A coupled SPH-DEM model for fluid-structure interaction problems with free-surface flow and structural failure

A coupled SPH-DEM model for fluid-structure interaction problems with
free-surface flow and structural failure
Ke Wu a
, Dongmin Yang a,⇑
, Nigel Wright b
a
School of Civil Engineering, University of Leeds, LS2 9JT, UK
b
Faculty of Technology, De Montfort University, LE1 9BH, UK
a r t i c l e i n f o
Article history:
Received 30 May 2016
Accepted 5 August 2016
Available online 1 October 2016
Keywords:
Discrete Element Method
Smoothed Particle Hydrodynamics
Fluid-structure interaction
Free surface flow
Fracture
a b s t r a c t
An integrated particle model is developed to study fluid-structure interaction (FSI) problems with frac-
ture in the structure induced by the free surface flow of the fluid. In this model, the Smoothed Particle
Hydrodynamics (SPH) based on the kernel approximation and particle approximation is used to model
the fluid domain in accordance with Navier-Stokes equations and the Discrete Element Method (DEM)
with a parallel bond model is used to represent the real solid structure through a hexagonal packing
of bonded particles. Validation tests have been carried out for the DEM model of the structure with defor-
mation and fracture failure, the SPH model of the fluid and the coupled SPH-DEM model of FSI without
fracture, all showing very good agreement with analytical solutions and/or published experimental and
numerical results. The simulation results of FSI with fracture indicate that the SPH-DEM model developed
is capable of capturing the entire FSI process from structural deformation to structural failure and even-
tually to post-failure deformable body movement.
Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction
Fluid-structure interaction (FSI) is a phenomenon in which a
flexible structure suffers pressure from the surrounding fluid flow
to give rise to deformation, and conversely, the fluid pressure field
and flow is affected by the moveable or deformable structure. As a
result, the whole interactive process is repeated continuously until
the deformation of solid structure remains unchanged.
FSI is a common engineering problem, for instance, the blade of
wind turbine is bent in the flapwise direction due to aerodynamic
loading [1], the presence and flow of a red blood cell in capillary or
arteriole has a significant effect in understanding its microcircula-
tion and regulation [2], the flow pattern of blood in the heart
affects the performance of a heart valve [3], in the event of
flooding/landslide-building interaction the flood/rock flow poten-
tially gives rise to the collapse of buildings [4], injected CO2 inter-
acts with reservoir and caprock underground and the interaction
can result in unforeseen leakages [5], and vibration-induced buck-
ling is observed in the carbon nano-tubes (CNTs) when delivering
fluid flow [6], among many others. This study is concerned with
the FSI process involving free surface flow and structural failure,
thus a brief review of existing FSI models and their capability or
penitential to model such process is discussed first.
1.1. Current numerical modelling approaches for fluid-structure
interaction
FSI problems usually involve flow nonlinearity and multi-
physics which are too complex to be solved by analytical methods,
and a small number of numerical models have been developed in
recent years. Although there are various numerical methods being
developed and applied to simulate the separate behaviour of fluid
and structure, combined methods for FSI are still limited. The chal-
lenge of coupling two methods for FSI largely depends on the nat-
ure of their discretisation. Conventional mesh-based methods such
as the finite difference method (FDM), the finite element method
(FEM) and the finite volume method (FVM) discretise the domain
into individual meshes. The reliance on mesh makes the treatment
of discontinuities (e.g., wave breaking, cracking and contact/sepa-
ration) difficult because the path of discontinuities may not coin-
cide with the mesh lines. Remeshing techniques can ensure the
discontinuities evolve along the mesh lines but at the expense of
reduced computational efficiency and degradation of numerical
accuracy. In comparison to conventional mesh-based methods,
meshfree (or meshless) methods are intended to approximate
mathematic equations in the domain only by nodes without being
http://dx.doi.org/10.1016/j.compstruc.2016.08.012
0045-7949/Ó 2016 Elsevier Ltd. All rights reserved.
⇑ Corresponding author.
E-mail address: d.yang@leeds.ac.uk (D. Yang).
Computers and Structures 177 (2016) 141–161
Contents lists available at ScienceDirect
Computers and Structures
journal homepage: www.elsevier.com/locate/compstruc

connected by meshes. If the nodes are particles that carry physical
properties (e.g., mass) and the system is simulated by the evolution
of the particles’ trajectory and the particles’ properties, then this
type of method is usually called a particle method. Typical particle
methods are molecular dynamics (MD), Discrete Element Method
(DEM), Smoothed Particle Hydrodynamics (SPH), Immersed Parti-
cle Method (IBM) and Lattice Boltzmann Method (LBM). It should
be noted that in LBM the particles are only allowed to move along
the predefined lattices, so it is in some ways a mesh-based particle
method. In the meshfree particle methods of MD, SPH and DEM, a
contact detection algorithm as well as an interaction law is
required to define the particle interaction. The contact detection
algorithm is used to determine whether two particles are interac-
tive, and once they interact, then the interaction law must be used
to calculate the interaction forces. In previous research, LBM and
SPH are mainly used for simulating fluid flow [7,8] whilst DEM is
mainly used for simulating granular flow [9] and solid fracture
[10]. Coupled models like SPH-SPH [11], SPH-DEM [12], IBM
[13,14] and LBM-DEM [15] have been developed for fluid-
structure or fluid-particle interactions.
As FSI involves two phases, i.e., fluid and solid, the numerical
methods for each can be the same or different. As the interface
between the fluid and solid structure is evolving in space and time,
the numerical models of FSI can be classified as Eulerian-Eulerian,
Eulerian-Lagrangian and Lagrangian-Lagrangian. In general, an Eule-
rian method discretises the space into a mesh and defines the
unknown values at the fix points, while a Lagrangian method tracks
the pathway of each moving mass point. Communications between
the mathematical frameworks for fluid and structure are realised
through a fluid-structure interface.
The Eulerian-Eulerian models tend to use an Eulerian FDM to
treat both fluid and structure boundaries on fixed meshes to avoid
mesh reconstruction. This is able to handle large deformation and
free movement of the structure in the fluid as well as the contact
between structures. However, this comes at the price of high com-
putational costs and additional discretisation errors since the
interface is only tracked implicitly by the solution itself. Special
techniques have to be used to link the material points between
the reference framework and the current framework [16,17].
The Eulerian-Lagrangian models solve the Eulerian form of the
Navier-Stokes equations for fluid on a fixed grid using a finite vol-
ume method, e.g., computational fluid dynamics (CFD), and track
the moving body (structure) in a Lagrangian fashion. A typical
example is the CFD-FEM model [18–21]. An alternative, the Arbi-
trary Lagrangian-Eulerian method (ALE), was developed to allow
arbitrary motion of grid/mesh points with respect to their frame
of reference by taking the convection of these points into account.
However, for large translations and rotations of the solid or inho-
mogeneous movements of the mesh points the fluid elements tend
to become ill-shaped, which reflects on the accuracy of the solu-
tion. Remeshing, in which the whole domain or part of the domain
is spatially rediscretised, is then a common strategy. The process of
generating mesh multiple times during a computation can, how-
ever, be a very troublesome and time consuming task. In particular
the contact of the elastic structure with the boundary is not possi-
ble within a monolithic formulation using simple ALE coordinates
without remeshing techniques [22].
Even though some remedies have been used to minimise those
limitations [23,24], the features such as large deformation, free
surfaces and deformable boundaries are still great challenges in
coupled CFD-FEM models and conventional Eulerian-Eulerian
methods and Euler-Lagrangian methods can only solve FSI prob-
lems where the structure is immersed in the fluid field and
deforms without any fracture. On the other hand in the meshfree
methods, the identification of free surfaces, moving interfaces
and deformable boundaries can be handled straightforwardly
[25]. Due to those evident advantages in meshfree methods, some
research efforts have been focused on coupling meshfree methods
with CFD [12] or FEM [26,27], and even developing coupled mesh-
free models such as SPH-SPH [11], SPH-DEM [12,28,29] and LBM-
DEM [26].
The presence of free surface flow in the concerned FSI problems
makes SPH preferred to remain in the coupled model to be devel-
oped. Among the above models the coupled SPH-FEM model [30],
the coupled SPH-SPH model [11,31] and the coupled SPH-DEM
[12,28,29] are Lagrangian-Lagrangian schemes. These models are
capable of simulating the free-surface flow and dynamic boundary
problems involved in FSI problems, but the kernel functions used
Nomenclature
F the contact force
K the stiffness of a particle
U the contact displacement
V the contact velocity
l the friction coefficient
b the critical damping ratio
b the moment-contribution factor
m the particle mass
q the density of a particle
g the body force acceleration
x the position vector of a particle
M the resultant moment acting on a particle
I the principle moment of inertia of a particle
d the element thickness
m Poisson’s ratio of the material
E Young’s modulus of the particles
R the radius of a parallel bond
A the cross-sectional area of a parallel bond
W the kernel function
h the smoothing length for a kernel function
N the number of particles within the support domain of a
kernel function
r the distance between two particles
k the constant related to particle search radius
q the ratio of distance to smoothing length
P the pressure of SPH particles
c the speed of sound
P the viscosity term in the SPH governing equations
R the anti-clump term in the SPH governing equations
Superscripts and subscripts
n the normal component of a vector
s the shear component of a vector
A particle A in a particle pair
B particle B in a particle pair
i particle i in a particle pair
j particle j in a particle pair
d the dashpot in the DEM model
0 the reference value of a physical variable
b the bending moment
t The twisting moment
142 K. Wu et al. / Computers and Structures 177 (2016) 141–161

in SPH for solid structure lack a physical representation of fracture,
not to mention further complications such as the permeation of
fluid in the porous or fractured zones of solid structure and the
large deformation in FEM is still under numerical challenges. In
the coupled SPH-DEM models developed in [12,28] the structures
are treated as rigid bodies thus the interaction between structure
and fluid is not fully studied and the deformation and fracture of
structure has not been achieved. Even if the structures in FSI prob-
lems with free surface flow is represented by SPH or FEM, to the
authors’ best knowledge, none of those models is capable of deal-
ing with fracture or crack initiation in the structure part during the
FSI process.
The FEM as a traditional mesh-based method and its extended
versions play an important role in dealing with solid fracture or
structural failure problems [32,33]. Phantom-Node method [34]
was also incorporated into FEM through integration of overlapped
elements in order to handle crack kinematics, but the crack-tip
enrichment is still challenging and its flexibility is comprised when
crack growth is the only focus. Therefore coupling FEM with SPH
for modelling fluid induced structural failure during the FSI process
would become even more challenging.
Another method referred as continuous/discontinuous defor-
mation analysis (CDDA) [35] was developed to account for fracture
by employing a link element to connect two adjacent elements as a
virtual crack extension. Alternatively, meshfree methods [36,37] as
a promising technique in recent years have been applied in mod-
elling of fracture. The development of test and trial function with
a sign function can model cracks with arbitrary movement [14].
Rabczuk [13] used immersed particle method treated in fluid and
structure, in which a Kirchhoff–Love shell theory is adopted, to
model FSI with crack propagation. A cubic/quartic polynomial basis
[38] was used in meshfree particle methods, but without taking
the gradient of a kernel function to model cracks the polynomial
functions used for solid structure lack a physical representation
of fracture unlike the traditional constitutive laws described in
solid mechanics. Even though these methods are promising in
dealing with fracture, it is difficult to extend them for modelling
more complicated Fluid-Particle-Structure Interaction (FPSI) sys-
tem where the particle-particle interaction, particle-structure
interaction and particle-fluid interaction has to be considered.
As another type of meshfree methods, DEM, has recently been
successfully applied to model the fracture of solids such as ceram-
ics [10], concrete [39] and even composite materials [40]. The par-
ticles in DEM are bonded together and the crack initiation and
propagation is treated as the progressive breakings of bonds. The
crack pattern is automatically determined without any need of
re-meshing and can be dynamically visualised during the simula-
tion process. DEM model does not require the formulation of com-
plex constitutive laws that are essential in FEM model, while it
requires calibration with measured macro-scale results to deter-
mine the micro-scale particle and contact parameters that will pre-
dict the macro-scale response. Therefore, DEM is practical for
studying general features of the statics and dynamics of fracturing,
like the crack shape, global structural failure due to the collective
behaviour of many interacting cracks as well as the dynamic insta-
bility of cracks during their propagation.
1.2. Motivation and objectives
This paper aims to present a new approach based on fully mesh-
free particle methods of SPH and DEM to handle the FSI problems
with free surface flow and/or structural failure. One of the objec-
tives of this research is to develop an advanced FSI model for inves-
tigating the failure mechanism of infrastructures (e.g. bridges and
buildings) during the flooding events. For example, it is becoming
more public concerned that the recent failure of transport bridges
(particularly those historic and listed masonry bridges that are still
widely in service in the UK) due to flooding have caused enormous
impact on local transportation and it is timely and financially
costly to get them repaired/rebuilt. Proactive reinforcing or
strengthening techniques are thus preferred in order to make the
bridges more resistant to the scouring and buoyance effects caused
by the flood. To address this problem, interdisciplinary knowledge
of geotechnical, hydraulic and structural engineering are required,
and it also raises a demand on a robust and reliable computer
model to predict the interaction between soil, flood and bridge.
Thus a numerical model for fluid-particle-structure (FPSI) interac-
tion would be extremely helpful for assessing the risk of bridge
collapse and also assisting the development of dedicated strength-
ening technique to prevent the failure of the bridge at risk.
The SPH-DEM model presented in this paper is the first step of
developing a unified particle model for general FPSI problems in
engineering with a principal application in flooding caused bridge
failure. The coupled SPH-DEM model will be able to capture either
the deformation or the fracture events in the solid structure
induced by the free surface flow of the fluid. In this approach,
the SPH based on the Navier-Stokes equations is used to model
the fluid domain. The DEM is used to represent the solid structure
through a dense packing of bonded particles which allows defor-
mation and/or fracture. Similar approaches have already been
adopted for modelling ceramics [10] and concrete [41]. As the
interaction between discrete particles can be naturally taken into
account by DEM, the coupled SPH-DEM presented in this paper
for FSI has the potential of being easily extended to model the
interaction between fluid, particles and structure simultaneously,
and applied to address the FPSI problems in engineering as such
discussed above. As both fluid and structure components are rep-
resented in the same framework, the coupling between SPH and
DEM can be easily achieved, and more importantly they can be
computationally accelerated for large scale simulations by using
GPU technique which has been already successful for individual
SPH and DEM models. In addition the DEM can deal with discrete
particles through contacts as well as continuous structures through
bonded particles, thus the coupled SPH-DEM model is applicable to
FSI problems and also Fluid-Particle-Structure Interaction (FPSI)
problems. The coupled model is first applied in the FSI problems
before being extended to the FPSI problems.
This paper is organised as follows: First, individual DEM and
SPH models are developed and validated against theoretical and
existing numerical/experimental results; An interface model is
then proposed to couple the DEM and SPH models, which is then
validated against a standard FSI test; Finally, the coupled SPH-
DEM model is applied to investigate a more complex FSI problem
in which the fracture failure in the structure is allowed due to
increasing fluid pressure.
2. DEM model of a structure
The discrete or distinct element method (DEM) was initially
proposed by Cundall for studying the discontinuous mechanical
behaviour of rock by assemblies of discs and spheres [42]. Upon
the development of DEM in recent decades, its applications have
been extended in various engineering research fields such as gran-
ular flow [39,40] and fracture of materials and structures including
rock [35], ceramics [10], concrete [41] and composites [36]. DEM is
a Lagrangian method in which the target material is represented by
particles that can interact with their neighbours. Every single par-
ticle is tracked in DEM throughout the entire time history of sim-
ulation, thus the field variables of a particle are updated in every
timestep according to the interaction with neighbour particles that
are in ‘direct’ or ‘indirect’ contact. The contact between two parti-
K. Wu et al. / Computers and Structures 177 (2016) 141–161 143

cles in DEM is typically represented by a spring and a dashpot in
both normal and tangential directions, as well as a frictional ele-
ment as shown in Fig. 1. By direct contact, the two particles phys-
ically touch or overlap with each other. Two particles could be
considered as in indirect (or distance) contact when their distance
is within certain range [43]. The indirect contact can enable long-
range interaction between particles in a way similar to the Van
der Waals forces between molecules according to a potential func-
tion in Molecular Dynamics (MD). This indirect contact feature will
be adopted to account for the interaction between particles within
the smoothing length in Smoothed Particle Hydrodynamics (SPH)
for the fluid part that will be discussed later. In this section, the
structural part will be modelled by DEM using particles in direct
contact. It should also be noted that particles in DEM can be rigid
or deformable and can have a complex shape, e.g., elliptical. In this
paper, 2D rigid particles with a circular shape are considered, and
the Particle Flow Code in two dimensions (PFC2D 5.0) is employed
as the simulation platform [44].
In a DEM the relative displacement (see Fig. 2) between two
contacting particles is fed into a force-displacement law to update
the contact forces. The calculated contact forces are then applied in
the law of motion, Newton’s Second Law, to determine the parti-
cle’s acceleration which is then used to update the particle velocity
as well as the particle position. These two laws are applied repeat-
edly to form the whole calculation cycle of DEM. Therefore, DEM is
particularly suitable for simulating the dynamic behaviour of dis-
continuous system, in which the movement of every particle is
recorded and analysed over each time step.
2.1. Force-displacement law
The force-displacement law for a contact between two particles
without the presence of a bond is usually based on contact
mechanics theory such as Hertz linear contact theory [45]. When
a bond is assigned to the contact, the overall force-displacement
of the bonded particles is a combination of particle and bond prop-
erties. As fracture of bonds, which could induce pure particle-
particle contact on the cracked surfaces, will be allowed in some
of the simulations presented in this paper, the force-
displacement law for pure particle-particle contact is briefly
described first, followed by the constitutive law of the bond. More
details are available in the literature [43,46].
At the contact between two unbonded particles, the contact
force vector is further resolved into normal and shear components
with respect to the contact plane (see Fig. 2) as follows:
F ¼ Fn
þ Fs
ð1Þ
where Fn
and Fs
denote the normal and shear components,
respectively.
The magnitude of the normal force is the product of the normal
stiffness at the contact and the overlap between the two particles, i.
e.,
Fn
¼ Kn
Un
ð2Þ
where Kn
is the normal stiffness and Un
is the overlap.
The shear force is calculated in an incremental fashion. Initially
the total shear force is set to zero upon the formation of contact
and then in each timestep the relative incremental shear-
displacement is added to the previous value in last timestep:
Fs
¼ Fs
þ DFs
ð3Þ
DFs
¼ ÀKs
DUs
ð4Þ
DUs
¼ Vs
Dt ð5Þ
where Ks
is the shear stiffness at the contact, DUs
is the shear com-
ponent of the contact displacement, Vs
is the shear component of
the contact velocity and Dt is the timestep.
In addition, the maximum allowable shear contact force is lim-
ited by the slip condition:
Fs
max ¼ ljFn
j ð6Þ
where l is the friction coefficient at the contact.
In cases where a steady-state solution is required in a reason-
able number of cycles, the dashpot force acting as viscous damping
is grouped into the force-displacement law to account for the com-
pensation of insufficient frictional sliding or no frictional sliding. In
line with spring forces, the dashpot force is also resolved into nor-
mal and shear components at the contact:
Fd
n ¼ 2bn
ffiffiffiffiffiffiffiffiffiffi
mKn
p
dn ð7Þ
Fd
s ¼ 2bs
ffiffiffiffiffiffiffiffiffiffi
mKs
p
ds ð8Þ
m ¼
mAmB
mA þ mB
ð9Þ
where d in superscript denotes dashpot, A and B in subscript denote
the two particles in the contact pair, b is the critical damping ratio
and d is the relative velocity difference between two particles in
contact.
When a bond is created between two particles, the normal and
shear components of the bond force are included in the force-
displacement law. It is noted that normal bond force is first exam-
ined to see if the tensile-strength limit is exceeded. If a bond is still
present in the tension state, the shear-strength limit is enforced for
second iteration. When the bond is broken the bond force is dimin-
ished in the force-displacement law. Details of the fracture of
bonds in DEM will be discussed in Section 2.3.Fig. 1. 2D representation of contact between two particles in DEM.
B
A d
R[A]
R[B]
xi
[B]
xi
[A]
ni
xi
[C]
Un
contact plane
Fig. 2. Two particles in direct contact W.

2.2. Law of motion
The motion of each particle in each timestep is governed by
Newton’s Second Law in terms of translational and rotational
motions as follow
Translational motion:
Fi ¼ mð€xi À giÞ ð10Þ
Rotational motion:
Mi ¼ I _xi ð11Þ
where i in subscript is the indicial notation with respect to coordi-
nate system, Fi is the resultant force, m is the total mass of particle,
gi is the body force acceleration vector, _xi is the acceleration vector
of a particle, Mi is the resultant moment acting on a particle, I is the
principal moment of inertia of the particle, and _xi is the angular
acceleration about the principal axes.
The leap-frog method is used to update the position of the par-
ticle. First the relationship between the acceleration and velocity is
defined by
_x
ðtÞ
i ¼
1
Dt
€x
tþDt
2ð Þ
i À _x
tÀDt
2ð Þ
i

ð12Þ
_xðtÞ
i ¼
1
Dt
x
tþDt
2ð Þ
i À x
tÀDt
2ð Þ
i

ð13Þ
Then Eqs. (12) and (13) are substituted into Eqs. (10) and (11)
respectively and the velocity at time t þ Dt
2
À Á
is resolved as:
_x
tþDt
2ð Þ
i ¼ _x
tÀDt
2ð Þ
i þ
F
ðtÞ
i
m
þ gi
!
Dt ð14Þ
x
tþDt
2ð Þ
i ¼ x
tÀDt
2ð Þ
i þ
M
ðtÞ
i
I
!
Dt ð15Þ
Finally the position of the particle is updated accordingly:
x
ðtþDtÞ
i ¼ x
ðtÞ
i þ _x
tþDt
2ð Þ
i Dt ð16Þ
2.3. Contact models in DEM
Particles in DEM can be bonded together at contacts and sepa-
rated when the bond strength or energy is exceeded. Therefore it
can simulate the motion of individual particles and also the beha-
viour of a structure which is formed by assembling many particles
through bonds at contacts.
The advantage of DEM is that the two bonded particles can be
separated and thus form a crack at the contact point once the frac-
ture criterion of the bond is satisfied. In a DEM model, elementary
micro scale particles are assembled to form the structure with
macroscopic continuum behaviour determined only by the
dynamic interaction of all particles. Unlike the conventional FEM
that is based on the traditional continuum mechanics and provides
stress and displacement solutions by solving a global stiffness
matrix equation, DEM is discontinuous and the information of each
particle element and contact is recorded individually and updated
dynamically. Thus, DEM is convenient to deal with local behaviour
of a material by defining local models or parameters for the spec-
ified particles and contacts. Subject to external loading, when the
strength or the fracture energy of a bond between particles is
exceeded, flow and disaggregation of the particle assembly occur
and the bond starts breaking.
Particles can be packed in a regular (e.g. hexagonal or cubic in
2D) or random form. When they are packed in a hexagonal form
in plane stress condition, as shown in Fig. 3, the relationship
between the elasticity of the constructed structure and the stiff-
ness of the contacts can be derived as [47]:
Kn ¼
Ed
ffiffiffi
3
p
ð1 À mÞ
ð17Þ
Ks ¼
Edð1 À 3mÞ
ffiffiffi
3
p
ð1 À m2Þ
ð18Þ
where Kn and Ks are the contact stiffness in normal and shear direc-
tions respectively, E is the Young’s modulus, d is the element thick-
ness and m is the Poisson’s ratio.
As illustrated in Eq. (18), there is a constraint of (1 À 3m) term
on the right-hand side of equation in which the value of Poisson’s
ratio needs to be smaller than or equal to 0.33 so as to guarantee a
positive value of Ks.
A bond in DEM can be regarded as a glue to stick two particles
together, and linear parallel bond is special bond (see rectangular
box indicated in Fig. 3) that can be decomposed into linear model
and parallel bond model which are acting in parallel. The bond is
broken when the strength limit of bond is exceeded [40,43] and
after that only the linear model is active. Upon the use of a linear
parallel bond model, the contact stiffness Ki is the result of combi-
nation of both particles’ stiffnesses and bond stiffness according to
the following formulation [43]:
Ki ¼ Aki þ ki ð19Þ
A ¼ 2 R d ð20Þ
ki ¼
k
½AŠ
i k
½BŠ
i
k
½AŠ
i þ k
½BŠ
i
ð21Þ
where R and A are the radius and cross-sectional area of the bond,
respectively, ki is the parallel bond stiffness and ki is the equivalent
stiffness of two contacting particles. In this study the radius of the
bond is the same as the particle radius. If two particles have the
same normal and shear stiffness, ki is simplified as:
ki ¼
k
½AŠ
i
2
¼
k
½BŠ
i
2
ð22Þ
It can be assumed that the parallel bond stiffness is much larger
than the particles’ stiffness, thus the forces are predominantly
passed through parallel bonds, i.e. ki ¼ 0:01Aki,
Ki % Aki ð23Þ
Thus the parallel bond stiffness is determined by combining
Eqs. (19) and (20) with Eq. (23).
According to the nature of parallel bond model, bond strength is
the only criterion to determine the fracture of a structure. When
Fig. 3. Hexagonal packing of discrete particles with parallel bonds.

the structure is under pure tension, the bond strength can be
derived in terms of ultimate tensile strength and Poisson’s ratio
[41]:
f
crit
n ¼
Rdrult
2ð1 À mÞ
ffiffiffi
3
p
À
m
ffiffiffi
3
p

ð24Þ
f
crit
s ¼
Rdrult
2ð1 À mÞ
ð1 À 3mÞ ð25Þ
rcrit
n ¼
f
crit
n
2Rd
ð26Þ
rcrit
s ¼
f
crit
s
2Rd
ð27Þ
where f
crit
n and f
crit
s are maximum normal and shear forces acting on
the parallel bond, rcrit
n and rcrit
s are critical tensile and shear stresses.
It should be noted that the above derivation is only valid for 2D sim-
ulations in plane stress condition.
During the simulation, the parallel bond forces in normal and
shear directions are updated at each timestep through the force-
displacement law:
fn ¼ AknDdn ð28Þ
fs ¼ ÀAksDds ð29Þ
rn ¼
fn
A
þ b
MbR
I
¼ knDdn þ b
MbR
I
ð30Þ
rs ¼
jfsj
A
þ
0; ð2DÞ
b MtR
I
; ð3DÞ

¼ ksDds þ
0; ð2DÞ
b Mt R
I
; ð3DÞ

ð31Þ
where Ddn and Dds are the relative normal-displacement increment
and the relative shear-displacement increment respectively, Mb is
the bending moment, Mt is the twisting moment and b is the
moment-contribution factor. It should be noted that b in Eqs. (30)
and (31) is set to be zero in order to match those derived formula-
tions in Eqs. (26) and (27).
Then the strength limit is enforced to examine if the gained
stresses exceed the threshold value of critical stresses. If the
tensile-strength limit is exceeded (i.e. rn P rcrit
n ), then the bond
is broken in tension, otherwise, shear-strength limit is enforced
subsequently and the bond is broken in shear if rs P rcrit
s . Once
two particles are in unbonded state, parallel bond model is not
active any more, but the linear particle-particle contact model,
which force-displacement law is described in Section 2.1, is then
activated to account for the collision of particles. More details
about parallel bond can be found in [43,46].
As seen from Eqs. (30) and (31), parallel bond is behaved lin-
early and the plastic deformation is not taken into consideration
here. As for plastic or adhesive materials, several alternative mod-
els may be used by considering more complicated constitutive
behaviour. One of them is the contact softening model [47] which
is a bilinear elastic model and is similar to cohesive zone model
(CZM) in continuum mechanics. In this study the structure is con-
sidered as elastic.
2.4. DEM modelling of structural deformation and fracture
To verify the capability of DEM in modelling the structure part
in later FSI simulations, a tip-loaded cantilever beam test is studied
in this section. Comparisons of deflections, stress distributions and
final failure load are made to carefully evaluate the accuracy of the
DEM approach in modelling structural deformation and fracture.
The material properties of cantilever beam are shown in Table 1
and the configuration of the beam is shown in Fig. 4. The left end of
cantilever is clamped and the other side of cantilever is under an
increasing upward force F to give rise to a deflection.
The deformations of cantilever under three sets of upward
forces, 50 N, 500 N and 5000 N, applied on the right bottom tip
of cantilever are compared with analytical solutions [48] in Table 2.
The deflection is measured when the model reaches an equilibrium
state that the ratio of the unbalanced force (i.e. the sum of contact
force, body force and applied force) to the sum of body force and
applied force is extremely small, e.g. 1 Â 10À7
. The results from
DEM model and analytical solution are almost fully matched with
acceptable small errors which may be due to the fact the load is
applied at the centre of the particle not the exact edge of the real
beam.
In addition, the same test of cantilever beam at load 5000 N is
carried using FEM software ABAQUS in order to compare the stress
distribution. The element size in FEM is the same as the particle
radius in DEM. It can be seen from Fig. 5 that the distribution of
stress component r11 in FEM is nearly identical to the one in
DEM. The maximum stresses in both methods are also very close
with an error of 0.24%. This further confirms that the DEM model
can accurately predict the structural deformation.
To test the failure of the beam, the incremental loading
approach used above is replaced by assigning a constant and very
small upward velocity v ¼ 0:02 m=s to the particle at the right bot-
tom end of the beam. This small loading velocity is chosen to
ensure the structure under quasi-static loading condition till the
final failure [43]. The simulation is stopped immediately once a
bond breaking occurs. The obtained force at right bottom end of
the beam is compared with analytical solution according to:
rult ¼
PL
Z
ð32Þ
Z ¼
bh
2
6
ð33Þ
where Z is section modulus, b is the thickness of beam which is unit
in 2D simulations. In this test, the cantilever beam is assumed to fail
when maximum stress is equal to ultimate tensile strength. In
Table 3 the maximum applied load obtained from the DEM model
shows good agreement with the applied load computed from Eqs.
(32) and (33). It should be note that the DEM prediction is slightly
higher the theoretical one which is calculated under the assumption
that the beam is still perfectly straight at failure.
3. SPH model of fluid
Smoothed Particle Hydrodynamics (SPH) is also a Lagrangian
particle method which was initially used in astrophysical simula-
tions [49] and later extensively applied in fluid hydrodynamics
[50,51]. The applications of SPH range from multi-phase flow
Table 1
The list of material and particle properties.
Material properties Values
Density, q (kg/m3
) 2800
Ultimate tensile strength, rult (MPa) 310 Â 106
Young’s modulus, E (N/m2
) 70 Â 109
Poisson’s ratio, m 0.33
Particle radius, R (m) 0.0005
Bond radius, R (m) 0.0005
Cantilever length, L (m) 0.201
Cantilever height, h (m) 0.006196

[51], quasi-incompressible flows [52], heat transfer and mass flow
[50] and so on.
3.1. Interpolation of a function and interpolation of the derivative of a
function
The formulation of SPH is made up of two key steps, kernel
approximation and particle approximation. In the first step, the
typical integral forms of a function is given by the multiplication
of an arbitrary function and a smoothing kernel function, and its
derivative are described by simply substituting fðxÞ with r Á f ðxÞ
and finally formatted as:
fðxÞ ¼
Z
X
fðx0
ÞWðx À x0
; hÞdx
0
ð34Þ
r Á fðxÞ ¼
Z
s
fðx0
ÞWðx À x0
; hÞ Á~ndx
0
À
Z
X
fðx0
Þ Á rWðx À x0
; hÞdx
0
ð35Þ
In the second step, the integral representation of the function
and its derivative is approximated by summing up the values of
influential surrounding particles and this step is usually called par-
ticle approximation, as shown in Fig. 6. Only the particles located
in the support domain of kernel function with a radius of kh are
taken to account in particle approximation. As a result, the final
forms of Eqs. (34) and (35) are approximated as:
fðxiÞ ¼
XN
j¼1
mj
qj
fðxjÞWij ð36Þ
r Á fðxÞ ¼
XN
j¼1
mj
qj
fðxjÞ Á riWij ð37Þ
where i and j in subscript denote particle i and j, N is the number of
particles within the support domain of the kernel function, m is the
mass of the particle and q is the density of the particle.
SPH is adaptive as the field variable approximation is performed
at each timestep based on a current local set of arbitrarily dis-
tributed particles. Because of this adaptive nature, the formulation
of SPH is not affected by the arbitrariness of particle distribution.
Therefore, it is attractive in treating large deformations, tracking
moving interfaces or free surfaces, and obtaining the time depen-
dent field variables like density, velocity and energy.
3.2. Kernel
Up to now, various kernel functions have been developed and
used in the SPH method [25], among which the most widely used
are the cubic spline kernel function [53] and the Wendland kernel
function [54].
Cubic spline À Wðr; hÞ ¼ Ch
ð2 À qÞ3
À 4ð1 À qÞ3
for 0 6 q 6 1
ð2 À qÞ3
for 1 q 6 2
0 for q 2
8

:
ð38Þ
0.006196 m
F
0.201 m
Fig. 4. Configuration of cantilever under single point load in DEM.
Table 2
Deflections for the tip-loaded cantilever beam test.
Load
(N)
Deflection in DEM
model (m)
Deflection in analytical
solution (m)
Error
(%)
50 9.7533EÀ05 1.002417EÀ05 2.78
500 9.7533EÀ04 1.00247EÀ04 2.78
5000 9.7533EÀ03 1.002286EÀ03 2.76
Fig. 5. Distribution of stress r11 in cantilever beam at load 5000 N.

where q ¼ jrj=h. jrj is the distance between two particles, h is the
smoothing kernel length associated with a particle and the normal-
isation is ensured by setting up the constant Ch to be 15=ð14ph
2
Þ in
two dimensions.
Wendland Wðr; hÞ ¼ Ch
ð2 À qÞ4
ð1 þ 2qÞ for 0 6 q 6 2
0 for 2
(
ð39Þ
where Ch in two dimensions is normalised to be 7=ð64ph
2
Þ.
Static tank tests are carried out using SPH with both kernel
functions. According to the results shown in later section, the sim-
ulation using Wendland kernel show more orderly distribution of
particle than cubic spline kernel, as a result, the Wendland kernel
is chosen for all simulations in this study.
3.3. SPH modelling of incompressible fluid flow
With the application of SPH, Navier-Stokes equations in the
form of partial differential equations (PDEs) were transformed into
ordinary differential equations (ODEs) through kernel approxima-
tion and particle approximation.
For continuity equation, the rate of change of density, in Navier-
Stokes form is given by:
Dq
Dt
¼ Àqr Á v ð40Þ
In SPH form, the continuity equation becomes:
Dqi
Dt
¼
XN
j¼1
mjvij
@Wij
@xb
i
ð41Þ
In addition, the density of particle can be directly calculated by
summing up all the particles’ mass together since the integration
of density over the entire problem domain is exactly the total mass
of all the particles:
qi ¼
XN
j¼1
mjWij ð42Þ
However, the summation density approach is influenced by the
boundaries where the domain of the kernel function is partly trun-
cated, and the non-zero surface integral is directly the result of
truncation. One of the accuracy improvements has been proposed
to normalise Eq. (42) by summing up the kernel function over the
surrounding particles [55]:
qi ¼
PN
j¼1mjWij
PN
j¼1
mj
qj

Wij
ð43Þ
In concern with the discontinuity at boundary or interface, the
density integrated by Dqi
Dt
can assure the preservation of discontinu-
ity all the time with much less computational cost. Therefore, con-
tinuity density approach is the default one to calculate particle
density.
Before solving the momentum equations, the inclusive pressure
term should be calculated to account for the artificial compressibil-
ity assumed in the incompressible flow. In the SPH method, each
particle is driven by a pressure gradient which is based on local
particle density through an equation of state. However, there is
no equation of state for incompressible flow in which the pressure
is obtained through continuity and momentum equations. In order
to calculate the pressure term in the momentum equations in
incompressible flows, the concept of artificial compressibility
was proposed to consider what is theoretically an incompressible
fluid as weakly compressible fluid [52]. A feasible quasi-
incompressible Tait equation of state for incompressible flow is
applied as follows:
P ¼ B
q
q0
c
À 1

ð44Þ
where c is a constant taken to be 7 in most circumstances, q0 is the
reference density and B is the pressure constant. The subtraction of
1 on the right-hand side of Eq. (44) is to remove the boundary effect
for free surface flow [25].
The value of pressure constant is important to keep the density
fluctuation as small as possible. The density fluctuation can be
defined as:
jq À q0j
q0
% M2

v2
max
c2
s
ð45Þ
where M is the Mach number, vmax is the maximum velocity and cs
is the speed of sound.
The formulation of the speed of sound at reference density is:
c2
s ¼
cB
q0
ð46Þ
If the speed of sound is assumed to be 10 times of the maximum
velocity, the pressure constant can be worked out then:
B ¼
100q0
c
v2
max ð47Þ
In the same way as the transformation of continuity equation
from Navier-Stokes form to SPH form, the moment equation in
SPH form is described as:
dvi
dt
¼
Xn
j¼1
mj
Pi
q2
i
þ
Pj
q2
j
þ Pij
!
rWij ð48Þ
where Pij is the viscosity term.
There is a wide variety of derivation of the viscosity term [56],
and the first one derived as an artificial viscosity is based on the
consideration of strong shocks [57]:
Pij ¼
Àaclijþbl2
ij
qij
if vij Á rij 0
0 otherwise
8

:
ð49Þ
where a and b denotes the artificial viscosity coefficient respec-
tively, lij ¼ 1=2ðli þ ljÞ and qij ¼ 1=2ðqi þ qjÞ. As it has been a
common practice to use an artificial viscosity in compressible SPH
formulations for better accuracy in the simulation of shock wave,
this viscosity form will not be taken into consideration here.
Table 3
Maximum applied load for the tip-loaded cantilever beam test.
Analytical DEM Error
Maximum applied load P (N) 9868.66 10322.1 4.595%
Fig. 6. Particle approximations for particle i within the support domain kh of the
kernel function W. rij is the distance between particle i and j, s is the surface of
integration domain, X is the circular integration domain, k is the constant related to
kernel function and h is the smooth length of kernel function.

Instead, another viscosity form including physical viscosity of parti-
cle derived in [58] is adopted in this study:
Pij ¼ mj
ðli þ ljÞrij Á rWij
qiqj r2
ij þ 0:01h
2
vij ð50Þ
where 0:01h
2
in the denominator is meant to avoid singularity.
Apparently Eq. (50) can approximate the viscosity term physi-
cally and it is also useful for dealing with multiphase problems
where densities at interface are not identical. This will become
more important when discrete particles are incorporated in the
present SPH-DEM model in the future to enable the FPSI
simulations.
3.4. Completeness and tensile instability
Even though SPH is an increasingly promising numerical
method, several difficulties have been encountered in recent dec-
ades. The first difficulty is the completeness of SPH which is the
ability of the approximation to reproduce specified functions and
other ones are the rank deficiency and the tensile instability that
manifests itself as a bunching of nodes. This unphysical phe-
nomenon, which is normally due to tensile instability, could reduce
resolution and even cause numerical errors during the simulation.
To extend applications of SPH into a wide range of fluid dynamics
problems, a series of modifications and corrections have been
introduced to improve the approximation accuracy.
In terms of completeness, there are two approaches for approx-
imating the continuity equation: one is density summation
approach and the other one is continuity density approach. The
density summation approach conserves the mass since the integra-
tion of density over the entire support domain is equal to the total
mass of all the particles. However, this approach suffered edge
effect, namely boundary particle deficiency where the support
domain is not fully filled with particle at the edge of fluid domain,
as a result, the density is smoothed out to cause some spurious
results. Randles and Libersky [59] proposed the normalisation of
the summation density approach with the SPH summation of the
smoothing function itself over the surrounding particles to
improve the accuracy of approximation. In this study, the continu-
ity density approach was applied instead of the density summation
approach to introduce velocity difference into the discrete particle
approximation as the usage of the relative velocities in anti-
symmetrized form serves to reduce errors arising from the particle
inconsistency problem [25].
The rank-deficiency is defined as that the number of integration
points is less enough so that the solution to the underlying equilib-
rium equation becomes non-unique. Even though some research-
ers [60,61] proposed to eliminate the rank-deficiency by
introducing additional integration points (e.g. stress points) at
other locations than the SPH centroids, the increased computa-
tional effort associated with the additional integration points ren-
ders this approach less efficient, and a precise guideline as to how
many additional stress points are needed is missing. In this study,
several smooth lengths were tested to find out the optimal value of
smooth length in order to keep the rank deficiency as minimum as
possible.
The original updated Lagrangian formulation of SPH, which is
also termed Eulerian SPH, suffers from the so-called tensile insta-
bility, in which leads to the clumping of particles. In a Lagrangian
formulation the kernel approximation is performed in the initial,
undeformed reference coordinates of the material [14]. In the
application of Lagrangian kernel, the tensile instability is absent,
however, rank deficiency still exists. In this study, the instability
can be removed by using an artificial stress which, in the case of
fluids, is an artificial pressure. An anti-clump term was introduced
to be added into the momentum equations to prevent particles
from forming into small clumps due to unwanted attraction [62]:
dvi
dt
¼
Xn
j¼1
mj
Pi
q2
i
þ
Pj
q2
j
þ Pij þ Rij
!
rWij ð51Þ
Rij ¼
v2
max
c2
s
Pi
q2
i
þ
Pj
q2
j

Wij
WðDPÞ
4
ð52Þ
where vmax ¼ 1
10
cs, and DP is the initial particle spacing.
Another remedy also applied in this model is to correct the rate
of the change of particle position in order to keep particles move
orderly in the high speed flow:
Support domain of
kernel function
SPH particle i
Solid boundary
Fig. 7. Truncation of particle support domain by a boundary.
Boundary
particles
SPH particle i
Support domain of
kernel function
Boundary particles have
interaction with SPH particle i
Fig. 8. Boundary particles and their interaction with SPH particles.
Fig. 9. Initial configuration of the static tank test (SPH particles in black colour and
boundary particles in white colour).
Table 4
SPH parameters used for the static tank test.
Parameters Values
Boundary particle spacing (m) 0.0025
SPH particle spacing (m) 0.005
Particle number 1278
Kernel function Cubic spline/Wendland
Kernel smooth length (m) 0.005
Fluid density (kg/m3
) 1000
Fluid viscosity (PaÁs) 8.9 Â 10À4
Time step (s) 0.000004
Physical time (s) 1.0

dri
dt
¼ vi þ e
X
j
mjðvj À viÞ
qij
Wij ð53Þ
where the second term on the right hand side of Eq. (53) is the cor-
rection factor, the value of e is problem-dependent as large e can
slow down the particle velocity unphysically.
3.5. No-slip boundary
When a SPH particle is approaching a boundary (see Fig. 7), its
support domain overlaps with the problem domain, consequently
its kernel function is truncated partially by the boundary and the
surface integral is no longer zero. Theoretically only particles
located inside the support domain are accounted for in the summa-
Time Cubic spline kernel Wendland kernel
(a)
t=0.2s
(b)
t=0.4s
(c)
t=0.6s
(d)
t=0.8s
(e)
t=1.0s
Fig. 10. Particle distribution during a period time of 1.0 s using two different kernel functions.

tion of the particle interaction, but there are no particles existing in
the truncated area beyond the solid boundary. Different remedies
have been proposed recently to rectify boundary truncation. The
normalisation formulation of density approximation was derived
to satisfy the normalisation condition and ensures the integral of
kernel function over the support domain is unity [55]. In compar-
ison with kernel re-normalisation, the application of virtual or
ghost particles is widely used to replace the solid boundary and
to produce a repulsive force in order to avoid wall penetration
[52]. The interaction force between a boundary particle and an
SPH particle could be in Lennard-Jones form [52], in which the
SPH particles are repelled within a cut-off distance, but Lennard-
Jones form is highly dependent on the problem being simulated.
In order to have a simple, robust as well as reliable, interaction
between boundary and SPH particles, in this study two-layers of
fixed boundary particles are placed as solid boundaries, which
are initialised with a reference density of SPH particles, but their
density and other parameters such as position and velocity are
all fixed and not evolved with the parameter variance of SPH par-
ticles. In order to produce sufficient repulsive forces, the distribu-
tion of boundary particles is denser than the distribution of SPH
particles as shown in Fig. 8.
3.6. Smoothing length
Smoothing length can vary in space and time, and this is useful
in modelling of compressible flow to ensure the number of particle
within kernel is more or less constant. The formulation of smooth-
ing length is:
hi ¼ r
mi
qi
1=d
¼ rDp ð54Þ
where r is the constant with a typical value of 1.3 and d is the
dimensionality of the simulation.
As the mass of particle in SPH is assumed to be constant, the
smoothing length associated with particle volume should vary
accordingly with density. However, the density fluctuation in
incompressible flow is minor [63], thus using variable smoothing
length with extra computational cost is not expected to produce
significantly different results. Therefore, smoothing length is cho-
sen to be constant with 1:25Dp in this study for easy numerical
implementation and saving computational time. Note that it would
be better to use variable smooth length to improve the numerical
accuracy in the future, however, as a first step in developing this
SPH-DEM model, constant smooth length is used for simplicity.
3.7. Numerical implementation of SPH
In this study, the SPH theory above is implemented in PFC2D
v5.0 using C++. The indirect contact feature is adopted to enable
the particle interaction in SPH. PFC2D 5.0 as a DEM software pack-
age which has many features that can be directly utilised for SPH
simulations such as a particle search scheme and a time integra-
tion scheme. A particle search scheme is based on a Linked-list
algorithm to sub-divide the particles within different cells and par-
ticles are identified through a linked list. PFC2D 5.0 uses a leapfrog
technique for numerical integration to update field variables at
0.142 m
0.293m
0.35m
0.584 m
SPH particles
Boundary particles
Fig. 11. 2D SPH representation of the dam-break test.
Table 5
SPH parameters for the dam-break test.
Parameters Values
Kernel function Wendland
Kernel smooth length (m) 0.005/0.00625/0.0075
) 1000
Fig. 12. Initial density of SPH particles with an assumption of artificial compressibility.

each particle. The workload for coding SPH and later SPH-DEM
coupling is significantly reduced by making use of those two fea-
tures. A detailed flow chart of implementing SPH and its coupling
with DEM is shown later in this section. As the codes are written
in C++, they are portable for other open source DEM codes for
SPH-DEM simulations without much modification.
3.8. Sensitivity study and validation of the SPH model
To show the success of implementing SPH into PFC code for
modelling the fluid flow in later FSI simulations, the SPH model
is carefully assessed through a series of numerical tests including
a static tank test to see the performance and adaptability of differ-
ent kernel functions and a dam break test to evaluate the effects of
different smoothing length and particle resolution.
3.8.1. SPH simulation of static tank test with different kernel functions
A simple static stank test with an initial cubic packing of parti-
cles is set up in a30 mm Â 30 mm tank, as shown in Fig. 9. Two dif-
ferent kernel functions are investigated, i.e. cubic spline kernel and
Wendland kernel. Under only gravity the particle distribution was
observed for a time period of 1.0 s.
Each SPH particle is initialised with a hydrostatic pressure in
accordance with the particle’s position and its reference density.
(a)
t=0.2s
(b)
t=0.4s
V (m/s)
V (m/s)
(c)
t=0.6s
(d)
t=0.8s
(e)
t=1.0s
V (
V (
V (
m/s)
m/s)
m/s)
Time Experiment [66] MPS [64] SPH, h=1.25×∆p
Fig. 13. Results from experiment [66], MPS [64] and SPH with h = 1.25 Â Dp for a time period of t = 1.0 s.

The density of each particle is then updated through equation of
state in Eq. (44). The particle spacing Dp is 0.005 m and the mass
of particle in 2D simulation is described as:
mi ¼ qiðDpÞ2
ð55Þ
where mi is the mass of fluid particle.
The smoothing length is initially set as h ¼ 1:0 Â Dp for all cases
and the maximum velocity of particle is assumed to be
vmax ¼
ffiffiffiffiffiffiffiffiffi
2gD
p
, where D is the depth of the fluid and g is the gravi-
tational acceleration, 9:81 m=s2
. All the material and numerical
properties are listed in Table 4.
Fig. 10 shows the particle distribution at 0.2 s time interval for
cubic spline kernel and Wendland kernel, respectively. It can be
seen that the particle distribution for the test using Wendland ker-
nel nearly remains the same as the original particle distribution
and only a small disorder is found at the corners of fluid, which
is due to the boundary/interface deficiency. For the test using cubic
spline kernel, the particles are packed orderly as well, but the par-
Time
(a)
t=0.2s
(b)
t=0.4s
(c)
t=0.6s
V (m/s)
V (m/s)
V (m/s)
h=1.0×∆p h=1.25×∆p h=1.5×∆p
(d)
t=0.8s
(e)
t=1.0s
V (m/s)
V (m/s)
Fig. 14. SPH simulations with three different smoothing length for a time period of t = 1.0 s.

ticle distribution is not cubic any more, it and more likely becomes
hexagonal after 0.2 s. Even though the reason for this difference is
not clear, both tests show good particle distribution without any
particle cluster. Considering the Wendland kernel seems to pro-
duce better form of particle distribution, it will be used for all
the simulations later.
3.8.2. SPH simulation of dam break test with different smoothing
length
The case of a collapsing water column has been used in SPH
studies [64,65], therefore it is utilised here to validate the imple-
mented SPH model. Besides the validation of SPH model, the effect
of different smoothing length, h ¼ 1:0 Â Dp, h ¼ 1:25 Â Dp and
h ¼ 1:5 Â Dp, is also examined. The geometry of the case is
depicted in Fig. 11 and the simulation parameters are listed in
Time Coarse Medium Fine
(a)
t=0.2s
V (m/s)
(b)
t=0.4s
(c)
t=0.6s
(d)
t=0.8s
(e)
t=1.0s
V (m/s)
V (m/s)
V (m/s)
V (m/s)
Fig. 15. SPH simulations with three different particle resolutions for a time period of t = 1.0 s.
Table 6
Particle resolutions in the dam-break test.
Parameters Coarse Medium Fine
Boundary particle spacing (m) 0.003 0.0025 0.002
SPH particle spacing (m) 0.006 0.005 0.004
Particle number 2087 2743 4098

Table 5. The water column is initially adjacent to the left wall and
is supported by a wall that is instantaneously removed when the
experimental test starts. The water is thereby released into a dry
channel. The SPH particles are initialised with hydrostatic pressure
in accordance with the position and the density of each SPH parti-
cle derived by reversing Eq. (44). The distribution of density for the
SPH particles is displayed in Fig. 12.
Density change
rate
Particles density
update
Particles position
and velocity
update
Particles position
and velocity
update
Repulsive forces
Bond forces,
repulsive forces,
and external forces
Internal forces,
viscous forces and
external forces
Neighbour Particle search
(Linked-list scheme)
Set-up parameters including particle
search radius, bond stiffness and
timestep, etc.
Generate SPH, DEM and boundary
particles, and assign parallel bonds for
DEM particles
Bond breaking?
Delete bond
End of calculation?
Output results
NO
YES
t+ Δt
YES
NO
SPH
particles
DEM
particles
Boundary
particles
Fig. 16. Computation ﬂowchart for coupling DEM and SPH.
0.005 m
0.079m
0.14m
SPH particles
DEM particles
Boundary particles
Fig. 17. Conﬁguration of 2D elastic gate test in a coupled SPH-DEM model.
Table 7
Parameters for SPH-DEM modelling of the elastic gate test.
Parameters Values
DEM particle size (m) 0.00125
Kernel function type Wendland
Kernel smooth length (m) 0.0021875
) 1000
Gate density (kg/m3
) 1100
Gate elastic modulus (MPa) 12.0
Gate Poisson’s ratio 0.33

In Fig. 13, the SPH simulation results with a smoothing length of
h ¼ 1:25 Â Dp are compared with experimental images as well as
the numerical simulations using the moving particle semi-
implicit method (MPS) for a time period of 1.0 s with a time inter-
val of 0.2 s [64]. The collapsing water runs along with bottom wall
with an increasing velocity at the leading edge at 0.2 s (see
Fig. 13a), and the accelerated water is then blocked by the right
vertical wall thereby moving upwards at 0.4 s (see Fig. 13b). At
0.6 s, the SPH particles tend to reach the highest position with los-
ing momentum energy which is offset by gravitation acceleration
and then these SPH particles fall down to hit other SPH particles
which still move along with bottom wall. At 1.0 s, the movement
t
t
t
Time
(a)
t=0.0s
(b)
t=0.08s
(c)
t=0.16s
(d)
t=0.24s
Experiment [11] SPH-SPH [11]
V (m
V (m
V (m
V (m
SP
/s)
/s)
/s)
/s)
H-DEM
(e)
t=0.32s
(f)
t=0.40s
V (m/s)
V (m/s)
Fig. 18. Comparisons between experimental, SPH-SPH and SPH-DEM results of elastic gate test with a time period of 0.40 s.

of reflected SPH particles is gradually restricted by the left vertical
wall. In general all the simulated flow patterns of water in SPH
agree well with experiment and MPS.
Fig. 14 shows the numerical results with different smoothing
length. The flow patterns are almost identical before 0.6 s. After
0.6 s, those fluid particles repelled back by right vertical wall are
gradually mixing with incoming fluid particles that are approach-
ing right vertical wall. Due to this expected phenomenon, the sim-
ulations with longer smoothing length can search more
surrounding particles to more accurately represent the fluid pro-
file. It is apparent that results with h ¼ 1:25 Â Dp and
h ¼ 1:5 Â Dp showed a good match in fluid profile from the begin-
ning to 1.0 s. In addition to the fluid flow profile, computational
cost is another determining factor. More computational cost is
required for the simulation with h ¼ 1:5 Â Dp, and it produces bet-
ter results than h ¼ 1:0 Â Dp (particularly at time 1.0 s, see
Fig. 15e) but similar with h ¼ 1:25 Â Dp. Therefore, smoothing
length h ¼ 1:25 Â Dp is a better choice for numerical accuracy
and computational efficiency and it is chosen for the rest of numer-
ical simulations.
3.8.3. SPH simulation of dam break test with different particle
resolutions
In this section the dam break test is simulated again using the
same kernel function (Wendland kernel) and smoothing length
(h ¼ 1:25 Â Dp) but three different particle resolutions, i.e. particle
spacing Dp. The particle spacing of 0.005 m used before is chosen
as a sample data, and two more different particle spacing, one is
finer whilst the other one is coarser, are investigated for compar-
isons. The data for three particle resolutions are presented in
Table 6, and the simulation results for physical time 1.0 s are
shown in Fig. 15.
It is evident in Fig. 15 that the results for particle spacing of
Dp ¼ 0:004 show the best fluid flow profile and even capture the
void at time t ¼ 0:8 s and the curved wave at time t ¼ 1:0 s. Usually
finer particle resolutions would give better results with more fluid
flow profile details but at a cost of computational time. Therefore it
is essential to balance the numerical accuracy and the computa-
tional cost, which highly depends on the kind of results that are
expected to achieve. In this study, Wendland kernel and smoothing
length in h ¼ 1:25 Â Dp are determined to be applied in the follow-
ing simulation of fluid-structure interaction, and the particle reso-
lution will be adjusted in accordance with the testing problem at a
reasonable computational cost.
4. Coupled SPH-DEM model for FSI
4.1. Methodology
In dealing with interface between SPH and DEM particles, the
interactive force could be defined differently in accordance with
flow configurations (e.g. fluid-structure interaction flow or fluid-
particle interaction flow). In the fluid-particle system, the force
acting on a single DEM particle is the summation of DEM-DEM
contact force, drag force, buoyancy force and gravity force
[29,67]. Conversely, as DEM particles are bonded together in
fluid-structure system, the interaction force between SPH and
DEM particles can be determined by Newton’s Third Law of Motion
under a non-slip condition [12,68]. In this study of fluid-structure
interaction, the SPH particles for fluid domain and the DEM parti-
cles for solid domain are coupled together by using fluid-structure
interaction under Newton’s Third Law in which the forces on the
solid from the fluid and the forces on the fluid from the solid are
equal in magnitude but opposite in direction. The interaction
forces between SPH particles and DEM particles follow Eq. (34).
The density and the pressure for DEM particles stay unchanged
at all times, and only their velocity and position evolve with time.
It is worth noting that special care needs to be taken with the
numerical technique when considering DEM particles in Eq. (25).
For a structure composed of bonded DEM particles, the force from
fluid to solid acts only on the surface layer of the structure, hence
the inner DEM particles that are included in the support domain of
a SPH particle should not be taken into account [69]. Even though
this special technique is physically correct, the forces acting on the
inner DEM particles are too small as they are close to the edge of
support domain. Therefore, this technique is not used in this study
yet and it will be undertaken in future to improve the coupled SPH-
DEM model. The flow chart of coupling SPH and DEM is schemat-
ically described in Fig. 16.
4.2. Validation of the coupled SPH-DEM model
The validation test for the coupled SPH-DEM model is to simu-
late the water flow in the elastic gate problem for comparison with
experimental data and numerical results using a coupled SPH-SPH
method [11]. The initial configuration is illustrated in Fig. 17 and
the simulation parameters for this validation case are listed in
(a) Water level behind the gate (b) Water level 5cm far from the gate
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Waterlevel,y(m)
Time, t (s)
Experiment [11]
SPH-DEM
SPH-SPH [11]
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
Waterlevel,y(m)
Time, t (s)
Experiment [11]
SPH-DEM
SPH-SPH [11]
Fig. 19. Water levels at different time.
SPH particles
DEM particles
Boundary particles
Fig. 20. 2D representation of FSI with fracture.

Table 7. The top end of elastic gate in purple is ﬁxed and the other
end is free to move. The bonded DEM particles representing the
elastic gate are distributed in a hexagonal pattern and particle
and bond stiffness is determined according to Eqs. (19)–(21) and
(23). The SPH particles for water are initialised with hydrostatic
pressure, and there is no pre-existing stress and deformation for
bonded DEM particles.
A comparison of numerical results from the present SPH-DEM
model against experimental data and numerical results from the
coupled SPH-SPH model is shown in Fig. 18. Compared to the
Time Particles velocity (m/s) σ22 in the structure (Pa)
(a)
t = 0.04s
(b)
t = 0.08s
(c)
t = 0.12s
(d)
t = 0.16s
(e)
t = 0.20s
V (m/s)
V (m/s)
V (m/s)
V (m/s)
V (m/s)
Fig. 21. SPH-DEM modelling of FSI with fracture.

experiment snapshots, the deformation of the elastic plate and the
vertical displacement of free surface of water are generally well
predicted by the coupled SPH-DEM model. The maximum defor-
mation of the plate is at top end and behaves almost as a rigid body
without deformation at bottom end of the plate. It should be noted
that after removing the hammer, which ﬁxes the elastic plate
against water pressure immediately in the experiment, water leak-
age besides the elastic plate is observed. Owing to this leakage, the
water pressure acting on elastic plate is lower than that in the SPH-
DEM simulation and consequently the vertical displacement of the
free surface in the SPH-DEM simulation is larger than the
experimental results. In comparison with both experimental and
SPH-SPH results, both larger deformation and higher vertical
displacement of the free surface in SPH-SPH and SPH-DEM results
make sense without water leakage. Due to the decreasing hydrody-
namic pressure of water, the deformation of the plate tends to be
(f)
t = 0.24s
(g)
t = 0.28s
(h)
t = 0.32s
(i)
t = 0.36s
(j)
t = 0.40s
V (m/s)
V (m/s)
V (m/s)
V (m/s)
V (m/s)
Fig. 21 (continued)

smaller after 0.16 s in the SPH-DEM model, and as a result the ver-
tical displacement of free surface changes more slowly.
In Fig. 19 the water levels behind the gate and the water level
5 cm far from the gate are quantitatively recorded to represent
the evolution of free surface with time. Due to the water leakage
in the experiment, the flow rates calculated from both SPH-SPH
and SPH-DEM models are slightly higher than the experimental
data, which results in a faster decrease of water level. The SPH-
SPH model and SPH-DEM model show good agreement throughout
the entire test process in terms of plate deformation and vertical
displacement of free surface.
5. Coupled SPH-DEM model for FSI with fracture
In order to demonstrate the versatility of the coupled SPH-DEM
model in simulating fluid-structure interaction in this section, an
FSI problem with fracture is presented. The same configuration in
the validation test of coupled SPH-DEM model is used, as shown
in Fig. 20, but the elastic plate is clamped at the bottom end and
free to move at the top end. In addition, the fixed plate in pink is
removed in this case, and the material strength limit is lowered
in order to allow for a fracture to occur due to the pressure of
the water. Simulation parameters in Table 7 are used here. In this
case, the boundary particles not only produce repulsive forces to
SPH particles but also need to respond to the fractured bonded
DEM particles when they contact. As the momentum energy of
fractured bonded DEM particles is much greater than an individual
SPH particle, the two-layer boundary particles cannot produce
large enough force to impede the penetration of fractured bonded
DEM particles. To solve this issue, two different kinds of repulsive
forces are created from every single boundary particle to act on the
SPH particles and DEM particles, i.e., the smaller repulsive forces
are only designated for SPH particles while fractured bonded
DEM particles receive other greater repulsive forces.
In Fig. 21(a) and (b), the largest stress (stress component r22) is
found near to the bottom end of elastic plate before the occurrence
of fracture due to the maximum bending moment induced by the
water pressure. At time around 0.12 s, the strength limit of elastic
plate is exceeded and consequently the elastic plate breaks into
two parts and then the fractured part moves towards the left
boundary wall under the forces produced by the SPH particles of
water. With the modification of the boundary particles in handling
the approaching bonded DEM particles, wall penetration is fully
avoided. Due to the vibration of the elastic plate, the flow pattern
of water is highly affected to cause flow fluctuation which leads to
some irregular movements of certain individual particles or small
clusters of particles. After 0.16 s, the fractured structure is pushed
away to approach the left solid boundary. When the plate moves
along with bottom solid wall, the stresses acting on bond are
negligible as no significant deformation is observed. This coupled
SPH-DEM model used in FSI with fracture is not experimentally
validated yet, but these results demonstrate its capabilities.
6. Conclusions
A 2D coupled SPH-DEM model for fluid-structure interaction
problems has been proposed and developed. In this model, SPH
based on Navier-Stokes equations is used to model the fluid phase
while DEM with a bond feature is used for the solid phase. In deal-
ing with the fluid-solid interface, Newton’s Third Law is applied
with the same magnitude in both phases, but in opposite direc-
tions. As both SPH and DEM are Lagrangian particle methods, no
special treatment is required to define the fluid-structure interface,
even in the presence of large deformation and/or fracture of struc-
ture. The contact between SPH and DEM particles is automatically
detected in accordance with a particle search radius that is twice
the smooth kernel length. When a smoothed particle is approach-
ing a fixed boundary and its support domain is intersected with the
boundary, two-layer boundary particles are placed in the position
of solid boundary to produce repulsive forces to handle the kernel
truncation.
The individual DEM model and SPH model has been validated
by comparison with analytical, experimental and other numerical
results. A tip-loaded cantilever beam has been chosen to validate
the DEM model with bonded particles for predicting structural
deformation and fracture. A typical dam-break test with dry bed
was considered to validate the SPH model for predicting free sur-
face flow. After the validations of both DEM and SPH models, the
coupled SPH-DEM model was then validated against a typical
fluid-structure interaction problem where a thin and long elastic
plate interacts with free surface flowing fluid. Finally the coupled
SPH-DEM model is extended to include the occurrence of struc-
tural fracture by allowing the bonds in the DEM model to break
and the broken structure to move under fluid pressure.
The obtained results have shown satisfactory predictions in
terms of flow pattern, structure deformation and velocity contour,
although some improvements such as particle distribution density,
no-slip condition, skin layer of solid structure, sensitivity studies in
different kernel functions and smooth kernel lengths are still
required. In the future, the coupled model will be expanded from
2D to 3D simulation. The case for fluid-structure interaction with
fracture will be validated through laboratory experiment and fur-
ther improvements for this coupled model will be made to enable
the simulations of real engineering problems.
Acknowledgements
The authors would like to thank Mr. Sacha Emam (ITASCA, USA)
and Dr. Paul Brocklehurst (Lancaster University, UK) for their use-
ful suggestions on the C++ programming in PFC 5.0, Prof. Thomas
Lewiner (Pontifical Catholic University of Rio de Janeiro, Brazil)
for his advice on the SPH coding, and Dr. Yong Sheng (University
of Leeds, UK) for providing the PFC 5.0 software license for this
research. The first author would like to acknowledge the School
of Civil Engineering, University of Leeds for financial support of this
PhD research project.
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