SPH modelling in coastal engineering

E-proceedings of the 36th IAHR World Congress
28 June – 3 July, 2015, The Hague, the Netherlands
SPH MODELLING IN COASTAL ENGINEERING
ALEJANDRO J.C. CRESPO(1), CORRADO ALTOMARE(2) (3), JOSE M. DOMINGUEZ(1), TOMOHIRO SUZUKI(2) (4), TOON VERWAEST(2)
& MONCHO GÓMEZ-GESTEIRA(1)
(1)
Environmental Physics Laboratory, Vigo University, Ourense, Spain,
[email protected]
(2)
Flanders Hydraulics Research, Antwerp, Belgium,
[email protected]
(3) Department
(4) Department
of Civil Engineering, Ghent University, Ghent, Belgium,
[email protected]
of Hydraulic Engineering, Delft University of Technology, Delft, The Netherlands,
[email protected]
ABSTRACT
The present work aims to demonstrate the accuracy of SPH to quantify the wave forces on coastal defences such as
storm return walls. The SPH-based DualSPHysics code is employed to generate and propagate waves and to study the
wave impact on the structure. First, the numerical model has been validated using physical model test data in terms of
water surface elevation and wave forces showing a very good agreement with relatively short time series of random
waves. Once wave loading on coastal structures is proven to be accurately reproduced by DualSPHysics, several
numerical improvements are introduced to increase the numerical efficiency. First, variable resolution was considered to
use higher resolution in a region, while the rest of the numerical domain was solved using coarse refinement. On the
other hand, surface wave and velocity field from deep water can be accurately and more efficiently simulated using a
wave propagation model coupled with DualSPHysics. Validations and computational times of all the presented numerical
approaches are compared and discussed.
Keywords: wave force, SPH, meshless approach, variable resolution, SWASH
1.
INTRODUCTION
The design of coastal defences requires a proper assessment of actions exerted by the sea waves on the structures,
such as wave overtopping, wave forces and pressures. The numerical modelling can represent a useful and
complementary tool to physical tests in order to cope with those aforementioned phenomena. Its main advantage is its
ability to simulate any scenario without building expensive physical models. Numerical models do not suffer from scale
effects, and numerical simulations can provide physical data that can be difficult or even impossible to measure in a real
model. Finally, numerical modelling can reduce the number of physical tests required, which translates to significant
savings in money and time.
The aim of this work is to prove the suitability of the numerical method named Smoothed Particle Hydrodynamics (SPH)
to deal with coastal engineering problems. SPH is a meshless particle method that has been widely used in the
simulation of free-surface flows and fluid-structure interaction and has been developed rapidly during the last decade for
its application in engineering. Traditional computational fluid dynamics techniques such as volume-of-fluid methods (VOF)
have been used to study wave-structure interactions (Kleefsman et al., 2005). However, Eulerian numerical methods,
such as those based on finite element methods, finite difference discretisation and the finite volume technique, require
expensive mesh generation and have severe technical challenges associated with implementing conservative multi phase schemes that can capture the nonlinearities within rapidly changing geometries. Instead of using a mesh, the SPH
method uses a set of interpolation nodes placed arbitrarily within the fluid. This gives several advantages in comparison
to mesh-based methods when simulating nonlinear flow phenomena. Thus, the emergence of meshless schemes has
provided a much needed alternative, including SPH that is becoming now popular.
The DualSPHysics code (Crespo et al., 2015) has been developed to use SPH for real engineering problems.
DualSPHysics is open source and can be freely downloaded from www.dual.sphysics.org. The first rigorous validation of
DualSPHysics code was presented in Crespo et al. (2011) and more details about the implementation of DualSPHysics
can be found in Domínguez et al. (2013a), (2013b). The code was applied to study the run-up on a real armour block
breakwater in Altomare et al. (2014a) and to estimate wave forces comparing with physical model tests in Altomare et al.
(2015). Note that some of the experimental data used in Altomare et al. (2015) will be also employed in this work.
In general, all those problems involve large domains that should be solved with fine resolution, which makes the model
expensive in terms of computational requirements. Meshless particle methods such as SPH are still considered to be
slower than mesh-based methods due to the large number of interactions for each particle at each time step. This is the
reason why SPH codes should be optimized and accelerated as much as possible. DualSPHysics has been designed to
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E-proceedings of the 36th IAHR World Congress,
28 June – 3 July, 2015, The Hague, the Netherlands
be run on multi-core CPUs, which is a relatively common resource, but also on Graphics Processing Units (GPUs). The
GPU technology has experienced a rapid development during the last few years and constitutes a fast and cheap
alternative to classical computation on CPUs. Hence, hardware acceleration, such as the GPUs has reduced the
execution time which is one of the bottlenecks for numerical modelling and for SPH in particular. Simulations with more
than one million particles on a single GPU card exhibit speedups of up to two orders of magnitude compared to a singlecore CPU. The simulations performed in this work were executed using a GPU card installed on a personal computer.
However, coastal problems may involve processes at different spatial and temporal scales, from wave propagation, that
starts from the wave generation offshore, to the impact of overtopping flows on the coastal structures; furthermore to
represent the whole duration of storm events is often required. For all the above mentioned reasons a GPU acceleration
might be not enough. Therefore novel numerical techniques are applied in this work to improve the efficiency of the SPH
model while maintaining the same level of accuracy. The first approach consists of using variable resolution, which is
commonly applied in mesh-based methods. Thus, higher resolution is used in the region of interest (for example, close to
the structure) while the rest of the numerical domain can be solved using coarser resolution. The second is the coupling
of SPH with a model that propagates waves from offshore to coastline and SPH only simulates the wave-structure
interaction.
Physical model tests performed in Ghent University are used to validate DualSPHysics. The aim is to prove the capability
of SPH to quantify the sea wave impact on coastal defences such as storm return walls. Experimental data (water surface
elevation and wave forces) are compared with numerical results. Once this first validation is performed, the mentioned
improvements (variable resolution and coupling with wave propagation models) are applied to show the achieved
numerical accuracy and efficiency. The results of the different numerical approaches are compared and discussed.
2.
SMOOTHED PARTICLE HYDRODYNAMICS MODEL
The Smoothed Particle Hydrodynamics (SPH) method uses particles to represent a fluid and these particles move
according to the governing dynamics. These particles are nodal points where physical quantities (e.g. position, velocity,
density, pressure) are computed as an interpolation of the values of the neighbouring particles. When simulating freesurface flows, the Lagrangian nature of SPH allows the domain to be multiply-connected, with no need of a special
treatment of the surface, making the technique ideal for studying violent free-surface motion. SPH has been used to
describe a variety of free-surface flows (wave propagation over a beach, plunging breakers, impact on structures and
dam breaks).
2.1 SPH formulation
The mathematical fundamental of SPH is based on integral interpolants. Any function F can be computed by the integral
approximation. This function F can be expressed in a discrete form based on the particles. Thus, the approximation of the
function is interpolated at particle a and the summation is performed over all the particles within the region of compact
support of the kernel:
[1]
F( r)   F( r')W ( r  r ', h)dr '
F( ra )   F( rb )W ( ra  rb , h )
b
mb
[2]
b
where the volume associated to the neighbouring particle b is mb/ρb, with m and ρ being the mass and the density,
respectively.
The kernel functions W must fulfil several properties (Monaghan, 1992), such as positivity inside the area of interaction,
compact support, normalization and monotonically decreasing with distance. One option is a quintic kernel (Wendland,
1995) where the weighting function vanishes for inter-particle distances greater than 2h, using q=r/h:
4
q

W  q    D  1    2q  1
2

0 q 2
[3]
In the classical SPH formulation, the Navier-Stokes equations are solved and the fluid is treated as weakly compressible
(e.g. see Gómez-Gesteira et al., 2012). The conservation laws of continuum fluid dynamics, in the form of differential
equations, are transformed into their particle forms by the use of the kernel functions.
The momentum equation proposed by Monaghan (1992) has been used to determine the acceleration of a particle (a) as
the result of the particle interaction with its neighbours (particles b) but using a variational approach :
 P  Pa

d va
  mb  b
 Π ab   a Wab  g
dt
ρ

ρ
b
b
a


[4]
being v velocity, P pressure, ρ density, m mass, g=(0,0,-9.81) ms-2 the gravitational acceleration and Wab the kernel
function that depends on the distance between particle a and b. Πab is the viscous term according to the artificial viscosity
proposed in Monaghan (1992).
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E-proceedings of the 36th IAHR World Congress
28 June – 3 July, 2015, The Hague, the Netherlands
The mass of each particle is constant, so that changes in fluid density are computed by solving the conservation of mass
or continuity equation in SPH form:
dρa
  mb vab   a Wab
dt
b
[5]
In the weakly compressible approach, the fluid is treated as weakly compressible and Tait’s equation of state is used to
determine fluid pressure based on particle density. The compressibility is adjusted so that the speed of sound can be
artificially lowered; this means that the size of time step taken at any one moment (which is determined according to a
Courant condition, based on the currently calculated speed of sound for all particles) can be maintained at a reasonable
value.
The Symplectic time integration algorithm (Leimkuhler, 1996) was used in the present work and a variable time step was
calculated, involving the CFL (Courant-Friedrich-Lewy) condition, the force terms and the viscous diffusion term.
2.2 Variable resolution in SPH
Variable resolution is usually employed in Eulerian numerical schemes in order to improve the efficiency of the simulation
without reducing the accuracy. The splitting algorithm by Vacondio et al. (2013) is implemented in DualSPHysics. The key
idea is to modify dynamically the particle sizes by splitting individual particles to obtain good resolution only where it is
needed. To increase resolution in the area of interest, one particle is split into several daughter particles. A fixed
refinement pattern is used to define the relative position of daughter particles and their masses. The hexagonal
refinement pattern (7 daughter particles) is adopted according to Vacondio et al. (2013) where the masses of the
daughter particles were obtained by minimising the error between the refined and unrefined local density field. The
summation of the masses of the seven daughters is the mass of the original particle and the velocities of the daughter
particles are defined equal to the velocity of the original one. Thus this dynamic particle splitting procedure guarantees
both mass and momentum conservation.
The domain is now discretized using particles with different smoothing length and therefore a variable h (ha≠hb)
implementation of the momentum and mass equations is derived:
d va
1
[6]
Pa  bWab (hb )  Pb   aWab (ha ) g
  mb
dt
ρb  ρa
b
dρa
  mb v ab   bWab (hb )
dt
b
[7]
Vacondio et al. (2013) proved that this new formulation is accurate and computationally efficient.
2.3 SPH coupled with wave propagation model
The main idea is to reduce the SPH domain to the area of interest close to the coast where wave-structure interaction
takes place. Thus, the water surface elevation and the velocity field from deep water can be accurately and more
efficiently simulated with a wave propagation model.
The Simulating WAve till SHore (SWASH) model is a time domain model for simulating non-hydrostatic, free-surface and
rotational flows. Wave propagation models as SWASH have been proven to be able to simulate accurately surface wave
and velocity field from deep water and with satisfactory results both in the open ocean and in nearshore but they are not
suitable to deal with coastal structures with an arbitrary shape such as a parapet since it is a depth integrated model.
The governing equations are the shallow water equations including a non-hydrostatic pressure term. The SWASH model
uses sigma coordinates in the vertical direction and the number of the fluid layer can be changed in the calculation. By
introducing the layers, SWASH can maintain frequency dispersion even in a deep water condition. A full description of the
numerical model, boundary conditions, numerical scheme and applications are given in Zijlema et al. (2011).
Suzuki et al. (2011) demonstrated that this model produces satisfactory results for both wave transformation and wave
overtopping for shallow foreshore topography using one layer in their one-dimensional calculation. This numerical model
is a strong tool for the estimation of wave transformation since it is not demanding in terms of computation resources due
to the depth averaged assumption and parallel computation capability even though it is a time-domain model. Previous
works on the SWASH-DualSPHysics hybridization were presented by Domínguez et al. (2014) and Altomare et al.
(2014b), where the technique was validated against physical model results of wave propagation and run-up over sandy
beaches.
The coupling between DualSPHysics and SWASH has been developed as a one-way coupling at this stage: it means that
the information is passed from SWASH to DualSPHysics and not otherwise. SWASH is in fact run before DualSPHysics
and the output from SWASH in a selected coupling location is passed as input frozen information to DualSPHysics. This
might represents a limitation especially for high reflective structures because it is not possible to treat with re-reflected
waves inside the DualSPHysics domain: in such cases the time necessary to the re-reflective waves to reach again the
examined coastal structure has to be estimated.
The hybridization procedure can be summarised in four steps (Figure 1):

A model domain with SWASH is built to get only incident waves so a flat bottom is added at the location, where the
coupling point is defined, with a sponge layer to absorb all the reflection.

SWASH is executed in a multi-layered approach to provide only incident wave characteristics in the coupling point.
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E-proceedings of the 36th IAHR World Congress,
28 June – 3 July, 2015, The Hague, the Netherlands



A moving boundary (MB) acting as a non-conventional wave generator is defined in DualSPHysics.
The velocity provided by SWASH in the hybridization point is passed to DualSPHysics.
The time history of the displacement in each point or layer of the propagation model is reconstructed starting from
the velocity information and interpolated along the vertical.
The so-calculated movement is passed to the SPH moving particles.
DualSPHysics is executed only in the close area to the coast.


SWASH domain
SWASH output: Total wave = incident wave
Actual bathymetry
SPH domain
Flat bottom + sponge layer
Coupling point
Figure 1. Sketch of the model domain and hybridization point.
3.
EXPERIMENTAL CASE
The same physical test used in Altomare et al. (2015) is also presented here to validate DualSPHysics. The aim of the
experimental campaign was to assess the overtopping flow rates and wave loadings on the dikes and storm return walls
in the Blankenberge Marina (Belgium). In this physical test the forces are exerted by waves that approach the existing
dike with a crest berm on a parapet wall. The dimensions of the physical test are shown in Figure 2. A 1:5 model scale
was used for the physical model tests in this case. The simulated wave conditions use a significant wave height
Hm0=0.101m and peak period Tp=2.683s.
Wavemaker
WG
Structure
2.15 m
0.64 m
3.2 m
0.8 m
Berm
23.5 m
Figure 2. Numerical model setup of the Blankenberge Marina test cases with a parapet wall.
During the experiment, wave elevation surface and force exerted onto the parapet wall were recorded. Figure 2 shows the
location of wave gauge WG (x=3.63 m) at the beginning of the berm and the parapet wall (represented in red) where
forces will also be numerically computed. Those experimental signals are used to characterize the wave propagation and
wave loading.
4.
VALIDATION
DualSPHysics is now used to simulate the experimental case. The numerical flume is built to resemble the physical flume
at Ghent University in layout and dimensions. Note that the DualSPHysics-piston will move accordingly to the theoretical
piston displacement as in the physical model tests. The SPH simulation is performed in two dimensions.
A total physical time of 55 seconds are simulated with DualSPHysics. The initial interparticle distance is dp=0.01m which
leads to 135,883 particles in the numerical domain.
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E-proceedings of the 36th IAHR World Congress
28 June – 3 July, 2015, The Hague, the Netherlands
Different instants of the simulation are depicted in Figure 3.
Figure 3. Different instants of the simulation with DualSPHysics. The colour of the particles represents the velocity value.
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E-proceedings of the 36th IAHR World Congress,
28 June – 3 July, 2015, The Hague, the Netherlands
Numerical wave surface elevation is compared with experimental signal (WG) in Figure 4. The agreement proves that
waves are numerically generated in an accurate way using DualSPHysics.
0.10
EXP
SPH
0.08
0.06
eta (m)
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
10
15
20
25
30
35
40
45
time (s)
Figure 4. Comparison between numerical and experimental water surface elevation (eta).
On the other hand, the times series of the experimental wave forces and the numerical ones obtained with DualSPHysics
are plotted in Figure 5. The hydrostatic force corresponding to the initial still water level was removed. Results prove the
accuracy of DualSPHysics to estimate wave loadings.
200
EXP
SPH
175
150
Force (N)
125
100
75
50
25
0
-25
-50
20
25
30
35
40
45
50
55
time (s)
Figure 5. Comparison between numerical and experimental wave forces.
5.
NUMERICAL EFFICIENCY
DualSPHysics has been validated using physical data test so the capability of the code has already been shown. In this
section, the two numerical improvements (splitting algorithm and coupling with SWASH) are applied in order to improve
the efficiency of the simulation but maintaining the same reliability. Wave forces obtained with new numerical approaches
are computed again and computational runtimes are also measured to discuss about the achieved speedups.
The two numerical improvements (described in section 2.2 and 2.3) are represented in Figure 6 for a better
understanding. The top snapshot corresponds to the instant time=30s using the former DualSPHysics code. As
mentioned before an initial particle size of 1cm was used in the simulation.
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E-proceedings of the 36th IAHR World Congress
28 June – 3 July, 2015, The Hague, the Netherlands
SPH
dp = 0.01 m
SPHSPLITTING
dp ≈ 0.01 m
dp = 0.02 m
SPHCOUPLING
dp = 0.01 m
Figure 6. Same instant of the simulation (time=30s) using original DualSPHysics (top), splitting (middle) and coupling (bottom).
The frame in the middle of Figure 6 corresponds to the simulation where splitting procedure was applied. In this case a
coarse resolution is used for the entire domain (dp=0.02m) which means that around 4 times less particles are initially
created comparing with dp=0.01m. However, higher resolution is desirable in the area close to the dike and the parapet
wall, therefore splitting is applied and particles within that area are split into 7 daughters with new values of masses and
smoothing lengths and with a new dp=0.0113m (dp≈0.01m). The different size of the particles and the area where
splitting was applied can be observed in Figure 6.
The numerical configuration of the coupling can be observed in the bottom of Figure 6. The coupling point is located at
x=10.5m and the DualSPHysics domain was reduced. SWASH is first executed to simulate incident waves. The execution
time of SWASH is negligible comparing with SPH runtimes so it will not be considered later in the efficiency discussion.
The velocity provided by SWASH at the coupling point is passed as input to define the motion of the SPH-piston. Hence,
the particles of the piston are initially created at x=10.5m (the coupling point) and they move to create the same waves
that were first propagated with SWASH. The length of the SPH domain is here reduced to the 60% of the original one.
Figure 7 shows a more detailed view of the SPH particles at time=30s with the three numerical approaches (SPH is the
basic implementation; SPHSPLITTING is the case with variable resolution, and SPHCOUPLING when using the hybrid model). It
is worth noting that the same velocity field is achieved for the different options.
SPH
SPHSPLITTING
SPHCOUPLING
Figure 7. Same instant of the simulation (time=30s) with a zoom close to the dike and the parapet wall. Colour represents velocity values
of the particles.
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E-proceedings of the 36th IAHR World Congress,
28 June – 3 July, 2015, The Hague, the Netherlands
Now, the wave force exerted onto the parapet wall is also computed using SPH SPLITTING and SPHCOUPLING and compared
with basic SPH implementation. The times series of the numerical wave forces computed with original DualSPHysics and
with the variable resolution approach are plotted in Figure 8. A very good agreement is observed showing that the
splitting procedure provides the same accuracy as the previous SPH results.
200
SPH
175
SPH_SPLITTING
150
Force (N)
125
100
75
50
25
0
-25
-50
20
25
30
35
40
45
50
55
time (s)
Figure 8. Comparison between numerical wave forces obtained with SPH and SPHSPLITTING.
Figure 9 plots the wave force computed with only DualSPHysics modelling and the numerical force computed with the
coupling SWASH+DualSPHysics. Differences between the signals are only observed after 40 seconds when re-reflected
waves became important. That is the time that the first incoming waves impact against the wall, are reflected in the
direction of the SPH piston, impacts also with the piston and travels in the initial incoming direction to hit again the wall.
This re-reflection is a problem that needs to be solved in the coupling model in order to simulate longer tests.
200
SPH
175
SPH_COUPLING
150
Force (N)
125
100
75
50
25
0
-25
-50
20
25
30
35
40
45
50
55
time (s)
Figure 9. Comparison between numerical wave forces obtained with SPH and SPHCOUPLING.
The numerical comparison is summarised in Figure 10 where the times series of the three numerical signals are
compared with the experimental wave forces. All numerical results are in agreement with the physical test. Hence, the
proposed numerical improvements preserve the same accuracy to estimate wave loadings.
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E-proceedings of the 36th IAHR World Congress
28 June – 3 July, 2015, The Hague, the Netherlands
200
EXP
SPH
SPH_SPLITTING
SPH_COUPLING
175
150
Force (N)
125
100
75
50
25
0
-25
-50
20
25
30
35
40
45
50
55
time (s)
Figure 10. Comparison between experimental and numerical wave forces obtained with SPH, SPHSPLITTING and SPHCOUPLING.
.
Finally, numerical efficiency is analysed to determine the performance of the original DualSPHysics code and the
achieved speedups with the splitting procedure and the coupling model. The case under study is simulated both on a
CPU and a GPU to analyse the performance of DualSPHysics code. The simulation took 42.40 minutes using a GPU
(GeForce GTX Titan) and more than 31 hours using multi-core CPU (4 cores of Intel Xeon E5-2640). Therefore the Kepler
GPU card is 44 times more efficient that the CPU machine. The achieved performance can be compared to a large
cluster machines of CPUs. The performance of the models using the proposed numerical improvements (splitting and
coupling) are shown in Table 1. The execution time of the SWASH model
Table 1. Numerical performance (number of particles, computational times and speedups) with SPH, SPHSPLITTING and SPHCOUPLING.
Interparticle distance
Number of particles
Simulated steps
Simulation runtime
Speedup (vs SPH)
SPH
SPHSPLITTING
SPHCOUPLING
0.01m
135,883
275,165
42.40 min
1.00x
0.02m
35,670 to 44,484
245,793
18.77 min
2.26x
0.01m
80,594
248,994
26.11 min
1.62x
The initial number of particles in the basic SPH simulation of 135k is reduced to 35k using variable resolution (but 8,800
new daughter particles are created) and to 80k in the SPH domain of the coupling model. Both approaches accelerate the
simulation; splitting code took only the 44.27% of the time of first SPH simulation and the coupling took the 61.56%. The
speedups can be increased if an area closer to the wall is defined with SPH SPLITTING or if the coupling point moves further
with SPHCOUPLING.
6.
CONCLUSIONS
The SPH-based DualSPHysics code is employed to study the wave impact onto the coastal structures. Physical model
tests performed in Ghent University are used to validate DualSPHysics. Experimental data such as water surface
elevation and wave forces are compared with numerical results. DualSPHysics is proven to quantify the sea wave impact
on coastal defences such as a vertical wall with parapet.
Two novel numerical techniques are applied in this work to improve the efficiency of the SPH model while maintaining the
same level of accuracy; variable resolution with higher resolution in the region of interest and the coupling of SPH with a
the wave propagation model SWASH that propagates waves from offshore to coastline and SPH only simulates the
wave-structure interaction.
Once the code has been validated and the accuracy of the numerical results has been assessed, the efficiency of using a
GPU was also shown accelerating 44 times the CPU execution.
The capability of the particle refinement procedure was demonstrated reducing the computational time by a factor of 2.26
in comparison with the SPH simulation with uniform resolution and of the same accuracy. The use of the coupling model
SWASH+DualSPHysics reduced the domain length and also the computational time to 60%. Both numerical
improvements accelerate the simulation showing the same degree of accuracy when computing wave forces against the
wall.
Finally DualSPHysics model can be proposed as complementary tool to physical model experiments for a preliminary
design of the coastal defences.
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E-proceedings of the 36th IAHR World Congress,
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ACKNOWLEDGMENTS
The authors acknowledge the Department of Civil Engineering of Ghent University to have provided all the data regarding
the designs and analysis carried out for the harbours of Blankenberge. This work was partially financed by Xunta de
Galicia under project Programa de Consolidación e Estructuración de Unidades de Investigación Competitivas (Grupos
de Referencia Competitiva) funded by European Regional Development Fund (FEDER), and by Ministerio de Economía y
Competitividad under Project BIA2012-38676-C03-03.
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