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FronTier and Applications to Convection Dominated Problems
Xiaolin Li, Wurigen Bo, James GlimmAnd the FronTier team
Department of Applied Math and StatisticsSUNY at Stony Brook
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Outline of the talk1. Benchmark examples of FronTier2. Convection Dominated Problems3. Other problems4. Coupling FronTier with other packages5. Challenges to FronTier
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A quotation from Albert Einstein
1. Stony Brook, AMS Department, galaxy cluster (over 500 processors)
2. Stony Brook, CEAS, Seawulf cluster
3. New York Blue: 103.22 teraflops
Major ComputingResources:
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如果我们选择了最能为人类的福利而劳动的职业,我们就不会为它的重负所压倒,因为这是为全人类所做的牺牲,那时我们所感到将不是一点点自私而可怜的欢乐,我们的快乐属于千万人,我们的工作并不显赫一时,但将永远存在,而面对我们的骨灰,高尚的人们将会洒下热泪
卡尔.马克思
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The Discrete Mesh Representation in Front Tracking
Volume fillingrectangular mesh(Eulerian Coord.)
(N-1) dimensionalLagrangian mesh(interface)
A 3D InterfaceA 2D Representation
Y
X
(i,j)
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FronTierA software package based on
the front tracking method
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Front Tracking: Scientific applications to interfacial physics, the methodology and software
1. 2006-2009, over 36,000 new lines of code, removal of hundreds of run-time bugs, set up benchmarks.
2. Collaborations with ANL, BNL, LLNL, ORNL, PNNL, Oxford.3. Over 20 publications and many conference presentations.4. Implemented iMesh interface, support SciDAC applications.5. Interoperability with Hypre and PETSc, coupling with parabolic and
elliptic solvers, implicit and Crank-Nicolson and incompressible solver.
Front tracking was first introducedBy Richtmyer in 1950’s. It was Used by Moretti for computationIn aerodynamics problems.
Extension to 2D and 3D startedBy J. Glimm and O. McBryanIn 1980’s at Courant Institute.
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Interface Topological Changes
Algorithms to handle topological changesGrid free tracking (GF)Grid based tracking (GB)Locally grid based tracking (LGB)
Tangled interface
GF
LGBGB
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Robust Locally Grid Based (LGB) Untangle
AdvantageLocal, it is suitable for large scale computing.Robust, It generates topologically valid surface mesh.
A robust algorithm to reconnect a grid based surface mesh with a grid free surface mesh
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Interface Topological ChangesGrid based tracking is robust but too diffusive.Challenge: Robustness of the algorithm is crucial for large scale computing.
Grid based tracking Grid free tracking
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Zalesak’s slotted disk (2D)
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Interface Topological ChangesGrid based tracking is robust but too diffusive.Challenge: Robustness of the algorithm is crucial for large scale computing.
Grid based tracking Grid free tracking
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Resolution Test
FronTier can provideResolution of surfaceDown to 1/200 of meshspacing
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模拟维模拟模拟晶体结晶过程Resolution independent of mesh
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Front tracking reversal test of interface in the deformation velocity field
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2D Reversal Test
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Front tracking reversal test of interface in 3D deformation velocity field
646464 ××
128128128 ××
0.0=t 0.3=t5.1=t
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VOF by Kothe (2005)
0.3=t5.1=t 5.1=t 0.3=t
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模拟维模拟模拟晶体结晶过程Deformation of 3D surface mesh
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2D Topological Bifurcation
FronTier is equippedWith robust and Efficient bifurcationFunctions for meshesRepresenting thePropagating front
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模拟维模拟模拟晶体结晶过程Topological merging of 3D surface mesh
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ExamplesInterface bifurcation and merging are commonly observed in multiphase flow
mesh bifurcation in a curvature dependent surface propagation
mesh merging in a droplet collision simulation
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εκ−= 0vvn
模拟维模拟模拟晶体结晶过程Geometry dependent interface velocity
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Convection dominated PDE
• Hyperbolic equation• Hyperbolic-elliptic (NS equation)• Parabolic equation• Hyperbolic-parabolic (convection-diffusion)
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FronTier Applications to the study of Rayleigh-Taylor Instability
(Coupling with hyperbolic Euler’s equation for gas dynamics)
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Incompressible Rayleigh-Taylor instability
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FronTier application: chaotic mixing
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FronTier application: chaotic mixing
Chaotic mixing is not only important to ICF, but also a test of large scale FronTierapplication to petascale computing. We have implemented a load balanced parallel algorithm and ran up to 1024 processors on New York Blue. Collaboration with B. Cheng, John Grove, and D. Sharp at LANL.
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3D Turbulent Mixing
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FronTier Applications to the study of Richtmyer-Meshkov Instability
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Inertial Confinement Fusion (ICF)
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Simulation of Spherical RM
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FronTier Applications to BiodieselJet Simulation
3D simulation for primary jet breakup
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模拟三维内燃机喷嘴Ask not what the earth can do for us, ask what we can do for the earth
American consumes about 200 billion gallons per year,a 10% saving will be 20 billion gallon amounts to morethan 40 billion dollars, not to mention the benefit to theenvironment.
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Burning biofuel in an engine
1. Understanding the difference between conventional fuel and biofuel: a) Ethanol used to blend with gasoline for automobile; b) Biodiesel blend with distillate petroleum fuels such as diesel, kerosene and heating oils for use in diesel engines, boilers possibly turbines.
2. How to burn cleanly and uniformly: The prerequisite for an optimized combustion is an optimized fuel injection spray.
3. What will affect the formation of spray: The physical properties of the fuel, the geometry of orifice and the application of pressure, the dynamics of the fluids through the nozzle.
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Surface mesh evolution in jet simulation
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Verification: Kelvin-Helmholtz Instability
0.03130.02050.0094
80160320
FT/GFMMesh per mode
The relative errors of the growth rate
Comparison with the dispersion relation dispersion relation
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Verification: Rayleigh Instability
0.28530.17020.0672
0.13960.06070.0321
51020
FT/GFM (3D)
FT/GFM (2D)
Number of cells on radius
The relative errors of the growth rate
Comparison with the dispersion relation dispersion relation
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Parallel Performance of FT
157.1159.81638416×16×64256×256×1024
157.1158.2819216×16×32256×256×512
157.1157.5409616×16×16256×256×256
157.1157.1204816×16×8256×256×128
Ideal(s)Time to solution(s)nCoresPartitionGrid
Performance of LGBJet simulation300-3million TrianglesBluegene/L 4096 cores
Weak scalingRayleigh InstabilityBluegene/L
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High Pressure Diesel Injection
Importance: The fuel injection process plays an important role in the combustion and emissions performance of diesel engines.Experimental challenge: droplets in the spray from the breakup obscure the liquid core.Numerical challenge:
large range of spatial and temporal scaleslarge density ratio between fuel and airtopological changes of the jet surface
Figure source: Baumgarten, Atomization and Sprays, 16, (2006)
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Nozzle Flow Simulation
Cavitation ModelA cavitation bubble is modeled as an interface between vapor and fuel.A cavitation bubble is dynamically inserted in the center of a rarefaction wave of critical strength.
Simulation setup
2D simulation 3D simulation Nozzle radius (R): 0.1mmGrid: 40/RFuel density: 0.66 g/cm3
Reynolds number: 30,000
Injector geometry: Macphee et.al. , Science, 295, (2002)
reservoir
nozzle
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Nozzle Flow Simulation
As the flow reaches a statistically steady state, mean velocity, turbulence length scale and integral turbulence length scale are studies at the nozzle outlet
Calculated turbulence properties at the nozzle outletIntegral turbulence length scale (Λ) = 0.38RTurbulence intensity (Ti) = 0.025
Vorticity
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Surface color is rendered by penetration velocity.
Simulation is performed with 4096 cores in BG/L at BNL.
Calculation time is about 12 days.
liquid film
Intact jet core
ligament
droplet
46*A. Lefebvre, Atomization and Sprays, 1989
Droplet Diameter Distribution
Droplet diameter is approximated from surface area d=(A/π)1/2.Sauter mean diameter (SMD) = Σd3/Σd2.The probability density function of droplet diameter distribution can be fitted by a log-normal distribution, which is consistent with experimental observations for high pressure injectors*.
time evolution of droplet diameter distribution
droplets from the simulation
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1.23
1.22
Simulation
MMD/SMDMMD (μm)
SMD (μm)
1.224.525.520.22320.030R
1.223.724.519.52660.0250.38R
Correlation*Correlation*Simulation
Number of droplets from simulations
TiΛ
Droplet Diameter Distribution
* P.K. Wu, Atomization and Sprays, 3, (1993) MMD=Mass Medium Diameter
Λ=0.38R Ti=0.025 Λ=R Ti=0.030
Two simulations are performed with different turbulence inflow conditions.
69.05/3
3/22
'0
0 7604.01−
⎟⎠⎞
⎜⎝⎛ Λ
=⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ Λ
⎟⎟⎠
⎞⎜⎜⎝
⎛+
Λ dWe
SMDvUSMD
l
g
ρρ
An experimentally validated correlation*
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3D Simulation of a Real Fuel Injection
All parameters are from an experiment performed by Parker*
nozzle radius (R) 0.1mmgrid 20/Rfuel density 0.66 g/cm3
gas density 0.0165 g/cm3
fluid viscosity 0.013 Poisesurface tension 24 mN/m2
Reynolds number 20,300Weber number 2.2×106
Ohnesorge number 0.073Density ratio 40
* P. Parker, Atomization and Sprays, 8, (1998)
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FronTierand incompressible fluid
Navier-Stokes solver(Hyperbolic-Elliptic)
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Incompressible fluid solver
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Incompressible Rayleigh-Taylor instability on Atwood number
(From left: 0.82,0.33,0.14)
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Incompressible Rayleigh-Taylor instability on Reynold number
(from left: 14,140,1400)
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Incompressible code in 3D
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FronTieron Fluid-structure interaction
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Fluid-Rigid body interaction
Example: shock-rigid body (with biased center of mass) interaction
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Fluid-Rigid body interaction
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Wind power generator platform
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FronTier and Parachute Simulation
1. Motivation2. Structure Dynamics (SD)3. Computational Fluid Dynamics (CFD)4. Couple Simulations
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Three Stages of Parachute Dynamics
1. Deployment2. Inflation3. Terminal Descent
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The Inflation stage of parachute
We pick up this phase for simulation because
1. Canopy geometry changes dramatically2. Canopy experiences largest force and is most stressed3. Parachute undergoes biggest deceleration4. A phase of “to be, or not to be”.
Mathematical and computational challenges
1. Process is highly nonlinear2. Requires robust geometry handling3. Needs accurate SD and CFD coupling
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Previous Studies
K. R. Stein, R. J. Benney andT. E. Tezduyar, Journal of Aircraft, 2001.1. Unstructured grid, FEM2. Static mesh (133,097 nodes)3. Terminal descent
M. Accorsi, J. Leonard, R. BenneyAnd Keith Stein, AIAA Journal, 1999.1. Inflation study,2. SD simulation3. Fluid effects are approximated
Youngsam Kim and Charlie Peskim,Computer and Fluids, 20091. Penalty immersed boundary method2. Full 3D SD-CFD simulation
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What is missing in FronTier
1. Incompressible solver: the speed of parachute is much less than sound speed (except for parachute of re-entry vehicle).
2. Mono component surface representation and functions: parachute surface is a fully immersed non-manifold surface.
3. Constrained surface propagation in fluid velocityfield: inflexible fabric or slightly flexible elasticsurface.
4. Collision handling: different from topologicalbifurcation.
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The strategy of FronTier Parachute Simulation
1. Computational fabric modeling• Inflexible model• Linear spring model• Post-buckling response
2. Surface tension modeling
3. Collision and friction4. PDE coupling with immersed boundary (canopy)
• Penalty immersed boundary method (Y. Kim)• Immersed interface method (Z. Li)
dstsXxtstx )),((),(),( −= ∫ δFf
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The Linear Spring Model
⎪⎩
⎪⎨⎧
<
≥−=
Lx
LxLxkE
ij
ijijs
0
)(21 2
Using the energy function:
⎪⎩
⎪⎨
⎧
<
≥−=
∂∂
−=∝Lx
Lxxx
Lxk
xEfa
ij
ijij
ijijs
iiti
0
)(
The acceleration and force on vortex:
65
Constrained and unconstrained curve propagation
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Test simulations of flow around canopy
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FronTier Applications to solute precipitation and crystal formation
68
Mathematical Model
s
esn
ess
ess
CCkdtdnv
dtCCkdn
CCkn
C
CDtC
ρ
ρ)(
)(
)(
−==
−=
−=∂∂
Δ=∂∂
The equation forsolute diffusion:
The interface propagation:
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模拟维模拟模拟晶体结晶过程One Dimensional Solute Precipitation
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Numerical discretization
1,2
)()2(
1)(2
)()2()(
11
1111
−=−+
=
+⎟⎟⎠
⎞⎜⎜⎝
⎛ −−
−+=
++
++++
drh
xChxCS
Sh
CCkh
xChxCxC
sn
sn
s
esn
sn
sn
sn
χχ
ρ
A semi-implicit front state solver
With correction of curvature
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模拟维模拟模拟晶体结晶过程
0.320=aD
Two Dimensional Solute Precipitation
0Cs =ρ
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模拟维模拟模拟晶体结晶过程Two Dimensional Solute Precipitation
640=aD
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模拟维模拟模拟晶体结晶过程Two Dimensional Solute Precipitation
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模拟维模拟模拟晶体结晶过程Three Dimensional Solute Precipitation
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Precipitation with flow
CDCvtC
Δ=∇⋅+∂∂ )(
The equation for solute diffusion:
Where v is solved from the Navier-Stokes equation
76
Convection diffusion model
77
FronTierApplications to Other Problems
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Pricing of American Option
( )
1
,)0,(,0)0,(
,0)(21
2
222
=∂
∂
−=
>−=≤=
−=
=−+∂∂
−∂∂
−∂∂
SC
ESC
ESESSCESSC
tT
CDSCS
SCSC
f
f
τ
γσγτ
The Black-ScholeEquation:
The interface Condition at all time:
Initial Condition:
79
Exercise point condition
1
1
11
1
,
,,
)(
+
−
+−
+
→
i
ii
fiii
nf
SSS
SSSS
S τAssume at
We need to find:
1+nτ
From B-S PDE:
From degenerated B-S (ODE):
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One Dimensional American Option
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Simulation of Cell Migration
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Interoperation of FronTierwith other software
libraries
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1. PETSc: TOPS supported linear solver library2. HDF4: To convert data into movies with color map3. GD: To produce interface movies4. VisIt: Visualization package by LLNL5. Weblinks:
http://sitsec.ams.sunysb.edu/trac/wiki/FronTierhttp://www.mcs.anl.gov/petsc/petsc-as/ http://www.hdfgroup.org/products/hdf4/http://www.boutell.com/gd/ https://wci.llnl.gov/codes/visit/about.html
84
Challenges to the front tracking method
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Parallel load balancing
Like AMR, FronTier has encountered greatObstacle in load balancing and parallel scaling. One important development is adaptive partition load balancing.
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FronTier and AMR
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Space-time cells
Conservative tracking
88
FronTier for Dummies
1. Download and test runhg clone http://sitsec.ams.sunysb.edu/hg FronTiercd FronTierbuild.sh [–n –d –g]makecd example or application directoriesmakerun
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Implementation
2. Start up/* Initialize basic computational data */
FrontInitStandardIO(argc,argv,&f_basic);FrontReadRegularDomainInfo(in_name,&f_basic);FrontStartUp(&front,&f_basic);FrontInitDebug(in_name);FrontInitVelo(&front,&velo_func_pack);FrontReadTimeControlInfo(in_name,&front);Problem specific initializations
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Implementation
4. Time loopFrontProp(front);FrontSetTimeStep(front);Interior solver time step;for (;;){
FrontProp(front);Interior solver;FrontSetTimeStep(front);Interior solver time step;Output or exit;
}
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User Query Functions
Int Dimension(interface);Int NumOfPoints(curve);Int NumOfPoints(surface);Int NumOfPoints(intfc);Int NumOfCurves(intfc);Int NumOfSurfaces(intfc);Int NumOfTris(surface);Int NumOfTris(intfc);
ArrayOfPoints(curve, coords);ArrayOfPoints(intfc, coords);ArrayOfCurves(intfc, curves);ArrayOfCurves(intfc, coords, vertex_indices);ArrayOfSurfaces(intfc, surfaces);ArrayOfTri(surface, tris);ArrayOfTri(surface, coords, vertex_indices);ArrayOfTri(intfc, tris);
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Thank you