Study of the SPH method for the simulation of regular and ......In this thesis, the SPH method, as implemented in LS-Dyna , is studied for the simulation of free surface water ow,

Study of the SPH method for simulation

of regular and breaking waves

Joren Pelfrene

Promotoren: prof. dr. ir. Wim Van Paepegem , prof. dr. ir. Jan Vierendeels

Begeleider: ir. Kameswara Sridhar Vepa

Masterproef ingediend tot het behalen van de academische graad van

Master in de ingenieurswetenschappen: Werktuigkunde-Elektrotechniek

Vakgroep Toegepaste materiaalwetenschappen

Voorzitter: prof. dr. ir. Joris Degrieck

Vakgroep Mechanica van Stroming, Warmte en Verbranding

Voorzitter: prof. dr. ir. Roger Sierens

Faculteit Ingenieurswetenschappen

Academiejaar 2010–2011

i

Study of the SPH method for simulation

of regular and breaking waves

Joren Pelfrene

Promotoren: prof. dr. ir. Wim Van Paepegem , prof. dr. ir. Jan Vierendeels

Begeleider: ir. Kameswara Sridhar Vepa







Faculteit Ingenieurswetenschappen


Permission for use of content

The author gives the permission to use this thesis for consultation and to copy parts of it

for personal use. Every other use is subject to copyright law, more specifically the source

must be extensively specified when using from this thesis.

Joren Pelfrene, August 2011

i

Toelating tot bruikleen

De auteur geeft de toelating deze scriptie voor consultatie beschikbaar te stellen en delen

van de scriptie te kopieren voor persoonlijk gebruik. Elk ander gebruik valt onder de

beperkingen van het auteursrecht, in het bijzonder met betrekking tot de verplichting de

bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uit deze scriptie.

Joren Pelfrene, augustus 2011

ii

Acknowledgements

The completion of this thesis has only been possible with the help of several people.

Firstly, I would like to thank prof. dr. ir. Wim Van Paepegem en prof. dr. ir. Jan

Vierendeels for giving me the opportunity to work on this fascinating subject. I am espe-

cially appreciative to prof. Van Paepegem for always being supportive for the continuation

of the thesis during my stay in Portugal.

I owe much gratitude to my supervisor Sridhar Vepa, who provided the necessary guidance

and was available whenever I had questions, whether at the university in Ghent or on a

long-distance call.

I also want to thank prof. dr. ir. Carlos Guedes Soares and prof. dr. ir. Marc Vantorre

who gave me the opportunity to spend one year of my studies abroad, which has been a

valuable experience.

Finally, many thanks go to my parents who have always helped and supported me, for the

long run, and in particular during the last month of writing this thesis back in Ghent.

Joren Pelfrene, August 2011

iii

Study of the SPH method for thesimulation of regular and breaking waves

by

Joren Pelfrene



Promotoren: prof. dr. ir. Wim Van Paepegem, prof. dr. ir. Jan Vierendeels

Scriptiebegeleiders: ir. Kameswara Sridhar Vepa





Faculteit Ingenieurswetenschappen en Architectuur

Universiteit Gent


Abstract

In this thesis, the SPH method, as implemented in LS-Dyna, is studied for the simulationof free surface water flow, with a focus on regular and breaking waves. The aim is tocreate a framework for the numerical modelling of sideways wave impact. First, the dambreak problem is studied as a validation test case to identify the merits and demerits ofthe method and model. Then, a numerical wave flume is designed for the simulation ofregular waves, and finally, breaking waves are modelled by two types of generation. Ingeneral, it is found that the SPH solver in LS-Dyna is able to simulate free surface flow,and to capture the main features of plunging breaking waves. However, improvements tothe code can be made, in particular to avoid the much-recurring artificial clustering.

Keywords

Smoothed Particle Hydrodynamics (SPH), Free Surface Flow, Breaking Waves, Dam

Break

Study of the SPH method for simulation of regularand breaking waves

Joren Pelfrene

Supervisor(s): Kameswara Sridhar Vepa, Wim Van Paepegem, Jan Vierendeels

Abstract— In this article, the SPH method, as imple-mented in LS-DYNA, is studied for the simulation of freesurface water flow, with a focus on regular and breakingwaves. The aim is to create a framework for the numericalmodelling of sideways wave impact. Dam break is studied asa validation test case. A numerical wave flume is designedfor the simulation of regular waves, and finally, breakingwaves are modelled by two types of generation. In general, itis found that the SPH solver in LS-DYNA is able to simulatelaminar free surface flow, and to capture the main featuresof plunging breaking waves. However, improvements to thecode can be made, in particular to avoid artificial clustering.

Keywords— Smoothed Particle Hydrodynamics (SPH),Free Surface Flow, Breaking Waves, Dam Break

I. INTRODUCTION

THE origin of this thesis is found in the contributionof Ghent University to the FlanSea project, which

aims to generate electricity from sea waves, at a price thatcan compete with classical methods for energy genera-tion. For this purpose a new type of wave energy converter(WEC) has been developed, consisting of a free-floatingbuoy, termed point absorber, that is connected to the seabottom by a cable. Up and down movement of the pointabsorber is then transformed to electricity.

Ghent University was responsible for the design of thepoint absorbers, which are to be able to withstand also themost rough sea condition. The most critical load on thebuoy is found when wave impact, either downward or side-ways, occurs. While downward slamming has frequentlybeen simulated by means of a drop test, sideways waveimpact has not.

Numerical modelling of free surface water flow is dif-ficult, especially when large deformations and fragmenta-tion are likely to be present. Meshed methods, both Eu-lerian and Lagrangian, can be used, but are seen to sufferfrom significant numerical errors. The SPH method rep-resents the state of a system by a set of discrete particleswithout fixed connectivity, followed in a Lagrangian man-ner, and no special approach to the free surface is needed.

The present study aims to put up a framework for the

simulation of sideways wave impact by exploring the SPHtechnique for numerically generating regular and breakingwaves. This is performed in the commercially availableLS-DYNA software, which includes both finite-elementand SPH solvers, and enables fluid-structure interaction(FSI).

II. VALIDATION TEST: DAM BREAK

A test case is executed to identify the merits and demer-its of the model and method. The dam break problem isan idealised case where a cuboid of water is exposed togravity as from t = 0 s. A variation of this test case isproposed by SPHERIC [1], modelled after an experimentat MARIN, where a box was placed at the center of the do-main. Detailed experimental data for the water level andpressure measurements is made available by SPHERIC.Figure 1 shows that the evolution of the water level at asensor location in the reservoir shows very good corre-spondence between the SPH model and the experimentaldata as long as the flow is laminar. After the initial stagegreat divergence is seen in the curve. This can, in part, beascribed to the SPH model being single-phase, which im-pedes the simulation of air-entrapment in a foam layer atthe water surface.

Fig. 1. Water level at a height probe in the reservoir

For the pressure data, it was found that the water pres-sure on a fixed surface is not easily obtained for the SPH in

LS-DYNA. A very fine particle size is required, and eventhen the results show only rough correspondence. Artifi-cial clustering, a typical numerical error in SPH, is foundto be responsible for extra noise in the pressure results.Clustering occurs when two or more particles clump to-gether to form one big particle with high mass and den-sity. The origin for this unphysical behaviour is found inthe smoothing kernel function having an inflection point,whereby the particles experience a maximum in the forceby which they repel each other.

III. SIMULATION OF REGULAR WAVES

A numerical wave flume is designed for the simulationof regular waves. Two special boundary conditions areapplied: waves are generated by a rotating paddle wave-maker, hinged at the bottom, and the water wave energyis absorbed by a sponge layer. This concept is illustratedin figure 2. Effective wave absorption is achieved in LS-DYNA by assigning a mass-proportional damping to theSPH particles in the sponge layer. For the reference modeldepicted in figure 2, it is seen that a steady state progress-ing wave field is reached after 5 to 6 wavemaker periods.

0

2

x(m)

y(m)

5 10 20 2515

Sponge layerFluid domain

1

Fig. 2. Concept of numerical wave flume. Snapshot taken whensteady state is reached for waves with period T = 2 s

The wave height generated in the vicinity of the wave-maker corresponds fairly well to the value predicted byGalvin’s wavemaker theory [2]. In the reference model,a wave height of 0.55m is reported, whereas the theoryexpects 0.58m. However, further downstream, a loss ofwave heigth is noted, with a value as low as 0.30m beforethe sponge layer.

IV. SIMULATION OF BREAKING WAVES

Two types of models are tested for wave breaking. Inthe first type, an overturning wave front is created period-ically in the direct vicinity of the wavemaker, much like abreaking bore. The second type generates a wave, whichbreaks in a plunging manner farther downstream as a re-sult of the bottom geometry. For both types, the breakingpoint occurs as expected from the kinematic breaking cri-terion, and the simulations show that the SPH method inLS-DYNA is able to capture the main processes involved

in the surf zone. These include splash-up, reverse break-ing, downburst and the formation of vortical structures.Figure 3 presents a snapshot of the splash-up in an SPHmodel of the first type, in comparison with a photographfrom the experiments by Li and Raichlen [4].

Fig. 3. Splash-up in photograph from Li and Raichlen (above)and in snapshot of the SPH model: particles in black belongto the domain of the original plunging jet (below)

Toothpaste jets are seen to form at the boundary, asa narrow, straight line of particles in black in figure 3.These jets consist of a long layer of clustered particles,which move along the bottom in an unphysical way, dueto smoothing effects and constrainment to stay in contactwith the boundary. This is not seen in the similar model ofDalrymple and Rogers [3], where artificial clustering is notpresent, owing to the use of a quadratic smoothing kernel.

V. CONCLUSION

It is found that the SPH solver in LS-DYNA is able tosimulate laminar free surface flow, and to capture the mainfeatures of plunging breaking waves. However, improve-ments to the software can be made, in particular to avoidartificial clustering by use of a monotonically decreasingkernel function, or inclusion of CSPH.

REFERENCES

[1] Reza Issa and Damien Violeau, Test Case 2: 3D-Dambreaking,SPHERIC ERCOFTAC, 2006

[2] R.G. Dean and R.A. Dalrymple, Water Wave Mechanics for En-gineers and Scientists, World Scientific Publishing Company Inc.,1991

[3] R.A. Dalrymple and B.D. Rogers, Numerical Modeling of WaterWaves with the SPH Method, Coastal Engineering, 53:141-147,2006

[4] Y. Li and F. Raichlen, Energy balance model for solitary waverunup, Journal of the Waterways, Ports and Coastal EngineeringDivision, 129(2):47-59, 2003

Studie naar de SPH methode voor simulatie vanregelmatige en brekende golven

Joren Pelfrene

Supervisor(s): Kameswara Sridhar Vepa, Wim Van Paepegem, Jan Vierendeels

Abstract—In dit artikel wordt de SPH methode, zoals geımplementeerdin LS-DYNA, bestudeerd voor het simuleren van vrije vloeistofoppervlak-stroming, met speciale aandacht voor regelmatige en brekende golven. Hetdoel is het creeren van een raamwerk voor het numeriek modelleren vanzijwaartse golfimpact. Het ’brekende dam’ probleem wordt bestudeerd alsvalidatietest. Een numerieke golfgoot is ontworpen voor simulatie van re-gelmatige golven, en brekende golven zijn gemodelleerd met twee types vanopwekking. Algemeen werd bevonden dat de SPH code in LS-DYNA instaat is om laminaire vrije vloeistofoppervlak-stroming te simuleren, en omde voornaamste processen van neerslaande brekende golven vast te leggen.Niettemin is de gebruikte software voor verbetering vatbaar, voornamelijkwat betreft artificieel clusteren.

Keywords— Smoothed Particle Hydrodynamics (SPH), Vrije Vloeistof-oppervlak Stroming, Brekende Golven, Brekende Dam

I. INLEIDING

DE oorsprong van deze thesis ligt in de deelname van deUniversiteit Gent aan het FlanSea project, dat de energie-

winning uit zeegolven beoogt. Hiervoor werd een nieuw typegolfenergieconvertor ontworpen, dat bestaat uit een vrij drij-vende boei die door middel van een kabel met de zeebodemwordt verbonden. Op- en neerwaartse beweging van de boeiwordt dan omgezet naar elektriciteit.

De Universiteit Gent was hierbij verantwoordelijk voor hetontwerp van de boeien, die ook tegen de ruwste weercondi-ties bestand dienen te zijn. De meest kritische belasting voorde boei stelt zich wanneer golfimpact, ook slamming genaamd,zich voordoet. Terwijl neerwaartse golfimpact reeds vaak ge-simuleerd werd, bij wijze van een vallende constructie op eenstilstaand wateroppervlak, is dit niet zo voor zijwaartse golfim-pact.

Numeriek modelleren van zwaartekrachtgolven is echter nieteenvoudig, vooral wanneer grote vervorming en fragmentatievan het vloeistofoppervlak voorkomen. Mesh-gebaseerde me-thoden, zowel Euleriaans als Lagrangiaans, kunnen toegepastworden, maar belangrijke numerieke fouten werden hierbij on-dervonden. In de SPH methode wordt de toestand van een sys-teem voorgesteld door een stelsel van discrete partikels zonderonderlinge connectiviteit. De beweging van deze deeltjes wordtgevolgd op Lagrangiaanse wijze, en er is geen speciale aanpakvan het vrije vloeistofoppervlak benodigd.

Deze studie beoogt een raamwerk op te zetten voor de simula-tie van zijwaartse golfimpact door het onderzoeken van de SPHtechniek voor numeriek opwekken van regelmatige en brekendegolven. Dit wordt uitgevoerd met de commerciele software LS-DYNA, waarin zowel eindige-elementen- als SPH-code begre-pen zijn, en waarmee gekoppelde interactie van vloeistof enstructuur gesimuleerd kan worden.

II. VALIDATIETEST: BREKENDE DAM

Een validatietest wordt uitgevoerd om de verdiensten enmoeilijkheden van het model en de methode te identifiren. Hetbrekende dam probleem is een geıdealiseerd geval, waarbij eenbalkvormig watervolume blootgesteld wordt aan de zwaarte-kracht vanaf t = 0 s. Een variatie op deze test is voorgestelddoor SPHERIC [1], gemodelleerd naar een experiment doorMARIN, waarbij een star blok in het centrum van de testtank ge-plaatst werd. SPHERIC heeft hierbij gedetailleerde experimen-tele gegevens voor de waterhoogte- en drukmetingen beschik-baar gesteld. Figuur 1 toont aan dat een zeer goede overeen-komst tussen het SPH model en het experiment behaald wordtvoor de evolutie van het waterniveau ter hoogte van een sensorin het reservoir, zolang de stroming laminair is. Na de beginfasewordt echter een grote afwijking vastgesteld in de grafiek. Ditkan deels toegeschreven worden aan het feit dat het SPH modelslechts een enkele fase modelleert, waardoor schuimvorming ophet vloeistofoppervlak niet gesimuleerd kan worden.

Fig. 1. Waterniveau ter hoogte van een sensor in het reservoir

Er werd bevonden dat goed vergelijkbare gegevens voor dewaterdruk op een vast oppervlak niet eenvoudig te verkrijgenzijn voor de SPH modellen in LS-DYNA. Een zeer fijn grid isvereist, en zelfs dan vertonen de resultaten slechts ruwe over-eenkomst. Een typische numerieke fout, artificieel clusteren ge-naamd, is verantwoordelijk voor extra ruis op deze reultaten.Clusteren komt voor wanneer twee of meer SPH deeltjes samen-klitten om een groot partikel te vormen met grote massa en hogedensiteit. De oorzaak voor dit probleem ligt in het gebruik vaneen kernel-functie met een buigpunt, waardoor de partikels eenmaximum ondervinden in de kracht waarmee ze elkaar aantrek-ken of afstoten.

III. SIMULATIE VAN REGELMATIGE GOLVEN

De regelmatige golven worden gesimuleerd door middel vaneen numerieke golfgoot. Twee bijzondere randcondities wordenhierbij toegepast: de golven worden gegenereerd door een rote-rende peddel, die scharniert omheen een punt op de bodem vande tank, en de watergolfenergie wordt geabsorbeerd door eennumerieke sponslaag. Dit concept is geıllustreerd in figuur 2.In LS-DYNA is de golfabsorptie bereikt door het toepassen vaneen massa-proportionele demping op de SPH deeltjes die tot desponslaag behoren. Voor het referentiemodel in figuur 2 wordteen steady state voor lopende golven bereikt na 5 tot 6 periodesvan de golfopwekker.

0

2

x(m)

y(m)

5 10 20 2515

Sponge layerFluid domain

1

Fig. 2. Momentopname van de numerieke golfgoot, genomen wanneer steadystate bereikt wordt voor lopende golven met periode T = 2 s

De golfhoogte, die in de omgeving van de golfpeddel gegene-reed wordt komt vrij goed overeen met de waarde die voorspeldwordt door de golfopwekkingstheorie van Galvin [2]. In hetreferentiemodel wordt een golfhoogte van 0.55m vastgesteld,terwijl de theorie 0.58m vooropstelt. Desalniettemin wordt ver-der stroomafwaarts een verlies aan golfhoogte bemerkt, met eenwaarde van 0.30m net voor de sponslaag.

IV. SIMULATIE VAN BREKENDE GOLVEN

Twee modeltypes zijn getest voor het breken van zwaarte-krachtgolven. Bij het eerste type wordt een omrollend golffrontperiodiek opgewekt in de directe nabijheid van de golfpeddel.Het tweede type genereert een golf die verder stroomafwaartsbreekt, ten gevolge van de bodemgeometrie. Voor beide typeswordt bevonden dat het breekpunt plaats vindt zoals voorspelddoor het kinematische brekingscriterium, en de simulaties tonenaan dat de SPH methode in staat is om de voornaamste proces-sen in de brekingszone vast te leggen. Dit behelst het opspattenna neerkomen van het omrollende golffront, terugbreken, neer-storten naar de bodem en de vorming van wervelstructuren. Infiguur 3 is een momentopname voorgesteld van het opspatten ineen SPH model van het eerste type, in vergelijking met een fotovan de experimenten van Li en Raichlen [4].

In de simulatie wordt gemerkt dat een onfysische, langs debodem kruipende straal van partikels gevormd wordt; als eensmalle, rechte lijn van deeltjes in zwart in figuur 3. Deze straalbestaat uit een lange laag van samengeklitte deeltjes en is te wij-ten aan een samenspel van artificieel clusteren, smoothing effec-ten en de restrictie van behoud van contact met de bodem. Ditverschijnsel wordt niet gezien in het gelijkaardige model vanDalrymple en Rogers [3], waar artificieel clusteren niet voor-komt omwille van het gebruik van een kwadratische kernelfunc-tie.

Fig. 3. Opspatten in een foto door Li en Raichlen (boven) en in een momentop-name van het SPH model: deeltjes in zwart behoren tot het domein van hetoorspronkelijk omrollende golffront (onder)

V. CONCLUSIE

Er wordt bevonden dat de SPH-code in LS-DYNA in staatis om laminaire vrije vloeistofoppervlak-stroming te simuleren,en om de voornaamste processen bij golfbreking vast te leg-gen. Niettemin is de gebruikte software voor verbetering vat-baar, voornamelijk wat betreft artificieel clusteren.

REFERENCES

[1] Reza Issa and Damien Violeau, Test Case 2: 3D-Dambreaking, SPHERICERCOFTAC, 2006

[2] R.G. Dean and R.A. Dalrymple, Water Wave Mechanics for Engineers andScientists, World Scientific Publishing Company Inc., 1991

[3] R.A. Dalrymple and B.D. Rogers, Numerical Modeling of Water Waveswith the SPH Method, Coastal Engineering, 53:141-147, 2006

[4] Y. Li and F. Raichlen, Energy balance model for solitary wave runup, Jour-nal of the Waterways, Ports and Coastal Engineering Division, 129(2):47-59, 2003

Contents

1 Introduction 1

1.1 Sustainable energy: the FlanSea project . . . . . . . . . . . . . . . . . . . . 1

1.2 Water wave simulation by SPH . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Goal of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Literature Study 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Description of water waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Regular waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.2 Waves in coastal waters . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Smoothed Particle Hydrodynamics (SPH) . . . . . . . . . . . . . . . . . . . 13

2.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 SPH formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Solving the discretised Navier-Stokes equations . . . . . . . . . . . . 18

3 Implementation in LS-Dyna 22

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Material definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Contact definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Validation test: elliptic drop . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.1 Background for the elliptic drop test . . . . . . . . . . . . . . . . . . 26

3.4.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.4 Density variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5 Validation test: dam break . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5.1 The MARIN experiment . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5.2 Comparison to experimental results . . . . . . . . . . . . . . . . . . 36

3.5.3 Comparison to analytical description . . . . . . . . . . . . . . . . . . 45

3.5.4 Errors at the boundary . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Simulation of regular waves 51

4.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

ix

Contents

4.1.2 Wave generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.3 Wave absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.4 Implementation of the model . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.1 Behaviour of the model . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2 Wave Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Simulation of breaking waves 65

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3.1 Breaking point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3.2 Evolution of a plunging breaking wave for Model I . . . . . . . . . . 70

5.3.3 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3.4 Evolution of a plunging breaking wave for Model II . . . . . . . . . . 75

6 Conclusion 77

Bibliography 79

List of Figures 82

List of Tables 85

x

List of acronyms

CFD Computational Fluid Dynamics

CISPH Corrected Incompressible SPH

CSPH Corrected SPH

FEM Finite Element Method

ISPH Incompressible SPH

SPH Smoothed Particle Hydrodynamics

VOF Volume Of Fluid

WEC Wave Energy Converter

XSPH The ’X’-variant of SPH

xi

Chapter 1

Introduction

1.1 Sustainable energy: the FlanSea project

Over the last century, fossil fuel and nuclear energy have powered the economies of the

modern world. But at the same time the consciousness grew that these classic energy

sources pose great dangers, whether on global or local scale, the latter of which was

demonstrated dramatically earlier this year. Therefore, in the last decades, the search for

a clean, renewable energy as an alternative to fossil fuel and nuclear energy has gained

interest by a large, worldwide community.

Wave energy is one type of sustainable energy with great potential, but is not easily cap-

tured. It was not until the mid-1990s that large-scale installations have been built in front

of the ocean coasts, where the energy density is the highest. In these waters, the wave

energy converters (WECs) require a very rigid design to withstand all forces of nature

acting upon them.

However, the situation is different for calmer seas, which is the focus of the Flanders

Electricity from the Sea (FlanSea) project, that is coordinated by Ghent University. Log-

ically, the possible energy proceeds are smaller there, but this also goes for the cost of the

equipment.

The FlanSea project makes use of point absorbers, illustrated in figure 1.1, to convert wave

energy into electricity. The point absorbers can be described as large buoys that follow

the wave motion. A special winch is built into the buoy to facilitate attachment to the

sea bottom by means of a cable.

Of course, also the composite shell of a point absorber needs to be able to withstand rough

sea conditions, which remains one of the main challenges for WECs.

1

Chapter 1. Introduction

Figure 1.1: Point absorber WEC concept

1.2 Water wave simulation by SPH

For the estimation of the pressures acting on the buoy, the possibility of numerical simu-

lation forms a useful and inexpensive tool. Since impact on the water surface, also termed

slamming, poses the most critical condition for a floating structure at sea, many studies

have been conducted to investigate wave loads on floating structures. The simulation of

such a situation could be achieved by use of a software package that enables fluid-structure

interaction (FSI).

Modelling of highly complex free surface flows with large deformation and fragmentation,

such as wave breaking, is not easily achieved in a grid based method. Problems with mesh

entanglement and determination of the free surface have been encountered. Smoothed

Particle Hydrodynamics (SPH), as a meshfree, Lagrangian method, can in these cases be

a good alternative to the classic CFD methods. The SPH method has been shown to be

able to perform well in the simulation of free surface water flow, for both regular waves

(e.g. Groenenboom and Cartwright, [9]) and wave breaking (e.g. Dalrymple and Rogers

[4], Khayyer [15]). An example of the latter is shown in figure 1.2

Figure 1.2: SPH simulation of an overturning water front with plunging jet [18]

In this thesis, the commercial software package LS-Dyna by Livermore Software Tech-

2

Chapter 1. Introduction

nology Corporation (LSTC) is used. This software has an SPH solver implemented and

enables coupling with FEM for the structural domain. To the author’s knowledge, no pre-

vious SPH simulations of water waves have been performed in LS-Dyna. Most researchers

in the field of SPH use experimental codes, developed for research only, such as the freely

available SPHysics code. The spread of the SPH method has also been energised by the

SPH European Research Interest Community (SPHERIC).

1.3 Goal of the thesis

The aim of this thesis is to explore the capabilities and limitations of the SPH method for

the simulation of free surface water flow, as implemented in LS-Dyna.

The performance of all simulations is assessed by comparing with theoretical description,

experimental data and results from previous studies using SPH, as found in literature.

Models for regular and breaking waves are designed and proposed for further study, which

is to include the fluid-structure interaction with a point absorber WEC.

1.4 Structure of the thesis

First, a review of the theoretical background for this thesis is given in Chapter 2. This

consists of two parts: the physics of regular waves and breaking waves, and the concepts

of SPH. Considerations in the build-up of an SPH model in LS-Dyna are discussed in

Chapter 3, and two test cases are simulated to reveal the main virtues and flaws of the

code. Chapter 4 presents an SPH model for regular waves as a numerical wave flume with

wave generation by a paddle and wave energy absorption. In Chapter 5, plunging breaking

waves are simulated and the post-breaking processes are identified. In the final chapter,

conclusions are drawn on the SPH implementation in LS-Dyna and for creating models

to simulate regular and breaking waves.

3

Chapter 2

Literature Study

2.1 Introduction

The sea waves, whether in deep oceans or close to shore, are a phenomenon of nature

that has been studied since early ages. The very first attempt at describing a water wave

theory was performed by Isaac Newton in his ’Principia’. He proposed an analogy with

oscillations in a U-tube, from which he could correctly deduce that the wave frequency

must be proportional to the inverse of the square root of the wavelength. In this chapter,

an overview of various analytical wave theories and a description of breaking waves is given,

as well as an introduction to the concepts of SPH, which enables an effective modelling of

the free water surface.

2.2 Description of water waves

2.2.1 Regular waves

The linear wave theory was developed by Airy in 1845. In his theoretical model, incom-

pressible fluid and irrotational flow is assumed, which allows using a velocity potential

φ(x, t). Then, all wave-induced velocities are given by the relation:

v = −∇φ(x, t) (2.1)

The velocity potential itself needs to satisfy the Laplace equation (2.2), as a formulation of

the continuity equation with the incompressibility assumption, in the fluid domain. The

coordinate system consists of x- and y-axes in the horizontal plane and the z-axis in the

vertical. The domain is taken as a rectangle with the ocean bottom at z = −h(x, y) and

a free surface z = ζ(x, y, t).

∇2φ(x, t) = 0 (2.2)

4

Chapter 2. Literature Study

Boundary conditions exist at the bottom, the free surface and the lateral sides. At the

bottom, the flow is taken as zero through the boundary (equation 2.3). At the free surface,

which moves in response to pressure in the fluid, two boundary conditions are required.

1) to locate the boundary (kinematic, equation 2.4), and 2) to prescribe constant pressure

(dynamic, equation 2.5).

∂φ

∂z= 0 at z = −h(x, y) (2.3)

∂ζ

∂t=∂φ

∂zat z = ζ(x, y) (2.4)

∂φ

∂t+ gζ = −p0

ρ0at z = ζ(x, y) (2.5)

For regular waves over an even bottom, the lateral boundary condition reduces to stating

that the wave is subject to periodicity, both in time (eq. 2.6) and space (eq. 2.7, when x

is the direction of wave propagation).

φ(x, y, z, t) = φ(x, y, z, t+ T ) (2.6)

φ(x, y, z, t) = φ(x+ λ, y, z, t) (2.7)

The linear Airy theory can be used under the assumption that the wave amplitude is small

and the waveform is invariant in time and space. Then, the domain for the free surface

boundary conditions simplifies to z = 0.

A solution for the velocity potential for a highly simplified case is found by means of

seperation of variables. Hereby, the dispersion relation (2.8) is provided, which relates the

wavelength λ, wave period T and water depth h:

ω2 = gk tanh kh (2.8)

where the angular frequency ω = 2π/T and the wave number k = 2π/λ.

The profile of the free surface is found to be sinusoidal. Moreover, from the solution of the

velocity potential, the water particle paths can be derived and are found to be closed and

of elliptical form. This means no mass transport of water takes place. However, energy

transport does occur, with a theoretical average of E = 18ρgH

2λ per wave per unit width.

The linear or small-amplitude wave theory provides a useful first approximation to the

wave motion. Ocean waves are generally not small in amplitude, and from an engineering

point of view it is usually the large waves that are of interest since they result in the largest

forces. To provide a better fitting description of the larger amplitude wave mechanics, in

5


both deep and shallow water, several extended wave theories have been developed, each

with their own range of applicability.

For deep water, Stokes extended the linear theory to include non-linear terms by using a

perturbation approach (1847), developed as a power series of ε = kζA. This solution is

expected to converge as more and more terms are included in the expansion. In general,

the perturbation expansion for the velocity potential φ(x, t) may be written:

φ = φ1 + εφ2 + ε2φ3... (2.9)

In this expansion φ1 is the first-order theory (linear theory), φ2 the second-order theory

and so on.

As a result, the Stokes wave profile is characterised by higher crests and flatter troughs,

predicting a wave form that is asymmetrical about the still water line. Also, the nonlinear

terms cause the water particle path to be no longer closed. There is a constant component

in their horizontal velocity, which introduces mass transport.

The linear dispersion relation is still valid to second order, and both wavelength and

celerity are independent of the wave height to this order. At higher orders, they do

depend on wave height, and therefore, with a given wave period, will increase with higher

waves.

Over the last decades, variations of the Stokes perturbation at much higher orders have

been developed, oriented to computer implementation. Dean (1965) used the stream func-

tion wave theory, which was computationally simpler than the velocity potential theory.

The stream function ψ(x, t) is defined in a similar manner as the velocity potential, but

it exists only for two-dimensional flow. The stream function is orthogonal to the velocity

potential, such that the flow velocity can be expressed as: v = ∇× ψ.

Both the stream function and the velocity potential are constructed of sums of sine func-

tions that satisfy the Laplace equation. In the theory of Stokes the corrections to the

linear theory are successive; every higher-order correction is obtained on the basis of the

previously obtained lower-order corrections. Dean’s method for the stream function, how-

ever, determines the coefficient of each higher order term so that a best fit is obtained

to the theoretically posed dynamic free surface boundary condition. The stream function

representation has succesfully predicted the wave phenomena observed in some laboratory

wave studies, and thus it may describe naturally occurring wave phenomena [5].

Neither the theory of Stokes or the theory of Dean performs very well in shallow water.

For such conditions, the cnoidal theory (Korteweg - de Vries, 1895) should be used (see

6


section 2.2.2). A representation of the applicability ranges for the main wave theories is

given in figure 2.1.

A notable expansion to the Dean method is Fenton’s Fourier series wave theory (1988).

This theory satisfies field equations and boundary conditions to a specified level of accuracy

and is valid for both deep and shallow water depths. The Fourier coefficients are found

numerically from simultaneous algebraic equations by satisfying two non-linear free surface

boundary conditions and the dispersion relationship. Finding the coefficient requires that

water depth, wave period, wave height and the depth-averaged mass transport velocity

are specified.

Comparisons with other numerical methods and experimental data show that Fenton’s

theory agree consistently and better than results from other theories for a wide range of

wave height, period and water depth.

2.2.2 Waves in coastal waters

The nearshore coastal region can be identified as the region between the shoreline and a

fictive offshore limit where the depth is so large that it no longer influences the waves.

This depth depends on the height of the wave motion itself, and can, in a simplification,

be estimated as half the wavelength.

In shallow water, the particle motion becomes more and more horizontally oriented and, as

the depth over amplitude ratio becomes too small, the Stokes and stream function theories

are no longer applicable. Various shallow water equations can be derived by assuming the

pressure to be hydrostatic so that vertical water particle accelerations are small, and by

imposing a horizontal velocity on the flow to make it steady with respect to the moving

frame. If u is the vertically averaged horizontal velocity, these equations become [11]:

∂ζ

∂t+

∂

∂x[(h+ ζ) u] = 0

∂u

∂t+ u

∂u

∂x+ g

∂ζ

∂x=

1

2h∂3h u

∂x2∂t− 1

6h2 ∂3u

∂x2∂t

(2.10)

Equations 2.10 are termed the Boussinesq equations. They are one-dimensional shallow

water equations with corrections for the vertical accelerations under the wave.

In the nonlinear Stokes’ and Dean’s stream function wave theories the perturbation is made

in terms of the wave steepness ε = kζA. In the case of shallow water depth, corrections to

the linear theory need to be applied to account for finite-depth effects. The cnoidal wave

7


Figure 2.1: Ranges of suitability for various wave theories (Le Mehaute 1976)

theory is designed for this purpose, and constructed in a very similar manner to Stokes

theory and stream function theory. This theory was developed by Korteweg and de Vries

(1895) as an analytical solution to the Boussinesq equations (2.10), using power series of

the small parameter β = ζAh to develop the velocity potential. The resultant free surface

elevation does not consist of harmonic waves, but of cnoidal waves, which are expressed

in terms of the Jacobian elliptic function cn. Typically, as the water depth decreases, this

results in sharper wave crests and flatter troughs, similar to the nonlinear effect of wave

steepness. Figure 2.2 gives an image of the typical wave profile shape in different wave

theories. The solitary wave is included as a special case, being a cnoidal wave with infinite

wavelength.

In recent times, many researchers have developed modifications to the Boussinesq equa-

8


Figure 2.2: Wave profile shape of different progressive gravity waves [6]

tions to improve various characteristics of the corresponding wave, yielding extended

Boussinesq equations or Boussinesq-type equations. An important contribution has been

made by Madsen and Sorensen (1992) with the ability to extend the model from shallow

water into deeper water. They were also able to include wave breaking into the model by

introducing a roller, a separate body of fluid on the wave surface.

Due to these and many other extensions to the basic Boussinesq equations, along with

the advent of numerical simulation, Boussinesq modelling in the time domain has become

a widespread alternative to models solving the Navier-Stokes equations for problems con-

sidering wave transformation in coastal engineering. The models are able to effectively

simulate nearshore effects such as shoaling, refraction, diffraction, breaking and boundary

reflection, and can simulate wave absorption by use of a numerical sponge layer.

Maximum wave height

Both the Boussinesq and Stokes wave theories make possible to predict the highest possible

wave on a constant depth. Stokes (1880) predicted theoretically that a wave would become

unstable at the crest and break if the water particle velocity at the crest were to become

higher than the wave celerity. This is often referred to as the kinematic breaking criterion.

Furthermore, Stokes showed that for an irrotational wave would break, the surface in the

proximity of the crest attains an opening angle less than 120°. However, this solution is

only valid at the crest of the wave and says nothing about the rest of the wave shape, and

9


thus the wave height.

In deep water, a theoretical limit can be found for the maximum wave steepness (H/λ)max.

Values have been obtained by various authors and are listed in table 2.1. In this table,

the last result is considered the most accurate.

Table 2.1: Maximum wave steepness [33]

Author year (H/λ)max

Mitchell 1893 0.142

Havelock 1918 0.1418

Longuet-Higgins 1975 0.1412

For shallow water depth, the maximum possible wave height can be expressed by (H/h)max.

Table 2.2 lists the results obtained over the years. Again, the last value is considered the

most accurate. It should be noted that all waves of constant form are strictly symmetri-

cal with respect to the vertical lines that can be drawn through the crests and troughs,

whereas waves breaking in the surf zone always attain a certain skewness as they ap-

proach breaking. Therefore, the symmetrical waves considered here are not necessarily on

the verge of breaking. Nonetheless, the value obtained by McCowan is most commonly

used as a breaking criterion for a first estimate of the breaking position of waves on a

sloping beach.

Table 2.2: Maximum wave height to depth ratio [33]

Author year (H/λ)max

Boussinesq 1871 0.73

McCowan 1895 0.78

Gwyther 1900 0.83

Davies 1952 0.83

Packham 1952 1.03

Fenton 1972 0.85

Longuet-Higgins & Fenton 1974 0.8261

Wave breaking

Breaking waves in coastal waters were classified by Galvin (1968) according to the form

of the wave at breaking. An image of the profile shape of the breakers in the surf zone

10


(extending from the shore to the seaward point of breaking) is given in figure 2.3. Three

main types can be identified:

1. Spilling breakers are found on mildly sloping beaches. At breaking, their wave crest

becomes unstable and cascades down the shoreward face of the wave, forming an

air-entrained ’roller’ that rides on top of the wave.

2. On steeper beaches, plunging breakers occur. At the onset of breaking, they have

become very skew with a vertical front face. Then, the crest shoots forward as a

jet and plunges down into the base of the wave, generating a high splash-up shortly

after touchdown.

3. Surging waves occur on very steep beaches and are characterised by a narrow surf

zone and high reflection. When close to the beach, the water depth in front of the

wave rapidly becomes very small. The front-face then becomes near-vertical and the

toe of the wave shoots forward to run up the beach.

As a fourth type, Galvin named collapsing waves, which are a combination of plunging

and surging breaker types.

In spilling and plunging breaker types, essentially all the incident wave energy is turned

into turbulence and dissipated. In the surging waves, turbulence generation is limited.

Only part of the wave energy is dissipated, while the remaining energy gets reflected and

carried back. Thus, surging breakers on a steep slope can be seen as the transition to full

reflection on a vertical wall.

Breaking criteria

Incipient breaking can be defined in several ways. Commonly, it is identified as the point

where the wave height is maximum. Other qualitative definitions include: the point where

the front face becomes vertical for plunging breakers, or the point just prior to the forming

of foam for spilling breakers. Many studies have been performed to provide criteria to

predict the onset of wave breaking, but none of them has been universally accepted in

practical applications [27]. These criteria can be categorised into three types: geometric,

kinematic and dynamic criteria.

The geometric criteria often describe the limiting wave height in terms of the maximum

wave steepness or the breaker depth index Hbr/h. A heuristic interpolation formula (equa-

tion 2.11) for arbitrary h/λ by Miche (1944) is frequently quoted in literature.

Hbr

λ= 0.142 tanh(kh) (2.11)

11


Figure 2.3: Three types of wave breaking on beaches. Small figures denote different stages of the

breaking process [33]

As mentioned before, the earliest breaking criterion was the value found by McCowan for

the maximum stable wave height to water depth ratio (see table 2.2). Weggel further

refined this criterion by taking in account the influence of the beach slope and wave

period [36]. From laboratory data on waves breaking on smooth, plane slopes, he derived

expression 2.12 for the breaker depth index. The parameters a and b are empirically

determined functions of the slope tanβ. For low-steepness waves, the breaker index is

bounded by the value of 0.78 over an even bottom, and twice this value as the beach slope

approaches infinity (as the sum of incident and perfectly reflected wave components).

Hbr

h= b− aHbr

gT 2(2.12)

Dalrymple et al. formulated a breaking criterion, based on the conservation of energy flux

and including the deep water wave angle cos θ0 [3]. The criterion is given by equation 2.13,

where H0 is the deep water wave height, and C0 the deep water wave celerity.

Hbr =

(0.78

g

)(1/5)(h02C0 cos θ0

2

)(2/5)

(2.13)

However, field observations and experimental investigation have shown that great deviation

on these geometric criteria is possible ([11], [27]). The differences to the theoretical values

can often be ascribed to crest instabilities that cause the wave to break immediately. For

12


the criterion by Dalrymple et al., it appeared from laboratory tests that the breaker height

was underestimated by approximately 12% [5].

The kinematic breaking criterion is usually formulated as the ratio of horizontal water

particle velocity at the wave crest to the wave celerity. Incipient breaking occurs when this

ratio becomes greater than unity. This criterion appears very reasonable, but in fact great

discrepancy on its validity exists among experimental studies [27]. In some experiments, a

ratio greater than unity was observed at the moment of breaking. In others, substantially

smaller ratios were reported.

A third type of breaking criterion, the dynamic criterion, is related to the particle acceler-

ation near the wave crest. Longuet-Higgins (1963) theoretically showed that the downward

acceleration near the crest of a regular wave equals 0.5g for wave breaking. This criterion,

due to the difficulty of measurement of acceleration, has not been sufficiently examined.

A recent experimental investigation on deepwater wind waves by Oh et al. [27] hints that

this criterion is applicable in the presence of wind action.

2.3 Smoothed Particle Hydrodynamics (SPH)

As computer technology grew, various methods have been developed for solving the Navier-

Stokes equations in discrete formulation. However, effective simulation of free surface

waves is a challenge for most numerical methods, especially when the wave height is large

or when wave breaking occurs. In general, when the deformation of a material boundary

is studied using a numerical method, it is common to contrast Eulerian and Lagrangian

description. The Eulerian approach uses a fixed grid and has difficulty tracking the bound-

aries (or interfaces) of the various materials described in the simulation model. The La-

grangian approach on the other hand uses a comoving grid so that material boundaries

coincide with element boundaries. When large deformations and fragmentation are likely

to be present, a meshfree method can be favoured over a meshed grid-based method (La-

grangian FEM, FDM), where strains in the order of unity and above cause mesh tangling

and the computation to fail eventually.

SPH was developed as one of the earliest meshfree particle methods (MPM) by Monaghan

and Gingold [7]. The method was originally aimed to model problems in astrophysics, and

has since been adapted into a number of fields. Being a Lagrangian, meshfree method, SPH

uses a set of discrete particles without fixed connectivity as the computational framework

on which the governing equations are resolved.

13


2.3.1 Governing Equations

At any time t and position x the state of a continuous and locally homogeneous fluid is

defined when the speed v and any other two thermodynamic variables are specified. In

unsteady motion, the fluid cannot strictly be in thermodynamic equilibrium and it will

be necessary to define the thermodynamic properties of (infinitesimally) small individual

fluid particles of which the fluid may consist. These fluid particles can move freely through

one another under the influence of applied forces or other externally imposed changes at

the boundaries.

In the Lagrangian or spatial description the fluid properties are described as seen by

an observer moving with the fluid particle. This approach differs from the Eulerian or

material description in the use of the total time derivative as a combination of the local

and convective derivative. The total time derivative operator for a Lagrangian moving

frame is defined for an arbitrary function in equation 2.14

Df

Dt≡ ∂f

∂t+ v ·∇f (2.14)

The governing equations for determining these motions are the Navier-Stokes equations:

the continuity-equation (2.15a), momentum-equation (2.15b) and energy-equation (2.15c).

These equations are not solvable in an analytical way.

In the following notation the Greek superscripts are used to denote the coordinate direc-

tions, and the summation is taken over repeated indices.

Dρ

Dt= − ρ ∂v

β

∂xβ(2.15a)

Dvα

Dt=

1

ρ

∂sαβ

∂xβ+ f (2.15b)

De

Dt=

sαβ

ρ

∂vα

∂xβ+ f ·v (2.15c)

Where:

f is the external force per unit of volume

v is the velocity vector and x the position vector

s is the total stress tensor, which is made up of two parts. One part of the isotropic

pressure p, and the other of the viscous stress ταβ.

sαβ = − p ∂αβ + ταβ (2.16)

14


Here, ∂αβ is the Kronecker delta.

For Newtonion fluids the shear stress is proportional to the shear strain rate through

the dynamic viscosity µ.

ταβ = µ

[∂vβ

∂xα+∂vα

∂xβ− 2

3(∇ ·v) ∂αβ

](2.17)

2.3.2 SPH formulation

The SPH formulation can be divided in two key steps: the integral representation or kernel

approximation of field functions and the particle approximation.

The integral representation of a field function (equation 2.18) is given by the integration

of the multiplication of an arbitrary function and the smoothing kernel function (also

called smoothing function, kernel function or simply kernel) W (x−x′, h) in the integration

domain Ω. Figure 2.4 shows a concept image of the smoothing kernel and support domain.

f(x) =

∫Ωf(x′)W (x− x′, h) dx′ (2.18)

where h is the smoothing length defining the influence area of the smoothing function W .

Figure 2.4: Representation of the smoothing kernel and support domain. While the support

domain is shown here as a circle, it is in fact a sphere in 3 dimensions. [35]

15


The smoothing function plays a very important role in SPH approximations, as it de-

termines the accuracy of the function representation and efficiency of the computation.

The kernel function can be constructed taking in account a number of conditions and will

always have the following properties [21]:

1. The smoothing function is normalised:∫ΩW (x− x′, h) dx′ = 1 (2.19)

2. There is a compact support for the smoothing function:

W (x− x′, h) = 0, for∣∣x− x′

∣∣ > κh (2.20)

The dimension of the compact support is defined by the smoothing length h and a

scaling factor κ that determines the spread of the smoothing function.

Using this condition, the entire problem domain is localised as integration over the

support domain of the smoothing function. Therefore, the integration domain Ω can

be, and usually is, taken as the support domain.

3. W (x− x′, h) is non-negative for any x′ within the support domain. This is strictly

not a condition for the kernel, but necessary to achieve physically meaningful (stable)

results in hydrodynamic computations.

4. The smoothing value for a particle decreases monotonically with the distance to the

particle.

5. With the smoothing length approaching zero, the kernel approaches the Dirac delta

function:

limh→0

W (x− x′, h) = ∂(x− x′) (2.21)

6. The smoothing function should be an even function

7. The smoothing function should be sufficiently smooth to obtain better approxima-

tion.

Many different kinds of smoothing functions can be constructed. Some frequently used

functions are: the Gaussian kernel, the cubic, quartic and quintic spline kernels, the

quadratic kernel function (Johnson, 1996 [14]; used for simulation of high velocity impact).

The commercial software LS-Dyna, which was used for simulations throughout this thesis,

employs following cubic B-spline smoothing function [10]:

W (x, h) =1

h(x)dΘ(x) (2.22)

16


where d is the number of space dimensions (2 or 3) and Θ(x) is the cubic B-spline function

defined by:

Θ(x) = C ×

1− 3

2 |x|2 + 3

4 |x|3 for |x| ≤ 1

14(2− |x|)3 for 1 < |x| ≤ 2

0 for 2 < |x|

(2.23)

with C a constant for normalisation, depending on the number of space dimensions. A

plot of the function Θ(x) is given in figure 2.5.

x

Θ(x)

Figure 2.5: Function used for the smoothing kernel in LS-Dyna, given for a 1-dimensional situ-

ation.

The second key step in SPH formulation is the particle approximation, which enables the

system to be represented by a finite number of particles that carry an individual mass and

occupy an individual space.

The continuous integral representations can be transformed into discrete form as a summa-

tion over all particles in the support domain with radius κh. The particle approximation

for a particle i can be written as:

f(xi) =N∑j=1

mj

ρjf(xj) ·Wij (2.24)

where Wij = W (xi−xj , h) is the smoothing function centered on particle i and evaluated

at particle j.

17


Thus, the value of a particle i is approximated using the weighted average of the function

values at all the particles within the support domain of particle i.

The particle approximation for the spatial derivative of the function is

∇ · f(xi) = −N∑j=1

mj

ρjf(xj) ·∇Wij (2.25)

where

∇Wij =xi − xjrij

∂Wij

∂rij(2.26)

With these formulae it can be seen that the SPH method uses particles to represent

material and as a computational frame. There is no need for predefined connectivity

between these particles.

2.3.3 Solving the discretised Navier-Stokes equations

The smoothing kernel and particle approximation can be used for discretisation of partial

differential equations (PDE’s). The SPH formulation is derived by discretising the Navier-

Stokes equations spatially, thus leading to a set of ODE’s which can be solved via time

integration.

The derivative of density of particle i may be solved from the continuity equation (this is

the so-called continuity density approach), as in equation 2.27. Optionally renormalisation

of the density field can be performed.

DρiDt

=

N∑j=1

mj(xβi − xβj ) ·

∂Wij

∂xβi(2.27)

The updated velocity may be obtained from the discretisation of the momentum equation.

However, in the application of the discretised momentum equation a number of formula-

tions can be used (and are available in LS-Dyna).

With a symmetric kernel function, such as the cubic B-spline kernel, the most commonly

used formulation is given by the following equation:

DvαiDt

=N∑j=1

mj

(sαβiρ2i

+sαβjρ2j

)·∂Wij

∂xβi(2.28)

Another possible choice is given in equation 2.29. This is called the ´fluid formulation’

and gives better results than other SPH formulations when fluid material is present [10].

18


This formulation for the momentum equation and a similar one for the energy equation

will be used throughout this thesis.

DvαiDt

=

N∑j=1

mj

(sαβiρi ρj

·∂Wij

∂xβi−

sαβjρi ρj

·∂Wji

∂xβj

)(2.29)

The particle approximation of the energy equation as used in LS-Dyna is given in the

following equation:

DeiDt

= −pi + Πij

ρ2i

N∑j=1

mj(vi − vj) ·∂Wij

∂xβi(2.30)

The term Πij denotes the Monaghan artificial viscosity term [26]. This term is added to

the physical pressure to allow shocks to be simulated. If not, the simulation will result

in unphysical oscillations around the shocked region. The role of the artificial viscosity

is to smoothen the shock over several particles and to allow the simulation of viscous

dissipation, the transformation of kinetic energy to heat. Also, it prevents unphysical

penetration for particles approaching each other.

For solving compressible flows the particle motion is calculated from the pressure gradient,

while the pressure for each particle is computed from local particle density and internal

energy through the equation of state. In the SPH formalism a theoretically incompressible

flow will be treated as weakly compressible. This facilitates an equation of state to be

used to determine fluid pressure, which is much faster than solving an equation such as

the Poisson’s equation.

In many SPH formulations the relationship between density and pressure is assumed to

follow the expression 2.31, which was first applied to SPH by Monaghan to model free

surface flows for water [25].

p = B

[(ρ

ρ0

)γ− 1

](2.31)

where ρ0 is the reference density, γ ' 7 and B can be taken as the initial pressure.

In LS Dyna, the equation of state is connected to the material model (*MAT_001_ELASTIC_FLUID)

and is given by:

p = K ln

(ρ

ρ0

)(2.32)

where K is the material bulk modulus (for water: K = 2.22 · 109 N/m2)

19


It should be noted that a Poisson equation for computation of the fluid pressure can

also be implemented in SPH. This approach is called the Incompressible SPH method.

The strict ISPH method has been succesfully used to simulate numerous free surface

flow problems, such as wave breaking [31] and wave-structure interaction [8]. The ISPH

formulation enforce the incompressibility of flow and preserve linear momentum, but not

angular momentum. This has shown to be a major drawback when simulating violent free

surface flows, such as wave breaking and post-breaking.

Therefore, recently a corrected ISPH formulation (CISPH) to preserve also the angular

momentum in free surface flow problems has been developed [15] and has shown to reach

good accuracy in the simulation of various types of breaking waves.

In addition, particles are moved with the following equation:

dxidt

= vi + ε

N∑j=1

mj

ρj(vi − vj)Wij (2.33)

The last term, including the parameter ε ' 0.5, is the XSPH correction of Monaghan

[24], which ensures that neighbouring particles are moving with approximately the same

speed. When applied to incompressible flow, the XSPH helps to keep the particles more

orderly. With compressible flow this effectively prevents particles with different velocities

from occupying nearly the same location.

For explicit time integration schemes the Courant-Friedrichs-Lewy (CFL) condition is a

necessary (but not sufficient) condition for convergence while solving PDE’s numerically.

In essence, it states that the numerical domain of dependence must include the analytical

domain of dependence to assure that the scheme can access the information required to

form the solution. The CFL condition requires the time step to be proportional to the

smallest spatial particle resolution, which in SPH is represented by the smoothing length.

Again, various formulations for the calculation of the time step are possible. LS-Dyna

uses following expression:

∂t = CCFL · mini

(hi

ci + |vi|

)(2.34)

where CCFL is a numerical constant.

The calculation cycle, used by LS-Dyna, is given in figure 2.6.

20


Velocity / positions LS-DYNA

Smoothing length SPH

Sorting SPH

Pressure, thermal energy, stresses LS-DYNA

Density, strain rates SPH

Particle forces SPH

Contact, boundary conditions LS-DYNA

Accelerations LS-DYNA

Start

Figure 2.6: LS-Dyna calculation cycle for SPH particles, [10]

21

Chapter 3

Implementation in LS-Dyna

3.1 Introduction

The SPH method used in this study was implemented within the LS-Dyna commercial

software package, from Livermore Software Technology Corporation (LSTC). LS-Dyna

allows mesh-based and mesh-free techniques such as SPH to exist and interact in one

simulation. Thus, fluid-structure interaction (FSI) with deformable bodies can be treated

by the software code. To guide the users, LSTC provides a manual and various example

models, as well as papers from user’s conferences on their website. In this chapter, it is

intended to explain the reasoning behind the construction of an input file for the solver,

and to examine the results of two validation tests: the elliptic drop test, first conducted

for SPH by Monaghan, and a dam break test, as described by the SPHeric community.

It can be noted that the SPH techniques are still in a state of development, with ad-

vancements and improvements regularly reported in scientific literature. Therefore, a

commercial software code might not always include all features from recent developments.

3.2 Material definitions

In LS-Dyna a variety of element types can be used in modelling. Aside from the SPH

particles, solid, shell and cable elements are used in this thesis. A material model needs to

be specified for all elements. Furthermore, they are characterised by following properties:

SPH elements consist of only one node, on which all element properties are centered.

Their initial density needs to be specified and from the initial particle spacing the

elements are given a certain mass. This mass stays constant throughout the simu-

lation, whereas particle density and occupied volume may deviate from the initial

value. Also, a constant (CSLH) applied to the smoothing length is to be defined.

CSLH is the same factor that is usually named κ, as in section 2.3, and, together

with the smoothing length h determines the size of the support domain.

22

Chapter 3. Implementation in LS-Dyna

The SPH processor in LS-Dyna uses a variable smoothing length [22]. The initial

smoothing length h0 is computed for each particle by taking the maximum of the

distances between neighbouring particles. The smoothing length then varies in time

according to following equation:

dh(t)

dt= h(t)∇ ·v (3.1)

The smoothing length decreases when particle concentration is high and increases

when few particles are around. Thus, h varies to keep the same number of particles

within its support domain.

Solid elements are three-dimensional particles with 8 nodes. They can be employed

to form the mesh for different types of numerical modelling applications, but here

they are used only as Lagrangian elements of a solid material.

Shell elements are similar to solid elements, but consist of only 4 nodes and are given

a certain thickness. In this work, they are also used as solid material, Lagrangian

elements.

Cable elements are part of the family of beam elements. They connect 2 nodes, be-

longing to other parts and can exert forces between these two nodes. Their thickness

is defined through the cable section area.

Material models in LS-Dyna

For the solid and shell type of elements, a wide variety of material models are available in

LS-Dyna. Both rigid and deformable solid materials can be applied, as well as different

types of foam, tissue, fluids, . . . 1.

However, since the emphasis of this work is on the study of the fluid behaviour of the

SPH elements, solid and shell elements were only used in a rigid material formulation as

boundaries for the fluid and as floating bodies.

There is a special material model to be assigned with cable elements. This model requires

the specification of the Young’s modulus and material density for the cable. It may be

clear that cable elements are not rigid and only exert force in tensile condition.

For the SPH elements, modelled as water, the choice is down to two material models.

Either the fluid material (*MAT_001_ELASTIC_FLUID) can be used, which is designed to

model fluids, or the null material (*MAT_009_NULL). The null material can be used to

model any type of homogeneous and isotropic fluid and solid. An equation of state needs

1A useful tool for material selection can be found under http://app.d3view.com/d3mat/index

23

http://app.d3view.com/d3mat/index


to be assigned in conjunction, for which a limited number of options are available. The

Gruneisen equation of state is a good choice when applied to fluids. For the fluid material,

the equation of state comes as a built-in function, given by equation 2.32.

Both material models have been employed and compared in the validation tests from

sections 3.4 and 3.5. However, it was found that the flow of water with these two materials

shows great similarity. Since *MAT_001_ELASTIC_FLUID is designed for the simulation of

fluids in LS-Dyna, this material model has been opted for initially.

Speed of sound

The classic SPH method uses artificial compressibility and pressure is determined by the

use of an equation of state. This is the approach used in LS-Dyna also. The alternative

would be the use of a Poisson equation, which results in an implicit scheme that requires

an iterative solution and thus more computations per time step. On the other hand, the

artificial compressibility means that the SPH simulates the propagation of sound through

the fluid [25]. Then, from the CFL condition for the stability of the computation, it is

seen that much shorter timesteps are needed. A decrease of the total computation time,

is therefore often achieved by artificially reducing the speed of sound. Monaghan advises

on this subject that a lower bound for the speed of sound c can be taken so that the Mach

number stays below M ≈ 0.1 in the simulation.

In LS-Dyna, the speed of sound can be manipulated directly when the Gruneisen equation

of state is used together with the null material *MAT_009. When using the fluid material

*MAT_001, the choice of the bulk modulus determines the speed of sound through the

relation:

c =

√K

ρ(3.2)

The physical values for these parameters are K = 2.22GPa and c = 1498m/s, provided

that a water density ρ = 1000 kg/m3 is considered.

3.3 Contact definitions

In the classic SPH technique a laborious composition of boundary particles and ghost

points is used. Within LS-Dyna there is no need for such a construction. The software

allows mesh-based and mesh-free methods to co-exist and interact in one simulation. For

practical use, the SPH fluid particles can be bordered by three different types of bound-

aries.

24


The first type are the Lagrangian mesh elements of solid material, such as the shell

and solid elements that were discussed in the previous section. When functioning as

a rectangular, rigid wall, only one element is needed. When a deformable object is

modelled, a mesh needs to be composed, for which the resolution is to be decided

on case-by-case basis.

Since the SPH particles are not interconnected, only one-way type of contact defin-

tions are applicable in which the SPH is always defined to be the slave and the

elements are defined to be the master. The interaction between the SPH and other

elements is defined using penalty based contact algorithms. An appropriate con-

tact type is the simple *CONTACT_NODES_TO_SURFACE card, where the slave nodes

are checked for penetration through the master surface ([22]). When a node is in

contact with the surface, a restoring force is applied to prevent further penetration.

This force is proportional to the penetration distance into the shell or solid element

and acts in the direction normal to the master surface. The restoring force is defined

by ([10]):

f = k dn (3.3)

In this equation, d is the penetration distance, n is the surface normal vector and k

is a penalty factor, comparable to a spring constant. The constant k should be set

large to minimise penetration, but not so large that the CFL condition for stability

is violated. In this thesis, default values suffice.

For the handling of mesh-free particle methods and disjoint meshes, where prediction

of the contact location may be difficult or impossible, the automatic contact option

is recommended. These contacts are non-oriented, which means penetration coming

from either side of a shell element can be detected. The automatic contacts determine

the contact surface by projecting normally from the shell mid-plane a distance equal

to half the contact thickness. For this reason, the contact thickness for the slave

particles should be defined as equal to the particle spacing. Then, the particles

adjacent to the boundary keep a distance to the shell elements of half the particle

spacing, thus preserving the volume of water within the basin. If no contact thickness

is defined, a drop of the water depth by slightly more than half the particle spacing

can be observed.

The *RIGIDWALL option provides a simple way to treat the contact between the

SPH nodes and a rigid surface. The standard rigidwall part consists of an infinite

plane without any thickness, mass or material characteristics defined. Optionally,

the surface area can be made finite, or the part can be given a predefined motion in

25


time. However, no contact thickness can be defined for this element, which means

the same decrease of water depth is seen when used as the boundaries for a water

basin. Nonetheless, the rigidwall part provides an easy solution to create a border

that is not to be crossed by the particles.

A third type of boundary that can be applied to continuum domains modelled with

SPH particles is the *BOUNDARY_SPH_SYMMETRY_PLANE. As the name says, this allows

the definition of a symmetry plane, and does so by creating ghost particles, as in

figure 3.1.

Figure 3.1: Symmetry plane boundary condition for SPH in LS-Dyna

3.4 Validation test: elliptic drop

In this section, the evolution of an elliptic drop will be examined as a first verification of

the accuracy of the SPH, as included in LS-Dyna. The same test has previously been

carried out by Monaghan in his first work on free surface flow [25] and by Vaughan to

perform a first validation of the SPH code Marian [34].

3.4.1 Background for the elliptic drop test

The simple test consists of a 2-dimensional circular water drop with a predefined velocity

field at the initial state, chosen so that the drop will conserve its elliptic shape in time.

Under the assumption of an incompressible fluid in irrotational flow, the problem can be

26


described by a set of ordinary differential equations. The solution to these equations can

be obtained using numerical techniques and comprises the time evolution of the semi-

major and semi-minor axes of the ellipse. Thus, performing this validation test has the

advantage of comparing the results against the theoretical solution and the results obtained

by researchers in the field of SPH.

Figure 3.2: Particle positions for Monaghan’s elliptic drop after 0.8ms (left) and 8.2ms (right)

[25]

Theoretical description

A complete derivation of the ODEs that define the problem can be found in [34]. Here, a

brief outline of the equations and considerations for the theory is given.

The coordinates of all the points within the domain of the ellipse can be written as a

function of the parameters r and θ, where 0 ≤ r ≤ 1 and 0 ≤ θ < 2π. They are given by:

x = a(t)r cos θ (3.4a)

y = b(t)r sin θ (3.4b)

In the above relations a(t) and b(t) denote the length of the semi-major and semi-minor

axes of the ellipse as a function of time. Differentiation with time yields expressions for

the velocities within the ellipse, which ensure the drop remains elliptic with time varying

axes a and b. This condition is given by following equations:

27


vx =x

a

da

dt(3.5a)

vy =y

b

db

dt(3.5b)

The fluid must satisfy the continuity equation, which reduces to ∇ ·v = 0 for an incom-

pressible fluid. For the ellipse, this reduces to the condition that the product a(t) b(t) is

constant. This can also be seen when noting that conservation of mass is equal to conser-

vation of volume in an incompressible fluid. Thus, considering a 2-dimensional drop, the

area of the ellipse should be preserved. This area amounts to πab.

The momentum equation can be written for irrotational flow, enabling the use of a velocity

potential. Together with the requirement that the pressure is constant on the surface of

the drop, it can be deduced that:

dB(t)

dt= B(t)2 ω

2 − b(t)4

ω2 + b(t)4(3.6)

In equation 3.6 ω = a b is a constant, as stated earlier. The function B(t) is defined by

the relation:

db(t)

dt= B(t) b(t) (3.7)

Initial conditions

In the test, as performed by Monaghan and Vaughan, the drop initially has a circular

shape. The radius of the circle is 1m and the particles are centered around the point

with coordinates (0, 0). The particles are given an initial velocity, the value of which is

based on their initial coordinates. The initial velocity field is given by vx = −100x and

vy = 100 y. The pressure is initially set to 0Pa for the whole domain.

In the equations, these initial conditions correspond to b(0) = 1 and B(0) = 100. Also,

ω = 1 throughout the computation. Due to the velocity field, it is seen that the semi-major

axis of the ellipse coincides with the y-axis.

3.4.2 Model description

An image of the particle positions in the initial state is given in figure 3.3. In the simulation,

1976 particles were used, distributed on a rectangular array, lying within the radius of a

circle. The particle spacing is 0.04m in both directions and each particle has a mass of

1.60 kg, which remains constant during the whole simulation. In the material card, the

initial density is taken as ρ = 1000 kg/m3 and the bulk viscosity is given its real value for

28


water. This means the speed of sound in the fluid is c = 1500m/s, as was the case in

the simulations of Monaghan and Vaughan. Also, there would be little reason to opt for

a smaller value for the sound speed in this model, since the initial velocity almost reaches

10% of the actual speed of sound.

Table 3.1 summarises the main characteristics of the model.y

(m)

x (m)

0

-1 10

-1

1

Figure 3.3: Particle positions in the initial stage of the elliptic drop simulation

It can be seen in the model that the y-coordinate of the top-most particle at the initial stage

is not y = 1m, but y = 0.98m instead. This is because the node of each particle is located

at its centre. This node represents the whole particle as an element in the model, and its

exact shape at each stage is unknown. However, it is known that the particle occupies

a volume that corresponds to its mass (slightly fluctuating with its density). Thus, a

correction of half the particle spacing should be applied to the y-coordinate of the top

particle when looking for the value of the major radius b. Due to the uncertainty concerning

the exact boundary position, this correction of 0.02m is an estimate and therefore causes

a slight inaccuracy in the results.

29


Table 3.1: Characteristics of the elliptic drop simulation

number of particles 1 976

particle size 0.04m× 0.04m

CSLH 1.2

material model *MAT_001_ELASTIC_FLUID

speed of sound 1500m/s

simulation time 10ms

CPU time small, O(1 s)

3.4.3 Results

In figure 3.4 three images are given for the evolution of the elliptic drop. Monaghan,

in [25], remarks that, in his model, the particle configuration preserves a smooth outer

boundary. This cannot be said from the simulation in LS-Dyna where the boundary of

the ellipse looks more ragged in the later stages of the simulation.

The explanation for this difference may be that in Monaghan’s model the XSPH correction

was applied, while LS-Dyna uses a very basic variant of SPH. Recalling from section 2.3.3,

the XSPH helps to keep the particles more orderly in an incompressible fluid. This is also

confirmed by the observations of Vaughan, who performed simulations with and without

the XSPH, as well as Corrected SPH (CSPH) for enhanced consistency. This comparison

is shown in figure 3.5

(a) t = 0.8ms (b) t = 3.8ms (c) t = 8.2ms

Figure 3.4: Particle positions for the evolution of an elliptic drop

30


Figure 3.5: The effects of XSPH and CSPH variants on particle positions for the evolution of an

elliptic drop [34]. All snapshots are taken at time t = 8.2ms.

However, looking at the evolution of the semi-major axis gives a very good correspondence

to the theory. The results for the values of the major radius b are listed in table 3.2 for the

same points in time that are observed in the foregoing simulations (as in [25] and [34]).

The error in the values obtained for LS-Dyna model and listed in table 3.2 is always less

than 0.6% from the theoretical result, and in the same order of magnitude as recorded in

previous studies. Therefore, it can be said the model is running accurately.

Table 3.2: SPH and theoretical values of the major radius b. Results from Monaghan [25] and

Vaughan [34] are included.

Time (ms) Theory Monaghan [25] Marian [34] LS-Dyna

0.8 1.083 m 1.086 m 1.082 m 1.082 m

2.0 1.219 m 1.217 m 1.218 m

3.8 1.439 m 1.44 m 1.437 m 1.436 m

5.0 1.595 m 1.590 m 1.591 m

7.6 1.945 m 1.91 m 1.937 m 1.934 m

8.0 1.999 m 1.985 m 1.987 m

3.4.4 Density variation

Monaghan [25] notes that the deviation from the reference density ∂ρ can be predicted

from following relation:

∂ρ

ρ=v2

c2(3.8)

31


For the elliptic drop simulations, the maximum initial speed is v = 100m/s and the speed

of sound is taken as c = 1500m/s. From equation 3.8 it is seen that the density fluctuations

are less than 0.5%. The maximum density deviation from the reference density, as recorded

in LS-Dyna, is plotted in figure 3.6. The figure shows that a higher variability is found.

Vaughan comments that this is due to artificial clustering [35]. Although the effect found

here is less pronounced than in Vaughan’s simulations, where density variations up to 5 %

were recorded.

Figure 3.6: Maximum density fluctuation for the elliptic drop simulation.

Artificial clustering is a type of numerical error that is characterised by groups of particles

that are unphysically close, surrounded by a region with too few particles. Schussler and

Schmitt [30] found that clustering can occur when the kernel function has a maximum in

its derivative. This is the case for the cubic spline kernel (given by equation 2.22) as used

in LS-Dyna. When two particles are at a distance corresponding to the inflection point in

the kernel function, the forces with which they repel each other reaches a maximum and

then decreases as they come closer. Clustering is not found in the CSPH variant, which

uses a kernel gradient correction that is able to rule out the effect.

32


3.5 Validation test: dam break

The SPH European Research Interest Community (SPHERIC) has defined several bench-

mark test-cases for SPH. For the study of free surface water flow, a simulation of a bursting

dam is proposed. The test, defined by Kleefsman [16], is made to reproduce a dambreaking

experiment, which was performed at MARIN. This experiment has provided data for the

pressures, as well as for the water depth as a function of time at a number of locations.

Moreover, the breaking dam problem in general has been studied as a solution of potential

flow theory, and a number of researchers in the field of SPH have used the dam break

problem as a validation case for their code.

3.5.1 The MARIN experiment

The experiment was conducted at the Maritime Research Institute Netherlands (MARIN)

in a large tank of dimensions 3.22 × 1 × 1m with an open roof. Before the experiment

starts, a fixed volume of water with water depth h = 0.55m is contained in the right part

of the tank, which is closed by a door. At t = 0 s in the experiment, the door is instantly

opened by releasing a weight and the water starts to flow into the empty part of the tank.

Inside the tank, a box has been placed, meant to be representing a container on the deck of

a ship. On this box eight pressure sensors are placed to take measurements, four on top of

the box and four on the side facing towards the upcoming water. Also, four height probes

are installed in the tank. Figure 3.7 shows a representation of the tank at the instant after

releasing the door, with the positions of the height probes and pressure sensors.

A detailed description of the tank dimensions and positions of the door, sensors and probes

can be found in the test case description by SPHeric [13]. This document is accompanied

by the experimental results for all measurement positions up to t = 6 s and a video of the

experiment with a timestep of 0.04 s between the frames.

Figure 3.7: Initial state (left) and measuring positions in the dambreak experiment [16]

33


Kleefsman has previously simulated this dambreaking test-case with a Volume-Of-Fluid

(VOF) method [16]. He used a grid of 236×76×68 cells and the fluid is initially at rest in

the right part of the domain. In this simulation a very good correspondence was achieved

between the model and the experiment in both water level and pressure results. Some

snapshots of the MARIN experiment, as well as Kleefsman’s simulation are given in figure

3.10. A comparison of results for the water level is given in figure 3.8, pressure results are

given in figure 3.9.

(a) H4, in the tank (b) H2, in the reservoir

Figure 3.8: Vertical water heights, from Kleefsman’s VOF model and the MARIN dam break

experiment [16]

(a) At sensor P1 (b) At sensor P3

Figure 3.9: Water pressures at the front face of the box, from Kleefsman’s VOF model and the

MARIN dam break experiment [16]

34


(a) t = 0.40 s

(b) t = 0.56 s

(c) t = 0.72 s

Figure 3.10: Snapshots of a dam break simulation with a box in the flow, taken from Kleefman’s

VOF model (left) and the MARIN dam break experiment (right) [16]

35


3.5.2 Comparison to experimental results

3D simulation

A three-dimensional model was created in LS-Dyna to reproduce the dam break test case

as described in [13]. Apart from the geometry definition, also the number of particles and

the relevant physical quantities are prescribed, as given in table 3.3. The grid is to be

made of 67 × 54 × 30 particles, initially at rest as a rectangular cuboid in the right part

of the tank.

Table 3.3: Physical quantities in the dambreaking test case

gravity acceleration g = 9.81m/s2

reference density ρ = 1000 kg/m3

kinematic viscosity ν = 1.1

Furthermore, the configuration of the box and boundary walls to be modelled with fixed

SPH particles is specified in the description by SPHeric. However, LS-Dyna offers the

option to simply construct the boundaries with rigid shell elements. Also the box is

modelled using shell elements. This is done so to retrieve pressure data for fixed locations,

such as the pressure sensors in the experiment. LS-Dyna can give the force on a surface

element in the output, from where the pressure can be calculated by dividing by the area

of the element. The box shell elements have a size of 20mm × 20mm, which is in the

same order of magnitude as the SPH particles.

The 3D model of dam break used in this comparison has following characteristics:

Table 3.4: Characteristics of the 3D dam break simulation


particle size 18.3mm× 18.5mm× 18.3mm

CSLH 1.1



simulation time 6 s

CPU time 21hrs, 40min

In a comparison with the images from the MARIN experiment, the 3D simulation in LS-

Dyna, shown in figure 3.11, appears to agree well. The time instants at which the surge

36


front reaches the box and the back wall correspond. Also the high splash-up and spray

when hitting the box is seen, and, unlike the VOF model, is not bounded by an upper

limit in the vertical direction.

(a) t = 0.40 s

(b) t = 0.56 s

(c) t = 0.72 s

Figure 3.11: Snapshots of the 3D dam break simulation with a box in the flow in LS-Dyna

37


This correspondence in free surface flow is confirmed by the results for the water height

at probes H2 and H4, as seen in figure 3.12. Certainly in the beginning phase, before the

box is reached by the surge front, an almost exact match for both H4 and H2 is achieved

in the SPH model. However, it can also be seen that in the graphs some peaks in the

experimental curve are completely missed in the simulation, although the general trend is

preserved nonetheless.

For a good interpretation, it could be worthwile to map out how these curves were acquired,

since the used method causes some uncertainty. These considerations are:

The curve for the simulated results was obtained by tracking all particles having an x-

coordinate within a range of 0.015m on either side of the probe location (x = 2.638m

for H4 and 0.992m for H2). Then, the particle with the highest z-coordinate, and

still belonging to the main body of water, is found and this z-coordinate is added to

a table. This is repeated for every timestep.

The method does not take in account the difference in water level in y-direction, so

that some of the three-dimensional character of the SPH result is lost in the curves.

This may be the cause that the dips in the graph for H2 (from t = 2.6 s to 4.7 s) are

not followed by the curve for the SPH simulation. Examining the animated result

more closely indicates that this may indeed be the case.

In selecting the value for the water level to be put in the graph, only particles

belonging to the main body of water are taken in account. Spray is not considered,

and for high splash-up it is not always clear whether the particles are still attached

to the bulk of the water.

Neither the experimental curves, nor Kleefsman, nor SPHeric specify how the water

level was evaluated, or which type of sensors were used. In these curves, spray or

high splash-up could be the cause for some of the peaks.

From the video images of the experiment, it can be seen that an air-entrained layer

is formed on top of the free surface. Since air is not included in the SPH model, this

is not seen in the simulation. The foam layer is found to be certainly responsible for

the peak at 3.9 s in the curve for H4, and most likely also for the peaks in the H2

curve.

Altogether, it can be seen from figures 3.12 and 3.8 that the SPH model in LS-Dyna is

well capable of simulating the gravity-driven free surface flow of the dam break problem,

but performs less accurate than Kleefsman’s VOF model when the flow is turbulent. The

38


results hint that the SPH model could be improved when air is also taken into account in

the domain. However, this would highly increase the complexity of the model.

(a) Water level at probe H4 in the tank

(b) Water level at probe H2 in the reservoir

Figure 3.12: Comparison of vertical water heights

39


The pressures at the sensor locations P1 and P3 are given in figure 3.13. The pressures

in the graph were obtained by the particle force in contact with the shell element at the

considered location, divided by the element area A = 4 . 10−4m2.

(a) Pressure at P1

(b) Pressure at P3

Figure 3.13: SPH pressure results at P1 and P3 for the 3D dam break model in LS-Dyna

40


Clearly, the pressures obtained in LS-Dyna do not compare well to the experimental

pressures in figure 3.9. A very fragmented curve is seen with high peaks shortly after the

box is hit. The first peak after impact is substantially higher in the simulation: 31.5 kPa

versus 11.2 kPa for P1, and 13.6 kPa versus 6.4 kPa for P3.

A possible explanation for the bad correspondence could be that, since the particle size

is still quite coarse, the individual particle forces can be very high and the surface of the

shell element is not constantly hit by a particle, or is hit by several particles at the same

time, when artificial clustering has occurred.

2D simulation

To further study the pressure results, a 2D model of the same problem was made with a

much refined grid. The main characteristics of this model are listed in table 3.5. The shell

elements for the box are given a size of 4.0mm.

Table 3.5: Characteristics of the 3D dam break simulation


particle size 4.0mm× 4.0mm

CSLH 1.2



simulation time 6 s

CPU time 87hrs

Looking first at the pressure results in figure 3.14, the image is still very fragmented, but

a better correspondence is achieved. In both P1 and P3 it is found that the value for the

first pressure peak corresponds to the experimental value. Moreover, also an increase in

pressure is seen at the end of the simulation, albeit with a delay in comparison with the

experiment. The increase in pressure here has its origin in a bore, which is returning from

the tank wall, directed towards the front face of the box, where the pressure sensors are

located. The delay is found to be due to the alteration to 2D, which impedes the water

to flow back before the void behind the box is filled.

41


(a) Pressure at P1

(b) Pressure at P3

Figure 3.14: SPH pressure results at P1 and P3 for the 2D dam break model in LS-Dyna

42


The curves for the water level at H2 and H4 are plotted in figure 3.16, and some surprising

results can be seen. In the beginning of the simulation, the water level at H4 drops fast,

then increases to drop again soon after. This is repeated a number of times. In the

animated result, it is seen that the upper free surface first lowers, while staying more or

less horizontal. At t = 0.16 s, when the first minimum in the curve for H4 is reached, the

upper free surfaces slopes, which then causes the water level at H4 to rise slightly. This

behaviour is not observed in the 3D model. Comparing to the experimental results, it

is also unphysical, and no explanation could readily be found why this occurs in the 2D

simulation only.

The high peak in the water level at H4 is caused by a very high splash-up, as shown in

figure 3.15. Overall, the flow is found to be act more violently in the 2D simulation, which

could be expected since the water cannot pass by the sides of the box. The lateral width

of the box in 3D is 0.4m, which is 40% of the tank width. Considering this, the effect on

the splash-up looks plausible.

Figure 3.15: Splash-up at H2, t = 1.50 s

43


(a) Water level at probe H4 in the tank

(b) Water level at probe H2 in the reservoir

Figure 3.16: Comparison of vertical water heights for the 2D simulation

44


3.5.3 Comparison to analytical description

In 2009, Korobkin presented a description of the gravity-driven flow and free surface in

the initial stage of dam breaking by the method of matched asymptotic expansions [17].

In the analysis, dam break is taken as a 2-dimensional problem, and an infinite body of

water of water depth h behind the bursting dam is considered. A summary of the main

principles and results is given here. However, the asymptotic solution is not valid in the

vicinity of the intersection point between the initially vertical free surface and the bottom

of the tank.

A coordinate system O(x, y) can be defined where the x-axis is coincident with the initial,

horizontal free surface and the y-axis with the initial, vertical free surface. Thus, the side

wall x = 0, −h < y < 0 represents a dam, as shown in figure 3.17. Initially, the pressure

distribution is hydrostatic: p(t = 0) = −ρ0gy, where ρ0 is the fluid density.

Figure 3.17: Initial flow region, [17]

Assuming incompressible fluid and irrotational flow, the liquid flow is described in non-

dimensional variables by the velocity potential φ′(x′, y′, t′). The non-dimensional variables

are introduced as follows:

x = hx′, y = h y′, t = T t′,

φ = ghT φ′ and p = ρ0gh p′, where T =

√h/g

The velocity potential needs to satisfy the Laplace equation 3.9 and the boundary condi-

tions, given by equations 3.10 to 3.14. The functions X ′(y′, t′) and Y ′(x′, t′) represent the

part of the free surface which is initially vertical and initially horizontal respectively.

∇2φ′(x′, y′, t′) = 0 (3.9)

45


p′ = 0 at y′ = Y ′(x′, t′) and x′ = X ′y′, t′) (3.10)

∂φ′

∂y′=∂Y ′

∂x′∂φ′

∂x′+∂Y ′

∂t′at y′ = Y ′(x′, t′) (3.11)

∂φ′

∂x′=∂X ′

∂y′∂φ′

∂y′+∂X ′

∂t′at x′ = X ′(y′, t′) (3.12)

∂φ′

∂y′= 0 at y′ = −1 (3.13)

φ′ → 0 for x′ → +∞ (3.14)

The initial conditions are given by:

φ′(x′, y′, 0) = 0 (3.15)

X ′(y′, 0) = 0 (3.16)

Y ′(x′, 0) = 0 (3.17)

In a first-order outer solution, the free surface is described by equations 3.18 and 3.19.

Transforming back to dimensional variables and translation of the origin enables to plot

the free surface as a function of time and compare the free surface profile against the 2D

simulation in LSDyna, as shown in figure 3.18. In this comparison, also the second-order

outer solution for the free surface, as given in [17], is taken into account. However, it is

found that the second-order solution only adds a minor correction to the curves, in the

order of 1mm for the considered case.

X ′(y′, t′) =1

πt2 ln

[tan

π

4(1 + y)

](3.18)

Y ′(x′, t′) = − 2

πt2 arctan

[exp

(−πx

2

)](3.19)

46


(a) t = 20ms

(b) t = 36ms

(c) t = 60ms

(d) t = 160ms

(e) t = 200ms

Figure 3.18: Snapshots of the 2D dam break simulation in LS-Dyna in comparison with the

analytical solution for the free surface at the initial stage of dam break

47


It is seen that the free surface of the simulated flow and the theoretical solution match

fairly well initially. The agreement holds for about 40ms, from where the deviations

quickly become larger and the theoretical result finally breaks down after some 200ms.

The analytical solution was indeed meant to describe the dambreaking flow in the initial

stage only, but Korobkin does not specify what is meant by this in terms of a restriction

on (dimensionless) time. Nor is a validation against experimental results given in [17].

Although, on this subject, it could be argued that an experiment does not give reliable

information about the very initial stage, since a door needs to be mechanically removed,

which disturbs the initial flow.

Also the assumption of an infinite water reservoir in the theory can be considered. This

causes the water level for the theoretical result to be always slightly higher than in the

simulation, where a fixed amount of water is contained in the tank.

Concluding, even though some correspondence can be spotted at the initial stage of dam

break, it would be wrong to draw conclusions on the validity of the SPH model, solely

based on this analytical description.

3.5.4 Errors at the boundary

A note on the material model

The 2D and 3D reference models in the previous sections use the *MAT_001_ELASTIC_FLUID

as a material model for water. Closer examination of the fluid at the boundary shows that,

after 0.26 s in the simulation, the contact with the wall is temporarily lost for a number

of particles. Figure 3.19 show a gap at the boundary, taken at t = 0.36 s, when the effect

is the largest.

Figure 3.19: Boundary contact failure in 2D dam break simulation, using

*MAT 001 ELASTIC FLUID, at t = 0.360 s

A similar looking error is frequently encountered in literature (see figure 3.20). A wedge-

shaped gap in the beginning of the simulation is seen for SPH and XSPH, in the models

48


of Monaghan [25] and Vaughan [34]. The explanation given states that particles at the

top corner of the fluid experience a boundary force, which keeps the fluid further away

than at the base. This is due to the hydrostatic pressure being smaller near the top than

at the bottom.

(a) Monaghan [25] (b) SPHysics

Figure 3.20: Loss of contact with the boundary for 2D dam break simulations with particle

boundary definition

However, the models in LS-Dyna use a different contact definition than those in figure

3.20. Moreover, when the *MAT_009_NULL is used for the fluid, this problem is not en-

countered in the simulation. The reason why this does not occur here is not clear, but it

may be attributable to a different treatment of the viscosity. Unfortunately, no detailed

information is given on this subject in the LS-Dyna manuals.

Temporary voids at the bottom of the tank

Another type of unphysical ’void’ is seen for both material models and is caused by the

combination of artificial clustering and smoothing. This is illustrated in figure 3.21

When the fluid impacts the box, the SPH particles experience a sudden acceleration, which

causes a high amount of particles in the vicinity of the box to be drawn together and form

clusters. These small groups of particles can then be considered as one super-particle with

high mass and density. Where clusters in the bulk of the fluid can still move, this is more

problematic for those in contact with the boundaries. These particles are constrained to

remain next to the boundary and are completely stopped by the presence of the box. This

causes all boundary particles to pile up and form one long, clustered layer with zero speed

at the bottom.

The particles just above this layer are not constrained to keep contact and their velocity

is determined by smoothing. Below them lies the layer with zero velocity, and above them

the rest of the fluid moves with a high velocity in x-direction. This asymmetry causes

them to not only be dragged along by the bulk of the fluid, but also obtain an upward

49


Figure 3.21: Clustering of particles at the boundary leads to a temporary void near the bottom

velocity.

Eventually, this results in the tearing of a hole in the grid. This void expands in x-direction

with boundary particles being added to the clumped layer. Finally, the hole is closed again

by gravity, but the layer of clustered particles at the bottom remains.

An error with a similar cause is reported in section 5.3.2, as the so-called toothpaste jets.

50

Chapter 4

Simulation of regular waves

In the previous chapter, the behaviour of SPH simulations in LS-Dyna was investigated

by means of some test cases. It was confirmed that the SPH model in LS-Dyna compares

well to other SPH models found in literature and is able to realistically simulate gravity-

driven free surface flow. Some typical problems could be identified and difficulties in the

results were found.

This knowledge forms the basis for the creation of models to simulate regular and breaking

waves, as explicated in the current and the next chapter.

4.1 Model description

4.1.1 Introduction

In LS-Dyna, the Lagrangian water particles, when subject to gravity, always need to be

contained in a basin of some sorts. In a number of other SPH software codes, algorithms

have been included to mimic a smaller fragment of an infinite domain length.

For example, in PAM-CRASH, the lateral boundaries can be coupled to move along with

the particle motion along the boundary [1]. Lastiwka et al. [19] developed a boundary

condition that enables the simulation of flow through permeable boundaries in SPH. These

boundaries are associated with an inflow and outflow zone in which particles are created or

removed. In SPHysics, it is possible to make use of yet another type of boundary condition:

when a particle crosses over the lateral domain boundary it disappears, only to appear

again at the other end of the domain, with the same properties [23]. This is illustrated in

figure 4.1, where fluid particles and the solid box are crossing over boundaries.

No such methods can be used in LS-Dyna. Therefore, it is chosen to model a domain

similar to a laboratory wave flume, rather than the sea itself. Such wave flumes are

characterised by their length and a finite water depth. They have a wave generator system

on one end of the tank and a wave energy absorption system at the other end.

51

Chapter 4. Simulation of regular waves

Figure 4.1: SPH particles crossing over boundaries in SPHysics. Fringes are given for horizontal

velocity.

4.1.2 Wave generation

In laboratory wave flumes, waves are excited by the use of a wavemaker. The classic

method exists of a wave paddle, either a piston type, or a hinged flap type. Dean and

Dalrymple [5] note that the preferred type of paddle is the one whose motion most closely

matches the undisturbed particle trajectories. This means the piston type is the better

choice for generating waves in shallow water and the flap type is preferable in deep water.

The emphasis of this chapter is on regular waves in ’deep’ water, so a choice is made on

a flap type wavemaker, hinged at the bottom.

Figure 4.2: Wavemaker paddles: piston type and flap type, [5]

A different method for wavemaking, that has previously been applied to SPH, consists

of a moving tank bottom. This technique has been described and implemented in PAM-

CRASH by Groenenboom and Cartwright [9]. The boundary condition applied to the

tank floor mimics the motion of a deepwater wave at the depth of the floor, corresponding

to linear Airy wave theory. This motion excites the water wave in the SPH particles

continuously along the length of the tank.

However, this method has not been explored during the course of this thesis.

52


Figure 4.3: Wave generated by a moving floor in PAM-CRASH. Pressure contours and selected

particle trajectories are shown. [9]

4.1.3 Wave absorption

Without any special treatment of the second boundary wall, the incoming water waves

get reflected on the impermeable wall. This results in a partially standing wave profile

instead of the smoothly propagating progressive wave that was intended. The problem

of boundary reflection can be avoided in several ways, some of which were mentioned

in section 4.1.1. However, those concepts can not be implemented in LS-Dyna, and a

different method needs to be applied. Again similar to a laboratory wave tank, wave

energy absorption can be simulated in a numerical wave flume.

In Boussinesq wave modelling, the sponge layer technique is widely utilised for absorbing

boundaries. Theoretically, the sponge layer is designed to absorb all energy arriving at

the boundary within the fluid domain. However, this ideal boundary condition is rarely

achieved.

The sponge absorber is implemented by adding damping terms with artificial damping

coefficients to the momentum equation. These damping coefficients are usually determined

by empirical methods or numerical experiments from previous studies. Moreover, the

damping needs to be evaluated case by case. Hsu [12] found that the values for the

damping coefficients increase with increasing water depth.

For SPH models, a sponge layer has recently been described by Omidvar [28] for the open-

source code SPHysics. This is illustrated in figure 4.4.

In SPHysics, the absorbing layer is implemented by adjusting the Riemann solver approx-

imation for particle-particle interaction in the sponge layer domain. In this domain, a

parameter is gradually decreased to zero, so that the absorption is stronger when closer to

the impermeable boundary. In this way, all incident wave energy is effectively absorbed.

Since LS-Dyna is not an open-source code, an absorbing layer for SPH can not be imple-

mented in the same way as described by Omidvar. Therefore, a different definition of the

sponge layer is proposed here for SPH models in LS-Dyna, combining ideas from both

the above descriptions.

The absorbing layer is made of SPH particles with the same properties and spacing as the

53


Figure 4.4: Concept sketch of a model with wavemaker and absorbing sponge layer, by Omidvar

[28]

fluid particles. The only difference to the normal fluid particles is that an additional damp-

ing is applied to the sponge layer. This is done by the keyword card *DAMPING_PART_MASS,

which defines a mass-proportional damping to all particle motions [22]. When computing

the acceleration (at the end of the SPH calculation cycle, see figure 2.6), a damping force

is taken into account:

ma = P − F − F damp (4.1)

In this equation P is the external load vector, F is the internal load vector and F damp is

the force vector due to system damping. This damping force is defined as:

F damp = Ds mv (4.2)

where Ds is the damping coefficient. Optionally, this damping coefficient can be given a

different value for the x- and y-directions (and z-direction). However, this is not made use

of here.

The correct damping coefficient to be applied for efficient wave energy absorption is found

in an iterative way and differs from case to case. If the damping is too high, the particles in

the sponge layer show little movement and the incident wave gets reflected on the sponge

layer itself, causing a partially standing wave in the fluid domain. In the other case, the

sponge layer particles move along too much with the wave motion and the wave is reflected

on the impermeable wall.

This leads to thinking that an optimal damping coefficient exists, where all incident wave

energy gets fully absorbed. However, such a value would not easily be found by a method

of trial and error. Therefore, the damping coefficient is tuned to an acceptable level, where

the standing wave character in the wave progression can no longer be seen by the naked

eye.

54


4.1.4 Implementation of the model

A numerical wave flume, having a water depth of 2m, is modelled. At the left boundary,

a rigid, flap-type wavemaker is moved with a period T = 2 s, and an angular amplitude

θA = 0.14 rad. The total tank length is 25m, corresponding to four times the deepwater

wavelength. The absorbing sponge layer is spread over 1 wavelength, as in figure 4.5.

Ds = 2.5 is found to be an appropriate value for the damping coefficient. Particle mass is

m = 0.125 kg.

40 particles are taken along the vertical direction, which corresponds to the grid resolution

for SPH wave models found in literature, such as the one in figure 4.3. The particle spacing

in x- and y-direction is equal, resulting in a total 20 000 particles.

Figure 4.5: Concept sketch for the numerical wave flume in LS-Dyna

The model characteristics are listed in table 4.1 and table 4.2, where the model properties

are also given in dimensionless form to enable comparison of the results for different

simulations.

The stroke S is defined in the same way as in figure 4.2, and is computed by: S = 2h sin θA.

Table 4.1: Setup characteristics of the numerical wave flume


particle size 50mm× 50mm

CSLH 1.2

material model *MAT_009_NULL


simulation time 100 s

CPU time 37hrs

55


Table 4.2: Geometric characteristics of the numerical wave flume reference model

Period T (s) 2

Water depth h(m) 2

θA(rad) 0.14

h/λ0 0.320length of fluid domain

λ03

length of absorption layerλ0

1

S/λ0 0.089

Ds 2.5

4.2 Results

4.2.1 Behaviour of the model

In figure 4.6 selected snapshots of the progressing waves at the beginning of the simulation

are shown. The waves form directly at the wavemaker and then propagate towards the

sponge absorber. A steady state progressing wave movement over the three wavelengths

in the fluid domain is reached after some 5 to 6 periods of the wavemaker. In figure 4.9,

the total and internal energy is shown, and it can be seen that steady state sets in after

approximately 10 s.

Figure 4.7 shows trajectories of selected particles at the free surface when steady state has

already been reached. The particle paths in the fluid domain are not closed, and show

resemblance to the particle trajectories shown in [9] (see figure 4.3), where this is referred

to as the Stokes drift.

The particles in the absorption layer better keep position, which was the purpose. The

damped particles at the interface between the fluid domain and the sponge layer move

along with the wave motion. This movement decreases further in the sponge layer, and

the particles close to the impermeable wall show little or no displacements.

Due to the drift, a number of particles crosses over to the side of the sponge layer after some

time, as in figure 4.8. However, it takes a long time until the interface at the free surface

has significantly been moved towards the boundary wall. Moreover, it is not a violent

mixing and the water waves are still efficiently absorbed, thus preserving the progressing

wave motion.

56


(a) t = 1.5 s

(b) t = 6.6 s

(c) t = 10.8 s

Figure 4.6: Snapshots of the beginning stage of wave generation in SPH simulation

Figure 4.7: Particle trajectories and pressure contours for steady-state progressive waves in SPH

simulation at t = 19.6 s

Figure 4.8: Progressive waves in SPH simulation at t = 100 s with mixing between fluid domain

and sponge layer

From figure 4.9 it can be seen that the mean kinetic energy in the system decreases slightly.

This can be ascribed to the deformation over time of the interface between the fluid domain

and the sponge layer. As shown in figure 4.8, the contact surface between the two domains

57


Figure 4.9: Total and internal energy in SPH wave generation

has become larger, and the sponge layer actively dampens more movement from the fluid

domain.

4.2.2 Wave Height

The wave height H, defined as twice the wave amplitude, is dependent of the wave paddle

motion and the water depth.

In wavemaker theory, the steady-state wave height is analytically determined by solving

the generation of waves as a potential flow problem. The periodical displacement of the

wave paddle is then taken as a boundary condition. Such a wavemaker theory is described

in detail by Schaffer [29], for a wide range of wave paddles.

However, the theory assumes a domain of infinite length, which is not the case here.

Furthermore, the potential flow theory is based on the assumption that the fluid is in-

compressible, and therefore does not take into account the dispersion of sound waves in

the fluid. Due to the artificial incompressibility, pressure waves can not be avoided in the

SPH simulations. Moreover, the sponge layer is effective in absorbing the water waves,

but this is not true for pressure waves. Generated at the wave paddle, they get fully

reflected on the impermeable walls and the tank floor. Therefore, the wavemaker theory,

as a solution to potential flow theory would not be appropriate for comparison with the

SPH simulation.

A simple theory to predict the wave height in the vicinity of the wave paddle was developed

for shallow water by Galvin (1964, and described in [5]). He reasoned that the water

displaced by the wavemaker equals the crest volume of the propagating wave. This is

illustrated in figure 4.10.

58


Figure 4.10: Shallow water wavemaker theory by Galvin, [5]

The water displacement by the flap-type waveboard is given by:

V1 =1

2S h =

1

22h2 sin θA (4.3)

The crest volume of the generated wave is given by:

V2 =

∫ λ/2

0

H

2sin(kx) dx =

H

k(4.4)

Then the wave height can be calculated by:

H = k h2 sin θA (4.5)

For the observed case, this results in H = 0.58m. In figure 4.11, the vertical particle

motion at the free surface in the vicinity of the wave paddle is shown. It is seen that the

wave height generated at the wavemaker is approximately 0.55m.

59


Figure 4.11: Y-coordinate of the SPH particles at the free surface in the vicinity of the wave

generator

However, the steady state wave height further away from the wavemaker assumes a lower

value, as in figure 4.12. In the third wavelength of the fluid domain, the wave height is

seen to be approximately 0.30m. This loss of the wave amplitude as the wave propagates

has also been noted by other researchers and is not easily explained. The early simulations

of Cartwright and Groenenboom [1] showed some loss of wave height when the wave is

generated by the classic method of a wavemaker. However, they dismissed the concerning,

since the focus of their research was elsewhere. Vaughan [34] has studied SPH for the

simulation of shoaling and wave breaking. He found that these processes did not develop

in the expected way, mainly because of a loss in wave amplitude. But no explanation is

given as to why this loss in wave height occurs.

Thus, the cause for the loss in wave amplitude would require further investigation. An

interesting test case could be the propagation of a single, solitary wave over an even,

horizontal bottom. A loss of wave amplitude has also been reported in some experimental

results of a solitary wave [32]. However, this test was not performed within the duration

of this thesis.

60


Figure 4.12: Y-coordinate of the SPH particles at the free surface in the third wavelength of the

fluid domain for progressing waves in steady state

For further use of the model, the dependency of the steady-state wave height to the

water depth and paddle stroke is investigated. This is done by the testing of models with

the same main characteristics as the reference model (see table 4.1). For the geometric

characteristics all but one are kept the same as the reference model (see table 4.2). This

parameter can either be the dimensionless water depth or the dimensionless paddle stroke.

The characteristics of all models used in the test are listed in tables 4.3 and 4.4.

Table 4.3: Geometric characteristics of the numerical wave flumes tested for depth dependency

of the wave height

Model Depth 1 Depth 2 Reference model Depth 3 Depth 4

Period T (s) 2 2 2 2 2

Water depth h(m) 1 1.5 2 3 4

θA(rad) 0.283 0.187 0.140 0.093 0.070

h/λ0 0.160 0.240 0.320 0.480 0.640

S/λ0 0.089 0.089 0.089 0.089 0.089

Ds 2.5 2.5 2.5 2.5 2.5

61


Table 4.4: Geometric characteristics of the numerical wave flumes tested for paddle stroke de-

pendency of the wave height

Model Stroke 1 Stroke 2 Reference Stroke 3 Stroke 4

Period T (s) 2 2 2 2 2

Water depth h(m) 2 2 2 2 2

θA(rad) 0.094 0.125 0.140 0.157 0.173

h/λ0 0.320 0.320 0.320 0.320 0.320

S/λ0 0.060 0.080 0.089 0.100 0.110

Ds 2.5 2.5 2.5 2.5 2.5

The results for the wave height are given in table 4.5 and table 4.6. The trend of the wave

height is shown in figure 4.13.

Additionally, it is found that the damping coefficient Ds = 2.5 is in the acceptable range

for all models used in this test.

Table 4.5: Depth dependency

h/λ0 H/λ0

0.160 0.019

0.240 0.035

0.320 0.048

0.480 0.058

0.640 0.061

Table 4.6: Stroke dependency

S/λ0 H/λ0

0.060 0.034

0.080 0.043

0.089 0.048

0.100 0.051

0.110 0.054

62


(a) Depth dependency (b) Stroke dependency

Figure 4.13: Depth and stroke dependency of the steady state wave height in the SPH numerical

wave flume

4.2.3 Final remarks

1. It is found that reduction of the speed of sound is not to be recommended for

wave generation models. Even though the velocities stay quite low, the free surface

profile looks very distorted in the initial stage of the simulation when the speed of

sound is reduced. Only after some time in the simulation, a progressing wave can be

observed, but the density variation is still very large, as shown in figure 4.14. Density

fluctuations over 1% are noted already in the beginning of the simulation, whereas

the density in the simulation with c = 1489m/s deviates less than 0.1%. Over time,

the maximum deviation becomes larger in certain positions near the bottom and the

wave paddle, also for c = 1489m/s, due to artificial clustering.

For c = 150m/s, the steady state sets in after some 10 periods of the wave paddle.

Thus, the advantage of reduction in CPU time is counteracted by the time to reach

steady state and the overall quality of the simulation.

2. Following what was concluded from the dam break test case, the null material was

implemented for this simulation. However, at first, the *MAT_001_ELASTIC_FLUID

was used. From comparison of the simulations, no significant difference between the

two material models could be noted, not even at the boundary or contact behaviour.

Surely, the particle positions at each output frame are not exactly the same, but no

dissimilarity can be seen for the contours of velocity, pressure, etc.

3. In a recent paper, Omidvar [28] explores the possibility of local refinement of the

grid to reduce the number of particles. This was not investigated for LS-Dyna here,

but can be of interest when a certain area in the domain is to be examined in a

63


Figure 4.14: Particle density contours at t = 7.5 s of wave generation for SPH with reduced speed

of sound c = 150m/s.

more detailed way. In this way, a reduction of the CPU time can be achieved, while

keeping higher refinement in the studied area.

The locally refined grid is achieved by using a variable particle mass distribution

with small mass for the region where higher resolution is required and particles of

larger mass elsewhere. An example of a grid with two different levels of refinement

is given in figure 4.15. Typically, the particle mass ratio between two levels is 1 : 4.

Figure 4.15: Particle distribution of a still water problem at (a) the initial state and (b) t = 10 s,

from [28]

64

Chapter 5

Simulation of breaking waves

5.1 Introduction

In the breaking of waves, a large amount of energy can be released that results in a

turbulent flow of the water in the surf zone. This presents a heavy load to any structure

located in the breaker line and can lead to breaking rocks apart or moving large volumes

of sediment, causing erosion at a beach. For these reasons, it is worthwile to explore

the possibilities of numerical simulation to gain better understanding of the mechanisms

involved in wave breaking. SPH, as a Lagrangian particle-based method, is well suitable to

model free surface flow, and should therefore have a certain advantage over other numerical

methods.

This chapter focusses on the simulation of plunging breakers, since they present the heav-

iest load to a structure, such as a ship or a floating buoy. These breaking waves release

most of their energy at once, with a violent impact when the plunging jet hits the trough

in front of the wave.

SPH modelling of breaking waves in the literature

The numerical simulation of wave breaking and post-breaking phenomena with the SPH

method has challenged several authors. Monaghan (1994) was the first to model a very

short, steep wave onto a sloping beach [25]. He succeeded at generating a weakly plunging

breaker by use of a piston-type wavemaker.

Landrini et al. used SPH particles to simulate a breaking bore, driven by a piston with

constant speed, over a flat bottom surface [18]. In this manner, a plunging jet is created,

which induces splash-up and the formation of vortical structures. This is illustrated in

figure 5.1, showing the counter rotating vortices that move downward from the point of

impact.

65

Chapter 5. Simulation of breaking waves

(a) Comparison of the jet to BEM (b) Vortices in post-breaking

Figure 5.1: Breaking bore, driven by a piston, from [18]

Vaughan modelled a shallow water solitary wave on a plane sloping beach [34]. While he

expected a plunging breaker to develop at a certain point, the wave either spilled or was in

the process of surging, which was difficult to distinguish due to low resolution. Vaughan

suggests a number of possible reasons for this failure:

Particles are smoothed over the kernel support domain. When the wave is in shallow

water, the crest is formed by a small number of particles, and the cumulative effect of

smoothing causes an almost uniform velocity over a significant number of particles.

The wave has lost a lot of height when it reaches the expected breaking point. This

loss of height is the most significant reason why it doesn’t break as expected.

A toothpaste jet is observed at the boundary. This is a layer of particles that runs

parallel to the beach. The weight of the crest causes them to be squeezed, much like

toothpaste from a tube, forming an unphysical jet that flows up the beach.

Artificial clustering occurs, and might contribute to the failure.

Vaughan states that both the smoothing effect and the toothpaste jets can be reduced by

a better grid resolution, at the cost of higher CPU times.

Another simulation was carried out by Dalrymple and Rogers in their code SPHysics [4].

They describe a periodic wave that breaks as a plunger on a sloping beach. The results

shown in the publication encompass the identification of the stages of a wave in the surf

zone, the vortical structures in post-breaking and the pressure recorded at a point in the

surf zone. Selected snapshots of the evolution of the plunging wave are shown in figure 5.2.

In this figure, dark colored particles are associated with the plunger, and lighter particles

with the trough in front of the wave.

A similar model was implemented for a solitary wave by Khayyer, using Corrected In-

compressible SPH (CISPH) [15]. The simulation reproduces a wave flume experiment by

Li and Raichlen [20] and it is shown that CISPH performs significantly better than the

foregoing ISPH method at tracking the water surface.

66


(a) Overturning wave front (b) Splash-up

(c) Reverse breaking (d) Downburst

Figure 5.2: Snapshots of a plunging breaking wave on a beach, from [4]

5.2 Model description

In all simulations discussed in the previous section, apart from Vaughan’s, it is seen that

the wave breaking occurs at a point close to the wavemaker. The wave breaks primarily

because of the very large displacement of the piston-type wave paddle, and not because of

the influence of the bottom surface. Also, the waves in these models are generated in very

shallow waters, which is less interesting from a practical point of view when the breaking

wave loads on a floating structure are to be studied.

Therefore, two different models are presented. The first resembles a model found in

literature, since these are better documented. The second model uses a bottom step,

instead of a plane sloping beach, to quickly reduce the water depth. The waves are

generated at a flap-type wavemaker and would not break over an even, horizontal bottom.

The bottom step causes them to break as a plunger, in the part of the domain where the

water is deep.

Model I

Dalrymple and Rogers [4] describe a breaking wave on a beach of slope 1:13.5. The

wave period is 1.4 s and the initial water depth is 0.16m, meaning very shallow water:

h/λ0 = 0.05. The wave is created by a paddle and has an offshore wave height of 0.08m

(it is not clear how the piston motion was determined). In all, they used 97 915 particles

in the simulation, including the boundary particles.

67


The setup proposed here is made to resemble Dalrymple and Rogers’ model, based on the

data given in [4]. An image of the initial state is given in figure 5.3, and the main charac-

teristics are listed in table 5.1. The stroke of the piston-type wavemaker was determined

by trial and error to produce a clearly visible plunging jet.

Figure 5.3: Initial state of Model I for wave breaking

The number of particles used here is in the same order of magnitude as in the models

by Khayyer and Landrini, even though this is less than in Dalrymple and Roger’s model.

However, with this coarser grid, it is seen from table 5.1 that the CPU time is already

quite high.

Model II

In this model, the waves are generated in relatively deep water (h/λ0 = 0.64) by a flap-

type paddle. The wavemaker period is T = 2 s. The paddle stroke is large, which causes

the waves to be generated with a high skewness. However, these waves do not break over

a flat, horizontal bottom, as can be seen from figure 5.5. To induce breaking before the

wave loses too much height (as was the case in Vaughan’s simulation [34]), a bottom step

of 2.75m is built in to cause a sudden decrease of water depth. The wave then moves

further shoreward on a beach of slope 1:8, as shown in figure 5.4.

Figure 5.4: Initial state of Model II for wave breaking

68


Figure 5.5: Steady-state waves in deep water (h = 4m), generated by a flap-type paddle with

period T = 2 s and stroke S = 1.27m

Table 5.1: Characteristics of the models for wave breaking

Model I Model II

number of particles 15 600 12 639

particle size 5mm× 5mm 0.05m× 0.05m

CSLH 1.2 1.2

material model *MAT_009_NULL *MAT_009_NULL

speed of sound 1489m/s 1489m/s

simulation time 7 s 16 s

CPU time 45hrs 4hrs 40min

Water depth at paddle 0.16m 4m

Wave period 1.4 s 2 s

Paddle stroke 0.135m (piston) 1.27m (flap)

5.3 Results

The breaking of a wave on a beach is a complicated turbulent process, for which a fully

theoretical approach is not possible. Most of the investigations on wave breaking are

experimental or numerical, both of which are based on various simplifications. In addition

to this, none of the models found in literature are described in a sufficiently detailed

manner to be exactly reproduced here. Therefore, the discussion of the results of the

simulations in LS-Dyna will be mostly qualitative.

5.3.1 Breaking point

According to Stokes, the breaking point of a plunging breaker can be identified as the

point where the front face of the wave crest becomes vertical before a plunging jet shoots

forward. This position is shown for both models in figure 5.6. It can be noted that

69


geometric breaking criteria to predict the breaking position can not be applied to either of

the models. For Model I, this can’t be done because the wave is generated as a bore that

breaks close to the wavemaker. Model II, on the other hand, has a bottom step, whereas

the breaking criteria assume a plane, sloping beach.

The kinematic breaking criterion states that, at the point of breaking, the water particle

velocity at the crest becomes greater than the wave celerity. In shallow water, the celerity

may be approximated by C =√gh, which amounts to C = 1.25m/s for Model I. For

Model II, the deepwater wave celerity is C = 3.12m/s. It is found that the kinematic

criterion is fulfilled in both simulations, as can be seen in figure 5.6.

(a) Model I

(b) Model II

Figure 5.6: Contours of the x-velocity at the breaking point

5.3.2 Evolution of a plunging breaking wave for Model I

After the front face of the crest has assumed vertical position, a jet shoots forward from

the top of the crest, creating large splash-up when it plunges down in the trough in front

of the wave. Christensen and Deigaard [2] have described the splash-up as a very complex

process, that is responsible for the generation of large-scale vortical structures and plays

a role of importance in the viscous dissipation of wave energy. Due to large deformations

and splashing, the simulation of this process is found to be very difficult for grid-based

numerical methods (as in their LES-model).

70


Figures 5.7 and 5.8 illustrate the various stages in plunging wave breaking and splash-up,

as seen from the simulation. This is done for the results of Model I, in a qualitative

comparison with photographs taken from the laboratory experiments of Li and Raichlen

[20]. Snapshots are shown for the fourth wave in the simulation, which has its breaking

point at t = 4.85 s. In each panel of figures 5.7 and 5.8, the SPH snapshots are colored

to show which particles are associated with the plunger (black), and which are associated

with the fluid in front of the wave (blue).

The same development of the free surface flow can be identified for Model II, which is

performed in section 5.3.4.

(a) t = 4.99 s

(b) t = 5.02 s

(c) t = 5.05 s

(d) t = 5.11 s

Figure 5.7: Snapshots of plunging breaking, from 4.99 s to 5.11 s - qualitative comparison with

photographs from [20]

Touchdown of the overturning wave front (fig. 5.7c) initiates a disturbance in the free

71


surface, causing the initially smooth surface to become rough. In this phase of breaking,

a system of counter-rotating eddies is formed. It is seen that the plunging jet of water is

largely reflected, which corresponds to the photographs, and to the field observations of

Svendsen [33].

As the wave moves shoreward, the shape of the reflected jet changes and reverse breaking

occurs (fig. 5.8c). A downburst is created when the jet from reverse breaking hits the

water surface, and thumps down the vortices generated by the initial touchdown. In figure

5.9, it is seen that a number of particles, originally belonging to the plunging jet, is dragged

down towards the bottom by a pair of counter-rotating eddies.

Furthermore, it is noted that a higher concentration of artificial clustering is found at the

center of the vortices.

(a) t = 5.18 s

(b) t = 5.39 s

(c) t = 5.59 s

Figure 5.8: Snapshots of plunging breaking, from 5.18 s to 5.59 s - qualitative comparison with

photographs from [20]

Also the toothpaste jets, described by Vaughan [35], can be recognised in figures 5.7 and

5.8. This is a boundary effect where the bottom layer of particles is squeezed out from

under the wave.

To prevent the collapse of the water column, a sufficient pressure gradient is required

72


(a) Colored particle plot

(b) Velocity vector plot

Figure 5.9: Downburst due to reverse breaking impact, at t = 5.88 s

to oppose gravity. In order to generate a pressure gradient, a density gradient must

be generated. Where a particle has a lot of neighbours, each needs to move only a small

distance to cause an increase in density. Particles at the boundary have limited neighbours

and thus more movement is required from the surrounding particles. This causes them

to be more susceptible to clustering. In fact, the layer can be considered as one long

cluster, which constrains the particles in the layer to remain at the boundary. When the

high-pressure zone under the wave crest propagates through the fluid, this results in the

acceleration of the whole bottom layer.

As an effect of both smoothing and clustering, two remedies to avoid toothpaste jets can

logically be found:

1. A monotonically decreasing smoothing kernel effectively impedes the artificial clus-

tering, since this is caused by an inflection point in the kernel.

2. An increase of the number of particles used in the simulation may reduce the smooth-

ing effects.

It is seen that no toothpaste jets are formed in the simulation of Dalrymple and Rogers

(figure 5.2). In their paper, it is mentioned that a quadratic smoothing kernel, which has

no inflection point, was used in the simulation [4]. Unfortunately, LS-Dyna does not offer

this possibility.

73


5.3.3 Pressure

Dalrymple and Rogers show the time history of pressure, recorded at a fixed point in the

surf zone (the exact location is not specified). Pressure peaks are observed whenever a

breaking event occurs, as shown in figure 5.10a. A similar image can be obtained from

the LS-Dyna simulation of Model I, when the pressure of a particle close to the bottom

(but not in the clustered layer) is plotted. This particle is located under the wave trough

where the plunging jet impacts the water surface. It can be seen from figure 5.10 that the

pressure peaks are in the same order of magnitude as in the simulation by Dalrymple and

Rogers. Also, it is seen that the pressure in fig. 5.10b decays slower than in fig. 5.10a.

This can be attributed to the fact that a moving particle is observed, instead of a fixed

point. After touchdown of the plunger, the particle is dragged along by the wave, in the

high-pressure zone.

(a) SPHysics simulation by Dalrymple and Rogers [4]

(b) LS-Dyna simulation

Figure 5.10: Time history of pressure (kPa) at a point in the surf zone

74


5.3.4 Evolution of a plunging breaking wave for Model II

Figure 5.11 illustrates the different stages and processes in wave breaking for the results

of Model II. It is seen that the waves break in a weakly plunging manner, and particles

quickly fill the (normally air-entrained) void. The plunging jet and splash-up jets collapse

much sooner than for Model I and reverse breaking can barely be noticed. Therefore, the

vortices seen in this simulation are also weak and do not travel all the way to the bottom

of the beach.

(a) Overturning wave front (b) Touchdown of the plunger

(c) Initiation of splash-up (d) Reverse breaking

(e) Velocity vector plot (f) Beach runup as a periodic bore

Figure 5.11: Snapshots of the evolution of plunging breaking for Model II

It is seen that the resolution at the plunger is low, even though the amount of particles

used here is comparable to other model. A smaller particle size would certainly enhance

the accuracy of free surface tracking. And, as seen from the dam break test case, an even

smaller particle size is required when pressure data on a floating buoy are to be recorded.

However, this would result in a great increase of the computational cost. Local refinement

of the grid, as designed by Omidvar [28], is not an option here, since water particles travel

great distances in the surf zone and too much mixing between smaller and bigger particles

would occur.

75


To conclude, an image of the 3D simulation of a spherical buoy in the surf zone is presented

in figure 5.12.

(a)

(b)

Figure 5.12: 3D simulation of a spherical buoy in the surf zone

76

Chapter 6

Conclusion

In this thesis, the SPH method was studied for gravity-driven free surface flow, with a

focus on regular and breaking waves. This was performed by use of the commercially

available software LS-Dyna, which features a solver for SPH and enables Fluid-Structure

Interaction (FSI).

Two common validation test cases were executed to establish that the model operates

well, and to reveal some of the difficulties and errors. The elliptic drop test consists of

finding the solution to a system of ordinary differential equations. Both theoretical data

and the SPH results from various researchers are available for comparison, and it was

found that LS-Dyna is able to perform well at simulating the deformation of the water

drop. Nevertheless, the model is of a lower quality compared to those found in literature

that include XSPH and CSPH.

Another common test is the simulation of dam break; an idealised case where a cuboid

of water is exposed to gravity as from t = 0. A variation of this test case is proposed by

SPHERIC, modelled after an experiment at MARIN, where a box was placed at the center

of the domain. Comparison to the experimental results showed that the SPH model in

LS-Dyna performs very well as long as the flow stays laminar. After the water impacts

the box, it was seen that the general trends in the development of the free surface are

followed, but the curves for the water level show great divergence. This was found to be

due to the SPH model being single-phase, in this case meaning that air-entrapment can

not be simulated, and to some indistinctness in the method used for obtaining the water

level results.

Furthermore, it was found that results for the water pressure on a fixed surface are not

easily obtained for the SPH in LS-Dyna. On this point, the model underperforms in

comparison to Kleefsman’s VOF model [16], even though correspondence can be seen.

The pressure results are also seen to be disordered by artificial clustering. This is a typical

numerical error in SPH, that can arise when a smoothing kernel function with an inflection

77

Chapter 6. Conclusion

point is used. Then, there exists a maximum in the force by which particles repel each

other. When two particles are driven to each other, they can clump together and form

a cluster. In the dam break test, it was seen that the combination of clustering and

smoothing effects temporarily causes a large, unphysical void near the tank floor.

For the simulation of regular waves a numerical wave flume was designed, which incorpo-

rates an absorbing sponge layer to effectively absorb the incoming water wave energy by

damping the particle motion. It was shown that the wave height generated in the vicinity

of the wavemaker corresponds fairly well to the value expected by Galvin’s wavemaker

theory, but a loss of wave amplitude is experienced further downstream. This loss of wave

height could not readily be explained, nor was detailed information found in the literature,

where the same evolution has been encountered. A test case is suggested for further study.

Two types of models were tested for wave breaking. In the first type, an overturning wave

front is created periodically in the direct vicinity of the wavemaker, much like a break-

ing bore. The second type generates a wave, which breaks in a plunging manner farther

downstream as a result of the bottom geometry. It is seen that the first type generates

waves that break more violently, and would be more suitable to simulate a freak wave.

The second type, on the other hand, offers a more convenient geometry for a study of the

response of a floating structure, but would need a much higher resolution for accuracy of

free-surface tracking and pressures on a buoy. However, this would highly increase the

computational cost.

For both types, the breaking point occurs as expected from the kinematic breaking cri-

terion and the simulations show that the SPH method in LS-Dyna is able to capture

the main processes involved in the surf zone. These include splash-up, reverse breaking,

downburst and the formation of vortical structures.

A peculiar problem, named toothpaste jets, is seen to form at the boundary. These jets

consist of a long layer of clustered particles, which move along the bottom in an unphysical

way, due to smoothing effects and constrainment to stay in contact with the boundary.

This is not seen in the similar model of Dalrymple and Rogers [4], where artificial clustering

is not present, owing to the use of a quadratic smoothing kernel.

In general, it is seen that the SPH-solver in LS-Dyna is capable of simulating gravity-

driven free surface flow to some success, but great improvements to the software could be

made. The SPH code, as implemented in LS-Dyna, is a very basic one, which does not

include any of the variants or corrections found in literature. The software offers little

options for the adjustment of parameters and boundary conditions for SPH. Mainly the

restriction to a cubic spline smoothing kernel lies at the basis of many errors, since it is

responsible for artificial clustering, which was seen in every model, and disturbs the flow

wherever it shows up.

78

Bibliography

[1] B. Cartwright. Motion prediction of ships and yachts by smoothed particle hydrody-

namics. 2nd high performance yacht design conference, Auckland, February 2006.

[2] E.D. Christensen and R. Deigaard. Large eddy simulation of breaking waves. Coastal

Engineering, 42:53–86, 2001.

[3] R. A. Dalrymple and R.A. Eubanks, W.A. Birkemeier. Wave-induced circulation in

shallow basins. Journal of the Waterways, Ports and Coastal Engineering Division,

103, 1977.

[4] R.A. Dalrymple and B.D. Rogers. Numerical modeling of water waves with the SPH

method. Coastal Engineering, 53:141–147, 2006.

[5] R. G. Dean and R. A. Dalrymple. Water wave mechanics for engineers and scientists,

volume 2 of Advanced Series on Ocean Engineering. World Scientific Publishing

Company, Incorporated, 1991.

[6] Zeki Demirbilek and Vincent C. Linwood Vincent. Coastal Engineering Manual,

chapter II-I, Water wave mechanics. USACE, 2008.

[7] R. A. Gingold and J. J. Monaghan. Smooth particle hydrodynamics: theory and

application to non-spherical stars. Mon. Not. R. Astron. Soc., 181:375–389, 1977.

[8] H. Gotoh and S. Shao. SPH-LES model for numerical investigation of wave interaction

with partially immersed breakwater. Coastal Engineering, 46:39–63, 2004.

[9] P. Groenenboom and B. Cartwright. SPH prediction of free surface waves and the in-

teraction with objects. 5th European Conference on Computational Fluid Dynamics,

Lisbon, June 2010.

[10] John O. Hallquist. LS-Dyna Theory Manual. Livermore Software Technology Cor-

poration, 2006.

[11] Leo H. Holthuijsen. Waves in Oceanic and Coastal Waters. Cambridge University

Press, 2007.

79

Bibliography

[12] Tai-Wen Hsu. On the damping coefficients of sponge layer in boussinesq equations.

Wave Motion, 41:47–57, 2005.

[13] Reza Issa and Damien Violeau. Test Case 2: 3D-Dambreaking. SPHeric ERCOFTAC,

2006.

[14] G. R. Johnson. SPH for high velocity impact computations. Computer methods in

applied mechanics and engineering, 139:347–373, 1996.

[15] A. Khayyer. Corrected incompressible SPH method for accurate water-surface track-

ing in breaking waves. Coastal Engineering, 55:236–250, 2008.

[16] K.M.T. Kleefsman. A volume-of-fluid based simulation method for wave impact prob-

lems. Journal of Computational Physics, 206:363–393, 2005.

[17] A. Korobkin and O. Yilmaz. The initial stage of dam break. Journal of Engineering

Mathematics, 63:293–308, 2009.

[18] M. Landrini. SPH simulation of onshore bores with counter rotational vortical struc-

ture. 20th international workshop on water waves and floating bodies, 2005.

[19] M. Lastiwka. Permeable and non-reflecting boundary conditions in SPH. Interna-

tional Journal for Numerical Methods in Fluids, 61:709–724, 2009.

[20] Y. Li and F. Raichlen. Energy balance model for breaking solitary wave runup.

Journal of the Waterways, Ports and Coastal Engineering Division, 129(2):47–59,

2003.

[21] G. R. Liu and M. B. Liu. Smoothed Particle Hydrodynamics, a meshfree particle

method. World Scientific Publishing Company, Incorporated, 2003.

[22] Livermore Software Technology Corporation. LS-Dyna keyword user’s manual, 2007.

[23] Gesteira, Rogers, Dalrymple, Crespo and Narayanaswamy. User guide for the

SPHysics code, September 2010.

[24] J. J. Monaghan. On the problem of penetration in particle methods. Journal of

Computational Physics, 82:1–15, 1989.

[25] J. J. Monaghan. Simulating free surface flows with SPH. Journal of Computational

Physics, 110:399–406, 1994.

[26] J. J. Monaghan and R. A. Gingold. Shock simulation by the particle method of SPH.

Journal of Computational Physics, 52:374–381, 1983.

[27] Sang-Ho Oh. Experimental investigation of breaking criteria of deep water waves

under strong wind action. Applied Ocean Research, 72:235–250, 2005.

80

Bibliography

[28] P. Omidvar. Wave body interaction in 2D using smoothed particle hydrodynamics

(SPH) with variable particle mass. International Journal for Numerical Methods in

Fluids, 2011. Published online before inclusion in an issue.

[29] H.A. Schaffer. Second-order wavemaker theory for irregular waves. Ocean Engineer-

ing, 23(1):47–88, 1996.

[30] M. Schussler and D. Schmitt. Comments on smoothed particle hydrodynamics. As-

tronomy and Astrophysics, 97:373–379, 1981.

[31] S. Shao. Simulation of wave overtopping by an incompressible SPH method. Coastal

Engineering, 53:723–735, 2006.

[32] G. Simarro. Viscous effects on the propagation of solitary waves in a wave tank: a nu-

merical model. Ocean wave measurement and analysis, 5th international symposium,

Waves 2005, Spain, 2005.

[33] Ib A. Svendsen. Introduction to Nearshore Hydrodynamics, volume 24 of Advanced

Series on Ocean Engineering. World Scientific Publishing Company, Incorporated,

2006.

[34] G.L. Vaughan. Simulating breaking waves using SPH. Doctoral thesis, University of

Waikato, Hamilton, 2005.

[35] G.L. Vaughan. Completeness, conservation and error in SPH for fluids. International

Journal for Numerical Methods in Fluids, 56:37–62, 2008.

[36] J.R. Weggel. Maximum breaker height. Journal of the Waterways, Harbors and

Coastal Engineering Division, 98:529–548, 1972.

81

List of Figures

1.1 Point absorber WEC concept . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 SPH simulation of an overturning water front with plunging jet [18] . . . . 2

2.1 Ranges of suitability for various wave theories (Le Mehaute 1976) . . . . . . 8

2.2 Wave profile shape of different progressive gravity waves [6] . . . . . . . . . 9

2.3 Three types of wave breaking on beaches. Small figures denote different

stages of the breaking process [33] . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Representation of the smoothing kernel and support domain. While the

support domain is shown here as a circle, it is in fact a sphere in 3 dimen-

sions. [35] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Function used for the smoothing kernel in LS-Dyna, given for a 1-dimensional

situation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 LS-Dyna calculation cycle for SPH particles, [10] . . . . . . . . . . . . . . . 21

3.1 Symmetry plane boundary condition for SPH in LS-Dyna . . . . . . . . . . 26

3.2 Particle positions for Monaghan’s elliptic drop after 0.8ms (left) and 8.2ms

(right) [25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Particle positions in the initial stage of the elliptic drop simulation . . . . . 29

3.4 Particle positions for the evolution of an elliptic drop . . . . . . . . . . . . . 30

3.5 The effects of XSPH and CSPH variants on particle positions for the evo-

lution of an elliptic drop [34]. All snapshots are taken at time t = 8.2ms. . 31

3.6 Maximum density fluctuation for the elliptic drop simulation. . . . . . . . . 32

3.7 Initial state (left) and measuring positions in the dambreak experiment [16] 33

3.8 Vertical water heights, from Kleefsman’s VOF model and the MARIN dam

break experiment [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.9 Water pressures at the front face of the box, from Kleefsman’s VOF model

and the MARIN dam break experiment [16] . . . . . . . . . . . . . . . . . . 34

3.10 Snapshots of a dam break simulation with a box in the flow, taken from

Kleefman’s VOF model (left) and the MARIN dam break experiment (right)

[16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.11 Snapshots of the 3D dam break simulation with a box in the flow in LS-Dyna 37

82

List of Figures

3.12 Comparison of vertical water heights . . . . . . . . . . . . . . . . . . . . . . 39

3.13 SPH pressure results at P1 and P3 for the 3D dam break model in LS-Dyna 40

3.14 SPH pressure results at P1 and P3 for the 2D dam break model in LS-Dyna 42

3.15 Splash-up at H2, t = 1.50 s . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.16 Comparison of vertical water heights for the 2D simulation . . . . . . . . . 44

3.17 Initial flow region, [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.18 Snapshots of the 2D dam break simulation in LS-Dyna in comparison with

the analytical solution for the free surface at the initial stage of dam break 47

3.19 Boundary contact failure in 2D dam break simulation, using *MAT 001 ELASTIC FLUID,

at t = 0.360 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.20 Loss of contact with the boundary for 2D dam break simulations with par-

ticle boundary definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.21 Clustering of particles at the boundary leads to a temporary void near the

bottom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 SPH particles crossing over boundaries in SPHysics. Fringes are given for

horizontal velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Wavemaker paddles: piston type and flap type, [5] . . . . . . . . . . . . . . 52

4.3 Wave generated by a moving floor in PAM-CRASH. Pressure contours and

selected particle trajectories are shown. [9] . . . . . . . . . . . . . . . . . . . 53

4.4 Concept sketch of a model with wavemaker and absorbing sponge layer, by

Omidvar [28] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 Concept sketch for the numerical wave flume in LS-Dyna . . . . . . . . . . 55

4.6 Snapshots of the beginning stage of wave generation in SPH simulation . . 57

4.7 Particle trajectories and pressure contours for steady-state progressive waves

in SPH simulation at t = 19.6 s . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.8 Progressive waves in SPH simulation at t = 100 s with mixing between fluid

domain and sponge layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.9 Total and internal energy in SPH wave generation . . . . . . . . . . . . . . 58

4.10 Shallow water wavemaker theory by Galvin, [5] . . . . . . . . . . . . . . . . 59

4.11 Y-coordinate of the SPH particles at the free surface in the vicinity of the

wave generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.12 Y-coordinate of the SPH particles at the free surface in the third wavelength

of the fluid domain for progressing waves in steady state . . . . . . . . . . . 61

4.13 Depth and stroke dependency of the steady state wave height in the SPH

numerical wave flume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.14 Particle density contours at t = 7.5 s of wave generation for SPH with

reduced speed of sound c = 150m/s. . . . . . . . . . . . . . . . . . . . . . . 64

4.15 Particle distribution of a still water problem at (a) the initial state and (b)

t = 10 s, from [28] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

83

List of Figures

5.1 Breaking bore, driven by a piston, from [18] . . . . . . . . . . . . . . . . . . 66

5.2 Snapshots of a plunging breaking wave on a beach, from [4] . . . . . . . . . 67

5.3 Initial state of Model I for wave breaking . . . . . . . . . . . . . . . . . . . 68

5.4 Initial state of Model II for wave breaking . . . . . . . . . . . . . . . . . . . 68

5.5 Steady-state waves in deep water (h = 4m), generated by a flap-type paddle

with period T = 2 s and stroke S = 1.27m . . . . . . . . . . . . . . . . . . . 69

5.6 Contours of the x-velocity at the breaking point . . . . . . . . . . . . . . . . 70

5.7 Snapshots of plunging breaking, from 4.99 s to 5.11 s - qualitative compar-

ison with photographs from [20] . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.8 Snapshots of plunging breaking, from 5.18 s to 5.59 s - qualitative compar-

ison with photographs from [20] . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.9 Downburst due to reverse breaking impact, at t = 5.88 s . . . . . . . . . . . 73

5.10 Time history of pressure (kPa) at a point in the surf zone . . . . . . . . . . 74

5.11 Snapshots of the evolution of plunging breaking for Model II . . . . . . . . 75

5.12 3D simulation of a spherical buoy in the surf zone . . . . . . . . . . . . . . 76

84

List of Tables

2.1 Maximum wave steepness [33] . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Maximum wave height to depth ratio [33] . . . . . . . . . . . . . . . . . . . 10

3.1 Characteristics of the elliptic drop simulation . . . . . . . . . . . . . . . . . 30

3.2 SPH and theoretical values of the major radius b. Results from Monaghan

[25] and Vaughan [34] are included. . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Physical quantities in the dambreaking test case . . . . . . . . . . . . . . . 36

3.4 Characteristics of the 3D dam break simulation . . . . . . . . . . . . . . . 36

3.5 Characteristics of the 3D dam break simulation . . . . . . . . . . . . . . . 41

4.1 Setup characteristics of the numerical wave flume . . . . . . . . . . . . . . 55

4.2 Geometric characteristics of the numerical wave flume reference model . . . 56

4.3 Geometric characteristics of the numerical wave flumes tested for depth

dependency of the wave height . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 Geometric characteristics of the numerical wave flumes tested for paddle

stroke dependency of the wave height . . . . . . . . . . . . . . . . . . . . . 62

4.5 Depth dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.6 Stroke dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.1 Characteristics of the models for wave breaking . . . . . . . . . . . . . . . 69

85

Documents

Study of the SPH method for the simulation of regular and ......In this thesis, the SPH method, as implemented in LS-Dyna , is studied for the simulation of free surface water ow,