27 PAZ BONET a Corrected Smooth Particle Hydrodynamics Formulation

8/12/2019 27 PAZ BONET a Corrected Smooth Particle Hydrodynamics Formulation

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A corrected smooth particle hydrodynamics formulationof the shallow-water equations

Miguel Rodriguez-Paz, Javier Bonet *

Civil and Computational Engineering Centre, School of Engineering University of Wales Swansea, Swansea SA2 8PP, UK

Accepted 30 November 2004

Abstract

A shallow-waters formulation based on a variable smoothing length SPH method is presented. This new formulation

of the SPH equations treats the continuum as a Hamiltonian system of particles where the constitutive relationships for

the materials are introduced via an internal energy term. Some of the advantages of the new SPH formulation are evi-

dent in the solution of the shallow-water equations for expanding flows. The shallow-waters approach incorporates the

terrain into the equation of motion through terrain properties evaluated using SPH methodology. Several examples are

presented on the simulation of breaking dams on different geometries. A comparison with the analytical solutions is also

included.

2005 Elsevier Ltd. All rights reserved.

Keywords: Shallow-waters; SPH; Breaking dam; Free surface flows

1. Introduction

During the last decade, a number of changes in glo-

bal climate and other man-induced changes in nature

such as deforestation and pollution, have triggered envi-

ronmental problems, in particular water related issues:

floods and mudslides. On the other hand, the manage-

ment of water resources is also a main area of research,

which includes the prediction of storm surges, hazardprediction of dam breaks, sediment transport and coast-

al tides. Flow models that realistically represent the

physical properties of the flow and the complex topo-

graphic that are found in regions where debris ava-

lanches occurrence is high, can help in the hazard

prediction of such phenomena and help to mitigate their

destructive power. Numerical techniques are a viable

alternative when the phenomenon is difficult to repro-

duce in the laboratory due to many factors, such as,

its scale and magnitude.

Many of the hydrodynamic models for reservoirs and

tidal predictions are based on the solution of the depthaveraged shallow-water equations using finite differences

or finite element procedures. Other numerical methods

used for the solution of the shallow-water equations

for bore wave propagation include the finite element

method and more recently finite volume methods [14].

Most of these methods are based on elements or cells,

with the dependence on the grid and mesh refinement

to resolve the complex topography and evolving flow

features. However, most of the techniques that have

0045-7949/$ - see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2004.11.025

* Corresponding author. Tel.: +44 1797 295 689; fax: +44

1792 295 676.

E-mail addresses: [email protected] (M. Rodri-

guez-Paz),[email protected](J. Bonet).

Computers and Structures 83 (2005) 13961410

www.elsevier.com/locate/compstruc
mailto:[email protected]:[email protected]:[email protected]:[email protected]


2/15

been developed based on the shallow-water equations,

assume a small gradient of the terrain and do not con-

sider a vertical component of the velocity.

Meshless methods, and in particular Smooth Particle

Hydrodynamic (SPH) methods have already been used

to simulate free surface flows [58]. SPH is a robust

numerical technique introduced by Lucy and Gingold[9,10]. Since its introduction in astrophysics the method

has been applied to simulate problems with complicated

physics such as multiphase flows [11] and high strain

dynamics. SPH has also been applied to simulate geo-

physical flows like debris flows [12]and ice fields[13].

In this paper, a novel variational formulation of the

Lagrangian shallow-water equations is presented. This

formulation uses the variable smoothing length ap-

proach for SPH developed by Bonet[14]and which will

be also presented in the following sections. This new

methodology is intended to deal with problems of flows

over a steep and non-uniform general terrain, like in thecase of avalanches and debris flows.

As the algorithm presented is explicit and solves only

two components of the 3-D space, the storage require-

ments are minimum and can be implemented on per-

sonal computers or small workstations. This fact

presents some importance to practising engineers and

geophysicists who may wish to experiment with the

code.

2. General assumptions

The shallow-water assumption is based on a 2-D plan

view projection of the problem domain. In this way, for

the case of a SPH discretisation of the resulting 2-D do-

main, each particle represents a column of fluid of a cer-

tain height. These particles move according to the

topography of the terrain but always in a direction tan-

gent to the terrain.

Consider a plan view of a terrain as shown inFig. 1.

The terrain is represented by a general function H(x,y),

which gives the height at each point. The continuum is

discretised with a system of Lagrangian particles, in

which each particle represents a column of water of

height ht with constant mass m, which moves over theterrain. The basic assumption is that the velocity

through all the height of the vertical column is uniform

and parallel to the terrain. This implies that the instan-

taneous spatial variation of ht is small. The motion of

the Lagrangian particles is then followed in time.

The motion of the columns of water is constrained to

follow the terrain. This implies that the globalz position

of the bottom of each column is given by (see Fig. 2)

z Hx;y 1

Differentiating with respect to time, the vertical compo-

nent of the velocity can be evaluated as

vz rH v 2

where $His the gradient of the terrain at the current po-

sition occupied by the column and v = (vx, vy) is the 2-D

velocity vector containing the x and y velocity compo-

nents of the column. The unknowns of the problem

are therefore the x and y co-ordinates of every particle

at each time and the height of the water column ht.

Alternatively, instead of the column heightsht, it is also

possible to use a related variable given by the 2-D pro-jected density of the fluid q, that is the amount of mass

per unit of area. Given that the fluid in motion will be

assumed to be incompressible, this density and the col-

umn height are simply related by

q htqw 3

whereqw denotes the constant 3-D density of the fluid.

Using this variable instead of the column height renders

the problem formally analogous to a standard 2-D SPH

formulation of a compressible fluid.

In order to derive the governing equations, a

variational approach is used [14]. This is based on the

Fig. 1. Discretisation of the fluid with a system of Lagrangian

particles that move on top of a general terrain.

Fig. 2. Velocity components of each column of water.

M. Rodriguez-Paz, J. Bonet / Computers and Structures 83 (2005) 13961410 1397


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expression of the total energy of the system as a function

of the particle positions and will therefore require the

evaluation of the projected density in terms of such par-

ticle positions.

3. Density evaluation

3.1. Standard SPH

In a standard SPH formulation with constant

smoothing length, the density at a particle is smoothed

over a sub-domain that is defined by a circle of radius

2h, where h is the smoothing length of a kernel function

W(seeFig. 3) [6].

If the particles have a mass mI, the discrete system of

particles has a density function of the form

^qx XI mIdxxI 4where d is the Dirac delta. In order to reconstruct the

smooth continuum, the kernel-smoothing concept can

be applied to the discrete density(4)to give a continuum

density approximation over a domain D as

qx

RD

qx0Wxx0dx0RDWxx0dx0

5

whereWx 0 x; h Wx; x0; hrepresents a suitable kernel

function[16]. Substituting Eq.(4)into the above expres-

sion and noting the Diracs delta properties, gives

qI

PJmJWIxJ; hIR

DWIxJ;hIdV

6

Typically, the kernel function is scaled so that the inte-

gral in the denominator becomes one. However, in order

to deal with rigid boundaries, Bonet et al. [14,15] have

introduced a function gamma as

cIxI; hI

ZD

WIx; hI dV 7

and substituting it into Eq. (6)gives

qI

PJmJWIxJ; hI

cIxI; hI 8

In the standard SPH formulation and in the absence of

boundaries the gamma function is cI(xI, hI) = 1. This is

no longer valid for particles that are within a distanceto a rigid boundary that is less than 2h. The evaluation

of the integral defined by Eq.(7)can be complicated for

the case in which a particle is within certain distance

from a rigid boundary. Kulasegaram et al. [15]propose

a numerical method to approximate this integral. In the

following sections, however, we assume there are no ri-

gid boundaries, and therefore c = 1. This is appropriate

for many shallow-water applications.

3.2. Variable-h SPH

The evaluation of the density using a constantsmoothing length is normal practice in SPH. In the case

of fluids with certain compressibility, however, a simple

case of uniform expansion or contraction can only be ex-

actly represented if the smoothing length is allowed to

vary. In the shallow-water approximation, the fluid will

follow the terrain and its projected 2-D density will ex-

pand or contract according to the height of the water

column as shown by Eq. (3). A variable smoothing

length is therefore needed in order to maintain accuracy

of the solution. In general, h must change according to

[16]

qhdm constant q0hdm0 9

where dm is the number of space dimensions, q0, h0 are

the initial density and smoothing length, respectively,

and q and h are the current values of density and

smoothing length for any particle. This gives an equa-

tion for the instantaneous smoothing length h for a par-

ticle Ias

hIh0q0qI

1=dm10

It is important to note that the above equation for the

density is implicit as the density qI is itself a functionof hI. In particular, the density is evaluated according

to Eq. (6), which for the case where there are no rigid

boundaries (i.e., c = 1) gives

qIXJ

mJWIxJ; hI 11

Note that this is now a non-linear equation for qI due

to the above dependency of hI on qI. Fortunately, the

equation for each particle are not coupled and can there-

fore be solved independently. A simple NewtonRaph-

son iteration to achieve this is described in Section 5.1

below.Fig. 3. Particle interpolation and kernel function.

1398 M. Rodriguez-Paz, J. Bonet / Computers and Structures 83 (2005) 13961410


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4. Equations of motion: non-dissipative case

Consider now the fluid described by a system of SPH

particles that are located in a 2-D Cartesian space (x,y),

the position of each particle defined by the vector xIand

the horizontal component of the velocity of the particle

by vI. Each particle I will represent a column of waterwith a total mass mI, which will remain constant during

the motion and hence, conservation of mass will be

ensured.

The EulerLagrange equations of motion of the sys-

tem of particles are, in the absent of dissipative forces,

e.g., bottom friction and viscosity effects [17]:

d

dt

oL

ovI

oL

oxI0 12

where the Lagrangian functional L is defined in terms of

the kinetic energy Kand potential energy p, as

L K p 13

and noting that p is only a function of the positions of

particles xI, Eq.(12)can be written as

d

dt

oK

ovI

oK

oxI

op

oxI14

The expressions for the kinetic and potential energies are

presented in the following sections.

4.1. Kinetic energy

The kinetic energy of the system of particles can be

approximated as the sum of the kinetic energy of eachparticle, which with the help of Eq. (2), gives

K1

2

XI

mIvI vI v2z; vz vI rHI 15

It is important to notice the termv2z vI rHI2

, which

takes into account the vertical component of the veloc-

ity. This term is often neglected in other shallow-water

approaches, since it is considered to be too small for

small terrain gradients. In order to be able to deal with

large gradients in the terrain, it is important to retain it

throughout the derivation of the equations of motion.

4.2. Potential energy

The potential energy of each particle is calculated at

the centre of gravity of each water column, i.e.,H 12ht.

Hence, the total potential energy of the system of parti-

cles can be expressed as the sum of the potential energy

of each particle

pXI

mIgHI1

2

X1

mIght;I 16

where g denotes the gravity acceleration. The first

term represents the external energy term, whilst the sec-

ond term can be interpreted as the pseudo-internal

energy,

pext XI

mIgHI 17

pint XI mI1

2ght;I 18The internal energy component can be expressed in

terms of an internal energy per unit mass w(q) as

pint XI

mIwqI 19

where

wqI 1

2g

qIqw

20

This expression matches that presented in [14] for gen-

eral fluid applications. The corresponding pressure term

defined asp q2dwdq

becomes the well-known height inte-

grated hydrostatic pressure, as usual in the case of shal-

low-water applications, and is given by

p q2dw

dq

1

2gqwh

2t 21

4.3. Evaluation of inertial forces

Eq.(14)can be re-written in the more usual format of

Newtons second law as

d

dt

oK

ovI

oK

oxI|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}inertial forces

FI TI; FI

opext

oxI ; TI

opint

oxI

22

where the left-hand side represents the inertial forces of

the system,FIand TIare the external and internal forces

of the system, respectively, which will be discussed in de-

tail in the section below.

The equation of motion can also be expressed as

IIFI TI 23

where the inertial forces IIcan be evaluated with the help

equation(15)as

II d

dt

oK

ovI

oK

oxI24

and substituting the expression for Kgives after simple

algebra:

II d

dtmIvI mIvI rHIrHI mIvI rHIkIvI

25

wherekIis the curvature tensor of the surface H(x,y) at

the position occupied by particle Iand is given by

k rrH 26



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Note that the last term in Eq. (25)could be interpreted

as the centripetal acceleration of the column of water.

4.4. External forces

The external forces can be evaluated with the help of

Eqs.(22) and (17) to give

FI opext

oxI mIgrHI 27

4.5. Internal forces

The internal constitutive forces are calculated using

the procedure described in [13] and are given by the

expression

TIopint

oxI28

The evaluation of this term, in general, requires the con-

stitutive definition of the material. For the case of fluids

with no viscosity, the internal forces are calculated using

Eqs.(19) and (11) as

TIopint

oxI

XJ

mImJpJaJq

2J

rWJxI; hJ pIaIq

2I

rWIxJ;hI

29

where a is a correction factor that emerges in the vari-

able-h formulation (for details see [14]):

aI XJ

mJrIJdWI

drIJ30

andp denotes the hydrostatic pressure given by Eq.(21).

4.6. Acceleration

An expression for the acceleration of the particles can

be found by substituting Eqs.(25), (27) and (29)into Eq.

(23)and solving for aIgives, after some simple algebra:

aI

gvI kIvI tI rHI1 rHI rHI rH

I tI 31

wheretI= TI/mI. Eq.(31)includes a term for the curva-

ture of the terrain as well as one term for the gradient of

the terrain. It can be noted that for the case of a flat ter-

rain ($H= 0; k = 0), Eq.(31)becomes aI=tI. In gen-eral, Eq. (31) can be easily implemented in an explicit

time integration scheme. For the examples included in

this paper, the values of$Hand kIcan be easily evalu-

ated finding the derivatives of the equation that defines

the surface for the terrain. For more general applica-

tions, an interpolation of the terrain from given grid

points will be necessary.

5. Numerical implementation

5.1. Numerical evaluation of density

As previously mentioned, Eq.(11)is non-linear inqI.

In order to solve it, a NewtonRaphson solution proce-

dure can be used. Let us define a residualRqkI for eachiterative value kof the density as

RqkI qkI

XJ

mJWIxJ; hkI 32

where the smoothing lengthh is a function of the density

through Eq.(10)as

hkI h

0I

q0I

qkI

!1=dm33

and h0I and q0I are the initial values of hI and qI,

respectively.

A simple possible iterative solution for qIis given by

qk1I

XJ

mJWIxJ;hIqkI 34

However, a much better option is to use a Newton

Raphson solution by the iteration of

qk1I q

kI

RkI

dRdq

kI

35

Substituting Eq.(32)gives the iterative solution as

qk1I q

kI

qkI

PJmJWIxJ; h

kI

dRdq

kI

36

where the derivative of the residual is calculated as

dR

dq 1

XJ

mJdWI

dhI

dhI

dqI

1XJ

mJ

qIdmWIdm rIJ

dWI

drIJ

37

After simple algebra, this last equation becomes

dRdq

1 1qI

XJ

mJWIxJ; hI aIdmqI

38

Finally, substituting Eq.(38)into Eq.(35), gives

qk1I q

kI 1

RkI dm

Rk

I dmakI

" # 39

which is the NewtonRaphson approximate solution for

the density equation at iteration k+ 1.

In order to start up the iteration process an initial

guess is required. For this purpose, the equation for

the rate of change of density [18] can be integrated in

time to give



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_qIqIdm

aI

XJ

mJrWIxJ; hI vJ vI 40

which enables a simple guess of the density at step n + 1

to be evaluated in terms of the density at step n as

q0

I;n1 q

I;nekn 41

where

kdmDt

aI

XJ

mJvJ vI rWIxJ;hI

" # 42

5.1.1. Convergence

In order to stop the NewtonRaphson iterative pro-

cedure, a tolerance that is within the machine precision

must be defined. Convergence is achieved when

jRk1I j

qkI6 e 43

Typical values in a double precision machine are

e= 1015, which is usually achieved within a few itera-

tions. A cheaper alternative is to evaluate the density

using Eq. (41) and the value of the smoothing length

from the previous time step.

Once the new values for the density have been evalu-

ated, new heights of the water columns can be obtained

using Eq.(3) or

ht;IqIqw

44

It is important to mention that by evaluating the density,new values for the smoothing length and the correction

factor a are also calculated in each iteration. The up-

dated values of the smoothing lengths are then used in

all the subsequent SPH interpolations within the time

step.

5.2. Time integration scheme

The equation of motion is assembled using Eqs.(23)

(31)and in order to update the position of particles, an

explicit time integration scheme is used, namely the leap-

frog method, defined as

vn1=2I v

n1=2I Dta

nI 45

xn1I xnI Dt

n1vn1=2I 46

where

Dt1

2Dtn Dtn1 47

Due to the explicit nature of the scheme, the Courant

FriedrichsLewy (CFL) stability criteria must be

satisfied. This implies that the time step size must be less

than

Dt CFL hmin

maxcI kvIk; 0 6 CFL 6 1.0 48

where c is the wave speed of propagation or speed, de-

fined as[19]

cI ffiffiffiffiffiffiffiffiffight;Iq 49andhminis the minimum smoothing length of the system

of particles. The magnitude of the velocity is also consid-

ered. Although this equation should provide time steps

that would satisfy the stability condition, in the numer-

ical examples presented in the following section, the

CFL factor considered was


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Fig. 4. Dimensions of the channel and the dam.

Fig. 5. SW-SPH results for t= 0.01 s and t = 0.10 s. Colours indicate depths. (For interpretation of colour in this figure legend the

reader is referred to the web version of this article.)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.05 0.15 0.25 0.35 0.45 0.55 0.65

depth

h

h @ x=2.0m

Analytical Solution

Depth at gate

t (s)

Fig. 6. SW-SPH results vs. analytical solution for depth at x = 2.0 m.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.901.00

0.05 0.15 0.25 0.35 0.45 0.65

(g*h)^0.5

Vel @ x=2.0m

Analytical Solution

Velocity at gate

t (s)

0.55

Fig. 7. SW-SPH results vs. analytical solution for velocity atx= 2.0 m.



8/15


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t= 0.64 s. The SW-SPH results are shown inFig. 7, com-

pared to the value predicted by the analytical solution. In

the same manner, the velocity of the fluid at the point of

the gate (x= 0.0) should remain constant and equal to 2/3

c0, where c0 ffiffiffiffiffiffiffigh0

p 3.1314 m s1. The results are

shown inFig. 7, normalized with respect to c0.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Analytical Solution

SW-SPH

t=0.40 s

x pos

depth(m)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Analytical Solution

depth(m)

x pos

SW-SPH

t=0.60 s

Fig. 10. Comparison of the profile for t = 0.40 s and t = 0.60 s. The dots represent the SW-SPH solution and the continuous line the

analytical solution.

Fig. 11. Lateral view of the collapse of a cylindrical column of water. From left to right:t = 0.0 s, t = 0.10 s, t = 0.20 s and t = 0.30 s.



10/15

Using the formula given in Ref.[20], the position of a

particle at the front is given by

x 3ffiffiffiffiffigh

p v02

ffiffiffiffiffiffiffigh0

p t 50

Ifh is known for a particle located towards the front, the

position of the particle given by the program is com-

pared with the one provided by Eq. (50). Fig. 8 shows

the results for the front of the flow. The numerical re-

sults match very well those given by the analytical solu-

tion (Figs. 9 and 10).

6.4. Cylindrical dam on a horizontal plane

In this example a cylindrical column of water col-

lapses on a horizontal plane. The column has 1 m in

diameter and 1 m in height. In this case the properties

of the terrain are k = 0, $H= 0 (flat terrain). The results

are shown inFigs. 11 and 12. It can be seen that the ini-tial circular configuration is perfectly kept throughout

time, which would not be the case for a constant

smoothing length approach, as it was pointed out in

the previous chapter. A total of 5133 particles were used.

Fig. 12. Top view of the breaking dam.



11/15

The results are displayed as a surface, using standard

matlab graphics.

6.4.1. Constant-h results comparison

The same example was solved using a constant-hcor-

rected SPH formulation of the shallow-water equations.

As shown in Fig. 13, the results of the constant-h ap-

proach are not able to simulate the problem, as the par-

ticles near the expanding boundary of the circle have less

neighbour particles as the simulation continues. On the

other hand, the results for the SW-SPH approach pre-

sented in this paper show a uniform distribution of the

particles, without losing the circular shape. The same

distribution of particles was used in both cases and no

bottom friction was included.

In this example the terrain was considered as per-

fectly smooth, i.e., no bottom friction.

6.5. Dam in triangular channel

In this example a channel with triangular cross-sec-

tion is used to represent the terrain on which the flow

will move. The geometric properties and dimensions

for the initial set-up are shown in Fig. 14. The problem

consists of the instantaneous breaking of a dam curtain,which makes the whole water volume move down the

steep hill. The sequence is indicated in Fig. 17.

For this case the gradient and the curvature of the

terrain are defined as

rH0.4

0.4 signy

; k 0

The other parameters employed for this simulation are-

g= 9.806 m s2;q0= 1000 kg m3; no viscosity and bot-

tom friction were considered.

A total of 3793 particles were used in the simulation.

The results are shown in Fig. 15. The initial configura-

tion is a dam that contains the water in a triangular

channel. At time t = 0.0 s the curtain is removed instan-

taneously and the water starts moving down the chan-

nel. A top view is presented for better clarity and the

colours of the graphs represent the depth of the fluid

over the channel.

Once again, it is important to mention that the mesh-

less nature of the methodology allows for large changes

in the geometry of the domain of the fluid to take

place without the need of re-meshing. This is also possi-

ble due to the variational consistent equations for the

density, which in a way work as an adaptive procedure

by changing the smoothing length for each particle as

changes in particle distribution occur throughout the

simulation.

6.6. Steep parabolic channel

A more complex geometry is used in this case to

model the surface on which the fluid moves. It consists

of a surface defined by

z y2

2Rmx

whereRis a radius for the parable and mis a slope in the

x-direction. In this simulation R= 1.1 m and m= 40.

The geometric properties, gradient and curvature of

the terrain are therefore

Fig. 13. (a) Standard SPH for a collapsing column of water, (b) variational SPH.

Fig. 14. Cross-section for the channel.



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rH0.839

0.909y ; k

0.0 0.0

0.0 0.909 And the material parameters are g= 9.806 m s2; q0=

1000 kg m3. In Fig. 16, the results of the simulation

are shown in top views, i.e., XYplane.

Fig. 15. Breaking dam in a triangular channel, the colours indicate the depth of the fluid over the channel. (For interpretation of

colour in this figure legend the reader is referred to the web version of this article.)



13/15

The terrain is represented as contour curves. The

cylindrical column of water is released at time t= 0.0 s.

This column of water has a diameter of 60 cm and an

initial constant coordinate for the free surface of

Z= 0.10 m. A total of 4595 particles were used in this

simulation. The colours of the graphs represent in this

case the depth of the flow over the channel.

The results are for times t = 0.0 s, t = 0.50 s, t = 2.0 s

and t = 4.0 s, respectively. It is important to notice that

there are some parts of the fluid which have almost nildepth, indicate by the darker blue area, representing

wetted regions of the channel. The simulation was car-

ried out on a PC with 256 MB of physical RAM with 2

Pentium III processors with a clock speed of 550 MHz.

The total time for the simulation up to t= 4 s was

achieved in just a few hours.

6.6.1. Eccentric case

Considering now the same geometric properties of

the terrain used in the last example but now the column

of water is initially placed in an eccentric manner on the

channel. The purpose of this is to test the ability of the

method to deal with the curvature of the terrain. The

column of water was supposed to be a cylinder of

0.50 m of diameter, 0.25 m of height with centre at

x= 0.25, y = 0.50.

Fig. 17shows the results of the simulation. It is clear

how the fluid moves along the channel in a direction dri-

ven by its geometry. A sudden break initiates the flow

and then the fluid finds the centre of the channel and

starts moving down the slope along it. A total of 4100

particles were used to represent the fluid. The simulationwas run up to a time t = 8.0 s.

Taking this example as reference, if the same spacing

used for the initial distribution of particles used for the

fluid were to be used to mesh the entire channel, the

number of particles would be approximately over

187,000. The approach followed by other numerical

techniques is based on gridding the whole terrain and

then track the flow, with wet and dry cells. This SW-

SPH approach, however, discretises only the fluid and

the interaction of the terrain friction and any other dis-

sipative effect are included as forces affecting the motion

of each particle, acting at the position of that particle.

Fig. 16. Breaking dam in a parabolic channel. Dam is symmetrically located along the centre of the channel. The colours show the

depths of the flow for different times. (For interpretation of colour in this figure legend the reader is referred to the web version of thisarticle.)



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7. Concluding remarks

A new methodology for the numerical simulation of

shallow-water-like fluids over a general terrain was pre-

sented. The new set of equations is based on the varia-

tional SPH formulation presented by Bonet et al. [14].

The formulation has been denominated: variational

SPH formulation of Lagrangian shallow-water equa-

tions. Although the new formulation includes some of

the standard shallow-water equations assumptions, it

incorporates a new treatment of the terrain, as it allows

more general terrains to be considered. The resulting

technique shows a lot of potentiality for problems deal-

ing with breaking dams, flooding, debris flows [18,21],

avalanches and tidal waves, among others. Numerical

results show good agreement with analytical solutions.

Fig. 17. Breaking dam eccentrically in a parabolic channel. A total of 4100 particles were used in the simulation. The graphs show the

depth of the flow at different times. Initial depth 0.25 m.



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The results show that the technique is robust and stable,

which is in agreement with the previous work showing

that this type of particle methods are stable in the pres-

ence of compressive pressure values[22]. Given that the

pressure in the shallow-water model is proportional to

the height squared, its compressive nature is always

physically and numerically assured. Another advantageof the method over traditional numerical techniques that

use grids is that there is no need for a mesh for the entire

terrain; only the fluid is discretised with particles. This

enables the method to be implemented in serial machines

such as personal computers, since the memory require-

ments are vastly reduced. At present the cost of the

implementation appears high. This is very possibly due

to the type of searching technique used to determine par-

ticle neighbours, which is based on the Alternating Dig-

ital Tree method [23]. It is likely that other types of

searching methodologies may be more efficient for this

type of application and should be explored in the future.The method can also incorporate more sophisticated

constitutive models for different materials and bottom

friction forces. The method shows the potential to be ap-

plied in the simulation of geophysical flows and can help

in the hazard prevention for natural disasters.

Acknowledgment

Financial support from EPSRC through grant GR/

R72013 is gratefully acknowledged.

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