Fast free-surface detection and level-set function
deﬁnition in SPH solvers
S. Marronea,c, A. Colagrossia,∗, D. Le Touz´eb, G. Grazianic
aINSEAN,The Italian Ship Model Basin, Roma, Italy
bFluid Mechanics Lab., ´
Ecole Centrale Nantes / CNRS, Nantes, France
cDepartment of Mechanics and Aeronautics, University of Rome “La Sapienza”
The present paper proposes a novel algorithm to detect the free surface in
particle simulations, both in two and three dimensions. Since the proposed
algorithms are based on SPH interpolations their implementation does not
require complex geometrical procedures. Thus the free-surface detection can
be easily embedded in SPH solvers, without a signiﬁcant increase of the CPU
time. Throughout this procedure accurate normal vectors to the free surface
are made available. Then it is possible to deﬁne a level-set function algorithm
which is presented in detail. The latter allows in-depth analyses of three-
dimensional free-surface simulations by using standard visualization tools,
including internal features of the ﬂow. The algorithms proposed for detecting
free-surface particles and deﬁning the level-set function are validated on
simple and complex two- and three-dimensional ﬂow simulations. The
usefulness of the proposed procedures to post-process and analyze complex
ﬂows are illustrated on realistic examples.
In recent years the SPH method has been successfully applied to problems
involving free-surface ﬂows with fragmentation. In order to analyze ﬂows
with complex free-surface patterns (fragmentations, air entrapment, etc.)
and to face a larger range of problems it is required to know which particles
belong to the free surface. This detection can also be required for the
enforcement of suitable boundary conditions along the free surface (surface
∗Corresponding author: firstname.lastname@example.org
Preprint submitted to Journal of Computational Physics January 11, 2010
tension, isothermal condition, etc) in order to deal with diﬀerent physical
phenomena and ﬂow behaviors. Dilts  developed an algorithm for the
free-surface tracking that can detect free-surface particles in a robust and
reliable way and that is applicable to any meshless method. However it is
quite diﬃcult to implement, particularly in its extension to three-dimensional
In this work a novel algorithm for free-surface detection is presented. Such
a scheme, based on the properties of the SPH kernel, is easy to implement
both in two and three dimensions, and computationally cheap. The accuracy
of the method is comparable to that of the method proposed by Dilts. It is
possible, indeed, to catch small cavities of diameter as small as 2h (hbeing
the smoothing length) and ﬂuid elements with dimension smaller than h(like
jets and drops). Thanks to these valuable features, the proposed algorithm
can be used at each time-step of the simulations, without an appreciable
increase of the CPU-time.
Moreover, free-surface detection permits strong improvement of the post-
processing phase, particularly in three-dimensional simulations with complex
ﬂow features. In fact, if one uses merely a SPH output, ﬂow analysis is
problematic since data are known on scattered points, and it is diﬃcult
to obtain contour plots, slices and iso-surfaces. Such an analysis can
be performed in a straightforward way using standard tools if data are
interpolated on a regular grid. In this context it is useful to deﬁne a level-set
function among the grid nodes. This function permits to distinguish between
nodes inside and outside of the ﬂuid domain. To deﬁne the level-set function
the detection of the free-surface particles is required as a ﬁrst step. Besides
its utility for ﬂow analysis the deﬁnition of this function may be useful to
extend remeshing techniques, see e.g. , to free-surface ﬂow problems.
In the present paper the algorithm for free-surface detection is described
and validated in sections 2 and 3, for both two- and three-dimensional
cases. To assess the accuracy of the algorithm, the validation has ﬁrst been
performed using simple geometries, and then on complex ﬂow cases. Finally,
in section 4, we describe the procedure to deﬁne a level-set function which
can be useful to interpolate ﬂow data on a regular grid. Post-processing
results are shown to illustrate the proposed algorithm capabilities.
2. 2D algorithm
2.1. Algorithm details
The algorithm is composed of two steps: in the ﬁrst one the properties of
the renormalization matrix, deﬁned by Randles and Libersky , are used to
ﬁnd particles next to the free surface. This ﬁrst step strongly decreases the
number of particles that will be processed in the second step. In the second
step the algorithm, by means of geometric properties, detects particles that
actually belong to the free surface and evaluates their local normals.
In order to validate it the algorithm has been applied to simple geometries
as well as to a dam break problem (see e.g. ) which sketch and initial
conditions are displayed in ﬁgure 1. The complex free surface behavior of
the impact ﬂow simulated is displayed in ﬁgure 2. In this ﬁgure the plotted
rectangles delimit the zones which are enlarged in ﬁgure 3to highlight the
ﬂow complexity there, and which are the most challenging for the detection
The method used to perform the ﬁrst step of the algorithm was proposed
by Doring ; it exploits eigenvalues of the renormalization matrix  deﬁned
B(xi) = X
where ∆Vjis the volume of the j-th particle and Wj(xi) is the
interpolating kernel centered on particle jand evaluted in the position xi.
The spatial derivatives of Wj(xi) are referred with respect to the position xi.
LW= 5.366 H
L = 2H
H = 600 mm
Figure 1: Sketch and initial condition of the dam-break problem considered, see e.g. .
0 1 2 3 4 5
1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
t(g/H)½= 6.001 |u|/(gH)½
0 1 2 3 4 5
1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
t(g/H)½= 9.413 |u|/(gH)½
0 1 2 3 4 5
1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
t(g/H)½= 10.275 |u|/(gH)½
Figure 2: Diﬀerent time instants of the impact ﬂow after the dam-break. The rectangles
delimit the zones enlarged in ﬁgure 3.
Doring showed that the value of the minimum eigenvalue, λ, of the
matrix B−1depends on the spatial organization of the particles jin the
neighborhood of the considered calculation point i. When going away from
the ﬂuid domain this eigenvalue λtends theoretically to 0 while inside this
domain λtends theoretically to 1. This allows to determine regions of the
ﬂuid domain where the free surface can lie or not. Let us deﬁne Nas the set
of all the ﬂuid particles and F⊂Nas the subset of particles belonging to the
free surface. Then, computing λfor each particle, it is possible to further
deﬁne three complementary subsets: Ecomposed by particles belonging to
thin jets and drops, characterized by low values of λ;Icomposed by interior
particles far from the free surface, characterized by high values of λ; and B
composed by particles which are close to the free surface or are in regions of
the domain where particles are not uniformly spread. Particles belonging to
the last subset are characterized by intermediate values of λ. We have that
N=E∪I∪B. The free-surface particles subset Fis thus composed of all the
particles of subset Eand a part of the elements of subset B.
Figure 3: Values of the minimum eigenvalue λof the matrix B−1. Red particles belong
to subset E; green particles belong to subset I; blue particles belong to subset B.
To identify these subsets it is possible to deﬁne threshold values of λ
which depend on the considered kernel. Here we use a renormalized gaussian
kernel shape, see , with a support radius equal to 3h, where his the
smoothing length and is equal to 1.33 dx.dx is the average particle spacing
which means that in two dimensions a particle has ≃dx2for volume and an
average number of neighbors equal to 50. All the results and the conclusions
presented in this work have to be considered valid only for the aforementioned
h/dx ratio. Several tests have been performed to set the proper thresholds for
λ. Being ithe particle under examination, the values found are the following:
i∈B⇐⇒ 0.20 < λ ≤0.75
i∈I⇐⇒ 0.75 < λ
In this way the ﬁrst step of the algorithm computes the minimum
eigenvalue for each particle and gives a ﬁrst rough detection of the free
surface. This operation has a very low computational cost especially if the
renormalization matrix is already computed in the SPH scheme, as e.g. in
the formulation proposed in . In ﬁgures 3one can observe the result of
this ﬁrst detection. Particles next to the free surface and near cavities are
correctly detected but this also happens for some particles within internal
ﬂuid regions characterized by non-uniform distribution. Conversely, particles
which belong to drops and thin jets are easily captured by the lower threshold
and directly identiﬁed as free-surface particles (red coloured).
In the second step of the algorithm, a more precise and reliable control is
performed on particles belonging to Bin order to complete the free surface
detection. The proposed method is based on the fact that, inside the ﬂuid
domain, the sum of the kernel gradient over neighbors is very close to zero.
When a particle, instead, is near the free surface, such sum is a good
approximation of the local normal nto the free surface, see . Since the
accuracy of the evaluation of this vector depends on the particle disorder, it is
possible to get a more accurate evaluation by using again the renormalization
n(xi) = ν(xi)
|ν(xi)|;ν(xi) = −B(xi)X
Figure 4: Sketch of the regions used in the algorithm.
This is a standard way to improve accuracy of the SPH interpolation within
the ﬂuid domain, see e.g. . Here we use the same principle to improve the
accuracy of the evaluation of the local normal. Once this vector nis known,
it is possible to deﬁne a region of the domain like the one sketched in ﬁgure 4.
The algorithm then checks whether or not at least one neighbor particle lies
in this region, hereinafter referred to as scan region. If no neighbor is found
inside it, the candidate particle belongs to the free surface. It must be noted
that inside the ﬂuid domain this region cannot be void since h= 1.33 dx.
This control is carried out by the algorithm in the following way. We
denote by i∈Bthe particle that is under examination and by j∈Nthe
neighbor of iwhich is included in a 3hradius distance. We deﬁne also the
point Tat distance hfrom iin the normal direction and the unit vector τ
perpendicular to n. The conditions to assess whether particle ibelongs or
not to the free surface are therefore:
∀j∈Nh|xji| ≥ √2h, |xjT |< hi⇒i /∈F
∀j∈Nh|xji|<√2h, |n·xjT |+|τ·xj T |< hi⇒i /∈F
otherwise i ∈F
where notation Aij =Ai−Ajhas been used. If the ﬁrst condition is true it
means that the neighbor under examination is in the dark grey region (S1)
in ﬁgure 4while, if the second condition is true, it means that the neighbor is
in the pearl grey region (S2). The two regions together form the scan region.
If any neighbor is located in the scan region it means that there is no
cavity in the normal direction, or that this cavity has a diameter less than
2h, so that particles inside it are deeply interacting and it is not a true cavity.
Therefore through this process we are tracking the free surface of cavities of
diameter larger than 2h. More details on the geometry used for the scan
region are given in appendix A.1.
It can be noticed that for some particle distributions, e.g. for thin jets,
the sum of the kernel gradient can go to zero, giving wrong values for vector
n. However, in this circumstance the eigenvalue λwill be very low and the
particle will hence have already been detected as a free-surface particle in
the ﬁrst step of the algorithm. This ﬁrst step has thus two functions: it is an
eﬃcient selection allowing to quickly perform the second step, and a tool to
detect particles belonging to jets and drops which could hardly be detected
by the second step of the algorithm.
h/a ×10−27.52 3.76 1.88 0.94 0.63 0.38
ε2.16 1.23 0.68 0.34 0.23 0.13
Table 1: Average relative angle error between the analytically computed normals and the
one evaluated by equation 3.h/a denotes the ratio between the smoothing lenght and the
Figure 5: Left: free-surface detection of an elliptic ﬂuid domain. Right: free-surface
detection in an elliptic cavity of minor axis equal to 2h. Red particles are those detected
by the proposed free-surface algorithm.
In the left part of ﬁgure 5the algorithm is validated for an elliptic ﬂuid
domain, with 0.661 eccentricity. Since the evaluation of the free-surface
normals is critical for the algorithm eﬀectiveness, the ones calculated by
the algorithm are compared to the analytical values. In this comparison
a uniform distribution of particles had to be used inside the ﬂuid domain
in order to assign exact volumes to the particles. Results are reported in
table 1 in terms of average relative angle error. The convergence is close to
In the right part of ﬁgure 5an elliptic cavity with minor axis equal to 2h
was introduced in the ellipse. This cavity has thus the minimum dimension
which the algorithm is able to detect.
After this simple test, the algorithm is further assessed on the complex
ﬂow situations presented in ﬁgure 3. The free-surface particles detected
by the algorithm are plotted in pink in ﬁgure 6. Despite the geometrical
complexity of these conﬁgurations, the proposed method is able to provide a
very good qualitative estimation of the particles which form the boundary.
In particular one can observe that approximately circular cavities of diameter
Figure 6: Results after the application of free-surface detection algorithm: free-surface
particles are displayed in pink with their normals; in the bottom part small cavities of 2h
diameter are shown. A circle with dimension 2his reported for comparison.
just larger than 2hare well detected.
Even though the algorithm requires a cycle on the neighbors for each
particle involved in the second step, it has still a very low computational
cost. Actually, all but a few percents of the particles are ﬁltered out in the
The CPU time cost of the algorithm is lower than 5% of the total cost
of a standard SPH calculation, where the latter includes the calculation of
the neighbor list and the summations for the continuity and momentum
equation. It is the computation of the renormalization matrix which takes
the most part of the CPU time required by the algorithm (about 90%).
Hence, if the renormalization matrix is already evaluated in the numerical
scheme, see e.g.  and , the increase of CPU time due to free-surface
detection is absolutely negligible. Moreover, if the algorithm is used only
for visualization or analysis purposes, it has to be applied only periodically
(typically every 100 time steps), and its cost becomes again negligible.
3. 3D algorithm
3.1. Algorithm details
The extension of the algorithm to the third dimension is rather
straightforward and easy to implement. The algorithm has still the same
h/r ×10−215.0 7.52 3.76 1.88
ε×10−214.0 4.42 1.84 0.77
Table 2: Average relative angle error between the analytically computed normals and the
one evaluated by equation 3for a spherical ﬂuid domain. h/r denotes the ratio between
the smoothing lenght and the radius of the sphere.
structure as the two-dimensional one: it is composed of a ﬁrst step where
particles far from the free surface are ﬁltered out, and of a second step to
reﬁne the detection only to the particles belonging to the free surface. In
the ﬁrst step the minimum eigenvalue λof the renormalization matrix is
again needed for each particle. Again, two thresholds are used which are the
same as in two dimensions. These thresholds are valid for the renormalized
gaussian kernel with support radius equal to 3hand h/dx = 1.33 (this means
an average number of neighbors equal to 266).
While the ﬁrst step is formally unchanged in the extension to three
dimensions, the second step is slightly modiﬁed. The vector nis still
evaluated through equation 3but the conditions which deﬁne the region
to scan become:
∀j∈N|xji| ≥ √2h, |xjT |< h⇒i /∈F
∀j∈Nh|xji|<√2h, arccos n·xji
otherwise i ∈F
Therefore, in an intuitive way, the triangular region in ﬁgure 4becomes
a cone in three dimensions, while the semicircle becomes a hemisphere.
In the three-dimensional case it is more complex to test the algorithm
and assess its accuracy. Indeed, unlike in 2D cases, in 3D problems even
a qualitative evaluation of the particles belonging to the free surface could
be quite diﬃcult. In particular cavities and jets are generally blurred since
particles are spread in the space in a disordered way.
In order to overcome this problem the algorithm was tested on a particle
distribution with a free surface which is known a priori. According to this
strategy two tests were performed. In the ﬁrst one particles are arranged
to form a sphere with a spherical cavity inside, as sketched in left panel
of ﬁgure 7. In order to have a regular distribution, particles are placed on
concentric spheres with radius increasing by dx, where dx is cubic root of the
Figure 7: Left panel: sketch of the tested geometry. Right panel: corresponding set of
particle volume. On each sphere, particles are equispaced with a distance
approximatively equal to dx. The cavity inside the sphere has a diameter
equal to 4dx ≃3h. This quantity is actually slightly larger than 2hwhich
is the limit size for detecting a cavity. However, choosing a smaller cavity
radius was not possible in practice. Indeed, this would have resulted in
having too few particles distributed on the cavity surface, resulting in a bad
approximation of their volumes, and consequently providing an inappropriate
test of the algorithm. The particle distribution is shown in the right panel
of ﬁgure 7. In such a view it is not possible to detect the cavity inside the
The result given by the algorithm is presented in ﬁgure 8. The two
detected free surfaces (the ones from the inner and outer spheres) are shown
separately to better evidence the normals. We notice that free-surface
particles and normals are correctly evaluated both in the cavity and along the
external surface. Similarly to the 2D case, in table 2 calculated free-surface
normals are compared to analytical values in terms of average relative angle
error for a spherical ﬂuid domain. Again, the convergence rate is close to
In order to assess the capability of the algorithm to capture cavities of
dimension equal to 2h, a more complex test was performed. As displayed
in the left panel of ﬁgure 9, a toroidal cavity was created inside the sphere.
The width of the torus is 2hand the torus ring diameter is 14dx. Again,
Figure 8: Left: detected free-surface particles and normals on the spherical cavity
boundary. Right: detected free-surface particles and normals on the outer sphere.
when showing the whole particle set, it is not possible to distinguish the
toroidal cavity inside the sphere (see right panel of 9). The free-surface
particles detected by the algorithm are shown in ﬁgure 10. Also in this case
free-surface particles and normals are correctly evaluated.
To illustrate the method capabilities on an actual complex 3D situation
we show the results of the application of the algorithm on an impact ﬂow (see
e.g.  and ). The skecth of the problem geometry is displayed in ﬁgure
11. In ﬁgure 12 the whole particle distribution and the detected free-surface
particles are shown for a time instant just after the impact. Anyway in these
plots only a few ﬂow features can be analyzed. Therefore a useful tool to
improve the ﬂow analysis is proposed in the next section.
4. Deﬁnition of a level-set function
The interpolation of particle data onto a regular grid can be useful both
for allowing an in-depth analysis of the ﬂow features through adequate post-
processing, or for remeshing the particles during free-surface ﬂow simulations.
In order to perform this procedure it is necessary to locate the free surface
across the grid. In other words, one needs to separate nodes inside the ﬂuid
domain from those outside. This can be done from the knowledge of the free-
surface particle subset F. Let us consider a regular Cartesian grid of spatial
Figure 9: Left panel: sketch of the second tested geometry. Right panel: corresponding
set of particles.
Figure 10: Left: detected free-surface particles and normals on the toroidal cavity
boundary. Right: detected free-surface particles and normals on the outer sphere.
resolution dx, and which encloses all the computational domain. For each
node Nclose to free surface particles of subset F, the nearest free-surface
particle FNis detected and the scalar quantity dN FNis evaluated through:
Figure 11: Sketch of the problem geometry of a 3D impact ﬂow against a tall structure
(see e.g.  and ): side view (left panel) and top view (right panel). An initial layer of
water 1 cm deep exists on the bottom of the tank.
Figure 12: 3D impact against a tall structure after a dam-break. Left: whole particle
distribution over the ﬂuid domain; the colors represent the pressure. Right: free-surface
particles detected by the algorithm at same time instant.
where nFNis the normal to the free surface evaluated in FN. For each node
it is now possible to deﬁne a level-set function φ(xN):
dNFN/2h−2h < dNFN<2h
This function is positive inside the ﬂuid, negative outside it and equal to 0
along the free surface where dN FN= 0. More precisely, under the assumption
that the actual free-surface location is at a distance dx/2 from the center of
mass of particles belonging to F, the value of φon the free surface has to be
φ=−dx/4hand not zero anymore.
-1.00 -0.50 0.00 0.50 1.00
Figure 13: Contour plot of function φfor the particle distribution shown in ﬁgure 3.
Free-surface particles (black dots) and free-surface contour level (red dash-dotted curve).
At the computational level, the procedure to evaluate φcan be executed
in a fast manner. At each node we ﬁrst identify the particles present within
a 2h-distance. We denote N2hthis subset of particles through which we can
evaluate the function φ(xN) as:
where λFNis the minimum eigenvalue of the matrix B−1(see eq. 1) for the
particle FN. For λFN<0.1 the nearest free-surface particle for the node N
is a solitary particle. In such a case the vector nFNis null (see eq. 3) and
consequently the scalar product dN FNis meaningless. Therefore, the latter
is directly substituted with the distance |xN−xFN|. Further, to smooth out
the function φon the whole mesh, a Gaussian ﬁlter on the nodes is performed
once φis evaluated by equation (8).
In ﬁgure 13 the contour plots of function φfor the two time instants of
ﬁgure 3are shown. The free surface is represented by the red dash-dotted
line which corresponds to the contour level φ=−dx/4h, in close agreement
with the free-surface particles positions if they were shifted by dx/2.
In 3D simulations the interpolation on a regular mesh clearly brings
higher beneﬁts. Indeed, the visualization and the ﬂow analysis of a 3D
SPH simulation is generally quite diﬃcult. This is highlighted in ﬁgure 14
where a spherical ﬂuid domain with three concentric toroidal cavities and a
small spherical cavity of 3hdiameter is considered. Even though only the
free-surface particles are shown (left panel), it is obvious how diﬃcult it is to
detect the geometry of the ﬂuid domain. On the right side of the same ﬁgure,
the iso-surface φ=−dx/4hrepresenting the free surfaces conversely gives
a clear representation of the ﬂuid domain, thanks to transparency features
which are a standard visualization tool on a regular mesh.
Figure 14: Spherical ﬂuid domain with three toroidal cavities and a spherical cavity of 3h
diameter. Left: free-surface particles. Right: iso-surface φ=−dx/4h.
Some interpolation results of the impact ﬂow shown in ﬁgure 12 are
presented in the following. This operation has been carried out with a
Moving-Least Square (MLS) interpolator that exactly interpolates a linear
ﬁeld on a regular grid from scattered points, see e.g. . The representation
of the free surface is given by the iso-contour φ=−dx/4hin ﬁgure 15. In
particular, the pressure distribution during the impact is evidenced through
contours interpolated on a vertical plane inside the domain. Finally, the
last plots of ﬁgure 15 show how it is possible to analyze ﬂow internal details
through the reconstruction of the free surface as an iso-surface. In particular,
entrapped bubbles and a large tube cavity due to a strong wave breaking are
An eﬃcient algorithm capable to detect the free surface in SPH methods
has been proposed in 2D and 3D simulations. It is composed of two stages.
The ﬁrst stage consists in detecting the particles composing the free surface.
From this information, it is possible to deﬁne a level-set function throughout
the domain in a second stage. This function can be used to interpolate ﬂow
quantities on a cartesian grid, which makes possible the visualization and
analysis of ﬂow features using standard visualization tools. The two stages
have been carefully validated on test cases of increasing complexity, up to
real 3D impact ﬂow.
On these test cases it has been shown how the method allows detecting
ﬂow details such that thin jets, drops, entrapped air structures, etc. as well as
to recover contour slices, iso-surfaces, etc. in an eﬃcient way. The algorithm
proposed in the present paper could be exploited to impose speciﬁc conditions
on the free surface, or to extend remeshing techniques to free-surface ﬂow
Figure 15: 3D Flow impact against a tall structure after a dam-break. Top: two views of
the free surface represented by iso-surface φ=−dx/4h. Middle: free-surface and pressure
contours interpolated on a vertical plane at two diﬀerent time instants. Bottom: two
views of the free-surface iso-surface at a later stage of the ﬂow evolution; on the right plot
a detail of the cavity generated in front of the column is given (seen from below).
B= Renormalization Matrix
x= Spatial position
W = Kernel function
∆V= Particle volume
λ= Minimum eigenvalue of B−1
N= Set of all the ﬂuid particles
F= Set of the particles belonging to the free surface
E= Set of particles with λ≤0.20
B= Set of particles with 0.20 < λ ≤0.75
I= Set of particles with 0.75 < λ
h= Kernel smoothing lenght
dx = Average particle spacing
n= Unit vector normal to the free surface
τ= Unit vector tangential to the free surface
φ= Level set function
This work was partially supported by Programma Ricerche INSEAN
2007- 2009 and Programma di Ricerca sulla Sicurezza funded by Ministero
Infrastrutture e Trasporti.
The research leading to these results has received funding from the
European Community’s Seventh Framework Programme (FP7/2007-2013)
under grant agreement n225967 “NextMuSE”.
A.1. Motivation for the shape of the scan region adopted
The choice of the shape of the scan region, composed by a half circle
and a half square rather than by a simple circle for instance, is explained in
the following. In top sketch of ﬁgure 16 a uniform distribution of particles
is considered. In such a conﬁguration only the ﬁrst row of particles should
be detected as free surface. In the left part of the sketch the scan region
is reported for a particle belonging to the second row. Two particles are
present in the scan region which means that the second row is not detected
as free surface. On the right part of the same sketch circles with diameter
equal to hhave been reported in order to show that also the arc-method
proposed in  gives the same result. In the bottom sketch of ﬁgure 16
Figure 16: Sketch of the scan region applied on uniform distribution (top panel) and on
free-divergence stretched distribution (bottom panel). On the right a comparison with the
arc method, see .
a divergence-free stretching is applied to the previous particle distribution.
Due to this stretching, the horizontal distances between the particles are
equal to √2dx while the vertical ones are equal to dx/√2. There is hence a
ratio 2 between these two distances. We can consider it a limiting case: for a
ratio 2 or higher, the considered particle of the “second row” belongs to the
free surface, for a lower ratio it does not belong to it. Using the procedure
described in  an arc of 12ois not covered by other neighbor circles; hence
the particles of the second row are considered as free-surface particles. This
occurs also using the proposed algorithm since the scan region is now empty.
If a fully circular shape was used as scan region, the particles of the second
row would not be detected as free-surface particles. Other shapes could be
adopted as scan region. Anyway, in order to correctly detect the free surface,
two requirements have to be satisﬁed: a) the detection of circular cavities of
diameter equal to 2h, b) in the example discussed above the second row of
particles in bottom plot of ﬁgure 16 has to be detected as free-surface.
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