Video Games Fluid Flow Simulations Towards...
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UPTEC-F-14008 Examensarbete 30 hp Mars 2014 Video Games Fluid Flow Simulations Towards Automation Smoothed Particle Hydrodynamics Ann Johansson
Transcript of Video Games Fluid Flow Simulations Towards...
UPTEC-F-14008
Examensarbete 30 hpMars 2014
Video Games Fluid Flow Simulations Towards Automation Smoothed Particle Hydrodynamics
Ann Johansson
Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student
Abstract
Video Games Fluid Flow Simulations TowardsAutomation
Ann Johansson
A complete understanding of the cooling process when hot rolling steel is essential to understanding how the quality of the steel is connected to the cooling. This is why it is of great interest to simulate this process. However traditional CFD methods are too expensive in terms of CPU time. Knowing that video games successfully simulate fluids in reasonable time, those methods could be useful for simulating the cooling process in steel manufacturing. This would mean a loss in accuracy that could be acceptable. In this thesis different methods used for fluid simulations have been studied. The Smoothed Particle Hydrodynamics (SPH) method has been chosen. The method has been implemented for simulating the cooling process in MATLAB, which is a matrix operation based programming tool. Convincing results have been achieved for a big scale, but problems still remain for an implementation on a small scale.
Sponsor: ABB AB, Corporate Research, Västerås ISSN: 1401-5757, UPTEC F 14008Examinator: Tomas NybergÄmnesgranskare: Per LötstedtHandledare: Rebei Bel Fdhila
Sammanfattning
En full förståelse för kylningsprocessen vid hetvalsning av stål är väsentlig för att förstå hur
kvaliteten på stålet är kopplat till kylningen. Det är därför av stort intresse att simulera den
här processen. Däremot är traditionella CFD metoder alldeles för dyra i termer av CPU tid.
Men vi vet att dataspelsutvecklare har lyckats simulera flöden i rimlig tid. Dessa metoder
skulle kunna vara användbara för att simulera kylningsprocessen vid ståltillverkning. Detta
skulle innebära en förlust i noggrannhet som skulle kunna vara acceptabel. I det här
examensarbetet har olika metoder för att simulera flöden studerats. SPH (Smoothed Particle
Hydrodynamics) metoden har valts ut. Metoden har implementerats för att simulera
kylningsprocessen i MATLAB, som är ett matrisbaserat programmeringsverktyg. Trovärdiga
resultat har uppnåtts, men problem kvarstår för implementering på liten skala.
Uppsala University
Master Thesis
Video Games Fluid Flow SimulationsTowards Automation
Author:
Ann Johansson
Supervisors:
Rebei Bel Fdhila
Anders Daneryd
A thesis submitted in fulfilment of the requirements
for the degree of Master of Science in Engineering Physics
in the
Division of Scientific Computing
Department of Information Technology
March 2014
“We can do anything we want to do if we stick to it long enough.”
Helen Keller
UPPSALA UNIVERSITY
Abstract
Faculty of Science and Technology
Department of Information Technology
Master of Science in Engineering Physics
Video Games Fluid Flow Simulations Towards Automation
by Ann Johansson
A complete understanding of the cooling process when hot rolling steel is essential to
understanding how the quality of the steel is connected to the cooling. This is why it
is of great interest to simulate this process. However traditional CFD methods are too
expensive in terms of CPU time. Knowing that video games successfully simulate fluids
in reasonable time, those methods could be useful for simulating the cooling process in
steel manufacturing. This would mean a loss in accuracy that could be acceptable. In
this thesis different methods used for fluid simulations have been studied. The Smoothed
Particle Hydrodynamics (SPH) method has been chosen. The method has been imple-
mented for simulating the cooling process in MATLAB, which is a matrix operation based
programming tool. Convincing results have been achieved for a big scale, but problems
still remain for an implementation on a small scale.
Acknowledgements
Thanks to Rebei Bel Fdhila, Anders Daneryd and Lokman Hosain at ABB and Per
Lotstedt at Uppsala University for the support and help in completing this thesis.
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Contents
Abstract ii
Acknowledgements iii
Contents iv
List of Figures vi
Abbreviations viii
Physical Constants ix
Symbols x
1 Introduction 1
2 Problem Setting 3
3 Theory 5
3.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4 Methods 6
4.1 Why SPH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5 Smoothed Particle Hydrodynamics 9
5.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.3 History of SPH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.4 Solving the Navier-Stokes equation using SPH . . . . . . . . . . . . . . . . 14
5.4.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.4.2 Pressure force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.4.3 Viscosity force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.4.4 External forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.4.4.1 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.4.4.2 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . 18
5.4.5 A simple Navier-Stokes solver . . . . . . . . . . . . . . . . . . . . . 20
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Contents v
6 Implementation 22
6.1 Fast Nearest Neighbour search . . . . . . . . . . . . . . . . . . . . . . . . 22
6.1.1 Hash tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.1.2 Our own neighbour search algorithm . . . . . . . . . . . . . . . . . 23
6.1.3 Neighbour search algorithm . . . . . . . . . . . . . . . . . . . . . . 25
6.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.2.1 Leap-Frog Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.3 Boundary problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.4 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.4.1 Smoothing length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.4.2 Viscosity coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.4.3 Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.4.4 Gas stiffness constant . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.4.5 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.5 Implementation issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.5.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.5.2 Particles ”stick together” . . . . . . . . . . . . . . . . . . . . . . . 32
6.5.3 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.5.4 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7 Rendering 34
7.1 Our implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7.2 Surface tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7.3 Marching cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7.4 Point Splatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7.5 Ray Casting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
8 Results 36
8.1 Code structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
8.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
8.2.1 Water jet straight down . . . . . . . . . . . . . . . . . . . . . . . . 38
8.2.1.1 With Surface Tension . . . . . . . . . . . . . . . . . . . . 38
8.2.1.2 Without Surface Tension . . . . . . . . . . . . . . . . . . 39
8.2.2 Water jet hitting an inclining plane . . . . . . . . . . . . . . . . . . 41
8.2.3 Dam break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8.2.4 Water jet hitting a wall . . . . . . . . . . . . . . . . . . . . . . . . 43
8.2.5 Water jet in small scale . . . . . . . . . . . . . . . . . . . . . . . . 44
9 Discussion 45
10 Conclusion 47
A Kernels 48
Bibliography 49
List of Figures
2.1 (a) 3D schematics of the Runout Table. (b) 2D schematics of water jetsshown from the side. (c) Photo taken of the Runout Table. . . . . . . . . 4
4.1 The left figure illustrates the Eulerian viewpoint with a grid and the rightfigure illustrates the Lagrangian viewpoint with particles. . . . . . . . . . 6
5.1 Dirac delta function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 The three smoothing kernels from left to right; Wpoly6, Wspiky andWviscosity.The thick lines show the kernels, the thin lines their gradients and thedashed lines their Laplacians [1]. . . . . . . . . . . . . . . . . . . . . . . . 12
5.3 High density, hence repulsive pressure force. . . . . . . . . . . . . . . . . . 16
5.4 Low density, hence attractive pressure force. . . . . . . . . . . . . . . . . . 16
5.5 Rest density, hence no pressure force. . . . . . . . . . . . . . . . . . . . . . 16
5.6 Surface tension is caused by an imbalance of forces at the surface of theliquid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.7 Colour field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.1 Example of a hash table used for storing phone numbers [2]. . . . . . . . . 23
6.2 Simulation area is divided into three-dimensional boxes with side h. . . . 24
6.3 All neighbours will be found in the 3 × 3 × 3 boxes that constitutes thecurrent box and the adjacent boxes. . . . . . . . . . . . . . . . . . . . . . 24
6.4 The neighbours of the particle are inside the red circle, all the particlesinside the green square will be potential neighbours. . . . . . . . . . . . . 25
6.5 Arrays boxId and usedBoxes illustrated. . . . . . . . . . . . . . . . . . . . 26
6.6 Illustration of the Leap-Frog integration scheme, x is position, v velocityand t time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.7 Kernel function plotted with smoothing length h = 1.0 and h = 1.5. . . . 29
6.8 To the left, particles edge to edge, with a density lower than the densityof water. To the right, particles that inter-penetrate, with a density closerto the density of water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.9 To the left, three layers of particles initialized close after each other. Tothe right, three layers of particles initialized further apart from each other. 31
6.10 From left to right; the particles first experience an attraction force, inthe next time step they have advanced and are too close to each other,inducing a very high pressure force causing them to repel from each other. 32
7.1 The different cube configurations used in the Marching cubes algorithm [3]. 35
8.1 Main code structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
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List of Figures vii
8.2 Simulation including surface tension at times; t = 0.0s, t = 0.36s, t =0.72s, t = 1.08s, t = 1.44s and t = 1.80s. . . . . . . . . . . . . . . . . . . . 39
8.3 Simulation not including surface tension at times; t = 0.0s, t = 0.36s,t = 0.72s, t = 1.08s, t = 1.44s and t = 1.80s. . . . . . . . . . . . . . . . . 40
8.4 Simulation of inclining plane at times; t = 0.27s, t = 0.54s, t = 0.81s,t = 1.08s, t = 1.35s and t = 1.62s. . . . . . . . . . . . . . . . . . . . . . . 41
8.5 Simulation of a dam break at times; t = 0.0s, t = 0.3s, t = 0.6s, t = 0.9s,t = 1.2s, t = 1.5s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8.6 Simulation of a jet hitting a wall at times; t = 0.36s, t = 0.72s, t = 1.08s,t = 1.44s, t = 1.80s and t = 2.16s. . . . . . . . . . . . . . . . . . . . . . . 43
8.7 Simulation in small scale at times; t = 0.00s, t = 0.04s, t = 0.08s,t = 0.12s, t = 0.16s and t = 0.20s. . . . . . . . . . . . . . . . . . . . . . . 44
Abbreviations
CFD Computational Fluid Dynamics
SPH Smoothed Particle Hydrodynamics
ROT RunOut Table
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Physical Constants
Gravitational acceleration g = 9.81 m/s2
Dynamic viscosity of water µ = 1.003× 10−3 Pa · s
Density of water ρ = 998.29 kg/m3
Surface tension coefficient water/air σ = 0.0728 N/m
ix
Symbols
p pressure Pa
m mass kg
r distance m
h smoothing length m
f force N
u velocity m/s
µ dynamic viscosity Pa·s
ρ density kg/m3
σ surface tension coefficient N/m
x
1. Introduction
In high quality steel production the cooling process is of great importance for the steel
composition. A full understanding of the cooling is essential and the modelling, simula-
tion and prediction of the process are vital to improving it.
The cooling is the last part of the steel production process and is performed right before
the steel is coiled up. It occurs on the so called Runout Table (ROT), where the hot
steel is cooled by being sprayed with water jets.
Computational Fluid Dynamics (CFD) is often used in Research and Development
(R&D) where there is a need for high accuracy. Because of the complexity in sim-
ulating fluids, high accuracy means computationally heavy and very time consuming
simulations. Simulating larger parts of the Runout Table in the cooling process with
several water jets and important 3D effects, becomes too expensive in terms of CPU
time and can not be handled in an R&D industrial project.
Improving the cooling process could mean adjusting parameters in the problem setting.
However, testing this in reality is very expensive, therefore it is preferable to first use
simulations. This way, unnecessary tests can be avoided. Nevertheless, accurate simula-
tions require enormous amounts of computations resulting in extremely slow run times.
These sorts of simulations are currently used in the software Fluent, as an example
simulating one water jet can take almost a week.
On the other hand we know that developers in the gaming community have achieved real-
time fluid flow simulations with convincing results but with a loss in accuracy. Knowing
that industrial design, monitoring or control, often do not need the high accuracy that
the CFD methods can provide, the video games techniques to simulate and predict the
flow can be sufficient for many industrial processes.
1
Introduction 2
There are mainly three different aspects when speeding up computations: the software,
the hardware and parallelism/cloud computing. The focus in this thesis is the software.
The goal of this project is to explore the techniques used in video games when simulating
flow and to see to what extent they can be used to model the cooling process of the
Runout Table.
The main objectives of this thesis are to:
1. Investigate which numerical methods are used in video games when simulating
fluids, in particular when simulating liquids.
2. Choose a method suitable for simulating the cooling process on the ROT.
3. Implement the chosen method for a simplified version of the ROT.
4. Add complexity to the implementation to approach the real-world problem.
2. Problem Setting
When hot rolling steel is manufactured, the steel is heated then pressed together to
a thin plate, cooled down and finally rolled up. The cooling occurs on the so called
Runout Table, (ROT), which is shown in Figure 2.1. The upper image shows a 3D
schematics illustrating several water jets. The lower left image shows a 2D schematics
with two water jets and the lower right image shows a picture taken of the ROT. In
these images some of the different parameters are illustrated, for example the distance
between water jets, the height above the surface and the diameter of the inlet. All the
different parameter values could be adjusted in order to get a better result. Taking all
the parameters into account there are a great number of ways they can be coupled.
When cooling the steel, the most important aspect is the quality of the steel. Ideally,
the cooling will lower the temperature as much as needed, with as little water as possible
and without risking deterioration of the quality of the steel.
Ultimately the goal is to simulate the ROT in its entirety, i.e. several water jets, in
3D with thermodynamic effects. This thesis will scale down the problem and focus
on implementing a simulation with one water jet in 3D excluding temperature. More
specifically, the main problem setting will be one water jet hitting a plane surface from
25 cm above the surface. Other problem settings will also be simulated in order to
investigate the accuracy of the implementation. Those settings will be a water jet
hitting an inclining surface, a water jet hitting a wall and a dam break.
3
Problem Setting 4
Figure 2.1: (a) 3D schematics of the Runout Table. (b) 2D schematics of water jetsshown from the side. (c) Photo taken of the Runout Table.
3. Theory
The simulation of fluids plays an important role in a lot of different areas. It is used for
predicting weather, designing airplanes and cars, studying blood flow, studying magne-
tohydrodynamics and for developing video games. Almost all fluid simulations are based
on the governing equations called the Navier-Stokes equations.
3.1 Navier-Stokes equations
In the 19th century, Claude Navier and George Stokes formulated the famous Navier-
Stokes equations that describe the dynamics of fluids. They are derived from applying
Newton’s second law, conservation of momentum, to fluid motion and coupled with
the conservation of mass. The basic formulation of the Navier-Stokes equations for an
incompressible fluid is
ρ
(∂u
∂t+ u · ∇u
)= −∇p+ µ∇2u + f (3.1)
∇ · u = 0, (3.2)
where ρ is the density, u velocity, p pressure, µ viscosity and f external forces. The
second equation is the continuity equation for incompressible flow, which is derived
from the conservation of mass. In the general case these equations are impossible to
solve analytically. Therefore a lot of different numerical methods have been developed
to solve the Navier-Stokes equations.
5
4. Methods
There are two main categories of methods when numerically solving the Navier-Stokes
equations; grid-based Eulerian methods and particle-based Lagrangian methods. In the
Eulerian viewpoint, there is a fixed grid and the fluid is observed flowing in and out of
the grid cells. In the Lagrangian viewpoint the particles themselves and their movements
are observed. These concepts are illustrated in figure 4.1.
Figure 4.1: The left figure illustrates the Eulerian viewpoint with a grid and the rightfigure illustrates the Lagrangian viewpoint with particles.
A fluid is a continuum and the Navier-Stokes equations describe the motion of that
continuum. Since, in most cases, they are impossible to solve analytically numerical
methods must be used. This means that a discretization is necessary. One way of
discretizing is dividing the domain into a grid, which is what is done in Eulerian meth-
ods. Another approach is dividing the fluid into particles, which is done in Lagrangian
methods.
6
Methods 7
In areas such as physics, engineering and mathematics there is a need for high accuracy
and so accurate numerical methods are used. We will briefly discuss three of these
namely; Finite Differences, Finite-Element and Finite Volume Methods[4]. All of them
are grid-based Eulerian methods.
In Finite Differences partial spatial and temporal derivatives appearing in the equations
are approximated through Taylor series. These methods are typically employed only on
Cartesian geometries.
Finite-element methods require a 2D or 3D mesh. At each mesh element, a basis function
is used. It should locally describe the solution of the weak form of the governing equation
to be approximated. The finite-element method aims to minimize the difference between
the exact solution and the collection of basis functions.
The most commonly used method in science and engineering when simulating flow is
the Finite Volume Method (FVM), where the principle is local conservation. To solve
the equations numerically the computational domain is divided into cells. The partial
derivatives expressing a conservation principle can be rewritten at each cell as an alge-
braic contribution. The governing equation is reformulated at each computational cell
into a set of linear algebraic equations, which may be solved in an iterative manner.
In video games however there is no need for high accuracy, instead good performance
and plausible results are of importance. When it comes to liquid simulations the most
common methods are Eulerian grid-based methods, Smoothed Particle Hydrodynamics
(SPH) methods, Vorticity-based methods and Lattice-Boltzmann methods. These meth-
ods originated in the computational fluid dynamics community but have steadily been
adopted by the graphics community.
The Eulerian grid-based methods that are used in video games are mostly based on
Jos Stam’s Stable Fluids method [5]. This method uses a semi-Lagrangian advection
technique and implicit integration of viscosity to provide unconditionally stable behavior.
To be able to use an Eulerian grid-based method for simulating liquid, a surface tracking
method is also needed, such as the Level Set Method (LSM) or the Marker and Cell
(MAC) method.
Methods 8
Vorticity-based methods use particles representing small vortices, called vortons, which
induce a velocity field which in turn moves the vortons. To model the fluids motion it
solves the vorticity equation instead of the momentum equation [6].
Instead of solving the Navier-Stokes equations, which describe the macroscopic prop-
erties of a fluid, the Lattice-Boltzmann method solves the Boltzmann equation which
describes the microscopic behavior of a fluid. The method provides a first-order explicit
discretization of the Boltzmann equation in a discrete phase space. The simulation re-
gion is divided into a Cartesian grid of cells, each of which only interacts with cells of
its direct neighborhood [7].
Smoothed Particle Hydrodynamics is a mesh-free Lagrangian method. The principal
idea is to use a so called smoothing kernel, which approximates the Dirac delta function,
to interpolate properties of particles. This means that a particle’s physical properties
depend on the particles that lie within a certain distance from that particle.
4.1 Why SPH
The Smoothed Particle Hydrodynamics method was chosen for this project because it
has a number of advantages. Since it uses particles to represent the fluid there is no
need for the convective term in the Navier-Stokes equations. All particles represent a
constant mass, therefore mass conservation is ensured. As opposed to Eulerian methods
there is no need for a specific surface-tracking method and the particles can spread freely
in space, not bounded by a grid. It is also reasonably easy to understand and implement
and it is the most popular method in computer graphics for physically based simulation
algorithms [8].
5. Smoothed Particle
Hydrodynamics
Smoothed Particle Hydrodynamics (SPH) was developed in 1977 and was originally used
for astrophysical computations [9], but has since then been adopted in many different
areas [10]. It is a mesh-free Lagrangian method and as such models fluids by particles.
SPH is used to obtain approximate numerical solutions of the equations of fluid dynamics
by simulating the fluid with particles. Mathematically the particles can be seen as
interpolation points from which the properties of the fluid can be calculated. Physically
the particles can be seen as material particles. The particles can carry concrete entities
such as mass, position and velocity. They can also carry field quantities such as density,
pressure and temperature.
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Smoothed Particle Hydrodynamics 10
5.1 Derivation
The following subsection covers the derivation of the interpolating function used in SPH.
We know that the Dirac delta function, δ(x), (see figure 5.1),
Figure 5.1: Dirac delta function.
has the property that any function, A(x), can be evaluated at any point, x, as
A(x) =
∫A(x′)δ(x− x′)dx′. (5.1)
We introduce the smoothing kernel function,W (r, h), where r is position and h is the
smoothing distance. The kernel function is a normalized, positive and symmetric func-
tion
∫W (r, h)dr = 1 (5.2)
W (r, h) ≥ 0 (5.3)
W (r, h) = W (−r, h) (5.4)
and it approaches the Dirac delta function as h→ 0,
limh→0
W (r, h)→ δ(r). (5.5)
We replace the Dirac delta function in (5.1) with the kernel function. Thus, we have
that any scalar function, A(r), can be approximated at any point,r, by
A(r) =
∫A(r′)W (r− r′, h)dr′. (5.6)
Smoothed Particle Hydrodynamics 11
We discretize (5.6) by using the midpoint rule and instead of an integral we get the
summation
A(r) =∑j
A(rj)W (r− rj , h)∆Vj . (5.7)
We replace ∆Vj bymj
ρj, where mj is the mass of particle j and ρj is the density of particle
j and get
A(r) =∑j
mjAjρjW (r− rj , h). (5.8)
This is the interpolating function that is used in SPH when evaluating any scalar quan-
tity. If we are interested in the gradient or the Laplacian, since mj , ρj and Aj are
independent of r, we would get for the gradient
∇A(r) =∑j
mjAjρj∇W (r− rj , h) (5.9)
and for the Laplacian
∇2A(r) =∑j
mjAjρj∇2W (r− rj , h). (5.10)
These summations are over all particles j but only the particles within a radial distance
h will contribute to the sum. This is because the kernel function W will go to zero for
r ≥ h. This will reduce the computational cost from O(N2) to O(mN) where N is the
number of particles and m is the number of neighbours within a distance h.
Smoothed Particle Hydrodynamics 12
5.2 Kernel functions
Many different Kernel functions exist and we will here introduce three kernel functions
that are discussed in [1];
Wpoly6(r, h) =315
64πh9
(h2 − ||r||2)3 0 ≤ ||r|| ≤ h
0 otherwise
(5.11)
Wspiky(r, h) =15
πh6
(h− ||r||)3 0 ≤ ||r|| ≤ h
0 otherwise
(5.12)
Wviscosity(r, h) =15
2πh3
−||r||32h3
+ ||r||2h2
+ h2||r|| − 1 0 < ||r|| ≤ h
0 otherwise
(5.13)
These kernel functions, their gradients and their Laplacians can be seen in Figure 5.2.
Note the different scales on the y-axis. The kernel function Wpoly6 is used in the general
Figure 5.2: The three smoothing kernels from left to right; Wpoly6, Wspiky andWviscosity. The thick lines show the kernels, the thin lines their gradients and the
dashed lines their Laplacians [1].
case. It has been noted that if Wpoly6 is used for the pressure term, which includes the
gradient of the kernel function, particles tend to cluster together under high pressure.
This is because the gradient of Wpoly6 goes to zero for small distances and hence the
repulsive force disappears. Therefore the spiky kernel, which has a gradient that does
not approach zero when r → 0, is used for calculating pressure. We see in Figure 5.2
that the Laplacian of Wpoly6 becomes negative for small values of r. If Wpoly6 is used for
the viscous term, which includes the Laplacian, negative values can appear and increase
Smoothed Particle Hydrodynamics 13
the relative velocity thereby introducing energy to the system. This is non-physical
and to avoid it the viscosity kernel, which has a positive definite Laplacian, is used for
the viscous term. The kernels, its gradients and Laplacians are explicitly written in
Appendix A.
5.3 History of SPH
The Smoothed Particle Hydrodynamics method was introduced by Gingold and Mon-
aghan and independently by Lucy in 1977 when they were working on astrophysical
problems [9]. They derived the equations of motion of fluid dynamics using a kernel
estimation technique to estimate probability densities from sample values. Using this
technique for interpolation yielded an estimation of a function at any point using the
values of the function at the particles. This estimate could be differentiated exactly
provided the kernel function was differentiable. This meant the gradient terms in the
equations of fluid dynamics could be written in terms of the properties of the particles.
Because of the close relation to statistical ideas the method was described as a Monte-
Carlo method. However, in subsequent papers it was discovered that the errors were
much smaller than the predicted probability estimates.
Originally the numerical schemes proposed did not conserve linear and angular momen-
tum exactly but gave good results for the astrophysical problems that were considered
too difficult for the techniques available. In 1982 Gingold and Monaghan improved
the basic SPH algorithm to conserve linear and angular momentum exactly using the
particle equivalent of the Lagrangian for a compressible non-dissipative fluid [9].
Although SPH was originally introduced for use in astrophysical problems with com-
pressible flow. It has since been used in simulations of incompressible or nearly incom-
pressible flow. The technique has steadily been adopted in a number of different areas.
Some examples are simulations including elasticity and fractures, multifluid problems,
non-Newtonian fluids, virtual surgery and molecular dynamics [10].
SPH was introduced to the computer graphics community by Stam and Fiume to de-
pict fire and other gaseous phenomena [11]. Desbrun later used SPH to animate highly
Smoothed Particle Hydrodynamics 14
deformable bodies [12]. Muller et al. extended Desbruns method focusing on the sim-
ulation of fluids [1]. They also designed special purpose kernels and proposed a way to
model surface tension forces.
5.4 Solving the Navier-Stokes equation using SPH
The basic formulation of the Navier-Stokes equations for an incompressible fluid is
ρ
(∂u
∂t+ u · ∇u
)= −∇p+ µ∇2u + f (5.14)
∇ · u = 0 (5.15)
where u ·∇u is the convective term. Since we are using a particle system the convective
term is not needed because the flow will carry the particles along. The continuity
equation, which states the conservation of mass, is omitted because the particles will
carry the mass hence ensuring its conservation. So we are left with
ρ∂u
∂t= −∇p+ µ∇2u + f. (5.16)
The three terms on the left hand side represent the pressure force, the viscous force and
external forces. We denote these forces by F;
F = fpressure + fviscosity + fexternal (5.17)
and we get for particle i;
ai =duidt
=Fiρi
(5.18)
where ai is the acceleration of particle i. We integrate the right hand side to get the
velocity. Once we have the velocity we can integrate again to get the position. The
algorithm is simply computing these three forces for each particle, summing them up
and integrating.
Smoothed Particle Hydrodynamics 15
5.4.1 Density
The density is computed according to SPH as
ρi =∑j
ρjmj
ρjW (ri − rj , h) (5.19)
ρi =∑j
mjW (ri − rj , h), (5.20)
where ρj is the density for particle j and mj is the mass for particle j. The summation
is over all particles j within a radial distance h of particle i.
5.4.2 Pressure force
The ideal gas law states that
pV = nRT (5.21)
where p is pressure, V is volume per unit mass, n is number of gas particles in mol,
R is the universal gas constant and T is the temperature. In an isothermal fluid with
constant mass the right hand side will be kept constant and we replace it by a gas
stiffness constant k. The equation then becomes
pV = k (5.22)
p1
ρ= k (5.23)
p = kρ (5.24)
where ρ is the density. Desbrun and Cani suggest subtracting a “rest density” from the
density in the pressure equation, giving
p = k(ρ− ρ0) (5.25)
where ρ0 is the density when the fluid is at rest [12]. This new pressure equation will
introduce a pressure force that is repulsive when the density is higher than the rest
density, attractive when the density is lower than the rest density and zero if the fluid
is at rest. This is illustrated in figure 5.3, 5.4 and 5.5 [13].
Smoothed Particle Hydrodynamics 16
Figure 5.3: High density, hence repulsive pressure force.
Figure 5.4: Low density, hence attractive pressure force.
Figure 5.5: Rest density, hence no pressure force.
The pressure force is the gradient of pressure;
fp = −∇p. (5.26)
Smoothed Particle Hydrodynamics 17
So the pressure force at particle i in SPH notation becomes
fpi = −∑j 6=i
pjmj
ρj∇W (ri − rj , h). (5.27)
This pressure force is not symmetrical, which is non-physical since two interacting par-
ticles affect each other with the same pressure in opposite direction. There are two
different ways to correct this, one way is to replace the pressure in the equation by the
arithmetic mean of the pressure of particle i and particle j. This gives the symmetric
pressure force
fpi = −∑j 6=i
pi + pj2
mj
ρj∇W (ri − rj , h). (5.28)
Another solutions is to use the following equation
fpi = −ρi∑j 6=i
(pjρ2j
+piρ2i
)mj∇W (ri − rj , h), (5.29)
which we will now derive. We start by rewriting the gradient of a function A divided by
the density ρ as
∇(A
ρ
)=
1
ρ∇A− A
ρ2∇ρ. (5.30)
Rearranging gives us1
ρ∇A = ∇
(A
ρ
)+A
ρ2∇ρ. (5.31)
In SPH the expression becomes
1
ρ∇A =
∑j
Ajρj
mj
ρj∇W (r− rj , h) +
A
ρ2
∑j
ρjmj
ρj∇W (r− rj , h) (5.32)
∇A = ρ∑j
(Ajρ2j
+A
ρ2
)mj∇W (r− rj , h). (5.33)
We now replace A in equation 5.33 by −pi and get
−∇pi = −ρi∑j
(pjρ2j
+piρ2i
)mj∇W (ri − rj , h). (5.34)
However, according to Colin et al. [14] the arithmetic mean gives more accurate differ-
entials, hence (5.28) will be used for the pressure force.
Smoothed Particle Hydrodynamics 18
5.4.3 Viscosity force
Viscosity is a measure of the resistance to deformation by shear stress. For liquids, this
is often referred to as the thickness. The viscosity force in the Navier-Stokes equation is
fviscosityi = µ∇2u(ri). (5.35)
In SPH this becomes
fviscosityi = µ∑j 6=i
ujmj
ρj∇2W (ri − rj , h). (5.36)
This viscous force is not symmetrical which is physically inaccurate. It can be adjusted
by replacing the velocity of particle j with the velocity difference ui − uj . This is accu-
rate since the viscosity only depends on the velocity difference between the interacting
particles. The viscosity force therefore becomes
fviscosityi = µ∑j 6=i
(ui − uj)mj
ρj∇2W (ri − rj , h). (5.37)
5.4.4 External forces
External forces are all the forces affecting the fluid particles that are not explicitly in
the Navier-Stokes equation.
5.4.4.1 Gravity
The gravitational force fgravity acts on all particles straight downwards and is simply
added to the other forces.
fgravity = ρig (5.38)
where g is the gravitational acceleration.
5.4.4.2 Surface Tension
Surface tension occurs at the interface between fluids. Inside the liquid, water molecules
attract each other with equal forces and balance each other out. An imbalance of that
Smoothed Particle Hydrodynamics 19
attraction occur on the water surface where the surface molecules will only be attracted
by the molecules inside the fluid hence inducing a net force inward, see figure 5.6. The
surface tension force works to minimize the surface area. This is why liquid droplets
tend to form spheres at zero gravity. The net force tends to minimize curvature, which
is why the greater the curvature, the higher the force.
Figure 5.6: Surface tension is caused by an imbalance of forces at the surface of theliquid.
Muller et al [1] use the surface tension force
fsurface = σκn = −σ∇2csn
|n|(5.39)
where σ is the surface tension coefficient which depends on the fluids that form the
surface, in this case water and air, κ is the curvature of the surface, n is the surface
normal field and cs is the smoothed colour field. The colour field is a field quantity that
is 1 where there are particles and 0 elsewhere. This is illustrated in Figure 5.7.
Figure 5.7: Colour field.
Smoothed Particle Hydrodynamics 20
The SPH formulation of this quantity, using (5.8), becomes
cs(r) =∑j
mj1
ρjW (r− rj , h). (5.40)
The surface normal field, n, is obtained by taking the gradient of the smoothed colour
field
n = ∇cs (5.41)
using (5.9) and the divergence of n measures the curvature of the surface
κ = −∇n|n|
= −∇2c
|n|. (5.42)
The surface tension force will only apply to surface particles and a particle is identified
as a surface particle if
|n(ri)| > l, (5.43)
where l is a threshold parameter. The surface tension force is normally not a part of the
Navier-Stokes equation because it is regarded as a boundary condition.
5.4.5 A simple Navier-Stokes solver
Using the derived equations for density, pressure, pressure force and viscosity force a
simple Navier-Stokes solver would look as follows;
� Step 1: Compute density
ρi = ρ(ri) =∑j
mjWpoly6(ri − rj , h) (5.44)
� Step 2: Compute pressure
pi = p(ri) = k(ρ(ri)− ρ0) (5.45)
Smoothed Particle Hydrodynamics 21
� Step 3: Compute interaction forces
fpressurei = −∇p(ri) = −∑j
mjpi + pj
2ρj∇Wspiky(ri − rj , h) (5.46)
fviscosityi = µ∇2u(ri) =∑j
mjuj − uiρj
∇2Wviscosity(ri − rj , h) (5.47)
Sum up Fi = fpressurei + fviscosityi + fexternali (5.48)
� Step 4: Solve ODEs
vi =Fiρi, xi = vi (5.49)
� Step 5: Go to step 1
The ODEs in step 4 can be solved with any numerical integration technique; leap-
frog would be a good choice because it is of second order and does not require many
calculations.
6. Implementation
The implementation of the Smoothed Particle Hydrodynamics method was based on
pseudo code taken mainly from Muller’s Particle-Based Fluid Simulation for Interactive
Applications [1] and from Kelager’s Lagrangian Fluid Dynamics Using Smoothed Particle
Hydrodynamics [13].
The SPH method was implemented in MATLAB. The particle values such as density,
velocity and force were stored in arrays or matrices. Each particle is uniquely identified
by its index value. For example particle i has density dens(i) and velocity v(:, i), where
v is a 3 × N matrix where the first row represents the x-values, the second row the
y-values and the third row the z-value. Each column thus represents a particles x-, y-
and z-values of its velocity.
The simulation will run through a predetermined number of iterations, store the value
of the particle positions at the end of each iteration and then plot the particles at each
time step.
6.1 Fast Nearest Neighbour search
In particle simulations, the interacting forces between particles are computed in order
to calculate the force each particle will experience and then use that force to move
the particle in space. All particles will affect each other with forces but the further
away a particle is the less the effect will be. So to reduce the computational cost only
the particles closest, the neighbours, will be taken into account and the others will be
ignored. A particle is considered a neighbour if it is within a radial distance, h, called the
smoothing length. The problem of finding the neighbours can be very time consuming
so an efficient algorithm is needed.
22
Implementation 23
In each iteration in the algorithm the neighbours need to be found and stored because
they will be used in the computation of the densities and the forces.
6.1.1 Hash tables
Finding the nearest neighbours can typically be done using hash tables. The general
idea of a hash table is to use a hash function that takes a key as input and returns an
index to a bucket in the hash table, where the value is stored, thus mapping keys to
values, see figure 6.1. The risk with using hash tables are hash collisions, i.e. when two
keys are mapped to the same value.
Hash tables was the first method implemented in this project, using the coordinate
values of each particle as a hash key. The method is described in [15]. However to avoid
the risk of hash collisions and to save space from storing an unnecessary large hash table
another approach has been implemented.
Figure 6.1: Example of a hash table used for storing phone numbers [2].
6.1.2 Our own neighbour search algorithm
We divide our three dimensional simulation area into cells or boxes with a side length
equal to the smoothing length, h, as seen in figure 6.2. We number the boxes from left
to right and bottom up so all the boxes have a unique id number. The particles will
be associated to the boxes they are in by their coordinates, meaning all particles will
lie in a certain box. Since the neighbours of a particle will lie within a distance h, all
neighbours will be either in the same box or in an adjacent box. This means that you
Implementation 24
Figure 6.2: Simulation area is divided into three-dimensional boxes with side h.
are sure to find all neighbours in the 27 boxes that constitute the adjacent boxes and
the current box, see figure 6.3. This also means that instead of saving the neighbours of
Figure 6.3: All neighbours will be found in the 3 × 3 × 3 boxes that constitutes thecurrent box and the adjacent boxes.
every particle it suffices to save the neighbours of every used box. When the neighbours
need to be found for computations, simply check amongst the stored particles which
ones are within a distance h. Note that these boxes are imaginary and are not created
and stored.
We illustrate this idea of neighbours and boxes and particles in 2D in figure 6.4. All the
boxes that contain particles will be used boxes, in the figure these boxes are shown with
a yellow background. We want to find the neighbours of the particle in the middle. All
its neighbours are inside the red circle. However all the particles in the neighbouring
boxes, i.e. all the particles inside the green square, will be stored as potential neighbours.
When the neighbours are to be found for computations, the distance between the particle
in question and the potential neighbours will be computed and compared against the
smoothing length h.
Implementation 25
Figure 6.4: The neighbours of the particle are inside the red circle, all the particlesinside the green square will be potential neighbours.
6.1.3 Neighbour search algorithm
This neighbour search can be summarized as:
� For each particle i, get box id and store in array ‘boxId’.
� Check which boxes are used by getting the unique values in the array ‘boxId’ and
store in the array ’usedBoxes’. See figure 6.5.
� Create matrix ‘neighb’ for storing neighbours, where the number of rows equal
the number of used boxes and the number of columns equal a maximum allowed
number of neighbours.
� For each used box get neighbours by computing the ids of the adjacent boxes and
comparing them against the array ’boxId’. Store the neighbours in matrix ‘neighb’.
Each row will now contain all the neighbours of a specific box.
In the algorithm used for solving the Navier-Stokes equation we need to loop over all
particles when computing the density and the forces. Instead of looping over all particles
from start to end, we loop over all used boxes and then we loop over all the particles in
each particular box. This is efficient because all the particles in the same box will have
almost the same neighbours.
In this solution there is a maximum number of neighbours each box can store. This
maximum number is a user parameter initialized in the beginning of the simulation. If,
for some reason, the maximum number of neighbours is attained before having checked
Implementation 26
Figure 6.5: Arrays boxId and usedBoxes illustrated.
all the neighbouring boxing, the algorithm will be satisfied with the neighbours it has. To
make sure that all the particles in one box are neighbours, the center box is checked first.
This is needed because it is crucial that the particles are neighbours with themselves
for the summation that occurs in the density computation. It is important to note a
consequence of this, which is that particle i can have particle j as a neighbour, but it
does not necessarily mean that the reverse is true.
6.2 Integration
After having computed the forces for each particle we have the acceleration and we need
to integrate in time in order to get the velocity and then integrate again to get the
position. That is, we need to solve the ODEs
vi =Fiρi
and xi =viρi. (6.1)
When integrating numerically there are different integration schemes that can be used.
Some common schemes are the Explicit Euler Scheme, the Implicit Euler Scheme, the
Implementation 27
Verlet Scheme and the Leap-Frog Scheme. Leap-Frog was chosen since it is of second
order accuracy and it requires few evaluations per step.
6.2.1 Leap-Frog Scheme
In Leap-Frog the positions and the velocities are updated at interleaved time points.
They “leapfrog” over each other. For example if the positions are updated at integer
time steps and the velocities are updated at integer plus a half time steps. This is
illustrated in figure 6.6.
Figure 6.6: Illustration of the Leap-Frog integration scheme, x is position, v velocityand t time.
First the velocity at time t+ 12∆t is computed
vt+ 12
∆t = vt− 12
∆t + ∆tat (6.2)
then the position at time t+ ∆t is computed using the velocity at time t+ 12∆t
rt+∆t = rt + ∆tvt+ 12
∆t. (6.3)
The first iteration requires the velocity at negative time −12∆t, this velocity is given by
one Euler step.
v− 12
∆t = v0 −1
2∆ta0. (6.4)
To get the velocity at time t we take the average of vt+ 12
∆t and vt− 12
∆t
vt =vt− 1
2∆t + vt+ 1
2∆t
2. (6.5)
Implementation 28
6.3 Boundary problems
In this implementation, the main problem setting has been one simple water jet origi-
nating from an inlet above a planar surface and pointing straight down. The boundaries
have not been very complex.
There are different ways of dealing with the boundaries. One way is ghost particles or
boundary particles. They are implemented particles that represent the boundaries and
sometimes even the air. The idea is to simulate the air or the surface of the boundary
with particles that will affect the water particles the same way as the water particles
affect each other. In [16] Hagit Schechter and Robert Bridson propose a Ghost SPH
scheme where they create a narrow layer of particles in the surrounding air and solid.
Another possible way to deal with the boundaries is to use the Lennard-Jones potential
at the boundaries. This will attract particles far away and repel particles too close. The
risk is to have non-physical solutions as water tends to “stick” to surfaces when there is
a no-slip condition.
A very easy solution the boundary problem is to simply put the perpendicular component
of the velocity to zero or reflect a small percentage when a particle hits the surface. The
amount that is reflected is specified by the coefficient of restitution, cr. This solution
was chosen in this implementation. In each iteration after having moved the particles
by updating their positions, their coordinates are checked against the boundaries. If a
particle has hit the surface its perpendicular component is reflected by a small percentage
and if a particle is outside the simulation domain it is removed.
6.4 Parameter values
6.4.1 Smoothing length
The smoothing length, h, is of great importance in the Smooth Particle Hydrodynamics
method. Because the SPH interpolation formula use a kernel function that increases as
r decreases, the closer a particle is the more it will affect the sum. If the smoothing
length is too small the particles will have too few neighbours or no neighbours at all,
which will lead to imprecise results. On the other hand, the smoothing length may not
Implementation 29
be too large either. If h is too large the kernel function will be stretched out and the
particles close will not affect the sum as much. This is illustrated in figure 6.7, where the
higher red line corresponds to h = 1.0 and the dotted black line corresponds to h = 1.5.
It is clear from the plot that the greater the smoothing length is the less important the
particles at small distances will be.
Figure 6.7: Kernel function plotted with smoothing length h = 1.0 and h = 1.5.
In [9] Monaghan suggests the smoothing length,
h = σ
(m
ρ
)1/d
, (6.6)
where d is the number of dimensions and σ is a constant ∼ 1.3. There is however no
absolute best way to decide the smoothing length.
6.4.2 Viscosity coefficient
The viscosity used here, µ, is the dynamic viscosity. For water the dynamic viscosity is
µ = 1.003× 10−3. (6.7)
However, we will use a viscosity that is about a 1000 times higher than the physically
correct value. The reason for this is damping and numerical stability. Physically, the
viscosity term simulates the viscous behaviour of a fluid. Numerically, it damps the
Implementation 30
simulation and provides stability to the system. It is common to add a damping term
to the velocity when integrating. This is omitted and instead the only damping will be
from the viscosity.
6.4.3 Mass
In this implementation all particles have the same mass. It is determined from the rest
density, ρ0, and from the particle radius, r,
massparticle = ρ0V = ρ04πr3
3, (6.8)
where V is the volume of a particle.
6.4.4 Gas stiffness constant
We stated in 5.4.2 that the gas stiffness constant is
k = nRT, (6.9)
where n is the number of gas molecules in mol, R is the universal gas constant and
T is the temperature in Kelvin. The universal gas constant is R = 8.3144 JmolK , the
molar mass for water is 0.018 kgmol and 20 °C in Kelvin is 293.15K. This gives for our gas
constant
k =8.3144 · 293.15
0.018
[J mol K
kg mol K
]= 135, 409
[J
kg
]= 135, 409
[m2
s2
]. (6.10)
A gas stiffness constant of that size would require an extremely small integration time
step to avoid numerical instabilities. This constant is typically set in the order ∼ 1−10.
If k increases the time step, dt, must decrease.
6.4.5 Density
The rest density, ρ0, was set to the density of water
ρwater = 998.29 kg/m3. (6.11)
Implementation 31
It is worth to note that to obtain a density for the particles close to the real density
of water the particles must be allowed to inter-penetrate. This means that the distance
between the centre of two particles is allowed to be smaller than two radii. This concept
is illustrated in figure 6.8, where the particles on the left will all have density smaller
than the density of water and the particles on the right, which are closer, will have a
density much closer to that of water.
Figure 6.8: To the left, particles edge to edge, with a density lower than the density ofwater. To the right, particles that inter-penetrate, with a density closer to the density
of water.
6.5 Implementation issues
6.5.1 Initialization
The main problem setting is a water jet situated above a plain surface. The particles
are initialized in each iteration step in a circle at the top of the simulation area with an
initial velocity directed downwards. If the time step is small, the velocity is small and
new particles are initialized each iteration, the particles initialized in the previous step
will not have moved noticeably before the next set of particles are initialized. This is
illustrated in figure 6.9.
Figure 6.9: To the left, three layers of particles initialized close after each other. Tothe right, three layers of particles initialized further apart from each other.
If the particles are initialized too closely the simulation will be unstable due to the fact
that the pressure force is inversely proportional to the distance between the particles,
Implementation 32
r. If the particles are situated too close to each other the pressure force will explode.
This can be avoided by either initializing the particles more rarely or by increasing the
time step. However the time step can not be too large. If two particles are at a distance
from each other where they will experience an attraction force and the time step is large
the particles will be too close in the next iteration. This will induce a great repulsive
pressure force and the particles will be completely repelled. See figure 6.10.
Figure 6.10: From left to right; the particles first experience an attraction force, inthe next time step they have advanced and are too close to each other, inducing a very
high pressure force causing them to repel from each other.
6.5.2 Particles ”stick together”
If, for some reason, two particles end up in the same position i.e. they have the exact
same coordinates they will no longer be separable. The forces acting on the particles
depend on the distance between them, so if that distance is zero the force will be zero
and the particles will behave as one. The reason why the particles stick together is
unclear. An easy way to deal with this problem is to simply move one of the particles
slightly when it is detected that they are in the same place.
6.5.3 Scaling
SPH can not simply be rescaled in space and time. It requires parameter tuning, which
can be difficult. The parameters that need to be tuned are mainly the radius of the
particles, r, the time step, dt, the smoothing length, h, and the gas constant, k. This
has been a particularly difficult problem, since there are no absolute guidelines to how
to determine these parameters.
The main problem setting to be modelled was a water jet hitting a plane surface from
a distance of 25 cm above the surface. The jet originating from an inlet with diameter
3 cm. To be able to fit enough particles inside the inlet the particle radius was set to
Implementation 33
∼ 1− 3 mm. These small sizes meant a need for a really small time step, in the order of
∼ 0.0001 s. However the simulations with these parameter values didn’t look right and
were difficult to stabilize. Therefore the problem was scaled up 10 times. With an inlet
diameter of 30 cm, particle radius of 1 cm and a time step of 0.003 s the simulations
looked right and were stable.
6.5.4 Incompressibility
The biggest issue with the SPH method is the level of compressibility. Originally the
method was used for computations of compressible fluids and has since been adopted
for incompressible flow. For implementations in video games the compressibility is of
no concern as long as the simulations look convincing. However when used for other
purposes the compressibility can be a problem. Efforts have been made to enforce
incompressibility or to minimize it. For example, in [17] Markus Becker and Matthias
Teschner present a weakly compressible form of the SPH method for fluid flow based on
the Tait equation. Enforcing incompressibility means adding extra computations and
usually a smaller time step.
7. Rendering
7.1 Our implementation
In this thesis the rendering has been done by using the built-in MATLAB function plot. No
specific rendering technique has been implemented. This is because the implementation
was done in MATLAB, which does not provide any suitable graphics tools for rendering
water in a real-world manner.
There are however, different methods available for rendering fluid surfaces and we will
here describe a surface tracking technique and some of the most common techniques
available for rendering fluid surfaces.
7.2 Surface tracking
The colour field, cs, and its gradient field, n = ∇cs, introduced in Section 5.4.4.2 can be
used to track the surface. A particle is identified as a surface particle if
|n(ri)| > l, (7.1)
where l is a threshold parameter.
7.3 Marching cubes
One way of visualizing the surface is by rendering an iso surface of the colour field, cs.
If you have a field with scalar values, rendering an iso surface means you show a surface
34
Rendering 35
where the scalar values are the same. Each point in space is either inside or outside of
the surface.
Marching Cubes is a contouring algorithm to create surfaces of constant scalar value
in 3D [18]. The algorithm proceeds through the scalar field, taking eight neighbour
locations at a time. The eight values form the eight corners of an imaginary cube. Then
the polygon/polygons needed to represent the part of the iso surface that passes through
the cube are determined. The individual polygons are then fused into the desired surface.
The concept of marching cubes are illustrated in figure 7.1, where the yellow corners of
the cubes represent points inside the surface. The rest of the corners are outside of the
surface and surface elements are shown inside the cubes.
Figure 7.1: The different cube configurations used in the Marching cubes algorithm[3].
7.4 Point Splatting
Point Splatting is a technique where every volume element is splatted onto the viewing
surface in back to front order. These splats are rendered as disks whose properties
(colour and transparency) vary diametrically in normal (Gaussian) manner.
7.5 Ray Casting
The basic idea of ray casting is to determine the value of each pixel in the image by
sending a ray through the pixel into the scene. Then evaluate the data encountered
along the ray using some specified function in order to compute the pixel value.
8. Results
The goal with this project has been to implement a method used in video games fluid
flow simulations. When simulating flow in video games there are two important aspects.
The simulations need to be visually convincing and they need to be fast. It is of little or
no concern that simulations are physically accurate. When implementing this method
the focus has been mainly on the visual aspect, i.e. whether or not the simulations look
accurate.
8.1 Code structure
The structure of the code is illustrated in figure 8.1. The main function runSPH.m
initializes all the parameters, sets up the problem setting and calls the function SPH.m.
SPH.m initializes all the arrays containing mass, density, velocity etc. and starts the
iteration. In each iteration step new particles are initialized, the density, pressure and
forces are computed, the integration is performed, the limits are checked and the particles
positions are saved. There is a predetermined number of iterations. When they are
finished, the coordinate values are returned to runSPH.m which plots the particles and
stores the simulation.
36
Results 37
Compute Forces
Compute Pressure
Compute Density
Get Neighbours
Initialize Particles
SPH.m
runSPH.m
Compute Surface Tension
Integrate
Check Limits
Save Output
Plot Result
Figure 8.1: Main code structure.
Results 38
8.2 Simulations
The simulations in this section have used mainly the parameter values shown in table
8.1. If other values were used it is indicated in the text. All simulations have been
performed on a personal computer with an Intel Core i7-3630QM processor (4 cores),
2.40 GHz, 12.0 GB RAM and 64-bit operating system, Windows 8.1.
Description Symbol Value Unit
Rest Density ρ0 998.29 kg/m3
Particle Radius r 0.01 m
Inlet Radius R 0.15 m
Time Step dt 0.006 s
Dynamic Viscosity µ 3.5 Pa · s
Gas Stiffness Constant k 3 m2/s2
Coefficient of restitution cr 0.001 []
Surface Tension σ 0.0728 N/m
Smoothing Length h 1.3 · (m/ρ0)1/3 m
Initial Velocity v0 5 m/s
Surface Threshold l 7.065 []
Height above surface H 2.5 m
Table 8.1: Parameter values used in simulations.
8.2.1 Water jet straight down
8.2.1.1 With Surface Tension
This simulation was done with 40, 000 particles during 500 iterations. With dt = 0.006s
that is a total real-world time of 3.0s. The simulation took 8, 613s, that is 2 hours, 23
minutes and 33 seconds. In figure 8.2 the plots are shown from time t = 0.0s, t = 0.36s,
t = 0.72s, t = 1.08s, t = 1.44s and t = 1.80s.
The same simulation was run again after having used the built-in tool MATLAB Coder to
mex the files. This means that the files are first translated to C code and then compiled.
The compiled files are then used in the simulations. This is done in order to get a better
Results 39
performance. Because MATLAB code is compiled at runtime it is usually slower to run
than compiled C code. After mexing, the simulation took 7, 089s, that is 1 hours, 58
minutes and 9 seconds.
A simple check for accuracy was done using the equation
s = v0t+1
2at2, (8.1)
where s is the distance travelled, v0 the initial velocity, a the acceleration and t the
time. By noting the time it took for the first particle to hit the bottom surface then
using this to compute the distance travelled. This distance was calculated to be 2.54m.
In comparison with the actual distance 2.50m, the computed distance is off by 1.6%.
Figure 8.2: Simulation including surface tension at times; t = 0.0s, t = 0.36s, t =0.72s, t = 1.08s, t = 1.44s and t = 1.80s.
8.2.1.2 Without Surface Tension
In order to see how much the surface tension affects the simulation, the same simulation
was done once more but having removed the surface tension force as well as all the
computations it requires. This simulation was also done with 40, 000 particles during
500 iterations. With dt = 0.006s this is a total real-world time of 3.0s. The simulation
took 8, 616s, that is 2 hours, 23 minutes and 36 seconds. So there is no speed up in
comparison with the simulation with surface tension. In figure 8.3 the plots are shown
from time t = 0.0s, t = 0.36s, t = 0.72s, t = 1.08s, t = 1.44s and t = 1.80s. Comparing
the plots from the simulation with and without surface tension, there is very little
Results 40
difference in the two. This is most likely due to the fact that the water spreads out so
there is no thick fluid that produces an unambiguous surface.
For consistency this simulation was also done after mexing and the simulation time was
7, 372s, that is 2 hours, 2 minutes and 52 seconds.
Figure 8.3: Simulation not including surface tension at times; t = 0.0s, t = 0.36s,t = 0.72s, t = 1.08s, t = 1.44s and t = 1.80s.
Results 41
8.2.2 Water jet hitting an inclining plane
The problem setting was changed so that the bottom surface is subject to an inclination
angle, α, in this case α = 1°. This simulation was done with 40, 000 particles during
500 iterations. With dt = 0.006s that is a total real-world time of 3.0s. The simulation
took 5, 520s, that is 1 hours, 32 minutes and 0 seconds. After having mexed the files the
simulation was performed and took 4, 546s, that is 1 hours, 15 minutes and 46 seconds.
However at time t = 1.764s all the particles have left the simulation area. In figure 8.4
the plots are shown from time t = 0.27s, t = 0.54s, t = 0.81s, t = 1.08s, t = 1.35s and
t = 1.62s.
Figure 8.4: Simulation of inclining plane at times; t = 0.27s, t = 0.54s, t = 0.81s,t = 1.08s, t = 1.35s and t = 1.62s.
8.2.3 Dam break
This simulation was done with 11, 560 particles during 500 iterations. With dt = 0.003s
that is a total real-world time of 1.5s. The simulation area was a box of size 0.5× 0.5×
0.5m. The simulation took 3, 255s, that is 54 minutes and 15 seconds. After having
mexed the files the simulation was performed and took 1, 108s, that is 18 minutes and
28 seconds. In figure 8.5 the plots are shown from time t = 0.0s, t = 0.3s, t = 0.6s,
t = 0.9s, t = 1.2s and t = 1.5s.
To get a sense of how compressible the water is, the simulation was run for 2000 iterations
and at the end checking the height of the water level. Using this to compute the density
Results 42
as
ρ =m ·NV
, (8.2)
where m is the particle mass, N the total number of particles and V the volume of the
water. The density was computed to be 1940 kg/m3, which is 1.94 times as much as the
density of water, ρwater = 998.29 kg/m3. This clearly shows that the compressibility is
an issue in this implementation.
Figure 8.5: Simulation of a dam break at times; t = 0.0s, t = 0.3s, t = 0.6s, t = 0.9s,t = 1.2s, t = 1.5s.
Results 43
8.2.4 Water jet hitting a wall
This simulation was done with 40, 000 particles during 500 iterations. With dt = 0.006s
that is a total real-world time of 3.0s. The velocity was 10m/s and the smoothing length
was h = 1.69 · (m/ρ0)1/3. The simulation took 7, 421s, that is 2 hours, 3 minutes and 41
seconds. After having mexed the files the simulation was performed and took 5, 647s,
that is 1 hours, 34 minutes and 7 seconds. In figure 8.6 the plots are shown from time
t = 0.36s, t = 0.72s, t = 1.08s, t = 1.44s, t = 1.80s and t = 2.16s.
Figure 8.6: Simulation of a jet hitting a wall at times; t = 0.36s, t = 0.72s, t = 1.08s,t = 1.44s, t = 1.80s and t = 2.16s.
Results 44
8.2.5 Water jet in small scale
This simulation was done with 50, 000 particles during 1000 iterations. With dt =
0.0002s that is a total real-world time of 0.2s. The particles were initialized every sixth
iteration. The particle radius was r = 0.001m and the inlet radius was R = 0.015m and
the height above the surface was H = 0.25m. The coefficient of restitution was set to
zero, cr = 0.0. The simulation took 21, 576s, that is 5 hours, 59 minutes and 36 seconds.
After having mexed the files the simulation was performed and took 17, 446s, that is 4
hours, 50 minutes and 46 seconds. In figure 8.7 the plots are shown from time t = 0.00s,
t = 0.04s, t = 0.08s, t = 0.12s, t = 0.16s and t = 0.20s.
Figure 8.7: Simulation in small scale at times; t = 0.00s, t = 0.04s, t = 0.08s,t = 0.12s, t = 0.16s and t = 0.20s.
Comparing these plots with the ones in figure 8.2 and figure 8.3 it is clear that the
simulation in the small scale is more turbulent and not as smooth. Most likely this is
due to a poor choice in parameter values. It is worth to note that the smaller the particle
radius is, the smaller the time step needs to be and more iterations are needed so the
simulation will take longer time. Because the simulations become so slow it is very time
consuming to test different parameter values and the effect they have on the simulations.
Further investigation of the parameter values for this problem size will probably result
in more accurate simulations.
9. Discussion
In this thesis the SPH method has been implemented as it is used in video games.
Realistic results have been achieved for larger scales. However, it has been difficult to
achieve plausible results in smaller scales. This is because some parameters need to be
tuned, but there are no guidelines to know how they should be chosen.
So we have noted that we cannot simply have the same ratio and rescale our problem
setting. This means that the method is not linear. In order to understand how different
parameter values affect the simulation it would be wise to change one parameter at a
time and observe the result or derive some theoretical support for the scaling.
In particle methods such as SPH, the most time consuming part of an implementation is
usually the neighbour search. Typically the neighbour search is done using a hash table.
However, using hash tables always includes a risk of hash collisions and you need to store
a big hash table. To avoid this our own neighbour search algorithm was implemented.
In order to decide whether or not the SPH formulation, as it is used in Video Games
today, is suitable for the simulation of the ROT a deeper analysis of the accuracy needs
to be made. A relevant way to do this would be to simulate the same problem setting
using an SPH implementation and comparing with the currently used software Fluent.
Since the SPH method does not ensure incompressibility, this needs to be taken into
account in a future implementation.
One reason that the simulations are so time consuming in our implementation is the size
of the arrays. The size is fixed and is equal to the total number of particles used in the
simulation. However in all the examples, except for the dam break, the particles that
leave the simulation area are removed. This means that the array size is unnecessarily
large, because all the particles will not be in the simulation at the same time. In MATLAB
45
Discussion 46
the arrays can be of variable size, but not if the code is to be mexed, i.e. translated to
C code.
The purpose of investigating alternative methods for simulating the flow of the ROT
was to explore if an increase in performance is possible. If performance is of importance
in these types of simulations, it is recommended to use the computational power of
the Graphical Processing Unit, GPU. Using the GPU will most likely speed up the
simulation substantially. The algorithm could probably also achieve an additional speed
up by exploiting the parallelism of the SPH method.
10. Conclusion
The purpose of this project was to investigate whether or not the techniques used in
video games for simulating fluids could be used for simulating the cooling process of the
Runout Table (ROT). The method chosen and implemented was the Smoothed Particle
Hydrodynamics (SPH) method. The implementation did not reach a level of complexity
where it would make sense to compare the simulation against those in the currently used
software Fluent. This is mainly due to lack of time. Because of this we can not come to
the conclusion that SPH would be useful for simulating the ROT nor that it wouldn’t
be. We can only conclude that more research is needed.
It has been noted that the SPH method is not easily rescaled. In order to rescale an
implementation the parameter values need to be adjusted and there are no guidelines to
follow in order to know how to choose good parameter values.
However, comparing the simulation time of our implementation with that of Fluent, our
implementation is somewhere between 10− 100 times faster.
SPH is definitely a promising method for simulating fluid and has been used in a wide
variety of implementations and there are different techniques and methods when im-
plementing SPH. When applied in the field of science the accuracy is more important
and the performance is secondary, so the implementation is more focused on accuracy.
However, when applied in Video Games the performance is utmost important but the
accuracy is less important and so the focus is on performance. How much worse the
accuracy is remains to be investigated.
47
A. Kernels
Wpoly6(r, h) =315
64πh9
(h2 − ||r||2)3 0 ≤ ||r|| ≤ h
0 otherwise
(A.1)
∇Wpoly6(r, h) = − 945
32πh9
(h2 − ||r||2)2r 0 ≤ ||r|| ≤ h
0 otherwise
(A.2)
∇2Wpoly6(r, h) = − 945
32πh9
(h2 − ||r||2)(3h2 − 7||r||2) 0 ≤ ||r|| ≤ h
0 otherwise
(A.3)
Wspiky(r, h) =15
πh6
(h− ||r||)3 0 ≤ ||r|| ≤ h
0 otherwise
(A.4)
∇Wspiky(r, h) = − 45
πh6
r
||r||(h− ||r||)2 0 < ||r|| ≤ h
0 otherwise
(A.5)
∇2Wspiky(r, h) =90
πh6
1
||r||(h− ||r||)(h− 2||r||) 0 < ||r|| ≤ h
0 otherwise
(A.6)
Wviscosity(r, h) =15
2πh3
−||r||32h3
+ ||r||2h2
+ h2||r|| − 1 0 < ||r|| ≤ h
0 otherwise
(A.7)
∇Wviscosity(r, h) =15
2πh3
r(−3||r||3
2h3+ 2
h2− h
2||r||3
)0 ≤ ||r|| ≤ h
0 otherwise
(A.8)
∇2Wviscosity(r, h) =45
πh6
(h− ||r||) 0 ≤ ||r|| ≤ h
0 otherwise
(A.9)
48
Bibliography
[1] Matthias Muller, David Charypar, and Markus Gross. Particle-based fluid simula-
tion for interactive applications. In Proceedings of the 2003 ACM SIGGRAPH/Eu-
rographics symposium on Computer animation, pages 154–159. Eurographics Asso-
ciation, 2003.
[2] URL http://upload.wikimedia.org/wikipedia/commons/7/7d/Hash_table_3_
1_1_0_1_0_0_SP.svg.
[3] URL http://upload.wikimedia.org/wikipedia/commons/a/a7/
MarchingCubes.svg.
[4] Bengt Andersson, Ronnie Andersson, Love Hakansson, Mikael Mortensen, Rahman
Sudiyo, and Berend van Wachem. Computational Fluid Dynamics for Engineers.
Cambridge University Press, 2012.
[5] Jos Stam. Stable fluids. In Proceedings of the 26th annual conference on Computer
graphics and interactive techniques, pages 121–128. ACM Press/Addison-Wesley
Publishing Co., 1999.
[6] Michael J Gourlay. Fluid simulation for video games. Intel online articles, 2012.
[7] Youngmin Kwak. Advanced liquid simulation techniques for computer graphics ap-
plications. PhD thesis, University of Southern California, 2010.
[8] Bart Adams and Martin Wicke. Meshless approximation methods and applications
in physics based modeling and animation. In Eurographics 2009-Tutorials, pages
213–239. The Eurographics Association, 2009.
[9] J J Monaghan. Smoothed particle hydrodynamics. Reports on Progress in Physics,
68(8):1703, 2005. URL http://stacks.iop.org/0034-4885/68/i=8/a=R01.
49
Bibliography 50
[10] J.J. Monaghan. Smoothed particle hydrodynamics and its diverse applica-
tions. Annual Review of Fluid Mechanics, 44(1):323–346, 2012. doi: 10.1146/
annurev-fluid-120710-101220. URL http://www.annualreviews.org/doi/abs/
10.1146/annurev-fluid-120710-101220.
[11] Jos Stam and Eugene Fiume. Depicting fire and other gaseous phenomena using
diffusion processes. In Proceedings of the 22nd Annual Conference on Computer
Graphics and Interactive Techniques, pages 129–136. ACM, 1995.
[12] Mathieu Desbrun and Marie-Paule Gascuel. Smoothed particles: A new paradigm
for animating highly deformable bodies. In Computer Animation and Simulation’96,
pages 61–76. Springer, 1996.
[13] Micky Kelager. Lagrangian fluid dynamics using smoothed particle hydrodynamics.
Master’s thesis, University of Copenhagen, 2006. URL http://image.diku.dk/
projects/media/kelager.06.pdf.
[14] Fabrice Colin, Richard Egli, and Feng Ying Lin. Computing a null divergence
velocity field using smoothed particle hydrodynamics. Journal of Computational
Physics, 217(2):680–692, 2006.
[15] Matthias Muller, Bruno Teschner, Matthias Heidelberger, Danat Pomeranets, and
Markus Gross. Optimized spatial hashing for collision detection of deformable ob-
jects. Technical report, Computer Graphics Laboratory, ETH Zurich, Switzerland,
2003.
[16] Hagit Schechter and Robert Bridson. Ghost SPH for animating water. ACM Trans-
actions on Graphics (TOG), 31(4):61, 2012.
[17] Markus Becker and Matthias Teschner. Weakly compressible SPH for free surface
flows. In Proceedings of the 2007 ACM SIGGRAPH/Eurographics symposium on
Computer Animation, pages 209–217. Eurographics Association, 2007.
[18] Will Schroeder, Ken Martin, and Bill Lorensen. An Object-Oriented Approach To
3D Graphics. Prentice Hall, 1997.
