PECOSPredictive Engineering and Computational Sciences
Full System SimulationsAlgorithms and Uncertainty Quantification
Roy H. Stogner
The University of Texas at Austin
October 12, 2011
Roy H. Stogner FSS October 12, 2011 1 / 26
Outline
1 Introduction
2 Forward Uncertainty PropagationConvergence
3 Goal-Oriented RefinementTheoryCurrent Results
Roy H. Stogner FSS October 12, 2011 2 / 26
Introduction
Traditional Validation
CalibrationData
CalibrationProcess
ModelParameter(s)
Calibrated
withunknown
parameter(s)
ModelEvaluation
Observables
ExperimentalValidation Data
Stage 1: Maps Calibration Data to Validation Observables
Challenge
Roy H. Stogner FSS October 12, 2011 3 / 26
Introduction
Bayesian Validation in the Context of the QoI
ValidationData
Model Evaluation
ModelModel
Evaluation
Model EvaluationBayesian
Inference
CalibrationData
ModelParameter(s)
Calibrated
withunknown
parameter(s)
Stage 1: Maps Calibration Data to QoI
Stage 2: Maps Validation Data to QoI - Traditional Validation is Embedded
withparameter(s)
Challenge
m
mc
Qc
Qv
m
M(Qc, Qv) < γ
Model not rejected
Model rejected
Yes
No
Intervene
Bayesian Inference
Roy H. Stogner FSS October 12, 2011 4 / 26
Introduction
Atmospheric Reentry Problem• High enthalpy aerothermochemistry, hypersonic flow
• Surface pyrolysis, ablation, radiation
• Unreliable models (turbulence, ablation, carbon chemistry)
Roy H. Stogner FSS October 12, 2011 5 / 26
Introduction
Calibrated Uncertainty Quantification
Experimental Data
QUESO
FIN-SlibMeshBoost
Turbulence
SHOCKINGMUTATION
GSL
Chemistry/Shocktube
Plug FlowABLATION1D
GSL
Ablation
Calibration/Validation
FIN-SIn
vers
ion
Forw
ard
QUESO/Dakota
Multiphysics Analysis
MASA
QoI
GRVYExternal Libraries
LDVFlow
Measurements
NASA EAST Shocktube
Measurements
Molecular Beam (O2)Heated Flow Titration (N2)
Roy H. Stogner FSS October 12, 2011 6 / 26
Introduction
Ablator Nitridation Uncertainty
m′′N,c = −
√kbT
2πmNρyNβN (T )
Substrate
Virgin M
aterial
C100H89
.4O1
7.8N8(SiO2)64
.2
Shock Layer
Ma = 31
qradChar
Pyrolys
is Zone
qre−rad
m gh g
Pyrolys
is
Gas
Flow
Nsi=1 Jihi
P
s
m((
ch=ρchs
qchem=
Boundary Layer
qcond
Nitridation coefficient βN
• Value from initial literature survey: 0.3
• Disagreement, uncertainty range: (0.00003, 0.4)
• Highest predicted submodel uncertainty contribution
• Calibrated mean: 0.0024
• Calibrated std dev: 0.0005
Roy H. Stogner FSS October 12, 2011 7 / 26
Introduction
Nitridation Coefficient Calibration
Laboratory Investigation of Active Carbon Nitridation by Atomic Nitrogen,Zhang et. al.
Roy H. Stogner FSS October 12, 2011 8 / 26
Introduction
Reaction Chemistry Uncertainty
k = A
(T
T0
)n
e−EaRT
N + e− ↔ N+ + 2e−
O + e− ↔ O+ + 2e−
N2 +N ↔ 2N +N
N2 +N2 ↔ 2N +N2
N2 + e− ↔ 2N + e−
NO +O ↔ O2 +N
N2 +O ↔ NO +N
TN2 = (T(1−q)tr T q
ve)
TO2 = (T(1−q)tr T q
ve)
Uncertain Reaction Rates• Arrhenius pre-exponential
uncertainty: +/- 1 OOM
• Strong output sensitivities toN2 +O, NO +O reactions
• Joint calibration
• Details: Marco Panesi’s talk
Roy H. Stogner FSS October 12, 2011 9 / 26
Introduction
Reaction Chemistry Calibration
Electric Arc Shock Tube (EAST) Spectroscopy
Roy H. Stogner FSS October 12, 2011 10 / 26
Introduction
Turbulence Model Uncertainty
A Priori Uncertainty• Algebraic (Baldwin-Lomax) model, no transition model• Scalar “Turbulence augmentation” factor
I Uncertainty range: (0, 1.5)
• Second greatest contribution to output uncertainty
Calibrated Uncertainty• Spalart-Allmaras PDE-based model
• Joint calibration, 8 uncertain parameters
• Multi-model Bayesian validation
• Details: Todd Oliver’s talk
Roy H. Stogner FSS October 12, 2011 11 / 26
Introduction
Turbulence Model Calibration
Bowersox Supersonic BL data
400
450
500
550
600
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
u [
m/s
]
y [m]
Luker (2000)
Direct Numerical Simulation
Future UT experiments: near-wall Particle Image Velocimetrymeasurements
Roy H. Stogner FSS October 12, 2011 12 / 26
Forward Uncertainty Propagation
Latin Hypercube and Calibration
LHS• Quantile bins in each parameter
• 1 sample per bin
• Reduce variance from additiveresponse components
• Calibrated joint PDFs are notseparable tensor products!
LHS+MCMC• LHS for uncalibrated variables
• SRS from each calibrated jointdistribution
Roy H. Stogner FSS October 12, 2011 13 / 26
Forward Uncertainty Propagation Convergence
Off-baseline Samples
0 50 100 150 200 250Iteration N
108
109
1010
1011
1012
Uns
tead
yR
esid
ual
ISS Offbaseline Convergence - Dataset 270
10−10
10−9
10−8
10−7
Tim
eS
tep
[s]
‖du/dt‖∞∆t
Convergence• Large initial transients• Secondary transient spike
I Change propagation?I No tertiary spikes
• 6 OOM convergence stallI Vibrational energy
stabilizationI Reaction stabilization?
Roy H. Stogner FSS October 12, 2011 14 / 26
Forward Uncertainty Propagation Convergence
Off-baseline QoI Convergence
0 50 100 150 200 250Iteration N
10−6
10−5
QoI
Valu
e
ISS Offbaseline Convergence - Dataset 270
10−10
10−9
10−8
10−7
Tim
eS
tep
[s]
QoI∆t
Ablation Rate Convergence• Immediate change from
baseline
• Less rapid change until transientspike
• New value after transient spike
• Within tolerance: ∼ 250 timesteps
Roy H. Stogner FSS October 12, 2011 15 / 26
Forward Uncertainty Propagation Convergence
Calibrated Forward Propagation Results
UQ Output• More than 2× lower mean ablation mass loss than with uncalibrated
submodels
• Primary driver: 100× lower nitridation coefficient than initial prior
UQ Performance• Latin Hypercube Paradox:
I Significant parameters are calibrated; LHS convergence is unavailableI Other parameters are insignificant; LHS convergence is irrelevant
• LHSD methods may still show some improvement over SRS
Performance• 10× improvement in transient convergence
• 4× increase in wall clock time
Roy H. Stogner FSS October 12, 2011 16 / 26
Goal-Oriented Refinement Theory
Adjoint Refinement Error Estimator
Error Estimators• eQ ≡ Q(uh; ξ)−Q(u; ξ)
• R(uh, z; ξ) = eQ −RQ +RRI RQ and RR: higher order, often quadratic in
∣∣∣∣u− uh∣∣∣∣.
• R(uh, zh; ξ) = 0
• Higher order approximation of z:I Project uh to a refined spaceI Jacobian calculation, linear adjoint solve on refined meshI Residual evaluation on refined mesh
• No nonlinear solve on refined mesh
• Asymptotically bounded effectivity
• Improved QoI estimates
• Element-by-element QoI contributions
Roy H. Stogner FSS October 12, 2011 17 / 26
Goal-Oriented Refinement Theory
Adjoint Residual Error IndicatorError Indicators• Efficiently bounding eQ via per-element terms
• From our error estimator,
R(uh, z; ξ) =∑E
RE( uh∣∣∣E, z|E ; ξ)
• zh is cheaper than higher order approximation of z
Ignoring higher order terms:
q − qh = −Ru(u, z − zh; ξ)(u− uh)∣∣∣q − qh∣∣∣ ≤ ||Ru||B(U ,V ∗)
∣∣∣∣∣∣u− uh∣∣∣∣∣∣U
∣∣∣∣∣∣z − zh∣∣∣∣∣∣V∣∣∣q − qh∣∣∣ ≤∑
E
∣∣∣∣REu
∣∣∣∣B(UE ,V E∗)
∣∣∣∣∣∣ u|E − uh∣∣∣E
∣∣∣∣∣∣UE
∣∣∣∣∣∣ z|E − zh∣∣∣E
∣∣∣∣∣∣V E
Roy H. Stogner FSS October 12, 2011 18 / 26
Goal-Oriented Refinement Theory
Adjoint Residual Error Indicator
AdjointResidualErrorEstimator Procedure
• Calculate equal-order adjoint solution zh
• Use existing (patch recovery) estimators for∣∣∣∣u− uh
∣∣∣∣ and∣∣∣∣z − zh∣∣∣∣ on each element
• Combine element-by-element
AdjointResidualErrorEstimator Limitations• Asymptotic overestimate• No estimation of
∣∣∣∣REu
∣∣∣∣I Would require local DenseMatrix inversion, multiplication, 2-norm
estimate
Roy H. Stogner FSS October 12, 2011 19 / 26
Goal-Oriented Refinement Current Results
Shock Simulation with Goal-Oriented AMR
Shock Hanging Nodes• Shock thickness ∝ h• No artificial transverse velocity
• No ringing, overshoot, instability
• Immediate δt reduction required
• Rapid δt growth possible
• Reconvergence: 6 OOM in10− 100 time steps
Fully automatic convergence from very coarse meshes?
Roy H. Stogner FSS October 12, 2011 20 / 26
Goal-Oriented Refinement Current Results
Viscous Boundary Layer with Goal-Oriented AMR
Boundary Layer Hanging Nodes• Stress test:
I Ungraded boundary layerI Underresolved viscous fluxes
• Valid initial refinement step• Subsequent refinements:
I Coarse elements currentlyoverestimate convective flux
I Fine element equilibriumtemperature drops
I Peak surface value QoIlocation moves
I Wasteful overrefinementdownstream
Roy H. Stogner FSS October 12, 2011 21 / 26
Goal-Oriented Refinement Current Results
Viscous Boundary Layer with Goal-Oriented AMR
Boundary Layer Improvements• Primal Stabilization:
I Full viscous terms in DCOI In testing, Benjamin Kirk
• Adjoint Regularization:I DCO-aware suberror
estimatesI Mesh-aware forcing term for
peak value QoI derivativesI Smoothed adjoint forcing term
for peak value QoI derivatives• Standard practice, a priori
graded boundary layer meshesI “Doctor, it hurts when I do
this”
Roy H. Stogner FSS October 12, 2011 22 / 26
Goal-Oriented Refinement Current Results
Shock Patch Recovery: Primal Error Estimation
Shock Layer Error:• True Error:
I Discontinuity CapturingOperator
I Artificial DiffusionI Partitioning Independent
• Local Error Estimates:I H1 Patch RecoveryI Partitioning DependentI Shock narrowing decreases
error, increases estimate!
Roy H. Stogner FSS October 12, 2011 23 / 26
Goal-Oriented Refinement Current Results
Shock Patch Recovery: Primal Error Estimation
Shock Layer Improvements• Improved Local Estimates:
I L2 Patch RecoveryI Partitioning-independent
Patch RecoveryI Multiphysics-aware weightingI Refined adjoint test functions
Roy H. Stogner FSS October 12, 2011 24 / 26
Goal-Oriented Refinement Current Results
Continuing Work
Uncertainty Quantification• Adjoint-enhanced control variate surrogates
I Uncertain parameters exposed to internal perturbation
Algorithms• Goal-oriented adaptivity verification
• AMR with coarse initial elements
• Hanging nodes on curved 3D boundaries
Optimizations• Analytic derivatives replacing finite differencing in boundary Jacobians
• Weighting load balancing in partitioning
• Caching common subcalculations
Roy H. Stogner FSS October 12, 2011 25 / 26
Goal-Oriented Refinement Current Results
Thank you!
Questions?
Roy H. Stogner FSS October 12, 2011 26 / 26
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