A SUBCELL BASED INTERFACE
RECONSTRUCTION METHOD TO
DEAL WITH FILAMENTS
ECCOMAS 2012 | Christophe Fochesato1, Raphaël Loubère2, Renaud Motte1, Jean Ovadia3
1CEA,
DAM, DIF, F-91297 Arpajon, France
2Institut
3Retired
de Mathématiques de Toulouse, CNRS, Université de Toulouse
fellow from CEA, CESTA, F-33114 Le Barp, France
| PAGE 1
Outline
I Introduction
Context / Problem / Proposed solution / Example
II Presentation of the method
Detection / Centroids / Sub-zones / Volume distribution /
Reconstruction
III Test cases
Infinite vertical filament / isolated fragment / finite vertical filament
IV Conclusions / Perspectives
| PAGE 2
I - Context
Bi-fluids hydrodynamical flows with interfaces
VoF interface reconstruction
Maximum possible resolution is not always sufficient
structure size < mesh cell size
| PAGE 3
I - Problem
Usual VoF methods use only one interface per mixed cell,
typically a straight line (PLIC)
numerical surface tension
generation of flotsam
reduces robustness
Youngs’ normal can be irrelevant : ~0 because of almost
symmetrical volume fractions on the stencil
Reconstruction is coarse whereas there is enough information to
do better with the same 9-cells stencil
| PAGE 4
I – Proposed solution : a subgradients method
The idea is to compute subgradients from a local stencil associated with
each corner (2x2 cells for instance) in order to reconstruct one interface
per subzone, with normals given by these subgradients
Method does not contain more physics
Another geometrical choice
with less numerical surface tension
avoiding to use irrelevant normal information
Localized algorithm in the code : no change in the numerical scheme
| PAGE 5
I - Example
A filament with our method
VoF / PLIC with Youngs’ normal
| PAGE 6
II – Method summary
Detection of marked cells with subgradients computation
Estimation of centroid if the fluid is just entering
Determination of subzones with centroid as center of subdivision
Distribution of fluid per subzone with priority
Youngs/PLIC reconstruction : one straight line per subzone
Computation of centroids
Advection of centroids
| PAGE 7
II – Detection of concerned cells
Computation of a volume fraction gradient per cell f c
The cell is marked
if normal is potentially not relevant : if
f c 
or
1
f c

9 c
if the fluid potentially passes through the cell
f7
f8
f9
f4
f5
f6
f1
f2
f3
if
f1 * f 2 * f 3  f 7 * f8 * f 9  0
or
f1 * f 4 * f 7  f 3 * f 6 * f 9  0
If marked cell, compute a gradient associated with each corner
on a 2x2 stencil
Confirmation that cell is marked
if gradient is less relevant
than subgradients
if
f c 
1

 f c   f k 
5
k
 8
| PAGE
II – Estimation of centroids
If the fluid is coming in the cell
no centroid information
compute Youngs/PLIC interface,
compute the volume centroid
else
use the advected centroids from the previous step or init
| PAGE 9
II – Determination of subzones
keep only relevant subgradients if
f k 
1

 f c   f k 
5
k

detect particular configurations : vertical filament
2 subzones, …
with relevant subgradients, we associate subzones
4 relevant subgradients
3 relevant subgradients
2 relevant subgradients
1 relevant subgradient
| PAGE 10
II – Distribution of the volume of fluid
In priority, fill in subzones
next to non empty cells
uniform distribution in volume
If it remains some fluid to distribute,
fill in other subzones
uniform distribution in volume
Correction if more volume of fluid than volume of the subzone
uniform distribution on other non full subzones
| PAGE 11
II – Distribution of the volume of fluid
In priority, fill in subzones
next to non empty cells
uniform distribution in volume
If it remains some fluid to distribute,
fill in other subzones
uniform distribution in volume
Correction if more volume of fluid than volume of the subzone
uniform distribution on other non full subzones
| PAGE 12
II – Reconstruction of interfaces
PLIC per subzone
Youngs’ algorithm
| PAGE 13
II – Computation of centroid and advection
Compute the centroid for reconstructed cell
for now, mean of centroid of each subzone
to be improved because different from real centroid
Advect the centroid
| PAGE 14
III – Code used for the tests
Cartesian grid
Gradients computation with Youngs’ Finite Difference formula
Subgradients computation with Finite Difference formula
Lagrange + remap scheme with direction splitted remapping
interface reconstruction on 1D-stretched cells for each
direction
exact intersection of transfer volume of the cell with
reconstructed interface gives transfer volume for the fluid
| PAGE 15
Advection of an infinite vertical filament
u
Good advection of filament
with our method
u
Youngs’ VoF reconstruction
does not move !
| PAGE 16
Advection of a fragment smaller than the cell
u
Advection of the fragment
with our method
too slow through mesh interfaces
u
Youngs’ VoF reconstruction
does not move !
| PAGE 17
Advection of a finite vertical filament
Better advection speed
u
Filament ends go slower than
the middle
Youngs’ VoF reconstruction moves !
u
Surprisingly well, but advection speed
too large
| PAGE 18
Advection of a filament with (u,v)
v = u/2
Filament is nearly preserved
with our method
v = u/2
Youngs’ VoF reconstruction
early leads to blobby flows !
Advection speed is quite good
| PAGE 19
Conclusions, perspectives
summary
A subgradient method to compute VoF/PLIC interfaces in subzones
At given mesh, better representation of thin structures of fluid
Not a subgrid physical model : method as geometrical as original VoF
with the same volume fraction field information
Ability to advect filaments and fragments proven on simple cases
in progress
fluid distribution into subzones
centroid estimation
to be tested on more and more complex cases
perspectives
… extension to 3D
… extension to unstructured mesh
… extension to more than 2 fluids
| PAGE 20
| PAGE 21
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