Full System Simulations - University of Texas at Austinroystgnr/FSS_rs.pdfPECOS Predictive...

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


Introduction

Traditional Validation

CalibrationData

CalibrationProcess

ModelParameter(s)

Calibrated

withunknown

parameter(s)

ModelEvaluation

Observables

ExperimentalValidation Data

Stage 1: Maps Calibration Data to Validation Observables

Challenge


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


Introduction

Atmospheric Reentry Problem• High enthalpy aerothermochemistry, hypersonic flow

• Surface pyrolysis, ablation, radiation

• Unreliable models (turbulence, ablation, carbon chemistry)


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)


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


Introduction

Nitridation Coefficient Calibration

Laboratory Investigation of Active Carbon Nitridation by Atomic Nitrogen,Zhang et. al.


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


Introduction

Reaction Chemistry Calibration

Electric Arc Shock Tube (EAST) Spectroscopy


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


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


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


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?



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



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


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



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



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


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?



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



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”



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!



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



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



Thank you!

Questions?


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