Robust Numerical Methods for Nonlinear Wave-Structure ... · interaction between nonlinear ocean waves and marine structures. oT achieve this goal, a exible order nite di erence potential

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Robust Numerical Methods for Nonlinear Wave-Structure Interaction in a Moving Frame of Reference

Stavros KontosDCAMM Special Report No. S217August 2016

Robust Numerical Methods forNonlinear Wave-Structure

Interaction in a Moving Frame ofReference

Stavros Kontos

Kongens Lyngby 2016

Technical University of Denmark

Department of Mechanical Engineering

Section of Fluid Mechanics, Coastal and Maritime Engineering

Nils Koppels Allé , building 404,

2800 Kongens Lyngby, Denmark

Phone +45 4525 2525

www.mek.dtu.dk

Summary (English)

This project is focused on improving the state of the art for predicting theinteraction between nonlinear ocean waves and marine structures. To achievethis goal, a �exible order �nite di�erence potential �ow solver has been extendedto calculate for fully nonlinear wave-structure interaction problems at forwardspeed.

The model utilises the e�ciency of �nite di�erence methods on structuredgrids and exploits the �exibility of a novel Immersed Boundary Method (IBM)based on Weighted Least Squares (WLS) for the approximation of the no-�uxboundary condition on the body surface. As a result, the grid generation is verysimple and the need for regridding when considering moving body problems isavoided. The temporal oscillations related to the IBM method and movingboundaries are minimized by su�cient spatial resolution and an increased time-step size.

The time-dependant physical domain is mapped to a time-invariant com-putational domain with a sigma transformation. For a smooth and continuoustransformation a C2 continuous free surface is required over the entire domain.Thus, an arti�cial free surface that respects this property is created in the inte-rior of the body using a seventh order polynomial.

The forward speed problem is formulated in a moving coordinate systemattached to the mean position of the body. Robust approximations for all com-binations of forward speed and wave velocity are obtained by expressing the freesurface boundary conditions in Hamilton-Jacobi form and using a Weighted Es-sentially Non-Oscillatory (WENO) scheme for the convective derivatives. Thelinear WENO weights are derived with a new procedure that is suitable for nu-merical implementation and avoids the limitations of existing tabulated WENOcoe�cients. Furthermore, a simpli�ed smoothness indicator that performs aswell as the tabulated versions is proposed. Explicit high-order Runge-Kuttatime integration and a Lax-Friedrichs-type numerical �ux complete the scheme.The solver was tested on the two-dimensional zero speed wave radiation prob-lem and the steady forward speed problem with satisfactory results and thus,the proof of concept for extending the methodology to three dimensions is es-tablished.

Summary (Danish)

Fokus for dette projekt er at forbedre "state of the art"indenfor beregning afinteraktion mellem ikke lineære havbølger og marine strukturer. For at opnådette mål, er en �eksibel ordens �nite di�erence potential strømnings løser blevetudvidet til beregning af fuldt ikke lineær bølge-struktur interaktion problemermed et legeme i fart.

Modellen udnytter e�ektiviteten af �nite di�erence metoden på et strukture-ret beregningsnet og �eksibiliteten af en ny Immersed Boundary Method (IBM),baseret på Weighted Least Squares (WLS), til approksimation af randbetingel-sen for ingen gennemstrømning på legemets over�ade. Som et resultat heraf,er beregningsnet genereringen meget simpel og der er ingen behov for gengene-rering af nettet omkring det bevægelige legeme. De tidsafhængige oscillationerrelateret til IBM metoden og bevægelige rande, er minimeret ved tilstrækkeligrummelig opløsning og anvendelse af et større tidsskrid.

Det tidsafhængige fysiske domæne er afbilledet ind i et tidsuafhængigt be-regnings domæne ved hjælp af en sigma transformation. For at opnå en glat ogkontinuert transformation, er en C2 kontinuert fri over�ade funktion nødvendigpå hele domænet. Derfor er en kunstig fri over�ade, der overholder denne egen-skab, lavet i det indre af legemet ved hjælp af et syvende ordens polynomium.

Problemet med et legeme i fart gennem stille vand er formuleret i et bevæ-geligt koordinatsystem, centreret omkring middelpositionen af legemet. Robustapproksimation af alle kombinationer af legemets fart og bølge hastigheder eropnået ved at udtrykke friover�ade randbetingelserne i Hamilton-Jacobi formog ved anvendelse af Weighted Essentially Non-Oscillatory (WENO) metodentil beregning de konvektive a�ede. De lineære WENO vægte er udledt med enny procedure, der er velegnet til numerisk implementering og er uden begræns-ningerne i de eksisterende tabulerede WENO koe�cienter. Endvidere, er ensimpli�ceret glatheds indikator, med lige så gode egenskaber som den tabulere-de, præsenteret. Beregningsmetoden er fuldendt med en eksplicit højere-ordensRunge-Kutta tidsintegration og en Lax-Friedrichs type numerisk �ux beregning.Løseren er testet på todimensionale problemer, et bølge radiations problem foret legeme uden fart og et problem med et legeme i fart gennem et stille hav.Resultaterne er tilfredsstillende og fungere som et proof of concept for udvidelseaf metoden til tre dimensioner.

Preface

This thesis is submitted in partial ful�lment of the requirements for the degreeof Doctor of Philosophy at the department of Mechanical Engineering, TechnicalUniversity of Denmark.

The project began on the September 1, 2013 and had a duration of threeyears. All work has been carried out in the Section for Fluid Mechanics, Coastaland Maritime Engineering under the supervision of Harry B. Bingham, AllanP. Engsig-Karup and Ole Lindberg.

The �nancial support for this work was provided by the Technical Universityof Denmark and by the EU-FP7 framework project �Energy E�cient Safe SHipOPERAtion� (SHOPERA).

Stavros KontosLyngby, 31-August-2016

Acknowledgements

I would like to express my gratitude to Harry B. Bingham for his guidance andpatience throughout my PhD studies. He always made time for answering myquestions and kept me motivated through the di�cult stages of the project. Iwould like to thank Ole Lindberg for sharing his experience, insight and enthu-siasm and Allan P. Engsig-Karup for his advice and his honest and invaluablefeedback on my work.

I would also like to thank my colleagues Ju-hyuck, Mostafa, Najmeh, Pelle,Robert, Sopheak, Torben and Yasaman for creating a friendly and stimulatingworking environment.

A special thanks to my family for their love and support and to Lisa forbeing a source of inspiration and for her constant encouragement.

Contents

Summary (English) i

Summary (Danish) ii

Preface iii

Acknowledgements iv

1 Introduction 11.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . 11.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Scienti�c contribution . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 5

2 The potential �ow ship motions problem 7

3 Numerical Solution 93.1 Sigma coordinate transformation and Finite Di�erence discreti-

sations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Weighted Least Squares, Immersed Boundary Method body bound-

ary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.1 Weighted Least Squares Approximations . . . . . . . . . . 12

3.4 Hamilton-Jacobi WENO formulation of the Free Surface Bound-ary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.1 A simpli�ed derivation procedure for the linear WENO

weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.2 A simpli�ed smoothness indicator for the nonlinear WENO

weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.3 The performance of the new smoothness indicator . . . . 203.4.4 AWENO scheme for stable and accurate evolution of non-

linear waves . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Methods for computing the forces on the body . . . . . . . . . . 36

CONTENTS vi

3.6 Body - free surface intersection tracking and construction of theinterior free surface . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6.1 Internal free surface construction . . . . . . . . . . . . . . 423.6.2 Intersection treatment and adding new free surface points 43

4 Numerical force oscillations caused by moving boundaries 484.1 Hybrid points method . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Ensuring mass conservation . . . . . . . . . . . . . . . . . . . . . 504.3 Surging dipole in an in�nite �uid . . . . . . . . . . . . . . . . . . 51

5 Forced motion �uid-structure interaction cases 585.1 Linearised case of a heaving cylinder . . . . . . . . . . . . . . . . 585.2 Fully nonlinear heaving cylinders with various cross-sections . . . 61

5.2.1 Small amplitude motion of circular cylinder . . . . . . . . 615.2.2 Large amplitude motion of a circular cylinder . . . . . . . 625.2.3 Large amplitude motion of an elliptic cylinder . . . . . . . 625.2.4 Large amplitude motion of a wedge shaped cylinder . . . 64

5.3 Fully nonlinear submerged heaving cylinder . . . . . . . . . . . . 64

6 Forward speed �uid-structure interaction cases 706.1 Steady streaming �ow around a translating submerged circular

cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7 Conclusions 72

A Derivation of the internal free surface polynomial 74A.1 Closed form coe�cients . . . . . . . . . . . . . . . . . . . . . . . 74A.2 Numerically optimized coe�cients . . . . . . . . . . . . . . . . . 75

B WENO coe�cients 76B.1 Numerical implementation of the linear weights . . . . . . . . . . 76B.2 Numerical implementation of the nonlinear weights . . . . . . . . 77

Bibliography 79

Chapter 1

Introduction

The goal of this PhD project is to develop robust, high-order accurate numer-ical methods for predicting the interaction between nonlinear ocean waves andmarine structures. This will serve as a �rst step for developing a highly e�-cient solver for the seakeeping and added resistance of ships in short waves. Apotential �ow solver combined with an immersed boundary technique for thestructure representation is used to achieve this goal. A high-order WeightedEssentially Non-Oscillatory (WENO) scheme ensures the stable approximationof the fully nonlinear free surface boundary conditions. This project is partially�nanced by the EU-FP7 framework project �Energy E�cient Safe SHip OPERA-tion� (SHOPERA) which has set aside substantial funding for the improvementof existing computational tools for the accurate and e�cient prediction of theadded resistance and the seakeeping and manoeuvring performance of ships inadverse sea conditions.

1.1 Background and motivation

Seakeeping considers the response of ships and marine structures under the in-�uence of waves. The hydrodynamic analysis of ships sailing in waves dates backto the pioneering work of Froude and Kelvin in the late 1800s, Froude (1861);Thompson (Lord Kelvin), with notable contributions from Michell, Havelock,Krylov and Haskind in the early 1900s, Michell (1898); Roberts (1971); Haskind(1953). More recent and comprehensive presentations can be found for examplein Ogilvie (1977); Newman (1978). The tools that naval architects employ forpredicting seakeeping consist of (Bertram and Couser (2014))

1. Model tests that are expensive and time consuming.

2. Full scale measurements on-board sailing ships that are however expen-sive and problematic due to external noise and a generally very impreciseknowledge of the environmental conditions.

1.1 Background and motivation 2

3. Linear frequency domain methods that are widely used due to their e�-ciency and acceptable accuracy when nonlinear e�ects are less important.

4. Weakly-nonlinear time-domain methods which are e�cient but only in-clude selected �easy to compute� nonlinear e�ects such as: Froude-Krilov,hydrostatic and viscous damping forces.

5. Fully nonlinear time domain methods that are increasingly used but arestill computationally expensive.

The �ow dynamics of the seakeeping problem are essentially governed by the in-compressible Navier-Stokes equations. Viscous e�ects give an important contri-bution to the resistance of the ship, but due to the very large Reynolds numbersinvolved (on the order of 109), viscosity can generally be neglected in predictingthe wave-ship interaction and dynamics. Thus, a potential �ow assumption is inmost cases a very good approximation. Special care is needed for accurate pre-diction of the roll motions that are in�uenced from nonlinear viscous dampingand for the simulation of the complex �ow that is detaching from the transom ofboats with large transom sterns. Despite the shortcomings of the di�erent meth-ods, most give reasonable results for the prediction of ship motions. However,the prediction of forces and particularly added resistance is more complicatedand in the case of short waves it is still very challenging.

The �resistance� of a ship indicates the propulsive power required to movethe ship at a given speed in calm water (Newman (1977)). The resistance incalm water consists of three components. One component due to friction actingin the boundary layer around the hull, a second component due to �ow separa-tion and large-scale vorticity shed into the wake, and a third component due tothe waves generated behind and along the vessel that are known as the Kelvinwave pattern, Newman (1977). Numerical models combined with towing tanktesting are used for predicting the calm water resistance with satisfying accu-racy. �Added resistance� is the term used to describe the extra propulsive powerrequired to keep the ship at this speed while travelling through a given sea-state.It is comprised of two components, one due to the wind and another due to theocean waves and subsequent unsteady motions of the ship. Added resistance isa manifestation of second order wave drift forces which are proportional to thesquare of the wave amplitude and the induced ship motions, Faltinsen (1990).Thus it is one component of a force which more generally acts in both the di-rection of travel (resistance) and perpendicular to the direction of travel (drift),as well as inducing a rotational moment (yaw), Faltinsen (1990). The addedresistance can become negative (propulsive) when the ship sails with the windand waves. A factor of fundamental importance when designing a ship engine,is the requirement that the ship must be able to maintain a certain minimumspeed when travelling against adverse sea conditions, for example to escape fromrunning aground during a storm, Bingham and Amini Afshar (2013). A conven-tional ship with plenty of power reserves and a service speed of approximately15 knots can safely maintain a minimum speed of 5 knots under very heavy seaconditions. For ultra-slow ships with service speeds of 5 to 6 knots it is criticalto determine the extra power needed to sail through a storm. Thus, the relativeimportance of an accurate prediction for added resistance becomes much largerwhen considering slow ships, Bingham and Amini Afshar (2013). Accuratelypredicting the added resistance of a ship in waves is a very challenging task, on

1.2 Literature review 3

both the experimental and the theoretical/computational level. Although muchprogress has been made in recent years towards predicting seakeeping and addedresistance using CFD tools, which theoretically capture all nonlinear and viscouse�ects, these calculations are still far too demanding to be used for practical shipdesign. Thus potential �ow-based methods must be relied on for practical cal-culations. Potential �ow theory for added resistance is based on a perturbationexpansion around the steady motion of the ship in calm water, and can be com-puted from the �rst-order solution alone. In the short-wave regime (wave length< 0.5 ship length) however, the induced motions of the ship are negligible, butnonlinear e�ects arising from steep and possibly breaking waves, as well as �areof the hull near the bow and stern, can be important. The available numericalmethods that can capture these nonlinear e�ects are still relatively immature.In addition, experimental measurement of the added resistance in short-wavesis particularly di�cult and error prone. Added resistance is a time-average ofthe oscillating longitudinal force and extracting this relatively small quantityfrom a signal with large extremes is very challenging. Furthermore, generationof short waves at laboratory scale typically introduces large uncertainties in thewaves themselves. Waves with high steepness are generally unstable and shortwaves with low steepness are subject to more spatial variation than long wavesdue to the variation in their transversal amplitude across the basin (SeakeepingCommittee (2012)). Thus there is a real lack of reliable experimental data bywhich to judge the accuracy of a particular numerical formulation or technique.

With fuel prices, emissions standards and environmental regulations ex-pected to increase dramatically over the next twenty to �fty years, the shippingindustry is strongly motivated to develop substantially more energy e�cientships than those in use today. The obvious recipe for e�cient shipping is acombination of ultra-slow speed and complimentary propulsion technologies.Additionally, most of the modern ships are very large, often exceeding 320 min length. Consequently, the majority of the waves they encounter when sailingin normal sea states are considered short. As a result, developing e�cient andaccurate tools for predicting the seakeeping and added resistance of slow shipsin short waves is very important.

1.2 Literature review

In recent years the development of time domain methods for solving the fullynonlinear potential �ow problem is gaining momentum. Compared to moretraditional frequency domain methods, time domain methods are more intuitivein computing the nonlinear motions and loads and in incorporating externalforces such as propulsion and control forces. However, the computational costof time domain methods is higher, and especially regridding the time-dependantfree surface is a signi�cant task. A short survey of recent fully nonlinear, time-domain methods is presented here. An extensive summary of the state of theart in methods used for seakeeping and added resistance calculations can befound in Seakeeping Committee (2012).

The time domain Green Function Methods (GFM) require only the panel-ization of the wetted hull geometry, however in body exact approaches a re-panelization and re-calculation of the system matrix is required at each time-step. Rankine Panel Methods (RPM) use a simpler distribution of singularities

1.3 Scienti�c contribution 4

in comparison with the GFM. However, they require the panelization of theentire free-surface and an additional numerical method to satisfy the radiationboundary condition. Zhang et al. (2011) developed a body exact RPM withimproved computational e�ciency by introducing a pre-corrected Fast FourierTransform method in the scheme. Berkvens and Zandbergen (1996) used aboundary integral method to compute the hydrodynamic forces on a heavingsphere. Tanizawa (1996) used a similar method to calculate large-amplitudewave induced body motions in 2D. You and Faltinsen (2012) simulated the in-teraction of moored �oating bodies with waves using a fully nonlinear RPMcombined with a numerical wave tank and damping zone. Peltzer et al. (2008)applied a 3D NURBS-based Rankine boundary element method to the multi-body seakeeping design optimisation problem of an o�shore platform. Ferrant(2001) developed a RPM and computed the wave runup on a bottom mountedcylinder subjected to nonlinear waves and current. Hybrid methods combinea sophisticated inner domain solution with a simpler outer domain solution toimprove the e�ciency of the calculations. Kjellberg et al. (2011) developed ahybrid method by combining a 2D wave tank with a 3D fully nonlinear bodyexact boundary element method. The 3D �ow is resolved in the vicinity of thebody using constant strength source panels. Monroy et al. (2009) predictedship motions in irregular head waves by using a Higher Order Spectral (HOS)potential �ow model for the incident waves and a Reynolds Averaged Navier-Stokes Equation (RANSE) solver for obtaining the di�racted wave �eld. Grecoet al. (2013) coupled a linear potential �ow solver with a Navier-Stokes solverto investigate the seakeeping problem of a vessel interacting with regular deepwater head sea waves.

1.3 Scienti�c contribution

The added resistance predictions of Boundary Element Methods show a reason-able agreement with experiments, however convergence has been challenging todemonstrate due to the quadratic scaling of the solution e�ort with increasingresolution. In the Technical University of Denmark a �exible order �nite dif-ference potential �ow solver for nonlinear waves in a �xed reference frame hasbeen developed by Bingham and Zhang (2007); Engsig-Karup et al. (2009). Thesolver has been extended to calculate for linear wave-structure interaction in amoving frame of reference by Lindberg et al. (2014); Amini Afshar (2014) andis implemented on massively parallel Graphical Processing Unit (GPU) archi-tectures by Engsig-Karup et al. (2011); Glimberg (2013). The solution e�ort ofthe method scales linearly with an increasing number of grid points and exhibitsexcellent parallel scalability. It is therefore attractive to extend the solver forcalculating fully nonlinear wave-structure interaction problems in a �xed and ina moving frame of reference.

Lindberg et al. (2014) introduced the complex body geometry in the struc-tured �nite di�erence grid using a novel Immersed Boundary Method (IBM)based on Weighted Least Squares (WLS) approximations to solve linear wave-structure interaction problems. In this thesis, the method is applied over theinstantaneous wetted body surface and therefore the capabilities of the solverare extended to fully nonlinear problems. The IBM method makes the gridgeneration very simple and e�cient when considering �ows with moving bound-

1.4 Structure of the thesis 5

aries, since the expensive procedure of regridding and projecting the solution tothe new grid is avoided. However, the method is prone to temporal oscillationscaused by truncation error di�erences between di�erent time steps. In the con-text of this thesis, this problem was investigated by considering a surging dipolein an in�nite �uid and was shown that its in�uence is minimized with su�cientspatial resolution and an increased time-step size.

Since the free surface is a moving boundary with an a priori unknown po-sition, it is convenient to map the time-dependant physical domain to a time-invariant computational domain with a change of variable in the vertical coor-dinate. The transformation requires a C2 continuous free surface over the entiredomain. In the case of a wave-structure interaction problem the free surface isinterrupted by the existence of the body. Therefore, we create an arti�cial freesurface in the interior of the body that is C2 continuous over its length and atthe intersections with the external free surface. The internal solution is givenas a seventh order polynomial. This strategy allowed to solve the fully nonlin-ear zero speed radiation problem for several �oating or submerged simple 2Dgeometries with satisfactory results. Good agreement with reference solutionsis found in the prediction of �rst order forces. The comparison of our results forsecond and third order forces with the experimental work of Yamashita (1977)is promising.

The forward speed problem is formulated in a moving coordinate systemattached to the mean position of the body. As a result convective terms areintroduced in the free surface boundary conditions. To obtain robust approx-imations for all combinations of forward speed and wave velocity we recastthe free surface boundary conditions in Hamilton-Jacobi form and propose aWeighted Essentially Non-Oscillatory (WENO) scheme that automatically han-dles the upwinding of the convective terms. A new procedure for deriving thelinear WENO weights that is suitable for numerical implementation is intro-duced based on Taylor series expansion. A simpli�ed smoothness indicator thatis computed numerically through already available stencil coe�cients is derivedand is shown to perform identically to the tabulated versions in the literature.The scheme is combined with a dissipative Lax-Friedrichs-type �ux to solve fornonlinear wave propagation in a moving frame of reference and is found to berobust and accurate for all ratios of frame of reference to wave propagation speedconsidered. Unfortunately, there is a lack of experimental or nonlinear numericalresults in the literature for bodies travelling with steady forward speed in 2D.Therefore, the solver is validated on a steady translating submerged cylindercase by comparing the steady waves downstream the cylinder with publishedresults. The comparison is good and helps to establish a proof of concept forextending the existing solution strategy to three-dimensional problems.

1.4 Structure of the thesis

In the second chapter the potential �ow theory ship motions problem is re-viewed. Chapter 3 is dedicated to the numerical methods used to solve theproblem posed in Chapter 2. The implementation of the body boundary con-dition with the IBM-WLS method is described in detail. The WENO weightsand the new smoothness indicator are derived and validated in a series of testcases. The Hamilton-Jacobi WENO formulation of the free surface boundary

1.4 Structure of the thesis 6

conditions follows and its performance is evaluated on the linear and non-linearforward speed problem. The numerical methods for computing the forces onthe body are reviewed. An in depth analysis of the body-free surface inter-section treatment and of the construction of the internal arti�cial free surfacefollows. Chapter 4 elaborates on the temporal discontinuities caused by mov-ing boundaries in combination with the IBM method. Chapter 5 presents fullynonlinear calculations of the zero speed radiation problem on simple 2D geome-tries. Chapter 6 contains forward speed results for the case of a fully submergedcircular cylinder. Chapter 7 includes some concluding remarks and suggestionsfor future research.

Chapter 2

The potential �ow shipmotions problem

In this Chapter we review the potential �ow theory for ocean wave interactionwith a sailing ship. A Cartesian coordinate system (x, z) = (x, y, z) is adoptedwith the z-axis vertical and x-axis in the direction of the ship's forward mo-tion. Bold variables will be used to indicate horizontal vector quantities. Thiscoordinate system is �xed to the mean position of the ship, and moves alongthe x-axis with speed U relative to an earth-�xed coordinate system (x0, z0),as shown in Figure 2.1. U is slowly-varying on a time-scale which is large com-pared to the motions induced by the incoming waves, so that all derivatives ofU are neglected. The �uid is bounded by the free-surface given by ζ(x, t) andthe sea bottom h(x), both measured from the still-water level z = 0. Assumingan incompressible �ow and neglecting viscosity, the �uid velocity is given bythe gradient of a scalar velocity potential φ(x, z, t) which satis�es the following

x

U

z

x0

z0

Figure 2.1: De�nition sketch of the seakeeping problem.

8

initial-boundary-value problem

∇2φ+ ∂zzφ = 0, in V (2.0.1a)

∂tζ +∇ζ(∇φ− w ∇ζ −U

)= w, on z = ζ (2.0.1b)

∂tφ+∇φ(

1

2∇φ−U

)− 1

2w2 (1 +∇ζ∇ζ) = −gζ, on z = ζ (2.0.1c)

∂zφ+∇h∇φ = 0, on z = −h (2.0.1d)

∂nφ = Vn on Sb (2.0.1e)

ζ(x, 0), & φ(x, 0) given (2.0.1f)

where ∇ = (∂x, ∂y) is the horizontal gradient operator, ∂ indicates partial di�er-entiation with respect to the subscripted variable, U = (U, 0) is the ship forwardvelocity vector and g is the gravitational acceleration constant. The �rst equa-tion is the Laplace equation and represents mass conservation. The next twoconditions are the kinematic and dynamic free-surface boundary conditions ex-pressed in terms of surface quantities φ(x, t) = φ(x, ζ, t) and w = ∂zφ|z=ζ . Thefourth and �fth conditions impose no-�ux through the solid bottom and themoving ship surface Sb(t), where Vn(x, z, t) represents the normal componentof velocity of a point on the ship surface and ∂n = n · (∇, ∂z) is the derivativein the direction normal to Sb with n = [nx, ny, nz] the outward normal vector.Finally, for a well-posed problem, initial conditions must be speci�ed for ζ andφ on the free-surface. For a freely-�oating ship, this hydrodynamic boundaryvalue problem must be coupled with Newton's law to solve simultaneously forthe motion of the �uid and the position of the ship Sb. In the context of thisthesis however, we will focus on a stable treatment of the free surface boundaryconditions, and take the position of the ship to be prescribed a-priori.

Chapter 3

Numerical Solution

The boundary value problem posed in (2.0.1) is solved numerically by map-ping the solution to a time-invariant computational domain, and approximatingthe continuous spatial derivatives by arbitrary-order �nite di�erence schemesas discussed in Bingham and Zhang (2007); Engsig-Karup et al. (2009). Thecomputational domain is a unit-spaced Cartesian mesh, and one layer of ghostpoints is distributed along all domain boundaries except the free-surface. Thedegrees of freedom associated with the ghost points are used to satisfy Neumannconditions on the �uid boundaries. The free surface boundary provides Dirichletconditions. The classical explicit fourth-order, four-stage Runge-Kutta schemeis used for time-stepping. The ship surface boundary condition is imposed us-ing an immersed boundary technique as described in Lindberg et al. (2014).The method is implemented on massively parallel Graphical Processing Unit(GPU) architectures where it exhibits a linear scaling of the solution e�ort withincreasing number of total grid points, and excellent parallel scalability Engsig-Karup et al. (2011). The above cited references demonstrate the accuracy ande�ciency of the method for nonlinear wave propagation in a �xed-frame of ref-erence, and linear wave-structure interaction in a moving frame of reference. Inorder to solve nonlinear wave-structure interaction problems in a moving frameof reference, we propose a �nite di�erence WENO scheme for treating the freesurface boundary conditions (2.0.1b) & (2.0.1c). The components of the fullscheme are presented in the following sections.

3.1 Sigma coordinate transformation and Finite

Di�erence discretisations

The free surface is a time-dependant moving boundary with an a priori unknownposition. Therefore, a change of variable in the vertical coordinate which mapsthe solution to a time-invariant domain is adopted for convenience. The non-

3.1 Sigma coordinate transformation and Finite Di�erence discretisations10

conformal σ-coordinate transformation is de�ned by

σ(x, z, t) =z + h(x)

ζ(x, t) + h(x)=z + h(x)

d(x, t), (3.1.1)

where d = ζ + h is the total thickness of the �uid layer. The Laplace problemin the computational domain becomes

∇2φ+∇2σφσ + 2∇σ · ∇φσ + (∇σ · ∇σ + σ2z)φσσ = 0, 0 ≤ σ < 1 (3.1.2a)

(σz +∇h · ∇σ)φσ +∇h · ∇φ = 0, σ = 0, (3.1.2b)

where the derivatives of the coordinate σ are expressed as

∇σ = (1− σ)∇hd− σ∇ζ

d, (3.1.3a)

∇2σ =1− σd

(∇2h− ∇h · ∇h

d

)− σ

d

(∇2ζ − ∇ζ · ∇ζ

d

)− 1− 2σ

d2∇h · ∇ζ − ∇σ

d· (∇h+∇ζ),

(3.1.3b)

σz =1

d. (3.1.3c)

The derivatives of σ are determined by evaluating the speci�c derivatives of hand ζ. The internal kinematics of the �ow are obtained via the chain rule

u(x, z) = ∇φ(x, z) = ∇φ(x, σ) +∇σφσ(x, σ), (3.1.4a)

w(x, z) = φz(x, z) = φσ(x, σ)σz. (3.1.4b)

For the spatial discretisation along the horizontal xy-axes, a grid of (Nx, Ny)

points is de�ned at which the free surface quantities ζ and φ are evolved. Peri-odic or Neumann boundary conditions are imposed at the horizontal boundariesof the domain. The transformed Laplace problem (3.1.2) is solved by de�ningNz points in the vertical direction below each horizontal free surface grid point.The grid points in the vertical are arbitrarily spaced between 0 ≤ σ ≤ 1, re-sulting in a structured grid with arbitrary spacing in each coordinate direction.Finite di�erence schemes are developed for approximating the �rst and secondderivatives in x and σ by using Taylor series expansion and inverting a smalllinear system at each grid point. A stencil of ns = a+b+1 points is used for theone-dimensional �rst and second derivatives in x and σ where a indicates thenumber of points to the right/top and β the number of points to the left/bot-tom from the point of interest. All interior points are centrally discretized witha = b apart from the points where a central stencil would extend beyond the lastcomputational point. In this case o�-centred stencils are used near non-periodicboundaries. One layer of �ctitious computational points is distributed alongthe structural boundaries of the domain (at the bottom and side walls) ensur-ing that both the Laplace equation and the Neumann condition are satis�ed atall boundary points. Mixed xσ- and yσ- derivatives are evaluated using a fullsquare stencil of n2s- points. Consequently, the formal accuracy of all derivativesis O(∆xns−1

∗ ) where ∆x∗ is the maximum grid spacing. The discretisation of(3.1.2) results in a rank N = NxNyNz linear system of equations

AΦ = b, (3.1.5)

3.2 Time integration 11

where A is the �nite di�erence coe�cient matrix, Φ is a vector of the unknownvalues of velocity potential at each grid point and b is a vector populated byzeros, excluding the points that correspond to inhomogeneous boundary condi-tions.

3.2 Time integration

To time-step the kinematic and dynamic free surface boundary conditions astandard explicit fourth order, four stage Runge-Kutta scheme is used. Thegeneral form of the Runge-Kutta 44 method for time stepping a �rst orderdi�erential equation

dy(t)

dt= f(y, t) (3.2.1)

is expressed as

k1 = ∆tf(tn, yn) (3.2.2a)

k2 = ∆tf(tn +1

2∆t, yn +

1

2k1) (3.2.2b)

k3 = ∆tf(tn +1

2∆t, yn +

1

2k2) (3.2.2c)

k4 = ∆tf(tn + ∆t, yn + k3) (3.2.2d)

yn+1 = yn +1

6k1 +

1

3k2 +

1

3k3 +

1

6k4. (3.2.2e)

3.3 Weighted Least Squares, Immersed Bound-

ary Method body boundary condition

To capture the complex structure geometry in the existing solver without com-promising its e�ciency and accuracy, a novel Immersed Boundary Method(IBM) based on Weighted Least Squares (WLS) approximations is used. Thismethod for imposing the body boundary condition is similar to the immersedboundary method developed by Liu et al. (2000) for elliptic equations andwas developed for linear wave-structure interaction problems by Lindberg et al.(2014).

A sign function is used to distinguish between points in the interior or theexterior of the body. The computational points are then classi�ed in four cat-egories. The points that belong to the �uid domain are marked as �uid points.The points in the interior of the body that belong to the �nite di�erence stencilof a �uid point are identi�ed as ghost points. The body boundary conditionis satis�ed on the body points that are located at the normal projection of theghost points on the body surface. Points that belong to the interior of the bodybut are not ghost points are marked as dummy points and their φ value is zero.The Laplace equation is only solved on �uid points. The extra degrees of free-dom associated with the ghost points are used to satisfy the body boundarycondition at the body points. A WLS stencil xi,j,k ∈ S is formed for eachbody point. The WLS stencil of each body point consists of the �uid pointsthat belong in a centred stencil around it plus the associated ghost point as

3.3 Weighted Least Squares, Immersed Boundary Method body boundary

condition 12

Fluid pointsGhost pointsBody pointsDummy points

Figure 3.1: Point classi�cation.

shown in Fig. 3.2. Points that lie on the inside of the body tangent plane at thebody point are excluded. This choice was initially motivated by the di�erentcharacteristics of the �ow on either side of thin parts of the body. However,tests on the dipole solution in an in�nite �uid showed that it always resultsin a more accurate solution. The half width of the stencil in each direction is∆S = ns + 1, where ns is the full �nite di�erence stencil width of the sameorder. The WLS method is then used to approximate the normal derivative ofthe body boundary condition ∂nφ = Vn.

3.3.1 Weighted Least Squares Approximations

The Weighted Least Squares Approximations method is a generalization of the�nite di�erence method that can be applied to an unstructured point set. TheLeast Squares �nite di�erence generalization was developed by Iliev and Tiwari(2003) and was improved by using uneven weighting by Zienkiewicz et al. (2005)

As mentioned, a WLS stencil is associated with every body point xB

S =

NS⋃k=1

xj(k) (3.3.1)

where NS is the size of the stencil and the j(k) index maps from the local indexk to the global index j. The �rst point in the stencil is the associated ghostpoint xj(1) = xG.

The Taylor series approximation of a function at all points in the stencil Sbased on the derivatives at xB and truncated at amax are

φj ≈amax∑|a|=0

(xj(k) − xB)a

a!(φ(a)xB

), ∀xj ∈ S (3.3.2)

where φ(a)xB is the numerical approximation of the a derivative at the body point

xB. The order of the approximation is at least p ≥ ∆S − 1− a. It is assumedthat the number of points in the stencil is greater than the number of terms in

3.3 Weighted Least Squares, Immersed Boundary Method body boundary

condition 13

Fluid points

WLS Stencil

Ghost point

Body point

Figure 3.2: WLS Stencil example for ∆S = 4 (2nd order).

the Taylor series. Collecting the Taylor series expansions results in the followingoverdetermined system Ad = b

amax∑|a|=0

(xj1−xB)a

a!

...amax∑|a|=0

(xjNS−xB)a

a!

φ(0)xB

φ(1)xB

...

φ(amax−1)xB

φ(amax)xB

=

φj1φj2...

φjNS−1φjNS

(3.3.3)

with NS equations and Np = 12 (p+ 1)(p+ 2) unknowns where p = |amax|.

The solution to this overdetermined system is found by solving the weightedquadratic minimization problem:

d = arg mindJ(d) (3.3.4)

where J(d) = rTWr and r = b −Ad. It should be mentioned that r is theresidual of the linear system of equations and W is a matrix with the chosenweights on the diagonal

W =

w1

. . .

wNS

. (3.3.5)

The minimization gives the weighted normal equations

ATWAd = ATWb. (3.3.6)

3.4 Hamilton-Jacobi WENO formulation of the Free Surface Boundary

Conditions 14

This system has a unique solution provided that the matrix ATWA is non-singular. The vector that contains the solution is

d =[ATWA

]−1ATWb. (3.3.7)

The coe�cients for the numerical approximation of the derivatives are containedin the matrix

C =[ATWA

]−1ATW . (3.3.8)

There is a lot of freedom in the choice of the weighting function. The LeastSquares method is obtained by using the identity matrix W = I. However,a choice of a monotone decreasing weighting function of the distance from thecentre point r =

√(xj − xB)2 results into an improved solution. We used the

exponential Gauss function suggested by Zienkiewicz et al. (2005)

w(r) = exp

(−r2

2σ2

)(3.3.9)

where σ is the variance of r.

3.4 Hamilton-Jacobi WENO formulation of the

Free Surface Boundary Conditions

This section presents a stable high-order �nite di�erence scheme suitable fortreating highly nonlinear wave-ship interaction problems in the context of a po-tential �ow approximation. Part of the analysis is published in Kontos et al.(2016). The ship motions problem is most conveniently formulated in a movingcoordinate system attached to the mean position of the ship. The convectiveterms thus introduced into the free surface boundary conditions can be robustlydiscretized using a one-point upwind-biased scheme in the linearized problem,Bingham et al. (2014); Amini Afshar (2014). In the nonlinear problem, however,simple upwinding is not robust for all combinations of ship speed and wave veloc-ity. In this section, we pose the free surface conditions in Hamilton-Jacobi formand adopt the Weighted Essentially Non-Oscillatory (WENO) scheme reviewedfor example by Shu (2009) to obtain a stable scheme. An alternative derivationprocedure of the linear WENO weights is presented based on Taylor series ex-pansion. This method is similar to that developed by Carlini et al. (2005), butis more suited to numerical implementation. A simpli�ed smoothness indicatorfor generating the nonlinear WENO weights is also proposed and shown to givenearly identical behavior to the tabulated versions in the literature for a seriesof test cases. This smoothness indicator is also straightforward to compute nu-merically and employs only already available stencil coe�cients. The resultantWENO scheme is tested on smooth and discontinuous problems and shown toconverge to the expected order of accuracy in all cases. The scheme is combinedwith high-order Runge-Kutta time integration and a Lax-Friedrichs-type �ux tosolve for nonlinear wave propagation in a moving frame of reference. The solveris tested at a representative set of wave speed to ship speed ratios and is foundto be stable in all cases. The WENO scheme is also generally less dissapativethan the equivalent order upwind-biased scheme and thus is a key componentfor extending the existing fast solver for linear wave-ship interaction problemsto nonlinear problems.


Conditions 15

i-2 i-1 i i+1

Substencil 0

Substencil 1

φ(1)−i

(a) Left-Biased stencil.

i-1 i i+1 i+2

Substencil 0

Substencil 1

φ(1)+i

(b) Right-Biased stencil.

Figure 3.3: WENO stencils and sub-stencils for r = 2.

3.4.1 A simpli�ed derivation procedure for the linear WENO

weights

Essentially Non-Oscillatory (ENO) schemes were originally introduced by Hartenet al. (1987) in the context of �nite volume solutions of hyperbolic conservationlaws. The core idea is to consider a multiplicity of interpolation stencils at eachpoint on the grid, and choose the stencil which gives the smoothest approxi-mation of the �ux. Finite di�erence versions of ENO schemes where proposedby Osher and Shu (1991) for treating equations in Hamilton-Jacobi form. Liuet al. (1996) realized that the multiple stencils involved in an ENO scheme couldbe combined to increase the order of the interpolation when the solution wassmooth, which produced the Weighted Essentially Non-Oscillatory (WENO)scheme. Jiang and Shu (1996) proposed improved weights and showed thatfrom an rth-order WENO scheme with r possible stencils, a (2r − 1)th-orderaccurate result could be obtained for smooth functions. Using symbolic manip-ulation software, they tabulated the required linear weights and demonstratedthe predicted accuracy for r = 2, 3, . . . , 7. Carlini et al. (2005) proved that theweights are positive for all orders. A detailed review of WENO schemes is givenby Shu (2009). We present here an alternative procedure for deriving the linearWENO weights which is straightforward to implement numerically and ensuresthe desired order of accuracy when the generating matrix is non-singular. As-sume that φ(x) is a Lipschitz continuous function in R1 with piecewise smoothderivatives. If the derivatives of φ have discontinuities, they are assumed to beisolated. We will present the derivation in terms of a uniform grid spacing ∆xand grid point values φi = φ(xi), with xi = i∆x for simplicity, but the exten-sion to non-uniform grid spacing is trivial. In practice, when non-uniform gridspacing is desired we prefer to to implement it by means of a coordinate trans-formation and derive the WENO derivative scheme on the uniformly-spacedcomputational grid to minimize the number of required coe�cients.

The WENO-r scheme develops a left- and a right-biased derivative approxi-mation at grid point i, each of which is based on a weighted sum of r sub-stencilapproximations, as illustrated in Fig. 3.3 for the case of r = 2. We will use thenotation

φ(n)i =

∂nφ

∂xn

∣∣∣∣x=xi

, (3.4.1)

and an additional superscript − or + to indicate the left- and right-biased


Conditions 16

approximations respectively.The derivation of �nite di�erence and linear WENO weights in Jiang and Shu

(1996) is done using divided di�erences. Another way to derive such schemesis by means of Taylor series expansion and the Vandermonde matrix, or moree�ciently using the recursion formulae of Fornberg (1988), see e.g. LeVeque, R.J. (2007). To illustrate the derivation procedure we will consider the left-biased,

WENO-r scheme derivative approximation φ(1)−i . The total stencil includes 2r-

points, and over this stencil we have r, r + 1-point sub-stencil approximations

which we denote φ(1)s,i , s = 0, . . . , r−1. The sub-stencil approximations are given

by

φ(1)s,i =

r+1∑k=1

Cs,k φs,k (3.4.2)

where Cs,k are the �nite di�erence weights and φs,k are the function values, allon stencil s. The exact derivative at grid point i can be written

φ(1)i = φ

(1)s,i +

∞∑j=1

Ss,j φ(r+j)i ∆xr+j−1 (3.4.3)

where Ss,j are the truncation error coe�cients of the derivative approximationon stencil s. The �nite di�erence coe�cients Cs,k can be found by buildingand inverting the Vandermonde matrix which expresses the Taylor series ex-pansion of the function at each stencil point around the value at grid point i.Similarly, by keeping additional terms in each row of the Vandermonde matrix,and combining these with the coe�cients Cs,k, any number of truncation errorcoe�cients Ss,j can be found.

We now seek r linear WENO weights ds such that

φ(1)−i =

r−1∑s=0

ds φ(1)s,i = φ

(1)i +O(∆x(2r−1)), (3.4.4)

i.e. the �nal form of φ(1)−i should be an order 2r − 1 accurate approximation

to the exact derivative φ(1)i . Clearly for a consistent scheme we must have

r−1∑s=0

ds = 1. (3.4.5)

This leaves r − 1 degrees of freedom which can be used to set the �rst r −1 truncation error terms to zero in (3.4.4). The resultant linear system ofequations based on the truncation error terms Ss,j is easily solved to give theds. As long as this system of equations is non-singular and the ds can be found,an additional r − 1 truncation error terms have been set to zero and the �nalapproximation for φ

(1)−i is by construction, order 2r − 1 accurate.

As an example, consider the case of r = 2 which is illustrated in Fig. 3.3.The two possible derivative approximations in this case are given by

φ(1)0,i =

1

∆x

(1

2φi−2 − 2φi−1 +

3

2φi

)(3.4.6a)

φ(1)1,i =

1

∆x

(−1

2φi−1 +

1

2φi+1

)(3.4.6b)


Conditions 17

r d0 d1 d2 d3 d4 d5 d6 d7 d8

2 13

23 - - - - - - -

3 110

35

310 - - - - - -

4 135

1235

1835

435 - - - - -

5 1126

1063

1021

2063

5126 - - - -

6 1462

577

2577

100231

25154

177 - - -

7 11716

7286

105572

175429

175572

21286

71716 - -

8 16435

566435

1962145

3921287

4901287

3922145

1966435

86435 -

9 124310

3612155

2054944

131677

5211436

3931354

1311354

837006

924310

Table 3.1: Linear WENO weights for r = 2, . . . , 9.

and the �rst truncation error terms are

φ(1)i = φ

(1)0,i −

1

3φ(3)i ∆x2 + . . . (3.4.7a)

φ(1)i = φ

(1)1,i +

1

6φ(3)i ∆x2 + . . . . (3.4.7b)

Thus we have the following linear system of equations to solve for the ds:[− 1

316

1 1

] [d0d1

]=

[01

](3.4.8)

The solution to this system is d0 = 1/3, d1 = 2/3, which agrees with thevalues found by Jiang and Shu (1996). The �nal result for the order 2r − 1approximation is in this case

φ(1)−i =

∑1s=0 ds φ

(1)s,i =

1

∆x

(1

6φi−2 − φi−1 +

1

2φi +

1

3φi+1

)(3.4.9)

which is exactly the 4-point, 3rd-order approximation of φ(1)i which is obtained

from a direct derivation of the coe�cients.This procedure can be applied for any choice of r, and in particular recovers

the coe�cients derived by Balsara and Shu (2000) for r = 3, . . . , 7. The linearWENO weights for r = 2, . . . , 9 are presented in Table 3.1 and the numericalimplementation of the derivation is found in Appendix B.

3.4.2 A simpli�ed smoothness indicator for the nonlinear

WENO weights

The essential idea of WENO is to develop nonlinear weights based on the lineards which are nearly identical to the ds when the function is smooth, but auto-matically turn o� one or more sub-stencils when dicontinuities are encountered.


Conditions 18

The nonlinear weights should choose the smoothest combination of stencils andwill thus reduce the order of accuracy to no less than r.

Following Liu et al. (1996); Jiang and Shu (1996) the nonlinear weights arede�ned as:

ωs =as∑r−1s=0 as

, s = 0, . . . , r − 1 (3.4.10)

with

as =ds

(ε+ βs)2, (3.4.11)

and the left approximation is then given by

φ(1)−i =

r−1∑s=0

ωs φ(1)s,i . (3.4.12)

The parameter ε > 0 is included to avoid dividing by zero, and as in Liu et al.(1996), we take ε = 10−6 here. The βs are the smoothness indicators whichmeasure the smoothness of the solution on each sub-stencil. When the functionis smooth, the βs will be nearly the same on all sub-stencils giving ωs ≈ ds.When a discontinuity is present on one stencil, then that βs will become rela-tively large forcing the associated weight to be relatively small. An alternativede�nition of as is given by Borges et al. (2008); Yamaleev and Carpenter (2009)

as = ds

(1 +

τ

ε+ βs

)=dsβzs, (3.4.13)

where

βzs =

(βs + ε

βs + τ + ε

). (3.4.14)

The factor τ is de�ned by Borges et al. (2008) as the absolute di�erence betweenthe leftmost and the rightmost smoothness indicators

τ = ‖β0 − βr−1‖ (3.4.15)

and alternatively by Yamaleev and Carpenter (2009) as the squared normalizedhighest order derivative on the full stencil of 2r points

τ = (φ(2r−1)i ∆x2r−1)2. (3.4.16)

The βzs are the normalized smoothness indicators βs including the higher orderinformation contained in τ . They are all approximately equal in smooth partsof the solution. The βzs coe�cient of a stencil that contains a discontinuity willbe larger than that of stencils where the solution is smooth. If we consider thecase where one of the stencils contains a discontinuity (e.g. stencil s=2) whilethe solution in stencils s = 0, 1 is smooth, it can be seen that the ratios betweenβz0 , β

z1 and βz2 are larger than when using the classical weights:

βzsβz2

=βsβ2

β2 + τ

βs + τ≥ βsβ2, β2 > βs, s = 0, 1. (3.4.17)

Therefore, the relative importance of stencil s = 2 is larger and the scheme isless dissipative. More detailed analysis can be found in Borges et al. (2008). The


Conditions 19

nonlinear weights as expressed by Yamaleev and Carpenter (2009) are found tobe the less dissipative for the applications of interest as demonstrated by fol-lowing numerical experiments, and will be therefore used in all numerical testsunless noted otherwise.

In Jiang and Shu (1996), the smoothness indicators are computed as the sumof the squares of scaled L2 norms of all the higher derivatives over the interval(xi− 1

2, xi+ 1

2)

βs =

r∑l=1

∫ xi+1

2

xi− 1

2

∆x2l−1(φ(l)s

)2dx. (3.4.18)

For up to r = 7, the smoothness indicators are derived using symbolic manip-ulation software and the resultant expressions are tabulated in Jiang and Shu(1996); Balsara and Shu (2000); Osher and Fedkiw (2002). Consider, for ex-ample, the case of r = 3 on the left-biased sub-stencil 0 where the smoothnessindicator is given as

β0 =13

12(u1 − 2u2 + u3)

2+

1

4(u1 − 4u2 + 3u3)

2(3.4.19)

where the divided di�erences uj are given by

uj =φi−r+j − φi−r+j−1

∆x. (3.4.20)

Expanding the divided di�erences in equation 3.4.19, the smoothness indicatorcan be written

β0 =13

12

(−φi−3 + 3φi−2 − 3φi−1 + φi

∆x

)2

+

1

4

(−φi−3 + 5φi−2 − 7φi−1 + 3φi

∆x

)2

.

(3.4.21)

This is equivalent to

β0 =13

12

(φ(3)

0,i− 12

∆x2)2

+(φ(2)

0,i− 12

∆x)2

(3.4.22)

which is very close to the sum of the squares of the second and third derivatives ofφ at the point i− 1

2 using sub-stencil 0, with each derivative scaled to have unitsof velocity. This observation motivates us to propose a simpli�ed smoothnessindicator given by

βs =

r∑l=2

(φ(l)s,i ∆xl−1

)2. (3.4.23)

This de�nition gives slightly di�erent weights on the higher derivatives comparedto the tabulated forms, and the derivatives are evaluated at grid point i ratherthan the position i−1/2; but it has the advantage of being relatively simple ande�cient to implement numerically and it only uses the already available stencilinformation. The numerical implementation is presented in Appendix B.


Conditions 20

0 0.2 0.4 0.6 0.8 1−8

−6

−4

−2

0

2

4

6

8

x

φ(x)φ(1)(x)

(a) Smooth φ(x).

−2 −1 0 1 2−1

−0.5

0

0.5

1

x

φ(x)φ(1)(x)

(b) Discontinuous φ(x).

Figure 3.4: Test functions

3.4.3 The performance of the new smoothness indicator

In this sub-section, we compare the convergence of the simpli�ed smoothnessindicator of Eq. (3.4.23) with the tabulated values from the literature for evalu-ating the derivative of a smooth and a discontinuous test function. The smoothtest function is given by:

φ(x) = sin(kx), −1 ≤ x < 1 (3.4.24)

and the discontinuous test function is:

φ(x) =

{− cos(π(x− 0.4))/π, −2 ≤ x < 0,

cos(π(x− 0.4))/π, 0 ≤ x ≤ 2(3.4.25)

where the o�set is introduced to avoid any fortuitous cancellation of errors. Thetwo test functions are plotted in Fig. 3.4. For these calculations, no �ux is used

to combine φ(1)−i and φ

(1)+i . On the left of the discontinuity the left biased

stencil is used and on the right of the discontinuity the right biased stencil isused, which simulates the use of an exact Riemann solver. In the case of thesmooth function there is no di�erence in the choice of right or left stencil. Therelative error in each case is de�ned by

ε =||φ(1)i − φ

(1)±i ||∞

||φ(1)i ||∞. (3.4.26)

The convergence of the two approximations for the smooth function is shownin Fig. 3.5 for r = 3, 4, . . . , 6; and we can see that the two smoothness indica-tors give nearly identical results and converge at the expected order of accuracyin all cases. We note that the accumulation of round-o� errors prevents therelative error from dropping below approximately 10−13 as can be seen by thelevelling o� of the high-order errors on re�ned grids. The results for the discon-tinuous function are shown in Fig. 3.6, and again the two methods give nearlyidentical results, providing the expected rate of convergence in all cases. Theseresults support the conclusion that the proposed smoothness indicator is ableto function as well as the tabulated values in controlling the WENO scheme.


Conditions 21

101

102

103

10−15

10−10

10−5

Nx

Relativeerrorǫ

Smooth function

βs : Simpleβs : Table5th order

(a) WENO 3 convergence.

101

102

103

10−15

10−10

10−5

Nx

Relativeerrorǫ

Smooth function


(b) WENO 4 convergence.

101

102

103

10−15

10−10

10−5

Nx

Relativeerrorǫ

Smooth function


(c) WENO 5 convergence.

101

102

103

10−15

10−10

10−5

Nx

Relativeerrorǫ

Smooth function


(d) WENO 6 convergence.

Figure 3.5: Relative error convergence on a smooth φ(x).


Conditions 22

101

102

103

10−10

10−8

10−6

10−4

10−2

100

Nx

Relativeerrorǫ

Discontinuous function


(a) WENO 3 convergence.

101

102

103

10−10

10−8

10−6

10−4

10−2

100

Nx

Relativeerrorǫ



(b) WENO 4 convergence.

101

102

103

10−15

10−10

10−5

100

Nx

Relativeerrorǫ



(c) WENO 5 convergence.

101

102

103

10−15

10−10

10−5

100

Nx

Relativeerrorǫ



(d) WENO 6 convergence.

Figure 3.6: Relative error convergence on a discontinuous φ(x).


Conditions 23

3.4.4 A WENO scheme for stable and accurate evolution

of nonlinear waves

The free surface boundary conditions in (2.0.1b) & (2.0.1c) are not in conser-vation form, but as they represent the projection of the spatially integrated 3DEuler equations onto a 2D surface, we should expect them to exhibit much ofthe hyperbolic character of the Euler equations. Physically, the developmentof steep gradients and discontinuities in the solution comes from wave break-ing which can be instigated by wave-wave interaction, wave-bottom interactionor wave-structure interaction. A number of models for treating wave breakingin the context of otherwise inviscous and irrotational solutions have been de-veloped over the years, mostly in the context of Boussinesq-type models andshallow water breaking, e.g. Schä�er et al. (1993); Kennedy et al. (2000), butalso for deep water wave-wave induced breaking, e.g. Kontos (2013). Whileour long-term goal is to develop a physics-based model for wave breaking, atthis point we will focus on a numerical scheme that is stable and accurate forsmooth solutions and does not break down when steep gradients and disconti-nuities arise in the solution. To this end, the above described WENO schemeis applied to the nonlinear wave propagation problem posed in Hamilton-Jacobiform, as outlined by e.g. Osher and Fedkiw (2002).

Hamilton-Jacobi equations are of the form:

∂tφ+H(∇φ) = 0 (3.4.27)

and they depend on (at most) the �rst derivatives of φ. Hamilton-Jacobi equa-tions are discretized spatially through the numerical Hamiltonian H which is aconsistent approximation of H(∇φ) such that for a smooth function φ:

H(φ−x , φ+x , φ

−y , φ

+y , φ

−z , φ

+z )→ H(∇φ), as (∆x,∆y)→ 0. (3.4.28)

In this section we will use the notation φ±x , φ±y to indicate left and right approxi-

mations to the partial derivatives of φ with respect to x and y. Various schemeshave been proposed in the literature for de�ning the numerical Hamiltonian,several of which we will consider here.

3.4.4.1 Lax-Friedrichs Schemes

The Lax-Friedrichs scheme was developed by Crandall and Lions (1984):

H = H

(φ−x + φ+x

2,φ−y + φ+y

2

)− ax

(φ+x − φ−x

2

)− ay

(φ+y − φ−y

2

)(3.4.29)

where ax and ay are dissipation coe�cients for controlling the amount of nu-merical viscosity, and in the case of a scalar equation are de�ned as:

ax = max |H1(φx, φy)|, ay = max |H2(φx, φy)| (3.4.30)

H1 and H2 are the partial derivatives of H with respect to φx and φy, respec-tively. Increasing a, increases the arti�cial dissipation and decreases the qualityof the solution. Therefore it is bene�cial to select as small as possible dissipationcoe�cients.


Conditions 24

A system of Hamilton-Jacobi equations is de�ned as

∂tq +H(∇q) = 0 (3.4.31)

where q is the solution vector

q =

[φ1φ2

]. (3.4.32)

To �nd the dissipation coe�cients we split the Hamiltonian into

H(∇q) = Hx(φ1,x, φ2,x) +Hy(φ1,y, φ2,y). (3.4.33)

In the x-direction the dissipation coe�cient ax is de�ned as the maximum ab-solute eigenvalue λ of the Jacobian matrix J of Hx:

ax = max(|λ1|, |λ2|) of J =

[ ∂∂φ1,x

∂∂φ2,x

]Hx. (3.4.34)

The dissipation coe�cient ay is derived in a similar manner.A modi�ed version called the Local Lax-Friedrichs Flux (LLF) was proposed

by Shu and Osher (1989) where the ax coe�cient at a grid point is determinedusing only the values of φ−x , φ

+x at this grid point. However, the ay coe�cient

are determined using also the neighbouring points that are part of the stencilused to determine φy.

Osher and Shu (1991) proposed an improved version called the Local LocalLax-Friedrichs Flux (LLLF) which has the least numerical dissipation of all theLax-Friedrichs schemes. In this scheme, at grid point (i, j) both ax and ay aredetermined using only the values of φ±x,i,j and φ

±y,i,j .

3.4.4.2 The Roe-Fix Scheme

Shu and Osher (1989) combined Roe's upwind method with an LLF entropycorrection. This method applied to Hamilton-Jacobi equations was dubbed theRoe Fix. As in the LLLF scheme, H1 and H2 are calculated using only thenodal values for φ±x and φ±y . The scheme is de�ned as:

H(φ+x , φ−x , φ

+y , φ

−y ) =

H(φ∗x, φ

∗y

), if H1 and H2 for φ

±x and φ±y

do not change sign,

H(φ+x +φ−

x

2 , φ∗y

)− ax

(φ+x−φ

−x

2

), else if H2 does not change sign

for φ±x and φ±y ,

H(φ∗x,

φ+y +φ−

y

2

)− ay

(φ+y −φ

−y

2

), else if H1 does not change sign

for φ±x and φ±y ,

HLLLF , else

(3.4.35)


Conditions 25

where φ∗x and φ∗y are determined by upwinding:

φ∗x =

{φ+x , if H1 ≤ 0,

φ−x , if H1 ≥ 0(3.4.36)

and

φ∗y =

{φ+y , if H2 ≤ 0,

φ−y , if H2 ≥ 0(3.4.37)

3.4.4.3 WENO discretization of the linear forward speed problem

As a �rst step, WENO will be applied to the linearised version of the problemon a horizontal bottom. The kinematic and dynamic free surface boundaryconditions in this case reduce to:

∂tζ − U∂xζ = ∂zφ (3.4.38)

∂tφ− U∂xφ = −gζ (3.4.39)

and the depth h is taken to be a constant. In Bingham et al. (2014); AminiAfshar (2014), the high-order �nite di�erence scheme described at the top ofthis Section was shown to be stable when the convective derivatives on the free-surface are computed using a one-point, upwind-biased scheme. Here we willcompare that solution to the proposed WENO scheme.

The free surface boundary conditions in the WENO formulation are ex-pressed as:

∂tζ +Hζ = ∂zφ (3.4.40)

∂tφ+Hφ = −gζ (3.4.41)

where Hζ = −U∂xζ and Hφ = −U∂xφ. The terms on the right hand side,

namely ∂zφ and −gζ, are considered as source terms. Expressing the free surfaceboundary conditions as a system of equations with solution vector

q =

[ζ

φ

](3.4.42)

we de�ne the Hamiltonian as

Hx(ζx, φx) =

[Hζ

Hφ

]. (3.4.43)

The Jacobian matrix of Hx is

J =

[−U 00 −U

](3.4.44)

with eigenvalues λ1 = λ2 = −U . Therefore the dissipation coe�cient ax isde�ned as

ax = max(|λ1|, |λ2|) = U. (3.4.45)


Conditions 26

The Lax-Friedrichs scheme applied to the kinematic free surface boundary con-dition results in the following approximation:

Hζ = −U ζ+x + ζ−x2

− U ζ+x − ζ−x2

= −Uζ+x (3.4.46)

Similarly for the dynamic free surface boundary condition:

Hφ = −Uφ+x (3.4.47)

It can be seen that in this simple linear case the WENO scheme becomes anupwind-biased scheme if the sub-stencils are weighted with the linear weights.The nonlinear weights should therefore mimic the linear ones for a smoothfunction.

i-2 i-1 i i+1 i+2 i+3

Full WENO 3 right biased stencil

i-2 i-1 i i+1 i+2 i+3 i+4

1-point upwind FD stencil, r = 3

Figure 3.7: WENO right biased stencil compared to 1-point upwind FD stencil(r = 3).

The performance of the one-point upwind FD scheme will be compared withthe 1-D WENO FD scheme on the well-known travelling wave solution in amoving frame of reference on a periodic domain:

ζ = <{H

2ei(kxR−ωt)

}(3.4.48)

φ = <{−igω

H

2

cosh[k(z + h)]

cosh(kh)ei(kxR−ωt)

}(3.4.49)

where xR = x + Ut, H is the wave height, L = 2π/k the wave length, T =2π/ω the wave period, and ω and k are related by the dispersion relation ω2 =gk tanh(kh). We note that the upwind-biased FD scheme applied here usesa total of 2r + 1 points to give an order 2r accurate scheme and it is thusformally one order higher than the corresponding WENO scheme. The twostencils are illustrated in Fig. 3.7 for the case of r = 3. In all cases, centered�nite di�erence schemes of order 2r are used for the solution of the Laplace


Conditions 27

problem and o�-centered schemes of the same order are used for computingthe vertical derivative ∂zφ. For these tests, a uniform grid spacing is used inboth directions. The LLLF scheme is used for the WENO scheme. The explicitfourth-order, four-stage Runge-Kutta scheme is used for time-stepping, and theCourant number based on the wave celerity c = L/T is �xed at Cr = 0.05. Thischoice is to reduce the time-stepping errors below the level of the spatial errorsin order to test the spatial schemes.

The forward speed is set to U = c/3, which simulates a ship at moderatespeed encountering a wave of it's own length. Each test was run for �ve periodsafter which the error in the free surface elevation was measured as:

ε∞ =‖ζexact − ζ5‖∞‖ζexact‖∞

(3.4.50)

where ζ5 is the computed surface elevation after exactly �ve wave periods. Bothapproximations are found to be stable for these test cases, as expected. Theconvergence results are shown in Fig. 3.8. In �gures 3.8a to 3.8c the WENOschemes are one order more accurate than expected. They perform equivalentlywith the upwind-biased schemes. This can be attributed to the fact that WENOis only used to solve a part of the problem and the rest of the problem is solvedwith FD schemes of order 2r. The non-linear weights are emulating the linearweights correctly which is an additional indicator that the smoothness indicatorsare working properly. The in�uence of the fourth order time-stepping schemeis evident �gure 3.8d. The accumulation of temporal errors limits the order ofconvergence. However, the WENO scheme performs as well as the one-pointupwind biased scheme.

3.4.4.4 WENO discretization of the non-linear forward speed prob-lem

The free surface boundary conditions can be expressed in the WENO formula-tion as:

∂tζ +Hζ = ∂zφ (3.4.51)

∂tφ+Hφ = −gζ (3.4.52)

whereHζ = ∂xζ

(∂xφ− ∂zφ ∂xζ − U

)(3.4.53)

and

Hφ = ∂xφ

(1

2∂xφ− U

)− 1

2(∂zφ)2 (1 + ∂xζ ∂xζ) (3.4.54)

The right hand side terms, namely ∂zφ and −gζ, are again considered as sourceterms. The Hamiltonian in the x-direction is de�ned as

Hx =

[Hζ

Hφ

](3.4.55)

and the Jacobian matrix of Hx is

J =

[φx − 2φz ζx − U ζx

−φ2z ζx φx − U

](3.4.56)


Conditions 28

101

102

103

10−10

10−8

10−6

10−4

10−2

Nx

Infinitynorm

errorǫ ∞

WENO3Upwind r = 36th order

(a) WENO 3 vs 6th order FD.

101

102

103

10−12

10−10

10−8

10−6

10−4

Nx

Infinitynorm

errorǫ ∞


(b) WENO 4 vs 8th order FD.

101

102

103

10−12

10−10

10−8

10−6

Nx

Infinitynorm

errorǫ ∞


(c) WENO 5 vs 10th order FD.

101

102

103

10−14

10−12

10−10

10−8

10−6

Nx

Infinitynorm

errorǫ ∞


(d) WENO 6 vs 12th order FD.

Figure 3.8: WENO vs one-point upwind Finite Di�erence.


Conditions 29

with eigenvaluesλ1 = λ2 = φx − φz ζx − U. (3.4.57)

The dissipation coe�cient ax is de�ned as

ax = max(|λ1|, |λ2|) = |φx − φz ζx − U |. (3.4.58)

The WENO scheme is only used in the free surface boundary conditions. The∂xζ and ∂xxζ terms required in the sigma transformation are computed froman order 2r centered �nite di�erence scheme.

To test the accuracy and stability of the scheme, a deep water wave withkh = 2π is considered. The wave height is 90% of the stable limit,H/L = 0.1273.The water depth is again taken to be constant. From this solution the initialconditions for ζ and φ are obtained from the stream function theory of Fenton(1988). The lateral boundary conditions are periodic. A uniform grid is usedin the horizontal direction and a cosine stretched grid in the vertical. The totalenergy in the �uid can be conveniently expressed in the usual way as

E =ρ

2

∫S0

(φ∂tζ + gζ2)dxdy (3.4.59)

where S0 is the still water plane. This expression is used to monitor the level ofenergy conservation of each calculation.

In the �rst test case the WENO and a centered FD scheme were compared forU = 0. The Courant number was �xed at Cr = 0.5 and a resolution of Nx = 64,Nz = 9 was used. The solution was allowed to propagate for ten periods. Theenergy conservation was monitored and is presented in the following plot. Inthis preliminary test the original WENO weights of Liu et al. (1996); Jiang andShu (1996) were used and the results are presented in Fig. 3.9.

0 2 4 6 8 10−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

t/T

(E−E

0)/E

0

Nx=64, Nz=9

LLLF - WENO3Roe Fix- WENO3

(a) WENO 3.

0 2 4 6 8 10−20

−10

0

x 10−5 Nx=64, Nz=9

t/T

(E−E

0)/E

0

No FilteringS-G(13,10) filter

(b) 6th order centered FD.

Figure 3.9: WENO vs FD. Energy conservation.

It can be seen that the 5th order WENO scheme with the original weights fromLiu et al. (1996); Jiang and Shu (1996) is signi�cantly more di�usive than the 6thorder centered FD scheme. The Roe-Fix scheme is slightly less di�usive than theLLLF scheme, so this scheme is chosen for all subsequent WENO calculations.


Conditions 30

101

102

103

10−5

10−4

10−3

10−2

10−1

100

Nx

Relativeerrorper

waveperiod

FD r=2FD r=3WENO3 JSWENO6 JS

Figure 3.10: WENO JS Liu et al. (1996); Jiang and Shu (1996) vs FD. Relativeerror per wave period.

The centered FD scheme eventually breaks down as a result of an instabilityassociated with a nonlinear shifting of the stability eigenvalues slightly o� theimaginary axis. This mild-instability can be prevented by applying a 13-point,10th order Savitzky-Golay type �lter once per wave period, as discussed byBingham and Zhang (2007); Engsig-Karup et al. (2009).

The same wave was propagated for �ve periods with a �xed Courant numberof Cr = 1. The centered FD simulations did not require �ltering since thesolution is stable for the �ve �rst periods. The forward speed is still zero. Therelative error per wave period was computed as:

‖ζexact − ζ5‖25‖ζexact‖2

(3.4.60)

Figure 3.10 shows that an 11th order WENO 6 scheme is required to matchthe performance of a 6th order FD scheme when the weights of Liu et al. (1996);Jiang and Shu (1996) are used due to excessive numerical di�usion. When theweights of Borges et al. (2008) and Yamaleev and Carpenter (2009) are usedthe numerical di�usion is signi�cantly decreased as shown in Figures 3.11 and3.12 respectively. The 5th order WENO 3 scheme performed almost as wellas the 6th order FD scheme with the Yamaleev and Carpenter (2009) weightsbeing slightly less di�usive in comparison with the Borges et al. (2008) weights.For the WENO simulations in this numerical test the Laplace problem and theapproximation of ∂zφ were computed using 6th order FD schemes.

In a general ship motions simulation, the ship will encounter waves whichpropagate in all directions and at all speeds. Thus the numerical scheme mustbe able to handle all possible ratios of wave celerity to ship speed. For the �naltest case, we thus evaluate the performance of the scheme for a representativeset of wave celerities spanning the entire range of possible ratios. The other


Conditions 31

101

102

103

10−4

10−3

10−2

10−1

100

Nx

Relativeerrorper

waveperiod

FD r=2FD r=3WENO3 BWENO 4 BWENO6 B

Figure 3.11: WENO B Borges et al. (2008) vs FD. Relative error per waveperiod.

101

102

103

10−4

10−3

10−2

10−1

100

Nx

Relativeerrorper

waveperiod

FD r=2FD r=3WENO3 YWENO4 YWENO6 Y

Figure 3.12: WENO Y Yamaleev and Carpenter (2009) vs FD. Relative errorper wave period.


Conditions 32

parameters of the problem are as for the previous cases. The Courant numberis again �xed at Cr = 0.5 and a resolution of Nx = 64, Nz = 9 is used witha uniform grid in x and a cosine stretched grid in z. The WENO 6 scheme iscompared with centered and upwind-biased 6th order �nite di�erence schemes.The di�erent cases are visualized in Figure 3.13. When U ≤ 0 upwinding isachieved by shifting the stencil one point to the left. For U > 0 the stencil isshifted one point to the right.

The energy conservation was monitored for each simulation and the resultsare presented in Figures 3.14a -3.14f. The centered scheme is always unstablefor convective problems (and therefore not plotted for clarity). The upwind-biased scheme can treat most of the cases, but the scheme will always becomee�ectively down-winded, and hence unstable, for waves which travel faster thanthe frame of reference (ship) and in the same direction as the frame of reference(ship). In contrast, the WENO scheme is always stable and is generally lessdissipative than the upwind-biased scheme.

These results suggest that the proposed WENO scheme is a good choice fortreating the nonlinear ship motions problem.


Conditions 33

0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14z

x

c

U = 4c

(a) U = 4c.

0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

z

x

c

U = -4c

(b) U = −4c.

0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

z

x

c

U = 2c

(c) U = 2c.

0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

z

x

c

U = -2c

(d) U = −2c.

0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

z

x

c

U = 1/10c

(e) U = 1/10c.

0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

z

x

c

U = -1/10c

(f) U = −1/10c.

Figure 3.13: Stream function wave in a moving frame of reference.


Conditions 34

0 2 4 6 8 10

−8

−6

−4

−2

0

2

4

x 10−3

t/T

(E−E

0)/E

0

WENO 4WENO 6Upwind

(a) U = 4c.

0 2 4 6 8 10

−8

−6

−4

−2

0

2

4

x 10−3

t/T

(E−E

0)/E

0

WENO 4WENO 6Upwind

(b) U = −4c.

0 2 4 6 8 10−4

−3

−2

−1

0

1

2

3

4x 10

−3

t/T

(E−E

0)/E

0

WENO 4WENO 6Upwind

(c) U = 2c.

0 2 4 6 8 10−4

−3

−2

−1

0

1

2

3

4x 10

−3

t/T

(E−E

0)/E

0

WENO 4WENO 6Upwind

(d) U = −2c.

0 2 4 6 8 10−1

0

1x 10

−4

t/T

(E−E

0)/E

0

WENO 4WENO 6Upwind

(e) U = 1/10c.

0 2 4 6 8 10

−2

−1

0

1

2x 10

−4

t/T

(E−E

0)/E

0

WENO 4WENO 6Upwind

(f) U = −1/10c.

Figure 3.14: WENO 6 vs FD. Energy conservation. Nx = 64, Nz = 9, kh =2π and H/L = 0.1273.


Conditions 35

3.4.4.5 The importance of the numerical Hamiltonian

Since the dissipative character of the WENO schemes is a combination of theWENO weights and of the �ux used in the Hamiltonian approximation thequestion naturally arises: is WENO's ability to select the smoothest possiblereconstruction stencil enough to stabilize the solution on it's own? To answerthis question, we experimented with a 'centered' WENO scheme that computesonly one approximation per grid point.

i-2 i-1 i i+1 i+2

Substencil 0

Substencil 1

Substencil 2

φ(1)i

Figure 3.15: Centered WENO stencil for r = 2.

The centered WENO uses only one stencil that includes all di�erent sub-stencilsof the left/right stencils of the normal WENO schemes. The sub-stencil approx-imations are weighted with the nonlinear weights to produce the �nal derivativeapproximation. In that way there is no need to combine the left and right ap-proximations through a �ux. The scheme was tested on the linear forward speedproblem and produces accurate and stable results. However, in the nonlinearcase the nonlinear weights alone were not able to stabilize the solution and the�ux formulation that upwinds the solution automatically appears to be criticalfor nonlinear stability.

3.4.5 Summary

A new way of deriving the WENO coe�cients has been presented together witha new smoothness indicator that adds �exibility to the schemes and is straight-forward to implement numerically without the need for tabulated coe�cients.The method has been tested on both the linear and the nonlinear wave prop-agation problem in a moving frame of reference which mimics a forward speedseakeeping problem. In the linear case WENO acts as an upwind-biased �nitedi�erence scheme and thus the additional overhead is not warranted. In thenonlinear case however, a simple upwind-biased scheme will always become ef-fectively down-winded and unstable when the wave speed in the direction ofthe ship motion is larger than the ship speed. The proposed high order WENOscheme detects this situation automatically and remains upwinded and stableunder all combinations of wave speed and ship speed. The dissipation intro-duced by the WENO scheme is generally as low or lower than the upwind-biasedscheme, though for high-accuracy, very high-order schemes may be desirable.

3.5 Methods for computing the forces on the body 36

3.5 Methods for computing the forces on the body

The total hydrodynamic pressure force on the body is

F =

∫∫Sb

pn dS. (3.5.1)

In a �xed frame of reference and by expressing the dynamic pressure withBernoulli's equation the force expression becomes

F = −ρ∫∫Sb

[∂φ

∂t+

1

2∇φ∇φ

]ndS, (3.5.2)

where ∇ = (∂x, ∂y, ∂z) indicates the 3D gradient vector in this context. In aframe of reference that is moving along the x-axis with speed U (3.5.2) becomes

F = −ρ∫∫Sb

[∂φ

∂t− U∇φ+

1

2∇φ∇φ

]ndS. (3.5.3)

When considering prescribed motion problems equations (3.5.2) and (3.5.3) areevaluated during the post-processing. At each time step the potential and it'sderivatives are interpolated on a uniform distribution of np points on the wettedbody surface using the WLS method. Since the wetted body surface changeswith time the location of the points used for the integration also changes. Thus,the partial derivative of φ is expressed as

∂φ

∂t=

dφ

dt− dx

dt

∂φ

∂x− dy

dt

∂φ

∂y− dz

dt

∂φ

∂z. (3.5.4)

The time derivatives are then computed by a standard �nite di�erence schemeof the same order as the rest of the solution.

Alternatively, Equation (3.5.2) can be recast as shown in Newman (1977) byusing the Gauss theorem ∫∫∫

V

∇f dV =

∫∫S

fn dS (3.5.5)

and the Transport theorem

d

dt

∫∫∫V

f dV =

∫∫∫V

∂f

∂tdV +

∫∫S

fV n dS. (3.5.6)

For the derivation of the alternative force expression we are surrounding thebody by a �xed control surface Sc that is exterior to the body surface Sb. Therate of change of the �uid momentum in the volume V(t) enclosed by these two

3.5 Methods for computing the forces on the body 37

surfaces is

ρd

dt

∫∫Sb+Sc

φndS = ρd

dt

∫∫∫V(t)

∇φdV (3.5.7a)

= ρ

∫∫∫V(t)

∇[∂φ

∂t

]dV + ρ

∫∫Sb+Sc

∇φ(V n) dS (3.5.7b)

= ρ

∫∫Sb+Sc

[∂φ

∂tn +∇φ(V n)

]dS (3.5.7c)

where the Gauss theorem is used in equations (3.5.7a), (3.5.7c) and the Trans-port theorem is used at equation (3.5.7b). Since on the �xed control surfaceSc V n = 0, we can interchange the time derivative and the surface integration.Consequently, the �rst and last integrals of equation (3.5.7) can be equatedseparately over Sc and Sb

ρd

dt

∫∫Sb

φndS = ρ

∫∫Sb

[∂φ

∂tn +∇φ(V n)

]dS. (3.5.8)

Using the boundary condition ∂φ/∂n = V n on the body surface we get

ρd

dt

∫∫Sb

φn dS = ρ

∫∫Sb

[∂φ

∂tn +

∂φ

∂n∇φ]dS (3.5.9)

Finally, adding (3.5.9) and (3.5.2) results in the alternative expression of thehydrodynamic pressure force

F = −ρ ddt

∫∫Sb

φn dS + ρ

∫∫Sb

(∂φ

∂n∇φ− 1

2∇φ∇φn

)dS (3.5.10)

The two integrals that are needed for evaluating (3.5.10) are de�ned over thewetted surface of the body and they are computed at each time step. Allintegrand quantities are evaluated using the WLS method on a uniform distri-bution of points on the wetted surface of the body. The time derivative of the�rst integral is computed using �nite di�erence as a post-processing step. Bothformulations of the force are equivalent and should produce the same result totruncation error. Implementing both of them and ensuring that they are inagreement serves as an extra validation test.

All integrals are evaluated numerically using Taylor integration Bingham(2011). In 2D the integral of a function f(s) over a segment of the body runningfrom s1 to s2 with length ds1 is

s2∫s1

f(s) ds = ds1

1∫0

f(ξ) dξ (3.5.11)

The continuous solution can be de�ned locally using Taylor series expansion as

f(ξ) =

p∑n=0

ξn

n!f(n)0 +O(ξp+1) (3.5.12)

3.6 Body - free surface intersection tracking and construction of the

interior free surface 38

where p is the order of the �nite di�erence scheme used and f(n)0 is the n-th

derivative of f with respect to ξ evaluated at ξ = 0. Substituting (3.5.12) in(3.5.11) and integrating term by term gives

s2∫s1

f(s) ds = ds1

p∑n=0

1

(n+ 1)!f(n)0 . (3.5.13)

By summing all contributions of the np − 1 segments of the body, the totalintegral is computed

I =

np−1∑j=1

dsj

p∑n=0

1

(n+ 1)!f(n)j . (3.5.14)

If the length of each segment is not known exactly it is calculated from theexpression for the arc-length by expressing the position (x, z) of the curve para-metrically as a function of ξ

dsj =

j∫j−1

√(dx

dξ

)2

+

(dz

dξ

)2

dξ. (3.5.15)

This integral is calculated with Taylor integration as well.

3.6 Body - free surface intersection tracking and

construction of the interior free surface

The boundary value problem (2.0.1) is solved numerically by mapping the solu-tion from the physical domain to a time-invariant computational domain with asigma transformation as discussed in subsection 3.1. The sigma transformationrequires ∇ζ, ∇2ζ to be smooth and continuous along the whole free surface.Therefore, we choose to create an arti�cial free surface in the interior of thebody that is C2 continuous at the intersection with the exterior free surface.An example of the merged exterior and interior free surfaces is shown in Figure3.17. The �rst step for constructing the internal free surface is to track the bodyfree-surface intersection points using the bisection iteration method between the�rst interior and exterior free surface points. Consequently, ζ, ζx, ζxx are ex-trapolated on each intersection point and serve as the boundary conditions forthe construction of the internal free surface. The methods used for tracking theintersection, constructing the internal free surface and introducing new �uidpoints on the external free surface as the body is moving are explained in detailin the following subsections. Summarizing, the following algorithm is executedat each Runge-Kutta stage:

1. Compute the position and the velocity of the body.

2. Construct the new interior free-surface (FS).

3. If a new exterior FS point is added, its ζ and φ values are computed byextrapolation.



1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

z

x

(a) Phsyical domain.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3−0.2

0

0.2

0.4

0.6

0.8

1

σ

x

(b) Computational domain.

Figure 3.16: Visualization of the sigma transformation.

26 28 30 32 34 36−7

−6

−5

−4

−3

−2

−1

0

1

2

x

z

ζζ interiorBody pointsGhost pointsFluid PointsDummy points

Figure 3.17: Example of the interior free surface.



4. Solve the Laplace problem by using the IBM-WLS methods for satisfyingthe body boundary condition.

5. Step forward in time the free-surface boundary conditions using WENOfor the spatial discretization (automatic upwinding). The procedure isvisualized in Figure 3.18.



26 28 30 32 34 36

−5

−4

−3

−2

−1

0

1

2

z

x

ζ

(a) Exterior ζ and φ are known.

26 28 30 32 34 36

−5

−4

−3

−2

−1

0

1

2

z

x

ζInternal ζ

(b) Outmost interior ζ points are extrapo-lated.

Figure 3.18: Step by step visualization of the construction of the internal freesurface.



26 28 30 32 34 36

−5

−4

−3

−2

−1

0

1

2 Find ζ, ζx, ζx,x

z

x

ζInternal ζIntersection points

(c) Body-free surface intersection is

tracked. Boundary conditions for the

interior free surface are extrapolated.

26 28 30 32 34 36

−5

−4

−3

−2

−1

0

1

2

z

x

ζInternal ζIntersection points

(d) The interior free surface is created .

Figure 3.18: Step by step visualization of the construction of the internal freesurface.

3.6.1 Internal free surface construction

Computing an arti�cial internal free surface that respects C2 continuity at theintersections is a challenging task. Furthermore, the arti�cial free surface has tobe constrained in the interior of the body and to be C2 continuous over its entirelength. An internal free surface shooting out of the body boundary violates thesigma-transform and causes the algorithm to break down. Discontinuities in its�rst two horizontal derivatives lead to a badly conditioned system matrix anda compromised solution that causes oscillations in the force signal. Generatinga well behaved internal free surface is therefore of fundamental importance forobtaining accurate results. We have experimented with several methods toovercome this problem and we present here the method we selected and itslimitations.

After the values of ζ, ζx, ζxx are determined at the intersection points, theinterior solution is given by a seventh order polynomial. Its coe�cients canbe obtained in closed form by using the six boundary conditions at the twointersections. The extra degrees of freedom are used to keep the polynomialwell behaved by minimizing its square distance to the straight line that connectsthe intersections. Alternatively, the polynomial coe�cients can be determinednumerically via constrained minimization of either the arc-length, or the integral



of the second or third derivative along the interior free surface while satisfyingthe boundary conditions at the intersections. The derivation is described indetail in the Appendix A.

As a test case we de�ne a sine as the physical free surface that is inter-rupted by the body boundaries at the two intersections (xL = −0.5, sin(xL))and (xR = 0.5, sin(xR)). The interior solution is computed with the four vari-ations of the polynomial. The �rst and second horizontal derivatives over theentire reconstructed free surface are computed with fourth order centred �nitedi�erence operators and the quality of the internal solution is evaluated. Threewave cases are considered that test the behaviour of the polynomials in long,medium and short waves relative to the body length. The results are presentedin Figures 3.19-3.21. The closed form is the worst performing of the four as itintroduces discontinuities at the intersections. The numerical constrained poly-nomials behave much better, although short wavelengths and large gradients atthe boundaries can prove challenging. We will proceed with the solution thatminimizes the arc length since it tends to overshoot less than the others and ismore probable to remain constraint inside the body boundary. For short wave-lengths local smoothing of the external free surface near the body boundariescould be used to avoid large oscillations of the internal solution.

3.6.2 Intersection treatment and adding new free surface

points

The intersection of the free surface with the body boundary is found by applyingthe bisection iteration method between the �rst internal and external free surfacepoints. There is always a ghost point associated with the intersection to ensurethat the body boundary condition is satis�ed at the complete wetted bodysurface with the IBM method. The ∂xζ term on the free surface boundaryconditions that is computed with WENO is based only on one-sided stencilsnear the body boundary. The WENO stencil only includes values of the externalfree surface points plus the �rst internal point. This is achieved by settingthe rest of the internal points to very large values with large variations, e.gζj+1 = (10(j + 1))10, j = 1, 2, . . . , where ζ1 corresponds to the �rst internalpoint.

The value of the �rst internal free surface point is calculated with polynomialextrapolation from the neighbouring exterior points. If the extrapolated valuelies outside the body, a new exterior free surface point is added and it's φ valueis also calculated with extrapolation. Since the Courant number with respectto the body velocity is Cr < 1, a maximum of one free surface point can beadded or subtracted per time-step on each side of the body. The disadvantageof this method is that when the free surface develops large gradients in thevicinity of the body (short waves or large amplitude of motion), more than oneextrapolated points can end up outside the body resulting to a free surface thatdoes not intersect with the body.

In an attempt to circumvent this problem an alternative method has beenapplied where a straight line is �tted to the �rst r external free surface pointsand is evaluated at the x location of the �rst internal point. If the point liesin the exterior of the body a new external point is added and the value of thenext point is calculated. After the �rst internal point is located, the one sided�lter described in Berland et al. (2007) is passed through that point and the



-2 -1 0 1 2

-1

-0.5

0

0.5

1

(a) The entire reconstructed free surface ζ.

-2 -1 0 1 2-2

-1

0

1

2

(b) ζx computed with fourth order centred oper-

ators.

-2 -1 0 1 2-20

-10

0

10

20

(c) ζxx computed with fourth order centred oper-

ators.

Figure 3.19: Long wavelength test case. The closed form solution is com-pared with numerically determined polynomials minimizing thearc length (Min. ds), or the integral of the second (Min. ζxx) orthird derivative (Min. ζxxx) along the interior free surface.



-2 -1 0 1 2-1.5

-1

-0.5

0

0.5

1

1.5


-2 -1 0 1 2-10

-5

0

5

10


ators.

-2 -1 0 1 2

-100

-50

0

50

100


ators.

Figure 3.20: Medium wavelength test case. The closed form solution is com-pared with numerically determined polynomials minimizing thearc length (Min. ds), or the integral of the second (Min ζxx) orthird derivative (Min. ζxxx) along the interior free surface.



-2 -1 0 1 2-20

-15

-10

-5

0

5

10


-2 -1 0 1 2

-60

-40

-20

0

20

40

60


ators.

-2 -1 0 1 2-2000

-1500

-1000

-500

0

500

1000


ators.

Figure 3.21: Short wavelength test case. The closed form solution is com-pared with numerically determined polynomials minimizing thearc length (Min. ds), or the integral of the second (Min ζxx) orthird derivative (Min. ζxxx) along the interior free surface.



neighbouring r − 1 external free surface points. A centred SG-type �lter ispassed through the rest of the external free surface points that lie within onewave length from the body. The smoothing of the free surface helps in obtaininga well behaved internal solution. However, it is di�cult to control when and howmany points will be added on the free surface within one time-step. When thefree surface in the vicinity of the body is smoothed, the �rst internal point canbe moved outside the body (or an external point can become internal). Thatcan happen at any stage of the Runge Kutta algorithm making it di�cult tokeep track of the points that are being evolved. Furthermore, the smoothingof the free surface makes the intersection point move arbitrarily on the bodysurface through the Runge-Kutta stages. Therefore, in the test cases presentedin this thesis the internal free surface points are computed with extrapolation.

New points can be added after the �rst or the third Runge-Kutta 44 stagethat are evaluated at tn + 1

2∆t and tn + ∆t respectively. The potential at thenew point is found by polynomial extrapolation and the right hand side of thenext stage is evaluated. Since there is no history for the new point, only theexternal free surface points of time tn are evolved. The new point value at thenew time is computed with polynomial extrapolation of the evolved points.

Chapter 4

Numerical force oscillationscaused by moving

boundaries

The Immersed-Boundary type methods are classi�ed in two types by Mittaland Iaccarino (2005). One type imposes the boundary condition on the physi-cal boundary indirectly by introducing a volumetric force that is spread in the�uid cells in the vicinity of the boundary, e.g. Peskin (1972); Goldstein et al.(1993). The other type imposes the boundary condition directly on the physicalboundary by modifying the computational stencil near the immersed bound-ary. Therefore these sharp interface methods achieve higher order of accuracycompared to the di�used interface methods. Our method is a sharp interfacemethod that resembles the Ghost-Cell Finite Di�erence Approach as in Mit-tal and Iaccarino (2005). In this approach the boundary condition is imposedthrough ghost cells. Ghost cells are cells in the interior of the body that haveneighbours in the �uid. For every ghost cell an interpolation scheme is createdthat implicitly incorporates the boundary condition. Advantages of the methodinclude its simple formulation and implementation, the low computational costand the high order of accuracy. One disadvantage is that the method is proneto temporal oscillations when a moving boundary is considered Berthelsen andFaltinsen (2008); Pan and Shen (2009); Liao et al. (2010); Luo et al. (2012).In Pan and Shen (2009) the magnitude of the oscillations was reduced by in-creasing the size of the time-step. Liao et al. (2010) reduced the oscillations byintroducing a forcing term within the solid body when solving the momentumequation, however the method is not easy to generalize for other applications.Luo et al. (2012) proposed a treatment for the oscillations that are caused by theinstantaneous change of the computational stencils around the body boundaryas points move in or out of the body in consecutive time-steps. We note thatin the literature the moving bodies were fully submerged in the �uid domain.In our applications the extra complexity of the free surface-body intersectionand the construction of the internal free surface arises. In this chapter the

4.1 Hybrid points method 49

Fluid pointsHybrid pointsGhost pointsDummy points

Figure 4.1: Point classi�cation in the hybrid points method. Second orderoperators are used.

problem of the numerical oscillations will be analysed and di�erent methods forsuppressing the oscillations will be discussed. The e�ect of the methods will beevaluated with the test case of a surging dipole in an in�nite �uid.

4.1 Hybrid points method

In moving boundary problems the role of a grid point varies with time as theimmersed boundary moves in the grid. A �uid point can become a ghost pointin the next time-step as the body moves over it, and in the following time-stepit can become a �uid point again as the body is withdrawn. Consequently, thestencil for the velocity potential at this grid point changes from a WLS stencilsolving the body boundary condition at time level n, to a �nite di�erence stencilfor the Laplace equation at time level n + 1. Both approximations are of thesame order but are associated with truncation errors with di�erent character-istics. This inconsistency in the truncation errors of the potential results to atemporal discontinuity as a point switches role in consecutive time-steps. Sincethe di�erence within the truncation errors of the two approximations is of thesame order as the numerical scheme used, the oscillations can be reduced by in-creasing the order of the schemes and using a very �ne resolution. However, thatwould result in an increased computational cost and thus an attempt to regular-ize the temporal discontinuity by adopting a smooth transition of computationalstencils (Luo et al. (2012)) is adopted. As already described, the computationalpoints are classi�ed as �uid points, ghost points, and dummy points. The �nitedi�erence stencil of �uid points in the vicinity of the immersed boundary caninvolve points that are located inside the solid body (ghost points). We willmark these special �uid points as hybrid points as shown in Figure 4.1 wherethe point classi�cation is illustrated for second order operators. Furthermore,

4.2 Ensuring mass conservation 50

we will try to ensure that the hybrid points undergo a smooth transition from aWLS to a �nite di�erence approximation as they move towards or away from theimmersed boundary. Two solutions, φ1 and φ2, are computed for every hybridpoint. We compute φ1 in the usual way by solving the Laplace problem. Forcomputing φ2 three di�erent approaches have been tried:

� Taking the normal from the hybrid point to the immersed boundary we�nd a body point. We construct a WLS stencil around this body pointthat includes the hybrid point, the closest ghost point to the body point,and �uid/hybrid points. We then solve the Neumann body boundarycondition to obtain the value at the hybrid point.

� We create a WLS stencil around the hybrid point that includes only �uidand hybrid points and we interpolate to �nd φ2.

� We solve the Neumann body boundary condition to �nd the potential onthe body point (intersection of the immersed boundary and the normalfrom the hybrid point). Then we interpolate the potential on the hybridpoint using a WLS stencil that includes hybrid points, �uid points andthe body point.

Since all three methods give similar results the �rst one is adopted. The �nalhybrid solution is computed as

φ = (1− α)φ1 + αφ2, (4.1.1)

where

α =(√dx2 + dy2 − δ)√dx2 + dy2

(4.1.2)

and δ is the distance of the hybrid point from the immersed boundary. Thefactor α speci�es the relative importance of the two solutions and thus thepotential at the hybrid nodes is a weighted average of a WLS and a �nitedi�erence approximation. If the distance of the hybrid point from the immersedboundary becomes larger than the grid spacing metric (δ >

√dx2 + dy2 ) the

factor α is set to zero and 0 ≤ α ≤ 1. In that way the weighting factor satis�esthe following principles.

� As the hybrid point moves away from the boundary and becomes a regular�uid point, α approaches zero.

� As the hybrid point moves close to the boundary and becomes a ghostpoint, α approaches one.

4.2 Ensuring mass conservation

To ensure that there is no loss or gain of �uid while the body is moving weexplored the idea of imposing both the Neumann body boundary condition andthe Laplace equation on the body boundary. Two approaches were implementedand tested.

In the �rst approach we use the extra degrees of freedom of the ghost pointsto impose both the Neumann body boundary condition and the Laplace equation

4.3 Surging dipole in an in�nite �uid 51

at the body points. The resulting system of equations is overdetermined sincethe number of equations is larger than the number of unknowns and is solvedfor φ with the Least Squares method.

The second approach is consistent with the treatment of the rest of the struc-tural boundaries of the domain (bottom and side walls). Thus the ghost pointsare used to satisfy the body boundary condition on the immersed boundaryand the body points are added as new degrees of freedom to satisfy the Laplaceequation. These strategies are compared in the next section.

4.3 Surging dipole in an in�nite �uid

To investigate the source of the jump discontinuities in the force signal and toevaluate the e�ect of the di�erent methods for decreasing them, the case of asurging dipole in an in�nite �uid is considered. The dipole potential is de�nedas

φ(x, z) =µ

2π

x− x0(x− x0)2 + (z − z0)2

, (4.3.1)

where µ is the moment of the dipole de�ned as

µ = −2R2πug1, (4.3.2)

and x0, z0 is the coordinates of the centre of the dipole at time t = 0. Thehorizontal velocity of the dipole is de�ned as

ug1 = −Aωsin(ωt) (4.3.3)

xg1 = x0 +Acos(ωt) (4.3.4)

The analytical added mass coe�cient for a circular two dimensional cylinder issurge is

a11 = πρR2. (4.3.5)

and the analytical expression for the hydrodynamic force on the body is

F11 = ug1a11 (4.3.6)

In the �rst test the original IBM formulation is used with no treatment appliedfor damping the oscillations. The two force formulations are compared, namelythe Bernoulli formulation (3.5.2) and the alternative formulation (3.5.10) thatwill be referred to as the Newman formulation, Newman (1977). The amplitudeof the motion is A = 0.3R and the Courant number is Cr = 0.2. The results arepresented in Figure 4.2. In the force signal of the low order scheme (ns = 3, 2ndorder) the discontinuities are very strong but they are damped signi�cantly whena 4th order scheme is used. That indicates that the main cause of oscillationsis the truncation error di�erences in consecutive time-steps. The two forceformulations agree to machine precision as expected.


0 1 2 3 4 5 6−0.5

0

0.5

t

Force

BernoulliAnalytical

(a) Nx = 27, ns = 3.

0 1 2 3 4 5 6−0.5

0

0.5

t

Force

NewmanAnalytical

(b) Nx = 27, ns = 3.

0 1 2 3 4 5 6−0.5

0

0.5

t

Force

BernoulliAnalytical

(c) Nx = 27, ns = 5.

0 1 2 3 4 5 6−0.5

0

0.5

t

Force

NewmanAnalytical

(d) Nx = 27, ns = 5.

Figure 4.2: Force signal vs time for increasing order of accuracy with Cr = 0.2.Two force formulations are compared.

The same test is repeated but now the original IBMmethod is compared withthe three strategies for damping the discontinuities. Second order is used for thiscomparison and the resolution for all four methods is constant. The computedforce for each simulation is plotted against time in Figure 4.3. Unfortunately, theHybrid points method did not solve the problem of truncation error di�erences.Mass conservation violation is not an issue since imposing both the Neumannboundary condition and the Laplace equation at the immersed boundary didnot improve the solution. The jump discontinuities actually got worse when thestrategy of imposing both conditions through the ghost points was used.


0 1 2 3 4 5 6

−0.4

−0.2

0

0.2

0.4

0.6

t

Force

BernoulliAnalytical

(a) Original formulation.

0 1 2 3 4 5 6

−0.4

−0.2

0

0.2

0.4

0.6

t

Force

BernoulliAnalytical

(b) Hybrid points.

0 1 2 3 4 5 6

−0.4

−0.2

0

0.2

0.4

0.6

t

Force

BernoulliAnalytical

(c) Laplace and Neumann BC - overdeter-

mined system.

0 1 2 3 4 5 6

−0.4

−0.2

0

0.2

0.4

0.6

t

Force

BernoulliAnalytical

(d) Laplace and Neumann BC - body points as

extra DOF.

Figure 4.3: Force signal vs time. Di�erent IBM implementations compared.

The convergence of the four IBM implementations was computed on thesame test case and the results are shown in Figures 4.4-4.5. The error wasmeasured as

ε = ||a11,exact − a11,numerical||. (4.3.7)

The expected order of convergence was achieved in all cases. The method inwhich mass conservation was ensured by solving an overdetermined system ofequations underperformed in comparison to the other three methods, resultingin higher errors. The errors and the rate of convergence of the original methodare almost the same as the ones obtained with the Hybrid points method andwith the mass conservation method with body points acting as extra degrees offreedom.

In Pan and Shen (2009) the magnitude of the oscillations was reduced byincreasing the size of the time-step. Our formulation behaves similarly althoughthe reason in not yet understood. When we used a high Courant number (Cr =0.8) the oscillations decreased signi�cantly. An example is shown in Figure 4.6where a low order scheme, with a coarse spatial resolution was used to simulatethe same test case. The comparison with Figure 4.2 where a Cr = 0.2 was used,reveals the great in�uence of the time-step to the quality of the solution.


101

102

10−2

10−1

Nx

Error

BernoulliNewman2nd order


101

102

10−2

10−1

Nx

Error


(b) Hybrid points.

101

102

10−2

10−1

Nx

Error



mined system.

101

102

10−2

10−1

Nx

Error



extra DOF.

Figure 4.4: Di�erent methods for damping the force oscillations compared forns = 3.


101

102

10−4

10−3

10−2

10−1

Nx

Error

BernoulliNewman4th order


101

102

10−4

10−3

10−2

10−1

Nx

Error


(b) Hybrid points.

101

102

10−4

10−3

10−2

10−1

Nx

Error



mined system.

101

102

10−4

10−3

10−2

10−1

Nx

Error



extra DOF.

Figure 4.5: Di�erent methods for damping the force oscillations compared forns = 5.

0 1 2 3 4 5 6−0.5

0

0.5

t

Force

BernoulliAnalytical

(a) Nx = 27, ns = 3.

0 1 2 3 4 5 6−0.5

0

0.5

t

Force

NewmanAnalytical

(b) Nx = 27, ns = 3.

Figure 4.6: Force signal vs time with Cr = 0.8. Two force formulations arecompared.


0 1 2 3 4 5 6−0.5

0

0.5

t

Force

Newman exact-random pointsAnalyticalNewman exact-same points

Figure 4.7: Integrands of Newman's force formulation computed using the an-alytical formulation and integrated numerically.

To stress the in�uence of the di�erences in truncation errors in successivetime-steps, and how challenging is to obtain a smooth solution, the force wascomputed numerically through Newman's formulation (3.5.2) but the integrandswhere calculated through the analytical solution (4.3.1). To clarify, the analyt-ical solution was used at each time-step to compute φ and its derivatives onpoints distributed on the body boundary. The integrals and the time derivativeof the force expression were calculated numerically with Taylor integration andthe �nite di�erence method respectively. When the distribution of points wasconstant during the simulation the resulting force was smooth. If the distribu-tion of points changed slightly during the simulation, mild oscillations occurredin the force as shown in Figure 4.7.

To rule out any in�uence of the WLS method on the jump discontinuities, weconverted the nonlinar terms of the force expressions into the normal-tangentialcoordinate system and used �nite di�erence to compute them. The normalcomponent is known exactly. The tangential component was computed with a1D �nite di�erence scheme based on the values of the potential on the bodypoints. The two methods are compared in Figure 4.8 and their di�erences areon the level of truncation error.

Concluding, the main source of the jump discontinuities when the bodyis fully submerged is truncation error di�erences between di�erent time-steps.The only e�ective way to dampen them at this point is to increase the spatialresolution, increase the time-step and use higher-order schemes. Both forceformulations perform identically to machine precision and there is no advantageof one over the other.


0 1 2 3 4 5 6−0.5

0

0.5

t

Force

Newman 1D FDNewmanAnalytical

Figure 4.8: Nonlinear terms of the force formulation computed with a 1D �nitedi�erence scheme vs the usual WLS approximation.

Chapter 5

Forced motion�uid-structure interaction

cases

The complete algorithm is tested on a series of forced motion test cases andis validated against benchmark results from the literature. There are severalchallenges in getting a smooth force signal that are connected with the ImmersedBoundary method and the arti�cial internal free surface. A detailed analysisfollows.

5.1 Linearised case of a heaving cylinder

As a �rst step the methodology is validated on the linearised radiation problemof a heaving half-submerged cylinder as by Read and Bingham (2012). Thebody displacement follows a Gaussian pro�le and introduces a Gaussian rangeof frequencies. The displacement pro�le is determined by the following analysis.The minimum wavelength that a given grid can resolve is λmin = 2∆x. Thedeep-water dispersion relation can then be used to determine the maximumfrequency associated with this minimum wavelength

fmax =1

2π

√gk =

√g

2πλmin

. (5.1.1)

The deep-water dispersion relation is used since kh > 2π in the case we areconsidering. In the frequency domain the amplitude of the body displacementis de�ned as a normalized Gaussian function centred at the origin

c(f) =1√2πs

exp

(− f2

2s2

), (5.1.2)

where s is the standard deviation in the frequency domain. In order to set themagnitude of c(fmax) equal to a de�ned proportion of c(0), s has to be de�ned

5.1 Linearised case of a heaving cylinder 59

as

s =

√− f2max

2ln(p), (5.1.3)

where p is the amplitude at fmax as a proportion of the maximum value. Thevalue of p is selected arbitrarily as 0.01% to ensure that the added mass anddamping pro�les associated with the coarsest grid appear correct. The non-normalized Gaussian function in the time domain, computed by taking theinverse Fourier transform of (5.1.3), is

ξ3(t) = exp(−2π2s2t2

)= exp

(− t2

2σ2

), (5.1.4)

where σ is the standard deviation in the time domain

σ =1

2πs. (5.1.5)

The Gaussian function is then o�set in the time domain by

t0 = 8σ =4

πs, (5.1.6)

to ensure that the value of the displacement is zero at t = 0:

ξ3(t) = exp[−2π2s2(t− t0)2

]. (5.1.7)

This displacement function is used for the simulation. The depth is de�ned ash = 2R where R is the radius of the cylinder. The period of the sample is setas ts = 6t0. This choice ensures that the sample is long enough to capture thefull response. The time step is de�ned as

∆t = C∆x

umax

, (5.1.8)

where C is the Courant number and umax is the celerity of the fastest wavecomponent. The overall length of the numerical tank is de�ned as

Lx = 1.25(umaxts + 2R), (5.1.9)

to ensure that the fastest wave component won't return to the body surfacebefore t = ts. The cylinder is located mid-length of the tank. The added massand damping coe�cients have been non-dimensionalized as

µ33 =a33

12πρR

2, (5.1.10)

ν33 =b33

12πρωR

2. (5.1.11)

The excitation pro�le in the time domain and the computed normalized forceresponse are visualized in Figure 5.1. The added mass and damping coe�cientsare calculated by taking the FFT of these signals

ω2a33(ω)− iωb33(ω) =F{F33(t)}F{ξ3(t)}

. (5.1.12)

The computed hydrodynamic coe�cients are compared with the analytical re-sults of Bai (1977) in Figure 5.2 and good comparison is found.

5.1 Linearised case of a heaving cylinder 60

0 0.5 1 1.5 2 2.5 3−3

−2

−1

0

1

2

3

4

t/t0

amplitude

ξ3/RF33/(

12πρgR2)

Figure 5.1: Method of excitation and computed force response in the linearisedheaving cylinder case.

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

kR

amplitude

µ33ν33µ33 exactν33 exact

Figure 5.2: Comparison of the analytical and numerical hydrodynamic coe�-cients of the linearised heaving cylinder case.

5.2 Fully nonlinear heaving cylinders with various cross-sections 61

5.2 Fully nonlinear heaving cylinders with vari-

ous cross-sections

In this section a comparison of our numerical model with the experimental workconducted by Yamashita (1977) is attempted. Yamashita (1977) tested severalforced heaving 2D cylinders with di�erent cross-sections including an ellipticalcylinder, a circular cylinder, a stern-section cylinder and a wedge. He obtainedthe force harmonics up to third order by �tting the force signal with the followingFourier series

Fz = F (0)z +

3∑n=1

F (n)z sin(nωt+ ψ(n)). (5.2.1)

The �rst order force, expressed in terms of the added mass and damping coef-�cients is non-dimensionalized as

µ33 =a33ρa2

, (5.2.2)

ν33 =b33

ρa2√g/a

, (5.2.3)

where a is the half beam of the cylinder. The higher harmonics are non-dimensionalized in the following form

F(2)z

2ρga2A2,

F(3)z

2ρga2A3, (5.2.4)

where A is the amplitude of the motion. The prescribed motion of the cylinder'scenter of mass is

zg3(t) = zg3(0) +D(t)A sin(ωt), (5.2.5)

where D(t) is a ramp function de�ned as

D(t) =1

2+

1

2tanh

(10t− tramp

2

tramp

). (5.2.6)

The cylinder is set gradually in motion to avoid high wave number instabilities.The ramp-up time is tramp = 3T . The computational domain length is Lx = 12λand the cylinder is located mid-length.

5.2.1 Small amplitude motion of circular cylinder

First a small amplitude motion of a circular cylinder is considered where A =0.01R. The half-beam a in this case is the cylinder's radius R. The angularfrequency is selected such that ω2R/g = 1. The sea bottom is at h = λ. Forcomparison with the results of Kent (2005) the cylinder radius is set to R = 1 m,the gravitational acceleration to g = 1 m/s2 and the density to ρ = 1 kg/m3.The added-mass and damping results for two discretizations Nf = λ/dx arecompared with reference values from the literature (Porter (1960); Frank (1967);Kent (2005)) in the following table:


µ33 ν33Nf = 40 0.601 0.390Nf = 60 0.603 0.396Porter (1960) 0.58 0.41Frank (1967) 0.62 0.40Kent (2005) 0.582 0.410

Table 5.1: Added-mass and damping comparison for A = 0.01R

where the added mass and damping coe�cients are non-dimensionalized as

µ33 =a33ρS0

, (5.2.7)

ν33 =b33

√Rg

ρS0, (5.2.8)

where S0 = 12πR

2 is the mean submerged area of the cylinder. The solution isconverged and the agreement with the literature is reasonably good.

5.2.2 Large amplitude motion of a circular cylinder

The same test case is repeated for a large amplitude motion A = 0.2R, anda comparison is made with the experimental work conducted by Yamashita(1977). The �rst order forces are also compared with the predictions of the the-oretical model of Tasai and Koterayama (1976) that is based on linear theory.The predictions of the weakly nonlinear model of Lee (1968) that is based on theperturbation method, were used as reference for the higher order forces . Theresults for the added mass and damping coe�cients are presented in Figure 5.3and the comparison of the higher order forces is made in Figure 5.4. For the �rstorder forces good agreement is found in the mid to high frequency range withboth the theoretical and the experimental results. For lower frequencies thereis a discrepancy with the experiments, mainly in the damping coe�cient. Thatcould be due to viscous e�ects that have a stronger in�uence on the dampingcoe�cient. Satisfactory accuracy is found in the prediction of the second har-monic. The third harmonic compares well with the experiments for mid to lowfrequencies. A deviation is found in the higher frequencies. Kashiwagi (1996)notes that this could be due to lack of accuracy in the experiments due to thesmall absolute values of the third harmonic.

It is noted that for the range of frequencies and amplitudes considered, theoscillations were suppressed by su�cient resolution and a Courant number ofCr = 0.8. The computed force signal was basically oscillation-free as shownin Figure 5.5. Unfortunately, for shorter waves or higher amplitudes the simu-lations break down due to very large gradients developing in the interior free-surface. If at any time the internal free surface develops a discontinuity in its�rst or second horizontal derivative the system matrix becomes ill-conditionedleading to a discontinuous force signal.

5.2.3 Large amplitude motion of an elliptic cylinder

An elliptic cylinder with half beam over draft ratio (a/d = 0.7) is heavedwith amplitude A = 0.2a. The results of the fully nonlinear Mixed Eulerian-


0 0.5 1 1.5 20

0.5

1

1.5

ω2

g R

amplitude

µ33 Yamashita (1977)µ33 Tasai & Koterayama (1976)µ33 computedν33 Yamashita (1977)ν33 Tasai & Koterayama (1976)ν33 computed

Figure 5.3: Hydrodynamic coe�cients comparison for a heaving �oating cir-cular cylinder with A = 0.2R.

0 0.5 1 1.50

0.5

1

1.5

ω2

g R

F(2)

z/(2ρgR

2A

2)

Yamashita (1977)Lee(1968)Computed

(a) 2nd harmonic.

0 0.5 1 1.50

0.5

1

1.5

ω2

g R

F(3)

z/(2ρgR

2A

3)

Yamashita (1977)Lee(1968)Computed

(b) 3rd harmonic.

Figure 5.4: Non-dimensional 2nd and 3rd harmonics for a heaving �oating cir-cular cylinder with A = 0.2R.

5.3 Fully nonlinear submerged heaving cylinder 64

0 1 2 3 4 5 6

−0.2

−0.1

0

0.1

0.2

0.3

t/T

Force

BernoulliNewman

Figure 5.5: Vertical force on a heaving circular �oating cylinder with A = 0.2R

and ω2Rg = 0.78.

Lagrangian potential �ow model of Kashiwagi (1996) are used for comparison ofthe �rst order forces additionally to Yamashita's experiments. The comparisonis shown in Figure 5.6 and good agreement is found with the numerical modelof Kashiwagi. The agreement with the experiments is relatively good with thelarger deviation found in low frequencies. The higher order forces are shownin Figure 5.7. The second order force is captured with satisfactory accuracy.There is a large scatter in the experimental measurements of the third orderforce but the general tendency is predicted by the numerical model.

5.2.4 Large amplitude motion of a wedge shaped cylinder

A wedge shaped cylinder with half beam over draft ratio a/d = 0.4 (see Figure5.8) was tested in heave with amplitude A = 0.2a. The computed added massand damping coe�cients are shown in Figure 5.9 and the higher order hydro-dynamic forces in 5.10. Good agreement with the model by Kashiwagi (1996)is found. The comparison of the added mass and damping coe�cients with Ya-mashita's experiments is satisfactory with the larger discrepancy found again inthe lower frequencies. The second order force is overpredicted compared to theexperiments. The qualitative tendency of the third order force is captured byour model but the magnitude is underpredicted.

5.3 Fully nonlinear submerged heaving cylinder

The radiation problem of a fully submerged heaving circular cylinder is consid-ered. Wu (1993) solved this problem analytically by linearising the free surfaceboundary conditions and satisfying the body boundary condition at its instan-taneous position. The radiated wave potential was expressed as a multipoleexpansion and the wave number of the radiated waves was assumed to satisfy


0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

ω2ag

amplitude

µ33 Yamashita (1977)µ33 Kashiwagi (1996)µ33 computed

(a) Added mass coe�cient.

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

ω2ag

amplitude

ν33 Yamashita (1977)ν33 Kashiwagi (1996)ν33 computed

(b) Damping coe�cient.

Figure 5.6: Hydrodynamic coe�cients for a heaving elliptic cylinder (a/d =0.7) with A = 0.2a.

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ω2ag

amplitude

F(2)z Yamashita (1977)

F(2)z computed

(a) 2nd harmonic.

0 0.2 0.4 0.6 0.8 1 1.20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

ω2ag

amplitude


F(3)z computed

(b) 3rd harmonic.

Figure 5.7: Non-dimensional 2nd and 3rd harmonics for a heaving elliptic cylin-der (a/d = 0.7) with A = 0.2a.

d

a

Figure 5.8: Wedge section with half beam over draft ratio a/d = 0.4.


0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

ω2ag

amplitude

µ33 Yamashita (1977)µ33 Kashiwagi (1996)µ33 computed

(a) Added mass coe�cient.

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

ω2ag

amplitude

ν33 Yamashita (1977)ν33 Kashiwagi (1996)ν33 computed

(b) Damping coe�cient.

Figure 5.9: Hydrodynamic coe�cients for a heaving wedge (a/d = 0.4) withA = 0.2a.

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

ω2ag

amplitude


F(2)z computed

(a) 2nd harmonic.

0 0.2 0.4 0.6 0.8 1 1.20

0.05

0.1

0.15

0.2

0.25

0.3

ω2ag

amplitude


F(3)z computed

(b) 3rd harmonic.

Figure 5.10: Non-dimensional 2nd and 3rd harmonics for a heaving wedge(a/d = 0.4) with A = 0.2a.


6 6.5 7 7.5 8 8.5 9 9.5 10

−0.4

−0.2

0

0.2z

x

6 6.5 7 7.5 8 8.5 9 9.5 10

−0.4

−0.2

0

0.2

z

x

6 6.5 7 7.5 8 8.5 9 9.5 10

−0.4

−0.2

0

0.2

z

x

Figure 5.11: Forced heaving motion for A = 1.75R, kR = 1. Successive snap-shots at t/T = 3.2, 3.4, 3.7.

the in�nite water linear dispersion relation k = ω2/g. Eight amplitudes of mo-tion were considered and the largest amplitude case A = 1.75R is visualized atFigure 5.11. The non-dimensional wave number was set to kR = 1. A numericalwave tank of Lx = 16 m and h = 3 m was used for the simulations, with thecylinder of radius R = 0.1 m located mid-length. The mean position of submer-gence of the cylinder was zg3(0) = −3R under the undisturbed free surface. Theramp function (5.2.6) was used, and the ramp-up time was set to tramp = 3T .After reaching a steady state, the time history of the non-dimensional verticalforce was �tted with the following Fourier series

FzρAπR2ω2

= F (0)z +

2∑n=1

F (n)z sin(nωt+ ψ(n)). (5.3.1)

In the very nonlinear cases where the amplitude was larger than A = 1.25R thefree surface on top of the cylinder developed saw-tooth instabilities. Althoughthese instabilities were dampened by the WENO scheme and the simulations didnot crash, the necessary ∇ζ, ∇2ζ sigma transform terms became discontinuousand caused oscillations in the force signal. A 11-point, 9-th order Savitzky-Golaytype �lter was applied once per time-step to the elevation and the free surfacepotential to prevent these instabilities from developing. After the application ofthe �lter a smooth force signal was obtained. The computed results of the mean,�rst and second harmonics of the force are compared with the analytical resultsof Wu (1993) in Figures 5.12-5.13. The agreement is relatively good for A/R <


0.5 1 1.5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

A/R

Fz(0)

ComputedWu (1993)

Figure 5.12: Non-dimensional zero-th (mean) harmonic of the vertical forcefor kR = 1.

1 but the computed solution deviates from the analytical solution for higheramplitudes. This could be accredited to the highly nonlinear waves (H/L = 0.11for A/R = 1.75) that are generated when the amplitude is increased and play asigni�cant role in the solution.


0.5 1 1.50.7

0.8

0.9

1

1.1

1.2

A/R

Fz(1)

ComputedWu (1993)

(a) 1st harmonic.

0.5 1 1.5−0.1

0

0.1

0.2

0.3

0.4

A/R

Fz(2)

ComputedWu (1993)

(b) 2nd harmonic.

Figure 5.13: Non-dimensional 1st and 2nd harmonics of the vertical force forkR = 1.

Chapter 6

Forward speed�uid-structure interaction

cases

6.1 Steady streaming �ow around a translating

submerged circular cylinder

The case of a submerged circular cylinder steadily translating through calmwater is considered. The steady waves generated downstream the cylinder arecomputed and a comparison is made with the fully nonlinear solution of Scullenand Tuck (1995) that is based on the Rankine panel method. The cylinder hasa non-dimensional radius of R/zg = 0.2, where zg is the submergence depthof it's centre beneath the undisturbed free surface. The numerical tank has alength of Lx = 160 m and the centre of the cylinder is located at xg = 0. ThreeFroude numbers are considered

F =U√gzg

= 0.4, 0.8, 1.2. (6.1.1)

The forward speed velocity U and gravity g are held constant at 1. The Froudenumber varies by changing the submergence depth of the cylinder. The freesurface pro�les for the three cases are shown in Figure 6.1 and the comparisonwith the solution of Scullen and Tuck (1995) is satisfactory. The highest wavesare found in the medium Froude number case and they are signi�cantly non-sinusoidal. Furthermore, the strong depression of the free surface ahead of thecylinder for low Froude numbers changes to a weak elevation as the Froudenumber increases.

6.1 Steady streaming �ow around a translating submerged circular cylinder71

−15 −10 −5 0 5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

x

z

ComputedScullen & Tuck (1995)

(a) Fr = 0.4

−15 −10 −5 0 5

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x

z


(b) Fr = 0.8

−15 −10 −5 0 5

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

x

z


(c) Fr = 1.2

Figure 6.1: Steady waves produced by a translating cylinder of R/zg = 0.2 atvarious speeds.

Chapter 7

Conclusions

A fully nonlinear potential �ow �nite di�erence wave-structure interaction solverin two dimensions has been developed. The body boundary condition is sat-is�ed with a novel IBM-WLS method that is very e�cient in terms of gridgeneration. The disadvantage of the method is that it requires high resolutionaround the body boundary for minimizing the truncation error di�erences be-tween consecutive time-steps that cause temporal discontinuities. However, inthree dimensional problems the size of the body is much smaller compared tothe size of the numerical wave tank and therefore we don't expect big sacri�cesin the overall e�ciency of the model.

A �nite di�erence WENO scheme that automatically handles the upwindingof the convective terms in the free surface boundary conditions of the forwardspeed problem was derived. An alternative derivation of the linear WENOweights that is simple to implement numerically was presented. Furthermore,a simpli�ed smoothness indicator that is evaluated numerically was proposedand was shown to perform equivalently with the tabulated versions found in theliterature in several test cases. The complete scheme was found to be robust forall combinations of wave speed to frame of reference speed and the dissipationintroduced was as low or lower than the equivalent order upwind biased scheme.

The solver was tested on the zero speed radiation problem and the forwardspeed steady problem using simple geometries. The �rst order forces were pre-dicted with satisfactory accuracy. The second and third order forces comparedless well with the experimental results, but in most cases the qualitative ten-dency was captured correctly.

The limitations of the solver for calculating wave-structure interaction prob-lems with very high amplitudes of motion or very short wave lengths are relatedto the construction of the internal free surface that is at the moment the weaklink of the algorithm. When the free surface develops steep gradients near thebody boundary it is very challenging to compute a well behaved internal freesurface that respects C2 continuity. In extreme cases the calculations breakdown. Milder discontinuities on the internal free surface or its �rst and sec-ond horizontal derivatives lead to jump discontinuities in the computed forces.

73

By �ltering the free surface near the body we obtained some preliminary re-sults for higher amplitudes of motion that look promising. However, these arenot presented here since the algorithm is not robust for every discretization.Further re�nement is needed in adjusting the strength of the �lter to applyonly the amount of numerical di�usion needed. Furthermore, �ltering near thebody boundary was found to in�uence signi�cantly the location of the body-free surface intersection. The intersection was found to move non physically inconsecutive time steps so further development and tuning is required. For veryhigh amplitudes of motion a wave breaking model will also need to be devel-oped. WENO is designed to apply the minimum numerical di�usion requiredto maintain a stable and non-oscillatory solution while keeping the shock pro-�les as sharp as possible. Therefore, a dissipative mechanism that captures thecorrect energy loss of wave breaking needs to be included. As a next step thefree motion problem should be considered where the solver needs to be coupledwith Newton's law.

To extend the solver to three dimensions a 2D internal free surface that isG2 continuous has to be computed in an e�cient way. This problem is verycommon in computer aided geometric design and computer graphics where sur-faces need to be blended by constructing a new surface with G2 continuity atthe boundaries. For example in Liu and Xu (2007) a sixth order geometric par-tial di�erential equation (PDE) is derived by minimizing a curvature integralfunctional. This PDE is then solved with the �nite di�erence method to obtaina surface with high order continuity. However, this procedure is de�nitely nottrivial and it might be worth considering either to abandon the sigma transfor-mation or to instead abandon the generation of the G2 continuous internal freesurface and instead simply evaluate all the necessary derivatives one-sided onthe �uid side, setting the internal values to zero.

Appendix A

Derivation of the internalfree surface polynomial

A.1 Closed form coe�cients

The internal solution ζ(x) is expressed as a seventh order polynomial in thelocal variable x running from xA = −1 to xB = 1 across the body

ζ(x) =

7∑i=0

cixi, xA ≤ x ≤ xB . (A.1.1)

We want to satisfy three boundary conditions at each intersection point, corre-sponding to the value, �rst and second derivative of ζ(x):

ζ(xA) =

7∑i=0

cia1,i = b1 (A.1.2a)

ζx(xA) =

7∑i=0

cia2,i = b2 (A.1.2b)

ζxx(xA) =

7∑i=0

cia3,i = b3 (A.1.2c)

ζ(xB) =

7∑i=0

cia4,i = b4 (A.1.2d)

ζx(xB) =

7∑i=0

cia5,i = b5 (A.1.2e)

ζxx(xB) =

7∑i=0

cia6,i = b6, (A.1.2f)

where for example a1,i = xiA and a2,i = ixi−1A for i = 0, . . . , 7.

A.2 Numerically optimized coe�cients 75

The two extra degrees of freedom are used to minimize the integral of thepolynomial's squared distance from the straight line ζ0(x) that connects xA andxB

δ =

∫ xB

xA

(ζ(x)− ζ0(x))2

dx (A.1.3)

where

ζ0(x) = ζ(xA) +ζ(xB)− ζ(xA)

xB − xA(x− xA). (A.1.4)

The minimum is given by setting the derivative of δ with respect to each extradegree of freedom to zero

∂δ

∂c6=

7∑i=0

cia7,i = 0 (A.1.5a)

∂δ

∂c7=

7∑i=0

cia8,i = 0. (A.1.5b)

By collecting Equations (A.1.2), (A.1.5) in a linear system we can solve for thepolynomial's coe�cients.

A.2 Numerically optimized coe�cients

The polynomial coe�cients can also be determined numerically by solving aconstrained minimization problem of either the arc-length, or the integral ofthe second or third derivative along the interior free surface while satisfying theboundary conditions at the intersections (ζ, ζx, ζxx). For the solution of thisproblem Matlab was used and speci�cally the fmincon function:

c=fmincon(f,c_0,[],[],[],[],[],[],nonlcon,options);

The closed form coe�cients c0 are used as the initial condition. We will com-pute the coe�cients c that minimize the function f subject to the boundaryconditions at the intersection points de�ned in nonlcon.

The boundary conditions on the internal free surface points are expressed asthe 6× 6 system of equations

Aζ = b. (A.2.1)

The vector b holds the boundary conditions at the intersection points xA andxB . The matrixA holds the one-sided �nite di�erence coe�cients of the interiorgrid points.

The function f is the metric that is being minimized and can be either thearc-length, or the integral of the second or third derivative along the interiorfree surface. The derivatives are computed with the �nite di�erence methodand the integrals with the trapezoid rule.

Appendix B

WENO coe�cients

B.1 Numerical implementation of the linear weights

The numerical implementation of the derivation of the linear WENO coe�cientsds is presented below as a Matlab function.

function [d,c,C]=WENO_d_coefficients(r,x,x0)%% Find the factors d(1:r) which combine the r first−derivative schemes% around x0 to give a k=2*r−1 accurate scheme using r−1 points to either% side.%k=2*r−1;%C=zeros(k,r);%% Build all the possible r−point derivative schemes around x0.%for j=1:r % loop over stencils

is=j:j+r;% Calculate the finite difference coefficientsc(j).c= FornbergInterpolate(x0,x(is),r+1,r);C(is,j)=c(j).c(2,:);

end%% Collect the r to k Taylor coefficients in the Taylor expansion of each% stencil point.%for m=r+1:k

for j=1:k+1M(m−r,j)=(x(j)−x0)^m/factorial(m);

endend%% Sum up the Taylor coefficients for each scheme as the first r rows of% the d−coefficient matrix.%A=M*C;

B.2 Numerical implementation of the nonlinear weights 77

%% Add in the condition that sum(d)=1%A(r,:)=ones(1,r);%% Build the RHS and solve for the d coefficients.%b=zeros(r,1); b(r,1)=1;d=A\b;

The �nite di�erence weights are calculated with the following function.

function c=FornbergInterpolate(x0,x,n,m)%% A matlab implementation of the algorithm of%% B. Fornberg, 1998. "Calculation of weights in finite difference% formulas". SIAM Rev. Vol. 40, No. 3, pp. 685−691.%% for calculating arbitrary−order FD schemes on arbitrary 1−D grids for% all applicable derivatives. The interpolations are found at the point% x0, using the function positions x(1:n), for derivatives 0:m, m<=n−1;% and returned in: c(1:m+1,1:n).%if m>n−1, error('Too many derivatives for this stencil, try again.');end%c1=1; c4=x(1)−x0;c=zeros(n,m+1);c(1,1)=1;for i=2:n

mn=min(i,m+1); c2=1; c5=c4; c4=x(i)−x0;for j=1:i−1

c3=x(i)−x(j);c2=c2*c3;if j==i−1

for k=mn:−1:2c(k,i)=c1*((k−1)*c(k−1,i−1)−c5*c(k,i−1))/c2;

endc(1,i)=−c1*c5*c(1,i−1)/c2;

endfor k=mn:−1:2

c(k,j)=(c4*c(k,j)−(k−1)*c(k−1,j))/c3;endc(1,j)=c4*c(1,j)/c3;

endc1=c2;

endreturn

B.2 Numerical implementation of the nonlinear

weights

The simpli�ed smoothness indicator allows for a simple numerical implementa-tion of the nonlinear WENO weights. The followingMatlab function calculatesthe derivative df at the evaluation point of the full 2r WENO stencil by eval-uating the nonlinear weights numerically. The input variables are the linearweights ds, the �nite di�erence coe�cients c of all available substencils, a vector

B.2 Numerical implementation of the nonlinear weights 78

f that contains the 2r function values and the uniform grid spacing dx. Fornon-uniform grids a coordinate transformation is used.

function [ df ] = WENO_dfdx1D(d,c,f,dx)%% Take the derivative df using the stencil f(1:2*r) and the% FD coefficients on all possible stencils held in the structure% c(1:r).c(1:r,1:r); together with the linear coefficients d(1:r)% which weigh the r first−derivatives to give an O(2*r−1) scheme.% The coefficients are assumed to have been built on a unit−spaced grid.%k=length(f); r=(k)/2;%% The eps value to keep the coefficients from becoming singular at% zeros of the derivatives.%epsr=10^−6;%for js=1:r % loop over the r stencils

%% Find the coefficients beta_r, alpha_r=d_r/(eps+beta_r)^2, and% the weights omega_r=alpha_r/sum(alpha) at this point on the grid.%% The smoothness indicators, beta_r are the sum of the squares of% all possible higher derivatives on the stencil (scaled and divided% by dx).beta(js)=0;for jd=2:r

beta(js)=beta(js)+(c(js).c(jd+1,:)*f(js:js+r,1)/dx)^2;end%alpha(js)=d(js)/(epsr+beta(js)).^2;% The first−derivative on this stencildf_r(js,1)=c(js).c(2,:)*f(js:js+r,1)/dx;

endalphaS=sum(alpha); omega=alpha/alphaS;% Calculate the derivative by combining the substencil approximations% with the non−linear weightsdf=omega*df_r;%end

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