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Abstract

Using some intrinsic features of the Smoothed Particle Hydrodynamics (SPH) schemes, an innovative algorithm for the initialization of the particle distribution has been defined. The proposed particle packing algorithm allows a drastic reduction of the numerical noise due to particle resettlement during the early stages of the flow evolution. Moreover, thanks to its structure, it can be easily derived starting from whatever SPH scheme and applies under the hypotheses that the fluid is weakly-compressible or incompressible as well. A broad range of numerical test cases proved this tool to be fast, robust and reliable also for complex geometrical configurations.
Particle packing algorithm for SPH schemes
A. Colagrossia,b,, B. Bouscassea, M. Antuonoa, S. Marronea,c
aCNR-INSEAN,The Italian Ship Model Basin, Rome, Italy
bCentre of Excellence for Ship and Ocean Structures, NTNU, Trondheim, Norway
cDepartment of Mechanics and Aeronautics, University of Rome “La Sapienza”
Abstract
Using some intrinsic features of the Smoothed Particle Hydrodynamics schemes
(SPH), an innovative algorithm for the initialization of the particle distribution has
been defined. The proposed particle packing algorithm allows a drastic reduction
of the numerical noise due to particle resettlement during the early stages of the
flow evolution. Moreover, thanks to its structure, it can be easily derived starting
from whatever SPH scheme and applies under the hypotheses that the fluid is
weakly-compressible or incompressible as well. A broad range of numerical test
cases proved this tool to be fast, robust and reliable also for complex geometrical
configurations.
Key words: Meshless methods, Smoothed Particle Hydrodynamics, Particle
initialization, Lagrangian Systems.
Introduction
In the Smoothed Particle Hydrodynamics scheme (SPH) the matter of how
initialize the particle positions plays a relevant role. If particles are not initially
set in “equilibrium” positions, they may resettle giving rise to spurious motions
which can strongly aect the fluid evolution.
Here, the acceptation of the word “equilibrium” deserves a clarification.
We refer to an equilibrium configuration as the set of particle positions which,
under static conditions, does not lead to particle resettlement. As proved in the
following, the spurious particle motion is caused by inaccuracies in the SPH
representation of the pressure gradient. Specifically, these inaccuracies largely
increase when the particle distribution is anisotropic and disordered. At worst, the
Corresponding author: andrea.colagrossi@cnr.it
Preprint submitted to Elsevier June 11, 2015
pressure gradient is unable to approximate the correct static conditions and non-
physical currents/vorticity are generated. Them, the aim of the present work is
to provide an algorithm which automatically gives the equilibrium configuration,
that is, the specific particle distribution for which the pressure gradient is accurate
and no particle resettlement occurs.
A part from few cases characterized by simple geometries, the equilibrium
configuration is not known “a priori”. Further, the generation of spurious
currents/vorticity may be particularly strong in presence of complex solid
boundary profiles (i.e. corners, bended bodies, etc.).
A possible solution is to start numerical simulations with a high numerical
damping term and leave a long enough time to make particle self-resettle in
equilibrium positions (see, for example, Monaghan [1]). Unfortunately, the
attainment of a stable configuration can require a very long evolution, this leading
to a large increase of computational costs. Moreover, the high damping used for
particle initialization does not exclude that a further resettlement occurs when the
actual simulation is started with a real viscosity term. This behavior has been
observed in some numerical simulations of Section 3.
In the SPH framework, the first attempt to define a proper algorithm for
particle initialization is due to Oger et al. [2] who adapted the Bubble method
described in Shimada [3] to SPH solvers. This algorithm is based on the use of
Van der Waals-like forces to place particles throughout the fluid domain. This
method proves to be quite fast, applies to general geometries and provides a
regular particle distribution. One of the weak points is that the particle positions
obtained through the Bubble algorithm may be not perfectly compatible with the
SPH static solution leading to a further resettlement.
Then, the key point to build a robust packing algorithm relies on the capability
of providing a regular particle distribution which is compatible with the SPH
scheme, that is, that satisfies the static conditions when the SPH scheme is used.
To this purpose a novel packing algorithm has been derived taking advantage of
some intrinsic features of the SPH schemes. Thanks to this, the proposed method
allows the attainment of a regular particle distribution compatible with the static
solution. Further, it can be easily derived starting from whatever SPH code and
applies to weakly-compressible or incompressible SPH schemes as well.
The paper is organized as follows: Section §1 introduces the SPH scheme and
gives an insight of the constitutive features which are used to build the packing
algorithm. Section §2 describes the proposed algorithm and highlights some
interesting aspects about its Lagrangian structure. Finally, Section 3 provides a
broad range of numerical test cases which prove the packing algorithm to be fast,
2
robust and reliable also for complex geometrical configurations.
1. Governing equations
Two dierent approaches can be adopted in the SPH framework to model
incompressible flows: the first relies on the direct use of the Navier-Stokes
equations while the second is based on the assumption that the fluid is barotropic
and weakly-compressible. Both approaches have benefits and drawbacks (for a
detailed description we address the reader to Monaghan [4] and Shao [5]) and lead
to the so-called incompressible SPH and weakly-compressible SPH schemes. The
main advantage of the Packing algorithm is that it applies to both schemes as well.
In the present case, we adopt a weakly-compressible SPH scheme and, therefore,
use the weakly-compressible Euler equations:
Dρ
Dt +ρ∇ · u=0,
Du
Dt =gp
ρ,
p=F(ρ),
Dr
Dt =u.
(1.1)
where rindicates a position inside the fluid domain ,gis the gravity acceleration
while u, ρ and pare respectively the velocity, the density and the pressure fields.
The fluid is assumed to be barotropic and Fis the state equation linking the density
field with the pressure one. The weakly-compressibility assumption is enforced by
choosing a sound velocity, c0=pdF/dρ, which is much larger than the maximum
expected velocity of the fluid (for more details we address the reader to the works
of Madsen & Sh¨
aer [6] and Monaghan [4]).
1.1. The SPH scheme
The SPH scheme is based on the filtering (smoothing) of any generic flow field
fwith a convolution integral over the fluid domain
hfi(r)=Z
f(r0)W(r0r;h)dV0(1.2)
W(r0r;h) is a weight function, which in practical applications must have a
compact support (r), and h(usually referred to as the smoothing length) is a
3
characteristic length of such support (see figure 1). From a physical perspective,
the smoothing length hrepresents the characteristic length of the domain of
influence of the fluid particle which is at the position indicated by r. A very
comprehensive review of the SPH framework can be found in reference [4].
The weight function W(r0r,h), called smoothing function or kernel in the
SPH terminology, is positive, centered in rand decreases monotonously with
the distance krr0k. This monotonous decrease gets to zero at the border of
its support (r). The kernel considered in the present work is supposed to be
isotropic in space, which is equivalent to being dependent only on the distance
s=kr0rk. The notation W(r0r;h) will be shortened hereinafter as W(r0r)
and the dependence on hwill be implicitly assumed. In the limit as the smoothing
length hgoes to zero, the kernel has to converge weakly to a Dirac “function”
and the original field of the convolution integral (1.2) has to be recovered. To this
purpose, the kernel Wmust integrate to one (see e.g. [8]), that is:
Z
W(r0r)dV0=1h>0.(1.3)
As extensively discussed in [8], such a property is not satisfied when the kernel
domain is not completely immersed inside the fluid domain. This, for example,
occurs in the neighborhood of the free surface F(see figure 1) where the kernel
domain is cut by the free surface. As a consequence, the lack of “mass” inside the
kernel domain implies that the integral in (1.3) is smaller than one.
The filtering formula (1.2) can be applied to the gradient of a generic function
h∇fi(r)=Z0f(r0)W(r0r)dV0(1.4)
(r)
r
∂ΩF
(r)
r
Figure 1: Configurations of the kernel support (r) with respect to the fluid domain
boundary.
4
with the prime on meaning that the derivatives are computed on the r0variable.
Equation 1.4 can be further analyzed if it is integrated by parts:
h∇fi(r)=Z
f(r0)W(r0r)dV0+Z
f(r0)W(r0r)n0dS 0(1.5)
In this expression, indicates in turn the derivatives with respect to the variable
rand n0is a unitary normal vector of pointing outwards . To obtain
this equation, the antisymmetry property of the kernel gradient (0W(rr0)=
−∇W(rr0)) has been used.
With this reformulation of equation (1.4), the gradient of any generic function
can be accessible from the knowledge of the function itself; this is the key point
of the SPH method. When the smoothing procedure is applied to the dierential
operators of the governing equations (1.1), shortening the notation hfi(r) by hfi,
we get:
Dρ
Dt +ρh∇ · ui=0,
Du
Dt =gh∇pi
ρ,
p=F(ρ),
Dr
Dt =u.
(1.6)
The consistency of system (1.6) for the modeling of Euler equations in the
presence of solid boundaries and free surfaces has been deeply investigated in
Colagrossi et al. [8]. In practical SPH methods, the following formulas are
generally used for the divergence of the velocity field and for the pressure gradient
(see for example [9] and [8]):
h∇ · ui=Zu0u· ∇ W(r0r)dV0(1.7)
h∇pi=Zp0+p· ∇ W(r0r)dV0(1.8)
When the SPH scheme is written at the discrete level, the fluid domain is
represented through Lagrangian fluid particles carrying the main fluid properties
(e.g. the velocity, pressure, density etc.). In this framework, the integrals in (1.7)
5
and (1.8) are replaced by summations over the fluid particles and the discrete SPH
scheme reads:
Dρi
Dt =ρiX
j
(ujui)· ∇iWi j Vj
Dui
Dt =g1
ρiX
j
(pj+pi)iWi j Vj+Ti,
pi=F(ρi),
Dri
Dt =ui,
(1.9)
Here, the subscripts indicate the quantities associated with the i-th and j-th fluid
particle. In the specific, Viis the particle volume, ρi=mi/Vi,miis the particle
mass and iindicates dierentiation with respect to the position ri. The term
Tiindicates an artificial viscous force per unit of mass. This term is generally
implemented in the SPH schemes for stability reasons (see, for example, [4]).
The main dierence with the continuous framework is that the relation (1.3)
is only approximately satisfied inside the fluid domain. This is due to local
unevenness in the particle distribution. For the analysis which follows, it is
convenient to introduce the following variables:
Γi=X
j
Wi j VjΓi=X
jiWi j Vj.(1.10)
Variables Γiand Γigive a “measure” of the unevenness in the particle
distribution. In fact, if the particle distribution is perfectly uniform, Γi=1 and
Γi=0 otherwise Γi<1 and Γi,0.
Further, the use of Γiand Γihelps understand the convergence of the
discrete dierential operators. In fact, two dierent kinds of errors are made
when the exact dierential formulas are substituted with the discrete smoothed
formulas. One kind is due to the interpolation procedure (errors proportional to the
smoothing length, h) while the other is caused by the approximation of continuous
integrals with finite summations (see, for example, [10]). In the latter case the
error decreases as the number of particles inside the kernel domain increases.
Then, if the mean number of particles in the kernel domain is large enough, this
error can be assumed to be smaller than O(h). Under this assumption and using
6
the results obtained in the Appendix A, the following expansions hold true:
h∇ · uii=X
j
(ujui)· ∇iWi j Vj= Γi(∇ · u)i+O(h).(1.11)
h∇pii=X
j
(pj+pi)iWi j Vj= Γi(p)i+2piΓi+O(h).(1.12)
Here, both Γiand Γiare responsible for a deviation from the exact dierential
operators. As shown in the following example, the greatest issue is caused by the
pressure gradient.
Let us assume we want to start a SPH simulation with hydrostatic conditions.
Then, we assign ui=0, ρi=ρ(ri) at t=t0and try to find the hydrostatic pressure.
The continuity equation is satisfied exactly while, as a consequence of (1.12), the
momentum equation gives:
Γi(p)i+2piΓiρiΦ = O(h),(1.13)
where Φis the gravitational potential. This expression clearly shows that, unless
Γi=1 and Γi=0, it is not possible to attain any hydrostatic solution. In this
context, Γiplays only a minor role since it just causes an increase/decrease of
the intensity of the correct pressure gradient. On the contrary, Γiis responsible
for an unbalance in both the intensity and the direction of the SPH dierential
operator. Further, Γidiverges like 1/hwhen the particle distribution is strongly
irregular. Then, the only way to get a good initialization of the SPH scheme is to
reduce the magnitude of Γias much as possible and recover the consistency of
the SPH pressure operator. This is the principal idea at the basis of the Particle
Packing Algorithm described in the next section.
2. Particle Packing Algorithm
The Particle Packing Algorithm is built on a simple idea: to use the SPH
features highlighted in the previous section to initialize the particle distribution
and minimize k∇Γk. This is made by observing that the vector w=−∇Γalways
points in the direction of the maximum lack of “mass” and maximum anisotropy
(see figure 2). Now, let assume to use it to move particles during the initialization.
If the fluid domain is bounded and particles are not allowed to escape from the
boundaries, wtends to fill all the asymmetries in the particle distribution and,
at the same time, it reduces as a consequence of the more regular distribution
7
Figure 2: lefy panel: sketch of the vector w=−∇Γin the neighborhood of a spatial anisotropy.
Right panel: sketch of the solid and fluid particles in the packing algorithm framework.
of particles themselves. Then, the final distribution would be the most regular
possible and kwk(that is, k∇Γk) would be minimized as requested.
The first step to build the Particle Packing Algorithm is to close the domain
boundaries. As a consequence, this implies that the free surface has to be treated
as a solid boundary. The domain boundary has to be modeled through fixed solid
particles, that is, particles with zero velocity and fixed positions. This approach
can be regarded as a special use of the frozen particles (for details see [11]) or
as a straightforward application of the fixed ghost particle technique proposed by
Marrone et al. [7]. A sketch of this procedure is displayed in the right panel
of figure 2. Note that particles do not need any specific rule to be positioned
inside the fluid domain nor inside the solid bodies. The second step consists in
assuming the density, the pressure and the volumes constant all over the fluid
domain. We indicate them through symbols ρ0,p0and V0respectively. Since
volumes are constant and the packing algorithm has to converge towards a static
solution, we neglect the continuity equation. Conversely, the momentum equation
of system (1.9) becomes:
Dui
Dt =βΓi+T(ζ)
i
Dri
Dt =ui
(2.14)
where β=2p00and Γi=PjWi j V0. The damping force can be chosen
independently from the adopted SPH scheme since T(ζ)
iis just used to ensure
8
the convergence of the Particle Packing Algorithm (more details are given in
Section 2.1 and in the Appendix B). Similarly to Monaghan [1], we choose a
linear damping term:
T(ζ)
i=ζuiwith ζ=αβ
V1/d
0
(2.15)
where dis the spatial dimension and αis a free dimensionless parameter. By
numerical simulations we found that a good choice for αranges between 1 ·103
and 5 ·103. Then, the Particle Packing system becomes:
Dui
Dt =βΓiζui
Dri
Dt =ui.
(2.16)
The initial conditions for the Particle Packing Algorithm are obtained by setting
all the particle velocities to zero and V0=Vtot/Npart where Vtot is the total fluid
volume and Npart is the total number of particles. The time-step adopted for the
present algorithm is:
t=CFL V1/d
0
β,(2.17)
where CFL =1.
As discussed in detail in the following sections, the system (2.16) tends to
converge as much as possible towards a steady state characterized by ui=0 and
Γi=0. When the fluid system is suciently close to this state, the Packing
Algorithm is stopped and the particle positions are used to initialize the SPH
simulations. Since the spatial distribution is very regular, the particle volumes
can be assumed to be identical. Then, the volume used for the initialization of
the SPH is V=V0. The initial particle pressure, p
i, is assigned by using the
analytical expression for the hydrostatic pressure and the particle positions. Then,
inverting the state equation, the initial density ρ
iis computed and, finally, the
particle mass is obtained through m
i=ρ
i/V. During the SPH simulations, the
particle masses are kept constant while the densities and the volumes are updated
using the continuity equation and the relation Vi=mii.
9
2.1. The Particle Packing system as a Lagrangian system
Let us consider (2.14) and neglect the damping force T(ζ)
i. Under this
hypothesis, the Particle Packing system is a Lagrangian system and its Lagrangian
function is L=TVwhere:
T=X
i
kuik2
2,V=β
2X
i
X
j
Wi j V0+βX
i
X
j
Wi j V0.(2.18)
Here the starred series indicates the summation over the fluid particles while the
barred one denotes the summation over the fixed solid particles. Symbols T
and Vare the specific kinetic energy and specific potential energy respectively.
According to the theory of dynamical systems, the total energy (that is, E=
T+V) keeps constant during the evolution and the system has stable equilibria
at the points where the potential attains its local minima. The addition of any
dissipative term (like, for example, that in (2.15)) forces the dynamical system
to converge towards the static (and stable) solution ui=0, Γi=0. This is a
fundamental point since it proves that for a given arbitrary geometry (modeled
through fixed ghost particles), the packing algorithm always converges towards
a stable particle configuration. All details on the Lagrangian structure of the
Packing algorithm are given in the Appendix B.
2.1.1. Stability and equilibria in R2
Let us consider the system (2.16) in R2. In this case, there are no boundaries
and, therefore, for ζ,0, the system has stable equilibria at the points where the
potential attains its local minima.
If the number of particles is finite, their spatial distribution cannot be
homogeneous nor isotropic and, therefore, Γ,1 somewhere in R2. Since
w=−∇Γpoints outward the fluid domain in the direction of maximum anisotropy,
particles start going away one from the other and their reciprocal distance grows.
This implies a decrease of W(krjrik) and, consequently, a decrease of the
potential energy. The final configuration predicts particles escaping at infinity
and corresponds to zero potential energy (that is, a local and absolute minimum
for the potential energy).
If the particle number is infinite, there exist only three regular tessellations
of R2which ensure Γ = 0. These configurations (which corresponds to local
extrema for the potential energy and, therefore, to equilibria of the particle system)
are the triangular, square and hexagonal tessellation of the plane. Note that they
are all invariant for rigid rotations and translations which, therefore, represent
“directions” of neutral stability.
10
-10 -5 0 5 10
-10
-5
0
5
10
y/dx
x/dx
-10 -5 0 5 10
-10
-5
0
5
10
y/dx
x/dx
h/dx = 1.4
t(β/V0)½= 6200
-10 -5 0 5 10
-10
-5
0
5
10
y/dx
x/dx
-10 -5 0 5 10
-10
-5
0
5
10
y/dx
x/dx
h/dx = 1.4
t(β/V0)½= 6200
-10 -5 0 5 10
-10
-5
0
5
10
y/dx
x/dx
-10 -5 0 5 10
-10
-5
0
5
10
y/dx
x/dx
h/dx = 1.4
t(β/V0)½= 6200
Figure 3: Cartesian, triangular and hexagonal tessellations (left column, from top to bottom)
and some snapshots of their evolution (right column, from top to bottom) for h/dx =1.4. The
circumferences indicate the domains of the adopted kernel functions.
From a practical and theoretical point of view, it is interesting to study the
stability of such equilibria. Since the theoretical analysis is very prohibitive, this
11
h/dx =0.8h/dx =1.4h/dx =2.0h/dx =2.6h/d x =3.2
Cartesian U S S U U
Triangular S U S S U
Hexagonal U U U U U
Table 1: stability of the tassellations. ‘S’ =Liapunov Stable, ‘U’ =unstable.
is made through numerical simulations. The first step consists in the construction
of a bounded fluid domain using one of the tessellations mentioned above.
Such a domain has to be large enough to assume that the most inner particles
are not influenced by . The boundary is fixed and modeled through fixed solid
particles. Note that the presence of a fixed boundary automatically eliminates the
“directions” of neutral stability (that is, rigid rotations and translations). After
the domain and its boundary have been built, particles are perturbed from their
equilibrium positions. To avoid that the viscosity influences the stability of the
tessellations under consideration, we put ζ=0. Under this hypothesis, system
(2.16) is Lagrangian (that is, it preserves the total energy, E) and, therefore,
only two cases are possible: particles keep oscillating around the equilibrium
configuration or they move towards other configurations. In the former case the
tessellation is said to be Liapunov stable otherwise unstable.
Figure 3 displays the tessellations (left column) along with dierent supports
of the kernel functions (circumferences). In all the simulations we used a
Wendland kernel function with h/dx =0.8,1.4,2.0,2.6,3.2 and the particle
positions have been perturbed from their stable configuration using a noise of
order dx/100. In the right column, some snapshots of the evolutions of the
dierent tessellations have been drawn for h/dx =1.4. In this specific case, the
Cartesian tessellation is Liapunov stable while the triangular and the hexagonal
ones are unstable. Table 1 briefly summarizes the results obtained. We found that
the hexagonal tessellation is always unstable while the stability of the Cartesian
and the triangular tessellations cannot be established “a priori”. Note that behavior
of a tessellation generally depends on the specific kernel used.
3. Applications
In the present section we show some applications of the particle packing
algorithm. We first deal with the initialization of hydrostatic conditions in
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complex geometrical configurations, then we show a dynamical problem which
evolves after the particle packing initialization. In all the simulations, the standard
SPH scheme has been used (see [4] for more details) and solid profiles have been
modeled through the fixed ghost particles described in Marrone et al. [7]. In any
case, the qualitative results obtained in the following also hold true for those SPH
scheme that implement frozen particles.
3.1. Trapezoidal tank
We first consider a trapezoidal tank like that drawn in figure 4 (His the filling
height) and study the influence of the particle initialization on the capability of the
SPH of simulating the hydrostatic solution. As stated in Section 2, the first step is
to “close” the fluid domain. This means that the free surface has to be substituted
by a solid boundary and modeled accordingly. This procedure is displayed in
figure 5 where the fluid domain has been initialized through a Cartesian grid (left
panel) and using the packing algorithm (right panel).
In the former case, the use of a Cartesian grid leads to the generation of
large spatial anisotropies along the inclined plane. Here, k∇Γk=O(1) and,
therefore, an intense particle resettlement is expected during the early stages of
the fluid evolution. Conversely, the particle packing algorithm eliminates the
spatial anisotropies and drastically reduces the magnitude of k∇Γk(whose order
of magnitude is about 1013H). Incidentally, we note that it is possible to derive
SPH schemes which may reduce the spatial anisotropies close to the solid profiles
through the use of special boundary conditions (see, for example, [12]). However,
these schemes are generally more complex than the standard SPH model and leads
to higher computational costs.
Figure 4: sketch of the trapezoidal tank.
13
0 0.5 1 1.5
0
0.5
1
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00
H ||∇Γ||
y/H
x/H
H
0 0.5 1 1.5
0
0.5
1
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00
H ||∇Γ||
y/H
x/H
Figure 5: trapezoidal tank (H/dx =25). Initialization using a Cartesian grid (left) and through the
particle packing algorithm (right).
0 500 1000 1500 2000 2500 3000 3500 4000
10-15
10-13
10-11
10-9
10-7
10-5
10-3
10-1
T/β
Niter
dx = H/25
dx = H/50
dx = H/100
Figure 6: trapezoidal tank. Evolution of the specific kinetic energy during the initialization through
the particle packing algorithm.
It is also interesting to analyze the dependence of the proposed algorithm
on the spatial resolution. In figure 6 the specific Kinetic energy of the packing
scheme (that is, T) is displayed for three dierent spatial resolutions versus the
number of iterations. The overall behavior during the initial stages is similar for
all the cases and the specific kinetic energy rapidly decreases to 107β. This
heuristically shows that the Particle Packing Algorithm is weakly influenced
by the adopted spatial resolution. Then, a further decrease occurs for longer
time depending on the specific resolution. In any case, particle are practically
motionless after 2,500 iterations when the order of magnitude of the specific
kinetic energy is less than 108β. This means that the equilibrium configuration
has been attained and that the packing algorithm can be stopped. Obviously, the
14
number of iterations required for the attainment of equilibria may vary according
to the specific problem at hand, to the choice of the kernel function (i.e., Gaussian,
cubic spline, quintic spline etc.) and to the adopted damping but is generally of
order 103.
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4 0.00 0.01 0.01 0.01 0.02 0.03 0.03 0.04 0.04 0.04 0.05
y
x
t(g/H)½= 10
|u|/(gH)½
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4 0.00 0.01 0.01 0.01 0.02 0.03 0.03 0.04 0.04 0.04 0.05
y
x
t(g/H)½= 10
|u|/(gH)½
Figure 7: hydrostatic solution for the trapezoidal tank (H/d x =50). Evolution using a Cartesian
grid (left) and after the initialization through the particle packing algorithm (right).
0 20 40 60 80 100
10-5
10-4
10-3
10-2
10-1
t(g/H)½
Max(|u|)/(gH)½
starting on a cartisian grid
after Particle Initialization
Figure 8: hydrostatic solution for the trapezoidal tank (H/dx =50): time evolution of the
maximum intensity of velocity.
15
When the particle initialization is complete, the hydrostatic solution is
assigned to the fluid domain (that is, hydrostatic pressure field and zero initial
velocity) and we start the simulation through the standard SPH scheme [4].
As shown in the left panel of figure 7, the initialization through the Cartesian
grid, because of the high values of k∇Γknear the sloping plane, leads to the
generation of high spurious currents (maxi|ui| ' 0.1gH at t=10 pH/g). On the
contrary, these are completely absent when the simulations is initialized through
the particle packing algorithm (right panel of the same figure).
A global measure of the particle resettlement phenomenon is easily obtained
by inspecting the evolution of the maximum intensity of velocity (that is, maxi|ui|)
during the simulation of the hydrostatic solution. As shown in figure 8, the
maximum intensity of velocity after the use of packing algorithm is at least two
orders of magnitude smaller than that predicted by the simulation which starts on a
Cartesian grid. In the latter case the particle motion still persists at t=100 pH/g
while the simulation after the packing algorithm is practically motionless (see
right panel of figure 7).
3.1.1. Particle Packing Algorithm versus Initialization through SPH with linear
damping term
As already mentioned in the Introduction, an alternative solution to reduce
particle resettlement is to start the SPH simulation using a high numerical
damping term and leave a long enough time to make particle self-resettle in
equilibrium positions. The actual numerical simulation starts after the equilibrium
configuration is attained. Here, we show that such a procedure (that is, the
initialization using the SPH scheme itself) only leads to minor improvements.
Following Monaghan [1], we use the standard SPH scheme with a linear
damping term. By definition, this is identical to the damping term adopted in the
Packing algorithm and, consequently, its coecient has been denoted by ζ. Apart
from these similarities, the physical meaning of the damping coecient used in
the SPH simulations is slightly dierent from that of the ζcoecient adopted
in the particle packing algorithm. In the latter case, the damping term has to be
regarded as an “inter-particle” dissipation while in the former case it represents
artificial dissipations. For this reason, the ζcoecient of the SPH simulations has
been made dimensionless using physical variables (specifically, the sound velocity
c0and the filling height H).
The evolution of the kinetic energy during the particle initialization through
the SPH scheme is displayed in figure 9 for dierent values of ζ. Incidentally
we highlight that it has not been possible to use values of ζsmaller than 0.02
16
0 2000 4000 6000 8000 10000
10-6
10-5
10-4
10-3
Niter
Ek/ρ0gH VTOT
ζH/c0= 1
ζH/c0= 0.02
ζH/c0= 0.1
Figure 9: kinetic energy evolution during the initialization with the standard SPH scheme for
dierent values of the damping coecient ζ.
0 20 40 60 80 100
10-5
10-4
10-3
10-2
10-1
t(g/H)½
Max(|u|)/(gH)½
d)
a)
a)
b)
c)
Figure 10: evolution of the maximum intensity of velocity after the particle initialization with the
standard SPH scheme (case a:ζ=1.0c0/H; case b:ζ=0.1c0/H; case c:ζ=0.02 c0/H) and
with the particle packing algorithm (case d).
since the SPH scheme was unstable. The attainment of the equilibrium becomes
faster and faster as the magnitude of the damping coecient increases. However,
this behavior does not correspond to a reduction of particle resettlement when the
actual simulation is started but, on the contrary, leads to the generation of larger
spurious currents which persist for very long times. This is briefly summarized in
figure 10 where the evolution of the maximum intensity of velocity is displayed
for the same cases of figure 9. It is also interesting to note that the use of
17
values of ζsmaller than 0.1 does not lead to any significant improvement for
long time evolution. Note that particles are still moving at t=100 pH/gsince
maxi|ui| ' 0.01 gH. Conversely, the initialization through the packing algorithm
lead to the correct equilibrium configuration and avoids any further resettlement
(see case dof figure 10).
Figure 11: sketch of the complex tank geometry.
3.2. A complex tank geometry
As a second example, we consider a complex geometry characterized by
bended profiles with dierent curvatures and by acute and obtuse solid angles (see
figure 11). Because of these features, the particle initialization of such a geometry
represents a very dicult problem.
The top panel of figure 12 displays the fluid evolution under hydrostatic
conditions after the initialization on a Cartesian grid. In this case, the generation of
spurious currents and vorticity near corners and bended profiles is very strong and
persists for long times. On the contrary, the use of the particle packing algorithm
eliminates such an undesirable behavior and gives a uniform particle distribution
which keeps stable for long times (middle panel of figure 12) . The evolution of
the kinetic energy confirms the findings above proving that, after the use of the
proposed algorithm, particles are almost motionless (bottom panel of figure 12).
18
0 1 2 3
0
0.5
1
1.5
0.00 0.01 0.01 0.01 0.02 0.03 0.03 0.04 0.04 0.04 0.05
y/H
x/H
t(g/H)½= 54
|u|/(gH)½
H
0 1 2 3
0
0.5
1
1.5
0.00 0.01 0.01 0.01 0.02 0.03 0.03 0.04 0.04 0.04 0.05
y/H
x/H
t(g/H)½= 54
|u|/(gH)½
H
0 20 40 60 80 100
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2 Ek/ρ0gH VTOT
t(g/H)½
after Particle Initialization
starting on a cartesian grid
Figure 12: hydrostatic solution for a complex tank geometry. Top: evolution after initialization on
a Cartesian grid. Middle: evolution after initialization through packing algorithm. Bottom: time
history of the kinetic energy.
19
3.3. A freely-floating problem
Here we consider a ship hull section floating in hydrostatic conditions. Under
such a hypothesis, the hull section should maintain motionless. However, because
of the particle resettlement, an unphysical deviation of the ship hull from the initial
position may be observed.
0 5 10 15 20 25
-4
-2
0
2
xG/dx
yG/dx
after Particle Inizializa tion
starting on a cartesia n grid
t(g/D)½= 90
t(g/D)½= 90
0 20 40 60 80
-2
-1.5
-1
-0.5
0
0.5
θ(degree)
after Particle Inizializa tion
starting on a cartesia n grid
t(g/D)½
Figure 13: freely floating of a ship hull. Left panel: motion of the mass center. Right panel: roll
motion.
-2 -1 0 1 2
-1.5
-1
-0.5
0
0.5
1
y/D
x/D
-2 -1 0 1 2
-1.5
-1
-0.5
0
0.5
1
y/D
x/D
-6 -4 -2 0 2 4 6 8
-2
0
2
4
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
y/D
x/D
|u| /(gD)½
t(g/D)½=70
-6 -4 -2 0 2 4 6
-2
0
2
4
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
y/D
x/D
t(g/D)½=70
|u| /(gD)½
Figure 14: freely floating of a ship hull section after initialization on a Cartesian grid (left column)
and with the particle packing algorithm (right column).
Similarly to the test cases studied in the previous sections, we initialize the
fluid domain using a Cartesian grid (top left panel of figure 14) and the packing
algorithm (top right panel of the same figure). The bottom panels display the
related SPH simulations. In this case, the spurious currents that generate in the
neighborhood of the hull because of the Cartesian grid force the ship hull to move.
20
Figure 13 shows the motion of the mass center (left panel) and the roll motion
(right panel) of the ship hull. Because of the reduction of the spurious currents,
the packing algorithm drastically reduces the unphysical ship motion ensuring the
attainment of the correct hydrostatic solution.
-1 -0.75 -0 .5 -0.25 0 0.25 0.5 0.75 1
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1y/R
x/R
-1 -0.75 -0 .5 -0.25 0 0.25 0.5 0.75 1
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1y/R
x/R
Figure 15: evolution of an elliptical drop. Initialization using a Cartesian grid (left) and the particle
packing algorithm (right).
3.4. Evolution of an elliptical drop
Let us consider a fluid domain which at the initial time is a two-dimensional
ball of radius R, subjected to the velocity field:
(u0(x,y)=A0x
v0(x,y)=A0y⇒ ∇u0= A00
0A0!.(3.19)
The initial pressure field is derived using the Poisson equation [13] and reads:
p0(x,y)=ρ0A2
0
2hR2(x2+y2)i.(3.20)
Assuming the flow to be inviscid, preserves an elliptical form during the
evolution and this form can be derived analytically (see [1] and [13] for details).
This domain is initialized using a Cartesian grid (left panel of figure 15) and the
proposed particle packing algorithm (right panel of the same figure). Since the
SPH is a Lagrangian scheme, particles move along stream lines. Consequently,
when particles are initially set on a Cartesian grid, the flow evolution given by
21
(3.19) leads particles to clump along straight lines (see left panel of figure 16).
This partially prevents the SPH solution to match with the analytical solution
for the domain boundary (dashed lines in figure 16). On the contrary, the flow
evolution after the use of the packing algorithm displays a more uniform particle
distribution and, consequently, leads to a better agreement with the analytical
solution (right panel of figure 16).
-3 -2 -1 0 1 2 3
-0.5
0
0.5
y/R
x/R
tA0= 2
-3 -2 -1 0 1 2 3
-0.5
0
0.5
y/R
x/R
tA0= 2
Figure 16: evolution of an initially circular patch of fluid using a Cartesain grid (left) and the
particle packing algorithm (right). Dashed lines indicate the analytical solution for the domain
boundary.
Conclusions
Using some intrinsic features of the SPH scheme, a novel packing algorithm
has been derived for the particle initialization. The proposed algorithm has
been validated again several tests cases proving to be robust, fast and reliable
also for complex geometrical configurations. As shown for the evolution of the
elliptical drop, the particle distribution obtained through the packing algorithm
may even avoid the formation of those filamentous structures that are caused by
the Lagrangian nature of the SPH.
Acknowledgements. The research leading to these results has received funding
from the European Community’s Seventh Framework Programme (FP7/2007-
2013) under grant agreement n. 225967 “NextMuSE”. This work was also
22
partially supported by the Centre of Excellence for Ship and Ocean Structures
of NTNU Trondheim (Norway) within the “Violent Water-Vessel Interactions and
Related”.
A. Interpolation formulas
Let us consider a generic scalar function fand the following convolution
integral:
h∇fi=Z0f0W dV0(A.21)
where, for the ease of notation, f0=f(r0) and W=W(r0r). Integrating by parts
and using the divergence theorem, we get:
h∇fi=Z0f0WdV0Z
f00W dV0=Z
f0Wn0dV0+Z
f0W dV0,
where nis the normal to . Note that the last integral has been obtained using
the identity 0W(r0r)=−∇W(r0r). Now, consider the following expansion:
f0=f+f·r0r+Okr0rk2(A.22)
Then, substituting into the first integral and noting that kr0rk=O(h), we find:
h∇fi=fZ
Wn0dV0+(f)kZr0rkWn0dV0+Z
f0W dV0+O(h)
Now, applying the divergence theorem once again and the property 0W(r0r)=
−∇W(r0r), it follows:
h∇fi=fΓ + (f)kZ0(r0r)kWdV0+Z
f0W dV0+O(h)=
=fΓ + fZ
W dV0+fZ
(r0r)⊗ ∇0W dV0+Z
f0W dV0+O(h)=
=fΓ+Γf− ∇fZ
(r0r)⊗ ∇W dV0+Z
f0W dV0+O(h).
This formula can be rewritten as follows:
h∇fi=Z
(f0f)W dV0+ Γ f− ∇fZ
(r0r)⊗ ∇W dV0+O(h).(A.23)
23
Then, using the expansion (A.22) once again, we finally obtain:
h∇fi= Γ f+O(h).(A.24)
In a similar way, we get:
hfi=Z
f0W dV0=fZ
W dV0+O(h)= Γ f+O(h).(A.25)
Now, basing on (A.23), we can write:
∇hfi=Z
f0W dV0=h∇fi+fΓΓf+fZ
(r0r)⊗ ∇W dV0+O(h)
Then, substituting (A.24) and (A.25), we finally get:
Z
(r0r)⊗ ∇W dV0= Γ 1+O(h).(A.26)
Substituting this formula back into (A.23), we obtain a consistent formulation for
h∇fi, that is:
h∇fi=Z
(f0f)W dV0+O(h).(A.27)
The expansion (1.11) for the divergence of the velocity field comes directly by
applying (A.27) and (A.24) on each component of u. Similarly, the expansion
(1.12) for the pressure gradient is obtained by summing 2 fΓto (A.27) and,
then, using the relation (A.24).
B. Lagrangian structure of the Particle Packing system
To show the Lagrangian structure of the Particle Packing Algorithm, we first
need to prove the following equation:
DWi j
Dt =ujui· ∇iWi j .(B.28)
Proof. Since Wij =W(si j ,h) where si j =krjrik, we get:
DWi j
Dt =W
si j
Dsi j
Dt =W
si j
(ujui)·(rjri)
si j
,(B.29)
24
and the last equality has been obtained using Dri/Dt =ui. Then, the equality in
(B.28) is obtained by using the following identity:
iWi j =(rjri)
si j
W
si j
.(B.30)
Multiplying system (2.16) by ui, we find:
D
Dt kuik2
2!=βX
j
ui· ∇iWi jV0ζkuik2.(B.31)
Then, applying the summation over the fluid particles (here indicated by the
starred summation), we get:
DT
Dt =βX
i
X
j
ui· ∇iWi jV02ζT,(B.32)
where T=Pikuik2/2 is the specific kinetic energy. The summation on the right-
hand side can be rearranged as follows:
X
i
X
j
ui· ∇iWi jV0=X
i
X
j
ui· ∇iWi jV0+X
i
X
j
ui· ∇iWi jV0=
=X
i
X
j
(ui+uj)
2· ∇iWi jV0+X
i
X
j
(uiuj)
2· ∇iWi jV0+
+X
i
X
j
(uiuj)· ∇iWi jV0,
where the barred series indicate summations over the solid particles. The last term
is obtained by noting that the velocity of the solid particles is zero (that is, uj=0).
Because of the symmetry in the double summation, the containing (ui+uj) is
identically null. Then, using the identity (B.28), we finally get:
X
i
X
j
ui· ∇iWi jV0=X
i
X
j
(uiuj)
2· ∇iWi jV0+X
i
X
j
(uiuj)· ∇iWi jV0=
=D
Dt
1
2X
i
X
j
Wi j V0+X
i
X
j
Wi jV0
.
25
Substituting this equality in (B.32), we obtain the total energy equation of the
Packing algorithm:
DE
Dt =2ζT,(B.33)
where E=T+Vand:
V=β
2X
i
X
j
Wi j V0+βX
i
X
j
Wi jV0,(B.34)
is the specific potential energy of the system. The first term on the right-hand
side represents the component coming from the fluid domain while the second one
accounts for the action of the solid boundary. If the damping eects are neglected,
the total energy of the system is conserved during the evolution.
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27
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This paper addresses the Particle Shifting Technique (PST) in the SPH schemes. Improving the accuracy of SPH schemes leads to particle clustering along the flow streamlines which turns to be detrimental for the simulations. PSTs aim at avoiding this adverse effect by slightly disordering the particles, allowing to retrieve a regular particle distribution within the kernel interpolation support. The gain in accuracy is such that this technique is now commonly adopted by the SPH practitioners, however the conditions that should be respected by a PST are not clearly discussed in the literature. In this paper, such conditions are exposed and their fulfillment by the main existing PSTs of the literature is analyzed. None of these existing PSTs fully satisfying these conditions, a novel PST is introduced. The proposed PST is validated for three different SPH schemes on 2D and 3D test cases, in presence of free-surface and solid boundaries.
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Description of Research Smoothed Particle Hydrodynamics (SPH) is a Lagrangian meshless method dealing with potentially violent simulations such as a wave breaking, or a dam break where Eulerian methods can be difficult to apply. Dealing with wall boundary conditions is one of the most challenging parts of the SPH method and many different approaches have been developed among (i) repulsive forces such as Lennard-Jones one, which is efficient to give impermeable boundaries but leads to non-physical behaviours, (ii) fictitious (or ghost) particles which provide a better physical behaviour in the vicinity of a wall but are hard to define for complex geometries and (iii) semi-analytical approach such as Kulasegaram et al. (2004) which consists of renormalizing the density field near a solid wall with respect to the missing kernel support area. The present work extends this last methodology, where we use intrinsic gradient and divergence operators which ensure conservation properties. This work will present three key advances implemented in the two-dimensional EDF SPH code SPARTACUS2D: • The time integration scheme used for the continuity equation requires particular attention, and as already mentioned by Vila (1999), we prove there is no point in using dependence in time of the particles' density if no kernel gradient corrections are added. Thus, by using a near-boundary kernel-corrected version of the time integration scheme proposed by Vila, we are able to simulate long-time simulations ideally suited for turbulent flow in a channel in the context of accurate boundary conditions. • All boundary terms issued from the continuous approximation are given by surface summations which only use information from a mesh file of the boundary. • In order to compute a correction of the smoothing kernel (the SPH interpolation function), Feldman and Bonet (2007) use an analytical value of the corrections which are computationally expensive whereas Kulasegaram et al. (2004) and De Leffe et al. (2009) use polynomial approximation which can be difficult to define for complex geometries. We propose here to compute the renormalisation term of the kernel support near a solid with a time integration scheme, allowing us any shape for the boundary. • The wall friction also requires particular attention and has been treated in a similar way as the gradient of pressure giving satisfactory results for a Poiseuille flow. Concerning turbulence models, some efforts have been made in the past by Violeau and Issa on the k-ε model (2007), but suffered from difficulties to enforce boundary conditions and compute accurately the strain rate near a wall. As a work in progress, we are currently developing analogous boundary conditions for the k-ε model. Below we plot the pressure field of a 2D still water test case and a dam break in a tank with a wedge for the Lennard-Jones repulsive force in (a), fictitious particles in (b), the new method in (c) and the new method with finer resolution in (d). We can thus conclude that the improvements allow a better reproduction of the pressure field next to the solid boundaries. Description of Industrial Relevance/Impact As a Lagrangian method, SPH offers a great tool to study environmental flows with large nonlinear deformations such as overtopping over a dike, dam breaks or flow over a flooding sluice. In these three examples, the fluid is interacting with a solid boundary and a physically correct representation of the influence of the wall on the flow is needed. The advances of this work are finally treating in a comprehensive manner many of the issues that until now had prevented more widespread uptake of the SPH method by industry.
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A smoothed particle hydrodynamics model with numerical diffusive terms, hereinafter referred to as δ-SPH [1] is used to analyze violent water flows. The boundary conditions on solid surfaces of arbitrary shape are enforced with a new technique based on fixed ghost particles. The violent impacts studied result from dam-break water flows striking obstacles of different shapes. The numerical results are validated against experimental data from the literature and solutions from a Navier–Stokes Level-Set solver. Predicted impact pressures are also compared with analytical solutions. The proposed scheme thus proves to be accurate and robust for the prediction of global and local loads of impact flows on structures.
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In this review the theory and application of Smoothed particle hydrodynamics (SPH) since its inception in 1977 are discussed. Emphasis is placed on the strengths and weaknesses, the analogy with particle dynamics and the numerous areas where SPH has been successfully applied.
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A truncation error analysis has been developed for the approximation of spatial derivatives in smoothed particle hydrodynamics (SPH) and related first-order consistent methods such as the first-order form of the reproducing kernel particle method. Error is shown to depend on both the smoothing length h and the ratio of particle spacing to smoothing length, Δx/h. For uniformly spaced particles in one dimension, analysis shows that as h is reduced while maintaining constant Δx/h, error decays as h2 until a limiting discretization error is reached, which is independent of h. If Δx/h is reduced while maintaining constant h (i.e. if the number of neighbours per particle is increased), error decreases at a rate which depends on the kernel function's smoothness. When particles are distributed non-uniformly, error can grow as h is reduced with constant Δx/h. First-order consistent methods are shown to remove this divergent behaviour. Numerical experiments confirm the theoretical analysis for one dimension, and indicate that the main results are also true in three dimensions. This investigation highlights the complexity of error behaviour in SPH, and shows that the roles of both h and Δx/h must be considered when choosing particle distributions and smoothing lengths. Copyright © 2005 John Wiley & Sons, Ltd.
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The fundamentals of the smoothed particle hydrodynamics (SPH) method and its applications in astrophysics are reviewed. The discussion covers equations of motion, viscosity amd thermal conduction, spatially varying resolution, kernels, magnetic fields, special relativity, and implementation. Applications of the SPH method are discussed with reference to gas dynamics, binary stars and stellar collisions, formation of the moon and impact problems, fragmentation and cloud collisions, and cosmological and galactic problems. Other applications discussed include disks and rings, radio jets, motion near black holes, supernovae, magnetic phenomena, and nearly incompressible flow.
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The SPH (smoothed particle hydrodynamics) method is extended to deal with free surface incompressible flows. The method is easy to use, and examples will be given of its application to a breaking dam, a bore, the simulation of a wave maker, and the propagation of waves towards a beach. Arbitrary moving boundaries can be included by modelling the boundaries by particles which repel the fluid particles. The method is explicit, and the time steps are therefore much shorter than required by other less flexible methods, but it is robust and easy to program.
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SPH is the shorthand for Smoothed Particle Hydrodynamics. This method is a Lagrangian method which means that it involves following the motion of elements of fluid. These elements have the characteristics of particles and the method is called a particle method. A useful review of SPH (Monaghan 1992) gives the basic technique and how it can be applied to numerous problems relevant to astrophysics. You can get some basic SPH programs from http://www.maths.monash.edu.au/~jjm/sphlect. In the present lecture I will assume that the student has studied this review and therefore understands the basic principles. In today's lecture I plan to approach the equations from a different perspective by using a variational principle.
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In this paper a truly incompressible version of the smoothed particle hydrodynamics (SPH) method is presented to investigate the surface wave overtopping. SPH is a pure Lagrangian approach which can handle large deformations of the free surface with high accuracy. The governing equations are solved based on the SPH particle interaction models and the incompressible algorithm of pressure projection is implemented by enforcing the constant particle density. The two-equation k–ε model is an effective way of dealing with the turbulence and vortices during wave breaking and overtopping and it is coupled with the incompressible SPH numerical scheme. The SPH model is employed to reproduce the experiment and computations of wave overtopping of a sloping sea wall. The computations are validated against the experimental and numerical data found in the literatures and good agreement is observed. Besides, the convergence behaviour of the numerical scheme and the effects of particle spacing refinement and turbulence modelling on the simulation results are also investigated in further detail. The sensitivity of the computed wave breaking and overtopping on these issues is discussed and clarified. Copyright © 2005 John Wiley & Sons, Ltd.