Den Danske Maritime Fond projekt: , DTU Mek; Post Doc, udvikling af software The Final Report. Report. Mostafa Amini Afshar Harry Bingham

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1 Den Danske Maritime Fond projekt: 4-88, DTU Mek; Post Doc, udvikling af software The Final Report Report Mostafa Amini Afshar Harry Bingham

2 Contents Project summary Introduction Background Objectives and deliverables of the project Structure of the report The computational framework and grid generation strategy The computational framework Grid generation for ships Capabilities of the model and selected validation results The steady wave resistance problem The radiation problem The diffraction problem The motion response of the ship Second order mean drift forces and added resistance Future perspectives Dissemination activities Publications from the project Presentations Industry contact and outreach Conclusions References 7

3 Project summary The objective of this project was to extend the OceanWave3D-Seakeeping software developed during the PhD project of Mostafa Amini Afshar during the period -4, and to demonstrate the ability of this software to produce converged results for the added resistance of real ships. In addition, two conference and two journal publications were to be prepared and submitted, and the code was to be released as an open source tool to assist the general maritime community. These objectives have all been achieved and one additional conference publication was also published. The relevant publications are cited in the bibliography, and a dissemination plan has been prepared to alert the the community to the availability of the open source tool. Introduction This final report summarizes the results of the two-year post-doc project financially supported by the Danish Maritime Fund (4-88). The project started on November, 4 and finished on October 3, 6. It was carried out by Mostafa Amini Afshar with guidance and support from Prof. Harry Bingham, both employed at the Department of Mechanical Engineering, Technical University of Denmark.. Background As a consequence of the recent global economic crisis, together with long-term pressures to reduce fossil-fuel use, ships are generally sailing slower. Current trends in limiting greenhouse gas and other emissions by ships suggest that this trend will continue and require new ships to be designed and optimized for lower speeds. The energy required to move the ship through the water is always the sum of the calm-water resistance and the added resistance due to wind and waves. For ships, the wave component is generally at least an order of magnitude larger than the wind component. While the calm-water resistance scales with a high-power (3 or more) of the ship speed, the wave added resistance tends to be relatively insensitive to the ship speed. Thus, the slower the ship sails, the larger the added resistance becomes relative to the total, and the more important it becomes to be able to predict this quantity accurately in order to ensure adequate power in adverse environmental conditions. Although most ship designers and builders have access to a seakeeping and added resistance prediction tool based on two-dimensional, strip theory methods, the accuracy and robustness of these calculations is questionably, especially at low-speed. Tools based on three-dimensional methods are much less commonly used, generally quite expensive and usually relatively difficult to use. Accurately predicting the added resistance of a ship in waves is a very challenging task, on both the theoretical and the computational level. Although much progress has been made in recent years towards predicting seakeeping and added resistance using CFD tools, which theoretically capture all nonlinear and viscous effects, these calculations are still far too demanding to be used for practical ship design. Thus we will need to rely on potential flow calculations for the foreseeable future. Potential flow theory for added resistance is based on a perturbation expansion around the steady motion of the ship in calm water. In practice however, there are a number of possible choices for this steady basis flow, leading to different formulations of the unsteady solution. In addition, there are several different ways to express the added resistance force in terms of the basic hydrodynamic solution. On the computational side, most existing solutions are based on the Boundary Element Method which places relatively low limits on the possible level of grid refinement, with the result that a convincing demonstration of convergence 3

4 has so far not been given. Further clouding the issue, is the fact that experimental measurement of the added resistance is also a very difficult and error prone task, as it requires extracting a relatively small average quantity from a signal with large extremes. Thus it can be difficult to judge which numerical formulation and solution technique actually gives the best prediction of reality. Over the past years, collaborative work between DTU-Mechanical and DTU-Compute has led to the development of a computational strategy for predicting the interaction between nonlinear ocean waves and marine structures [, 4, ]. This work applies a combination of highorder numerical methods and effective iterative solution schemes to produced an efficient wavestructure interaction solver. During the PhD project of Mostafa Amini Afshar, this strategy was implemented using the open source C++ library Overture [3], and applied to compute the seakeeping and added resistance of a ship. The resulting software was named OceanWave3D- Seakeeping. This code was validated for simple exact geometries like a floating hemisphere and the Wigley hull. The current project follows up on this work, extending the solver to include the far-field method for computing added resistance and to demonstrate convergence of the calculations for real ship geometries.. Objectives and deliverables of the project The goals of this project were to demonstrate that the above described DTU software provides converged results for the added resistance of real ships. The performance of the various linearization strategies and the different force formulations were investigated and compared, and the software was documented and released as an open source tool for the Maritime Community. In addition, three conference contributions and two journal publications were prepared and submitted to relevant meetings and high-impact journals..3 Structure of the report Following this introduction and background discussion, the computational framework of the solver and grid generation issues are briefly described in. Then a summary of the physical problems which can be solved using the software is presented in 3, along with extensive validation results. In 4, some perspectives for further developments of the model are provided which will extend its accuracy and range of applicability. In 5, the dissemination activities are described which serve to publicize the results from the project. Conclusions are drawn in 6. The computational framework and grid generation strategy This section describes the computational framework in which the OceanWave3D-Seakeeping code is built, and outlines the strategy adopted for inserting ship hull geometries into the solver.. The computational framework The OceanWave3D-Seakeeping code has been implemented using the open-source Overture framework [3]. This package is a C++ library for solving Partial Differential Equations (PDEs) on overlapping, structured, high-order finite difference grids. Overture is freely available at: and is targeted for the inux operating system. This package must be installed on the user s computer before the OceanWave3D-Seakeeping software can be run. 4

5 The OceanWave3D-Seakeeping package can be downloaded by typing the following command from a inux terminal window: git clone This will copy the package files onto the user s computer. Installation and user guides can be found in the sub-directory./ow3d-seakeeping/doc, with detailed instructions on how to install and use the software. A basic familiarity with the inux operating system and the fundamentals of marine hydrodynamics theory are required, in order for the user to successfully utilize the software. The package is under continuous maintenance and development, and the user can benefit from this by following the progress of the package and regularly updating the codes on his/her computer.. Grid generation for ships A considerable part of this post-doc project was devoted to developing a convenient way to generate suitable grids on real ship geometries. The basic strategy is to define a set (usually three is enough) of curvilinear, body-fitted volume grids around the ship hull, and embed these in a rectangular or cylindrical background grid. This process is carried out using the freely available grid generator Ogen [5]. The complete grid generation process can be divided in to the following steps: - Define the surface of the ship hull using a convenient CAD surface modelling program and save the geometry in igs format. - Use the Ogen tool to develop a set of grids over the surface of the ship. Usually three grids are sufficient, one at the bow, one at the stern and one over the mid-body. - Use Ogen to grow the generated surface grids into the volume. - Use Ogen to combine the local grids and embed them into the background grid. Figures -3 show examples of the generation process for two tanker hulls, one with a bulbous bow and one without. 5

6 Figure : Surface grid (left) and mid-section volume grid (right) for a slow-speed tanker hull with no bulb. Figure : Surface grid (left) and volume grid (right) for the bow-section of the KVCC tanker hull. Figure 3: Volume grid for the stern section (left) and the surface of the complete local overset grid (right) for the KVCC hull. 6

7 y. y F n = U g = F n = U g = x x (a) The KVCC tanker at F n =.3 (b) The Wigley hull at F n =.45 Figure 4: Sample Kelvin wave patterns for two ships 3 Capabilities of the model and selected validation results In this section we describe the different physical problems that can be solved by the OceanWave3D- Seakeeping software, and present an overview of results which validate the accuracy and convergence of the model. More details can be found in the publications from the project listed in Section 5. OceanWave3D-Seakeeping adopts a linearized potential-flow approximation, which allows us to decompose the complete wave-ship interaction problem into a number of sub-problems, each of which represents one individual forcing component out of the total. Once these canonical subproblems have been solved, they can be superposed in different ways to quickly and efficiently predict the response of the ship to any particular environmental conditions which might be found at sea. This is generally refereed to as the seakeeping problem. The canonical linear seakeeping solutions can also be combined to estimate the mean drift forces and the added resistance of the ship. These canonical sub-problems are reviewed in the following sub-sections, which also show convergence and validation results developed during the project. 3. The steady wave resistance problem The wave resistance problem considers the ship sailing at constant speed in a calm ocean. The motion of the ship generates the classical Kelvin wave pattern around the ship, some examples of which can be seen in Figure 4. These waves carry energy away from the ship which implies a corresponding wave resistance force on the hull. Combining the wave resistance with the frictional resistance gives an estimate of the total engine power needed to reach a given speed. In OceanWave3D-Seakeeping, the wave resistance solution is obtained as the steady-state limit of a transient problem where the ship is accelerated from rest to a constant speed. A validation example of the computed force on a submerged sphere is shown in Figure 5a, where we can see a good comparison with the semi-analytic solution of []. For the case shown here, the Froude number F n = U/ ga =.7, where U is the speed of the body, g is the gravitational acceleration and a is the radius of submerged sphere. As a second validation example, we compute the steady wave resistance of the Wigley hull over a range of Froude 7

8 ..8 OceanWave3D-Seakeeping Analytic - Wu and Taylor (988) 4 OceanWave3D-Seakeeping Measurements Fx ρgπa Fx ρu S 3. F n =.7 Fx ρgπa 3 = t g/a (a) Resistance force for the submerged sphere F n (b) Resistance force for the Wigley hull Figure 5: Validation results for the wave resistance problem a33 ρπa OceanWave3D-Seakeeping (Wu and Taylor, 988) b33 ρπa 3 ωe.6.4 OceanWave3D-Seakeeping (Wu and Taylor, 988) νa (a) Added mass νa (b) Damping Figure 6: The heave hydrodynamic coefficients for the submerged sphere, F n =.4 numbers and compare the results with experimental measurements in Figure 5b. In these plots, the resistance force F x is non-dimensionalized by the fluid density ρ the gravitational acceleration constant g and the wetted surface area of the ship S. 3. The radiation problem In the radiation problem, we imagine the ship sailing with constant forward speed and oscillating sinusoidally in one degree of freedom, and at one frequency ω. There is one radiation problem for each degree of freedom, i.e surge, heave pitch, sway, roll and yaw. The numerical solution is again found as the steady-state limit of a transient problem, where in this case the motion follows a pseudo-impulsive Gaussian form and the resultant forces on the hull are computed. Taking the Fourier transform gives the frequency-domain hydrodynamic coefficients: the added mass and damping matrices. Some validation results for the radiation problem are shown in Figures 6 & 8 where comparisons are made to semi-analytic results from [] and the experimental measurements [6]. In the first case, a submerged sphere of radius a, travelling at 8

9 Figure 7: Part of the overlapping grid for the RIOS bulk carrier a53 ρ V pp OceanWave3D-Seakeeping (DB) OceanWave3D-Seakeeping (NK) (Iwashita, et. al, 6) ν c pp =.89 b55 ρ V ωe pp OceanWave3D-Seakeeping (DB) OceanWave3D-Seakeeping (NK) (Iwashita, et. al, 6) ν c pp = ν pp (a) Added mass (b) Damping ν pp Figure 8: The heave radiation coefficients for the RIOS bulk carrier, F n =.8 Froude number F n = U/ ga =.4, oscillates in heave motion. Figure 6 shows the corresponding non-dimensional added mass a 33 and damping coefficients b 33. In the second example, the geometry of a bulk carrier is built (see Figure 7) and the hydrodynamic coefficients (a 53, b 55 ) are computed and presented in Figure 8. The ship has a forward speed defined by the Froude number F n = U/ g pp =.8, where pp is the length between perpendiculars. Note in the figures ν = ω e/g with ω e the encounter frequency. The submerged volume is denoted by V. The results are presented based on both the Neumaan-Kelvin (NK) and the double-body (DB) linearization. 3.3 The diffraction problem In the diffraction problem, we imagine that the ship is moving with the steady speed U, but fixed to its mean position, while encountering long-crested waves at one frequency ω, which are incident from one heading angle β. The angle β is measured from the direction of the ship s motion, so 9 β 8 represent waves from ahead of the beam (head waves) and β 9 represent waves from abaft the beam (following-seas). Once the excitation forces are computed for all frequencies and heading angles, they can be combined with a general short-crested wave spectrum to get the resultant forcing in a real sea-state. In OceanWave3D-Seakeeping, the diffraction problem is again solved as the steady-state limit of a pseudo-impulsive transient problem. In this case, the incident wave elevation and velocities take a Gaussian form and the resultant excitation forces are computed from the induced fluid pressures on the surface of the body. The frequency-domain excitation coefficients are then obtained via a Fourier transform. As for the radiation problem, the solver has been validated using the experimental data from [6] and the analytic solutions of []. For the 9

10 .3. OceanWave3D-Seakeeping (Wu and Taylor, 988a) β =.75π.5 OceanWave3D-Seakeeping (Wu and Taylor, 988a) β =.75π Re{X} ρπaa 3 ν. Im{X} ρπaa 3 ν ν a ν a (a) The real part (b) The imaginary part Figure 9: The wave excitation force on a submerged sphere, F n =.4 and β = 35.8 OceanWave3D-Seakeeping (Iwashita, et. al, 6)..5 OceanWave3D-Seakeeping (Iwashita, et. al, 6) X3 ρgabpp.6.4 X5 ρgab pp λ/ pp λ/ pp (a) The heave force (b) The pitch moment Figure : The wave excitation on the RIOS bulker, F n =.8 and β = 8 Figure : UYSSES concept tanker hull

11 ξ3 A 8 6 OW3D-Seakeeping (.57 - coarse) OW3D-Seakeeping (.7) OW3D-Seakeeping (.85) OW3D-Seakeeping (.95) OW3D-Seakeeping (. - fine) (Journee, 99) ξ5 πa 8 6 OW3D-Seakeeping (.57 - coarse) OW3D-Seakeeping (.7) OW3D-Seakeeping (.85) OW3D-Seakeeping (.95) OW3D-Seakeeping (. - fine) (Journee, 99) λ (a) The heave RAO λ (b) The pitch RAO Figure : The Response amplitude operators for the Wigley hull, F n =.3 and β = 8 ξ3 A OW3D-Seakeeping (.55 - coarse) OW3D-Seakeeping (.63) OW3D-Seakeeping (.74) OW3D-Seakeeping (.85) OW3D-Seakeeping (.9 - fine) Measurements (SSPA) ξ3 λ πa OW3D-Seakeeping (.55 - coarse) OW3D-Seakeeping (.63) OW3D-Seakeeping (.74) OW3D-Seakeeping (.85) OW3D-Seakeeping (.9 - fine) Measurements (SSPA) λ (a) The heave RAO λ (b) The pitch RAO Figure 3: The Response amplitude operators for the tanker, F n =.7 and β = 8 submerged sphere at F n =.4, the excitation forces in the sway direction X are given in Figure 9. In this case the incidence angle is β = 35 and ν = ω /g, where ω is the wave frequency. For the RIOS bulk carrier mentioned in the previous section, the wave excitation forces in heave and pitch (X 3 and X 5 ) have been computed using the seakeeping solver, see Figure. For this example, the ship is moving with Froude number F n =.8 and the incidence angle of the waves is β = 8. Here λ denotes the wave length, B is the ship breadth and A is the incident wave amplitude. 3.4 The motion response of the ship Once the radiation and diffraction force coefficients are known, they can be combined with the body inertia and hydrostatic restoring force coefficients to solve Newton s second law for the motion response of the ship. This solution has been validated both for bodies at zero forward speed, and for ships moving with steady forward speed. As an example, the motions of the Wigley hull moving with a Froude number F n = U/ g =.3 are calculated and compared with the measurements by [7]. Results for the magnitudes of the heave and pitch Response

12 .8.6 (a) OceanWave3D (near-field) OceanWave3D (far-field) WAMIT (b) OceanWave3D (near-field) OceanWave3D (far-field) Rw ρga D.4 Rw ρga B ka λ Figure 4: The wave drift force F n =. eft: A floating sphere. Right: The KVCC tanker Rw ρga B OW3D-Seakeeping (.57 - coarse) OW3D-Seakeeping (.7) OW3D-Seakeeping (.85) OW3D-Seakeeping (.95) OW3D-Seakeeping (. - fine) (Journee, 99) Rw ρga B OceanWave3D (near-field) OceanWave3D (far-field) Experiment (Journee 99) /λ (a) Convergence of the near-field computations (b) Near-field and far-field calculations λ Figure 5: Added resistance for the Wigley hull, F n =.3, β = 8 Amplitude Operators (RAOs) are shown in Figure. Here ξ 3 and ξ 5 are the complex response phasors in heave and pitch respectively, A is the incident wave amplitude and λ is the incident wave length. In order to show the convergence of the calculations, the results are plotted for 5 different grid resolutions. The same computations have been performed for the UYSSES concept tanker hull shown in Figure, and the heave and pitch RAOs are compared in Figure 3 to experimental measurements performed at SSPA in Sweden as part of the UYSSES project [9]. The convergence of the calculations with grid refinement is also shown here. 3.5 Second order mean drift forces and added resistance The mean wave drift force is a second-order quantity (proportional to A ), which can be calculated directly from the first-order solution. In the context of a ship with the forward speed, the longitudinal drift force is usually called the added resistance. The added resistance represents the additional shaft power needed to maintain a given speed in a particular sea-state. Having solved the first order radiation, diffraction and motion response problems discussed in the previous sections, the mean drift forces can be computed either based on near-field

13 4 4 ξ3 A 3 OW3D-Seakeeping (.6 - fine) OW3D-Seakeeping (.5) OW3D-Seakeeping (.4) OW3D-Seakeeping (.) OW3D-Seakeeping (.8 - coarse) ξ5 λ πa 3 OW3D-Seakeeping (.6 - fine) OW3D-Seakeeping (.5) OW3D-Seakeeping (.4) OW3D-Seakeeping (.) OW3D-Seakeeping (.8 - coarse) λ/ pp λ/ pp Figure 6: Heave and pitch RAOs for the KVCC hull, F n =.4, β = 8 pprw ρga B 5 OW3D-Seakeeping (.6 - fine) OW3D-Seakeeping (.5) OW3D-Seakeeping (.4) OW3D-Seakeeping (.) OW3D-Seakeeping (.8 - coarse) (Measurement) pprw ρga B 5 OW3D-Seakeeping (near-field) OW3D-Seakeeping ( far-field ) (Measurement) λ/ pp (a) Convergence of the near-field calculations λ/ pp (b) The near-field and the far-field calculations Figure 7: Added resistance for the KVCC hull, F n =.4, β = 8 pressure integration, or far-field momentum conservation. The main goals of this project where to complete the implementation of these two methods and demonstrate the convergence of the calculations. These goals have been achieved, and some sample calculations are shown in Figures 4 & 5. The first case considers the wave drift force on a floating sphere and the KVCC tanker model, both at zero forward speed. The left hand plot in Figure 4 compares the results to benchmark calculations using the commercial software WAMIT [8], while the right hand plot compares the near-field and far-field calculations for the KVCC tanker in head seas (β = 8 ). In these plots, R w represents the added resistance force in Newtons, A is the incident wave amplitude, D is the diameter of the sphere and, B are the length and beam of the ship respectively. In Figure 5, we show calculations for the the added resistance of the Wigley hull at F n =.3 and β = 8. In the left hand plot, the convergence of the near-field method is demonstrated and the results are compared to experimental measurements by [7]. The right-hand plot compares the converged results for the near-field and far-field calculations with each other and with the experiments. 3

14 Aconvergence study has also been carried out for the KVCC hull. The results for the heave and pitch RAOs are shown in Figure 6, while the added resistance results are plotted in Figure 7. 4 Future perspectives The OceanWave3D-Seakeeping software developed during this project, is a robust tool for predicting the seakeeping and added resistance of ships. It can, of course, also be applied to other types of floating offshore structures such as oil and gas platforms, floating wind-turbines, aquaculture cages or wave energy devices. There are however, a number of interesting lines of further development which we intend to pursue to improve the code. In this Section, we briefly outline our highest priority future work on the tool.. Flare effects: In the linearized approximation of the interaction between the ship and the ocean waves, the ship is assumed to intersect the waterline vertically. This is generally not a good approximation along the fore- and aft-bodies of a typical ship, so it can be important to include these effects in the analysis. This is particularly relevant in the shortwave regime where the incident wave length is less than about half the ship length, and for the largest ships in operation today ( 35m), this short-wave regime represents the majority of the ship s sailing time. Several approaches are available to partially or fully include these effects including: Using the local flare angle in the second-order terms of the waterline integral term in the waterline integral terms; Solving the complete second-order diffraction problem (in the time-domain) for each monochromatic incident wave case; Solving the complete nonlinear problem in either monochromatic waves or a prescribed sea-state.. Simplified ship gridding approaches: One of the weak points of the current solver is the sensitivity of the overlapping grid approach to the grid quality near the ship hull free-surface intersection. The solution to this problem is thought to be found by aligning the edges of each grid with the physical boundaries between the hull, free-surface and the symmetry plane. This approach is currently under development and testing. 3. Fully nonlinear wave-ship interaction: The main advantage of the adopted numerical approach to solving the wave-ship interaction problem is the flexibility of the method to move smoothly from linearized boundary conditions to second-order and fully nonlinear conditions, with relatively low additional computational effort per time-step. Robust numerical strategies have been developed in parallel projects over the past ten years, which can be implemented in the current solver to extend the solution to fully nonlinear wave-ship interaction. This work will be carried out in steps, first for the resistance and short-wave diffraction problems, then ultimately for the complete free-motions problem. Thanks to new funding provided jointly by the Danish Maritime Fund and the Orients Fund, much of this work will be carried out over the coming two years, resulting in great improvements to the OceanWave3D-Seakeeping tool and important contributions to the field of Marine Hydrodynamics. 5 Dissemination activities This section describes the dissemination activities which have accompanied the work carried out during this project, and describes ongoing activities aimed at encouraging people in the maritime industry to make use of the OW3D-Seakeeping software. 4

15 5. Publications from the project The following manuscripts have been published, or submitted for publication, during the course of the project:. Amini Afshar, M. and H. B. Bingham. Solving the linearized forward-speed radiation problem using a high-order finite difference method on overlapping grids. Applied Ocean Research, (in press). Amini Afshar, M. and H. B. Bingham. A stable, high-order finite difference method for estimating the wave resistance of ships. Journal of Computational Physics, (in press) 3. Amini Afshar, M., H. B. Bingham, and R. Read. Potential flow solver predicting the added resistance of ships in ocean waves. In DANSIS Research Seminar, Copenhagen, 5 4. Amini Afshar, M., H. B. Bingham, and R. Read. A high-order finite-difference linear seakeeping solver tool for calculation of added resistance in waves. In 3th International Workshop on Water Waves and Floating Bodies, 5 5. Amini Afshar, M., H. Bingham, W. D. Henshaw, and R. Read. Convergence of near-field added resistance calculations using a high order finite-difference method. In Proceedings of the 3th International Symposium on Practical Design of Ships and Other Floating Structures (PRADS6), 6 6. Amini Afshar, M., H. B. Bingham, and R. Read. OceanWave3D-Seakeeping: a linear seakeeping and added resistance tool based on high-order finite differences on overlapping structured grids., 6. git 5. Presentations In addition to the publications listed above, the results of the project have been disseminated through the following presentations: Presentation at the DNV-G workshop at Chalmers University of Technology, Gothenburg, Sweden (January 5) Presentation at the 3th IWWWFB in Bristol, UK (April 5) Presentation at the 3th PRADS conference in Copenhagen (September 6) Presentation at the DANSIS yearly seminar at DTU (May 5) ectures to the students in DTU course 4 Wave load on ships and offshore structures (Fall semesters 5 and 6) 5.3 Industry contact and outreach In order to help inform people in the maritime industry of the availability of the OW3D- Seakeeping software package, contact has been made to the following three organizations who have all agreed to establish links to the software, along with short descriptions of the tool, on their respective web sites. Danske Rederier: Per Winther Christensen. 5

16 Maritime Development Center: Kirsten Weede. Danske Maritime: Valdemar Ehlers. 6 Conclusions This final report has summarized the highlights of Mostafa Amini Afshar s post-doc project over the period //4-3//6, financed by the Danish Maritime Fund, project number (4-88). Motivated by the need to improve the safety and efficiency of ships, we have produced an open source seakeeping and added resistance tool and made it freely available to the maritime community. The work has been disseminated in two submitted journal publications, along with three conference contributions and several oral presentations. inks to the tool, along with brief descriptions, have been established on the web pages of Danske Rederier, The Maritime Development Center, and Danske Maritime; in order to promote and advertise the software among the relevant players in the Danish maritime industry. We are grateful for the support provided by the Danish Maritime Fund for this project, which has, by all measures, been a great success. 6

17 References [] Amini Afshar, M. Towards Predicting the Added Resistance of Slow Ships in Waves. PhD thesis, DTU Mechanical Engineering, 4. [] Bingham, H. B. and H. Zhang. On the accuracy of finite-difference solutions for nonlinear water waves. Journal of Engineering Mathematics, 58(-4): 8, 7. [3] Brown, D.., W. D. Henshaw, and D. J. Quinlan. Overture: An object-oriented framework for solving partial differential equations on overlapping grids. Object Oriented Methods for Interoperable Scientific and Engineering Computing, SIAM, pages 45 55, 999. [4] Engsig-Karup, A. P., H. B. Bingham, and O. indberg. An efficient flexible-order model for 3d nonlinear water waves. Journal of Computational Physics, 8(6): 8, 9. [5] Henshaw, W. D. Ogen: An overlapping grid generator for overture. AN Unclassified Report, pages , 998. [6] Iwashita, H., M. Kashiwagi, Y. Ito, and Y. Seki. Calculations of ship seakeeping in lowspeed/low-frequency range by frequency-domain rankine panel methods. Journal of the Japan Society of Naval Architects, 4:9 46, 6. [7] Journée, J. Experiments and calculations on 4 Wigley hull forms in head waves. Delft University of Technology, Report, (99), 99. [8] ee, C.-H. WAMIT theory manual. Massachusetts Institute of Technology, Department of Ocean Engineering, 995. [9] Ultraslowships. The 7th Framework Programme, UYSSES - Ultra Slow Ships Project, 3. [] Wu, G. and R. E. Taylor. Radiation and diffraction of water waves by a submerged sphere at forward speed. Proceedings of the Royal Society of ondon. A. Mathematical and Physical Sciences, 47(853):433 46,

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