Bayesian Inference For The Calibration Of DSMC Parameters

Bayesian Inference For The Calibration Of DSMC Parameters

James S. Strand and David B. GoldsteinThe 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 probabilities. 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. The ultimate goal of this work is to use experimental data to calibrate important DSMC parameters.

DSMC Method

Direct Simulation Monte Carlo (DSMC) is a particle based simulation method.

• Simulated particles represent large numbers of real particles.• Particles move and undergo collisions with other particles.• Can be used in highly non-equilibrium flowfields (such as strong shock waves). • Can model thermochemistry on a more detailed level than most CFD codes.

DSMC Method

Initialize

Move

Index

Collide

Sample

Performed on First Time Step

Performed on Selected Time Steps

Performed on Every Time Step

Create

DSMC Collisions

DSMC collisions are performed statistically. Pairs of molecules are randomly selected from within a cell, a collision probability is calculated for each selected pair, and a random number draw determines whether or not the pair actually collides.

DSMC Collisions


DSMC Collisions


Selection Possibilities

DSMC Collisions

If the pair does not collide, their properties are left unchanged and a new pair is selected. If the pair does collide, the properties of both colliding particles are immediately adjusted to their calculated post-collision values, and then a new pair is selected. The number of selections in a given cell is a function of the overall simulation conditions and of the conditions within that cell at that time step.

DSMC Collisions

If the pair does not collide, their properties are left unchanged and a new pair is selected. If the pair does collide, the properties of both colliding particles are immediately adjusted to their calculated post-collision values, and then a new pair is selected. The number of selections in a given cell is a function of the overall simulation conditions and of the conditions within that cell at that time step.

Variable Hard Sphere ModelThe VHS model allows the collision cross-section to be dependent on relative speed.

There are two relevant parameters for the VHS model, dref and ω. Both of these parameters are usually determined based on viscosity data.

𝝈𝑽𝑯𝑺=𝝅𝒅𝑽𝑯𝑺=𝝅𝒅𝒓𝒆𝒇 (𝒄𝒓 ,𝒓𝒆𝒇𝒄𝒓 )(𝝎−𝟏𝟐 )

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 allows cross-sections for chemical reactions to be derived from Arrhenius parameters.

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

Chemistry Implementation

Reaction cross-sections based on Arrhenius rates TCE model allows determination of reaction cross-

sections from Arrhenius parameters.

, the average number of internal degrees of freedom which contribute to the collision energy.

is the temperature-viscosity exponent for VHS collisions between type A and type B particles

𝜎 𝑟𝑒𝑓∧𝑇 𝑟𝑒𝑓 are both constants related ¿ the VHS collisionmodel𝜀=1 (𝑖𝑓 𝐴≠𝐵 )𝑜𝑟 2 (𝑖𝑓 𝐴=𝐵 )

σR and σT are the reaction and total cross-sections, respectively

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.

Reactions𝑘 (𝑇 )=𝚲𝑇𝜼𝑒−𝑬𝒂 /𝑘𝑇

R. Gupta, J. Yos, and R. Thompson, NASA Technical Memorandum 101528, 1989.

# Reaction Forward Rate Constants Backward Rate Constants

qreaction Λ η EA Λ η EA

1 N2 + N2 <--> N2 + N + N 7.968E-13 -0.5 1.561E-18 6.518E-47 0.27 0.0 -1.561E-18 2 N + N2 <--> N + N + N 6.9E-8 -1.5 1.561E-18 4.817E-46 0.27 0.0 -1.561E-18 3 O2 + N2 <--> O2 + N + N 3.187E-13 -0.5 1.561E-18 6.518E-47 0.27 0.0 -1.561E-18 4 O + N2 <--> O + N + N 3.187E-13 -0.5 1.561E-18 6.518E-47 0.27 0.0 -1.561E-18 5 NO + N2 <--> NO + N + N 3.187E-13 -0.5 1.561E-18 6.518E-47 0.27 0.0 -1.561E-18 6 N2 + O2 <--> N2 + O + O 1.198E-11 -1.0 8.197E-19 1.8E-47 0.27 0.0 -8.197E-19 7 N + O2 <--> N + O + O 5.993E-12 -1.0 8.197E-19 1.8E-47 0.27 0.0 -8.197E-19 8 O2 + O2 <--> O2 + O + O 5.393E-11 -1.0 8.197E-19 1.8E-47 0.27 0.0 -8.197E-19 9 O + O2 <--> O + O + O 1.498E-10 -1.0 8.197E-19 1.8E-47 0.27 0.0 -8.197E-19 10 NO + O2 <--> NO + O + O 5.993E-12 -1.0 8.197E-19 1.8E-47 0.27 0.0 -8.197E-19 11 N2 + NO <--> N2 + N + O 6.59E-10 -1.5 1.043E-18 2.0976E-46 0.27 0.0 -1.043E-18 12 N + NO <--> N + N + O 1.318E-8 -1.5 1.043E-18 2.0976E-46 0.27 0.0 -1.043E-18 13 O2 + NO <--> O2 + N + O 6.59E-10 -1.5 1.043E-18 2.0976E-46 0.27 0.0 -1.043E-18 14 O + NO <--> O + N + O 1.318E-8 -1.5 1.043E-18 2.0976E-46 0.27 0.0 -1.043E-18 15 NO + NO <--> NO + N + O 1.318E-8 -1.5 1.043E-18 2.0976E-46 0.27 0.0 -1.043E-18 16 N2 + O <--> NO + N 1.12E-16 0.0 5.175E-19 2.490E-17 0.0 0.0 -5.175E-19 17 NO + O <--> O2 + N 5.279E-21 1.0 2.719E-19 1.598E-18 0.5 4.968E-20 -2.719E-19

Collision Rates

Temperature (K)

CollisionRate(#/m

3 )

5000 10000 15000 20000 25000

1E+29

3E+29

5E+29

7E+29

N2 - N2 (DSMC)N2 - N2 (VHS)N2 - N (DSMC)N2 - N (VHS)N - N (DSMC)N - N (VHS)N - O (DMSC)N - O (VHS)O2 - NO (DSMC)O2 - NO (VHS)

VHS Collision Pairs

Reaction Rates – Dissociation

Temperature (K)

ReactionRate(#/m

3 )

5000 10000 15000 20000 25000

1.0E+28

3.0E+28

5.0E+28

7.0E+28N2 + N2 --> N + N + N2 (DSMC)N2 + N2 --> N + N + N2 (Arrhenius)N2 + N --> N + N + N (DSMC)N2 + N --> N + N + N (Arrhenius)O2 + N2 --> O + O + N2 (DSMC)O2 + N2 --> O + O + N2 (Arrhenius)NO + O --> N + O + O (DSMC)NO + O --> N + O + O (Arrhenius)

Dissociation Reactions

Temperature (K)

ReactionRate(#/m

3 )

5000 10000 15000 20000 25000

1E+28

3E+28

5E+28

N + N + N2 --> N2 + N2 (DSMC)N + N + N2 --> N2 + N2 (Arrhenius)N + N + N --> N2 + N (DSMC)N + N + N --> N2 + N (Arrhenius)O + O + O --> O2 + O (DSMC)O + O + O --> O2 + O (Arrhenius)N + O + N --> NO + N (DSMC)N + O + N --> NO + N (Arrhenius)

Recombination Reactions

Reaction Rates – Recombination

Reaction Rates – Exchange

Temperature (K)

ReactionRate(#/m

3 )

5000 10000 15000 20000 25000

2E+28

4E+28

6E+28

8E+28

N2 + O --> NO + N (DSMC)N2 + O --> NO + N (Arrhenius)O2 + N --> NO + O (DSMC)O2 + N --> NO + O (Arrhenius)NO + N --> N2 + O (DSMC)NO + N --> N2 + O (Arrhenius)NO + O --> O2 + N (DSMC)NO + O --> O2 + N (Arrhenius)

Exchange Reactions

Parallelization

• DSMC: MPI parallel. Ensemble averaging to reduce stochastic noise for

0-D relaxations. Adaptive load balancing for 1-D shock simulations.

• Sensitivity Analysis and MCMC: MPI Parallel Separate processor groups call DSMC subroutine

independently.

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 Parameters𝑘 (𝑇 )=𝚲𝑇𝜼𝑒−𝑬𝒂 /𝑘𝑇 10𝛼=𝚲

Throughout the sensitivity analysis, the ratio of forward to backward rate for a given reaction is kept constant, since these ratios should be fixed by the equilibrium constant.

# Reaction αmin αnom αmax Nominal Forward Rate Constants

Λ η EA 1 N2 + N2 <--> N2 + N + N -13.099 -12.099 -11.099 7.968E-13 -0.5 1.561E-18 2 N + N2 <--> N + N + N -8.161 -7.161 -6.161 6.9E-8 -1.5 1.561E-18 3 O2 + N2 <--> O2 + N + N -13.497 -12.497 -11.497 3.187E-13 -0.5 1.561E-18 4 O + N2 <--> O + N + N -13.497 -12.497 -11.497 3.187E-13 -0.5 1.561E-18 5 NO + N2 <--> NO + N + N -13.497 -12.497 -11.497 3.187E-13 -0.5 1.561E-18 6 N2 + O2 <--> N2 + O + O -11.922 -10.922 -9.922 1.198E-11 -1.0 8.197E-19 7 N + O2 <--> N + O + O -12.222 -11.222 -10.222 5.993E-12 -1.0 8.197E-19 8 O2 + O2 <--> O2 + O + O -11.268 -10.268 -9.268 5.393E-11 -1.0 8.197E-19 9 O + O2 <--> O + O + O -10.824 -9.824 -8.824 1.498E-10 -1.0 8.197E-19 10 NO + O2 <--> NO + O + O -12.222 -11.222 -10.222 5.993E-12 -1.0 8.197E-19 11 N2 + NO <--> N2 + N + O -10.181 -9.181 -8.181 6.59E-10 -1.5 1.043E-18 12 N + NO <--> N + N + O -8.880 -7.880 -6.880 1.318E-8 -1.5 1.043E-18 13 O2 + NO <--> O2 + N + O -10.181 -9.181 -8.181 6.59E-10 -1.5 1.043E-18 14 O + NO <--> O + N + O -8.880 -7.880 -6.880 1.318E-8 -1.5 1.043E-18 15 NO + NO <--> NO + N + O -8.880 -7.880 -6.880 1.318E-8 -1.5 1.043E-18 16 N2 + O <--> NO + N -16.951 -15.951 -14.951 1.12E-16 0.0 5.175E-19 17 NO + O <--> O2 + N -21.277 -20.277 -19.277 5.279E-21 1.0 2.719E-19

Scenario: 1-D Shock

• Shock speed is ~8000 m/s, M∞ ≈ 23.• Upstream number density = 3.22×1021 #/m3.• Upstream composition by volume: 79% N2, 21% O2.• Upstream temperature = 300 K.

1D Shock Simulation

X

Pressure

StreamwiseVelocity

InletBoundary SpecularW

all

Simulation is initialized with a bulk velocitydirected towards the specular wall at the rightboundary of the domain, with pre-shock density,temperature, and chemical composition.

Vbulk

X

Pressure

StreamwiseVelocity


all

ShockPropagatesUpstream

X

Pressure

StreamwiseVelocity


all

The shock is allowedto move a reasonabledistance away from thewall before any form ofsampling begins.

X

Upstream PressureSampling Region

InletBoundary

SpecularWallPressure

Downstream Pressure Sampling Region

X0.0

0.5

1.0

1.5 Normalized PressureBoxcar Averaged Normalized Pressure

InletBoundary

SpecularWall

Shock Position

X0.0

0.5

1.0

1.5 Boxcar Averaged Normalized Pressure (Time = t)Boxcar Averaged Normalized Pressure (Time = t + t)

InletBoundary

SpecularWall

Shock Position(time = t)

Shock Position(time = t + t)

sShock PropagationSpeed = VP = s/t

X0.0

0.5

1.0

1.5 Boxcar Averaged Normalized Pressure (Time = t + t)

InletBoundary

SpecularWallShock Position

(time = t + t)

Shock SamplingRegion

VSampling Region = VP

• We require the ability to simulate a 1-D shock without knowing the post-shock conditions a priori.• To this end, we simulate an unsteady 1-D shock, and make use of a sampling region which moves with the shock.

Results: Nominal Parameter Values

X (m)

Density(kg/m

3 )

-0.05 0 0.05 0.1 0.15 0.20.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030 BulkN2NO2ONO

Bulk

O2, NO

N

O

N2

X (m)

Temperature(K)

0 0.05 0.1 0.15 0.20

5000

10000

15000

20000

25000

TtransTrotTvibT

X (m)

N 2Temperature(K)

-0.01 0 0.01 0.020

5000

10000

15000

20000

25000TtransTrotTvibT

Quantity of Interest (QoI)

J. Grinstead, M. Wilder, J. Olejniczak, D. Bogdanoff, G. Allen, and K. Danf, AIAA Paper 2008-1244, 2008.

X

QoI

We cannot yet simulate NASA EAST or other shock tube results, so we must choose a temporary, surrogate QoI.

?

{𝑸𝒐𝑰 }={𝑸𝒐𝑰𝟏𝑸𝒐𝑰𝟐𝑸𝒐𝑰𝟑

⋮𝑸𝒐𝑰𝒏

}

Sensitivity Analysis - Scalar vs. Vector QoI

The density of NO (our QoI for this work) is a vector, with values over a range of points in space.

X (m)

NO(kg/m

3 )

0 0.05 0.1 0.15 0.20.0E+00

1.0E-05

2.0E-05

3.0E-05

4.0E-05

5.0E-05

6.0E-05

7.0E-05

For our analysis, each blue dotrepresents a single, scalar QoI.

Scalar QoI used in later schematicsdemonstrating calculation of correlationcoefficients and the mutual information.

Sensitivity Analysis - Methods

Two measures for sensitivity were used in this work.• Pearson correlation coefficients:

• The mutual information:

Both measures involve global sensitivity analysis based on a Monte Carlo sampling of the parameter space, and thus the same datasets can beused to obtain both measures.

log10(O2 + NO <--> O + O + NO)

NO(kg/m

3 ),atx=-0.00016m

-12 -11.5 -11 -10.50

2E-05

4E-05

6E-05 r = -0.0346r2 = 0.0012

log10(NO + N <--> N + O + N)

NO(kg/m

3 ),atx=-0.00016m

-8.5 -8 -7.5 -70

2E-05

4E-05

6E-05

r = -0.3006r2 = 0.0904

log10(O2 + N <--> NO + O)

NO(kg/m

3 ),atx=-0.00016m

-18.5 -18 -17.5 -170

2E-05

4E-05

6E-05

r = 0.6045r2 = 0.3655

Sensitivity Analysis: Correlation Coefficient

1

QoI

1

QoI

Sensitivity Analysis: Mutual Information


1

p( 1)

QoI

p(QoI)

0.2460.1850.1230.0620.000

p(1,QoI)


1

p( 1)

QoI

p(QoI)

0.1150.0860.0570.0290.000

p(1)p(QoI)


𝑰 (𝜽𝟏 ,𝑸𝒐𝑰 )=∫𝜽𝟏

∫𝑸𝒐𝑰𝒑 (𝜽𝟏 ,𝑸𝒐𝑰 )[𝒍𝒏( 𝒑 (𝜽𝟏 ,𝑸𝒐𝑰 )

𝒑 (𝜽𝟏)𝒑 (𝑸𝒐𝑰 )) ]𝒅𝑸𝒐𝑰 𝒅𝜽𝟏

Hypothetical joint PDF for case where the QoI is indepenent of θ1.

Actual joint PDF of θ1 and the QoI, from a Monte Carlo sampling of theparameter space.

Kullback-Leibler divergence0.2460.1850.1230.0620.000

p(1,QoI)

0.1150.0860.0570.0290.000

p(1)p(QoI)0.0080.0060.0040.0020.000

𝐩 (𝛉𝟏 ,𝐐𝐨𝐈 )[𝐥𝐧 ( 𝐩 (𝛉𝟏 ,𝐐𝐨𝐈)𝐩 (𝛉𝟏 )𝐩(𝐐𝐨𝐈)) ]

Hypothetical joint PDF for case where the QoI is indepenent of θ1.

Actual joint PDF of θ1 and the QoI, from a Monte Carlo sampling of theparameter space.

QoI

θ1

X (m)

NO(kg/m

3 )

0 0.05 0.1 0.15 0.20.0E+00

1.0E-05

2.0E-05

3.0E-05

4.0E-05

5.0E-05

6.0E-05

7.0E-05

For our analysis, each blue dotrepresents a single, scalar QoI.

Scalar QoI used in later schematicsdemonstrating calculation of correlationcoefficients and the mutual information.

X (m)

r2

MutualInformation

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1N2 + N --> N + N + N (r2)

NO + N --> N + O + N (r2)NO + O --> N + O + N (r2)N2 + O --> NO + N (r

2)N2 + N --> N + N + N (MI)NO + N --> N + O + N (MI)NO + O --> N + O + N (MI)N2 + O --> NO + N (MI)

X (m)

r2

MutualInformation

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1N2 + O --> NO + N (r2)

N2 + O --> NO + N (MI)

At this x-location, r2 is zerowhile the mutual informationis greater than zero.

Sensitivities vs. X

r2 vs. Mutual Information

log10(N2 + O --> NO + N)

NO(kg/m

3 ),atx=0.021m

-16.5 -16.0 -15.5 -15.0

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05r2 = 0.0000002MI = 0.0668

X (m)

NO(kg/m

3 )

-0.01 0 0.01 0.02 0.03 0.04 0.050

2E-05

4E-05

6E-05

8E-05

Lowest for N2 + O <--> NO + NNominal for N2 + O <--> NO + NHighest for N2 + O <--> NO + N

Sensitivities vs. X for ρNO as QoI

r2

MutualInformation

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1N2 + N --> N + N + N (r2)

NO + N --> N + O + N (r2)N2 + N --> N + N + N (MI)NO + N --> N + O + N (MI)

X (m)

log10(N2 + N <--> N + N + N)

NO(kg/m

3 ),atx=0.141m

-8 -7.5 -7 -6.50.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04 r2 = 0.3053

log10(NO + N <--> N + O + N)

NO(kg/m

3 ),atx=0.003m

-8.5 -8 -7.5 -70.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04 r2 = 0.2902

Variance of ρNO vs. X

X (m)

var(

NO)xr2(Normalized)

var( N

O)xMI(Normalized)

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

N2 + N <--> N + N + N (r2)

NO + N <--> N + O + N (r2)NO + O <--> N + O + N (r2)N2 + O <--> NO + N (r

2)NO + O --> O2 + N (r

2)N2 + N <--> N + N + N (MI)NO + N <--> N + O + N (MI)NO + O <--> N + O + N (MI)N2 + O <--> NO + N (MI)NO + O --> O2 + N (MI)

X (m)

var(

NO)xr2(Normalized)

var( N

O)xMI(Normalized)

0 0.005 0.010

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1N2 + N <--> N + N + N (r2)

NO + N <--> N + O + N (r2)NO + O <--> N + O + N (r2)N2 + O <--> NO + N (r

2)NO + O --> O2 + N (r

2)N2 + N <--> N + N + N (MI)NO + N <--> N + O + N (MI)NO + O <--> N + O + N (MI)N2 + O <--> NO + N (MI)NO + O --> O2 + N (MI)

Variance Weighted Sensitivities

Overall Sensitivities

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Ove

rall

Sens

itivi

ty (N

orm

alize

d)

Parameter #

r^2 (QoI_2)

MI (QoI_2)

N2 + N <--> N + N + N

r2

Mutual Information

NO + N <--> N + O + N

NO + O <--> N + O + O

N2 + O <--> NO + N

NO + O <--> O2 + N

O2 + O <--> O + O + O

Synthetic Data Calibrations: 0-D Relaxation

• Due to computational constraints, a 0-D relaxation from an initial high-temperature state was used for the synthetic data calibrations performed thus far.• 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.

• Sensitivity analysis of the type discussed earlier was used to identify parameters for calibration.

Time (s)

NO(kg/m

3 )

0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-060.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

Synthetic Data

Synthetic Data

• We once again use ρNO as our QoI.

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 of data.• The likelihood of a given set of parameters is calculated based on the mismatch between the data and the simulation results for that set of parameters.• One or more chains explore the parameter space, moving towards regions of higher likelihood.• Candidate positions are drawn from a multi-dimensional Gaussian proposal distribution centered at the current chain position. The covariance matrix of this Gaussian controls the average distance (in parameter space) that the chain moves in one step.

Time (s)

NO(kg/m

3 )

0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-060.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

Synthetic Data

Time (s)

NO(kg/m

3 )

0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-060.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

Synthetic Data2 Error Bars

𝒍𝒊𝒌𝒆𝒍𝒊𝒉𝒐𝒐𝒅=𝑷 (𝑫|𝜽 )= 𝟏

(𝟐𝝅𝝈𝟐)𝑵 𝒅𝟐𝐞𝐱𝐩[− 𝟏

𝟐𝝈𝟐∑𝒊=𝟏

𝑵 𝒅(𝝆𝑵𝑶 ,𝒅𝒂𝒕𝒂, 𝒊− 𝝆𝑵𝑶 ,𝒔𝒊𝒎𝒖𝒍𝒂𝒕𝒊𝒐𝒏 , 𝒊)𝟐 ]

Time (s)

NO(kg/m

3 )

0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-060.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

Synthetic Data2 Error BarsCandidate Results

MCMC Method - Likelihood

MCMC Method – Metropolis-HastingsEstablish

boundaries for parameter space

Likelihoodcandidate

< Likelihoodcurrent

Likelihoodcandidate

> Likelihoodcurrent

Select initial position

Run simulation at current position

Calculate likelihood for the current

position

Draw new candidate position

Run simulation for candidate position parameters, and

calculate likelihood

Accept or reject candidate

position based on a random number draw

Candidate position is accepted, and becomes

the current chain position

Candidate position becomes

current position

Current position remains

unchanged.

Candidate automatically

accepted

Candidate Accepted

Candidate Rejected

MCMC Method - Improvements

We use the PECOS-developed code QUESO, which implements two major additions to the basic Metropolis-Hastings algorithm. Both of these features can help improve the convergence of the method.• Delayed Rejection:

When an initial candidate position is rejected, a second candidate position is generated based on a scaled proposal covariance matrix.

• Adaptive Metropolis: At periodic intervals, the proposal covariance matrix

is updated based on the calculated covariance of the previously accepted chain positions.

H. Haario, M. Laine, A. Mira, E. Saksman, Statistics and Computing, 16, 339-354 (2006).

Chain Progression

log10(N2 + O <--> NO + N)

log 1

0(O2+N<-->NO+O)

-16.5 -16 -15.5 -15

-18.5

-18

-17.5

-17

log10(N2 + O <--> NO + N)

log 1

0(O2+N<-->NO+O)

-16.5 -16 -15.5 -15

-18.5

-18

-17.5

-17

StartingPosition

log10(N2 + O <--> NO + N)

log 1

0(O2+N<-->NO+O)

-16.5 -16 -15.5 -15

-18.5

-18

-17.5

-17

StartingPosition

CurrentPosition

log10(N2 + O <--> NO + N)

log 1

0(O2+N<-->NO+O)

-16.5 -16 -15.5 -15

-18.5

-18

-17.5

-17

StartingPosition

CurrentPosition

log10(N2 + O <--> NO + N)

log 1

0(O2+N<-->NO+O)

-16.5 -16 -15.5 -15

-18.5

-18

-17.5

-17

StartingPosition

CurrentPosition

log10(N2 + O <--> NO + N)

log 1

0(O2+N<-->NO+O)

-16.5 -16 -15.5 -15

-18.5

-18

-17.5

-17

StartingPosition

CurrentPosition

log10(N2 + O <--> NO + N)

log 1

0(O2+N<-->NO+O)

-16.5 -16 -15.5 -15

-18.5

-18

-17.5

-17

StartingPosition

CurrentPosition

log10(N2 + O <--> NO + N)

log 1

0(O2+N<-->NO+O)

-16.5 -16 -15.5 -15

-18.5

-18

-17.5

-17

StartingPosition

CurrentPosition

log10(N2 + O <--> NO + N)

log 1

0(O2+N<-->NO+O)

-16.5 -16 -15.5 -15

-18.5

-18

-17.5

-17

StartingPosition

Parameter values usedto generate synthetic data

Post-Calibration PDFs

log10(N2 + O <--> NO + N), log10(O2 + N <--> NO + O)

NormalizedPost-CalibrationPDF

-18.5 -18 -17.5 -17 -16.5 -16 -15.5 -150

0.2

0.4

0.6

0.8

1

N2 + O <--> NOO2 + N <--> NO + O

NominalParameterValues

Conclusions

• Global, Monte Carlo based sensitivity analysis can provide a great deal of insight into how various parameters affect a given QoI.• A great deal of computer power is required to perform this type of statistical analysis for DSMC shock simulations.

5,000 shocks were run for this sensitivity analysis. Required a total of ~320,000 CPU hours.

• MCMC can be used to calibrate at least some DSMC parameters based on synthetic data for a 0-D relaxation.• MCMC allows for propagation of uncertainty from the data to the final parameter PDFs.

Future Work

• Synthetic data calibrations for a 1-D shock with the current code.• Upgrade the code to allow modeling of ionization and electronic excitation.• Couple the code with a radiation solver.• Sensitivity analysis for a 1-D shock with the additional physics included.• Synthetic data calibrations with the upgraded code.• Calibrations with real data from EAST or similar facility.

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Bayesian Inference For The Calibration Of DSMC Parameters

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