functions of zero speed coefficients as follows:

Ajk(U, ωe) = Ajk(0, ωe) +U

ω2e

∂Bjk(0, ωe)

∂xfor k = 1− 4 (24)

Bjk(U, ωe) = Bjk(0, ωe) − U∂Ajk(0, ωe)

∂xfor k = 1− 4 (25)

Aj5(U, ωe) = Aj5(0, ωe) −U

ω2e

Bj3(0, ωe) +U

ω2e

∂Bj5(0, ωe)

∂x

+

(U

ωe

)2∂Aj3(0, ωe)

∂x(26)

Bj5(U, ωe) = Bj5(0, ωe) + U Aj3(0, ωe) − U∂Aj5(0, ωe)

∂x

+

(U

ωe

)2∂Bj3(0, ωe)

∂x(27)

Aj6(U, ωe) = Aj6(0, ωe) +U

ω2e

Bj2(0, ωe) +U

ω2e

∂Bj6(0, ωe)

∂x

−(

U

ωe

)2∂Aj2(0, ωe)

∂x(28)

Bj6(U, ωe) = Bj6(0, ωe) − U Aj2(0, ωe) − U∂Aj6(0, ωe)

∂x

−(

U

ωe

)2∂Bj2(0, ωe)

∂x(29)

The above equations are the basis for a ship hydrodynamic coefficient databasewhich gives coefficients at arbitrary forward speed based on stored coefficients forzero forward speed.

3 Hydrodynamic Forces for TransientMotions

The hydrodynamic forces for transient motions of floating bodies are discussed inseveral references. Mei [2] and Wehausen [3] give comprehensive overviews. Mc-Taggart [4] discusses transient motions of icebergs during drifting and collisionswith offshore structures. Magee [5] gives a good overview of transient hydrody-namic forces acting on ships, and includes forward speed effects. Fonseca andGuedes Soares [6] present theory and results for time domain ship motions basedon frequency domain coefficients.

For a floating body undergoing transient motions, the equations of motion can be


expressed as:

([M ] + [A(U,∞)]) {η(t)} + [b(U)]{

˙η(t)}

+

∫ t

−∞[K(U, t− τ)] {η(τ)} dτ

+ ([C] + [c(U)]) {η(t)} ={F I(t) + FD(t)

}(30)

where[b(U)] is the speed dependent damping matrix in the time domain,[K(U, t− τ)]is the retardation function matrix for delay timet − τ , and[c(U)] is the speed de-pendent stiffness matrix in the time domain. Terms in the above equation can beevaluated using results from frequency domain computations.

3.1 Retardation Functions

The retardation functionsKjk(U, τ) can be determined from frequency domainadded mass or damping coefficients as follows:

Kjk(τ) = − 2

π

∫ ∞

0

[Ajk(U, ωe) − Ajk(U,∞)] ωe sin ωeτ dωe (31)

Kjk(τ) =2

π

∫ ∞

0

Bjk(U, ωe) cos ωeτ dωe (32)

When selecting between the above two equations for evaluating retardation func-tions, it should be noted that damping coefficients approach their infinite frequencylimits much more quickly than do added masses. Given the problems with irregularfrequencies that can occur at higher frequencies, it was decided to compute re-tardation functions based on damping where possible. Fonseca and Guedes Soarespresent a similar approach based on hydrodynamic coefficients evaluated using striptheory.

It is convenient to express retardation functions in terms of speed-independent andspeed-dependent terms. For modes for which velocity potentials are assumed tobe independent of ship speed (i.e., surge, sway, heave, and roll), the retardationfunctions can be expressed as:

Kjk(U, τ) = K0jk(τ) + U KU

jk(τ) for k = 1− 4 (33)

The speed-independent retardation functions are evaluated using:

K0ij(τ) =

2

π

∫ ∞

0

Bjk(0, ωe) cos ωeτ dωe for k = 1− 6 (34)


Note that the above equation is valid for all six degrees of freedom. The speeddependent terms based on Equations (25) and (32) are:

KUjk(τ) = − 2

π

∫ ∞

0

[∂Ajk(0, ωe)

∂x− ∂Ajk(0,∞)

∂x

]cos ωet dωe

for k = 1− 4 (35)

For pitch and yaw, the velocity potentials are influenced by forward speed, and theretardation functions are expressed as:

Kjk(U, τ) = K0jk(τ) + U KU

jk(τ) + U2 KUUjk (τ) for k = 5− 6 (36)

The speed-dependent coefficients due to pitch motion are evaluated by:

KUj5(τ) = − 2

π

∫ ∞

0

[∂Aj5(0, ωe)

∂x− ∂Aj5(0,∞)

∂x

]cos ωeτ dωe

+2

π

∫ ∞

0

Bj3(0, ωe)sin ωeτ

ωe

dωe (37)

KUUj5 (τ) =

2

π

∫ ∞

0

∂Bj3(0, ωe)

∂x

cos ωeτ

ω2e

dωe (38)

When evaluatingKUj5 in Equation (37), the contribution from the heave velocity po-

tential is evaluated using heave damping rather than added mass because it has beenfound to give better numerical accuracy. Both Equations (37) and (38) have inte-grands with encounter frequencyωe or its square in the denominator. Fortunately,these integrands are not problematic asωe approaches zero because damping is zeroat zero encounter frequency.

Similarly, the speed-dependent coefficients due to yaw motion are evaluated by:

KUj6(τ) = − 2

π

∫ ∞

0

[∂Aj6(0, ωe)

∂x− ∂Aj6(0,∞)

∂x

]cos ωeτ dωe

− 2

π

∫ ∞

0

Bj2(0, ωe)sin ωeτ

ωe

dωe (39)

KUUj6 (τ) = − 2

π

∫ ∞

0

∂Bj2(0, ωe)

∂x

cos ωeτ

ω2e

dωe (40)

3.2 Hydrodynamic Damping and Stiffness

Hydrodynamic damping and stiffness terms are related to frequency domain valuesas follows:

bjk(U) = − U∂Ajk(0,∞)

∂xfor k = 1− 4 (41)


bj5(U) = − U∂Aj5(0,∞)

∂x+ U Aj3(0,∞) (42)

bj6(U) = − U∂Aj6(0,∞)

∂x− U Aj2(0,∞) (43)

cj5(U) = − U2 ∂Aj3(0, 0)

∂x(44)

cj6(U) = U2 ∂Aj2(0, 0)

∂x(45)

Note that the two stiffness terms arise from converting frequency domain addedmass terms to stiffness terms using multiplication by−ω2

e . The stiffness terms arebased on sway and heave added masses at zero frequency.

3.3 Wave Excitation Forces

Linear wave excitation forces in the time domain can be evaluated using the fre-quency domain forces presented in Reference 1. For a linear wave system, theincident wave elevation in the time domain can be expressed as:

ζI(x, y, t) = a cos [ωe t + εI − kI (x cos β − y sin β)] (46)

wherea is incident wave amplitude,εI is the phase of the incident wave crest (posi-tive for time lead) at the origin,kI is the incident wavenumber, andβ is the incidentwave direction. Wave excitation forces in the time domain can be expressed interms of frequency domain values as follows:

F Ij (t) =

∣∣F I(U, β, ωI)∣∣ cos

[ωet + εI + ε

(F I(U, β, ωI)

)](47)

FDj (t) =

∣∣FD(U, β, ωI)∣∣ cos

[ωet + εI + ε

(FD(U, β, ωI)

)](48)

whereF I(U, β, ωI) is the complex incident excitation force in the frequency do-main andFD(U, β, ωI) is the complex diffraction excitation force in the frequencydomain. The phases of the incident and diffracted excitation forces for incidentwaves with zero phase angles are given by:

ε(F I(U, β, ωI)

)= arctan

Imag{F I(U, β, ωI)

}Real{F I(U, β, ωI)}

(49)

ε(FD(U, β, ωI)

)= arctan

Imag{FD(U, β, ωI)

}Real{FD(U, β, ωI)}

(50)


4 CPF Hydroelastic Model Hull for SampleComputations of HydrodynamicCoefficients

To demonstrate results of computations this report uses the HALIFAX class, alsoknown as the Canadian Patrol Frigate (CPF). To enable comparison of motion pre-dictions with model test results, the loading condition for the present computationsis the deep departure condition for the CPF hydroelastic model, as described inReference 7. Note that a slightly different loading condition was used for samplecomputations of frequency domain coefficients in Reference 1.

Table 1 gives the main particulars. Three different hydrodynamic meshes have beengenerated, as shown in Figures 3 to 5. The panel colours in each figure representdepth below the waterline. Table 2 gives hydrostatic properties computed usingeach mesh. Preliminary hydrodynamic computations indicated that thex deriva-tive of added mass was somewhat sensitive to the limiting aspect ratio (lenght-wise/girthwise) used for panelling the hull. Limiting aspect ratios of 3 and 5 weretested, with a value of 3 being ultimately chosen due to better convergence of resultswith increasing number of panels.

Table 1: Main Particulars for HALIFAX Class Frigate, CPF Hydroelastic ModelDeep Departure Condition

Length,L 124.5 m

Beam,B 14.8 m

Midships draft,Tmid 4.97 m

Trim by stern,ts -0.04 m

Displacement,4 4655 tonnes (fresh water)

Vertical centre of gravity,KG 6.26 m

Dry roll radius of gyrationrxx 5.82 m

Dry pitch radius of gyrationryy 28.8 m

Dry yaw radius of gyrationrzz 28.8 m


Figure 3: Coarse Mesh for CPF Hydroelastic Model

Figure 4: Medium Mesh for CPF Hydroelastic Model


Figure 5: Fine Mesh for CPF Hydroelastic Model

Table 2: Hydrostatic Properties for Different Panel Meshes

Coarse Medium Fine

Nominal panel size (m2) 5.0 2.5 1.0

Number of panels on port sideNportp 224 433 1040

Volume (m3) 4465.8 4488.8 4500.6

LCB aft of FP (m) 64.663 64.665 64.678

CB wrt waterline (m) -1.882 -1.889 -1.893

Wetted surface area (m2) 1965.6 1969.4 1971.2


5 Improvements to Computations ofFrequency Domain Coefficients

The present approach for computing time domain hydrodynamic coefficients re-quires accurate and efficient computation of frequency domain coefficients. Thissection presents several improvements that have been made to computations of fre-quency domain coefficients subsequent to the work reported in Reference 1.

5.1 Maintenance of Points on Waterline DuringPanel Transformation

The wetted surface of a hull is represented using triangular and quadrilateral panels.As discussed in Reference 1, it is usually necessary to make slight modifications topoints of quadrilateral panels to ensure that all points lie in the same plane. Pan-elling computations have been changed such that a panel point located on the hullwaterline remains on the waterlinezwl = 0, with only x andy coordinates beingchanged to obtain a planar panel. This modification prevents numerical inconsis-tencies which can occur when panel points are located above the waterline.

5.2 Sign Correction to Speed Dependent AddedMass

As stated in Section 2, Equation 9 of Reference 1 contained a sign error for thespeed dependent term of added mass. Equation (6) gives the correct added mass.

5.3 Improved Accuracy for x Derivatives of VelocityPotential

Improved accuracy has been introduced for evaluating thex derivatives of velocitypotential. The following equation is still used for computing thex derivatives:{

∂φ

∂x

}= [H] {σ} (51)

where[H] is the influence matrix and{σ} is the vector of source strengths. Theinfluence matrix coefficients were previously evaluated as follows:

Hjk = −δjk1

2nx−j + (1 − δjk)

1

4π

∫Sk

∂(1/R(~xj, ~xs))

∂x~xj

dS


+1

4π

∫Sk

∂(1/R1(~xj, ~xs))

∂x~xj

dS

+1

4π

∫Sk

∂G0(~xj, ~xs)

∂x~xj

dS for ωe ≤ ωte (52)

Hjk = −δjk1

2nx−j + (1 − δjk)

1

4π

∫Sk

∂(1/R(~xj, ~xs))

∂x~xj

dS

− 1

4π

∫Sk

∂(1/R1(~xj, ~xs))

∂x~xj

dS

+1

4π

∫Sk

∂G∞(~xj, ~xs)

∂x~xj

dS for ωe > ωte (53)

wherenx−j is thex component of the outward normal on panelj, δjk is the Kroe-necker delta function,Sk is the surface of source panelk, R(~xj, ~xs) is the radiusfrom field point~xj to source panel point~xs, R1 is the radius from the field pointto the image of the source panel point,G0 is the frequency dependent Green func-tion relative to the zero frequency Green function,ωt

e is the transition encounterfrequency for indicating which form of the frequency dependent Green functionto use, andG∞ is the frequency dependent Green function relative to the infinitefrequency Green function. The selection of which of the above equations to usedepends upon a specified transition encounter frequencyωt

e. For the casej = kconsidering the flow from a source panel on itself, the above equations are basedon the following assumption:∫

Sk

∂(1/R(~xj, ~xs))

∂x~xj

dS ≈ −2 π nx−j for j = k (54)

Although this equation is a good approximation, it has been found that it should notbe used because computedx derivatives of velocity potentials can be very sensi-tive to small differences in influence coefficientsHjk. Consequently, the influencecoefficients are now evaluated by:

Hjk =1

4π

∫Sk

∂(1/R(~xj, ~xs))

∂x~xj

dS +1

4π

∫Sk

∂(1/R1(~xj, ~xs))

∂x~xj

dS

+1

4π

∫Sk

∂G0(~xj, ~xs)

∂x~xj

dS for ωe ≤ ωte (55)

Hjk =1

4π

∫Sk

∂(1/R(~xj, ~xs))

∂x~xj

dS − 1

4π

∫Sk

∂(1/R1(~xj, ~xs))

∂x~xj

dS

+1

4π

∫Sk

∂G∞(~xj, ~xs)

∂x~xj

dS for ωe > ωte (56)


5.4 Gauss Quadrature for Integration from SourcePanel

When integrating the Green function from a source panel, the frequency dependentportion was previously approximated using the centroid of the source panel. A newoption has been introduced for using Gauss quadrature (see Cook [8]) on sourcepanels for integrating the frequency dependent portion of the Green function. Thenew implementation has options for using 1, 4, or 9 quadrature points per sourcepanel.

5.5 Galerkin Method for Evaluation of VelocityPotential Influence Coefficients

Previous computations of influence matrices were based on the velocity potential atthe centroid of each panel. The Galerkin approach allows influence matrix terms tobe based on weighted values of the Green function evaluated at 1, 4, or 9 quadraturepoints on each field panel. Sclavounos and Lee [9] give a detailed discussion of thisapproach.

The accuracy of computed velocity potentials will increase with the number ofquadrature points used with the Galerkin method and for integrating the frequencydependent portion of the Green function from source panels. This increased accu-racy comes at the expense of higher computation times due to larger numbers ofcomputations of Green functions. Figure 6 shows heave damping and added massx derivative for two different sets of computations. The first set of computationsuses 4 quadrature points per panel applied to the Galerkin method and integrationof the frequency dependent term from source panels. The second set of computa-tions is based on centroid approximations. At higher frequencies (the first irregularfrequency and higher), the numerical results are sensitive to accuracy of influencecoefficients; thus, the higher order computations give better results, particularly forthex derivative of added mass.

5.6 Additional C Code for Faster Computations

As discussed in Reference 1, most of the present numerical predictions have beendeveloped in Python [10], a high-level, interpreted computer language. When de-veloping Python applications, it is common practice to determine which portionsare CPU intensive and to re-write them in C to obtain the performance of compiled,optimized code. Previously, the frequency dependent portion of the Green func-tion was written in C. New C code has been developed to evaluate the frequencyindependent terms of the Green function. In addition, a new C function has been


0 2 4 6

Encounter Frequencyωe (rad/s)

0

2

4

6

HeaveDamping

B33/(4√

g/L)

.........................................................4 point sources, 4 point Galerkin

. . . . . 1 point sources, 1 point Galerkin

.....................................................

....................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................................................................................................................

0 2 4 6


0

1

2

3

HeaveAdded Massx Derivative

U ∂A33/∂x

4√

g/L

............................

.............................................................................................

..

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................................................................................................................................

..........................................................................................................................................................................................................

Figure 6: Dimensionless Forces from Zero Speed Heave Damping and xDerivative for HALIFAX at 20 Knots, Medium Mesh, with Centroid and Four PointApproximations


developed to evaluate influence matrices for the frequency dependent portion of theGreen function given input vectors of panel points.

5.7 Removal of Irregular Frequencies from Database

Computations of hydrodynamic coefficients can be adversely influenced by irreg-ular frequencies, at which mathematical solutions of velocity potentials are incor-rect. McTaggart [11] and Ando [12] discuss the occurrence of irregular frequenciesin detail.

For modelling and simulation work, it is likely that hydrodynamic coefficients willtypically be determined using interpolation from a previously generated databaseof hydrodynamic coefficients. The accuracy of interpolated coefficients will be im-proved if the database does not have hydrodynamic coefficients adversely affectedby irregular frequencies. Fortunately, a method is available for indicating likely ir-regular frequencies which should be excluded from the database. As was presentedin Reference 1, radiation source strengths on the port side of a symmetrical hull canbe solved by satisfying the following:

[Deven]{σport

j

}=

{∂φport

j /∂n}

for j = 1, 3, 5 (57)[Dodd

] {σport

j

}=

{∂φport

j /∂n}

for j = 2, 4, 6 (58)

where[Deven] is the influence matrix for normal velocity on the hull port side dueto sources on the port side when velocity potential is an even function ofy,

{σport

j

}is the vector of source strengths on the port side for motion modej,

{∂φport

j /∂n}

is the vector of hull normal velocities on the port side, and[Dodd

]is the influence

matrix for normal velocity on the hull port side due to sources on the port side whenvelocity potential is an odd function ofy. The normal velocity influence matricesare given by:

[Deven] =[Dport,port

]+[Dport,star

](59)[

Dodd]

=[Dport,port

]−[Dport,star

](60)

where[Dport,port] is the influence matrix for normal velocities on the port side due tosources on the port side and[Dport,star] is the influence matrix for normal velocitieson the port side due to sources on the starboard side.

As suggested by Lee and Sclavounos [13], irregular frequencies are indicated byhigh condition numbers for the influence matrices[Deven] and

[Dodd

]for their re-

spective motion modes. The condition number for a matrix[D] can be computed asfollows [14]:

λ = ||[D]|| ×∣∣∣∣[D−1

]∣∣∣∣ (61)


where||[D]|| is the norm of matrix[D] and∣∣∣∣[D]−1

∣∣∣∣ is the norm of its inverse. Thenorm of [D] is given by:

||[D]|| = maxN∑

k=1

|Djk| (62)

and the norm of[D]−1 is evaluated in a similar manner. The most computation-ally intensive aspect of using the above equation is evaluation of the inverse[D−1].Fortunately, this is not a big obstacle when evaluating hydrodynamic coefficientsbecause the number of degrees of freedom is typically less than 2000. Further-more, the computed inverse can be used for computing source strengths using directmatrix multiplication.

Figures 7 to 12 show normalized hydrodynamic coefficients and matrix conditionnumbers for the HALIFAX using coarse, medium, and fine panel meshes. Thefigures demonstrate that local maxima of matrix condition numbers can be indica-tors of irregular frequencies. To reduce irregular frequency effects, envelopes ofmaximum allowable matrix condition numbers can be provided as input for numer-ical computations. For a given hull panel mesh, separate envelopes are providedfor longitudinal and lateral modes. If the computed matrix condition number fora given encounter frequency exceeds the specified envelope, then computations forthat encounter frequency are not used. To determine appropriate condition numberenvelopes, initial computations of hydrodynamic coefficients and condition num-bers are required, followed by examination of results such as those given in Figures7 to 12. Note that the condition number envelopes will depend upon the underwaterhull geometry, and to a lesser extent upon the hull panel size used.


0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedAddedMass

Ajj/ max(Ajj)

.........................................................Surge

............. ............. ....... Heave

. . . . . Pitch

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................

.................................................................................................................................. ............. ............. ............. .......................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ........

.........................................

.............................

0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedDamping

Bjj/ max(Bjj)

.........................................................Surge

............. ............. ....... Heave

. . . . . Pitch

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...............

...............

...............

...............

...............

...............

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................ ............. ............................................................................................................................................................

.................................................... .......................... ............. ............. ............. .............

........................................................................................................ ............. ............. ............. ............. .......................... ............. ............. ............. ............. ............. ............. ......

......................................

.........................................................

........

0 2 4 6


0

1000

2000

3000

ConditionNumberλeven

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................

...........................................................................................................................

.............................................................................................................................................................................................................................................................................................

.............................................................

................................................................

.....................................................

.............

Figure 7: Normalized Added Mass and Damping, and Normal Velocity ConditionNumber for HALIFAX Coarse Mesh, Longitudinal Modes


0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedAddedMass

Ajj/ max(Ajj)

.........................................................Sway

............. ............. ....... Roll

. . . . . Yaw

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................

............. ............. .......................... ............. ................................................................. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

......................................................

..............

0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedDamping

Bjj/ max(Bjj)

.........................................................Sway

............. ............. ....... Roll

. . . . . Yaw

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................

................................................................................................................................................................

............................................................................................................................................................................... ............. .............

......................................................................................................................................................................... ............. ............. ....................................... ............. ............. ............. ............. .............

............. .............................................................................. ............. ....................................... ............. .......................... ............. ............. ............. ............. ............. .............. .................................................................................

0 2 4 6


0

200

400

600

800

ConditionNumber

λodd

............................................................................................................................................

...............................................................

...........................................

...........................................................................................................................................................................................................................................................................................................

.........................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................................

.................................................................

.................................................................................................................

..........................................

...................................................................................................

Figure 8: Normalized Added Mass and Damping, and Normal Velocity ConditionNumber for HALIFAX Coarse Mesh, Lateral Modes


0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedAddedMass

Ajj/ max(Ajj)

.........................................................Surge

............. ............. ....... Heave

. . . . . Pitch

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................................................................................................................................

.................................................................................................................................. ............. ............. ............. .................................................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ...........

........................................................ ........

....

0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedDamping

Bjj/ max(Bjj)

.........................................................Surge

............. ............. ....... Heave

. . . . . Pitch

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................

......................................................................

............................................................................................................................................................ ............. ..................................................................................................................................

............. .................................................... ............. ............. ............. ............. ............. .................................................... ............. ............. ............. ............. ............. ............. ............. ............. .............

.............................................................

...............................................

0 2 4 6


0

1000

2000

3000

4000


........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................

...................................................................

............................................................................

..............................................................................................................

............................................................................................................................................................................................................................................................................................................................................

........................................................

Figure 9: Normalized Added Mass and Damping, and Normal Velocity ConditionNumber for HALIFAX Medium Mesh, Longitudinal Modes


0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedAddedMass

Ajj/ max(Ajj)

.........................................................Sway

............. ............. ....... Roll

. . . . . Yaw.............................

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................

............. ............. .......................... ............. ................................................................. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

.........................................................

...........

0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedDamping

Bjj/ max(Bjj)

.........................................................Sway

............. ............. ....... Roll

. . . . . Yaw

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................

....................................................................................................................................................................................................................

................................................................................................... ............. .........................................................................................................................................................................

............. ............. ................................................................. ............. .................................................... ............. ............. ........................................................................................... ............. ............. ............. ............. .......................... ............. .............. .....

..........................................................................

0 2 4 6


0

500

1000

1500

ConditionNumber

λodd

.................................................................................................................................................................

................................

......................................

......................................................................................................................................................................................................................................................................................................................

........

........

........

........

........

........

........

........

........

........

........

........

.................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................

..............................................

.................................

....

Figure 10: Normalized Added Mass and Damping, and Normal Velocity ConditionNumber for HALIFAX Medium Mesh, Lateral Modes


0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedAddedMass

Ajj/ max(Ajj)

.........................................................Surge

............. ............. ....... Heave

. . . . . Pitch

................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................. ............. ............. ............. ................................................................. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

............................................................

........

0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedDamping

Bjj/ max(Bjj)

.........................................................Surge

............. ............. ....... Heave

. . . . . Pitch

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................

......................................................................................................................................................................... ............. ......................................................................................................................................................................... ......

.............

.............

.............

.............

.............

............. .......................... .......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .....................................................

..................................................

0 2 4 6


0

1000

2000

3000

4000

5000


............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ .......................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................

........

........

........

........

........

........

........

........

........

........

.........................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................

Figure 11: Normalized Added Mass and Damping, and Normal Velocity ConditionNumber for HALIFAX Fine Mesh, Longitudinal Modes


0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedAddedMass

Ajj/ max(Ajj)

.........................................................Sway

............. ............. ....... Roll

. . . . . Yaw.............................

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................

............. ............. .......................... ............. .................................................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .........

........................................................

............

0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

NormalizedDamping

Bjj/ max(Bjj)

.........................................................Sway

............. ............. ....... Roll

. . . . . Yaw

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................................................... ............. .............

............................................................................................................................................................

............. ............. .................................................... .......................... ............. ....................................... ............. ............. ............. ............. ............. .......................... ............. ............. ............. ............. ............. ............. .............. ...............................................................................

0 2 4 6


0

500

1000

1500

2000

ConditionNumber

λodd

..........................................................................................................................................................................

.........................................

...................................

.................................................................................................................................................................................................................................................................................................

........

........

........

........

........

........

........

........

........

........

........

.................................................................................................................................................................................................................................................

........

........

........

........

........

........

........

........

.......................................................................................................................................................................................

................................................................................

......................................................................................................................................................................................................................................................................................................................................

Figure 12: Normalized Added Mass and Damping, and Normal Velocity ConditionNumber for HALIFAX Fine Mesh, Lateral Modes


5.8 Direct Evaluation of Incident Wave Forces inDatabase of Radiation and Wave ExcitationForces

Reference 1 introduced a database class of radiation and wave excitation forcesin the frequency domain. This database stored computed values for added mass,damping, incident wave force, and diffracted wave force. Using the stored values,added mass and damping could then be quickly evaluated through linear interpola-tion based on encounter frequency. Similarly, incident and diffracted wave forces inunit amplitude waves could be quickly evaluated through linear interpolation basedon ship speed, relative sea direction, and wave frequency.

The database class has been revised such that incident wave forces are now eval-uated directly rather than obtained by interpolation of stored forces. Direct evalu-ation of incident wave forces requires relatively little computation time; thus, thisrevision increases accuracy at very little computational expense. Furthermore, in-cident wave forces are usually greater in magnitude than diffracted wave forces, sothe improvement provides a significant increase in the accuracy of wave excitationforces.

5.9 Improved Efficiency for Evaluating DiffractedWave Potentials

A new computational approach has been introduced for evaluating diffracted wavepotentials. Recall that diffraction source strengths potentials are solved as follows:

[D] {σD} = −{

∂φI

∂n

}(63)

where[D] is the influence coefficient matrix relating hull normal velocity to sourcestrength,{σD} is the vector of hull diffraction source strengths, and{∂φI/∂n}is the vector of incident wave normal velocities. The above equation is based onsources distributed over the entire submerged portion of the hull (i.e., both port andstarboard sides). The incident wave potential is given by:

φI =i g a

ωI

exp [−i kI (x cos β − y sin β)] exp(kI zwl) (64)

Equation (64) is based on the convention of the wave crest being at thex− y originat timet = 0. Fluid velocities are given by the following derivatives:

∂φI

∂x= a ωI cos β exp [−i kI (x cos β − y sin β)] exp(kI zwl) (65)


∂φI

∂y= −a ωI sin β exp [−i kI (x cos β − y sin β)] exp(kI zwl) (66)

∂φI

∂z= i a ωI exp [−i kI (x cos β − y sin β)] exp(kI zwl) (67)

The incident wave elevation is given by:

ζI = a exp [−i kI (x cos β − y sin β)] (68)

When evaluating diffraction potentials, it was found that a significant amount oftime was often required for solving diffraction source strengths in Equation (63).The required solution time was noticeably greater than for the solution of radia-tion source strengths, which utilized hull symmetry. Both radiation and diffractionsource strengths are solved using components of the linear algebra library LA-PACK [15]. The computational time for solving a system of equations with LA-PACK is approximately proportional toN3, whereN is the number of unknowns.To improve computational speed for evaluation of diffraction potentials, the inci-dent wave potentials and corresponding diffraction potentials can be divided intocomponents which are even and odd functions ofy, thus taking advantage of shipsymmetry. The even and odd components of the incident wave potential and theirderivatives are:

φevenI =

i g a

ωI

exp(−i kI x cos β) exp(kI zwl) cos(kI y sin β) (69)

∂φevenI

∂x= a ωI cos β exp(−i kI x cos β) exp(kI zwl) cos(kI y sin β) (70)

∂φevenI

∂y= −a ωI sin β exp(−i kI x cos β) exp(kI zwl) cos(kI y sin β) (71)

∂φevenI

∂z= i a ωI exp (−i kI x cos β) exp(kI zwl) cos(kI y sin β) (72)

φoddI =

i g a

ωI

exp(−i kI x cos β) exp(kI zwl) i sin(kI y sin β) (73)

∂φoddI

∂x= a ωI cos β exp(−i kI x cos β) exp(kI zwl) i sin(kI y sin β) (74)

∂φoddI

∂y= −a ωI sin β exp(−i kI x cos β) exp(kI zwl) i sin(kI y sin β)(75)

∂φoddI

∂z= i a ωI exp (−i kI x cos β) i sin(kI y sin β) (76)

Using the even and odd components of incident wave potential, source strengthsonly have to be solved for the port side of the hull using the following:

[Deven] {σevenD } = {−∂φeven

I /∂n} (77)[Dodd

] {σodd

D

}=

{−∂φodd

I /∂n}

(78)


where{σevenD } is the vector of source strengths on the port side for the even compo-

nent of diffraction potential,{∂φevenI /∂n} is the vector of incident normal velocity

on the port side due to the even component of the incident wave potential,{σodd

D

}is

the vector of source strengths on the port side for the odd component of diffractionpotential, and

{∂φodd

I /∂n}

is the vector of incident normal velocity on the port sidedue to the odd component of the incident wave potential. Once the source strengthsare solved, the total diffracted potential on the hull can be determined by:{

φportD

φstarD

}=

[Eport,port Eport,star

Eport,star Eport,port

] {σeven

D

σevenD

}

+

[Eport,port Eport,star

Eport,star Eport,port

] {σodd

D

−σoddD

}(79)

where{φport

D

}is the vector of diffraction potentials on the port side of the hull,

{φstarD } is the vector of diffraction potentials on the starboard side of the hull,

[Eport,port] is the influence matrix of velocity potentials on the port side due tosources on the port side, and[Eport,star] is the influence matrix of velocity poten-tials on the port side due to sources on the starboard side.

The new procedure for evaluating diffraction source strengths requires approxi-mately one quarter of the computation time as the original procedure. Although2 sets of source strengths are now solved (even and odd), each set only has to besolved for half of the hull.

5.10 Interpolation of Inverse of Normal VelocityInfluence Matrix when Evaluating DiffractionPotentials

The database of radiation and diffraction potentials previously mentioned allowsrapid evaluation using interpolation of stored values. A comprehensive databaseencompassing full ranges of speeds, headings, and wave frequencies can requirediffraction computations for several thousand sets of conditions. The previous sub-section noted that diffraction potentials are evaluated using Equations (57), (58)and (79). When generating the database, the number of required Green functionevaluations is greatly reduced using the following interpolation scheme presentedin Reference 1:[

Dport,port(ωe)]

= wi

[Dport,port(ωe,i)

]+ wi+1

[Dport,port(ωe,i+1)

](80)[

Dport,star(ωe)]

= wi

[Dport,star(ωe,i)

]+ wi+1

[Dport,star(ωe,i+1)

](81)[

Eport,port(ωe)]

= wi

[Eport,port(ωe,i)

]+ wi+1

[Eport,port(ωe,i+1)

](82)[

Eport,star(ωe)]

= wi

[Eport,star(ωe,i)

]+ wi+1

[Eport,star(ωe,i+1)

](83)


wherewi andwi+1 are weights for linear interpolation andωe,i ≤ ωe ≤ ωe,i+1.When using the above interpolation scheme, a large amount of computational timecan still be required because the diffraction source strengths must still be solved(solution ofσeven

D andσoddD that satisfy Equations (57) and (58)) for each combina-

tion of ship speed, heading, and wave frequency. This solution time can be greatlyreduced by interpolation of the inverse of[Deven] and[Dodd] as follows:

[Deven(ωe)]−1 = wi [Deven(ωe,i)]

−1 + wi+1 [Deven(ωe,i+1)]−1 (84)[

Dodd(ωe)]−1

= wi

[Dodd(ωe,i)

]−1+ wi+1

[Dodd(ωe,i+1)

]−1(85)

Using the interpolated inverses, the source strengths are then quickly evaluated us-ing:

{σevenD } = [Deven(ωe)]

−1 {−∂φevenI /∂n} (86){

σoddD

}=

[Dodd(ωe)

]−1 {−∂φoddI /∂n

}(87)

The potential benefits of the above interpolation scheme become obvious when oneconsiders that a representative database of diffraction forces could include 7 shipspeeds (0, 5,. . ., 30 knots), 13 sea directions (0, 15,. . ., 180 degrees), and 37 wavefrequencies (0.20, 0.25,. . ., 2.00 rad/s), giving a total of 3367 seaway conditions.The same database could have 60 encounter frequencies (0.1, 0.2,. . ., 6.0 rad/s).

It should be noted that there is loss of accuracy when interpolating influence ma-trices using Equations (80) to (83). The interpolation of inverse matrices usingEquations (84) and (85) likely leads to additional loss of accuracy. Losses in ac-curacy will likely be greatest in the vicinity of irregular frequencies of encounter,at which solved potentials can be sensitive to small variations in influence matrixterms.

5.11 New Procedure for Computation of DiffractionForces from a Database

When predicting wave diffraction forces using previously computed values, the pro-cedure presented in Reference 1 uses interpolation to obtain wave diffraction forcein unit amplitude waves as a function of ship speed, heading, and wave frequency.The main disadvantage of this approach is that it does not adequately model thestrong influence of wave encounter frequency on wave diffraction force. To obtainmore accurate results, wave diffraction forces are now evaluated as follows:

FDj = a

(ωe FD,ωe

j

a+

U FD,Uj

a

)(88)


whereFD,ωe

j is the diffraction force component that is proportional to wave en-

counter frequency andFD,Uj is the diffraction force component that is proportional

to ship speed. Reference 1 gives the following equation for diffraction forces:

FDj = ρ ωe

∫Sb

(iφD − U

ωe

∂φD

∂x

)nj dS (89)

whereφD is the potential of the diffracted waves. The corresponding componentsfor Equation (88) are:

FD,ωe

j = ρ

∫Sb

i φD nj dS (90)

FD,Uj = − ρ

∫Sb

∂φD

∂xnj dS (91)

When creating a database of hull radiation and diffraction forces, each of the abovecomponents in unit amplitude waves are stored for combinations of specified shipspeeds, headings, and wave frequencies. When accessing values from the database,the above components are interpolated as functions of ship speed, heading, andwave frequency.

5.12 Faster Versions of LAPACK Routines forSolution of Systems of Linear Equations

When solving radiation and diffraction potentials, a large amount of CPU timeis devoted to solving systems of linear equations. Routines from the LAPACKlibrary [15] are used for solving systems linear equations. Previously, functionssolvelinear equationsandinversefrom the Python Numeric library [16] were usedfor solving linear equations and inverting matrices. The Python package SciPy,available at no cost from http://www.scipy.org, has versions of LAPACK routinesoptimized for PC hardware. The SciPy routinessolveandinv are now used insteadof the Numeric routinessolvelinear equationsandinverse.

Table 3 gives representative CPU times for solving hull added mass and damp-ing coefficients in the frequency domain for HALIFAX. The values in Table 3are based on average CPU times over a range of encounter frequencies. As ex-pected, the amount of CPU time for solution of Green functions is approximatelyproportional to(Nport

p )2. The CPU time required for matrix inversion or solutionof equations without matrix inversion is approximately proportional to(Nport

p )3,which is also expected. The optimized inversion and solution routines from SciPyare approximately three times faster than the corresponding routines from Numeric.Solution of velocities using matrix inversion takes approximately three times longerthan solution without matrix inversion; however, matrix inversion can be useful fordetermining matrix condition numbers, as discussed in Section 5.7.


Table 3: Typical CPU Times (seconds) Per Encounter Frequency for Prediction ofHull Added Mass and Damping Coefficients in Frequency Domain

Coarse mesh Medium mesh Fine mesh

Nportp = 224 Nport

p = 433 Nportp = 1040

Evaluation of Green functionsand derivatives

30 110 637

Inversion of influence matricesfor even and odd modes usingNumeric routineinverse

2.1 17 250

Inversion of influence matricesfor even and odd modes usingSciPy routineinv

0.47 2.9 45

Solution of velocity potentialsfor even and odd modes usingNumeric routinesolvelinear equations

0.46 4.8 81

Solution of velocity potentialsfor even and odd modes usingSciPy routinesolve

0.16 1.0 15


6 Hydrodynamic Forces in the FrequencyDomain for HALIFAX

To verify the improved methods presented in the previous section, hydrodynamicforces in the frequency domain have been computed for HALIFAX. The coarse,medium, and fine meshes presented in Section 4 have been used. Table 4 gives pa-rameters for generated databases of ship hydrodynamic coefficients in the frequencydomain. To suppress irregular frequency effects, limiting envelopes on matrix con-dition numbers were provided as input. These envelopes are shown in Figures 13and 14, and were developed by examining initial predictions of hydrodynamic co-efficients and matrix condition numbers presented in Figure 7 to 12

Table 4: Parameters for Databases of Frequency Domain Radiation and DiffractionForces

Wave encounter frequenciesωe 0.1, 0.2, 0.3,. . ., 6.0 rad/s

Ship speedsU 0, 5, 10, 15, 20, 25, 30 knots

Relative sea directionsβ 30, 150, 165, 180 degrees

Wave frequenciesωI 0.20, 0.25, 0.30,. . ., 2.0 rad/s

Table 5 gives required computation times for generating the databases of hydrody-namic forces in the frequency domain. As discussed in Section 5, wave diffractioncomputations can be performed more quickly but somewhat less accurately usinginterpolation of influence matrix inverses according to Equations (84) and (85). Thecomputational times given in Table 5 are based on the more accurate approach ofinterpolation of influence matrices rather than their inverses.

Table 5: Required CPU Time for Generating Database of Radiation and DiffractionComputations

Mesh CPU time

Coarse,Nportp = 224 0.7 hours

Medium,Nportp = 433 2.6 hours

Fine,Nportp = 1040 16.8 hours


0 2 4 6


0

1000

2000

3000

ConditionLimitλeven

.........................................................Fine mesh

............. ............. ....... Medium mesh

. . . . . Coarse mesh

. . . . . . . . . ......................... . . . . . . . . . . . . . . . . . . .

............. ............. ............. ............. ..................................................................................................................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 13: HALIFAX Limiting Condition Numbers for Longitudinal Modes Surge,Heave, and Pitch

0 2 4 6


0

500

1000

1500

ConditionNumber

λodd

.........................................................Fine mesh

............. ............. ....... Medium mesh


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .........

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 14: HALIFAX Limiting Condition Numbers for Lateral Modes Sway, Roll, andYaw


6.1 Damping and x Derivative of Added Mass

Figures 15 to 20 show damping and thex derivative of added mass for HALIFAX.These two quantities are shown because they are both used for determining retar-dation functions in the time domain. The plots of damping andx derivative ofadded mass have been non-dimensionalized to indicate their relative magnitudesfor a ship speed of 20 knots. This speed was selected because it is approachingthe maximum speed at which HALIFAX would typically transit in a moderate toheavy seaway. Damping coefficients from the 3 different meshes have very similarvalues. Thex derivatives of added mass are more sensitive to mesh size becausethey are small quantities arising from fore-aft asymmetry of the ship hull. The goodagreement between the medium and fine meshes indicates numerical convergencewith increasing number of panels.


0 2 4 6


0.00

0.01

0.02

0.03

0.04

0.05

SurgeDamping

B11

4√

g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh


............. ............................................................................................................................................................................................................................. ............. ............. ............. .......................... .................................................... ....................................... ............. ............. ............. ............. ............. .......................... .......................... ....................................... ............. ............. ....................................................

.......................... ....... ..

................................. . .... .... ............... .................... ...........

..... .... .....

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................

0 2 4 6


0.00

0.01

0.02

0.03

SurgeAddedMass

x Derivative

U ∂A11/∂x

4√

g/L

............. ............. ............. ............. ............. ............. ............. ....................................................

..........................

..........................

............. ............. ............. ............. .................................................................

............. ............. ............. .......................... .......................... ............. ........................................................................................................

.....................................................................................................................

.................

. . . . . . . . . . . . . . ... ......

..........................

............. . . ...

..... ............................ .

...............

....................................................................................................................................................................................................................................................................................

.................................................

..................................................................

..........................................................................................................................................................................

...............................................................................

......................................................................................................................................

...................................................................................................................

Figure 15: Dimensionless Forces from Zero Speed Surge Damping and xDerivative of Added Mass for HALIFAX at 20 Knots


0 2 4 6


0

1

2

3

SwayDamping

B22

4√

g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh


............. ............. ............................................................................................................................................................................................................................. ........................................................................................................ .................................................... ................................................................. ............. .......................................

............. ............. ............. .................................................... ..................................

............................... ................... . . ..

................................. ...............................

............ . .........

..... .................

..... ..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................

0 2 4 6


−0.5

−0.4

−0.3

−0.2

−0.1

0.0

SwayAddedMass

x Derivative

U ∂A22/∂x

4√

g/L

............. ............. ............. ............. ............. ............. ....................

...... ............. ............. .................................................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .................................................................

....................................................

.............

. . . . . .. . .. . . ..

.. . . . . . ...... . . . . . . . . . . . . . . . ..........

..... . . .. . . . ...

................................................................................................................................................................................................................................................................................................

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 16: Dimensionless Forces from Zero Speed Sway Damping and xDerivative of Added Mass for HALIFAX at 20 Knots


0 2 4 6


0

1

2

3

4

5

HeaveDamping

B33

(4√

g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh


.......................................................................................................................................................................................................................................... .........................................................................................................................................................................

.............................................................................. ............. ............. ............. ............. ............. ....................................... ............. ............. ............. ............. ............. ............. ............. .......................... .............

.............................................................. ..

.............. .......... . ... . .. . .. ... . . .. .. ..

. . . . . ..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 2 4 6


0.0

0.1

0.2

0.3

0.4

0.5

0.6

HeaveAddedMass

x Derivative

U ∂A33/∂x

4√

g/L............. ............. .......................... ............. ............. .............

............................................................................................................................................... ............. ..........................

....................................... ............. ............. .....................................................................

..................................................................................................................................................................... ...........................

.......................................................................................... .............

.................................................................................................................................................................................................................................................................... .............

. . . . . .. . . . . .............. .

........ . .........

..............................

..........

.................................................... ................

.......................................................................................................................................................................................

.....................................................................................................................................................................................................................................

............................................................................................................

...............................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................

Figure 17: Dimensionless Forces from Zero Speed Heave Damping and xDerivative of Added Mass for HALIFAX at 20 Knots


0 2 4 6


0.0

0.1

0.2

0.3

RollDamping

B44

4r244

√g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh


............. ............. ......................................................................................................................................................................................

............. ........................................................................................... ....................................... ............. ............. ........................................................................................................ ............. .......................... ............. ............. .......................... ............. ........ . . ..

............................ . . .......... . . . ......

.................. ...... . ..... . . . . . . . . . . . .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 2 4 6


−0.10

−0.05

0.00

0.05

0.10

RollAdded Massx Derivative

U ∂A44/∂x

4r244

√g/L

............. ............. ............. ............. ............. ..........................

..........................

..........................

....................................................

.......................... ............. ............. ............. ............. ............. .............

....................................... ............. ............. ............. ............. .............

........................................................................................... ............. .....

. . . . ....... .........

.....

........ . .. . . .....

....... ..

. .. ................................................ ... ......

...................................................................................................................................................................................................................

.....................................................

...............................................................................................

.....................................................................................................................................

..........................................................................................................................................................................................................................................................

.....................................................................................

Figure 18: Dimensionless Forces from Zero Speed Roll Damping and x Derivativeof Added Mass for HALIFAX at 20 Knots


0 2 4 6


0

2

4

6

PitchDamping

B55

4r255

√g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh


............. ................................................................................................................................................................................................... ............. ................................................................. ................................................................. ....................................... ............. ............. .............

........................................................................................................ ............. ............. ............. ............. ............. .................................................... ............. .

............

. ....................................... .... . ..............

...............................................

. ... ..

.. . . . . . ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................

0 2 4 6


−1

0

1

2

PitchAdded Massx Derivative

U ∂A55/∂x

4r255

√g/L

............. ............. .......................... ............. ............. ............. .

...................................... ............. ............. ............. ............. .......................... ............. ............. ....................................................

.............................................................................. ............. ......................

........................................................ ............. ..............................................................................

. . . . .... . . . .

......... ... . . . .... .. .

...........

.... ... ..

........................ ..............

.................................................................................................................................................

.............................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................

Figure 19: Dimensionless Forces from Zero Speed Pitch Damping and xDerivative of Added Mass for HALIFAX at 20 Knots


0 2 4 6


0

1

2

3

4

YawDamping

B66

4r266

√g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh


.......................... ...................................................................................................................................................................................................

.......................... ....................................... .................................................... .......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............. . . . ..

............................... ........................... . ..... . . .. . . .. . . .. . . . . ......................................................................

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 2 4 6


−2.0

−1.5

−1.0

−0.5

0.0

YawAdded Massx Derivative

U ∂A66/∂x

4r266

√g/L

............. ............. ............. ............. ............. ..........................

..........................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ........................................................

......... .............

. . . .... . . ...........

. . . . .. . .. . . . .

. . . . . . . . . . . .............. . . . . . .

. . . . ..........................................................................................................................

..............................................

.............................................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 20: Dimensionless Forces from Zero Speed Yaw Damping and x Derivativeof Added Mass for HALIFAX at 20 Knots


6.2 Wave Excitation Forces on HALIFAX

Figures 21 and 22 show wave excitation forces on HALIFAX for a ship speed of 20knots in stern quartering and bow quartering seas. All results are for the mediummesh using the following three computational methods, listing in order of decreas-ing computational time:

• normal velocity influence matrix[D(ωe)] computed directly at frequencyωe,

• normal velocity influence matrix[D(ωe)] obtained using interpolation from ad-jacent encounter frequencies in database (Equations (80) and (81)),

• inverse of normal velocity influence matrix[D(ωe)]−1 obtained using interpola-

tion from adjacent encounter frequencies in database (Equations (84) and (85)).

For the most part, the three approaches give excellent agreement. Discrepanciesare limited to when the matrix[D(ωe)]

−1 is interpolated for bow quartering seaswith wave frequencies in the vicinity of 1.0-1.2 rad/s. The associated encounterfrequencies are of the order 2-3 rad/s, where the first irregular frequencies occurfor longitudinal and lateral modes, and small errors in influence matrix terms canlead to large errors in computed solutions. In summary, the above results indicatethat reliable diffraction forces can be obtained when interpolating the influencematrix [D] as a function of encounter frequency. If the inverse matrix[D]−1 isinterpolated as a function of encounter frequency, noticeable inaccuracies can occurin the vicinity of irregular frequencies.


.........................................................Direct evaluation of[D(ωe)] ............. ............. ....... Interpolation of[D(ωe)]

. . . . . Interpolation of[D(ωe)]−1

0 0 0.5 1 1.5


0 1 2

Wave FrequencyωI (rad/s)

0

1

2

SurgeForce

|F1|4ga/L

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................

.....................................................................

...........................................................................

.......................................................................................

.......................................................................

..........................................

.......................................................................................................................................................................................................................................................

...........................................................................................

............. ...........................................................................

...............................................

......................................

..............

..............

.......

0 1 2


0 0 0.5 1 1.5


0

1

2

SwayForce

|F2|4ga/L

.............................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................................................................................................................................................................

......................................................................................................

.............................................................................................................................................................................................................................

....................................... ............. ............. ............. ............................

.............................

...................................

....................

0 1 2


0

10

20

30

40

HeaveForce

|F3|4ga/L

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................... ............. ............. ............. ............. ............. ............. ............. .............

......................................0 1 2


0.00

0.01

0.02

0.03

0.04

RollForce

|F4|4ga

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................

....................................................................................

.............................................................................

...............................................

..........................................................................................................................................................................................................................................

........................................................................................................ .......................................

............. ...................................

...........................................

.................

.....................................

........................

...........

...

0 1 2


0

1

2

3

4

5

PitchForce

|F5|4ga

........................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................

...........................................................................................

.....................................................................

...........................................................................................

............................................................................................................................................... .................................................................

............. ............. ............. .............

..................................

....................

0 1 2


0.0

0.2

0.4

0.6

YawForce

|F6|4ga

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................

..........................................................................................................

.............................................................................

...........................................

..........................................................................................................................................................................................................................................

....................................................

............. .......................................

......................................................

.....................................

................

.......

Figure 21: Wave Excitation Forces for HALIFAX, Medium Mesh, 20 knots, SternQuartering Seas at 30 degrees




0 1 2


0 1 2 3 4 5


0

1

2

3

SurgeForce

|F1|4ga/L

......................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................

...............................................................................................

.............................................................................................................................................................................................................................................................................................

.................................................... ......

.........................................................

............... .......................................................................

..................

........

0 1 2


0 1 2 3 4 5


0

1

2

3

4

SwayForce

|F2|4ga/L

....................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................

........................................................................... ............. .......................... ............. ............. .............

...................................

........................

0 1 2


0

10

20

30

40

HeaveForce

|F3|4ga/L

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................... ............. ............. ............. ............. ............. ...........

..........................................

.............0 1 2


0.00

0.02

0.04

0.06

RollForce

|F4|4ga

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............

..............

..............

..............

...............

...............

...............

...............

...............

............................................................................................................................................................................................................................

..............

..............

..............

....................................................................................................................................................

.....................................................................................................................................

................................................................................................................................

......................................................................................

..............................

............. ..........................................................................................................................................................................................................................................

......................................................................................................................................................................................

.....................................................................................................

....................................................................

.................................................................

........................................

........................

.......................................

..

..

.......................

...............

...........

..........

..........

0 1 2


0

2

4

6

PitchForce

|F5|4ga

................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................

..........................................................................................................................................................................

........................................................................................... ..................................................................................................................... .............

...........................................................................................

.......................... ............. .............

..................................

................

.......

0 1 2


0.0

0.2

0.4

0.6

0.8

1.0

YawForce

|F6|4ga

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................

..............................................................................................................

.....................................................................................

............. ...............................................................................................................................................................................................................................................................................

................................................................................ .................................................................

.................

........................................

.........................

Figure 22: Wave Excitation Forces for HALIFAX, Medium Mesh, 20 knots, BowQuartering Seas at 150 degrees


6.3 HALIFAX Motions in the Frequency DomainUsing New Computational Methods

Motions for HALIFAX in stern quartering and bow quartering seas are presentedin Figures 23 and 24. The motions based on directly computed hydrodynamic co-efficients agree very closely with those from the database with interpolation of in-fluence matrix[D] as a function of encounter frequency. As was observed for waveexcitation forces, some discrepancies occur in the vicinity of irregular frequencieswhen the inverse matrix[D]−1 is interpolated as a function of encounter frequency.When high accuracy is required from a database of diffraction forces, the resultsindicate that one should interpolate the influence matrix[D] rather than its inverse[D]−1 as a function of encounter frequency

For the motions in stern quartering seas in Figure 23, wave frequencies have beenexcluded that have encounter frequencies less than 0.1 rad/s. At wave frequen-cies with encounter frequencies below 0.1 rad/s, predicted motions in unrestrainedmodes (surge, sway, and yaw) are unrealistically large due to the absence of re-straining forces and very small inertia and damping forces. For practical com-putations, this problem can be avoided by a slight shift of wave frequency or byincluding viscous damping terms.




0 1 2


0 0 0.5 1 1.5


0

1

2

3

SurgeRAO

|η1|a

..................................................................................................................................................................................................................................................................................................................

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

....................................................

............. ..................................................................................................................................................................................................................................................................................................................................................

.................................................... ............. ............. ............. .............

.........................

..

..

..

..

..

..

..

.......................................

0 1 2


0 0 0.5 1 1.5


0

1

2

SwayRAO

|η2|a

.................................................

...............................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................

............. ............. ............. .................................................... ..................................................................................................................... ............. ............. ............. .............

.......

.....................................

0 1 2


0.0

0.5

1.0

HeaveRAO

|η3|a

......................................................................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................................

............. ............................................................................................................................................................ ............. ............. .....................

..... ............. .......................... .............

..........................................

0 1 2


0.0

0.2

0.4

0.6

RollRAO

|η4|k a

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................................................................................................................................................................................................................................................................................................................................

............. ............................................................................................................................................................................................................................. ............. .............

............................................................................................................................................... ............. ..........

........................................

.........................

0 1 2


0.0

0.5

1.0

PitchRAO

|η5|k a

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. .................................................................................................................................. ............. ............. ............. ............. ............. ............. ....

...................................0 1 2


0.0

0.1

0.2

0.3

0.4

0.5

YawRAO

|η6|k a

......................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................

....................................................

................................................................................................................................................................................................... ............. ............. ............. ..

........................

.....................

Figure 23: Frequency Domain Motions for HALIFAX from Direct and DatabaseComputations, Medium Mesh, 20 knots, Stern Quartering Seas at 30 degrees




0 1 2


0 1 2 3 4 5


0.0

0.2

0.4

0.6

0.8

SurgeRAO

|η1|a

...........................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................. ............. ............. ............. ............. ............. ............. .............

..................................0 1 2


0 1 2 3 4 5


0.0

0.1

0.2

0.3

0.4

SwayRAO

|η2|a

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................ ............. ............. ............. ............. ............. ............. .............

......................................

0 1 2


0.0

0.5

1.0

HeaveRAO

|η3|a

.........................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................

............. ............. ............. ...................................................................................................................................................................................... ............. ............. ............. ............. ............. ............. ..

............................................................

0 1 2


0

2

4

6

RollRAO

|η4|k a

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................... ............. ............. ............. ............. ............. ............. ............. ............. .............

............................................... .... ...

0 1 2


0.0

0.5

1.0

PitchRAO

|η5|k a

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. ............. ............................................................................................................................................... ............. ............. ............. ............. ............. ............. .............

.......................................

0 1 2


0.0

0.1

0.2

0.3

YawRAO

|η6|k a

........................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................

............. .............................................................................. ............. ............. ............. ............. ............. .............

............................................

Figure 24: Frequency Domain Motions for HALIFAX from Direct and DatabaseComputations, Medium Mesh, 20 knots, Bow Quartering Seas at 150 degrees


7 Hydrodynamic Coefficients in the TimeDomain for HALIFAX

Retardation functions presented in Section 3 have been computed for HALIFAXusing the frequency domain coefficients presented in the previous section. Thecomputed retardation functions are based on numerical integration of frequencydomain coefficients for an encounter frequency range of 0 to 6 rad/s with an incre-ment of 0.05 rad/s. At encounter frequencies beyond 6 rad/s, damping and thexderivative of added mass are assumed to be equal to their infinite frequency val-ues. Figures 25 to 30 show the resulting speed-independent and speed-dependentterms of retardation functions. As was done for frequency domain coefficients inSection 6, speed-dependent terms are given for a ship speed of 20 knots to indicatetheir magnitude relative to speed-independent terms.

Figures 25 to 30 show that the speed-independent terms are much greater than thespeed-dependent terms for HALIFAX travelling at 20 knots. Similar behaviourwas observed for frequency domain coefficients in Section 6. The medium and finemeshes give similar results; however, the coarse mesh gives significant differencesin speed-dependent terms. These differences can be attributed to the relatively smallmagnitude of speed-dependent terms and their sensitivity to computedx derivativesof velocity potentials.


0 5 10 15 20

Delay timeτ (s)

−0.1

0.0

0.1

0.2

0.3

SurgeRetardation

K011

4g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh


............................................................................................................................................................................................................................. .............

.......................... .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

................................... ..

... ... ..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...............................................................................................................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 5 10 15 20

Delay timeτ (s)

−0.02

0.00

0.02

0.04

SurgeRetardation

SpeedTerm

U KU11

4 g/L

..........................................................................................................................................................................................................................................

.....................................................................................................................

............. ...........................................................................................

....................................... .......

...... .......................... ...............................................

..... ............. ....................................................

.............................................

...........................................

..........................................

..........................................................................................

..................................

...................................................

.............

...............

.... .

..................

...........

...... .. .

................

..

...................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................

.......................................................................................................................................................................................

....................................................................................................................................................

................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................................................................................................................

Figure 25: HALIFAX Surge Retardation Contributions K011 and UKU

11 for U = 20knots


0 5 10 15 20

Delay timeτ (s)

−5

0

5

10

15

SwayRetardation

K022

4g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh


............................................................................................................................................................................................................................. .................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

................................. ........... . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 5 10 15 20

Delay timeτ (s)

−0.3

−0.2

−0.1

0.0

0.1

0.2

SwayRetardation

SpeedTerm

U KU22

4 g/L ...................................................................................................................................................................................................

........................................................................................................

..............................................................................................

..................................................

...................................................

....................................................

...........................................................................................

...........................................................................................

................................................................. .............

..............................................................................

................................

.........................................

.................

....... ......

. ........ . . . . .....

. . . ... .. . . . . . .... .

.............................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................

.........................................................................................................................................................................................................................................................

Figure 26: HALIFAX Sway Retardation Contributions K022 and UKU

22 for U = 20knots


0 5 10 15 20

Delay timeτ (s)

−5

0

5

10

15

20

HeaveRetardation

K033

4g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh


.......................................................................................................................................................................................................................................... .......................... .............

............. ............. ............. ............. .......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

........................................

.. .... .. . ..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 5 10 15 20

Delay timeτ (s)

−1

0

1

2

HeaveRetardation

SpeedTerm

U KU33

4 g/L

........................................................................................................

....................................................................................

......................................

..................................

.....................................................................

......... ..............................................

................................................................

........................................

.......................................................................

............. ............. ....................................... ........

..... ............. ............. ............. ................................

...........................................................

..

..

..

..

..

..

..

..

....................

.............

........

...................

.............

.............

..........

..................

.........

.......................................................................................................................................................................................................................

.....................................................................................................................................

...................................................................................................................

.............................................................................................................

..........................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 27: HALIFAX Heave Retardation Contributions K033 and UKU

33 for U = 20knots


0 5 10 15 20

Delay timeτ (s)

−1

0

1

2

RollRetardation

K044

4r244g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh

. . . . . Coarse mesh...............................................................................................................................................

............. ....................................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

............................ ..

... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..........................................................................................................................................................................................................................................................................................................................................................

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 5 10 15 20

Delay timeτ (s)

−0.2

0.0

0.2

0.4

0.6

RollRetardation

SpeedTerm

U KU44

4r244 g/L

................................................................. .................................................... .................

......... ............. ............. ..........................

.......................... ............. ............. .................................................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

.............................................

......................

.............

...........

................ .

...........

....... . ....

........ . . ..........

........................................................................................................................................................................................................................

...........................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 28: HALIFAX Roll Retardation Contributions K044 and UKU

44 for U = 20knots


0 5 10 15 20

Delay timeτ (s)

−10

0

10

20

30

PitchRetardation

K055

4r244g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh


......................................................................................................................................................................................

............. .......................... .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

...............................

.. ... ...

.. .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .

..............................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 5 10 15 20

Delay timeτ (s)

−2

0

2

4

6

8

PitchRetardation

SpeedTerm

U KU55

4r255 g/L

..............................................................................

.....................................................................................................................

.................................................................

............. .......................................

.......................... ............. ............. ............. ............. ............. .......................... ............. ............. ............. ............. ............. ............. ............. .............

....................................................

..

..

..

..

..

......................

.............

............

..........

.......

.................

.......

..........................

....................................................................................................................................................................................

................................................................................................................................

...............................................................................................................

.....................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 5 10 15 20

Delay timeτ (s)

−0.4

−0.2

0.0

0.2

0.4

0.6

PitchRetardation

SpeedTerm

U2 KUU55

4r255 g/L

......................................................................................................................................................................... ............. ............. .............

.............................................................................. .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .........

..... ........................ ................ . . .

. . ... . ...... . ... . . .. . . . .

. . . . . . . . . .

...................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................................

Figure 29: HALIFAX Pitch Retardation Contributions K055, UKU

55, and U2KUU55 for

U = 20 knots


0 5 10 15 20

Delay timeτ (s)

−10

0

10

20

YawRetardation

K066

4r266g/L

.........................................................Fine mesh

............. ............. ....... Medium mesh


.........................................................................................................................................................................

.......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

..........................

.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

................................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 5 10 15 20

Delay timeτ (s)

−0.5

0.0

0.5

1.0

YawRetardation

SpeedTerm

U KU66

4r266 g/L

......................................................................................................................................................................... ..........................

............. .......................... .......

...... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ........

..................................... . .....

.................

.............

................

.............

...... ..........

... .........................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 5 10 15 20

Delay timeτ (s)

−0.2

0.0

0.2

0.4

0.6

YawRetardation

SpeedTerm

U2 KUU66

4r266 g/L

............................................................................................................................................................ ..........................

..........................

.......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ...

.................................... ..

.. . . .... . . . . . .. ... . ... .. . . . .... . . . . .. .. . . .

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 30: HALIFAX Yaw Retardation Contributions K066, UKU

66, and U2KUU66 for U

= 20 knots


8 Time Domain Simulation of Ship Motions

The hydrodynamic forces presented in this report can form the basis for simulat-ing motions of an unappended ship. Ship motions can be computed using a timestepping procedure using the following derived from Equation (30):

([M ] + [A(U,∞)]) {η(t)} = − [b(U)]{

˙η(t)}

−∫ t

−∞[K(U, t− τ)] {η(τ)} dτ

− ([C] + [c(U)]) {η(t)}+{F I(t) + FD(t)

}(92)

The above system of equations enables computations of accelerations{η(t)} pro-vided that displacements and velocities are know up to timet. Hornbeck [17] dis-cusses several numerical methods that can be used for integration of ship motions inthe time domain. When selecting a suitable method, it is desirable to minimize thenumber of time steps at which forces need to be evaluated because of computationalrequirements. Another consideration is that when integrating displacements, bothfirst and second derivatives (velocity and acceleration) are available. It was decidedto use Taylor series for time-stepping of displacements and velocities. The Taylorseries equations include contributions due to the time derivatives of accelerations,which are estimated using a backward difference approach. The following equa-tions are evaluated in order to obtain displacements and velocities at timet + ∆tbased on motions evaluated up to timet:

∂ηj(t)

∂t=

23 ηj(t) − 16 ηj(t − ∆t) + 5 ηj(t − 2∆t)

12 t(93)

ηj(t + ∆t) = ηj(t) + ∆t ηj(t) +(∆t)2

2ηj(t) +

(∆t)3

6

∂ηj(t)

∂t(94)

ηj(t + ∆t) = ηj(t) + ∆t ηj(t) +(∆t)2

2

∂ηj(t)

∂t(95)

When originally implementing the above equations into the Python computer code,arbitrary time step sizes∆t were facilitated using linear interpolation of resultsfrom previous time steps. Subsequent profiling of code execution revealed that alarge amount of CPU was being devoted to interpolation of the retardation func-tions in Equation (92). To improve code efficiency, the code was re-written torestrict time step size to a single value input at the beginning of a simulation. Thisscheme gives much greater efficiency because retardation functions only need tobe interpolated at the very beginning of a simulation using the input time step size.This coding change has increased execution speed by a factor of approximately 30for linear time domain computations.


9 Comparison of Motions in the Time andFrequency Domains

To verify hydrodynamic coefficients and derived motions predicted in the time do-main, comparisons have been made with frequency domain motion predictions.The predictions are for the HALIFAX using the medium mesh, with 433 panels onthe port side of the hull. To give realistic roll motions, supplementary roll damp-ing of 2 × 107 Nm/(rad/s) has been used, which is representative of damping fromappendages and viscous effects. Similarly, supplementary yaw stiffness of2× 109

Nm/rad has been used, which is representative of the contributions of the skeg andrudder to coursekeeping stability.

A ship speed of 20 knots was selected for comparing frequency and time domainpredictions. This speed was considered sufficiently high to verify consistency ofspeed dependent terms. The time domain computations are based on a time stepsize of 0.1 s, which gives excellent numerical stability. On a PC computer with a800 MHz Pentium III processor, the time domain computations run approximately30 times faster than real time. Figures 31 and 32 show excellent agreement betweenpredictions in the frequency and time domains for both stern quartering and bowquartering seas. Note that the present theoretical approach deteriorates at very lowencounter frequencies; thus, the predictions are limited to conditions with encounterfrequencies≥ 0.1 rad/s. This limitation can be handled from a practical standpointby slightly shifting wave frequencies when travelling with forward speed in wavesaft of the beam.


.........................................................Time domain . . . . . Frequency domain

0 1 2


0.0 0.0 0.5 1.01.5


0

1

2

3

SurgeRAO

|η1|a

................................................................................................................................................................................................................................................................................................................

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................

..

..

..

..

..

..

..

.......................................

0 1 2


0.0 0.0 0.5 1.01.5


0.0

0.5

1.0

1.5

SwayRAO

|η2|a

...........................................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................

..............................

0 1 2


0.0

0.5

1.0

HeaveRAO

|η3|a

.........................................................................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................................

.........................................0 1 2


0.0

0.5

1.0

1.5

RollRAO

|η4|k a

.......................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................

.................................

........

0 1 2


0.0

0.5

1.0

PitchRAO

|η5|k a

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................0 1 2


0.0

0.1

0.2

0.3

0.4

YawRAO

|η6|k a

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................

................................................. .

Figure 31: Motions for HALIFAX from Frequency Domain and Time DomainComputations, Medium Mesh, 20 knots, Stern Quartering Seas at 30 degrees


.........................................................Time domain . . . . . Frequency domain

0 1 2


0 1 2 3 4 5


0.0

0.5

1.0

SurgeRAO

|η1|a

.....................................................................................................................................................................................................................................................................................................................................................................................................

........................... ... .0 1 2


0 1 2 3 4 5


0.0

0.1

0.2

0.3

0.4

0.5

SwayRAO

|η2|a

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................. . ..

0 1 2


0.0

0.5

1.0

1.5

HeaveRAO

|η3|a

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................... ...

0 1 2


0

1

2

3

RollRAO

|η4|k a

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................ ....

0 1 2


0.0

0.5

1.0

PitchRAO

|η5|k a

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................... . . ...

0 1 2


0.0

0.5

1.0

YawRAO

|η6|k a

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................... .. . . . .

Figure 32: Motions for HALIFAX from Frequency Domain and Time DomainComputations, Medium Mesh, 20 knots, Bow Quartering Seas at 150 degrees


10 Extension to Nonlinear Forces fromIncident Waves and Buoyancy

Time domain predictions presented thus far are based on linearized forces. It isrelatively simple to extend the work to include nonlinear evaluation of forces due toincident waves and buoyancy. Furthermore, these two force components have largeforce magnitudes relative to other components such as radiation forces.

For evaluation of nonlinear forces, it is necessary to introduce a new ship-basedcoordinate systemxS, yS, zS that rotates with the ship, as shown in Figure 33 Theorigin of this coordinate system is at the ship centre of gravity. When wave-inducedship motions are zero, the ship-based coordinate system is identical to the coordi-nate systemx, y, z based on translating earth axes presented in Figure 1.

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................

.................

.................

.................

.................

.................

.................

...........

........

........

........

........

........

........

........

........

........

........

.........................

...................zS

......................................................................................................... ................... xSCG•

..................................................................................................................................................................................................

..................................................................................................................................................................................................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................................................................................

.......................................................................................................................................

......................

............................................................................................................................yS

......................................................................................................... ................... xs•

...........................................................................................................................................................................

........................................................................................................................................................................

...............................................................................................................................................................................

..................................................................................................

...................................

..............................

.........................................................................................................

...................yS

............................................................................................................................

zS

CG•

Figure 33: Ship Referenced Coordinate System for Large Angular Motions

Force components in the ship-based coordinate system are easily evaluated as fol-lows:

F Sj (t) = −

∫Sb

p(t) nSj dS for j = 1− 6 (96)

wherenSj is the outward normal component for modej in the ship-based coordi-

nate system. The following equation can be used to obtain force components in


translating earth axes from force components computed using ship-based axes:

F1(t) = F S1 (t) cos η6 cos η5 − F S

2 (t) sin η6 cos η5 + F S3 (t) sin η5 (97)

F2(t) = F S1 (t) (cos η6 cos η5 sin η4 + sin η6 cos η4)

+ F S2 (t) (− sin η6 sin η5 sin η4 + cos η6 cos η4)

− F S3 (t) sin η4 cos η5 (98)

F3(t) = F S1 (t) (− cos η6 sin η5 cos η4 + sin η6 sin η4)

+ F S2 (t) (sin η6 sin η5 cos η4 + cos η6 sin η4)

+ F S3 (t) cos η4 cos η5 (99)

The above equations are based on rotations in the order of roll, pitch, and yawwhen going from the translating earth to ship-based coordinate systems. Similarly,the location of a point on the ship can be converted to a location in translating earthaxes as follows:

x(t) = η1 + xS cos η6 cos η5 − yS sin η6 cos η5 + zS sin η5 (100)

y(t) = η2 + xS (cos η6 cos η5 sin η4 + sin η6 cos η4)

+ yS (− sin η6 sin η5 sin η4 + cos η6 cos η4)

− zS sin η4 cos η5 (101)

z(t) = η3 + xS (− cos η6 sin η5 cos η4 + sin η6 sin η4)

+ yS (sin η6 sin η5 cos η4 + cos η6 sin η4)

+ zS(t) cos η4 cos η5 (102)

zwl(t) = z(t) + zCGwl (103)

Equations (102) and (103) are required for evaluating the location of points on thehull within the wave field. Recall thatzCG

wl is the height of the ship centre of gravityabove the water surface when the ship is at rest in calm water.

The incident wave elevation in the time domain is given by Equation (46). Thewave potential and its derivatives in the time domain are given by:

ΦI =g a

ωI

sin [kI (x cos β − y sin β) − ωe t − εI ] exp(kI zwl) (104)

∂ΦI

∂x= a ωI cos β cos [kI (x cos β − y sin β) − ωe t − εI ]

× exp(kI zwl) (105)∂ΦI

∂y= −a ωI sin β cos [kI (x cos β − y sin β) − ωe t − εI ]

× exp(kI zwl) (106)∂ΦI

∂z= a ωI sin [kI (x cos β − y sin β) − ωe t − εI ] exp(kI zwl) (107)


∂ΦI

∂t= −g a

ωI

ωe cos [kI (x cos β − y sin β) − ωe t − εI ]

× exp(kI zwl) (108)

Using the above terms, the combined pressure due to incident wave forces andbuoyancy can be evaluated as follows:

p(t) = −ρ

(∂ΦI

∂t− U

∂ΦI

∂x+ g zwl

)for zwl ≤ ζ (109)

p(t) = 0 for zwl > ζ (110)

Figures 34 to 36 show results of computations for verifying nonlinear buoyancyforce predictions. For small displacements, the nonlinear buoyancy predictionsshow close agreement with linear buoyancy forces. For large heave displacements,nonlinear buoyancy forces approach limiting values as the ship hull becomes com-pletely submerged (negative heave) or emerged (positive heave). The nonlinearroll buoyancy moments behave as expected, with symmetry about zero roll angleand vanishing righting moment at large roll angles. The nonlinear pitch buoyancymoments behave in a manner similar to the heave forces, with slope magnitudereducing when portions of the hull become completely submerged or emerged.


−8 −6 −4 −2 0 2 4 6

Heaveη3 (m)

−2

−1

0

1

2

3

HeaveBuoyancy

ForceFB

3 /4

.........................................................Linear buoyancy

............. ............. ....... Nonlinear buoyancy

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ........

Figure 34: Heave Buoyancy Forces for HALIFAX, Medium Mesh

−90 −60 −30 0 30 60 90

Roll η4 (deg)

−3

−2

−1

0

1

2

3

RollBuoyancy

ArmFB

4 /4(m)

.........................................................Linear buoyancy

............. ............. ....... Nonlinear buoyancy...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................

..........................

............. ............. ............. ............. ............. ....................................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ....................................... ............. ............. ............. ............. ..........................

..........................

.............

Figure 35: Roll Buoyancy Moments for HALIFAX, Medium Mesh


−10 −5 0 5 10

Pitchη5 (deg)

−60

−40

−20

0

20

40

60

PitchBuoyancy

ArmFB

5 /4(m)

.........................................................Linear buoyancy

............. ............. ....... Nonlinear buoyancy

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

Figure 36: Pitch Buoyancy Forces for HALIFAX, Medium Mesh


11 Comparison of Numerical Predictionsand Experimental Results for Heave andPitch

The software testing presented thus far has been limited to verification, includingcomparison of predictions in the frequency and time domains. Validation of shipmotion predictions generally requires comparison with measurements from modeltests and/or sea trials. The current motion predictions do not include viscous orappendage forces, which can significantly influence lateral plane motions; however,predicted vertical plane predictions can be validated using experimental data.

Figures 37 and 38 show comparisons of heave and pitch predictions with exper-imental results for the CPF hydroelastic model in head seas [7]. The numericalpredictions include the influence of restrained surge motions. The numerical pre-dictions show very good agreement with experimental motions. As was observedin Section 9, frequency domain predictions correspond very closely to linear timedomain predictions. The nonlinear predictions, which are for a wave steepnessH/λ of 1/20, vary noticeably from the linear predictions. For the current heave andpitch cases, it appears that including nonlinear buoyancy and incident wave forcesdoes not increase the accuracy of predictions. For intermediate wave frequenciesat a Froude number of 0.20, Figure 38 suggests that including nonlinearities canactually give less accurate heave predictions. This observation could be due tomodelling of larger amplitude waves using linear wave theory. To improve accu-racy of nonlinear predictions in regular waves, a new regular wave model will beimplemented using Stokes second order wave theory (see Sarpkaya and Isaacson[18]).

As mentioned in Section 9, the linear time domain simulations run approximately30 times faster than real time when simulating a full-scale ship using a time stepsize of 0.1 s. When nonlinear buoyancy and incident wave forces are included usingthe medium mesh, computational speed is reduced to approximately 5 times fasterthan real time.


Experiments Predictions

• H/λ = 1/30 .........................................................Linear time domain

4 H/λ = 1/20 ............. ............. ....... Nonlinear time domain,H/λ = 1/20

2 H/λ = 1/15 . . . . . Frequency domain

1.5 2.0 2.5 3.0 3.5

Non-dimensional Wave Frequency:ωI

√L/g

0.0

0.2

0.4

0.6

0.8

1.0

HeaveAmplitude

η3/a

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

.................................................. . .••

•••

••

•

•

•

444444

4

4

4

4

22

22

22

2

2

22

1.5 2.0 2.5 3.0 3.5


√L/g

0.0

0.2

0.4

0.6

0.8

1.0

PitchAmplitude

η5/ka

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .................................................................................................................................. ............. ............. ............. ............. ........

............................................................••

•

•

••

•

•••

44

4

4

44

4

4

4

4

2

2

2

2

22

2

22

2

Figure 37: Heave and Pitch for CPF Hydroelastic Model in Head Seas, Fn = 0.12


Experiments Predictions

• H/λ = 1/30 .........................................................Linear time domain

4 H/λ = 1/20 ............. ............. ....... Nonlinear time domain,H/λ = 1/20

2 H/λ = 1/15 . . . . . Frequency domain

1.5 2.0 2.5 3.0 3.5


√L/g

0.0

0.2

0.4

0.6

0.8

1.0

HeaveAmplitude

η3/a

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............................................................................................................................................... ............. ............. ............. ............. ............. ...........

.......................................................••

•••••

•

•

•

44

44

444

44

4

1.5 2.0 2.5 3.0 3.5


√L/g

0.0

0.2

0.4

0.6

0.8

1.0

PitchAmplitude

η5/ka

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ................................................................................................................................................................................................... ............. ............. ............. ............

..............................................................••

•

•

••

••

••

4

4

4

4

44

44

44

Figure 38: Heave and Pitch for CPF Hydroelastic Model in Head Seas, Fn = 0.20


12 Conclusions

New software has been developed for predicting hydrodynamic forces and result-ing motions in the time domain for an unappended ship hull in waves. The ship isassumed to travel at steady speed and heading relative to the waves. The hydrody-namic forces in the time domain are evaluated using previously computed valuesin the frequency domain. Several improvements have been made to the accuracy,efficiency, and robustness of hydrodynamic force predictions in the frequency do-main. Comparisons between predicted motions in the frequency and time domainsindicate consistency between the two approaches. Initial validation using experi-mental motions for heave and pitch indicates that predictions in the frequency andtime domains are giving good results. Nonlinear effects arising from incident waveand buoyancy forces can be included in time domain predictions.

In the near term, forces from rudders, bilge keels, and other appendages will beincorporated into numerical predictions of ship motions in waves. In the longerterm, the motions of a freely maneuvering ship in waves will be predicted.


References

1. McTaggart, K.A. (2002). Three Dimensional Ship Hydrodynamic CoefficientsUsing the Zero Forward Speed Green Function. (DRDC Atlantic TM2002-059). Defence Research and Development Canada - Atlantic.

2. Mei, C.C. (1983). The Applied Dynamics of Ocean Surface Waves, New York:John Wiley & Sons.

3. Wehausen, J.V. (1971). The Motion of Floating Bodies.Annual Review ofFluid Mechanics, 3, 237–268.

4. McTaggart, K.A. (1989). Hydrodynamics and Risk Analysis of IcebergCollisions with Offshore Structures. Ph.D. thesis. University of BritishColumbia.

5. Magee, A. (1994). Seakeeping Applications Using a Time Domain Method. InTwentieth Symposium on Naval Hydrodynamics, pp. 1–19. Santa Barbara.

6. Fonseca, N. and Soares, C. Guedes (1998). Time-Domain Analysis ofLarge-Amplitude Vertical Ship Motions and Wave Loads.Journal of ShipResearch, 42(2), 139–153.

7. McTaggart, K., Datta, I., Stirling, A., Gibson, S., and Glen, I. (1997). Motionsand Loads of a Hydroelastic Frigate Model in Severe Seas.Transactions,Society of Naval Architects and Marine Engineers, Vol. 105.

8. Cook, R.D. (1981). Concepts and Applications of Finite Element Analysis,New York: John Wiley & Sons.

9. Sclavounos, P.D. and Lee, C. (1985). Topics on Boundary Element Solutionsof Wave Radiation-Diffraction Problems. InFourth International Conferenceon Numerical Ship Hydrodynamics, Washington.

10. Lutz, M. and Ascher, D. (1999). Learning Python, Sebastopol, CA: O’Reilly &Associates.

11. McTaggart, K.A. (1996). Improved Boundary Element Methods for PredictingSectional Hydrodynamic Coefficients for Strip Theory Ship Motion Programs.(DREA TM 96/212). Defence Research Establishment Atlantic.

12. Ando, S. (2002). Lower Bounds to Irregular Frequencies in Wave-ShipInteractions for Engineering Applications. (DREA TM 2001-192). DefenceResearch Establishment Atlantic.


13. Lee, C.-H. and Sclavounos, P.D. (1989). Removing the Irregular Frequenciesfrom Integral Equations in the Wave-Body Interactions.Journal of FluidMechanics, 207, 393–418.

14. Hager, W.H. (1988). Applied Numerical Linear Algebra, Englewood Cliffs,New Jersey: Prentice-Hall.

15. Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Croz, J. Du,Greenbaum, A., Hammarling, S., McKenney, A., Ostrouchov, S., andSorensen, D. (1995). LAPACK Users’ Guide, Second ed. Philadelphia:Society for Industrial and Applied Mathematics.

16. Ascher, D., Dubois, P.F., Hinsen, K., Hugunin, J., and Oliphant, T. (2001).Numerical Python. (Technical Report UCRL-MA-128569). LawrenceLivermore National Laboratory. Livermore, California.

17. Hornbeck, R.W. (1975). Numerical Methods, New York: Quantum Publishers.

18. Sarpkaya, T. and Isaacson, M. (1981). Mechanics of Wave Forces on OffshoreStructures, Van Nostrand Reinhold.


Symbols

[A] added mass matrixa wave amplitudeB ship beam[B] frequency domain damping matrix[b] dynamic damping matrix[C] hydrostatic stiffness matrix[c] dynamic stiffness matrixCB centre of buoyancy[D] normal flow velocity influence matrixeven superscript denoting velocity potential is even function ofy{FD}

diffracted wave force vectorFD,U

j diffracted force component proportional toU for modej

FD,ωe

j diffracted force component proportional toωe for modej{F I}

incident wave force vectorF S

j force for modej in ship-based coordinatesFn Froude numberFP fore perpendicularG0 frequency dependent term of Green function relative to zero

frequency Green functionG∞ frequency dependent term of Green function relative to infinite

frequency Green functiong gravitational acceleration[H] x velocity influence matrix[K] retardation function matrix[K0] zero-speed component of retardation function matrix[KU ] component of retardation function matrix proportional toU[KUU ] component of retardation function matrix proportional toU2

KG height of centre of gravity above baselinekI incident wavenumberL ship length between perpendicularsLCB longitudinal centre of buoyancy[M ] inertial matrixnk outward normal componentk for translating earth axesnS

k outward normal componentk for ship-based axesNport

p number of panels on port side of hullodd superscript denoting velocity potential is odd function ofyp pressureport superscript denoting port side of hull


R distance from field point to sourceR1 distance from field point to source imagerxx roll radius of gyrationryy pitch radius of gyrationrzz yaw radius of gyrationSb surface of floating bodySk surface of hull panelkstar superscript denoting starboard side of hullTmid draft at midshipsts trim by sternU ship speedwi interpolation weight factorx, y, z translating earth axis coordinatesxS, yS, zS ship-based axis coordinates~xs source locationzCG

wl height of ship centre of gravity above water surface in calm waterzwl z coordinate relative to calm water surfaceβ wave direction relative to ship∆t time step sizeδjk Kroenecker delta functionεI phase lead of incident waveζI incident wave elevationηj motion displacement in modejλ matrix condition numberρ water density{σ} source strength vectorτ delay time for convolution integralΦI incident wave potential in time domainφD diffracted wave potential in frequency domainφI incident wave potential in frequency domainφk radiated wave potential due to motion modekφ′k normalized potential for modek at zero or infinite frequency limitωI incident wave frequencyωe wave encounter frequencyωt

e transition encounter frequency for Green function computation4 ship mass displacement


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Defence R&D Canada - Atlantic

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Hydrodynamic Forces and Motions in the Time Domain for an Unappended ShipHull

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McTaggart, Kevin A.

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May 2003

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This report describes numerical predictions of hydrodynamic forces and motionsin the time domain for an unappended ship hull in waves. The hydrodynamicforces in the time domain are evaluated using previously computed values in thefrequency domain, which are based on the three-dimensional Green function fora body with zero forward speed. Forward speed effects are included in both thefrequency and time domains using suitable approximations. Due to the underly-ing importance of numerical predictions of hydrodynamic forces in the frequencydomain, attention has been given to their accuracy, robustness, and efficiency.Comparisons between motion predictions in the frequency and time domainsshow excellent agreement, suggesting correct theoretical and numerical imple-mentations. Comparisons of numerical heave and pitch predictions with modeltest data for the frigate HALIFAX show very good agreement. Time domain sim-ulations of ship motion execute significantly faster than real time, indicating thatthe approach will be practical for modelling and simulation applications.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (technically meaningful terms or short phrases that characterizea document and could be helpful in cataloguing the document. They should be selected so that no security classificationis required. Identifiers, such as equipment model designation, trade name, military project code name, geographiclocation may also be included. If possible keywords should be selected from a published thesaurus. e.g. Thesaurus ofEngineering and Scientific Terms (TEST) and that thesaurus-identified. If it not possible to select indexing terms whichare Unclassified, the classification of each should be indicated as with the title).

heaverollpitchseakeepingship motionssurgetime domainswayyaw


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