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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”

Abstract

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.

1. Introduction

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: a.colagrossi@insean.it

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 [1] 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

simulations [2].

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. [3], 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

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 [4], 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. [5]) 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

algorithm.

The method used to perform the ﬁrst step of the algorithm was proposed

by Doring [6]; it exploits eigenvalues of the renormalization matrix [4] deﬁned

as:

B(xi) = X

j∇Wj(xi)⊗(xj−xi) ∆Vj−1

(1)

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.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

P/ρgH

LW= 5.366 H

L = 2H

H = 600 mm

Figure 1: Sketch and initial condition of the dam-break problem considered, see e.g. [5].

3

0 1 2 3 4 5

0

0.5

1

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

x/H

y/H

0 1 2 3 4 5

0

0.5

1

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

x/H

y/H

0 1 2 3 4 5

0

0.5

1

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

x/H

y/H

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

4

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.

0.75

0.20

λ

I

B

E

0.75

0.20

λ

2h

I

B

E

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 [5], 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∈E⇐⇒ λ≤0.20

i∈B⇐⇒ 0.20 < λ ≤0.75

i∈I⇐⇒ 0.75 < λ

(2)

5

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 [4]. 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 [4]. 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

matrix:

n(xi) = ν(xi)

|ν(xi)|;ν(xi) = −B(xi)X

j∇Wj(xi) ∆Vj(3)

Figure 4: Sketch of the regions used in the algorithm.

6

This is a standard way to improve accuracy of the SPH interpolation within

the ﬂuid domain, see e.g. [4]. 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

(4)

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.

7

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

semimajor axis.

2h

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.

2.2. Validation

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

quadratic.

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

8

0.75

0.20

λ

I

B

E

F

0.75

0.20

λ

2h

I

B

E

F

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

ﬁrst step.

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. [7] and [4], 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

9

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

|xji |<π

4i⇒i /∈F

otherwise i ∈F

(5)

Therefore, in an intuitive way, the triangular region in ﬁgure 4becomes

a cone in three dimensions, while the semicircle becomes a hemisphere.

3.2. Validation

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

10

X

1

Y

0.5 11.5

Z

0.5

1

1.5

4dx=3h

10dx

Figure 7: Left panel: sketch of the tested geometry. Right panel: corresponding set of

particles.

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

sphere.

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

quadratic.

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,

11

X

0.5

1

1.5

Y

1

Z

0.5

1

1.5 4dx

X

0.5

1

1.5

Y

1

Z

0.5

1

1.5

10dx

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. [8] and [9]). 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

12

Figure 9: Left panel: sketch of the second tested geometry. Right panel: corresponding

set of particles.

X

0.5

1

1.5

Y

0.5

1

1.5

Z

0.5

1

1.5

2h

X

0.5

1

1.5

Y

0.5

1

1.5

Z

0.5

1

1.5

25dx

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:

dNFN= (xFN−xN)·nFN(6)

13

Figure 11: Sketch of the problem geometry of a 3D impact ﬂow against a tall structure

(see e.g. [8] and [9]): 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):

φ(xN) =

−1dNFN≤ −2h

dNFN/2h−2h < dNFN<2h

+1 dNFN≥2h

(7)

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.

14

-1.00 -0.50 0.00 0.50 1.00

φ=-(dx/2)/2h

φ

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:

−1N2h=∅

+1 N2h6=∅,N2h∩F=∅

dNFN/2hN2h6=∅,N2h∩F6=∅, λFN≥0.1

−|xN−xFN|/2hN2h6=∅,N2h∩F6=∅, λFN<0.1

(8)

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

15

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. [10]. 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

clearly identiﬁable.

16

5. Conclusion

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

problems.

17

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).

18

6. Notation

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

Acknowledgments

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”.

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A. Appendix

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 [1] 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 [1].

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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 [1] 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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