Wavelet-based Spatiotemporal Multiscaling in Diffusion...
Transcript of Wavelet-based Spatiotemporal Multiscaling in Diffusion...
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Wavelet-based Spatiotemporal Multiscaling in Diffusion Problems with Chemically Reactive Boundary
Sreekanth Pannala
Srdjan Simunovic
StuartDaw
PhaniNukala
Oak RidgeNational
Laboratory
Rodney Fox
ZhaosengGao
George Frantziskonis
SudibMishra
PierreDeymier
Aditi Mallik
Krishna Muralidharan
Thomas O’Brien
DominicAlfonso
MadhavaSyamlal
Ames LaboratoryIowa State University
Universityof Arizona
National Energy Technology Laboratory
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Spouted bed coater (device scale)
Coated fuel particle(small scale)
• 0.5- to 1-mm particles • Coating encapsulates
fission products• Failure rate < 1 in 105
• Quality depends on surface processes at nm length scale and ns time scales
Links multiscale
mathematicswith petascale
computingand NE
Links multiscale
mathematicswith petascale
computingand NE
• Design challenge:Maintain optimal temperatures, species, residence times in each zone to attain right microstructureof coating layersat nm scale
• Truly multiscaleproblem: ~O(13) time scales,~O(8) length scales
• Coating at high temperature (1300–1500°C) in batch spouted bed reactor for ~104 s
• Particles cycle thru deposition and annealing zones where complex chemistry occurs
~10-3 m
~10-1 m
UO2
~10-3 m
Pickup zone (~10-6-10-2s)Pickup zone (~10-6-10-2s)
Si-CSi-C
Inner Pyrolitic C
Inner Pyrolitic C
Amorphous CAmorphous C
KernelKernel
Background 1: Nuclear fuel coating process – a specific example of gas-solid contacting device
Ballistic zone
Pickup zone (~10-6-10-2s)Pickup zone (~10-6-10-2s)
Transportreaction zone (~10-6-10-2s)
Transportreaction zone (~10-6-10-2s)
Hopperflow
zone (~s)
Hopperflow
zone (~s) Inlet gas
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Background 2: Multiphysics heterogeneous chemically reacting flows for energy systemsGoal: Building a suite of models for unprecedented capability to simulate multiphase flow reactors
• Through support from various DOE offices (FE, EERE, and NE) we have developed suite of models for unprecedented capability to simulate heterogeneous chemically reacting flows
• Hybrid methods to couple two physical models (e.g. MFIX DEM)
• Uncertainty quantification to probe only quantities of interest at smaller scales
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Background 3: Micro-mesoscopic modeling of heterogeneous chemically reacting flows over catalytic/solid surfacesGoal: Develop a multiscale framework for accurate modeling of heterogeneous reacting flows over catalytic surfaces
Y
X
x
x
Fractalprojection
Actualsurface
KMC
LBM
t viqi
T viqi
KMC contribution
LBM contributionCWM
x-yy
x
Procedure: Perform upscalingand downscaling using CWM
Compound Wavelet Matrix (CWM)
Lattice Boltzmann(LBM)
QM: ~1 nm KMC: ~1 μmLBM: ~1 mm
Coupling LBM/KMCKinetic Monte Carlo
(KMC)Density Functional Theory
(DFT)CloselycoupledCloselycoupled
Reactionbarriers
Reactionbarriers
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Compound Wavelet Matrix (CWM) for Multiscaling
• CWM is a wavelet based spatio-temporal operator which has different functions depending on the context− Compounding operation (combine information
from multiple scales)− Projection or transfer operations
• Up-scaling fine scale information to coarse scale fields
• Down-scaling coarse information to reconstruct fine scale fields
• Fits within the general HMM framework
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Simple illustration of the compounding process – Coarse and Fine Signal
2000 4000 6000 8000
-2
-1
1
2
100 200 300 400 500
-1
-0.5
0.5
1Coarse (8192 s)
Resolves the slow and intermediate frequency over a long duration
Resolves the intermediate and fast frequency over a short duration
Fine (512 s)
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Simple illustration of the compounding process – Decomposition
4 6 8 1 0 1 2 1 4 1 6
- 2 0
- 1 0
1 0
2 0
4 6 8 1 0 1 2 1 4 1 6
0 . 8 4 4
0 . 8 4 5
0 . 8 4 6
5 1 0 1 5 2 0 2 5 3 0
- 6
- 4
- 2
2
4
6
1 0 2 0 3 0 4 0 5 0 6 0
- 0 . 2
- 0 . 1
0 . 1
0 . 2
2 0 4 0 6 0 8 0 1 0 0 1 2 0
- 0 . 0 1
- 0 . 0 0 5
0 . 0 0 5
0 . 0 1
5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
1 0 0 2 0 0 3 0 0 4 0 0 5 0 0
- 0 . 0 0 0 0 2
- 0 . 0 0 0 0 1
0 . 0 0 0 0 1
0 . 0 0 0 0 2
2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0
- 1 ´ 1 0 - 6
- 5 ´ 1 0 - 7
5 ´ 1 0 - 7
1 ´ 1 0 - 6
5 0 0 1 0 0 0 1 5 0 0 2 0 0 0
- 4 ´ 1 0 - 8
- 2 ´ 1 0- 8
2 ´ 1 0- 8
4 ´ 1 0 - 8
1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0
- 2 ´ 1 0 - 9
- 1 ´ 1 0 - 9
1 ´ 1 0 - 9
2 ´ 1 0 - 9
4 6 8 1 0 1 2 1 4 1 6
- 6
- 4
- 2
2
4
6
4 6 8 1 0 1 2 1 4 1 6
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
5 1 0 1 5 2 0 2 5 3 0
0 . 9 0 7 2 4
0 . 9 0 7 2 5
0 . 9 0 7 2 6
0 . 9 0 7 2 7
1 0 2 0 3 0 4 0 5 0 6 0
- 0 . 2
- 0 . 1
0 . 1
0 . 2
2 0 4 0 6 0 8 0 1 0 0 1 2 0
- 0 . 0 0 7 5
- 0 . 0 0 5
- 0 . 0 0 2 5
0 . 0 0 2 5
0 . 0 0 5
0 . 0 0 7 5
5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
10-5
Coarse (8192 s)
Fine (512 s)
10-6
10-8
10-9
1
10-1
10-2
10-440962048
512
256128
64
,( , ) ( ) ( )f a bW a b f x x dxψ∞
−∞
= ∫
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Simple illustration of the compounding process – Clipping
4 6 8 1 0 1 2 1 4 1 6
- 2 0
- 1 0
1 0
2 0
4 6 8 1 0 1 2 1 4 1 6
0 . 8 4 4
0 . 8 4 5
0 . 8 4 6
5 1 0 1 5 2 0 2 5 3 0
- 6
- 4
- 2
2
4
6
1 0 2 0 3 0 4 0 5 0 6 0
- 0 . 2
- 0 . 1
0 . 1
0 . 2
2 0 4 0 6 0 8 0 1 0 0 1 2 0
- 0 . 0 1
- 0 . 0 0 5
0 . 0 0 5
0 . 0 1
5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
1 0 0 2 0 0 3 0 0 4 0 0 5 0 0
- 0 . 0 0 0 0 2
- 0 . 0 0 0 0 1
0 . 0 0 0 0 1
0 . 0 0 0 0 2
2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0
- 1 ´ 1 0 - 6
- 5 ´ 1 0 - 7
5 ´ 1 0 - 7
1 ´ 1 0 - 6
5 0 0 1 0 0 0 1 5 0 0 2 0 0 0
- 4 ´ 1 0 - 8
- 2 ´ 1 0- 8
2 ´ 1 0- 8
4 ´ 1 0 - 8
1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0
- 2 ´ 1 0 - 9
- 1 ´ 1 0 - 9
1 ´ 1 0 - 9
2 ´ 1 0 - 9
4 6 8 1 0 1 2 1 4 1 6
- 6
- 4
- 2
2
4
6
4 6 8 1 0 1 2 1 4 1 6
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
5 1 0 1 5 2 0 2 5 3 0
0 . 9 0 7 2 4
0 . 9 0 7 2 5
0 . 9 0 7 2 6
0 . 9 0 7 2 7
1 0 2 0 3 0 4 0 5 0 6 0
- 0 . 2
- 0 . 1
0 . 1
0 . 2
2 0 4 0 6 0 8 0 1 0 0 1 2 0
- 0 . 0 0 7 5
- 0 . 0 0 5
- 0 . 0 0 2 5
0 . 0 0 2 5
0 . 0 0 5
0 . 0 0 7 5
5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
10-5
Coarse (8192 s)
Fine (512 s)
10-6
10-8
10-9
1
10-1
10-2
10-440962048
512
256128
64Clipping
Clipping
2
1 2
1
, 2
1( ) ( , ) ( )s
s s f a bs
daf x W a b x dbc aψ
ψ∞
−∞
= ∫ ∫
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Simple illustration of the compounding process – Prolongation
4 6 8 1 0 1 2 1 4 1 6
- 6
- 4
- 2
2
4
6
4 6 8 1 0 1 2 1 4 1 6
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
5 1 0 1 5 2 0 2 5 3 0
0 . 9 0 7 2 4
0 . 9 0 7 2 5
0 . 9 0 7 2 6
0 . 9 0 7 2 7
1 0 2 0 3 0 4 0 5 0 6 0
- 0 . 2
- 0 . 1
0 . 1
0 . 2
2 0 4 0 6 0 8 0 1 0 0 1 2 0
- 0 . 0 0 7 5
- 0 . 0 0 5
- 0 . 0 0 2 5
0 . 0 0 2 5
0 . 0 0 5
0 . 0 0 7 5
5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
Fine (8192 s)
1
10-1
10-2
10-4 40962048
1024
4 6 8 1 0 1 2 1 4 1 6
- 6
- 4
- 2
2
4
6
4 6 8 1 0 1 2 1 4 1 6
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
5 1 0 1 5 2 0 2 5 3 0
0 . 9 0 7 2 4
0 . 9 0 7 2 5
0 . 9 0 7 2 6
0 . 9 0 7 2 7
1 0 2 0 3 0 4 0 5 0 6 0
- 0 . 2
- 0 . 1
0 . 1
0 . 2
2 0 4 0 6 0 8 0 1 0 0 1 2 0
- 0 . 0 0 7 5
- 0 . 0 0 5
- 0 . 0 0 2 5
0 . 0 0 2 5
0 . 0 0 5
0 . 0 0 7 5
5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
Fine (512 s)
1
10-1
10-2
10-4 256128
64
Replicate
16 times
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Simple illustration of the compounding process – Compounding
4 6 8 1 0 1 2 1 4 1 6
- 2 0
- 1 0
1 0
2 0
4 6 8 1 0 1 2 1 4 1 6
0 . 8 4 4
0 . 8 4 5
0 . 8 4 6
5 1 0 1 5 2 0 2 5 3 0
- 6
- 4
- 2
2
4
6
1 0 2 0 3 0 4 0 5 0 6 0
- 0 . 2
- 0 . 1
0 . 1
0 . 2
2 0 4 0 6 0 8 0 1 0 0 1 2 0
- 0 . 0 1
- 0 . 0 0 5
0 . 0 0 5
0 . 0 1
5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
Coarse (8192 s)
5 1 0 1 5 2 0 2 5 3 0
0 . 9 0 7 2 4
0 . 9 0 7 2 5
0 . 9 0 7 2 6
0 . 9 0 7 2 7
1 0 2 0 3 0 4 0 5 0 6 0
- 0 . 2
- 0 . 1
0 . 1
0 . 2
2 0 4 0 6 0 8 0 1 0 0 1 2 0
- 0 . 0 0 7 5
- 0 . 0 0 5
- 0 . 0 0 2 5
0 . 0 0 2 5
0 . 0 0 5
0 . 0 0 7 5
5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
Fine (8192 s)
40962048
512
4 6 8 1 0 1 2 1 4 1 6
- 2 0
- 1 0
1 0
2 0
4 6 8 1 0 1 2 1 4 1 6
0 . 8 4 4
0 . 8 4 5
0 . 8 4 6
5 1 0 1 5 2 0 2 5 3 0
- 6
- 4
- 2
2
4
6
1 0 2 0 3 0 4 0 5 0 6 0
- 0 . 2
- 0 . 1
0 . 1
0 . 2
2 0 4 0 6 0 8 0 1 0 0 1 2 0
- 0 . 0 1
- 0 . 0 0 5
0 . 0 0 5
0 . 0 1
5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
5 1 0 1 5 2 0 2 5 3 0
0 . 9 0 7 2 4
0 . 9 0 7 2 5
0 . 9 0 7 2 6
0 . 9 0 7 2 7
1 0 2 0 3 0 4 0 5 0 6 0
- 0 . 2
- 0 . 1
0 . 1
0 . 2
2 0 4 0 6 0 8 0 1 0 0 1 2 0
- 0 . 0 0 7 5
- 0 . 0 0 5
- 0 . 0 0 2 5
0 . 0 0 2 5
0 . 0 0 5
0 . 0 0 7 5
5 0 1 0 0 1 5 0 2 0 0 2 5 0
- 0 . 0 0 0 4
- 0 . 0 0 0 2
0 . 0 0 0 2
0 . 0 0 0 4
CWM (8192 s)
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Simple illustration of the compounding process – Reconstruction
2000 4000 6000 8000
-2
-1
1
2
1
1 2
1
0, , ,2 20
1 1( , ) ( ) ( , ) ( )s
P Pf a b f a b
s
da daf W a b x db W a b x dbc a c aψ ψ
ψ ψ∞ ∞ ∞
∞−∞ −∞
= +∫ ∫ ∫ ∫
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Prototype Problem: Building Block for Heterogeneous Surface Reactions
Schematic of a simple A B heterogeneous chemical reaction with various elementary steps modeled using Kinetic Monte Carlo (KMC) and Lattice Boltzmann Method (LBM).
1. Transport (LBM)2. Diffusion on surface (KMC)3. Absorption (KMC)4. Production of B (KMC)5. Desorption of B (KMC)6. Transport (LBM)
Ni system for Chemical Looping
2Ni + O2 => 2 O-(Ni), FastO-(Ni) => (NiO), Slow
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Example 1: 1D diffusion with reacting boundary point
Fine scales results are obtained from the fine solution method while coarse ones are obtained from the coarse method.
Space (x)
Time (t)T1 T2
X1
X2
Δt1
Δx1
Δt2
Δx2
Fine scale Coarse scale Overlap
Diffusion
Reactions
KMC
Deterministic
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
100 200 300 40001020304050
*Frantziskonis, Mishra, Pannala, Simunovic, Daw, Nukala, Fox, Deymier (IJMCE, 2006).
Results for Example 1*
• Successfully applied CWM strategyfor coupling reaction/diffusion system
• An unique way to bridge temporal and spatial scales for multiphysics/multiscale simulations
Transferring mean field
Transferring fine-scale statistics
Spec
ies
conc
entr
atio
n A
(0,t)
20 40 60 80 100 1200
20
40
60
80
100
Time, t
Coarse
Fine
Time, t
A(0
,t)
100 200 300 40001020304050
Transferring mean field
01020304050
A(0
,t)
Time, t
CWMreconstruction
100 200 300 400100 200 300 400
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
2D diffusion with reacting boundary plane
Evolution of reactants A and B
Coarse Coarse
FineFine
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
2D diffusion with reacting boundary plane
Reconstructed species profile (effect of overlap and thresholding)
T = 1O = 5
T = 2O = 2
T = 3O = 5
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
2D diffusion with reacting boundary plane
Rel
ativ
e Er
ror, ε(
n)
Number of Overlap Scales, n
Rel
ativ
e Er
ror, ε(
n)Number of Overlap Scales, n
CWM Error Diffusion Error
The error is dominated by the discretization errorsin solving the diffusion equation
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
2D diffusion with reacting boundary plane
A gain of six times
This is without any coarsening in space
Comparison of computational expense
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Dynamic CWM (dCWM): Dynamic coupling of coarse and fine methods
• Coupling of the dynamics of both coarse and fine methods for non-stationary problems (similar to gap-tooth method)
• Better exploration of phase-space due to inclusion of stochasticity from fast scales
• Long term behavior feedback to fast scales from coarse representation
coarse Nc
fine Nf
1≤ n ≤ N
Cn CWM
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Example 3: 1D diffusion with reacting boundary plane with dCWM
Nc = 16384; Nf = 2048; N = 8C
A
Time
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Example 3: 1D diffusion with reacting boundary plane with dCWM
Nc = 16384; Nf = 2048; N = 8
dCWM is able to capture the later-time fluctuations in the mean trajectory when there is competition between diffusion and reaction processes.
Mean Trajectory
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Work in Progress: Reactive Boundary with LBM
• Chemical reactions in the flow are represented by mass source on RHS
• Implementation of boundary conditions for reactive boundary− Transport from bulk fluid to boundary
(flux/Neumann)− Reaction (concentration/Dirichlet)− Transport from boundary to bulk fluid
(flux/Neumann)− Reactive term must reproduce correct
density change rates for reactants, and total heat/release absorption per surface area
• Development of new combined flow-species transport with non-reflecting boundary conditions (absorbing layer, extrapolation method)
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Work in Progress and Future Work• Generalize the process of constructing the CWM in the
overlapping scales− Energy matching− Smooth variation of cross-correlation across the bridging
scales− Invoke conservation laws?
• Thermal LBM with chemistry• LBM coupled with KMC and CWM• Coarsening of KMC in space• MTS comparison to dCWM for time coupling• Application to NiO system and other realistic systems• Parallel framework to couple multiphysics code
− to be released as open source− solicit contributions from other applied math and
computational science groups
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Thank you and any questions?
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Backup Slides
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Background 1: Fluidized beds are widely used for gas-particle contacting (one special case of multiphase flow reactors
• Nonlinear gas-solid drag promotes turbulent mixing
• Good mixing produces high conversion, product quality
• Nonlinearities also cause density waves (e.g., bubbles) that interfere with good mixing, promote attrition
Challenge:• Direct measurements are very
difficult• Need simulations to improve
design and operating strategies• Several orders of magnitude in both
temporal and spatial scales− from the surface particle processes
scales to the large scale mixing scales
Reactant Gas
Exhaust Gas
Particle Bed
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
Background 5: Challenges in having predictive simulations for HCR Flows• How do we rewrite the equations or the solution methods so
that only relevant information is propagated upward from fine-to coarse-scales (upscaling) and coarse- to fine-scales (downscaling) in a tightly coupled fashion?− Possible when clear separation of scales between the
multiphysics modules− New mathematics, theory and analysis − Unification of governing equations across several scales
• Lattice based methods across all scales?• If that is not possible, can we take the information from
different methods and perform this in an online/offline fashion with various degrees of coupling?− Widely practiced− Can this be generalized?
OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY
CWM limitations
• Wavelets are linear operators− Compounding only buys linear superposition across
scales− Not an issue with well-separated scales− For non-separated scales, this would imply that the
CWM process has to be performed frequently to ensure local quasi-linear correlation across bridging scales
• The process developed in this project is general and down the line wavelets can be replaced with any other suitable nonlinear transforms