Computationally efficient dimension reduction of ...
Transcript of Computationally efficient dimension reduction of ...
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 1
Computationally efficient dimension reduction of combustion chemistry
via Principal Components Analysis based domain partitioning
Federico Perini, Rolf D. Reitz
University of Wisconsin-Madison, USA
Frontiers in Computational Physics: Energy Sciences
Zurich, Switzerland
June 5, 2015
Acknowledgements
U.S. D.O.E. Office of Vehicle Technologies, PMs Leo Breton, Gupreet Singh
Sandia National Laboratories
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 2
Summary - Motivation
Combustion research has been guided in the last decade by advances in computer modeling
The success of advanced combustion strategies relies on local mixture reactivity crucial to controllability, but also emissions
Need to incorporate realistic combustion chemistry in multi-
dimensional simulations for quantitative predictions in ignition and pollutant formation
Long-term goal: Well-resolved multi-physics modeling of ICE flows, sprays, and chemistry
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 3
Chemical Kinetics in CFD simulations
Usually part of an operator-splitting scheme
Each cell is treated as an adiabatic well-stirred reactor “Embarassingly” parallel problem
Very stiff ODE system need for appropriate integrators
Only the overall changes in species mass fractions and cell internal energy are passed to the flow solver
Skeletal mechanism or on-the-fly reduction
Reduce number of integrations storage/retrieval, multi-zone approaches, clustering
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 4
Potential for speed-up
ODE system level (1 reactor)
finite volume domain level (N reactors)
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 5
A new Modern Fortran library for standalone reactor calculations or the simulation of chemically reactive systems of gaseous mixtures
SpeedCHEM chemistry solver
Solution of chemical kinetic ODE systems
,,,1,
1
,
1
,
ss n
i
riik
n
i
iik nkMM
,1
,, k
n
k
ikikii q
W
dt
dY r
,1
1
sn
i
i
i
i
v dt
dY
W
U
cdt
dT
mass cons.
energy cons. (adiabatic const. V)
Provide the fastest accurate kinetics for large-size mechanisms, but with focus
on efficiency for practical multi-dimensional simulations (ns 100-1000)
Linear scaling of the solution time vs. problem size
101
102
103
104
10-3
10-2
10-1
100
101
102
103
104
number of species
CP
U t
ime
[s]
SpeedCHEM ignition delay time calculation scaling
Direct dense Jacobian
SpeedCHEM, direct sparse
SpeedCHEM, Krylov
ns
ns
3
available at http://www.erc.wisc.edu/chemicalreaction.php Perini et al., Energy&Fuels 26 (8), 4804-4822, 2012 Perini et al., Comb Flame 161(5), 1180-1195, 2014
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
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Solution space reduction approach
The idea of grouping cells with similar reactivity is not new*
Usually based on -T maps for engine calculations
Search for similar cells based on proximity (neighbors, ROI) or clustering (k-means)
Chemistry is solved for at each cluster, then conservatively re-distributed
*Babajimopoulos et al., Liang et al., Barths et al., Shi et al., Puduppakkam et al. , Perini et al.
Chemistry has to be solved on each cell of the mesh
Reactively similar cells are identified using a clustering algorithm Their thermophysical properties are averaged into ‘cluster’ cells
Chemical Kinetics are solved for at the ‘cluster’ cells
Changes in composition due to the reactions are mapped back
Updated species compositions are sent back to the flow solver as source terms
• The -T is specific to single-component fuel kinetics;
• Even with extremely large mechanisms, chemically reacting environments converge to low-dimensional manifolds;
• Inner homogeneity of the cell clusters should be defined and treated rigorously, i.e. by a general model dimensionality reduction method.
Why develop a different (better) approach?
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 7
[1] PCA-based states space dimension reduction
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 8
1st dimension reduction of the chemistry domain
Principal Component Analysis of the states space
Given a homogeneous reactor state
the instantaneous state of the whole CFD domain is given by the matrix of states of all active cells:
1
21
s
s
ndT
nYYYT y
cncnd yyyyY 1 32
Hp: Chemistry states converge to low-dimensional manifolds
dnpc
nd pc ,
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 9
Principal Component Analysis of the states space
Principal Component Analysis (PCA) applies a linear transformation P
coordinates changed into a more convenient point of view:
where
is the matrix containing the first npc principal components.
Reduced dimensionality npc is based on a variance fraction threshold:
YΠPT
cpc nn
pcnpcnd πππΠ 21
If v = 0.01, 99% of the total state space variance in the domain is retained
1
1,var1,var
pcn
jvpc jn PP
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 10
Principal Component Analysis of the states space
1] Linearity of the Y P transformation easy to compute
2] All principal components are orthonormal special properties
3] They are sorted in descending order of variance
Variance-covariance matrix of a dataset:
T
cndd YYCY
1
component principaleach along of variances thecontains
diagonal is
of thecontains
YC
ΠΠCC
CΠ
P
YP
Y
then
nnthen
rseigenvecto
T
pcpc
ji,,cov Y
j,var Y
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 11
[2] kd-tree + k-means mapping into homogeneous clusters
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 12
Partitioning of the reduced states space
Aim: to group states into ‘clusters’ which retained variance fraction along each reduced dimension is less than v
[1] kd-tree partitioning of the dataset – O(log(nc))
Level 1: dimension 1 Level 2: dimension 2 Level 3: dimension 1 Etc…. 1
3 2
6 5 4
Node = median point of i-th dimension = split plane = subset
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 13
Partitioning of the reduced states space
[2] bounding-box-constrained k-means – O(2npc nc)
Leaves of the kd-tree are the final variance-bound boxes Find the optimal cluster centers to the dataset, subject to: Points belong to any bounding cluster centers to their leaf K-means cluster initialization: box vertices
Chemical Kinetics ODE solved over clusters = thermodynamic averages of their member states
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 14
[3] Forward Sensitivity Analysis based remapping to the full solution space
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 15
Linearization of the solution space
Chem. Kin. ODE integration (tt+t) = path in full states space. Per given t, a mapping function
For y small enough,
1y
0yK
2y3y
0y
1yΚ
2yK
3yK
tt
t
nsns dtt0
,,:11yyfyKK 00
tt
t
nsndttpc ,,: 0
1ppfpκκ 0
From the reduced space,
yyy 0,0, cp
0,cyK
y
0,cy
yyKyK
y
0,
0,0,
cyj
icp
y
K
Matrix of linearized mapping gradients
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 16
Sensitivity Analysis for linearized solution mapping
The linear mapping gradients correspond to first-order sensitivities of the ODE system w.r.t. the initial conditions
evaluated at the final integration time
IyS
ySyJySyS
y0,
,,,:,
0
t
tttt
y
K
ddyj
i
Dense system expensive, computed by CVODES together with the ODEs
defined a reduced sensitivity system, w.r.t. the initial PCs:
0pS
pSΠpyJpSpSpS
p0,
~
,~
,,~
,~
:,~
0 t
tttdp
ft
tp
Kk
i
nnpj
i
pcpc
O(ns2)
O(npc2)O(102)
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 17
Sensitivity Analysis for linearized solution mapping
The ODE system solution at the cluster centers is eventually remapped back to each member point in the full space as:
pppκpκy 0,0,0, ,~
pcpp tStt
The formulation is intrinsically mass- and element- conserving because the reduced sensitivity system is an orthonormal transformation of the Jacobian
0,py ttc y
ttp y
0,cy
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 18
Results
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 19
Code setup for engine combustion
CFD solver: KIVA-ERC
with improved sumbodels
Chemistry solver:
Parameter Value
RTOL 10-4
ATOL 10-15
Integrator CVODES
Sensitivity Analysis
Forward,
simultaneous,
same tolerances
(unactive in full
chemistry cases)
Phenomenon Sub-model
Spray breakup KH-RT instability, Beale
and Reitz
Near-nozzle flow Gas-jet theory, Abani et al.
Droplet collision ROI w/ extended outcomes,
Munnannur and Reitz
Wall film O’Rourke and Amsden
Evaporation Discrete Multi-Component,
Ra and Reitz
Turbulence RNG k-epsilon, Han and
Reitz
Combustion SpeedCHEM, Perini et al.
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 20
HCCI combustion in a light duty engine
Engine specifications
Bore x stroke [mm] 82.0 x 90.4
Unit displacement [cm3] 477.2
Compression ratio 16.4 : 1
Squish height at TDC [mm] 0.88
Bosch CRI2.2 Injector parameters
Sac volume [mm3] 0.23
Number of holes 7
Included angle [deg] 149
Nozzle diameter [mm] 0.14
Hole protrusion [mm] 0.3
Fuel properties for PLIF studies
Composition [mole fractions] 42% nC16H34
58% iso-C16H34
Fluorescent tracer [mass fraction] 0.5% 1-C11H10
Equivalent Cetane Number [-] 50.7
Fuel properties for ignition studies
US #2 diesel fuel CN=47
HCCI-operated GM 1.9L light duty engine with a Primary Reference Fuel
Experiments run in single cylinder conf. at UW-DERC lab (Dempsey et al., 2013)
3 reaction mechanisms 2d axisymmetric grid, 3.1k cells
ns = 110 nr = 550
ns = 1034 nr = 4236
ns = 47 nr = 142
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 21
[small] ERC PRF, ns=47, nr=142
All cases of v 110-2 do a pretty good job at capturing thermodynamic (pressure, HRR) quantities and species mass fractions (small!)
Means at least 99% of the total variance must be represented even in this simple case
-25 -20 -15 -10 -5 0 5 10 15 201
2
3
4
5
6
7
8
9x 10
6 ERC PRF, ns=47
crank angle [degrees ATDC]
pre
ssu
re [
Pa
]
0
50
100
150
200
hea
t rele
ase
[J
/deg
]
v = 1 10-3
v = 5 10-3
v = 1 10-2
v = 5 10-2
v = 1 10-1
full chem.
-25 -20 -15 -10 -5 0 5 10 15 2010
-8
10-7
10-6
10-5
10-4
ERC PRF, ns=47
crank angle [degrees ATDC]
CO
ma
ss f
ract
ion
[-]
10-20
10-15
10-10
NO
mass
fracti
on
[-]
v = 1 10-3
v = 5 10-3
v = 1 10-2
v = 5 10-2
v = 1 10-1
full chem.
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 22
[small] ERC PRF, ns=47, nr=142
full chemistry PCASA, v = 5e-3
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 23
[medium] ERC PRF-PAH, ns=110, nr=550
Similar high temperature ignition timing, higher LTHR peak
Similar performance as for the smaller mechanism
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-8
10-7
10-6
10-5
10-4
ERC PRF-PAH, ns=110
crank angle [degrees ATDC]
CO
ma
ss f
ract
ion
[-]
10-20
10-15
10-10
NO
mass
fracti
on
[-]
v = 1 10-3
v = 5 10-3
v = 1 10-2
v = 5 10-2
v = 1 10-1
full chem.
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2
3
4
5
6
7
8
9x 10
6 ERC PRF-PAH, ns=110
crank angle [degrees ATDC]
pre
ssu
re [
Pa
]
0
50
100
150
200
heat
rele
ase
[J/d
eg]
v = 1 10-3
v = 5 10-3
v = 1 10-2
v = 5 10-2
v = 1 10-1
full chem.
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 24
[large] LLNL, ns=1034, nr=4236
Detailed low-temperature chemistry is very well captured but no main ignition sensitivity to initial conditions, geometry simplification, mechanism validation only with zero- and one-dimensional models
From the PCASA reduction standpoint, the variance parameter seems insensitive to mechanism size as earlier, v 110-2
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2
3
4
5
6x 10
6 LLNL PRF, ns=1034
crank angle [degrees ATDC]
pre
ssu
re [
Pa
]
0
2
4
6
8
10
heat
rele
ase
[J/d
eg]
v = 1 10-3
v = 5 10-3
v = 1 10-2
v = 5 10-2
v = 1 10-1
full chem.
-25 -20 -15 -10 -5 0 5 10 15 20
10-20
10-15
10-10
10-5
LLNL PRF, ns=1034
crank angle [degrees ATDC]
CO
ma
ss f
ract
ion
[-]
10-16
10-14
10-12
10-10
10-8
OH
mass
fracti
on
[-]
v = 1 10-3
v = 5 10-3
v = 1 10-2
v = 5 10-2
v = 1 10-1
full chem.
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 25
# of clusters, # of principal components
Both increase monotonically with reduced variance fraction
Instantaneous # of clusters not necessarily correlated w/ # of PCs
Peak of PCs and cell clusters during cool-flame LTHR region!
-50 0 50 1000
5
10
15
ERC PRF-PAH, ns=110
crank angle [degrees ATDC]
cell
clu
ster d
imen
sion
s [-
]
v = 1 10-3
v = 5 10
-3
v = 1 10
-2
v = 5 10
-2
v = 1 10-1
-40 -20 0 20 40 60 8010
0
101
102
103
LLNL PRF, ns=1034
crank angle [degrees ATDC]
cell
clu
sters
[-]
v = 1 10
-3
v = 5 10-3
v = 1 10-2
v = 5 10-2
v = 1 10
-1
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 26
full 1e-3 5e-3 1e-2 5e-2 1e-10
2
4
6
8
10
9.57
2.07
1.090.70
0.48 0.50
9.77
2.27
1.290.91
0.68 0.69
tota
l w
all
tim
e [h
]
LLNL, ns=1034
spray
chemistry
advection
diffusion
communications
not mapped
CPU time performance
Chemistry dominates these sims. 9x speed-up for v = 5e-3
Peak in CPU time / timestep during low-temperature ignition
-50 0 50 1000
20
40
60
80
100
LLNL PRF, ns = 1034
crank angle [degrees ATDC]g
rid
in
dex
[m
s/ti
me-
step
]
v = 1 10-3
v = 5 10-3
v = 1 10-2
v = 5 10-2
v = 1 10-1
full chem.
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 27
ns
v
PCASA vs. full chem. accuracy
1.5
1.5
2
2
2
2
2
2.5
3 3.5 4 4
4.5
5
47 110 10340.1%
1%
10%
Performance vs. accuracy
2
10
20
20
30
40
50
60
70
8090
ns
v
PCASA vs. full chem. speed-up
47 110 10340.1%
1%
10%
Detailed mechanisms are more sensitive to the variance tolerance
tendt
n
i fulli
fulliPCAi
PCA
s
Y
YYerr
1 ,
,,
The ‘trade-off region’ is for 0.001<v<0.005
Speed-ups of the order of 10-20 times at this simple case
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 28
Conclusions
Computationally efficient solution of combustion chemical kinetics in multidimensional CFD codes achieved through dimension reduction:
1) Dimension reduction of domain states via PCA;
2) Variance-based partitioning of the reduced
space via kd-tree and k-means
3) Mass- and element-conserving remapping of the solution space based on Sensitivity Analysis with respect to the initial reduced states
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 29
Conclusions
Application of the proposed reduction approach to HCCI engine combustion showed up to 2 orders of magnitude speed-up with respect to the full chemistry approach, regardless of the mechanism size
Need to test the approach against larger-scale, parallel simulations of spray engine combustion
The proposed method can be applied to other problems characterized by large numbers of stiff ODE system solutions
University of Wisconsin -- Engine Research Center
Frontiers in Computational Physics: Energy Sciences – Zurich, Jun 3-5, 2015
slide 30
Acknowledgements
U.S. D.O.E. Office of Vehicle Technologies, PMs Leo Breton, Gupreet Singh
Paul C. Miles, Stephen Busch - Sandia National Laboratories