Sensitivity Analysis for DSMC
Simulations of HighTemperature Air Chemistry
James S. Strand and David B. Goldstein
The University of Texas at Austin
Sponsored by the Department of Energy through the
PSAAP Program
Predictive Engineering and
Computational Sciences
Computational Fluid
Physics Laboratory
Motivation – DSMC Parameters
• The DSMC model includes many parameters
related to gas dynamics at the molecular level, such
as:
 Elastic collision cross-sections.
 Vibrational and rotational excitation cross-sections.
 Reaction cross-sections.
 Sticking coefficients and catalytic efficiencies for
gas-surface interactions.
 …etc.
DSMC Parameters
• In many cases the precise values of some of these
parameters are not known.
• Parameter values often cannot be directly
measured, instead they must be inferred from
experimental results.
• By necessity, parameters must often be used in
regimes far from where their values were
determined.
• More precise values for important parameters
would lead to better simulation of the physics, and
thus to better predictive capability for the DSMC
method.
MCMC Method - Overview
• Markov Chain Monte Carlo (MCMC) is a method
which solves the statistical inverse problem in order
to calibrate parameters with respect to a set or sets
of experimental data.
MCMC Method
Establish
boundaries for
parameter space
Candidate
Rejected
Candidate
Accepted
Current position
remains
unchanged.
Candidate
position
becomes
current position
Select initial
position
Run simulation at
current position
Calculate
probability for
current position
Select new
candidate
position
Candidate position is
accepted, and becomes
the current chain
position
Accept or reject
candidate
position based
on a random
number draw
Probcandidate
< Probcurrent
Run simulation for
candidate position
parameters, and
calculate probability
Probcandidate
> Probcurrent
Candidate
automatically
accepted
Previous MCMC Results – Argon VHS Parameters
5.5E-10
5.5E-10
D ref (in meters)
6E-10
D ref (in meters)
6E-10
5E-10
5E-10
4.5E-10
4.5E-10
4E-10
4E-10
3.5E-10
3.5E-10
3E-10
0.5
0.6
0.7
0.8
0.9
3E-10
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
1
Omega
Omega
6E-10
5.5E-10
5.5E-10
D ref (in meters)
6E-10
D ref (in meters)
1
5E-10
5E-10
4.5E-10
4.5E-10
4E-10
4E-10
3.5E-10
3.5E-10
3E-10
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
Omega
1
3E-10
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
Omega
P. Valentini, T. E. Schwartzentruber, Physics of Fluids (2009), Vol. 21
1
Sensitivity Analysis - Overview
• In the current context, the goal of sensitivity
analysis is to determine which parameters most
strongly affect a given quantity of interest (QoI).
• Only parameters to which a given QoI is sensitive
will be informed by calibrations based on data for
that QoI.
• Sensitivity analysis is used here both to determine
which parameters to calibrate in the future, and to
select the QoI which would best inform the
parameters we most wish to calibrate.
Numerical Methods – DSMC Code
• Our DSMC code can model flows with rotational
and vibrational excitation and relaxation, as well as
five-species air chemistry, including dissociation,
exchange, and recombination reactions.
• Larsen-Borgnakke model is used for redistribution
between rotational, translational, and vibrational
modes during inelastic collisions.
• TCE model provides cross-sections for chemical
reactions.
Variable Hard Sphere Model
The VHS model allows the collision cross-section to be dependent
on relative speed, which is more physically realistic than the hard
sphere model.
There are two relevant parameters for the VHS model, dref and ω.
Internal Modes
• Rotation is assumed to be fully excited.

Each particle has its own value of rotational energy,
and this variable is continuously distributed.
• Vibrational levels are quantized.

Each particle has its own vibrational level, which is
associated with a certain vibrational energy based on
the simple harmonic oscillator model.
• Relevant parameters are ZR and ZV, the rotational
and vibrational collision numbers.



ZR = 1/ΛR, where ΛR is the probability of the rotational
energy of a given molecule being redistributed
during a given collision.
ZV = 1/ΛV
ZR and ZV are treated as constants.
Chemistry Implementation
Reaction cross-sections based on Arrhenius rates

TCE model allows determination of reaction crosssections from Arrhenius parameters.
𝑘 𝑇 = 𝑨𝑇 𝜼 𝑒 −𝑬𝒂 /𝑘𝑇
𝟏
𝜼
𝜞(𝜻 + 𝟓 𝟐 − 𝝎𝑨𝑩 )
𝝅
𝝈𝑹
𝒎𝒓
=
𝝈𝑻 𝟐𝝈𝒓𝒆𝒇 (𝒌𝑻𝒓𝒆𝒇 )𝜼−𝟏+𝝎𝑨𝑩 𝜞(𝜻 + 𝜼 + 𝟑 ) 𝟐𝒌𝑻𝒓𝒆𝒇
𝟐
𝟐 𝜺𝑨𝑻𝒓𝒆𝒇
𝟏
𝟐 (𝑬
𝜼+𝜻+
𝒄 − 𝑬𝒂 )
𝟏
𝟑 −𝝎
𝑨𝑩
𝟐
𝑬𝒄 𝜻+
σR and σT are the reaction and total cross-sections, respectively
𝜻 +𝜻
𝜻 = 𝐴 2 𝐵, the average number of internal degrees of freedom which
contribute to the collision energy.
𝜀 = 1 𝑖𝑓 𝐴 ≠ 𝐵 𝑜𝑟 2 𝑖𝑓 𝐴 = 𝐵
𝜎𝑟𝑒𝑓 and 𝑇𝑟𝑒𝑓 are both constants related to the VHS collision model
𝜔𝐴𝐵 is the temperature-viscosity exponent for VHS collisions
between type A and type B particles
k is the Boltzmann constant, mr is the reduced mass of particles
A and B, Ec is the collision energy, and Γ() is the gamma function.
𝟐
𝑘 𝑇 = 𝑨𝑇 𝜼 𝑒 −𝑬𝒂 /𝑘𝑇
Reaction #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Reactions
Reaction Equation
N2 + N2 --> N2 + N + N
N + N2 --> N + N + N
O2 + N2 --> O2 + N + N
O + N2 --> O + N + N
NO + N2 --> NO + N + N
N2 + O2 --> N2 + O + O
N + O2 --> N + O + O
O2 + O2 --> O2 + O + O
O + O2 --> O + O + O
NO + O2 --> NO + O + O
N2 + NO --> N2 + N + O
N + NO --> N + N + O
O2 + NO --> O2 + N + O
O + NO --> O + N + O
NO + NO --> NO + N + O
N2 + O --> NO + N
O2 + N --> NO + O
NO + N --> N2 + O
NO +O --> O2 + N
A
1.16E-08
4.98E-08
4.98E-08
4.98E-08
4.98E-08
3.32E-09
3.32E-09
3.32E-09
3.32E-09
3.32E-09
8.30E-15
8.30E-15
8.30E-15
8.30E-15
8.30E-15
9.45E-18
4.13E-21
2.02E-17
1.40E-17
η
-1.6
-1.6
-1.6
-1.6
-1.6
-1.5
-1.5
-1.5
-1.5
-1.5
0
0
0
0
0
0.42
1.18
0.1
0
EA
1.56E-18
1.56E-18
1.56E-18
1.56E-18
1.56E-18
8.21E-19
8.21E-19
8.21E-19
8.21E-19
8.21E-19
1.04E-18
1.04E-18
1.04E-18
1.04E-18
1.04E-18
5.93E-19
5.53E-20
0
2.65E-10
T. Ozawa, J. Zhong, and D. A. Levin, Physics of Fluids (2008), Vol. 20, Paper #046102.
Reaction Rates – Nitrogen Dissociation
2.0E+28
3
Reaction Rate (#/m -s)
1.7E+28
N2 + N2 --> N 2 + N + N (Arrhenius)
N2 + N2 --> N 2 + N + N (DSMC)
N + N2 --> N + N + N (Arrhenius)
N + N2 --> N + N + N (DSMC)
1.4E+28
1.1E+28
8.0E+27
5.0E+27
2.0E+27
0
5000
10000
15000
Temperature (K)
20000
25000
Reaction Rates – O2 and NO Dissociation
N2 + O2 --> N2 + O + O (Arrhenius)
N2 + O2 --> N2 + O + O (DSMC)
N + NO --> N + N + O (Arrhenius)
N + NO --> N + N + O (DSMC)
3
Reaction Rate (#/m -s)
1.5E+29
1.0E+29
𝝈𝑹 ≮ 𝝈𝑽𝑯𝑺
5.2E+28
2.0E+27
5000
10000
15000
Temperature (K)
20000
25000
Reaction Rates – NO Exchange Reactions
N2 + O --> NO + N (Arrhenius)
N2 + O --> NO + N (DSMC)
O2 + N --> NO + O (Arrhenius)
O2 + N --> NO + O (DSMC)
NO + N --> N2 + O (Arrhenius)
NO + N --> N2 + O (DSMC)
NO + O --> O2 + N (Arrhenius)
NO + O --> O2 + N (DSMC)
3
Reaction Rate (#/m -s)
2.0E+29
1.5E+29
𝝈𝑹 ≮ 𝝈𝑽𝑯𝑺
1.0E+29
5.2E+28
2.0E+27
5000
10000
15000
Temperature (K)
20000
25000
Parallelization
• DSMC:
 MPI parallel.
 Ensemble averaging to reduce stochastic noise.
 Fast simulation of small problems.
• Sensitivity Analysis:
 MPI Parallel
 Separate processor groups for each parameter.
 Large numbers of parameters can be examined
simultaneously.
0-D Relaxation, Pure Nitrogen
• Scenarios examined in this work are 0-D
relaxations from an initial high-temperature state.
• 0-D box is initialized with 100% N2.



Initial number density = 1.0×1023 #/m3.
Initial translational temperature = ~50,000 K.
Initial rotational and vibrational temperatures are
both 300 K.
• Scenario is a 0-D substitute for a hypersonic shock
at ~8 km/s.

Assumption that the translational modes equilibrate
much faster than the internal modes.
0-D Relaxation, Pure Nitrogen
50000
0.0050
0.0045
0.0040
30000
0.0030
Ttrans - N2
Trot - N2
Tvib - N2
Ttrans - N
 - N2
-N
20000
0.0025
0.0020
0.0015
10000
0.0010
0.0005
0
0
5E-07
1E-06
Time (s)
1.5E-06
0.0000
2E-06
3
0.0035
Density (kg/m )
Temperature (K)
40000
Quantity of Interest (QoI)
𝑸𝒐𝑰𝟏
𝑸𝒐𝑰𝟐
𝑸𝒐𝑰 = 𝑸𝒐𝑰𝟑
⋮
𝑸𝒐𝑰𝒏
J. Grinstead, M. Wilder, J. Olejniczak, D. Bogdanoff, G. Allen, and K. Danf, AIAA Paper 2008-1244, 2008.
Sensitivity Analysis - QoI
ωmin
ωnom
ωmax
dref,min
dref,nom
dref,max
ZR,min
ZV,min
ZR,nom
ZV,nom
ZR,max
ZV,max
Sensitivity Analysis – Type 1
ωmin
ωnom
ωmax
ω = ωmin
dref,nom
dref,min
dref,max
dref = dref,nom
ZR,min
ZR,nom
ZR,max
ZR = ZR,nom
ZV,min
ZV,nom
ZV = ZV,nom
ZV,max
ωmin
Sensitivity Analysis – Type 1
ωmin
ωnom
ωmax
ω = ωmax
dref,nom
dref,min
dref,max
dref = dref,nom
ZR,min
ZR,nom
ZR,max
ZR = ZR,nom
ZV,min
ZV,nom
ZV = ZV,nom
ZV,max
ωmin
ωmax
Sensitivity Analysis – Type 1
ωmin
ωmax
ωmin
ωnom
Δω = ωmax – ωmin
ωmax
Sensitivity Analysis – Type 1
𝜟𝑸𝒐𝑰𝟏
𝜟𝑸𝒐𝑰𝟐
𝜟𝑸𝒐𝑰 = 𝜟𝑸𝒐𝑰𝟑
⋮
𝜟𝑸𝒐𝑰𝒏
ΔQoI2
ΔQoI1
ΔQoI3
ΔQoIn
ωmin
ωnom
ωmax
𝑺𝒆𝒏𝒔𝒊𝒕𝒊𝒗𝒊𝒕𝒚 = 𝜟𝑸𝒐𝑰
Δω = ωmax – ωmin
𝑻
𝜟𝑸𝒐𝑰
Sensitivity Analysis – Type 2
𝜟𝑸𝒐𝑰𝟏
𝜟𝑸𝒐𝑰𝟐
𝜟𝑸𝒐𝑰 = 𝜟𝑸𝒐𝑰𝟑
⋮
𝜟𝑸𝒐𝑰𝒏
ωmin
ωnom
ωmax
Δω = (ωmax – ωmin)×0.10
𝑺𝒆𝒏𝒔𝒊𝒕𝒊𝒗𝒊𝒕𝒚 = 𝜟𝑸𝒐𝑰
𝑻
𝜟𝑸𝒐𝑰
Pure Nitrogen – Parameters
Parameter
Number
Parameter
Name
1
ω (N2-N2)
2
ω (N2-N)
3
ω (N-N)
4
dref (N2-N2)
5
dref (N2-N)
6
dref (N-N)
7
8
ZR
ZV
9
α1
10
α2
Meaning
Temperature-viscosity
exponent for N2-N2 collisions
Temperature-viscosity
exponent for N2-N collisions
Temperature-viscosity
exponent for N-N collisions
VHS reference diameter for
N2-N2 collisions
VHS reference diameter for
N2-N collisions
VHS reference diameter for
N-N collisions
Rotational collision number
Vibrational collision number
𝟏𝟎𝜶𝟏 = 𝑨𝟏 , the pre-exponential
constant for the reaction
N2 + N2 --> N2 + N + N
𝟏𝟎𝜶𝟐 = 𝑨𝟐 , the pre-exponential
constant for the reaction
N + N2 --> N + N + N
Minimum
Nominal
Maximum
0.5
0.68
1.0
0.5
0.665
1.0
0.5
0.65
1.0
2.00E-10 (m)
3.58E-10 (m)
5.00E-10 (m)
2.00E-10 (m)
3.35E-10 (m)
5.00E-10 (m)
2.00E-10 (m)
3.11E-10 (m)
5.00E-10 (m)
1
1
5
10
10
50
-8.94
(A1 = 1.16E-9)
-7.94
(A1 = 1.16E-8)
-6.94
(A1 = 1.16E-7)
-8.30
(A2 = 4.98E-9)
-7.30
(A2 = 4.98E-8)
-6.30
(A2 = 4.98E-7)
Pure Nitrogen – Results
1.00
0.25
0.77
≈≈
Normalized Sensitivity
0.20
α1
(N2 + N2  N2 + N + N)
α2
(N + N2  N + N + N)
0.15
dref (N2-N2)
ω (N2-N2)
0.10
ZV
Numerical
Parameters
0.05
ZR
0.00
1
2
3
4
5
6
7
8
9
10
Parameter
Sensitivity Analysis Type 1
11
12
13
Pure Nitrogen – Results
1.00
0.25
0.53
≈≈
α1
(N2 + N2  N2 + N + N)
Normalized Sensitivity
0.20
0.15
α2
(N + N2  N + N + N)
ZV
0.10
dref (N2-N2)
ω (N2-N2)
0.05
Numerical
Parameters
ZR
0.00
1
2
3
4
5
6
7
8
9
10
Parameter
Sensitivity Analysis Type 2
11
12
13
Pure Nitrogen – Results
Sensitivity Rank
1
2
3
4
5
6
7
8
Sensitivity Type 1
α2
α1
ω (N2-N2)
dref (N2-N2)
ZV
ZR
dref (N2-N)
ω (N2-N)
Sensitivity Type 2
α2
α1
ZV
ω (N2-N2)
dref (N2-N2)
ZR
dref (N2-N)
ω (N2-N)
|QoI| (K)
1500
 (N2-N2)
dref (N2-N2)
ZR
ZV
1
2
RF Seed
1000
500
0
5E-07
1E-06
Time (s)
1.5E-06
2E-06
0-D Relaxation, Five-Species Air
• Another 0-D relaxation from an initial hightemperature state.
• 0-D box is initialized with 79% N2, 21% O2.



Initial bulk number density = 1.0×1023 #/m3.
Initial bulk translational temperature = ~50,000 K.
Initial bulk rotational and vibrational temperatures are
both 300 K.
• Scenario is a 0-D substitute for a hypersonic shock
at ~8 km/s.

Assumption that the translational modes equilibrate
much faster than the internal modes.
Five-Species Air – Densities
0.005
Bulk
N2
N
O2
O
NO
3
Density (kg/m )
0.004
0.003
0.002
0.001
0
0
5E-07
1E-06
Time (s)
1.5E-06
2E-06
Five-Species Air – Translational
Temperatures
50000
Bulk
N2
N
O2
O
NO
Ttrans (K)
40000
30000
20000
10000
0
0
5E-07
1E-06
Time (s)
1.5E-06
2E-06
Five-Species Air - Parameters
𝑘 𝑇 = 𝑨𝑇 𝜼 𝑒 −𝑬𝒂 /𝑘𝑇
Reaction #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Equation
N2 + N2 --> N2 + N + N
N + N2 --> N + N + N
O2 + N2 --> O2 + N + N
O + N2 --> O + N + N
NO + N2 --> NO + N + N
N2 + O2 --> N2 + O + O
N + O2 --> N + O + O
O2 + O2 --> O2 + O + O
O + O2 --> O + O + O
NO + O2 --> NO + O + O
N2 + NO --> N2 + N + O
N + NO --> N + N + O
O2 + NO --> O2 + N + O
O + NO --> O + N + O
NO + NO --> NO + N + O
N2 + O --> NO + N
O2 + N --> NO + O
NO + N --> N2 + O
NO +O --> O2 + N
10𝛼 = 𝑨
αmin αnom αmax
-6.94
-6.30
-6.30
-6.30
-6.30
-7.48
-7.48
-7.48
-7.48
-7.48
-13.08
-13.08
-13.08
-13.08
-13.08
-16.02
-19.38
-15.69
-15.85
-7.94
-7.30
-7.30
-7.30
-7.30
-8.48
-8.48
-8.48
-8.48
-8.48
-14.08
-14.08
-14.08
-14.08
-14.08
-17.02
-20.38
-16.69
-16.85
-8.94
-8.30
-8.30
-8.30
-8.30
-9.48
-9.48
-9.48
-9.48
-9.48
-15.08
-15.08
-15.08
-15.08
-15.08
-18.02
-21.38
-17.69
-17.85
Anom
η
EA
1.16E-08
4.98E-08
4.98E-08
4.98E-08
4.98E-08
3.32E-09
3.32E-09
3.32E-09
3.32E-09
3.32E-09
8.30E-15
8.30E-15
8.30E-15
8.30E-15
8.30E-15
9.45E-18
4.13E-21
2.02E-17
1.40E-17
-1.6
-1.6
-1.6
-1.6
-1.6
-1.5
-1.5
-1.5
-1.5
-1.5
0
0
0
0
0
0.42
1.18
0.1
0
1.56E-18
1.56E-18
1.56E-18
1.56E-18
1.56E-18
8.21E-19
8.21E-19
8.21E-19
8.21E-19
8.21E-19
1.04E-18
1.04E-18
1.04E-18
1.04E-18
1.04E-18
5.93E-19
5.53E-20
0
2.65E-10
Five-Species Air - Results
Normalized Sensitivity
• We used only sensitivity analysis type 2 for the
five species air scenario.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
N2 + O  NO + N
Nitrogen
Dissociation
Reactions
Oxygen
Dissociation
Reactions
NO
Dissociation
Reactions
NO + N  N2 + O
Numerical
Parameters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Parameter
QoI = Ttrans,N
NO Exchange
Reactions
Five-Species Air - Results
Normalized Sensitivity
• We also tested sensitivity with respect to a second
QoI, the mass density of NO.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
N2 + O  NO + N
NO + N  N2 + O
Nitrogen
Dissociation
Reactions
Oxygen
Dissociation
Reactions
NO
Dissociation
Reactions
Numerical
Parameters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Parameter
QoI = ρNO
NO Exchange
Reactions
Five-Species Air - Results
Sensitivity
Rank
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
QoI = Ttrans,N
Equation
Reaction #
N2 + O --> NO + N
16
NO + N --> N2 + O
18
N + N2 --> N + N + N
2
N2 + NO --> N2 + N + O
11
N2 + N2 --> N2 + N + N
1
O + N2 --> O + N + N
4
N2 + O2 --> N2 + O + O
6
N + NO --> N + N + O
12
O2 + N --> NO + O
17
O + NO --> O + N + O
14
O2 + N2 --> O2 + N + N
3
N + O2 --> N + O + O
7
O2 + O2 --> O2 + O + O
8
O + O2 --> O + O + O
9
NO + N2 --> NO + N + N
5
-
QoI = ρNO
Equation
N2 + O --> NO + N
NO + N --> N2 + O
N2 + NO --> N2 + N + O
O2 + N --> NO + O
N + NO --> N + N + O
O + NO --> O + N + O
N + N2 --> N + N + N
N2 + O2 --> N2 + O + O
O + N2 --> O + N + N
N2 + N2 --> N2 + N + N
O2 + N2 --> O2 + N + N
N + O2 --> N + O + O
NO + NO --> NO + N + O
O + O2 --> O + O + O
NO + N2 --> NO + N + N
NO +O --> O2 + N
O2 + O2 --> O2 + O + O
O2 + NO --> O2 + N + O
Reaction #
16
18
11
17
12
14
2
6
4
1
3
7
15
9
5
19
8
13
Conclusions
Pure nitrogen scenario:


Sensitivities to reaction rates dominate all others.
ZR, ZV, and VHS parameters for N2-N2 collisions are
important in the early stages of the relaxation.
Five-species air scenario:



Sensitivities for the forward and backward rates for the
reaction N2 + O ↔ NO + N are dominant when using
either Ttrans,N or ρNO as the QoI.
NO dissociation reactions are moderatly important for
either QoI.
Nitrogen and oxygen dissociation reactions are
important only for the Ttrans,N QoI.
Future Work
• Perform calibration with synthetic data for the 0-D
relaxation scenarios.
• Perform synthetic data calibrations for a 1-D shock
with chemistry.
• Perform calibrations with real data from EAST or
similar facility.