A compressible fluid particle model based on the Voronoi ...

A <strong>compressible</strong> <strong>fluid</strong> <strong>particle</strong> <strong>model</strong>
<strong>based</strong> on the Voronoi tessellation:
Comparison with Smoothed
Dissipative Particle Dynamics
Mar Serrano
Pep Español
Departamento de Física Fundamental, UNED, Spain

Outline
1. Motivacion: Lagrangian techniques
2. What is a mesoscopic <strong>fluid</strong> <strong>particle</strong>?
Volume definitions: “soft”, “sharp”
3. Building the <strong>model</strong>:
- Voronoi Reversible <strong>fluid</strong>
- Voronoi Irreversible dynamics: Structure
- Fluctuations
4. Comparison performance SDPD/Voronoi
5. General Conclusions

Outline
1. Motivacion: Lagrangian techniques
2. What is a mesoscopic <strong>fluid</strong> <strong>particle</strong>?
Volume definitions: “soft”, “sharp”
3. Building the <strong>model</strong>:
- Voronoi Reversible <strong>fluid</strong>
- Voronoi Irreversible dynamics: Structure
- Fluctuations
4. Comparison performance SDPD/Voronoi
5. General Conclusions

Motivation
Complex <strong>fluid</strong>s
Coupling between the microstructure
and the hydrodynamic variables of the
<strong>fluid</strong>
The thermal fluctuations are
the responsible of the diffusive
processes
Lagrangian methods are adaptative to the complex boundary conditions
involved in complex <strong>fluid</strong>s dynamics
The main objective of this work is the formulation of Lagrangian
simulation techniques and hydrodynamic <strong>model</strong>s that are able to
represent simple and complex <strong>fluid</strong>s in the mesoscopic scale consistent
with Thermodynamics

SPH Discretization of continuum hydrodynamics in
Lucy, Astron. J. 1977
Monaghan ,An. Rev. terms of <strong>particle</strong>s transforming PDE into ODE
Astron. Astrophys. 1992
DPD Mesoscopic <strong>model</strong> of interacting <strong>fluid</strong> <strong>particle</strong>s
via conservative, dissipative, random forces
Hoogerbrugge Koelman,
Europhys. Lett. 1992
Español, Bonet
Europhys.Lett.
1997
SDPD
SDPD
VORONOI With the help of the GENERIC formalism we
Serrano, Español,
Phys. Rev. E 2001
Lagrangian techniques
How to include thermal fluctuations?
What is the volume? Different implementations?
Conservative forces? Thermodynamic behaviour?
Relation between transport coeff. and <strong>model</strong> parameters?
SDPD technique considered as <strong>fluid</strong> <strong>particle</strong> <strong>model</strong>s with “soft”volumes
Español, Revenga
Phys Rev E 2003
We extract the best: -fluctuations from DPD
-connection to NS from SPH
build a new mesoscopic <strong>fluid</strong> <strong>particle</strong> <strong>model</strong>
with “sharp” volumes. This <strong>model</strong> represents
the discrete Lagrangian fluctuating
hydrodynamics

Outline
1. Motivacion: Lagrangian techniques
2. What is a mesoscopic <strong>fluid</strong> <strong>particle</strong>?
Volume definitions: “soft”, “sharp”
3. Building the <strong>model</strong>:
- Voronoi Reversible <strong>fluid</strong>
- Voronoi Irreversible dynamics: Structure
- Fluctuations
4. Comparison performance SDPD/Voronoi
5. General Conclusions

Mesoscopic <strong>fluid</strong> <strong>particle</strong>
Microscopic Level
N
atoms MD
{ r } , { p } , { }
k k 0 m
Point <strong>particle</strong>s
dri() t pi()
t
=
dt m0
d () V(
{} i )
i t ∂
=−∑
dt ∂
r
p
r
j
j
Representing the
same physics
Course Graining
Representation M ≤ N
Mesoscopic Level
M Fluid <strong>particle</strong>s DPD
m, ε , v
{ }
{ } , { } V R
m
i
i i
Extensive <strong>particle</strong>s
i i i
Internal Energy
Volumen
A <strong>fluid</strong> <strong>particle</strong> is a small
thermodynamic subsystem
in local equilibrium
dRi( t)
dt
dVi( t)
dt
d ε i ( t)
dt
=
=
=
?
?
?

Different volume definitions
“Soft” volume DPD, SPH, SDPD
h
vi
i
V
e ij
j
v j
Mass Density
Volume of the
<strong>fluid</strong> <strong>particle</strong>
Wrh (, )
0 ≠ ∑ v i
i
( )
ρ = ∑mW R −R
i j i j
j
v
( { R } )
i k
=
Kernel
ρ
h
r
i
m
i
( { Rk}
)
“Sharp” volume
Volume of the
<strong>fluid</strong> <strong>particle</strong>
Ri
( { } ) v0
V
VORONOI
Voronoi
characteristic
function
∆( r−Ri) χ i(
r)
=
∆ r−R ∑
v R =∫ χ ( r) dr
i i
0 = ∑ v i
i
i
j
( ) j
r
−
2
σ
∆ ( r) =
e
2
2

Outline
1. Motivacion: Lagrangian techniques
2. What is a mesoscopic <strong>fluid</strong> <strong>particle</strong>?
Volume definitions: “soft”, “sharp”
3. Building the <strong>model</strong>:
- Voronoi Reversible <strong>fluid</strong>
- Voronoi Irreversible dynamics: Structure
- Fluctuations
4. Comparison performance SDPD/Voronoi
5. General Conclusions

Energy
Total internal energy
Postulate:
Reversible dynamics
ε = =
i
i i
In order to preserve
the energy...
2
m i
E( x) = + ε
2
ε ε
∑ V
Lagrangian movement
of the <strong>fluid</strong> <strong>particle</strong>
of constant
mass and entropy
m
( { k}
)
∑ ∑ R
i
( mS , , v )
i
i
dV
dt
i
Definition of the equation
of state of the <strong>fluid</strong>, ideal
gas, van der Waals...
dRi
= Vi
dt
dmi
= 0
dt
dSi
= 0
dt
∂ε
∂ε
j
=− =−
∂ ∂ ∑ R R
i j i

m
i
dVi
dt
This dynamics can be understood as a Molecular Dynamics,
where the internal energy of the system plays the role of a
i
=
∑
j
many body interaction effective potential.
U
∂v
j
( { Rk}
)
∂R
( { k} ) ε ε i { k}
ε R = = =
R
ef i
i i
i
{ }
p
j
∑ ∑
mS ,
v
( )
( )
ε ( ( ) ) ( { } ) ( { } )
j v j Rk vj Rk
v
∑
j Rk
=−∑
∂ε
∂ ∂ ∂
j
=
∂R ∂v
∂R∂R Thevolumeisa
function of the
<strong>particle</strong>’s positions...
j j
i
j
m , S
i i
i
Thermodynamic pressure
of the <strong>fluid</strong> <strong>particle</strong>
dvi ∂vdR i j ∂vi
= i. = i.
dt ∂R dt ∂R
∑ ∑ V
∀ j j ∀ j j
p
Lagrangian <strong>fluid</strong> <strong>particle</strong>
j
j

Reversible <strong>fluid</strong> <strong>particle</strong> <strong>model</strong>
dRi
() t = Vi
dt
dv
dt
i
dV
i m i
dt
dS
dt
i
=
∑
j
=
() t = 0
∂ v
∂R
∑
j
i
j
iV
∂ v
∂R
j
i
j
p
j
Euler equations (Inviscid <strong>fluid</strong>)
dρ(,) r t
dv( r=−
, t)
ρ(,)
r t ∇⋅v(,)
r t
dt = v( r,
t)
∇ ⋅v(
r,
t)
dt
dgr (,) t
=−gr (,) t ∇ ∇⋅v(,) rt − ∇
∇P(,)
rt
dtdv
( r,
t)
m = −v(
r,
t)
∇P(
r,
t)
ds(,) r t dt
=−s(,) r t ∇⋅v(,)
r t
dt
dS( r,
t)
dt
Discrete variables ... In continuum Continuum extensive fields fields
Discrete version
of
continuum operators
.
=
Specific Volume, momentum and Entropy
ANALOGY
Gradient
Divergence
−
0
∂v
f ≈ v∇f ∑ R
j
j
j ∂ i
∂v
∑
i f j
j ∂Rj
⋅ ≈v∇⋅
f

Voronoi
Ri
Aij
Rij
Rj
cij
eij
lim
σ →0
R Distance between two neighbouring <strong>particle</strong>s
ij
eij Unit vector in the line joining this two <strong>particle</strong>s
dr
Aij = R ∫ ij
χ ( ) ( )
v 2 i r χ j r
0 σ
Rij dr
⎛ R i + R j ⎞
cij = ∫ χ ( ) ( )
v 2 i r χ j r ⎜r-⎟ A 0
ij
σ
⎝ 2 ⎠
A ij
Area joining two neighbouring cells
c ij Parallel vector to the face ij giving the position
of the face center of mass with respect to the
medium point defined by these two cells
∂v
⎛ c i
ij e ⎞ ⎛ ij
cij
e ⎞ ij
=− Aij⎜ + ⎟+ δ ∑ ij Aij⎜ − ⎟
∂R ⎜
j Rij2
⎟ ⎜
i≠j Rij
2 ⎟
⎝ ⎠ ⎝ ⎠

Voronoi
Final Reversible
equations
dRi () t = Vi
dt
,
dmi = 0
dt
,
dSi
= 0
dt
dV
⎛ c i
ij e ⎞ ij
m i = ∑ Aij⎜
− ⎟(
Pi − Pj)
dt ⎜
j≠i Rij
2 ⎟
⎝ ⎠
The analogy is a quantitative relationship, because one can demonstrate
1 ∂v
1 ⎛ ⎞
j
cij eij
∇P
( R = ∑
i) ≈− ∑ P(
R ) Aij
⎜ − ⎟βi.
j
Rij = β
v
v ⎜
i j Rij
2 ⎟
i j ∂Ri
⎝ ⎠
For linear pressures P() r = [ α + βi. r]
( )
The result is exact for
arbitrary meshes !!
Linear consistency supports our idea that
<strong>fluid</strong> pressure forces are a discrete pressure gradient
Serrano, Español, Zúñiga, J. Statistical Physics, 2005

Outline
1. Motivacion: Lagrangian techniques
2. What is a mesoscopic <strong>fluid</strong> <strong>particle</strong>?
Volume definitions: “soft”, “sharp”
3. Building the <strong>model</strong>:
- Voronoi Reversible <strong>fluid</strong>
- Voronoi Irreversible dynamics: Structure
- Fluctuations
4. Comparison performance SDPD/Voronoi
5. General Conclusions

How can we incorporate in the <strong>model</strong> the dissipative dynamics in order to
keep as close as possible to the continuum Navier Stokes equations?
dv ∂v
= ∑ dt ∂R
i i
j j
dV
∂v
m = p +
i
j
i
dt j ∂Ri
j
dS
i Ti dt
dR
dt
i
Irreversible dynamics
=
= 0 +
V
i
-A e
∑
?
i.
V
j
?
We requiere expressions for the second derivatives,
ij
ij
dv
= v∇⋅v
dt
dv
2 ⎛ d −2
⎞
m =− v∇P+ ηv ∇ ∇ v+ ⎜η + ζ⎟v
∇(∇
∇(∇⋅v)
dt
⎝ d ⎠
dS 2
T = κv∇∇ ∇ ∇T+ 2 ηv ∇ ∇v: ∇ ∇ v+ ζv(∇
(∇
(∇⋅v)
dt
⎣ ⎦ ∑ v
⎣ ⎦
⎡ f⎤ ⎣
∇∇
∇∇
⎦
≈− ∑
f
11 ⎡ f ⎤ =− A ⎡ ij ij f ⎤
⎣
=− ∇i
⎦ ∑ ijeij e i
v ⎣ ⎦
Divergence
⎡ theorem ∇∇ ∇∇f⎤ A ⎡ ∇
∇f⎤
fij
i i
i j i j
ij ij
i
1
vi j
Aij
ee ij ij
Rij
ij
≈
2
R
ij
e
ij

Spatial derivatives SDPD/Voronoi
“Soft” volume SDPD
rc
vi
i
e ij
( { } )
∂v
∂ R
j
v j
1
∑≠
i j i
Volume
[ ] [ ( ) ] µ ν
'
µ ν µ
∇ ∇ f i=
−∑
v jWij
d + 2 eij
eij
−δ
ij fij
j
“Sharp” volume
Ri
VORONOI
Volume
v R =∫ χ ( r) dr
( { } ) v0
i i
∂v
⎛ c i
ij e ⎞ ⎛ ij
cij
e ⎞ ij
=− Aij⎜ + + δ ij Aij −
⎜
⎟
j Rij2
⎟ ∑ ⎜ ⎟
∂R ⎜
i≠j Rij
2 ⎟
⎝ ⎠ ⎝ ⎠
1 ⎛ cij
e ⎞ ij
∇f i=
− ∑ A ⎜
ij − ⎟
v ⎜
i j i R ⎟
≠ ⎝ ij 2 ⎠
1 Aij
⎡ f ⎤
⎣
∇∇
∇∇
⎦
≈− ∑ ee f
i v R
[ ] ( 2 2 ) '
∇ f i=
− vi
fi
+ v j f j Wije
[ ] ij
fij
v
1
v
i
⎛
⎜W
⎝
( { R } )
k
=
∑
j
W
i Rk '
'
= ij eij
−δ
ij∑W
2
jk
j vi
k ≠i
r
c
1
( R − R )
Español, Revenga, Phys. Rev. E, 2003
i
e
jk
⎞
⎟
⎠
j
i j ij
ij ij ij
Serrano, Physica A, 2005
i

Complete Voronoi
dynamics
dRi
() t = Vi
dt
dV
⎛ cij e ⎞ ij Aij
⎡ i
⎛ ⎛ 2 ⎞ ⎞ ⎤
m i = ∑Aij⎜ − ⎟Pij + − η ij + η 1 − + ζ ( ij⋅ ij) ij
dt ⎜
j i Rij 2 ⎟ ∑ ⎢ v
≠
j R
⎜ ⎜ ⎟
ij
d
⎟ e v e ⎥
⎝ ⎠ ⎣ ⎝ ⎝ ⎠ ⎠ ⎦
dS
A
i
ij ⎡ 1 2 1⎛ ⎛ 2 ⎞ ⎞
2⎤
= 0 + ∑ ⎢− κ Tij
+ η vij + η⎜1 − ⎟+
ζ ( ij ⋅ ij ) ⎥
dt j Rij 2 2
⎜
d
⎟ e v
⎣ ⎝ ⎝ ⎠ ⎠ ⎦
REVERSIBLE IRREVERSIBLE

2
m i
Ex ( ) = ∑ + ε ( mi, Si, vi)
i 2
V
POTENTIALS
∂Ex
( )
Sx () =∑Si
i ∂
⎛ ⎞
∂x
∂x
⎜ ⎟
d
⎛ ⎞
⎜
R
i ⎟ ⎛ ⎞⎜
⎟ ⎛ ⎞⎛⎞
⎜ dt ⎟ ⎜ ⎟ vk
δij
0 ⎜
∂
− p ⎟ ⎜ ⎟⎜⎟
k 0 0
d
⎜
0 1
⎜ ⎟ ⎟ ∑ ⎜
0
⎟⎜0
V
⎜ k ∂R
i
j ⎟
⎟
VV SV
⎜m= ⎜ δ 0 ⎟ ⎜ i ⎟ ∑ −1
ij
0 ⎜ ⎟+
∑ 0 M ij M ⎟ ij ⎜0⎟ ⎜ dt ⎟ j ⎜ ⎟ mj
j ⎜ VS SS ⎟
0
⎜ V ⎟ j
⎜ ⎟
⎜
0 M ij M ij 1
dS ⎟ ⎜ 0 0 ⎟⎜
i
T ⎟ ⎜ ⎟⎜⎟
⎜ ⎟ j
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
dt
⎝ ⎠⎜ ⎟ ⎝ ⎜
⎠⎝⎟
⎜ ⎟
⎜ ⎟ ⎝
⎠
⎠
⎝ ⎠
T
Lx ( ) =− Lx ( )
M( x) = M ( x)
≥0
OPERATORS
S( x)
A SV ij ⎡ ⎛ ⎛d−2⎞ ⎞ ⎤
M ij = ⎢− η vij + ( ij ij ) ij
R
⎜η⎜ ⎟+
ζ
ij
d
⎟ e ⋅v
e ⎥
∂S(
x)
⎣ ⎝ ⎝ ⎠ ⎠ ∂E(
x⎦)
L(
x)
⋅ = 0 PROPERTIES M ( x)
⋅ = 0
∂x
A VS ij ⎡ ⎛ ⎛d−2⎞ ⎞ ∂x
T T ⎤
M ij = ⎢− η v ij + η⎜ ⎟+
ζ ( ij ⋅ ij ) ij ⎥
R
⎜
ij
d
⎟ e v e
The <strong>model</strong> has a STRUCTURE ⎣ known ⎝ ⎝ as GENERIC
⎠ ⎠ ⎦
dEx
( )
Assure the
A SS
ij A ⎡ ij η 2 ⎛η( d − 2)
ζ ⎞ ⎤ 2
M ij =−
First
κ
Law
= 0
thermodynamical
Tij
+ ⎢ vdt ij + ⎜ Grmela + & ⎟Öttinger
( eij
⋅vij
) ⎥
Rij Rij
⎢⎣2 ⎝ 2d2 consistency
dSx
( ) Phys. Rev. ⎠ E 1997
⎥
Second Law
⎦
0
dt ≥

Outline
1. Motivacion: Lagrangian techniques
2. What is a mesoscopic <strong>fluid</strong> <strong>particle</strong>?
Volume definitions: “soft”, “sharp”
3. Building the <strong>model</strong>:
- Voronoi Reversible <strong>fluid</strong>
- Voronoi Irreversible dynamics: Structure
- Fluctuations
4. Comparison performance SDPD/Voronoi
5. General Conclusions

Including thermal fluctuations
⎛ ∂E(
x)
∂S( x)
∂ ⎞
dx = ⎜ L ( x)
⋅ + M ( x) ⋅ + kB⋅M ( x)
⎟dt+
dx
⎝ ∂x
∂x
∂x ⎠
T
Fluctuation-disipation theorem dxd x = 2 k M ( x)
dt
Energy conservation
∂Ex
( )
⋅ dx
= 0
∂x
Voronoi Model with fluctuations: Spontaneous
fluctuations
dx = ( ,0, dV, 0, , ) local stresses
i dS
i
and random heat
Mimic the irreversible
flux
processes:
heat conduction and shear and bulk dissipation
1 2
d
T
1/2
Pi
= ∑(
BdW ij ijeij + ZdW ij ij eij
)
⎛ A ⎞ ⎛A⎞ ij
ij
j
Bij ∝ ⎜ kBTiζZij ∝⎜ kBTiη ⎜
⎟
⎟
R ⎟ ⎜R⎟ ⎝ ij ⎠
1 1
⎝ ij ⎠
1 2
dS
T
i = ∑HdV ij ij − ∑( Bijeij ⋅ vijdWij+ Zijeij ⋅vijdWij
)
1/2
Ti
j 2T
⎛A⎞ i j
ij
Hij ∝ ⎜ kBTT i jκ
⎜
⎟
R ⎟
⎝ ij ⎠
B
1/2

Outline
1. Motivacion: Lagrangian techniques
2. What is a mesoscopic <strong>fluid</strong> <strong>particle</strong>?
Volume definitions: “soft”, “sharp”
3. Building the <strong>model</strong>:
- Voronoi Reversible <strong>fluid</strong>
- Voronoi Irreversible dynamics: Structure
- Fluctuations
4. Comparison performance SDPD/Voronoi
5. General Conclusions

y
Comparing the performance of
SDPD/Voronoi in a shear flow
Serrano, Physica A, 2005
External forcing:
x
Theoretical
Stationary
profile:
(f x,f y)=(G0sin( ky),0)
(v ,v )=(v 0 sin( ky),0)
s s ρG
x y 0
k
Method to meassure the effective
shear viscosity of the <strong>fluid</strong>
Check:
η
2
η
-Precision
- Spatial resolution
-Complexityofthealgorithms

The mesh becomes
disordered
Accuracy
SDPD hexagonal ordered configuration
SDPD square ordered configuration
SDPD disordered configuration
Voronoi ordered
Input theory

Stationary velocity profiles
Disordered
Voronoi SDPD
Ordered
Ordered
Disordered

Relative error in the stationary
velocity profile as a function of
the linear resolution
Both methods have a second order
spatial error
C.p.u. time as a function of
the total number of <strong>fluid</strong>
<strong>particle</strong>s
SDPD
SDPD
VORONOI VORONOI
SDPD Verlet list
Voronoi vertex recombination
The complexity of both
algorithms is linear

Outline
1. Motivacion: Lagrangian techniques
2. What is a mesoscopic <strong>fluid</strong> <strong>particle</strong>?
Volume definitions: “soft”, “sharp”
3. Building the <strong>model</strong>:
- Voronoi Reversible <strong>fluid</strong>
- Voronoi Irreversible dynamics: Structure
- Fluctuations
4. Comparison performance SDPD/Voronoi
5. General Conclusions

DPD-SPH/VORONOI: Advantages/disadvantages
Both <strong>model</strong>s are thermodynamic consistent....
-Conceptually: Voronoi is elegant. Parameter free compared to SPH/DPD.
-Computational efficiency: Both <strong>model</strong>s are comparable, because the
geometrical calculus in 2D Voronoi is compensated for in SDPD by the
Verlet list updates and a bigger number of interactions. SDPD is trivial to
extend 3D while the Voronoi implementation is more involved.
-The accuracy in thephysicalresultsofthetechniques: Theeffective
viscosities depend on the regularity of the mesh. SDPD gives better
results than Voronoi for disordered meshes because uses more neighbors’
information. In both techniques the accuracy can be mantained by
performing a retessellation or remeshing into ordered configurations.

Conclusions
-The discrete <strong>model</strong> formulated on physical grounds can be interpreted
as a Lagrangian discretization of the continuum Navier Stokes
equations that preserves First and Second law of thermodynamics.
Any physical reasonable <strong>model</strong> should be compatible with the
thermodynamic laws.
-In order to be able to include thermal fluctuations in any <strong>model</strong> we have
to be sure that the dissipative matrix M(x) of the irreversible dynamics is
positive definite.
-In order to be able to exploit the Lagrangian power of the techniques for
applications to complex <strong>fluid</strong>s some more theoretical work has to be
done to get good discrete versions of the second derivative operators
in non uniform grids.

Complex Boundary conditions
Modelling complex geometries in Voronoi <strong>fluid</strong> is extremely easy.
Flat wall boundary
Not upgrading the properties and
variables of the wall <strong>particle</strong>, but
including it in the force of the
<strong>fluid</strong>
WALL
FLUID
Modelling suspended objects
Jointing the monomers or <strong>fluid</strong>
<strong>particle</strong>s with springs or
including differences in
constitutive equations
and transport coefficients
POLYMERIC
CHAIN
MICELLE

2d implementation
Structure of the code is similar to Molecular Dynamic Code in PBC: F77
Initialization <strong>particle</strong>s:
independent variables: position, velocities, masses, entropies
dependent variables: temperature, internal energy, pressure, volume
For simplicity regular initial configuration for building up two matrixes
TOPOLOGY: conectivities among <strong>particle</strong>s
GEOMETRY: geometrical quantities involved in the tesellation
Moving or advancing the dynamics of the independent variables:
Explicit Time integration schemes: RK4, Trotter, Euler...
Updating the dependent variables.
The volume is a function of the <strong>particle</strong>s’ position that change.
TOPOLOGY CHECK: RECOMBINATION or new calculus
GEOMETRY CALCULUS
Meassuring the physical observables

TOPOLOGY INFORMATION
Eulertheoremfor2D systemin PBC: M-F+V=0
In Voronoi
F/V=3/2
V=2M Vertex Matrix
F=3M
Faces Matrix
We clasify (label) all vertex and faces and conectivities
i j
GEOMETRICAL INFORMATION
Fil v2 Fjl
With the topology matrixes we compute vertex coordinates, areas
(lenght of the poligonal faces) vectors c and e, volumes...
Of course, they must satisfy total volume conservation. l
a\b b=1 b=2 b=3 b=4 b=5 b=6 b=7 b=8 b=9 b=10 a\b b=1 b=2 b=3 b=4 b=5 b=6
Fij i j k l V1 V2 Fik Fil Fjk Fjl V1 i j k Fij Fik Fjk
TOPOLOGICAL Fik i j l RECOMBINATION
... V1 ... Fik Fij Fj. Fj.
Fil i l j ... V2 ... Fil Fij Flj Fl.
V2 i j l Fij
....
Fil Fjl
Fjl j l i ... V2 ... Fji Fj. Fli Fl.
Because the number of faces/vertexes in PBC Voronoi tesselation are
....
constant, we can compute the connectivities of the mesh at the same
time the positions (Voronoi centers) follow the Lagrangian dynamics. The
essential process in the topological change is given by vertex recombination.
Fik
v1
Fij
k
Fjk

OLD TOPOLOGY
Fik
Fil
v1
i j
v2
Time 1
k
Fij
l
Fjk
Fjl
Fik
Fil
CHECK TOPOLOGY for each FACE
Use old topologic information. CIRCUMCIRCLE CRITERIA
i
OK!!
v1
v2
k
l
Time 2
Fjk
j
Fij
Fjl
WRONG <strong>particle</strong> k
inside!!
Fik
Fil
i
k
Fkl
v1 v2
l
Fjk
Fjl
WRONG <strong>particle</strong> l inside!!
Time 3
CHANGE
MFACES AND MVERTEX!!
j

Example of recombination algorithm: shear alternative forcong is applied

Properties of the non dissipative Voronoi <strong>model</strong>
These equations preserve energy conservation
The Voronoi volume is invariant under
- translations of the <strong>particle</strong>s: Conserves tot. Lin. Mom. P= ∑ mVi
- rotations of the <strong>particle</strong>s: Conserves angular mom.
These equations can be considered as a Lagrangian discrete
version of the Euler Equations (inviscid Navier Stokes) (if the
velocity field is smoothed)
Other symmetries of the <strong>model</strong>: Invariance of the equations
under spatial traslations, temporal, Galilean transformations,
rotations,...
Just like the Euler equations!!
...Can our inviscid <strong>model</strong> predict or say something about
the statistical properties of stationary turbulence?
∑
L= R × mV
i
i
i i

Euler <strong>fluid</strong>: Acceleration Distributions
In fully developed turbulence, big accelerations often appear in
comparison to the Gaussian distribution with the same variance.
Simulation Simulation result result
We have perfomed MD simulations with
pair potentials like Lennard Jones, and
this long tails events do not appear!
Our <strong>model</strong> captures some features
observed in experiments... But we should
include viscosity
A. La Porta, et al. Nature 409,1017, 2001
Experimental
Experimental
result result

SDPD Overlapping
Input theory
s=3
s=2
dependance
s=1.38
s=
h/
λ