uncertainty propagation in materials processing materials process design and control laboratory...

UNCERTAINTY PROPAGATION IN UNCERTAINTY PROPAGATION IN MATERIALS PROCESSINGMATERIALS PROCESSING

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


Nicholas Zabaras

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected]: http://www.mae.cornell.edu/zabaras/



UNCERTAINTY IN THE MATERIALS WORLD


On April 14, 1912, the

Titanic, the largest, most complex ship afloat, struck

an iceberg and sank. This is

perhaps one of the all-time

great failures to correctly

modeling the interaction of uncertainty in

the environment

and the way it can couple with the dynamics of

a system.



THE COLUMBIA DISASTER – UNCERTAINTY IN THE MATERIALS WORLD


SOLUTIONS:Identify sources of uncertainties that contribute most to uncertainties in outcomes

FAIL-SAFE design or SAFE-FAIL design?

Uncertainty is not ignorance!

Robust design to avoid catastrophic failures

Microstructure

TEM




Uncertainty propagation in simulations

Uncertainty in initial state and microstructure Uncertainty in materials testing

Unc

erta

inty

pro

paga

tion

in

sim

ulat

ions

Mat

eria

l mod

el &

par

amet

ers

Mod

elin

g of

tool

beh

avio

r

Mod

el v

alid

atio

n

Uncertainty propagation in

simulations

Multi-stage processing –

Conditions between stages

Simulation error, round off

errors

UNCERTAINTY IN THE MICROSTRUCTURE-PROPERTY-PROCESSING TRIANGLEUNCERTAINTY IN THE MICROSTRUCTURE-PROPERTY-PROCESSING TRIANGLE

Properties

● Averaging principles● Pole figures => ODF● Model validationProcessing

● Friction Tool wear● Geometric parameters● Process parameters● Material interaction● Material composition




Cracked blade

FAILURE MODES:Creep, low-cycle fatigue (LCF) & high-cycle fatigue (HCF)

Design & Safety assessment require the establishment of significance of defects in components subject to creep and creep/fatigue loading.

FAILURE ANALYSIS & COMPONENT DIAGNOSTICS

Characterisation of creep crack initiation and growthInitiation time forcrack size of Crack growth rate

These procedures assume that creep crack initiation (CCI) and creep crack growth (CCG) rate are correlated by the creep fracture mechanics parameters

C*, which are estimated from the load – displacement diagram.

Material parametersC*, D ,

Predictions of the behavior of component needs to be evaluated considering the stochastic nature of the creep fracture mechanics parameters.




FAILURE ANALYSIS & COMPONENT DIAGNOSTICSFAILURE ANALYSIS & COMPONENT DIAGNOSTICS

Error distributions from statistical analysis of CCG rate tests.

Variation in due to uncertainties in estimates of D; and C*

Involves combination of variabilities and calculations need to be carried out for combinations of these material values to produce probabilities of times for crack initiationFurthermore, the load on the component is uncertain due to material, geometric and measurement uncertainties

Intl. J. Press. Vessels Piping 80 (2003) 585–595

Intl. J. Press. Vessels Piping 80 (2003) 585–595




EFFECTS OF UNCERTAINTY ON PERFORMANCE, COSTS & SAFETY

UNCERTAINTY UNCERTAINTY AIRWORTHINESS AIRWORTHINESS

The material properties as well as the geometrical dimensions must be within specified limits in order to maintain the desired fabrication efficiency and product quality.

It is important in structural and mechanical design to allow for the fact that uncertainties exist; These uncertainties have traditionally been catered through the use of factor of safetyEfficient and optimal performance cannot be achieved through a conservative design!

Conservative design of aircraft leads to enormous usage of exotic materials which are expensive

FAA CATASTROPHIC FAILURE PREVENTION RESEARCH PROGRAM

Deviation from design geometry and material composition – CATASTROPHIC FAILURE

Robust design to allow for inappropriate crew response to propulsion and control malfunctions

FAA Airworthiness R&D – widespread fatigue damage (WFD), corrosion, aircraft aging




SOURCES OF UNCERTAINTYSOURCES OF UNCERTAINTYQuestioning isotropy assumptionTypical stress-strain response depends on direction and alloy as seen in the picture.Uncertainties in1. Direction and property quantification2. Material characterization

Geometric uncertaintySample of experimentally observed statistics for an extrusion process for different lengths

of extrusion (Materials and Design 2001, 22, 267-275).



MODEL VALIDATION


SOURCES OF MISMODELING - Geometry - Component types - Component properties - Poor modeling capability

MODEL DATA VARIATIONS - Manufacturing tolerances - Residual stresses – due to re-assembly - Environmental effects – thermal effects - Microdynamic behavior - Testing methodlogies MODEL VALIDATION - Important for model development for use in decision making process. - Trust worthiness of models is inevitably questioned. - Difficulties: 1. Evaluation of response can present severe mathematical and numerical difficulties. 2. Statistical properties of the system are not known.

Advanced Materials Processing Laboratory, NorthWestern University

Prognosis- main ideaPrognosis- main idea

Current state

Geometry of the system

Presence of any defects like cracks, voids, corrosion.

Most recent maintenance details

Accelerated simulation of duty cycles

Fatigue modeling, Multi-scale physics models, capability to model the system as a whole or individual components

Residual stresses, duty environment (temperature, humidity, working stress, thrust), Defect distributions

Upper and lower bounds of the

quantities available

Continuum sensitivity analysis provides sensitivities of key parameters like component stress, strain levels and strain rates, crack propagation

Use FORM to get failure surface plot

and failure probability

More data available about initial state – Can employ more sophisticated techniques like Bayesian inference to generate the complete probability distribution of failure of the component in oncoming duty cycle







PROGNOSIS, LIFING AND RELIABILITYPROGNOSIS, LIFING AND RELIABILITY

• Past mission history

• Knowledge of system behavior

• Expert knowledge

DATABASE

• Simple maneuvers to assess system state

• Benchmark tests

• Feature extraction

• Ultrasonics

EXTERNAL TESTING

• Failure physics modeling

• Reliability predictions

• Evolutionary physics based model (BAYESIAN)

DIGITAL LIBRARY

• Reduced order models for various failure regimes

• Classification of current system state

Prediction of failure response surface

Fail Safe

Predicting performance

based on current state

• Short term predictions more accurate

• Evolutionary prediction based on Markov autoregressive chains for long range predictions

• Confidence intervals for predictions

• Update reduced order models for system analysis

• Update benchmarking schemes

POST PROCESSING MODULE




MODELING MATERIALS ACROSS LENGTH SCALESMODELING MATERIALS ACROSS LENGTH SCALES

Length Scales

Materials modeling spans 12 orders of magnitude in length and predominantly stochastic

Realistic simulation: large reduction of degrees of freedom required at each step

1 nm 1 m 1 mm 1 m

Electronic

Atomistic

Ph

ysic

sC

hem

istr

yM

ater

ials

En

gin

eeri

ng

Micro-

structural

Continuum

Property averaging

Interfacial energies

Inter-Atomic Potentials




INFORMATION THEORY FOR MATERIALS PROCESSESINFORMATION THEORY FOR MATERIALS PROCESSES

Model chosen based on

microstructure

Poly-phase material Pure metal

• Distance between distributions for various classes of microstructures- Kullback-Liebler distance/ Cross-entropy distance- Classification techniques (CART, MARS) algorithms- Support vector machine based classification of input microstructure.

Given input microstructure, how to choose material models from a class of available models

Lineal analysis of microstructure

photograph

Orientation distribution

function model

Dendritic




Continuum Model

Voronoi Tessellation

Model

Spatial Point Field

Model

Simulated 3D grain structure Description Uncertainty

Average Properties

Spheroid Model

3D DESCRIPTION UNCERTAINTY

MICROSTRUCTURE CLASSIFICATION

Digital Microstructure Library

Noisy Input Image

Classifier Uncertainty

Feature Detection Uncertainty




UNCERTAINTIES IN MICROSTRUCTURE IMAGINGUNCERTAINTIES IN MICROSTRUCTURE IMAGING

Final Goal

Robust recovery of the spatial (3D) microstructure using a model that utilizes knowledge about the 2D image errors, the data processing uncertainty and the known features of the material under observation.

3D data

2D data

Sensor Uncertainty Strategy Uncertainty

Full 2D Uncertainty

Refined 2D Uncertainty

Recovered 3D Uncertainty Models

Uncertainty propogation in 3D recognition from 2D images.(Ref. Sobh, T.M. and Mahmood, A.)Uncertainty propogation in 3D recognition from 2D images.(Ref. Sobh, T.M. and Mahmood, A.)

Image Spatial Structure




UNCERTAINTY IN FEATURE MAPPING

Find the uncertainty of mapping a specific 3D Find the uncertainty of mapping a specific 3D feature to a 2D pixel value.feature to a 2D pixel value.

The 3D feature is located at 2D pixel position (i,j) with The 3D feature is located at 2D pixel position (i,j) with probability pprobability p11, (i+1,j) with p, (i+1,j) with p22 etc. given that the registered etc. given that the registered

location is (l,m) such that plocation is (l,m) such that p11+p+p22..p..pnn = 1 assuming no = 1 assuming no uncertainty in feature recovery mechanism. The goal is to uncertainty in feature recovery mechanism. The goal is to

find the probabilities.find the probabilities.

Uncertainty in spatial (3D) reconstruction from 2D microstructure imaging

Probabilistic representation of 3D microstructure

Estimation of 3D uncertainties in the structure and motion of a material microstructure imaged in 2D. (Another Problem is the recovery of 3D translational velocity CDF for microstructure evolution from 2D data)

Uncertainties in Mapping 3D Microstructural Features to 2D Domain (Sensor Uncertainty)Uncertainties in Mapping 3D Microstructural Features to 2D Domain (Sensor Uncertainty)

Spatial structure Imaged structure




Uncertainty ProblemUncertainty Problem: Given a feature recovered from an image is in pixel position (x,y), the : Given a feature recovered from an image is in pixel position (x,y), the probability that the feature was originally at the position (x+1,y) with probability p1, (x+2,y) with probability that the feature was originally at the position (x+1,y) with probability p1, (x+2,y) with probability pprobability p22 etc. such that p etc. such that p11+p+p22+…+p+…+pnn = 1 due to noise in the images. The problem is to find = 1 due to noise in the images. The problem is to find

the probabilities.the probabilities.

Errors in Image ProcessingErrors in Image Processing

• Data is lost during compression/denoisingData is lost during compression/denoising

• Noise is amplified when derivatives are computed.Noise is amplified when derivatives are computed.

•Addition of new unrelated features in the imageAddition of new unrelated features in the image

Edge Loss

Feature Addition

PROBABILISTIC NATURE OF FEATURE RECOVERYPROBABILISTIC NATURE OF FEATURE RECOVERY




UNCERTAINTY: MICROSTRUCTURE EVOLUTION IMAGESUNCERTAINTY: MICROSTRUCTURE EVOLUTION IMAGES

Goal:Goal:

To generate microstructure evolution uncertainty estimates from a series of intensity images of the microstructure

Refining estimates of microstructure evolution

Eliminate unrealistic estimates

(Faulty estimates results from noise, errors or mistakes from the sensor acquisition process)

Eliminate using upper and lower bound on the distribution based on known properties of the microstructure under observation (worst case estimates of the microstructure evolution rate)




Training samples

Classification Scheme

Process class

Property Class

New Input Class

Outside Training space

Update Class

Data Mining

Young's Modulus = 75 GPa

Hot rolling

Deep drawing

Y. Modulus

Yield stress

ODF

Image

Pole Figures




DYNAMIC MICROSTRUCTURE LIBRARYDYNAMIC MICROSTRUCTURE LIBRARY

MICROSTRUCTURE REPRESENTATIONMICROSTRUCTURE REPRESENTATION




Automatic texture recognition

Machine Vision Algorithms

Reduced representation

Physical Models

Life Prediction

Stereology

Crystallographic Texture

Substructure (defects)

REPRESENTATION USING PCAREPRESENTATION USING PCA

Raw Image 32 x 32

Reconstructed Image using 20

coefficients

Eigen Basis

Input Image Snapshots

Reduced Description using PCA Texture reconstruction using PCA statistics

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

Statistics of Eigen coefficients

Image Generated from random coefficients using known statistics

Input Image




PHASE FIELD METHODPHASE FIELD METHOD

Phase Field

(Evolution of Field Variables)

Atomistic level Continuum ScaleMeso Scale

Thermodynamic variables

Free energy

Anisotropy

Mobilities

Interfacial energies

Continuum Deformation Problem

Couple Field variables & displacement gradients in local

free energy functions

For Complex Multi-component Systems

Model Reduction

Crystallographic

Lattice Parameters

Digital Library







Pole figure data – inversion to ODFPole figure data – inversion to ODF

Sources of error

•Infinitely many pole figures required.•Poor quality data around the periphery.•Mathematical error - Discretization of the fundamental region. •Indetermination errors-

A range of solutions in agreement with experimental resultsIncomplete pole figures - too small a region of measurement.

•Integration errors - Pole data is discrete.

LS problem




Pole figure data – inversion to ODFPole figure data – inversion to ODF

Sources of error - Statistical errors

•Definition of the ODF •In selecting the individual crystals from the sample. •Account for statistically distributed inhomogeneities in the material - Large crystals from castings could lead to large regions of distinctive deformation textures.•Counting statistics of measuring apparatus. •For a reasonable amount of data (high angular resolution), one must use very small aperture sizes which lead to reduced intensity – that further increases the error.

X-ray diffraction EBSD

Stochastic Simulation of Microstructure EvolutionStochastic Simulation of Microstructure Evolution



Materials Process Design and Control Laboratory

Reconstruct microstructure from

noisyexperimental image

Bayesian linear/nonlinear

regression MCMC Random fields

Realizations of stochastic process

Transition kernelFuture PDF ofmicrostructure

Prediction of property/adjustment of processing

EVOLUTION

Images from a diffusion process

Corresponding stochastic process

Physical Equation (eg. Heat Conduction)

Transition Kernel (eg. Heat Source)




Metal forming – sources of uncertaintyMetal forming – sources of uncertainty

Parameter uncertainties

• Forging velocity

• Lubrication – friction at die - workpiece interface

• Intermediate material state variation over a multistage sequence – residual stresses, temperature, change in microstructure




Metal forming – sources of uncertaintyMetal forming – sources of uncertainty

Shape uncertainty

• Die shape – is it constant over repeated forgings ?

• Intermediate material state variation over a multistage sequence – expansion / contraction of the workpiece

• Preform shapes (tolerances)

Small change could lead to unfilled die cavity




Metal forming – optimal designMetal forming – optimal design

Optimal preform

Initial guess

Preforming StagePreforming Stage Finishing StageFinishing Stage

InitialInitialDesignDesign

FinalFinalDesignDesign

UnfilledUnfilledcavitycavity

FullyFullyfilledfilledcavitycavity

How sensitive is the optimal design to shape and parameter uncertainties ?

Needs to specify robustness limits for optimal design parameters

http://www.mae.cornell.edu/zabaras/Publications/PDFiles/INVITEDTALKS/AFRL-Presentation-April-23-01/movies/dhump_2StageDieDesign.avi

http://www.mae.cornell.edu/zabaras/Publications/PDFiles/INVITEDTALKS/AFRL-Presentation-April-23-01/movies/dhump_2StageDieDesign.avi

http://www.mae.cornell.edu/zabaras/Publications/PDFiles/PRESENTATIONS/afosr-2003/movies/slide14_m1.avi

Importance of uncertainty in solidification Importance of uncertainty in solidification processprocess

Modeled as flow in media with variable porosity

Only a statistical description is possible macroscopically, thus

need to have a stochastic framework for analysis

Meso-scale (dendritic structures seen)

Structure of dendrites affect macroscopic quantity like porosity

Dendritic structure is a strong function of initial process conditions

Small perturbation in initial material concentrations, temperature, flow profile can significantly alter the dendritic profiles

“ Can we employ a multiscale stochastic formulation to model initial uncertainty and provide a statistical characterization for porosity”?

Typical dendritic structures obtained due to small

perturbations to initial conditions




Stochastic simulation of solidification Stochastic simulation of solidification processesprocesses




Solidification

Heat Transfer

Mass Transfer

PhaseChange

Fluid flow

solid Mushy zone

q

liquid ~ 10-2 m

(b) Microscopic scale

~ 10-4 – 10-5m

solid

liquid (a) Macroscopic scale

Complexities involved i.) Physical phenomenon across multiple length scales ii.) Multiple time and length scales involved iii.) Individual initial and boundary processes for transport processes iv.) Direct/Indirect coupling between transport processes v.) Widely varying properties in two phase mushy zone

Uncertainties involved I.) Randomness in transport properties, initial conditions, boundary conditions ii.) mold geometry, surface roughness iii.) Errors in experiments and measurement of quantities iv.) perturbations in nuclei generating and dendrite growth v.) simulation errors

Potential approaches i.) deterministic analysis: sensitivity, reliability bounds … ii.) probabilistic approaches: SSFEM …iii.) statistical inference + deterministic simulation: Bayesian …

One crucial problem:Permeability in mushy zone




Uncertainties in physical properties of the system and its surroundings: Thermal, thermodynamic and chemical

Uncertainties in the controlled process parameters: temperature gradient, cooling rate, pulling speed, partial pressure, stoichiometry, rate of rotation, etc.

Uncertainties due to environmental conditions: ambient temperature, radiation effects, g-jitters, etc.

Uncertainties in the dimensions of the apparatus

UNCERTAINTY IN CRYSTAL GROWTH

Melt flow in

HBG

Effect of uncertainty on crystal growth

Bridgeman growth

Czochralski growth

Micro gravityMicro gravityTerrestrialTerrestrial

Variational multiscale modeling Variational multiscale modeling




Large scale flow behavior

Sub grid scale flow phenomena

• Can be resolved using a finite element mesh

• Phenomena occurring at scales lower than the mesh scale are not captured

• Need to approximate the effect of sub grid phenomena on the resolves large scales

• Uncertainties introduced due to small scale phenomena are important

• Sub grid scales are not homogenous as considered in many computational techniques

Approximate fine scale model

Add contributions to large scale

solution VMS

Algebraic sub grid scale model

Sub grid solution is a function of a stochastic intrinsic time scale

Computational sub grid modeling

• Explicit models considered for sub grid based on bubble functions

• Fractal modeling of sub grid phenomena

Approaches for sub grid approximation

Stochastic fluid flow - Example Stochastic fluid flow - Example




(0,0)

(1,1)

u=v=0

u=v=0u=v=0

u=u(), v=0

Initially quiescent

fluid

Schematic of the problem computational domain

• Lid driving velocity is uniform between (0.9 and 1.1)

• Viscosity is 0.0025, hence mean Reynolds number = 400

Mean velocity evolution Std deviation of velocity evolution

magnitude of pressure at midplane

y

-0.04 -0.03 -0.02 -0.01 00

0.2

0.4

0.6

0.8

mean psdev p

Mid plane mean pressure and standard deviation

Stochastic algebraic subgrid scale (SASGS) Stochastic algebraic subgrid scale (SASGS) model model




(Momentum residual from large scales) x (stochastic subgrid momentum time scale)

(continuity residual from large scales) x (stochastic subgrid continuity time scale)

Large scale uncertainty

directly resolved

Subgrid scale uncertainty

modeled

Subgrid velocity and pressure solution are stochastic processes evolving with the large scale solution

Intrinsic time scales for momentum and continuity are stochastic quantities

Boundedness of time scales depends on the distribution of large scale solutions

Normal distribution for large scale velocity leads to unboundedness of algebraic stochastic time scales

Robust design - motivationRobust design - motivation

Hi – peformance computing

USER INTERFACE

Robust product

specifications

Control and reduced order

modeling

Stochastic optimization,

Spectral/Bayesian framework

Design database,

simulations and experiments

User update

Output design

Input

Modifications in objectives

• Starting with robust product specifications, you compute not only the full statistics

of the design variables but also the acceptable variability in the system parameters

• Directly incorporate uncertainties in the system into the design analysis

• Experimentation and testing driven by product design specifications

• Improve overall design performance




Are levels of uncertainty (PDFs) in other process conditions

tolerable?

Required product with desired material properties and shape with specified confidence (output PDFs)

Sensitivity analysis toolboxSensitivity of product w.r.t material data and other process conditions

Can we obtain the

PDFs by existent testing?

Update model PDFs and database (digital library)

Reference input and process

conditions PDFs

Are PDFs of design variables

technically feasible?

MATERIALS TESTING DRIVEN BY DESIGN ROBUSTNESS LIMITS

BAYESIAN INFERENCE

High performance computing

environment

Output PDFs obtained from

SSFEM analysis

Yes

Yes

Yes

Yes

Interface with digital library and expert advice to modify design objectives, material models, process models

No

No

No

Approach to robust design of materials Approach to robust design of materials processesprocesses




Various robust design statementsVarious robust design statements

Minimization of variance approach

Reliability type design

optimization

Complete stochastic

optimization • Integral defined over the sample space

• Avoids over-design problems

Objective

Probabilistic constraint

• Constraint can become highly nonlinear, use RSMs

• Based on extension to least squares approach

• Results similar to robust regression techniques




Stochastic optimization with spectral Stochastic optimization with spectral methodsmethods




Original PDF

Realization of Solution

PDF after perturbing the

design parameter vector

Design decision

Finite Vs Infinite dimensional optimization

Parametric representation of stochastic design variables

Non-parametric representation, design variables considered as

functions

Commonality – Sensitivity calculations

APPROACH APPROACH

• Solve the direct problem with guessed probability distributions of design variables

• Compute stochastic sensitivity with respect to each of the design variable

• Obtain gradient as a function of sensitivities

• Use CGM

• Solve the direct problem with guessed probability distributions of design variables

• Define an adjoint problem to obtain the gradient of objective in distributional sense

• Solve the continuum sensitivity problem

•Use CGM




BAYESIAN FRAMEWORK FOR MATERIAL PROPERTY ESTIMATION

Material properties (θ) of interest •mechanical• thermal• optical• electrical• magnetic• chemical

Micro-scale

Macro-scale

Material processing

Te

stin

g e

xp

erim

en

t• Destructive• Non-destructive

Input θInput θTesting system FTesting system F Measurement YMeasurement Y

Y = F(θ) + ω Uncertainty in measurement

A data driven model

PPDF

)()|()(

)()|()|(

pY pY p

pY pY p

LikelihoodLikelihood Prior PDF

A Bayesian statistical

inference model

Markov Chain Monte Carlo(MCMC)

Confidenceintervals

statistics




BAYESIAN FRAMEWORK FOR MATERIALS PROCESS CONTROL

Material processing

Key control variables(g) • thermal conditions (heat flux)• mechanical force• external field force• chemical reactions• processing speed•other mechanisms Material property

monitoringprocess

noise

measurementnoise

Numerical process modeling

Outliers3930

22

Get rid of polluted data

Data

min

ing

Statistical modelingof the uncertainties (noise)

p(g|Y,λ)Bayesian description

Filtered dataY

Hyper-parameterIn Bayesian model

MCMC samplerPosterior state space

exploration

Marginalpdfs

Pointestimate

Statistical modeling of uncertainties




BAYESIAN FRAMEWORK FOR ESTIMATION OF (NON-DESIGN) PROPERTIES

Uncertainty propagation and direct analysis of materials

processing

Estimation of PDFs of key parameters

Accumulated information

Previous experiment and simulation data

Manufacturing illustration …

Determine Bayesian inverse formulation (PPDF)

Prior distribution modeling

Spatial statistics

MCMC design (model reduction if necessary)

Optimal experiment design

Posterior state space exploitation

System parameters withsignificant uncertainty




BAYESIAN FRAMEWORK FOR ADAPTIVE DESIGN

Design Cycle

Stochastic design

framework

design solutionwith associated statistical feature

reliability and robustness

study

preliminary design

inputs

input updateupdated inputs Large

deviation ?

yes

nolikelihood new PPDF

treat as prior model

post design

MCMCmodel reduction

updated design solutionwith associated statistical feature

meta model

• desired material properties• reliability requirement• robustness requirement• direct processing model• optimization objective• system parameters• experimental data• simulation results• uncertainty characterization

Gradient based algorithms in stochastic Gradient based algorithms in stochastic spacesspaces




h

0

Known stochastic

fluxGuess for unknown stochastic

flux

h

0

Insulate known flux boundary

Apply perturbation to PDF of guess flux h

0

Insulate guess flux boundary

I

I

• Obtain PDFs of input data, boundary conditions• Obtain desired temperature response on the internal boundary I

Obtain difference between

desired and computed

temperature along I

Obtain gradient of objective as

value of adjoint along unknown flux

boundary

Continuum sensitivity –

perturbation of PDF of unknown

flux due to perturbation in

PDF of temperature

DIRECT

ADJOINTSENSITIVITY

Example problem – Stochastic inverse heat conduction

Spectral stochastic optimizationSpectral stochastic optimization




0 0.25 0.5 0.75 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.1 0.2-0.05

-0.025

0

0.025

0.05

Scale magnified

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Non-dimensional time

Mea

nflu

xan

dfir

stP

CE

term

0 0.25 0.5 0.75 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 MeanFirst PCE term+

Mean temperature readings + Large measurement noise level

+

Surprisingly accurate mean estimate !!

• In the presence of large errors, deterministic design problems require regularization

• Here error is considered as an inherent part of the model. Thus no regularization needed

• Large measurement errors lead to diffuse estimates of mean in deterministic case

• Not only mean estimate is accurate but also the stochastic method points to the fact that readings are useless for the initial transient (captured by standard deviation large compared to mean)

• Standard deviation large compared to mean

• Points to the fact that readings have a high noise level initially leading to faulty predictions




BAYESIAN INFERENCE FOR STOCHASTIC INVERSE PROBLEMS

Bayes’ Formula

)(

)()|()|(

Yp

θpθYpYθp

Input θInput θ Direct numericalSolver F

Direct numericalSolver F Measurement YMeasurement Y

noise ωNumerical

errorY = F(θ) + ω

Role of Prior PDF• Constrain likelihood• Incorporate known to a priori information• Regularize ill-posedness

Markov Random Field (MRF)

• in conjugate family of white noise• explore the spatial and temporal dependence• close related to Tikhonov regularization

}))((exp{)( ~ ji jiijWp

Driving force

process uncertainty

DAQ uncertaintyinitial uncertainty

Result-to-cause ?

Mathematical representation

Posterior distribution--- high dimension --- non-standard form--- only know up to a scaling constant--- intractable likelihood “black box” ! Need to known marginal PDF and posterior expectation of f(θ)

MCMC




BAYESIAN INFERENCE TO INVERSE HEAT TRANSFER PROBLEMS

--- True q in simulationq

0 0.4 0.8

1.0

t1.0

--- Normalized governing equation

2

2

xT

tT

1t 0 ,0 1x

0),0( xT 1x 0

0

LxxT

)(0

tqxT

x

,

,, 1t 0

1t 0,

Posterior mean estimate

x

q

d

L

Y (d,iΔt)

d = 0.3 Δt = 0.02 (n=50) dt = 0.04 (m=26) σ = 0.001, 0.005, 0.010 (2.5% Tmax)

Temperature prediction at d=0.5




AUGMENTED BAYESIAN MODELING IN INVERSE HEAT TRANSFER

True σ12

0.21 0.31 0.41 0.51 0.610

4

8

q (t = 0.16)

0.80 0.90 1.000

4

8

12

1.10

q (t = 0.40)

0.25 0.35 0.45 0.550

4

8

12

q (t = 0.64)

Guess of 2σ

0.10.20.30.40.50.60.70

2

4

6

q (t = 0.16)

0.60.70.80.91.01.11.20

2

4

6

q (t = 0.40)

0.10.20.30.40.50.60.70

2

4

6

q (t = 0.64)

Unknown σ

Marginal PDFs

Marginal PDFs

Pre-compliance design material

Compliance testing

To consider any design modifications for better

performance in customer operating conditions

Modified material properties are

uncertain

Uncertainty added due to

manufacturing variations

Currently produced

beams

Actual produced material properties and microstructure

may vary from those of compliance design

Manufacturing process

• Production acceptance testing in the presence of a

customer

•Test conditions adjusted to best simulate the customer operating conditions

• Prohibitively expensive• Testing facility has to be recorrelated for every new

production

Test conditions variations, measurement errors and

other undefined uncertainties

Compliance design material

Salient features

• Testing may be very expensive and must be minimized

• Uncertainties are introduced at each stage of the testing cycle




Simulation matching designSimulation matching design

What is the outcome if compliance testing

is done on new material products?

Can we predict the material performance under new operation

conditions

Data collected from previous production acceptance tests

and compliance tests

Can we estimate the newly manufactured material properties

along with their intervals of confidence

Information provided

• Precompliance design results for materials

• Physical or mathematical model relating the material properties and testing results

•Estimates of manufacturing variations

For more complicated systems, can we use the information in the

collected data to build a model ?

Can we explain the effect of change in

manufacturing variations on model

response?




Simulation matching design: Technical issuesSimulation matching design: Technical issues

Can we update the model as and when new test data

arrive instead of building the model over again?




PERSPECTIVES OF THE MPDC GROUPPERSPECTIVES OF THE MPDC GROUP

Robust design of deformation processes Couple materials process design with required materials testing selection

Develop and implement a spectral stochastic FEM approach to robust deformation process design

Quantify the propagation of uncertainty in material and process data and its effect on the computed designs

Develop mathematical tools to allow for trade-off between achievable design objectives, design reliability and limits of variability in materials and process data

Design across length scales: Propagation of uncertainty across length scales

Material propertyCharacterization byBayesian inference

ODF

Robust melt flow design and directional solidification of binary alloys




Develop an integrated approach to materials process design and materials testing selection: Materials testing driven by design objectives!

With given robustness limits on the desired product attributes, a virtual design simulator can point to the required materials testing that can obtain material properties with the needed level of accuracy



MCMCTesting

PERSPECTIVE OF THE MPDC GROUPPERSPECTIVE OF THE MPDC GROUP





Develop data mining algorithms for data filtering

Develop multi-level Bayesian posterior formulation for property testing

Develop efficient MCMC samplers for posterior state space exploration

Develop meta models based one data only using machine learning algorithms



Testing Data Mining MCMC

Statistical description of

material property






Develop and implement a spectral stochastic FEM approach to solidification and crystal growth problems

Quantify the propagation of uncertainty in process data due to surface roughness and model parameters

Study the effect of varying process parameters on microstructure





Effects of g-jitter under terrestrial and

micro gravity conditions

uncertainty propagation in materials processing materials process design and control laboratory...

Documents

robust design

extrusion process

safefail design

outcomesfailsafe design

control laboratorysolutions

control laboratoryon

simulations uncertainty

microstructure uncertainty