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Computers and Geotechnics 171 (2024) 106395
Available online 3 May 2024
0266-352X/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Research Paper
Simulating the uid–solid interaction of irregularly shaped particles using
the LBM-DEM coupling method
Mohammad Hassan Ahmadian , Wenbo Zheng
*
School of Engineering, University of Northern British Columbia, Prince George, British Columbia V2N 4Z9, Canada
ARTICLE INFO
Keywords:
Irregularly shaped particles
Lattice Boltzmann method
Discrete element method
Fluid–solid interaction
Coupling analysis
ABSTRACT
In this paper, the lattice Boltzmann method (LBM) and the discrete element method (DEM) are coupled to
simulate the interaction between the uid phase and irregularly shaped solid particles. For this purpose, the
geometry of realistic particle shapes is represented as clumps of overlapping spheres and then simulated through
a multi-sphere model in DEM and coupled with LBM using two open-access codes, LIGGGHTS and Palabos. The
accuracy of the coupling method with clumps is demonstrated by simulating several benchmark cases and
comparing them with the results from the literature. The coupled LBM-DEM method is then used to simulate the
collapse and transport of submerged granular particles with spherical shape and irregular shape, respectively, to
highlight the inuence of grain morphology in the solid–uid interaction. Compared with previous LBM-DEM
coupling for highly idealized non-spherical shapes, the research provides a more realistic computational
framework to capture the complex irregular particle shapes of geomaterials with clumps/multi-spheres, which is
useful for studying the underlying effect of particle shape on geotechnical issues such as internal erosions.
1. Introduction
The uid–solid interaction is a fundamental concept in geotechnical
engineering with practical importance in various issues involving water
and soils, such as scouring phenomena around offshore mono-pile
foundations (Nielsen et al., 2015) and internal erosion in embankment
dams (Foster et al., 2000; Teng et al., 2023). The interaction between
uids and sediment particles, characterized by a strong coupling effect,
plays a crucial role in not only shaping particle movements but also
inuencing the uid ow patterns (Stratford et al., 2005; Jansen and
Harting, 2011; Wang et al., 2023). The essence of uid–solid system
behavior lies at the microscopic scale, where the interaction between
solid particles and uid depends on parameters like uid velocity, solid
volume fraction, and particle shape. Such parameters eventually
contribute to the interaction in terms of forces and momentum ex-
change. As conventional experimental measurements are unable to
capture the intricate interplay of uid and particle interactions,
employing numerical modeling is more promising in studying the un-
derlying mechanisms involved in a uid–solid system (Lai et al., 2023).
For the effective simulation of uid–solid systems, a coupled
approach is necessary. This involves using different methods for the
uid and solid phases. Computational Fluid Dynamics (CFD) and the
Lattice Boltzmann Method (LBM) are commonly employed for the uid
phase. CFD mostly utilizes either nite element (L¨
ohner, 2008) or nite
volume (Moukalled et al., 2016) approaches to handle complex uid-
–solid particle interactions in multiphase conditions. LBM, in contrast,
models systems of ctitious particles adhering to macroscopic conser-
vation laws, which is benecial in managing complex boundary condi-
tions and parallel computing (Wang et al., 2012). Numerical modeling of
solid particles often uses the Discrete Element Method (DEM). DEM,
based on Newton-Euler equations, can model the ow of discrete solid
materials, dynamically simulate each particle’s trajectory, and accu-
rately track translational and rotational movements, as well as in-
teractions with other particles and boundaries (Cundall and Strack,
1979). Thus, the coupling approach enhances the strength of each
method for a more comprehensive understanding of uid–solid systems.
The development of coupling methods in uid–solid interactions has
been marked by several studies, starting through 2D simulations of
seepage ow (Chen et al., 2011), piping erosion (Lomin´
e et al., 2013),
sedimentation of circular particles (Zhang, 2014), and granular column
collapse (Soundararajan, 2015). The LBM-DEM coupling, in particular,
stands out for its potential in simulating uid–solid dynamics. This is
attributed not only to LBM’s efciency in parallel computing but also to
its versatility in adopting various coupling techniques, such as immersed
* Corresponding author at: 3333 University Way, Prince George, British Columbia V2N 4Z9, Canada.
E-mail addresses: ahmadian@unbc.ca (M.H. Ahmadian), wenbo.zheng@unbc.ca (W. Zheng).
Contents lists available at ScienceDirect
Computers and Geotechnics
journal homepage: www.elsevier.com/locate/compgeo
https://doi.org/10.1016/j.compgeo.2024.106395
Received 21 September 2023; Received in revised form 7 April 2024; Accepted 29 April 2024
Computers and Geotechnics 171 (2024) 106395
2
boundary (Feng and Michaelides, 2004) and momentum exchange
(Ladd, 1994),. Notably, Noble and Torczynski (Noble and Torczynski,
1998) introduced the concept of Immersed Moving Boundary (IMB),
later combined with LBM and DEM to simulate uid-particle ow. In this
approach, the collision of interface lattice distribution functions corre-
sponds to the solid fraction, while momentum exchange at the particle
boundary calculates the hydrodynamic force reaction (Zhang et al.,
2021). IMB’s compatibility with LBM’s parallel nature and its ease of
coding make it a practical choice. Moreover, LBM-DEM has seen ad-
vancements in stability and accuracy, especially with the adoption of a
multiple-relaxation-time collision operator (Rettinger and Rüde, 2022)
and enhanced solid fraction calculation (Wang et al., 2018). To date,
LBM-DEM simulations have been used to simulate a wide range of en-
gineering scenarios, such as sand production (Han and Cundall, 2013),
sediment settlement (Jiang et al., 2022), and internal erosion (Zhou
et al., 2020).
Most of the abovementioned LBM-DEM studies simulated the solid
phase as spherical particles. In the eld of granular materials, especially
those in geotechnical engineering, the irregular shapes of granular
particles have been recognized as a crucial factor affecting their defor-
mation, strength, and permeability (Zheng et al., 2021). These particles,
diverging from the traditional spherical form, encompass a wide array of
shapes (Zheng et al., 2019), which can be broadly categorized into
regular and irregular congurations. Regular non-spherical particles
include ellipsoids, polyhedral, polygonal, superquadric, etc., each of
which possesses unique geometric characteristics but is distinct from
irregular non-spherical particles found in geomaterials. The coupled
LBM-DEM methods have been used in some studies with non-spherical
particle shapes, primarily with regular non-spherical particles. Han
et al. (Han et al., 2007) studied polygonal particles within turbulent
ows using the LBM-DEM coupling method. Gardner and Sitar (Gardner
and Sitar, 2019) investigated the interactions of arbitrarily shaped
polyhedral particles using LBM-DEM. Polygonal discrete elements were
used to simulate the interactions between convex and concave polygonal
particles, broadening the scope of particle dynamics analysis (Wang
et al., 2021). In a more recent study, LBM-DEM was applied for simu-
lating hydraulic plucking using polyhedral rock blocks (Gardner, 2023).
As research progresses, a shift towards more realistic simulations of
irregular particle shapes becomes evident. In the areas of CFD-DEM
coupling, there have been recent developments on representing irreg-
ular particle shapes with signed distance eld (Lai et al., 2023) and
clumps (Shen et al., 2022). However, there is a research gap specically
focusing on 3D LBM-DEM simulations for irregularly shaped particles
foundation in geomaterials other than idealized regular non-spherical
shapes.
In this paper, an LBM-DEM coupling framework was developed to
simulate the interaction between uid and solid granular particles with
realistic irregular shapes. For this purpose, irregularly shaped particles
were represented by clumps using sub-particles bonded together. The
interaction between solid clumps with uid were modelled by coupling
3D LBM with DEM using two open-source codes named Palabos and
LIGGGHTS. Three benchmark cases were considered to validate our
model using clumps. Finally, to study the impact of realistic particle
morphology in granular ow simulations, the collapse and transport of a
granular column of irregularly shaped particles immersed in a uid was
simulated. Comparison of the results with that from regular shape par-
ticles (spheres) demonstrates the importance of particle shape in gran-
ular ow simulations. This study represents a step towards more realistic
modelling of granular ows with complex particle shapes using LEM-
DEM coupling.
2. LBM-DEM formulation
2.1. Lattice-Boltzmann method
The lattice Boltzmann method (LBM) is a mesoscopic scale uid ow
simulation method based on the kinetic theory (Chen and Doolen,
1998). In this method, the computational domain is divided into struc-
tured Cartesian lattice nodes and then a probability distribution function
(PDF) is dened on each node of the lattice. In other words, fi(x,t)is the
PDF of the point x at time t at direction i.
In contrast to conventional CFD methods that solve nonlinear Navier-
Stokes partial differential equations, the LBM takes a different approach.
In LBM, the Boltzmann equation describes how PDFs evolve throughout
time. Based on the BGK approximation, named after its contributors
(Bhatnagar-Gross-Krook (Bhatnagar et al., 1954), the well-known
Boltzmann equation turns into Eq. (1). As a result of this approxima-
tion, the nonlinear Boltzmann equation becomes an explicit linear
equation efcient for parallel computing.
fi(x+ciδt,t+δt)− fi(x,t) = − 1
τ
[fi(x,t) − feq
i(x,t)] (1)
The governing equation (Eq. (1)) consists of two main processes:
streaming and collision. On the left-hand side (LHS) of the equation, the
streaming process occurs. It involves the propagation of the PDFs from
one lattice node (x) (Fig. 1) to its neighboring node (x+ciδt) in the
direction i at the time t+δt, where ci represents the lattice velocity and
δt is the increment in time t.
On the other hand, in Eq. (1), the right-hand side describes the
occurrence of the collision process. In this process, the PDFs undergo a
linear relaxation to the equilibrium distribution functions (EDFs), feq
i(x,
t), using a single dimensionless relaxation time (
τ
). The EDF employed in
this context is the Maxwellian distribution, which uses the Taylor series
and can be expanded regarding the macroscopic uid velocity (uf)
(Aidun and Clausen, 2010).
Fig. 1. D3Q19 lattice structure.
Table 1
Values of ci, wi for a D3Q19 lattice structure.
PDFs, fi Lattice velocity, ci Weights, wi
f0 (0,0,0) 1/3
f1−f6 ( ± 1,0,0),(0,±1,0),(0,0,±1)1/18
f7−f18 ( ± 1,±1,0),( ± 1,0,±1),(0,±1,±1)1/36
M.H. Ahmadian and W. Zheng
Computers and Geotechnics 171 (2024) 106395
3
feq
i=wi
ρ
f[1+ci.uf
c2
s+(ci.uf)2
2c4
s−uf2
2c2
s](i=0,⋯,18)(2)
where wi is the weight factor related to the corresponding ci direction.
The speed of sound cs in lattice is 1/
3
√. Note that a D3Q19 lattice
structure, as shown in Fig. 1, is adopted in this study. The associated
lattice velocity and weight factors are shown in Table 1.
After calculating fi at each node, the uid density
ρ
f and velocity uf
can be reconstructed using the following equations according to the
conservation laws of mass and momentum (He and Luo, 1997):
ρ
f=∑
18
i=0
fi(3)
ρ
fuf=∑
18
i=0
cifi(4)
Moreover, the kinematic uid viscosity, vf, is a material property con-
stant and has the following relationship with the relaxation time, LBM
timestep, and lattice spacing (He and Luo, 1997):
vf=c2
s(
τ
−1
2)δ2
x
δt
(5)
Pressure as other macroscopic variable is related to uid density based
on the equation of state (He and Luo, 1997):
p=c2
s
ρ
f(6)
One of the primary sources of compressibility error in LBM arises from
the truncated terms of Taylor expansion of the equilibrium distribution
functions. In order to approximate an incompressible ow, the Mach
number (Ma) must remain below unity (Ma≪1). The necessity for
incompressibility in LBM simulations imposes constraints on selecting
the LBM relaxation time. In other words, while based on Eq. (5), the
relaxation time has a bottom threshold of 0.5, the higher relaxation time
should be set in a way to ensure that the LBM accurately captures the
behaviour of incompressible ows while minimizing compressibility-
related errors (Yang et al., 2019).
2.2. Discrete element method
In the discrete element method (DEM), granular materials are rep-
resented as a collection of individual solid particles, where interactions
between neighbouring particles are explicitly computed using simplied
contact laws (Sun et al., 2013). In this study, an open-source DEM code,
LIGGGHTS, was employed to simulate the interactions among solid
particles. In each iteration of the DEM process, Newton-Euler equations
determine the amount of particle translation and rotation. Each iteration
starts with particle–particle contacts assessment, and accordingly, the
corresponding forces are computed. Subsequently, particles’ trans-
lational and rotational velocities are updated. Afterward, new particle
positions will be determined based on Newton-Euler equations (Ding
and Xu, 2018).
Fig. 2 illustrates particle “a” (green) and particle “b” (red) are in
contact with each other, which δn denotes the overlap. The overlap can
be calculated using the appropriate methods specied in the literature
(Yang et al., 2019) as:
δn= (ra+rb−rab)n(7)
where ra and rb are radii of particles a and b, respectively. The distance
between two particles is represented by rab and n is the normal unit
vector directed toward particle centers.
To calculate the contact force between particles, constitutive contact
models can be used, and this study uses the spring-dashpot model. In this
case, the following equation gives the normal force (Cundall and Strack,
1979):
Fn=knδn+cnΔun(8)
where kn is stiffness and cn is damping coefcient, both in the normal
direction. Δun is the relative normal velocity of two interacting particles.
In the same manner, the tangential force is given as follows (Cundall and
Strack, 1979):
Ft= − kt∫tc
tc,0
Δutdt −ctΔut(9)
where kt and ct are corresponding stiffness and damping coefcient in
the tangential direction, and Δut denotes the relative tangential velocity
of two particles. The integral term in the equation represents the
increment of spring, which accumulates energy resulting from the
relative motion in the tangential direction between contacting particles.
This integral accounts for the deformation of the particle surface
(elastic) throughout the time the particles are in contact from tc,0 to tc.
The tangential force exerted between the particles acts in the opposite
direction of the displacement in the tangential direction. Additionally,
the tangential force magnitude is restricted by the Coulomb friction,
denoted as
μ
Fn. The value of
μ
corresponds to the smaller of the two
friction coefcients of the contacting particles, and it represents the
threshold at which the particles initiate sliding motion against each
other.
There are different proposed contact models that calculate contact
force (Fc=Fn+Ft) as a function of relative velocity and distance. In
our study, we adopted the Hertz contact model (Jing et al., 2019). Note
Fig. 2. Interaction of two particles in DEM a) when two particles are in contact, b) the spring-dashpot model.
M.H. Ahmadian and W. Zheng
Computers and Geotechnics 171 (2024) 106395
4
that the parameters, kn cn, kt, ct in Eqs. (8) and (9) are calculated from
Young’s modulus, coefcient of restitution, coefcient of friction, and
Poisson’s ratio using the equations in the reference (Di Renzo and Di
Maio, 2004). To update the velocities of a particle (both linear and
angular), Newton-Euler equations take into account the forces (contact
force Fc, gravity G, uid drag force Ff) and torques (contact torque Tc,
uid drag torque Tf) exerting on the particle. The linear velocity is
updated by considering the net force acting on the particle and dividing
it by its mass, while the angular velocity is updated by considering the
net torque acting on the particle and dividing it by its moment of inertia.
By applying Newton-Euler equations, the particle’s linear and angular
velocities can be appropriately adjusted to account for the combined
effects of these forces and torques.
ma=Fc+G+Ff(10)
I˙
ω
=Tc+Tf(11)
where m is mass and I is the moment of inertia of particles. a and ˙
ω
are
translational and rotational acceleration of particles, respectively.
Eventually, by using Eqs. (10) and (11) via the Verlet method (Verlet,
1967), the position and rotation of particles can be updated at each
timestep.
2.3. Immersed moving boundary method
The Immersed Moving Boundary (IMB) method, also known as a
partially saturated bounce-back method, operates based on the principle
of introducing a new collision operator Ω that depends on the solid ratio
ε
. While the ideal value of this ratio can be obtained through a geometric
analysis, it often demands signicant computational resources (Yang
et al., 2019). The IMB method contains several steps in identifying
partially saturated cells and then calculating their solid fraction (Wang
et al., 2017); in this study, this process is modied in the following order.
In this study, rst, each sphere is encased within its own bounding
box (Fig. 3). The core of this method involves calculating the solid
fraction for each cell of the bounding box in relation to the sphere’s
center. Specically, cells within a distance less than the radius from the
center are classied as solid (solid fraction =1). Conversely, cells
beyond the radius have a solid fraction of 0. In the case that the cell is
covered by more that two particles (Fig. 3), the solid fraction of the cell
will be calculated based on the particle that has higher solid fraction in
the cell. For the partially saturated cells, the method divides them into
smaller, equal-sized sub-cells as shown in Fig. 4 (denoted as nsub). Then
the cell decomposition method is employed (Owen et al., 2011), as
depicted in Fig. 4. An algorithm is applied to the centers of these sub-
cells to estimate the solid ratio,
ε
, by dividing the number of sub-cells
located inside the solid boundary over the total sub-cell number n3
sub
with one cell (Yang et al., 2019). The solid fraction in these cells is then
calculated by comparing the number of sub-cells within the solid
boundary to the total number of sub-cells, nsub .
An important aspect of this process is the assignment of particle IDs
to LBM cells. Cells that are fully or partially solid receive the ID of the
occupying particle, while purely uid cells are assigned an ID of zero.
This approach offers two signicant benets. Firstly, for cells in the
overlapping region of two spheres, the cell is assigned only one particle
ID, preventing the duplication of force and momentum calculations.
Secondly, as each cell carries a particle ID, calculating the total hydro-
dynamic force on each particle becomes more efcient – the code simply
sums the forces across cells sharing the same particle ID.
In uid cells where
ε
is equal to 0, the aforementioned uid hydro-
dynamic collision occurs, which is represented by the BGK collision
operator Ωf, as provided in the Eq. (1). In solid cells where the solid ratio
Fig. 3. Process of calculating solid fraction of cells covered by a clump of particles (the red box is the bounding box of each sub-particle). (For interpretation of the
references to colour in this gure legend, the reader is referred to the web version of this article.)
Fig. 4. Irregularly shaped particle modeled by clump (left gure) is mapped on an LBM grid (right gure), with a zoomed-in cell partially saturated by solids and
being divided into 5 sub-cells in each direction.
M.H. Ahmadian and W. Zheng
Computers and Geotechnics 171 (2024) 106395
5
is equal to 1, a collision operator given by Noble and Torczynski (Noble
and Torczynski, 1998), according to the concept of non-equilibrium
bounce-back, is employed. This collision operator, denoted as Ωs, is
given by the following expression:
Ωs
i=f−i(x,t)− feq
−i(
ρ
f,uf)+feq
i(
ρ
f,us)−fi(x,t)(12)
where us is the macroscopic velocity of the solid at the location of the
lattice node position. The subscript, −i, means the opposite direction of
i. The purpose of the solid collision operator (Ωs) is to enforce a no-slip
boundary condition between two phases of uid and solid. This is ach-
ieved by setting the PDF fi(x+ciδt,t+δt)equal to the equilibrium dis-
tribution function (EDF), feq
i(
ρ
f,us), plus the bounce-back of the non-
equilibrium part in the opposite direction (Yang et al., 2019). The
non-equilibrium bounce-back term ensures that the uid particles
reect off the solid surface with a velocity that corresponds to the
macroscopic velocity of the solid phase at that lattice node.
In case the cell is partially saturated, namely
ε
is between 0 and 1, a
weighting factor incorporates both collision operators into account.
Ω=BΩs+(1−B)Ωf(13)
where the weighting factor can be calculated based on the relaxation
time and the solid ratio as follows (Noble and Torczynski, 1998):
B(
ε
,
τ
) =
ε
(
τ
−1/2)
(1−
ε
) + (
τ
−1/2)(14)
Afterward, the following equations provide the hydrodynamic force and
torque at each node according to the lattice structure that was adopted
earlier (Strack and Cook, 2007).
Ff=∑
n
j=1
Bj∑
18
i=0
Ωs
ici(15)
Tf=∑
n
j=1[Bj(xj−xs) × ∑
18
i=0
Ωs
ici](16)
where xs is the center of mass of solid particles and xj is the jth lattice cell
coordinates. When the hydrodynamic force and torque are calculated
from Eqs. (15) and (16), and then are incorporated again in Eqs. (10) and
(11) to update solid particles’ velocity and location. By utilizing these
equations, the particle’s motion and position are adjusted based on the
calculated hydrodynamic forces and torques, ensuring the accurate
representation of the particle’s dynamics within the simulation.
2.4. Coupling scheme
The coupled LBM-DEM method used in this study utilizes Palabos
(Latt, 2021), an open-source C++ library for LBM, as the master pro-
gram. This master program calls an external library, LIGGGHTS (Kloss
et al., 2012), for DEM simulations (Seil and Pirker, 2017). The owchart
in Fig. 5 illustrates the coupling scheme employed in this study. The
computational cycle begins by generating DEM particles and then
initializing the uid eld. Subsequently, interactions of particles are
calculated using Eqs. (8) and (9).
In order to ensure stability in the DEM simulation, it is crucial for the
DEM time step (t) to be smaller than a critical time step (tcr) that can be
obtained by the stiffness and mass of the particles. Therefore, the
calculated critical DEM time step (tcr) is generally smaller than the time
step (t) used in LBM simulations, particularly for geotechnical engi-
Fig. 5. Flowchart of LBM-DEM coupling method (Yang et al., 2019).
M.H. Ahmadian and W. Zheng
Computers and Geotechnics 171 (2024) 106395
6
neering problems where soils and rocks exhibit high stiffness. To syn-
chronize the DEM simulations with LBM, Nsub DEM sub-cycles are per-
formed for each LBM evolution step, resulting in the following
relationship:
Δt=δt
Nsub
(17)
which means that for every LBM timestep, there are Nsub DEM sub-cycles
during which hydrodynamic force and torque are constant.
In an LBM-DEM simulation, once DEM sub-cycles (Nsub) are
completed, the updated positions of the particles are mapped onto the
lattice grid. This mapping process identies the lattice cells that are
covered by solid particles. For cells that are partially saturated, the solid
ratio is determined using the cell decomposition method, as depicted in
Fig. 4. Based on the type of the cell (uid, partially saturated, or solid),
the collision process happens. It is worth noting that the modied
collision operator is also present in Eqs. (15) and (16), enabling the
calculation of hydrodynamic forces and torques immediately after the
collision process, before the streaming step, thus ensuring high
computational efciency. Subsequently, the hydrodynamic forces are
given back to the DEM to determine the motion of the particles. The
resulting PDFs then stream to the neighbouring lattice nodes. Using the
redistributed PDFs, the uid density (
ρ
f) and velocity (uf) are updated
according to Eqs. (3) and (4). If necessary, the uid pressure eld can be
obtained using Eq. ((6). After all the steps are taken, one cycle of the
LBM-DEM simulation concludes, and the simulation continues until it
reaches the number of cycles that is preset.
2.5. Irregular shape simulation
The practice of modelling real particles as clumps of overlapping
spheres or clusters of non-overlapping spheres has been widely adopted
in many studies. There are a number of methods for clumps proposed in
the literature, including Wang et al. (Wang et al., 2007), Ferellec and
McDowell (Ferellec and McDowell, 2010), Favier et al. (Favier et al.,
1999), and Taghavi (Taghavi, 2011), to name a few. The used open-
source code, LIGGGHTS, does not have functions for generating
clumps. Therefore, this study adopted CLUMP (Angelidakis et al., 2021),
a MATLAB code that has modules to generate and export clumps based
on Euclidean distance transform.
In this clump-generating code, the particle morphology undergoes a
transformation from a surface mesh into a voxelated representation. The
user can set a minimum dimension to control the number of voxels
representing the particle. Within the voxelated image, all voxels corre-
sponding to the particle are assigned the value zero. The Euclidean
distance transform calculates the minimum distance between each zero
voxel to its nearest non-zero voxel. Afterward, the voxel that has the
Table 2
Parameters used in CLUMP algorithm from generating clumps.
Minimum
dimension
Minimum
radius
Minimum No. of
spheres
Maximum
overlap
400 0 100 95 %
Fig. 6. Illustration of original STL le (left column), corresponding clump (middle column) and the subsequent shape (right column) of particles after transformation
into clump.
Fig. 7. Monte Carlo algorithm for calculating center of mass and moment of
inertia in clumps.
M.H. Ahmadian and W. Zheng
Computers and Geotechnics 171 (2024) 106395
7
maximum value of Euclidean distance transform is set as the rst
inscribed sphere and the corresponding voxels are set as one. After that,
another Euclidean distance transform calculates the new voxelated
image. This process repeats until one of the two conditions is met. The
rst condition happens when the number of spheres passes the
maximum user-dened number of spheres. The second condition occurs
when the algorithm wants to generate a sphere that is smaller than the
user-dened minimum radius. Moreover, this approach allows for
generating overlapping spheres by selectively setting a certain per-
centage of voxels to one, rather than all, forming each new sphere. The
extent of overlap is controlled using a variable overlap, which can take
values within the range of [0, 1), providing exibility in managing the
degree of overlap among the spheres produced. For this study, all shapes
are represented by 100 sub-particles, and the rest of the setting is
brought in Table 2. Fig. 6 also provides the illustration of two particle
geometries after transformation into a clump, which will be used in
Section 3.
As the generated clumps consist of overlapping spheres, it is
important to correctly calculate the center of mass and moment of
inertia for every clump before the simulation starts. This calculation
occurs in DEM, and it involve the initial generation of an oriented
bounding box for a given multi-sphere clump (Fig. 7), followed by the
implementation of a Monte Carlo simulation process (Zhou et al., 2017).
Within this process, thousands of points (denoted as N) were randomly
distributed within the bounding box. Subsequently, points located
outside the clump were eliminated (red dots), while those within the
clump were retained (black dots). The number of points remaining in-
side the clump was denoted as n, and the ratio n/N was considered as an
approximate volume ratio representing the proportion of the clump
within the bounding box.
The next step involved treating the retained points as mass-based
voxels. Various kinematic parameters, including mass center and
inertia tensor, were then obtained by integrating these voxels in accor-
dance with their mathematical denitions (Zhou et al., 2017). Notably,
this approach offers a distinct advantage by excluding the inuence of
overlapping density. Consequently, it facilitates the extraction of true
kinematic parameters for the clump, providing a more accurate repre-
sentation of its mass distribution and inertial properties. This method-
ology is particularly advantageous in situations where obtaining precise
kinematic parameters without the interference of overlapping density is
Fig. 8. Sedimentation velocity of an irregularly shaped particle with different numbers of sub-particles in the clump. The shape of the actual particle is shown at
the side.
Fig. 9. Schematic of benchmark case domains a) single clump b) two clumps c) disc shape clump. All boundaries are considered as no-slip walls.
M.H. Ahmadian and W. Zheng
Computers and Geotechnics 171 (2024) 106395
8
Fig. 10. Comparison of current simulation and experimental results (ten Cate et al., 2002) of a single clump settling in ambient uid.
Fig. 11. Comparing the simulation results with the experiment (ten Cate et al., 2002) for settling trajectory of particles at various Reynolds numbers.
Fig. 12. Vertical velocity contour of a clump at different times (Re =11.6).
M.H. Ahmadian and W. Zheng
Computers and Geotechnics 171 (2024) 106395
9
crucial for the analysis of clump behavior.
To investigate the impact of the sub-particle number used in gener-
ating a clump, Fig. 8 illustrates the sedimentation velocity of the same
irregularly shaped particle using different numbers of sub-particles.
When only 20 spheres are used to generate the geometry of the irregu-
larly shaped particle, the terminal velocity is off by 0.005 m/s compared
with other results. However, as the number of particles increases, the
terminal velocity converges to a certain value. Therefore, in this study,
N =100 is used as the number of sub-particles needed to generate a
clump.
3. Validation cases
The code uses Palabos version 1.5 for LBM simulations and
LIGGGHTS version 3.8 as an external library for DEM simulations. Three
benchmark cases were simulated to demonstrate the accuracy of LBM-
DEM models using clumps. The three benchmark cases are: 1) settling
of a single clump, 2) settling of two clumps, and 3) settling of a disc-
shaped clump, all in an ambient uid. It is worth noting that all 3D
LBM simulations in this study used the D3Q19 lattice structure.
3.1. Settling of a single clump
For the rst benchmark case, the settling of a heavy single particle in
an ambient uid is simulated. As mentioned before, the single spherical
particle shown in Fig. 6 is regenerated using 100 sub-particles for a
clump. The simulation domain is shown in Fig. 9a. The diameter of the
clump is dp=15 mm, and its density is 1120 kg/m3.
For this simulation, the lattice resolution and relaxation times are 20
and 0.56, respectively. By considering the diameter of the clump particle
as the length scale and the particle’s terminal velocity as the charac-
teristic velocity, the simulation was performed for four Reynolds
numbers (Re), namely 1.48, 4.1, 11.6, and 31.9. The results were
compared to experimental data, as shown in Fig. 10.
In Fig. 10, it can be seen that for low Reynolds numbers, the velocity
of the particle rst increases until it reaches its terminal velocity, where
the buoyancy force is equal to gravity. Subsequently, as the particle gets
closer to the bottom, the velocity gradually decreases. For higher Rey-
nolds numbers, the same process is observed but the particle settles
faster. Overall, Fig. 10 demonstrates that the sedimentation velocity of a
spherical particle simulated by a clump is in good agreement with the
experimental results at each of the four Reynolds numbers (ten Cate
et al., 2002).
Fig. 11 presents another comparison between the coupling code re-
sults and the experiment. With the lower Reynolds number, it takes
much more time for the particle to reach the bottom; and as the Reynolds
number increases, the particle falls faster. Fig. 11 once again veries that
the code can accurately simulate both the velocity and the trajectory of
the particle with little discrepancies.
Fig. 12 provides an appreciation of the vertical velocity contour of
the x-plane that is located in the middle of the tank. The gure depicts
that as the particle gains more speed, the wake that occurs behind the
particle also gets bigger. The wake region has lower pressure than the
surroundings. Therefore, as the wake region grows, the pressure dif-
ference between the front and back of the particle increases, leading to a
higher drag force. In this regard, the particle loses its velocity as it ap-
proaches the bottom of the tank.
To further study the inuence of lattice resolution, Fig. 13 demon-
strates the effect of four different resolutions, with N =10, 15, 20, and
25, respectively. It is worth noting that the number N in each case
represents the number of cells in one diameter of sphere, which is 15 mm
in this case. In order to depict the difference between each grid size,
Fig. 13 only shows the last two seconds of the sedimentation, as the
discrepancy in this interval is more apparent. There is an apparent
Fig. 13. Study the effect of gird resolution on the simulation of sedimentation of a spherical particle in a tank of uid at Re =1.48.
Fig. 14. Settling velocity of two clumped particles in comparison with Dash
and Lee (Dash and Lee, 2015).
M.H. Ahmadian and W. Zheng
Computers and Geotechnics 171 (2024) 106395
10
discrepancy between N =10 and the rest, and N>=15 seems to give
satisfactory results. N =20 was used for the benchmark cases presented
in Section 3.
3.2. Settling two clumps
The interaction between two clumps falling from a certain height is
investigated, as shown in Fig. 9b. All boundaries are no slip wall. In this
case, the density of particles is 1350 kg/m3 with dp=12.7mm. The
lattice resolution and relaxation times are 20 and 0.51, respectively. In
addition, uid density and viscosity are 1195 kg/m3 and 0.0305 Pa.s,
respectively. Fig. 14 shows the simulation results in comparison with
those by Dash and Lee (Dash and Lee, 2015). Note that there are some
differences among the simulation results by different codes found in the
published literature. The following equations are used to non-
dimensionalize the time and velocity in Fig. 14.
tc=
dp/
ρ
b/
ρ
f−1g
√(18)
Uc=
ρ
b/
ρ
f−1gdp
√(19)
As shown in Fig. 14, when two particles start to fall (Fig. 15a), they
initially have same speed, but after some time the trailing particle gets
more speed since it is located at the wake of the leading particle also
called drafting (Fig. 15b). Then the particles start to contact each other
(kissing, Fig. 15c) until they begin to tumble and separate at a later stage
(Fig. 15d). While there is a slight discrepancy between the two results,
Fig. 14 shows that our simulation of clump particles can follow all three
stages of drafting, kissing, and tumbling, which are associated with this
benchmark. Note that in this simulation, in order to capture are afore-
mentioned three stages, a small offset is introduced to the initial position
of two clumps, in which the position of leading clump is (5.5dp+Δx,
5.5dp+Δy,35dp) and the position of trailing clump is (5.5dp,5.5dp,
37dp).
3.3. Settling of a disc-shaped clump
The third simulated benchmark case is the settling of a disc-shaped
clump. The shape of the disc shown in Fig. 6 is generated using a
clump of 100 sub-particles. The lattice resolution and relaxation times
are 15 and 0.55, respectively. Furthermore, the initial orientation of the
disc is set to 45◦as shown in Fig. 9c. According to the literature, the
settling process of a disc-shaped particle depends on the dimension of
the disc. Therefore, we used this point to bring another validation to our
code.
There are two non-dimensional numbers that characterize the
settling process of a disc-shaped particle, the Reynolds number and the
Fig. 15. Location of two clumps at different times.
Table 3
Non-dimensional number values for two cases of disc-shaped particle
benchmark.
I* Uz,mean (m/s)Rep
Case 1 0.0191 0.021 21
Case 2 0.0398 0.1087 267
Table 4
Characteristics of two cases related to disc-shaped particle benchmark.
ρ
p(kg/m3) d (mm) Box size (mm)
Case 1 1300 1 mm 6 ×6 ×15
Case 2 2700 2 mm 12 ×12 ×48
Fig. 16. Settling velocity of disc-shaped particle in two different cases.
Fig. 17. Z-velocity contour of settling disc-shaped clump at different times
(Case 2). (a) t =0.14 s (b) t =0.28 s (c) t =0.38 s.
M.H. Ahmadian and W. Zheng
Computers and Geotechnics 171 (2024) 106395
11
non-dimensional moment of inertia, with the following denition
(Willmarth et al., 1964).
I*=
πρ
pt
64
ρ
fd(20)
where t, d, and
ρ
p
are the thickness, diameter, and density of the disc,
respectively. The study presents that the falling of a disc in a uid de-
pends on the value of the two numbers (I* and Re) and could have either
stable or unstable falling patterns. Therefore, we specically set the
properties of the uid and the disc to simulate two scenarios, including
stable falling (case 1) and periodic oscillation (case 2). Table 3 provides
the value of the non-dimensional numbers for each case. Table 4 pro-
vides the characteristics of the domain for two cases. It should be
mentioned that in both cases, the thickness of the disc is 0.3 times the
diameter.
Fig. 16 shows the settling velocity versus time for the two cases,
which correctly simulated the falling pattern of the disc-shaped particle
consistent with those predicted by (Willmarth et al., 1964). Moreover,
Fig. 17 provides further illustration of a disc falling with the oscillating
pattern. As the gure shows, based on the non-dimensional number of
the particle in Case 1, the particle correctly follows a stable and non-
oscillating settling pattern. However, in Case 2, the particle has more
moment of inertia compared to Case 1, plus a higher Reynolds number,
both of which set the particle unstable and with a certain oscillating
frequency.
4. Immersed granular collapse
As a demonstration of the coupled methodology in geotechnical
engineering applications, the simulation of a column of irregular shape
particles collapsing in an ambient uid is presented. The dimension of
the domain is given in Fig. 18. No slip wall is applied to all domain
boundaries. The modelling properties of uid and solid particles are
given in Table 5. The irregularly shaped particle has the morphology
shown in Fig. 18 and has a sphericity of 0.96, an elongation 0.71, and a
atness of 0.94 based on the denitions in (Zheng et al., 2021). After
performing a grid sensitivity test with N =10, 15, and 20 for the column
collapse models with spheres, N =15 (number of cells within one clump
diameter) is chosen as the lattice resolution. The relaxation times is set
to 0.508, respectively. To generate the column of 1600 particles
(clumps), rst, particles stacked at one side of the domain under gravity
using only DEM (Li region). Then by removing the wall that holds the
particles to one side, the particles start to collapse to the other side of the
domain. The primary aim of this simulation is to demonstrate the
capability of LBM-DEM in simulating a system of irregularly shaped
particles. It is worth mentioning that the current model uses BGK
approximation for this demonstration case; the adoption of a two-
relaxation-time collision operator in LBM is recommended to enhance
the accuracy of ow modeling in porous media, which is also available
on Palabos.
The simulation results are shown in Fig. 19. First, the particles near
the front top of the column started to slump down, and larger uid ve-
locities at t =0.08 s were near the front face (Fig. 19a). As time passed,
some of the slumping particles reached the oor and created waves with
high velocities near the toe (t =0.16 s). Some particles near the top front
toppled over and induced relatively large velocities in their vicinity.
When t =0.39 s, the rear and middle portion of the column approxi-
mated their nal slope grade, while the front toe had particles with great
momentum and kept progressing forward. The uid near the toes had
some of the largest velocities around t =0.39 s and gradually decreased
afterwards as particles reached the farthest distance Lf=3.25 cm when t
is around 0.52 s. Note that the front portion of the nal deposition is
relatively at compared with the rear and middle portions of the gran-
ular slope. For comparison, the simulation is performed with spherical
particles. A similar collapsing process to those with the irregular shape
particles was observed. However, spherical particles slumped and
travelled relatively faster and induced bigger areas of larger uid ve-
locities near the toe. In addition, there is greater momentum in spherical
Fig. 18. Irregular particle (left) and domain dimensions for immersed column collapse (right). All boundaries are no slip walls.
Table 5
Characteristics of uid and solid.
Parameter Value
Particle Equivalent diameter of irregular shape particle 0.8 mm
Regular shape particle diameter (sphere) 0.8 mm
Density,
ρ
p 2468 kg/m3
Young’s modulus, E 10
9
Pa
Coefcient of restitution, e 0.65
Coefcient of friction,
μ
0.4
Poisson’s ratio 0.24
Fluid Density,
ρ
f 1000 kg/m3
Viscosity,
μ
f 0.001 Pa.s
M.H. Ahmadian and W. Zheng
Computers and Geotechnics 171 (2024) 106395
12
Fig. 19. Collapse of column of irregularly shaped particles at different times a) t =0.08 s, b) t =0.16 s, c) t =0.39 s, d) t =0.52 s and spherical particles at times e) t
=0.08 s, f) t =0.16 s, g) t =0.25 s, h) t =0.33 s.
M.H. Ahmadian and W. Zheng
Computers and Geotechnics 171 (2024) 106395
13
particles, and they stop when t is 0.33 s, while the case with irregular
clumps is still progressing. As shown in Fig. 20, the nal deposition
distance of spherical particles is Lf=3.45 cm, slightly larger than the
case with irregular clumps. Note that the cases presented here only serve
as a qualitative demonstration of the shape inuence on granular ma-
terials behaviors in the uid–solid interaction. More quantitative studies
with different particle shapes are warranted.
5. Conclusion
In this paper, an LBM-DEM coupling framework was established to
simulate the uid–solid interaction considering irregular solid particle
shapes. Irregular particles were represented as clumps and were
modelled in two open-source codes, LIGGGHTS and Palabos, for coupled
LBM-DEM simulations. Three benchmarks, including the settling of a
single clump, the settling of two clumps, and the settling of a disc-shaped
clump, were successfully simulated, and the results are consistent with
those in the literature. Notably, the simulated settling of two clumps was
satisfactory in duplicating the interaction of two clumps in the uid
domain, including drafting, kissing, and tumbling. In addition, the
simulation of a disc falling in the uid domain can adequately produce
stable falling and periodic oscillations that were consistent with the
theoretical predictions. The number of sub-particles equal to 100 for
representing an irregular particle as a clump in the established frame-
work seems to be sufcient in capturing the grain morphology and their
inuence on the uid–solid interaction.
The established methodology was employed to simulate the collapse
of a column of irregularly shaped particles. The simulation results can
reasonably reproduce the slumping and trending of granular particles
and the changes in ow velocity in the uid domain. The simulation
results were compared with the column collapse of spherical particles. It
was found that spherical particles possessed larger momentum, travelled
faster, and reached a longer run-out distance, which warrants the
consideration of realistic particle shape in granular ow when inter-
acting with uids.
To the best of the authors’ knowledge, this study represents the rst
time to couple 3D clumped DEM with LEM for simulating the interaction
of the uid with a system of non-regular irregularly shaped solid par-
ticles, which are widely found in geomaterials. The established meth-
odology is ready to be used for many other geotechnical engineering
problems, such as internal erosion in granular packs where particle
shape has an important inuence on pore structure and the related ne
migration.
CRediT authorship contribution statement
Mohammad Hassan Ahmadian: Writing – original draft, Valida-
tion, Methodology, Investigation. Wenbo Zheng: Writing – review &
editing, Validation, Investigation, Funding acquisition,
Conceptualization.
Declaration of competing interest
The authors declare the following nancial interests/personal re-
lationships which may be considered as potential competing interests:
Wenbo Zheng reports nancial support was provided by Natural Sci-
ences and Engineering Research Council of Canada.
Data availability
Data will be made available on request.
Acknowledgments
This work was supported by the University of Northern British
Columbia, and the Natural Sciences and Engineering Research Council
of Canada (RGPIN-2021-04215, ALLRP 581102-22).
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