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AbstractTwo boundary treatment methods were developed forincompressible flow simulations and fluid-structural interaction

problems using Smoothed Particle Hydrodynamics (SPH): 1) To

apply repulsive force on the boundary particles while keeping the

same particle spacing for inner fluid particles and wall boundaryparticles; 2) To use denser wall particles without any additional force.

The dam-breaking problem and another testing example are used to

demonstrate the performance of this method. Results obtained from

the present approach show reasonable agreement with experimental

data. The fluid pressure values obtained with SPH method is

investigated. Based on the result of the study, it can be concluded

that the present approach is reliable to simulate incompressible fluid

and the pressure value obtained can be used to solve fluid-structural

interaction problems.

KeywordsBoundary condition treatments, incompressible SPH,pressure prediction.

I. INTRODUCTIONThe Smoothed Particle Hydrodynamics (SPH) method is a

fully Lagrangian mesh-free method used widely in large

deformation problems such as fluid motions where the

continuum hydrodynamic equations are solved with a set of

interacting fluid particles [1], [2]. The original equations that

are discretised are those for a compressible viscous fluid.

When SPH is applied to simulate incompressible flows, there

are generally two ways to impose incompressibility: one is to

run the simulations in the quasi-incompressible limit by

assuming a small Mach number to ensure density fluctuations

within 1% [3]-[5], which is known as Weakly Compressible

Smoothed Particle Hydrodynamics (WCSPH); the other one iscalled truly Incompressible SPH (ISPH) in which

incompressibility is enforced by solving a Poisson equation at

every time step. The velocity divergence is set to zero as a

condition to ensure the incompressibility in this method [6]-

[8]. It is noted that incompressible condition also means that

the volume of each fluid particle should not change. Hence,

the incompressibility can be enforced by setting the volume of

Manuscript received May 11, 2011.

F. Sun is with the Fluid Structural Interaction research group of University

of Southampton, SO171BJ UK (e-mail: ffs1g09@soton.ac.uk).

M. Tan is with the Fluid Structural Interaction research group of

University of Southampton, SO171BJ UK (e-mail: mingyi@soton.ac.uk).

J. T. Xing is with the Fluid Structural Interaction research group ofUniversity of Southampton, SO171BJ UK, (e-mail: jtxing@soton.ac.uk).

each fluid particle as a constant in the simulation using

Lagrange multipliers [9]. Another way to enforce

incompressible fluid is to set the density variation and velocity

divergence to be zero. This method is used for multi-phase

fluid simulations to enforce the incompressibility [10]. All

those treatment methods, either setting density variation to be

zero or force the velocity divergence to be zero, require

additional consideration on the fluid density variation. In fact

the density of the fluid can be simply set to be a constant for

the incompressibility, and the zero velocity divergence can be

satisfied automatically [11]. This method provides a

straightforward approach to the incompressible fluid problem

and it is adopted in this paper.

It is well known that SPH does not have zeroth order

consistency in boundary area. On the boundaries, the failure of

SPH modelling is characterized by wall penetration of fluid

particles. Generally, there are three ways to prevent this fromhappening: 1) mirror particles [10], 2) repulsive forces [3] or

3) dummy particles [6], [12]. Usually, repulsive forces are

used in WCSPH whereas mirror particles need special

consideration on corners or curved surfaces. Hence, dummy

particles or ghost particles are preferred in ISPH method [13].

This paper focuses on investigation of boundary treatment

methods in order to improve the efficiency of SPH model for

incompressible flow simulations.

Ghost particles are useful to keep the symmetry

configuration of the particles near the wall. Therefore, the

kernel domain of the particles can remain complete and thephysical properties such as density can be calculated correctly.

However, when dealing with problems with complex solid

boundaries the ghost boundary treatment becomes difficult.

Taking compartment flooding as an example, water can fill

both inside and outside of the structure and at least two layers

of ghost particles need to be placed on the inside wall and

outside wall respectively. These ghost particles sometimes

overlap the true fluid particles, which causes inaccurate

neighbouring particles counting and results in a wrong

predictions. It is also difficult to treat the angled boundaries by

using ghost particles. Special consideration is required to

calculate the exact position of the ghost particle for the angled

points since the ghost particle position is important to ensure

Investigations of Boundary Treatments in

Incompressible Smoothed ParticleHydrodynamics for Fluid-Structural Interactions

Fanfan Sun, Mingyi Tan, and Jing T Xing

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that there is no fluid particle penetration.

Practically, as long as the density can be kept as constant,

preventing particles from penetrating the walls is the major

concern of these boundaries. Therefore, repulsive force can be

applied on the wall particles instead of using several lines ofdummy particles which not only increases computation time

but also complicates the model especially in fluid structural

interaction problems. Another boundary treatment using

denser wall particles is also investigated. With repulsive force,

all the particles can be maintained in a uniform arrangement

but the additional force may affect the pressure values

obtained. This problem can be overcome by using denser wall

particles, say half spacing wall particles, but it requires slightly

more computation time in this case. These two boundary

treatments can be chosen according to different situations.

Both the present boundary treatments allow efficient

simulations with complex solid boundaries and even provide anew coupling approach for fluid structural interactions in the

future.

II. NUMERICALMODEL2.1 SPH formulation

The SPH formulation is based on the theory of integral

interplant that uses kernel function to approximate delta

function. A physical property is obtained by the interpolation

between a set of points inside a certain area. These points

known as particles carry all the properties the fluid has, such

as mass and velocity. The basic idea of this method is toapproximate a function A(r) as [14]

),()( hWA

mA bab b

bba rrr =

(1)

A model of SPH formulated gradient term in the N-S equation

is employed to preserve linear and angular momentum [3]

abab b

b

a

aba W

PPmP

+= 22)

1(

(2)

where

ab

ab

ab

baaba

rW

rW

= xx (3)

2.2 Governing equations

The governing equations for incompressible continuum

including the conservation of mass and momentum are

presented in the following equations, respectively.

01

=+ vDt

D

(4)

P

Dt

D+=

11g

v(5)

Where t is the time, is density, g is the gravitational

acceleration, P is pressure, v is the velocity, is viscous stress

tensor and D/Dt refers to the material derivative. The

momentum equations include three driving force terms, i.e.,

body force, forces due to divergence of stress tensor and thepressure gradient.

2.3 Incompressible SPH and numerical formulationFor incompressible flow the mass density is a constant.

According to (4) the velocity divergence will be zero [11] .

0= v (6)

Splitting the momentum equation into two parts, one is with

the effect of body force and viscosity introducing an

intermediate velocity,

nnn

t

+=

+ 12/1g

vv(7)

Another is from pressure influence and the new velocity can

be updated based on the intermediate one obtained from

previous step

12/11

1 +++

= n

nn

Pt

vv(8)

Taking the divergence of (8) and substitute (6) so the

pressure Poisson equation can be derived

2/11 ++ = nnt

P v

(9)

The Poisson equation is formulated with SPH method

++

=+ b

n

ab

n

abbb

n

ab

ab

n

ab

n

abb mtr

rPm'

2'

2/1

22

1

WuW

(10)

Hence, the pressure can be updated by (10) implicitly.

The viscous force is computed in SPH form as

+=

baba

b

b

a

ab

a

Wm22

1

(11)

Where the stress tensor is related to the strain tensor. The

suffix a and b represent different particles.

+

==

i

j

j

ieffjiij

x

u

x

u (12)

The full derivative between two particles is first obtained

using finite difference before decomposing it into x and y

directions. Thus

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=

=

ab

j

b

j

a

ab

i

b

i

a

j

ab

ab

i

aj

i

r

xx

r

uu

x

r

r

u

x

u1

(13)

For Newtonian fluids such as water, the viscosity coefficient

eff (effective viscosity) has a constant value . Hence, the

SPH formulation of viscosity term can be written as [6]:

+

=

bab

baabaabb

a r

Wrm

222

2 )(4

uuu (14)

The quantities are updated following the steps

+

++= ++

bab

baabaabbn

a

n

ar

Wrm

gt 22212/1 )(4

uu

uu

(15)

+

=+

+

+

bab

ab

nab

nab

b

bab

n

abbn

a

r

rPm

mt

P

'

'2

22

1

2/1

1

W

Wu

(16)

abb

n

b

n

ab

na

na

PPmt '

2

1

2

12/11

Wuu

+=

++++

(17)

III. BOUNDARYCONDITIONS3.1 Free surface

Free surface particles are tracked down to set their pressure

to zero to simplify the dynamic surface boundary conditions

[15]. The following quantity is calculated to identify the free

surface particles

ababab

b

b wm

err '=

(18)

This value equals to 2 in 2-D applications or 3 in 3-D cases

when the smoothing domain is not truncated but it is far belowthese values for surface particles, a criterion used in this paper

is 1.6 in 2-D cases.

3.2 Wall boundary

The solid walls are simulated by particles which prevent the

inner particles from penetrating the wall. Conventionally, the

wall boundary conditions are modelled by fixed particles

exerting a repulsive force on inner fluid particles in weakly

compressible SPH method.

2

00

21

)(r

r

r

r

r

rDrf

pp

= (19)

It is zero when 0rr> so that the force is purely repulsive.

Mostly, 41 =p and 22 =p , D=5gH according to [3]. The

length scale 0r is taken to be the initial spacing between the

particles.

In incompressible SPH method, ghost particles which mirror

the physical properties of inner fluid particles are the usual

treatment of wall boundary conditions. These ghost particles

make smoothing domain complete for the near wall fluid

particles so that the consistence of SPH simulation near wall

boundaries is ensured. The repulsive force boundary treatment

as shown in Fig. 1 has not been used in incompressible SPH

method. Normally ghost particles are considered to be

necessary to avoid unphysically large density variation for thenear boundary particles. However, when the density of all the

particles is set to be a constant, ghost particles will not be

necessary any more.

Wall particles are involved in the Poisson equation, using

denser particles on the wall boundary can produce pressure to

keep the inner particles away from the boundary. A halfed

spacing is set on the wall particles compared with the inner

fluid particles as shown in Fig. 2. Alternatively, the simple

repulsive force treatment can be applied to save computational

time.

Fig. 1: Boundary treatment: using halfed spacing on wall

particles

Fig. 2: Boundary treatment: using repulsive force

IV. COURANTNUMBERCONDITIONSince this incompressible SPH method calculates pressure

implicitly and other properties explicitly, the size of time step

must be controlled in order to have stable and accurate results.

The following Courant condition must be satisfied [6]

max

1.0v

ht (20)

where h is the initial particle spacing andmaxv is the

maximum particle velocity in the computation. The factor 0.1

ensures that the particle moves only a fraction (in this case 0.1)

of the particle spacing per time step. Another constraint is

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based on the viscous terms [10]

/125.0

2

eff

ht (21)

where eff is the effective viscosity. The allowable time-step

should satisfy both of the above criteria.

V. TESTINGEXAMPLESWithout ghost particles, model with complex wall

boundaries can be simulated efficiently. An example is used to

test the two boundary treatments in simulating water flooding

into compartments as shown below in Fig. 3.

Fig. 3: water flooding compartment

The water column is set to be 2m high and 1m wide. It is

kept in a static state in the beginning, but suddenly a hole is

unblocked on the right wall and the water starts to flow out.

Simulation of the water flooding was recorded at time 0.4s,

0.6s and 1.8s as shown below in Fig. 4.

Fig. 4: testing case with halfed spacing for wall particle when

t=0.4s, 0.6s, 1.8s

Water flows violently after the unblocking of the hole. Thefront of water impacts the right side wall and comes back

impacting the second building. Some air bubbles are generated

during the process, fluid motion changes quickly. If ghost

particles are used to model the wall boundary, several layers of

these ghost particles must be placed on each side of the "small

structures". This will affect the fluid particles when flow fills

both sides of the structure. Some of the ghost particle may

overlap with the true fluid particle, which makes the counting

of the neighbouring particles inaccurate.

Results obtained using repulsive force applied on the boundary

particles are also displayed in Fig. 5 to make a comparison

with the denser wall particle treatment.

Fig. 5: testing case 1 using repulsive force on wall particles

when t=0.4s, 0.6s, 1.8s

Only small differences can be observed from the results

obtained based on these two different boundary treatments.

VI. PRSSUREINVERSITGATIONThe investigation of these two boundary treatments are

carried out with a 2-D dam break simulation. The model is set

up as shown in Fig. 6. The spacing of fluid particles is set to be0.01m, the overall height of the water column (H) is 0.6m and

its width (a) is 1.2m. The size of the solid container is 3.22m

long. The water column is kept in hydrostatic state in the

beginning. The flow starts when the right side gate is suddenly

opened so the water column collapsed. The pressure values (P)

at point 0.16m on the right wall are traced. The results are

obtained using ghost particles, half spacing wall boundary

particles and repulsive force boundary treatments and the

results are compared with experimental data provided by [16]

Fig. 6: Dam break

Water configuration with time is shown in Fig. 7

Fig. 7: Water configuration at time 0s,0.5s,0.75s,1.5s

Results obtained using different boundary treatments in SPH

method are compared with experimental data [16] as shown in

Fig. 8.

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Fig. 8: Results obtained from three different boundary

treatments compared with experimental data and another

numerical method

All these three boundary treatments find the first pressure

peak around the right time compared with experimental data.

The first peak values obtained from numerical method areslightly larger than observed in the experiment. Boundary

treatments with repulsive force and ghost particles gave similar

results, denser wall particle boundary treatment give slightly

higher peak value than the other two treatments. Except the

second peak value, the overall curves agree with experimental

data well. There is no obvious second peak pressure in the

simulations. This is perhaps because of entrained air effects

which are not predicted during the simulation. But compared

with other numerical method such as Navier-Stokes Solver

provided by [17], SPH gives closer values to the experimental

ones.

Fig. 9: Investigation of repulsive force boundary treatment

with different time stepping size

The Fig. 9 shows large variations of the first peak value

when using different time stepping sizes. The rest of the curves

are almost the same. When time stepping size is 0.0001s, the

results are fairly close to the experiment, which means that a

time stepping size between 0.0005s and 0.0001s is sufficiently

accurate for the simulation. This time stepping size value can

be obtained from Courant number condition.

Fig. 10: Investigation of halfed wall particle spacing treatment

with different time stepping size

From Fig. 10, it can be seen that similar to repulsive force

treatment, using time stepping size of 0.0001s provides better

results than using 0.0005s. However, the curves do not change

as much as the previous boundary treatment case. Further time

stepping size decreasing does not change the results

noticeably. And a second peak is predicted in this case when

using time stepping size of 0.00005s, which indicates that

using denser wall particles is a better boundary treatment to

obtain accurate results compared with repulsive force

treatment.

VII. CONCLUSIONSimpler boundary treatments can be used instead of

conventional ghost particles for incompressible SPH with

constant fluid density. Simulations with complex solid

boundaries can now be modelled without difficulty. Two

testing examples have been used to demonstrate the

application of these two boundary treatments. Model set-up

can be done more efficiently using one of these two boundary

treatments. They not only offer a simpler configuration for the

model but also produce better simulations. It is observed from

the pressure results analysis that incompressible SPH can

provide more accurate pressure value.

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