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Final Report”Added resistance in short waves”A project jointly funded by Den Danske Maritime Fond (2016-087) and Orients Fund (2016)

Mostafa Amini AfsharHarry Bingham

February 6, 2019

Contents

Project summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Objectives and deliverables of the project . . . . . . . . . . . . . . . . . . . . . 3

1.3 Structure of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The computational framework and grid generation strategy . . . . . . . . . . . . . . . 3

3 New extensions to the linear OceanWave3D-Seekeeping model . . . . . . . . . . . . . . 4

3.1 Generalized modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2 Wave excitation forces in followings seas . . . . . . . . . . . . . . . . . . . . . 5

3.3 Added resistance using the Kochin function . . . . . . . . . . . . . . . . . . . . 7

4 The nonlinear potential-flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1 Moving and deforming overlapping grid . . . . . . . . . . . . . . . . . . . . . . 9

4.2 Stream function theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3 Forced oscillation of a submerged circular cylinder . . . . . . . . . . . . . . . . 11

5 Future perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Dissemination activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6.1 Publications from the project . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

References 15

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Project summary

The objective of this project was to extend and build upon an existing linear potential-flow model forwave-structure interaction, OceanWave3D-Seakeeping [5]. The extensions include adding the capabil-ity to capture the effects of flexible ship motions and waves incident from behind the vessel; adding anew more robust far-field method for predicting added wave resistance; and finally, adding the abilityto solve for fully nonlinear wave-structure interaction. Of particular interest here is the estimation ofadded resistance in short waves, which is highly relevant for today’s large ships. Two journal papersand two conference contributions were to be prepared and submitted on the work. All these projectgoals have been achieved, and the the results have appeared in two published journal papers, with anadditional submission under review, and four accepted conference papers. Other appropriate dissemi-nation measures have also been taken to present the project to the broader network among industriesand scientists.

1 Introduction

This final report summarizes the results of a two-year post-doc project financially supported jointly bythe Danish Maritime Fund (2016-087) and The Orients Fund (2016). The project started on November1, 2016 and finished on October 31, 2018. It was carried out by Mostafa Amini-Afshar with guidanceand support from Prof. Harry Bingham, both employed at the Department of Mechanical Engineering,Technical University of Denmark.

1.1 Background

A reliable prediction of added resistance in short waves (relative to the ship length) is difficult to achieveusing linear seakeeping calculation tools. This is mainly due to the highly nonlinear character of thewave-ship interaction in the high-frequency, short-wave range. Conventionally the added resistance inthis range is calculated using simplified analytical or empirical relations, see for example [9], whichare then combined with more refined numerical calculations over the low-frequency (long-wave) range.The low-frequency results are usually computed using either 2D strip-theory or 3D linearized potential-flow solvers. This fact has motivated the current project to start developing a nonlinear potential-flowmodel for the purpose of calculation of added resistance in the high-frequency wave range.

1.2 Objectives and deliverables of the project

The goals of this project were to develop the mentioned nonlinear potential-flow framework for wave-structure interaction, and to add additional capabilities to the original linear seakeeping code. Theoutcomes of the project as deliverables were to be published in two journal submissions and two con-ference papers, together with presentation and dissemination opportunities for the relevant maritimeindustries.

1.3 Structure of the report

Following this introduction and background discussion, the computational framework of the solver andgrid generation issues are briefly described in §2. The improvements which have been carried out tothe linear seakeeping solver are presented in §3. Section §4 is devoted to explaining the framework forthe developed nonlinear potential-flow model. In §6, the dissemination activities are described whichserve to publicize the results from the project. Conclusions are drawn in §7.

2 The computational framework and grid generation strategy

The software is written in C++ and builds on the open-source library Overture[8]. The library hasfacilities to solve partial differential equations on overlapping grids using 4th-order finite-differenceapproximations. The grid generation strategy for the nonlinear solver is the same as for the original

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linear solver OceanWave3D-Seakeeping, and the only difference is regarding the grid deformation whichis explained in detail in Section §4. The linear solver was originally developed during Mostafa Amini-Afshar’s PhD project [1], and subsequently improved and extended during another DDMF fundedpost-doc project [2].

3 New extensions to the linear OceanWave3D-Seekeeping model

3.1 Generalized modes

Figure 1: Response amplitude operator for the first vertical bending mode (left) and vertical displace-ment midship (right) for the prismatic barge.

Figure 2: Hydrodynamic coefficients for the first vertical bending mode of a modified Wigley hull atvarious values of forward speed.

The larger the ship, the more flexible it becomes, which leads to lower natural frequencies andincreases the likelihood that these modes are excited by wave induced loading. More flexibility alsoleads to stronger interaction between the flexible modes and the rigid-body modes. In order to capturethese effects, we have extended the linear solver to include flexural ship motions via a Generalized modesapproach. Much of this development was carried out in connection with the Masters Thesis project of

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Matilde Andersen [6]. Some of the results of this extension have been presented and published in 32ndInternational Workshop on Water Waves and Floating Bodies [7], see also Figure 1 and Figure 2. Inthe first figure (left) the response amplitude operator (RAO) for the first vertical bending mode ofa prismatic hull is shown, while on the right we plot the vertical displacement of the hull’s midshipsection. These results are at zero forward speed, and are compared with the measurements and othernumerical codes. Figure 2 shows instead the hydrodynamic coefficients (added mass and damping)for a modified Wigley hull at several Froude numbers for the first vertical bending mode of the ship.Very few models exist in the litterature which are capable of treating this problem, and can accuratelycapture the effects of forward speed on a ship’s bending modes.

3.2 Wave excitation forces in followings seas

x

y

Figure 3: Part of the overlapping grid for the bulk carrier

The linear solver OceanWave3D-Seakeeping has also been extended to support the evaluation ofwave excitation forces, ship response and added resistance in following-seas (waves incident from abaftthe beam). In Figures 4 and 5, some calculations are compared with the experiments and boundaryelement codes for the motion response and the wave exciting forces for a bulk carrier in followingseas. Note also that part of the overset grid for this ship is shown in Figure 3. Extensive validationresults in following-seas, together with other features of the diffraction hydrodynamic problem whichare particular to our solver, have been published in the journal Applied Ocean Research journal [4].

5

0 1 2 3 4 50

0.05

0.1

0.15

0.2

λ/Lpp

|X5|/ρgABL2 pp

OceanWave3D-Seakeeping(Iwashita, et. al, 2016) - Num.

TiMIT - BEM.ω02 − ω03 boundary, λ/L = 0.204ω01 − ω02 boundary, λ/L = 0.814

0 1 2 3 4 5-π

-π/20

π/2

π

λ/Lpp

θ 0

Figure 4: Wave excitation pitch moment X5 for the bulk carrier Fr = 0.18, magnitude (left) phase(right)

0 1 2 3 4 50

0.5

1

1.5

2

λ/Lpp

|ξ 5|/k

A

OceanWave3D-Seakeeping - (NK)TiMIT - BEM.

(Iwashita, et. al, 2016) - Exp.ω02 − ω03 boundary, λ/L = 0.204ω01 − ω02 boundary, λ/L = 0.814

0 1 2 3 4 5-π

-π/20

π/2

π

λ/Lpp

θ 0

Figure 5: RAO in pitch ξ5 for the bulk carrier Fr = 0.18, magnitude (left) phase (right)

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3.3 Added resistance using the Kochin function

Finally, a second, and more robust, far-field method for computing the the wave added resistance fromfirst-order quantities has been implemented into OceanWave3D-Seakeeping as part of this project.This method uses the so-called Kochin function to compute the added resistance Rw based on a lineintegral over intervals of a wave-number like parameters m. The final form of the added resistance in

Figure 6: Overlapping grid for the floating spheroid

-π/2 -π/4 0 π/4 π/2

0.02

0.04

0.06

α

|H1(m

)|2/gL

5

m = k̄3+k̄2

2 + k̄3−k̄2

2 sinα

−5 0 5 10 15m

-π/2 -π/4 0 π/4 π/20

0.5

1

LI(α

)/ρgA

2B

2

0 10 20 30 40 50√κ̄2 −m2

0 1 2 3 4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

kL/2π

LR

w/ρgA

2B

2OW3D (near)

OW3D (far)The 1st integralThe 3rd integral

(Iwashita and Ohkusu, 1989)

;

Figure 7: Left: plots of the added resistance integrand and the Kochin function. Right: addedresistance for the fixed submerged spheroid at Fr = 0.20

-π/2 -π/4 0 π/4 π/2

0.2

0.4

0.6

α

|H1(m

)|2/gL

5

m = k̄3+k̄2

2 + k̄3−k̄2

2 sinα

−2 0 2 4 6 8 10m

-π/2 -π/4 0 π/4 π/2

10

20

30

LI(α

)/ρgA

2B

2

0 5 10 15 20√κ̄2 −m2

0 1 2 3 4

10

20

30

40

λ/L

LR

w/ρgA

2B

2

OW3D (near)

OW3D (far)

Figure 8: Left: plots of the added resistance integrand and the Kochin function. Right: addedresistance for the floating spheroid using only the middle integral for several Froude numbers from leftto right given by Fr = 0.0, 0.05, 0.15.

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0 0.5 1 1.5 2 2.5 30

4

8

12

16

λ/L

LR

w/ρgA

2B

2

OW3D (far)

Measurements (SSPA)

Figure 9: Part of the overlapping grid for a modern tanker (left), and the added resistance for Fr = 0.17based on the far-field and Kochin function method (right)

this formulation is:

Rw =ρ

4π

{−∫ k̄1

−∞+

∫ k̄3

k̄2

+

∫ ∞

k̄4

}κ̄ (m− k cosβ)√

κ̄2 −m2|H1(m)|2 dm , (1)

where H1(m) is the Kochin function, β the incident wave heading angle, k the incident wave numberand κ̄ = (ω −mU)2/g, with k̄i, i = 1, 2, 3, 4 the four roots of the equation κ̄ = ±m. The the addedresistance calculated using this Kochin function integral for a submerged and a floating spheroid areshown in Figures 7 and 8. Part of the overlapping grid for the floating spheroid is also shown inFigure 6. These results have also been presented and published in the 33rd International Workshopon Water Waves and Floating Bodies [3]. Calculations for the added resistance of a modern tankerhull based on this formulation are also shown in Figure 9 where they can be seen to compare well withexperiments.

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4 The nonlinear potential-flow model

As part of this project, we have developed a fully nonlinear potential-flow model for wave-structureinteraction problems using the same computational strategy as was applied to solve the linear sea-keeping problems described above. Some early results from this work have also been accepted forpresentation at the 34th International Workshop on Water Waves and Floating Bodies, April 2019,Newcastle Australia. In this section this nonlinear model is briefly explained and presented.

4.1 Moving and deforming overlapping grids

Sf

G1

G2

xz

Figure 10: Deformable grid methodology to simulate wave propagation.

In the linear version of OceanWave3D-Seakeeping, the computational grid is time-constant andcan thus be built once and then used for all subsequent time-steps without modification. This grid isbounded by the mean free surface plane, the submerged ship surface in still water and at its equilibriumposition, and the sea bottom. In the nonlinear model, on the other hand, the computational gridmust follow the exact instantaneous position of the free surface and the body surface at each stepof the calculation. To this end, we make use of the overlapping grid methodology and the built-inOverture feature of deformable component grids. As a first example, we consider the case of pure wavepropagation in a periodic domain. This grid is shown in Figure 10 which is made of 2 componentsG1 and G2. The only changes which occur in the first grid are associated with interpolation pointsin the overlapping region, as shown by the small black squares. On the other hand, G2 also changesits shape in order to follow the motion of the free surface Sf . The location of the free surface is usedas the starting curve for re-generation of the component grid G2. Hyperbolic Grid Generation is thenemployed to march the grid from the starting curve Sf downwards in the z-direction. The overlappingregion is then updated to complete the generation of the new grid. This strategy is very efficient, andleads to minimal extra computational effort compared to using a fixed grid.

To evolve the solution forward in time, the exact free-surface boundary conditions are integratedusing the explicit fourth-order Runge-Kutta scheme. The free-surface conditions are given by:

∂ζ

∂t= −∇ζ · ∇φ̃+ w̃ (1 +∇ζ · ∇ζ) , (2)

∂φ̃

∂t= −gζ − 1

2∇φ̃ · ∇φ̃+

1

2w̃2 (1 +∇ζ · ∇ζ) . (3)

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where the horizontal gradient operator is defined by ∇ = [∂/∂x, ∂/∂y]. Here the instantaneous free-surface position is ζ, and h is the water depth while φ̃ represents the velocity potential at Sf , andw̃ is the vertical component of the fluid velocity at the free-surface Sf . In order to compute w̃, andevolve the equations forward in time, we must solve for the full 3D velocity potential φ from massconvervation which is expressed by the Laplace equation:

∇2φ+ φzz = 0, −h < z < ζ (4a)

∂φ

∂z+∇φ · ∇h = 0, z = −h, (4b)

φ = φ̃, z = ζ, (4c)

along with suitable conditions on the horizontal truncation boundaries of the numerical domain (gen-erally either periodic or solid wall boundaries.)

We have validated this computational strategy first by simulating an exact, periodic nonlinearwave solution given by the stream function theory described below.

4.2 Stream function theory

For this case, the velocity potential on the free surface φ̃ and the free-surface elevation ζ are initializedusing the semi-analytical stream function theory solution of Fenton [10]. The computational grid forthis case is shown in Figure 10. Note that the problem is periodic in the horizonatal direction, soperiodic conditions are applied at the right and left boundaries so that the wave circulates through thecomputational domain. The code is run for several wave periods and the surface elevation is comparedwith the analytical results in Figure 11. The convergence of the model has been also investigated,and the results are presented in Figure 12. The results are shown for increasing wave steepness in therange H/λ = 0.02, 0.04, · · · 0.10, 0.11, 0.126. In all these cases kh = 2π. Here H is the wave height, λis the wave length and k = 2π/λ is the wave number. These results confirm the ability of the modelto treat highly nonlinear waves in a robust manner.

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.05

−0.025

0

0.025

0.05

x

ζ

Comp.Ana.

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.08

−0.04

0

0.04

0.08

x

ζ

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.1

−0.05

0

0.05

0.1

x

ζ

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.1

−0.05

0

0.05

0.1

x

ζ

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.1

−0.05

0

0.05

0.1

x

ζ

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.1

−0.05

0

0.05

0.1

x

ζ

Figure 11: Comparison between the computed free-surface wave elevation ζ with the analytical resultsfrom the stream function theory. The results are for increasing steepness, and are obtained after 5wave periods. The extreme case shown in the bottom right plot is for the wave at 80 % of the breakinglimit.

10

1 1010−4

10−3

10−2

10−1

100

N

Error

1 1010−4

10−3

10−2

10−1

100

N

Error

1 1010−3

10−2

10−1

100

N

Error

1 1010−3

10−2

10−1

100

N

Error

1 1010−3

10−2

10−1

100

N

Error

1 1010−3

10−2

10−1

100

N

Error

Figure 12: Convergence studies for the cases shown in Figure 11. The dashed lines in the plots show1 to 4 slope. Note that N = 1 represents the computational grid with Nx = 11 and Nz = 14 numberof grid points. The resolution study is performed by N = [1 1.5 1.7 2 2.5 4]. The plots in Figure 11are for the case with N = 4.

4.3 Forced oscillation of a submerged circular cylinder

Sf

SB

G1

G2

G3

x

z

η

Figure 13: The Deformable and moving overset grid. The grid is for the case where a submergedcircular cylinder is performing forced oscillation η in the z direction.

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As a final validation test, we consider the large amplitude, heaving motion of a submerged circularcylinder, and the resultant wave generation and loading on the structure. The computational gridfor this simulation is shown in Figure 13. Here, component grid G2 represents the cylinder withradius a at its initial position with a submergence depth of h̃ = 3a. The applied hydrodynamic forceon the body is calculated for a sinusoidal forced heaving motion with non-dimensional frequency ofνa = ω2/g = 1.0 (where ω is the radian frequency) and a range of different motion amplitudes η/a.One example of the calculated force is shown in Figure 14 (top right) for the largest amplitude caseof η/a = 1.75. By performing a harmonic analysis of the force signal, the amplitude of the mean forcealong with each harmonic componant with frequency nω, n = 1, 2, . . . can be obtained. The resultsof this computation are compared with the solution of [11] which is based on the linear free-surfaceconditions. These harmonics are also shown in Figure 14. These results demonstrate the ability of the

0 0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2−0.08

−0.04

0

0.04

0.08Mean

η/a

Fz/ρω2πa2η Computational

Analytical - Wu(1993)

0 10 20 30 40 50 60 70−2

−1

0

1

2

t√g/a

Fz(t)/ρω2πa2η

0 0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 20.6

0.8

1

1.21st Harmonics

η/a

Fz/ρω2πa2η

0 0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2−0.1

0

0.1

0.22nd Harmonics

η/a

Fz/ρω2πa2η

0 0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2−0.04

0

0.04

0.083rd Harmonics

η/a

Fz/ρω2πa2η

0 0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2−0.01

0

0.01

0.02

0.034th Harmonics

η/a

Fz/ρω2πa2η

Figure 14: The results of the harmonic analysis for the applied force on the submerged cylinderperforming forced oscillation in the vertical direction. The oscillation amplitudes are in the rangeη/a = 0.2, 0.4, · · · 1.0, 1.25, 1.5, 1.75.

model to capture highly-nonlinear wave-structure interaction, paving the way to a robust estimationof the nonlinear short-wave added resistance of ships.

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5 Future perspectives

These nonlinear results presented above represent a proof of concept which demonstrates the abilityof the model to treat nonlinear, large-amplitude motions of a floating structure and the resultantnonlinear wave interactions and loading. This nonlinear framework for wave-structure interactionprovides a flexible tool which can be used to compute nonlinear wave forces on marine structures.Although we did not manage to apply the method to a ship before the end of the current project,work continues on the model and we expect that 3D results for nonlinear diffraction of waves arounda ship-like structure will be available in the near future. Important advances have also been madeon the topics of flexible ship motions and predicting the response of a ship in following seas, both ofwhich will benefit the industrial and research community and lead to future publications which arealready in preparation.

6 Dissemination activities

This section describes the dissemination activities which have accompanied the work carried out duringthis project.

6.1 Publications from the project

The following manuscripts have been published, or submitted for publication, during the course of theproject:

1. Amini-Afshar, M. and H. B. Bingham. Solving the linearized forward-speed radiation problemusing a high-order finite difference method on overlapping grids. Applied Ocean Research, 69:220–244, 2017

2. Amini-Afshar, M. and H. B. Bingham. Pseudo-impulsive solutions of the forward-speed diffrac-tion problem using a high-order finite-difference method. Applied Ocean Research, 80:197–219,2018

3. Amini-Afshar, M., H. B. Bingham, and W. D. Henshaw. Stability analysis of high-order finite-difference discretizations of the linearized forward-speed seakeeping problem. Applied OceanResearch, (under review)

4. Amini-Afshar, M. and H. B. Bingham. Accurate evaluation of the Kochin function for addedresistance using a high-order finite difference-based seakeeping code. In 33rd International Work-shop on Water Waves and Floating Bodies, Guidel-Plages, France, 4-7 April, 2018

5. Amini-Afshar, M. and H. B. Bingham. Implementation of the far-field method for calculationof added resistance using a high order finite-difference approximation on overlapping grids. In32nd International Workshop on Water Waves and Floating Bodies, Dalian, China, 23-26 April,2017

6. Andersen, M. H., M. Amini-Afshar, and H. B. Bingham. Implementation of generalized modesin a 3d finite difference based seakeeping model. In 32nd International Workshop on WaterWaves and Floating Bodies, Dalian, China, 23-26 April, 2017

7. Amini-Afshar, M., W. D. H. Bingham, Harry B, and R. Read. A nonlinear potential-flow modelfor wave-structure interaction using high-order finite differences on overlapping grids. In 34thInternational Workshop on Water Waves and Floating Bodies, Noah’s on the Beach, Newcastle,NSW, Australia, 7-10 April, 2019

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6.2 Presentations

In addition to the publications listed above, the results of the project have been disseminated throughthe following presentations:

• Presentation at 19th DNV-GL workshop at The Technical University of Denmark, Kgs. Lyngby,Denmark (January 2018)

• Presentation at 20th DNV-GL workshop at KTH Royal Institute of Technology, Stockholm,Sweden (January 2019)

• Presentation at the 32th IWWWFB in Dalian, China (April 2017)

• Presentation at the 33rd IWWWFB in Brest, France (April 2019)

• Lectures to the students in DTU course 41222 “Wave load on ships and offshore structures”(Fall semesters 2017 and 2018)

• Presentation in Ship 4.0: Safe, Efficient, Sustainable and Autonomous A DTU status andoutlook (IDA Maritim ), March 4, 2019.

7 Conclusions

This final report has summarized the highlights of Mostafa Amini-Afshar’s post-doc project over theperiod 1/11/2016 - 31/10/2018, financed jointly by the Danish Maritime Fund and Orients Fund.The work has been disseminated in two published and one submitted journal publications, along withfour conference contributions and several oral presentations. The open source model OceanWave3D-Seakeeping has been improved and extended to include generalized modes, following seas and a morerobust far-field added resistance method. This makes the software even more useful for the mar-itime industry, as well as for people in Denmark and around the world who are engaged in maritimeresearch. Valuable test cases and examples have been generated to support DTU’s education of Mar-itime students. A nonlinear proof of concept has been produced which paves the way to an improvedunderstanding of the nonlinear interaction between short waves and large ships, and the influence thishas on the added resistance. We are grateful for the support provided by the Danish Maritime Fundand by the Orients fund for this project, which has, by all measures, been a great success.

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References

[1] Amini Afshar, M. Towards Predicting the Added Resistance of Slow Ships in Waves. PhD thesis,DTU Mechanical Engineering, 2014.

[2] Amini-Afshar, M. and H. Bingham. Den danske maritime fond projekt: 2014-088, DTU Mek; PostDoc, udvikling af software, Final Report. Technical report, DTU Mechanical Engineering, 2016.

[3] Amini-Afshar, M. and H. B. Bingham. Accurate evaluation of the Kochin function for added resis-tance using a high-order finite difference-based seakeeping code. In 33rd International Workshopon Water Waves and Floating Bodies (IWWWFB 2018), 2018.

[4] Amini-Afshar, M. and H. B. Bingham. Pseudo-impulsive solutions of the forward-speed diffractionproblem using a high-order finite-difference method. Applied Ocean Research, 80:197–219, 2018.

[5] Amini Afshar, M., H. B. Bingham, and R. Read. OceanWave3D-Seakeeping: a linear seakeepingand added resistance tool based on high-order finite differences on overlapping structured grids.,2016. https://gitlab.gbar.dtu.dk/oceanwave3d/ow3d-seakeeping.git.

[6] Andersen, M. H. Implementation of generalized modes in a 3D finite difference based seakeepingmodel. Master’s thesis, DTU Mechanical Engineering, 2017.

[7] Andersen, M. H., M. Amini-Afshar, and H. B. Bingham. Implementation of generalized modes ina 3d finite difference based seakeeping model. In 32nd International Workshop on Water Wavesand Floating Bodies (IWWWFB 2017), 2017.

[8] Brown, D. L., W. D. Henshaw, and D. J. Quinlan. Overture: An object-oriented framework forsolving partial differential equations on overlapping grids. Object Oriented Methods for Interoper-able Scientific and Engineering Computing, SIAM, pages 245–255, 1999.

[9] Faltinsen, M., O, J. Minsaaa, N. Liapis, and S. Skjørdal, O. Prediction of resistance and propulsionof a ship in a seaway. In Proc. of 13th Symp. On Naval Hydrodynamics, 1980.

[10] Fenton, J. The numerical solution of steady water wave problems. Comp. Geosci., 14(3):357–368,1988.

[11] Wu, G. Hydrodynamic forces on a submerged circular cylinder undergoing large-amplitude mo-tion. Journal of Fluid Mechanics, 254:41–58, 1993.

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