UNCERTAINTY PROPAGATION IN MATERIALS PROCESSING

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UNCERTAINTY PROPAGATION IN MATERIALS PROCESSING

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Title: UNCERTAINTY PROPAGATION IN MATERIALS PROCESSING

1
UNCERTAINTY PROPAGATION IN MATERIALS PROCESSING
Nicholas Zabaras
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering188 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu URL
http//www.mae.cornell.edu/zabaras/
Materials Process Design and Control Laboratory
2
THE TITANIC SINKING 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.
Materials Process Design and Control Laboratory
3
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? Robust design to avoid catastrophic
failures
Materials Process Design and Control Laboratory
4
UNCERTAINTY IN THE MICROSTRUCTURE-PROPERTY-PROCESS
ING TRIANGLE

Uncertainty propagation in simulations
Uncertainty in initial state and
microstructure
Uncertainty in materials testing

Processing

Uncertainty propagation in simulations
Material model parameters
Modeling of tool behavior
Model validation

Uncertainty propagation in simulations
Multi-stage processing
Conditions between stages
Simulation error, round off
errors

Materials Process Design and Control Laboratory
5
FAILURE ANALYSIS COMPONENT DIAGNOSTICS
Design safety assessments require the
establishment of significance of defects in
components subject to creep and creep/fatigue
loading.
FAILURE MODES Creep, low-cycle fatigue (LCF)
high -cycle fatigue (HCF)
Characterization of creep crack initiation and
growth
Initiation time for crack size of
Cracked blade
Crack growth rate
Predictions of the behavior of component needs to
be evaluated considering the stochastic nature of
the creep fracture mechanics parameters.
Materials Process Design and Control Laboratory
6
FAILURE ANALYSIS COMPONENT DIAGNOSTICS
Materials Process Design and Control Laboratory
7
EFFECTS OF UNCERTAINTY ON PERFORMANCE, COSTS
SAFETY
Materials Process Design and Control Laboratory
8
SOURCES OF UNCERTAINTY
Questioning isotropy assumption Typical
stress-strain response depends on direction and
alloy as seen in the picture.

Uncertainties in
Direction and property quantification
Material characterization

Geometric uncertainty Sample of experimentally
observed statistics for an extrusion process for
different lengths of extrusion (Materials and
Design 2001, 22, 267-275).
Materials Process Design and Control Laboratory
9
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 methodologies

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
Materials Process Design and Control Laboratory
10
Prognosis- main idea
Residual stresses, duty environment (temperature,
humidity, working stress, thrust), Defect
distributions
Upper and lower bounds of the quantities available
Use FORM to get failure surface plot and failure
probability
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
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
Continuum sensitivity analysis provides
sensitivities of key parameters like component
stress, strain levels and strain rates, crack
propagation
Materials Process Design and Control Laboratory
11
PROGNOSIS, LIFING AND RELIABILITY
DATABASE

Short term predictions more accurate
Evolutionary prediction based on Markov
autoregressive chains for long range predictions

Past mission history
Knowledge of system behavior
Expert knowledge

Predicting performance based on current state

Failure physics modeling
Reliability predictions
Evolutionary physics based model (BAYESIAN)

Safe
Fail
Prediction of failure response surface

Simple maneuvers to assess system state
Benchmark tests
Feature extraction
Ultrasonics

Reduced order models for various failure regimes
Classification of current system state

EXTERNAL TESTING

Confidence intervals for predictions
Update reduced order models for system analysis
Update benchmarking schemes

POST PROCESSING MODULE
DIGITAL LIBRARY
12
MODELING MATERIALS ACROSS 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
Property averaging
Continuum
Engineering
Interfacial energies
Micro- structural
Inter-Atomic Potentials
Atomistic
Materials
Electronic
Chemistry
Physics
Length Scales
1 nm 1 mm
1 mm
1 m
13
INFORMATION THEORY FOR MATERIALS PROCESSES
Given input microstructure, how to choose
material models from a class of available models
Model chosen based on microstructure
Orientation distribution function model
Lineal analysis of microstructure photograph
Poly-phase material
Dendritic
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.

14
3D DESCRIPTION UNCERTAINTY
Continuum Model
Average Properties
Simulated 3D grain structure
Description Uncertainty
Spatial Point Field Model
Voronoi Tessellation Model
Spheroid Model
15
MICROSTRUCTURE CLASSIFICATION
Noisy Input Image
Feature Detection Uncertainty
Classifier Uncertainty
Digital Microstructure Library
Materials Process Design and Control Laboratory
16
UNCERTAINTIES 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.
Image
Spatial Structure
Uncertainty propogation in 3D recognition from 2D
images.(Ref. Sobh, T.M. and Mahmood, A.)
Materials Process Design and Control Laboratory
17
UNCERTAINTY IN FEATURE MAPPING
Uncertainties in Mapping 3D Microstructural
Features to 2D Domain (Sensor Uncertainty)

Find the uncertainty of mapping a specific 3D
feature to a 2D pixel value.
The 3D feature is located at 2D pixel position
(i,j) with probability p1, (i1,j) with p2 etc.
given that the registered location is (l,m) such
that p1p2..pn 1 assuming no uncertainty in
feature recovery mechanism. The goal is to find
the probabilities.

Imaged structure
Spatial structure
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)
Materials Process Design and Control Laboratory
18
PROBABILISTIC NATURE OF FEATURE RECOVERY
Feature Addition
Edge Loss

Errors in Image Processing
Data is lost during compression/denoising
Noise is amplified when derivatives are
computed.
Addition of new unrelated features in the image

Uncertainty Problem Given a feature recovered
from an image is in pixel position (x,y), the
probability that the feature was originally at
the position (x1,y) with probability p1, (x2,y)
with probability p2 etc. such that p1p2pn
1 due to noise in the images. The problem is to
find the probabilities.
Materials Process Design and Control Laboratory
19
UNCERTAINTY MICROSTRUCTURE EVOLUTION IMAGES

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)
Materials Process Design and Control Laboratory
20
DYNAMIC MICROSTRUCTURE LIBRARY
Data Mining Young's Modulus 75 GPa
Training samples
New Input Class
Hot rolling
Process class
Image
Deep drawing
Classification Scheme
Y. Modulus
Property Class
ODF
Yield stress
Update Class
Outside Training space
Pole Figures
Materials Process Design and Control Laboratory
21
MICROSTRUCTURE REPRESENTATION
Materials Process Design and Control Laboratory
22
REPRESENTATION USING PCA
Texture reconstruction using PCA statistics
Reduced Description using PCA
Statistics of Eigen coefficients
Input Image Snapshots
Input Image
Eigen Basis
Reconstructed Image using 20 coefficients
Raw Image 32 x 32
Image Generated from random coefficients using
known statistics
Materials Process Design and Control Laboratory
23
PHASE FIELD METHOD
For Complex Multi-component Systems
Phase Field (Evolution of Field Variables)
Continuum Deformation Problem
Digital Library
Model Reduction
Thermodynamic variables Free energy Anisotropy Mob
ilities Interfacial energies
Couple Field variables displacement gradients
in local free energy functions
Crystallographic Lattice Parameters
Atomistic level
Continuum Scale
Meso Scale
Materials Process Design and Control Laboratory
24
Pole figure data inversion to ODF
LS problem

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 results
Incomplete pole figures - too small a region of
measurement.
Integration errors - Pole data is discrete.

Materials Process Design and Control Laboratory
25
Pole figure data inversion to ODF
EBSD
X-ray diffraction

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.

Materials Process Design and Control Laboratory

26
Stochastic Simulation of Microstructure Evolution
Materials Process Design and Control Laboratory
27
Metal 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

Materials Process Design and Control Laboratory
28
Metal 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
Materials Process Design and Control Laboratory
29
Metal forming optimal design
Preforming Stage
Finishing Stage
Unfilled cavity
Initial Design
Initial guess

Fully filled cavity
Final Design
Optimal preform
How sensitive is the optimal design to shape and
parameter uncertainties ? Needs to specify
robustness limits for optimal design parameters
Materials Process Design and Control Laboratory
30
Importance of uncertainty in solidification
process
Meso-scale (dendritic structures seen)
Typical dendritic structures obtained due to
small perturbations to initial conditions
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?
Modeled as flow in media with variable
porosityOnly a statistical description is
possible macroscopically, thus need to have a
stochastic framework for analysis
Materials Process Design and Control Laboratory
31
Stochastic simulation of solidification processes
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
Solidification
Heat Transfer
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
Materials Process Design and Control Laboratory
32
UNCERTAINTY IN CRYSTAL GROWTH
Effect of uncertainty on crystal growth
Terrestrial
Micro gravity
Bridgeman growth
Czochralski growth
Melt flow in HBG
Materials Process Design and Control Laboratory
33
Variational multiscale modeling
Large scale flow behavior

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

Sub grid scale flow phenomena

Uncertainties introduced due to small scale
phenomena are important
Sub grid scales are not homogenous as considered
in many computational techniques

Add contributions to large scale solution
VMS
Approximate fine scale model
Approaches for sub grid approximation

Computational sub grid modeling
Explicit models considered for sub grid based on
bubble functions
Fractal modeling of sub grid phenomena

Algebraic sub grid scale model
Sub grid solution is a function of a stochastic
intrinsic time scale
Materials Process Design and Control Laboratory
34
Stochastic fluid flow - Example
Mid plane mean pressure and standard deviation
Schematic of the problem computational domain
uu(q), v0
(1,1)

Lid driving velocity is uniform between (0.9 and
1.1)
Viscosity is 0.0025, hence mean Reynolds number
400

Initially quiescent fluid
uv0
uv0
(0,0)
uv0
Mean velocity evolution
Std deviation of velocity evolution
Materials Process Design and Control Laboratory
35
Stochastic algebraic subgrid scale (SASGS) model
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
(Momentum residual from large scales) x
(stochastic subgrid momentum time scale)
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
(continuity residual from large scales) x
(stochastic subgrid continuity time scale)
Materials Process Design and Control Laboratory
36
Robust design - motivation
Modifications in objectives
User update
Robust product specifications
Input
USER INTERFACE
Output design
Stochastic optimization, Spectral/Bayesian
framework
Design database, simulations and experiments
Control and reduced order modeling

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

Materials Process Design and Control Laboratory
37
Approach to robust design of materials processes
Output PDFs obtained from SSFEM analysis
Required product with desired material properties
and shape with specified confidence (output PDFs)
Are PDFs of design variables
technically feasible?
No
Yes
Sensitivity analysis toolbox
Sensitivity of product w.r.t material data and
other process conditions
Interface with digital library and expert advice
to modify design objectives, material models,
process models
Reference input and process conditions PDFs
Yes
High performance computing environment
Are levels of
uncertainty (PDFs) in other process conditions
tolerable?
No
Yes
Can we obtain the PDFs by
existent testing?
No
MATERIALS TESTING DRIVEN BY DESIGN ROBUSTNESS
LIMITS BAYESIAN INFERENCE
Yes
Update model PDFs and database (digital library)
Materials Process Design and Control Laboratory
38
Various robust design statements
Minimization of variance approach

Based on extension to least squares approach
Results similar to robust regression techniques

Objective
Reliability type design optimization
Probabilistic constraint

Constraint can become highly nonlinear, use RSMs

Complete stochastic optimization

Integral defined over the sample space
Avoids over-design problems

Materials Process Design and Control Laboratory
39
Stochastic optimization with spectral methods
Design decision Finite Vs Infinite dimensional
optimization
Non-parametric representation, design variables
considered as functions
Parametric representation of stochastic design
variables
PDF after perturbing the design parameter vector
Original PDF
APPROACH
APPROACH

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

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

Realization of Solution
Commonality Sensitivity calculations
Materials Process Design and Control Laboratory
40
BAYESIAN FRAMEWORK FOR MATERIAL PROPERTY
ESTIMATION
Confidence intervals
statistics
Markov Chain Monte Carlo (MCMC)
PPDF
Likelihood
Prior PDF
A data driven model
p
p

q
)
(
)
(
q
Y
p
p

(
q
µ

)
(
)
)

(
q
q
Y
Y
p
p
)
(
Y
A Bayesian statistical inference model
Uncertainty in measurement
Testing experiment
Y F(?) ?

Destructive
Non-destructive

Materials Process Design and Control Laboratory
41
BAYESIAN FRAMEWORK FOR MATERIALS PROCESS CONTROL
Material processing
Point estimate
Numerical process modeling
Filtered data Y
Marginal pdfs
Hyper-parameter In Bayesian model
Posterior state space exploration
Materials Process Design and Control Laboratory
42
BAYESIAN FRAMEWORK FOR ESTIMATION OF (NON-DESIGN)
PROPERTIES
Manufacturing illustration
Previous experiment and simulation data
Accumulated information
Uncertainty propagation and direct analysis of
materials processing
Statistical modeling of uncertainties
Prior distribution modeling
Spatial statistics
Determine Bayesian inverse formulation (PPDF)
MCMC design (model reduction if necessary)
Optimal experiment design
Estimation of PDFs of key parameters
Posterior state space exploitation
Materials Process Design and Control Laboratory
43
BAYESIAN FRAMEWORK FOR ADAPTIVE DESIGN
Design Cycle
preliminary design

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

design solution with associated statistical
feature
reliability and robustness study
Stochastic design framework
treat as prior model
yes
meta model
inputs
no
Large deviation ?
new PPDF
updated inputs
likelihood
input update
MCMC
model reduction
updated design solution with associated
statistical feature
post design
Materials Process Design and Control Laboratory
44
Gradient based algorithms in stochastic spaces

Obtain PDFs of input data, boundary conditions

Example problem Stochastic inverse heat
conduction

Obtain desired temperature response on the
internal boundary GI

Known stochastic flux
Guess for unknown stochastic flux
Continuum sensitivity perturbation of PDF of
unknown flux due to perturbation in PDF of
temperature
GI
Obtain difference between desired and computed
temperature along GI
Insulate known flux boundary
Apply perturbation to PDF of guess flux
Insulate guess flux boundary
GI
Obtain gradient of objective as value of adjoint
along unknown flux boundary
Materials Process Design and Control Laboratory
45
Spectral stochastic optimization
Surprisingly accurate mean estimate !!
Mean temperature readings Large measurement
noise level

Scale magnified

Standard deviation large compared to mean
Points to the fact that readings have a high
noise level initially leading to faulty
predictions

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)

Materials Process Design and Control Laboratory
46
BAYESIAN INFERENCE FOR STOCHASTIC INVERSE PROBLEMS
Materials Process Design and Control Laboratory
47
BAYESIAN INFERENCE TO INVERSE HEAT TRANSFER
PROBLEMS
--- True q in simulation
--- Normalized governing equation
q
Y (d,i?t)
1.0
q
x
1.0
0
0.4
0.8
t
d
L
Posterior mean estimate
Temperature prediction at d0.5
Materials Process Design and Control Laboratory
48
AUGMENTED BAYESIAN MODELING IN INVERSE HEAT
TRANSFER
Marginal PDFs
True s
Marginal PDFs
Guess of 2s
Unknown s
Materials Process Design and Control Laboratory
49
Simulation matching design
Uncertainty added due to manufacturing variations
Modified material properties are uncertain
Compliance design material
To consider any design modifications for better
performance in customer operating conditions
Manufacturing process
Compliance testing
Pre-compliance design material
Actual produced material properties and
microstructure may vary from those of compliance
design
Currently produced beams
Test conditions variations, measurement errors
and other undefined uncertainties
Materials Process Design and Control Laboratory
50
Simulation matching design Technical issues
Data collected from previous production
acceptance tests and compliance tests

Information provided
Precompliance design results for materials
Physical or mathematical model relating the
material properties and testing results
Estimates of manufacturing variations

What is the outcome if compliance testing is done
on new material products?
Can we estimate the newly manufactured material
properties along with their intervals of
confidence
Can we predict the material performance under new
operation conditions
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?
Can we update the model as and when new test data
arrive instead of building the model over again?
Materials Process Design and Control Laboratory
51
PERSPECTIVES OF THE MPDC GROUP