Moment-of-Fluid Interface Reconstruction Method for Multi-Material ...
Construction of multi-material interfaces from moment data Mark Christon
Rao Garimella
Mikhail Shashkov
Vadim Dyadechko
Blair Swartz
September 10–14, 2007
Talk layout 1) brief look at the Volume-of-Fluid (VoF) algorithms 2) new Moment-of-Fluid (MoF) strategy
• two-material case • Lagrangian remap: how to update the moment data • multi-material case
1
Volume-of-fluid (VoF) technology Two components of typical VoF method: 1) interface reconstruction (IR) algorithm
• input data: cell-wise material volume fractions • preserves (volume=mass) of each material • can separate two materials in a mixed cell • uses one linear segment per mixed cell 2) volume tracking scheme:
volume fractions are updated by calculating the material fluxes through the cell boundaries 2
VoF-IR resolution limit The evaluation of the interface normal from the volume fractions can not be accomplished without the data from the neighbors.
⇓ Resulting interfaces in adjacent mixed cells are not independent
⇓ VoF interface approximation can not resolve interface details smaller than the size of the cell cluster participating in the interface normal evaluation.
⇓ VoF-IR resolution limit ≈ 2 to 3 cell sizes 3
“zigzag” shape reconstruction true
Youngs
LVIRA
Swartz
4
Moment-of-Fluid (MoF) reconstruction assist VoF with material centroids
5
MoF interface reconstruction
ements
input data: a volume fraction (µ∗ ) and a centroid (x∗ ) Among all the subcells ω with a linear interface and the prescribed volume
Ω
|ω| = µ∗ |Ω|,
ω min
x∗
xc(ω)
find the one whose centroid xc(ω) is closest to the given centroid: || xc(ω) − x∗|| → min
local, dimension- and cell-shape- independent unique, stable, 2nd-order accurate 6
“zigzag” shape reconstruction true
MoF
LVIRA
Swartz
7
“constellation” shape reconstruction true
MoF
LVIRA
Swartz
8
Approximation error: circle of radius R 0
Youngs LVIRA Swartz MoF
log10 ∆Γ/R
−1
−2
replacements
Youngs −3 cements LVIRA −4 Swartz MoF
1
−5 2
−log2 (h/R) −6 log10 ∆Γ/R −7 −1
-18% 0
1
2
3
4
5
−log2 (h/R)
6
7
8
Approximation error: 2Lx2L square 0
Youngs LVIRA Swartz MoF
log10 ∆Γ/L
−1
−2
replacements −3
cements
−log2 (h/L) −4 log10 ∆Γ/L
1
Youngs −5 2
LVIRA Swartz −6 MoF −7 −1
-50% 0
1
2
3
4
5
−log2 (h/L)
6
7
8
Lagrangian remap how to update the moment data
11
Volume tracking
ements
Ωi
Ωi
Ωi,k-1
PSfrag replacements Ωi,k-1
1) trace the cell vertices back in
2) intersect the Lagrangian preim-
time; connect them in the proper
age with the underlying pure sub-
order with the straight segments
cells; calculate the total volume of the materials enclosed
12
Centroid tracking
Ωi
x∗i
Ωi
ements
PSfrag replacements Ωi
Ωi
x c (Ω i ) x∗i
x∗i,k-1
x c (Ω i )
x∗i,k-1
The centroid of any parcel of incompressible fluid moves very much like a Lagrangian particle d x (ω) = dt c
v(xc(ω)) + O(diam2ω)
13
Dynamic example reversible “vortex-in-a-box” field:
v((x, y), t) = t = 0,
"
+ sin2(πx) sin(2πy) − sin2(πy) sin(2πx)
t=T =8 1
0.75
0.75
0.5
0.5
0.25
0.25
0
0.25
0.5
0.75
cos(πt/T )
t = T /2 = 4
1
0
#
1
0
0
0.25
0.5
0.75
1
14
“Vortex-in-a-box” test, t=T/2 Youngs
LVIRA
Swartz
MoF
15
“Vortex-in-a-box” test, t=T Youngs
LVIRA
Swartz
MoF
16
The errors measured in the reversible vortex test −2
Youngs LVIRA Swartz MoF
−2.5
ag replacements −3
log10 ∆Γ
acements− log2 h −3.5 log10 ∆Γ Youngs
1
−4
LVIRA Swartz −4.5 MoF 1 −5 2 −5.5
4
2 4.5
5
5.5
6
− log2 h
6.5
7
-67% 7.5
8
17
MoF coupled with incompressible Navier-Stokes solver
−1.5
log10∆Γ
−2
−2.5
replacements
CFL=0.25 CFL=1.00 1 2
1.6
−3
−3.5
−4
4
4.5
5
5.5
6
−log2h
6.5
7
7.5
8
18
Multi-material MoF automatic material ordering
19
Multi-material MoF a single mixed cell with M > 3 materials
ω4∗
ω2∗
∗ ωp,2
PSfrag replacements ω3∗
ements
ω1∗
∗ ωp,4
∗ ωp,3 ∗ ωp,1
true partition
polygonal approximation 20
Automatic material ordering Like multi-material VoF MoF uses the two-material algorithm to separate materials one by one.
Unlike multi-material VoF MoF can determine the right material order automatically, by trying all possible material orders and selecting the one that results in the minimal defect of the 1st moment:
P m
2 ∗ 2 ∗ |ωm| || xc(ωp,m) − xm||2
→ min 21
Automatic material ordering true partition
MoF approximations obtained with all possible material orders
3
2
1
2
2
3
3
1 ∆M1= 3.89e-3
2
3
∆M1= 7.74e-3
1
3 2
1 ∆M1= 3.89e-3
1
∆M1= 7.75e-3
∆M1= 1.14e-2
3
1 2
∆M1= 1.14e-2
22
Examples of the MoF reconstruction
materials are separated one by one 23
Serial partitions
R
⇓
90 90 (0.5,0.5)
⇓
R
M! trial partitions
Serial partition: all materials can be separated one by one with twice-continuously-differentiable dissections.
Theorem: MoF approximation to any serial partition with sufficiently low interface curvature is 2nd-order accurate: ∆Γ = O(h2 /R)
24
Automatic material aggregation
instead of separating materials one by one, one can recursively separate the groups of materials 25
B-tree partitions
R
⇓
(0.5, 0.5)
60
R
R
⇓
R
M!(M-1)! trial partitions
B-tree partition: all materials can be separated with M twicecontinuously-differentiable nested dissections.
Theorem: MoF approximation to any B-tree partition with sufficiently low interface curvature is 2nd-order accurate: ∆Γ = O(h2 /R)
26
Concluding remarks Summary: • new two-material interface reconstruction technique • automatic processing of the the multi-material cells
Ongoing research: • Lagrangian remap with discrete velocities • stable multi-segment interface approximation • error-driven AMR for the MoF interface reconstruction
Publications & supplemental material: http://math.lanl.gov/∼vdyadechko/research 27
MoF Interface Reconstruction in 3D - Bolt-and-Nut H. Ahn and M. Shashkov, T-7, LANL
Rao Garimella
Mikhail Shashkov
Vadim Dyadechko
Blair Swartz
September 10–14, 2007
Talk layout 1) brief look at the Volume-of-Fluid (VoF) algorithms 2) new Moment-of-Fluid (MoF) strategy
• two-material case • Lagrangian remap: how to update the moment data • multi-material case
1
Volume-of-fluid (VoF) technology Two components of typical VoF method: 1) interface reconstruction (IR) algorithm
• input data: cell-wise material volume fractions • preserves (volume=mass) of each material • can separate two materials in a mixed cell • uses one linear segment per mixed cell 2) volume tracking scheme:
volume fractions are updated by calculating the material fluxes through the cell boundaries 2
VoF-IR resolution limit The evaluation of the interface normal from the volume fractions can not be accomplished without the data from the neighbors.
⇓ Resulting interfaces in adjacent mixed cells are not independent
⇓ VoF interface approximation can not resolve interface details smaller than the size of the cell cluster participating in the interface normal evaluation.
⇓ VoF-IR resolution limit ≈ 2 to 3 cell sizes 3
“zigzag” shape reconstruction true
Youngs
LVIRA
Swartz
4
Moment-of-Fluid (MoF) reconstruction assist VoF with material centroids
5
MoF interface reconstruction
ements
input data: a volume fraction (µ∗ ) and a centroid (x∗ ) Among all the subcells ω with a linear interface and the prescribed volume
Ω
|ω| = µ∗ |Ω|,
ω min
x∗
xc(ω)
find the one whose centroid xc(ω) is closest to the given centroid: || xc(ω) − x∗|| → min
local, dimension- and cell-shape- independent unique, stable, 2nd-order accurate 6
“zigzag” shape reconstruction true
MoF
LVIRA
Swartz
7
“constellation” shape reconstruction true
MoF
LVIRA
Swartz
8
Approximation error: circle of radius R 0
Youngs LVIRA Swartz MoF
log10 ∆Γ/R
−1
−2
replacements
Youngs −3 cements LVIRA −4 Swartz MoF
1
−5 2
−log2 (h/R) −6 log10 ∆Γ/R −7 −1
-18% 0
1
2
3
4
5
−log2 (h/R)
6
7
8
Approximation error: 2Lx2L square 0
Youngs LVIRA Swartz MoF
log10 ∆Γ/L
−1
−2
replacements −3
cements
−log2 (h/L) −4 log10 ∆Γ/L
1
Youngs −5 2
LVIRA Swartz −6 MoF −7 −1
-50% 0
1
2
3
4
5
−log2 (h/L)
6
7
8
Lagrangian remap how to update the moment data
11
Volume tracking
ements
Ωi
Ωi
Ωi,k-1
PSfrag replacements Ωi,k-1
1) trace the cell vertices back in
2) intersect the Lagrangian preim-
time; connect them in the proper
age with the underlying pure sub-
order with the straight segments
cells; calculate the total volume of the materials enclosed
12
Centroid tracking
Ωi
x∗i
Ωi
ements
PSfrag replacements Ωi
Ωi
x c (Ω i ) x∗i
x∗i,k-1
x c (Ω i )
x∗i,k-1
The centroid of any parcel of incompressible fluid moves very much like a Lagrangian particle d x (ω) = dt c
v(xc(ω)) + O(diam2ω)
13
Dynamic example reversible “vortex-in-a-box” field:
v((x, y), t) = t = 0,
"
+ sin2(πx) sin(2πy) − sin2(πy) sin(2πx)
t=T =8 1
0.75
0.75
0.5
0.5
0.25
0.25
0
0.25
0.5
0.75
cos(πt/T )
t = T /2 = 4
1
0
#
1
0
0
0.25
0.5
0.75
1
14
“Vortex-in-a-box” test, t=T/2 Youngs
LVIRA
Swartz
MoF
15
“Vortex-in-a-box” test, t=T Youngs
LVIRA
Swartz
MoF
16
The errors measured in the reversible vortex test −2
Youngs LVIRA Swartz MoF
−2.5
ag replacements −3
log10 ∆Γ
acements− log2 h −3.5 log10 ∆Γ Youngs
1
−4
LVIRA Swartz −4.5 MoF 1 −5 2 −5.5
4
2 4.5
5
5.5
6
− log2 h
6.5
7
-67% 7.5
8
17
MoF coupled with incompressible Navier-Stokes solver
−1.5
log10∆Γ
−2
−2.5
replacements
CFL=0.25 CFL=1.00 1 2
1.6
−3
−3.5
−4
4
4.5
5
5.5
6
−log2h
6.5
7
7.5
8
18
Multi-material MoF automatic material ordering
19
Multi-material MoF a single mixed cell with M > 3 materials
ω4∗
ω2∗
∗ ωp,2
PSfrag replacements ω3∗
ements
ω1∗
∗ ωp,4
∗ ωp,3 ∗ ωp,1
true partition
polygonal approximation 20
Automatic material ordering Like multi-material VoF MoF uses the two-material algorithm to separate materials one by one.
Unlike multi-material VoF MoF can determine the right material order automatically, by trying all possible material orders and selecting the one that results in the minimal defect of the 1st moment:
P m
2 ∗ 2 ∗ |ωm| || xc(ωp,m) − xm||2
→ min 21
Automatic material ordering true partition
MoF approximations obtained with all possible material orders
3
2
1
2
2
3
3
1 ∆M1= 3.89e-3
2
3
∆M1= 7.74e-3
1
3 2
1 ∆M1= 3.89e-3
1
∆M1= 7.75e-3
∆M1= 1.14e-2
3
1 2
∆M1= 1.14e-2
22
Examples of the MoF reconstruction
materials are separated one by one 23
Serial partitions
R
⇓
90 90 (0.5,0.5)
⇓
R
M! trial partitions
Serial partition: all materials can be separated one by one with twice-continuously-differentiable dissections.
Theorem: MoF approximation to any serial partition with sufficiently low interface curvature is 2nd-order accurate: ∆Γ = O(h2 /R)
24
Automatic material aggregation
instead of separating materials one by one, one can recursively separate the groups of materials 25
B-tree partitions
R
⇓
(0.5, 0.5)
60
R
R
⇓
R
M!(M-1)! trial partitions
B-tree partition: all materials can be separated with M twicecontinuously-differentiable nested dissections.
Theorem: MoF approximation to any B-tree partition with sufficiently low interface curvature is 2nd-order accurate: ∆Γ = O(h2 /R)
26
Concluding remarks Summary: • new two-material interface reconstruction technique • automatic processing of the the multi-material cells
Ongoing research: • Lagrangian remap with discrete velocities • stable multi-segment interface approximation • error-driven AMR for the MoF interface reconstruction
Publications & supplemental material: http://math.lanl.gov/∼vdyadechko/research 27
MoF Interface Reconstruction in 3D - Bolt-and-Nut H. Ahn and M. Shashkov, T-7, LANL
