Exploring the Implications of
Bayesian Approach to
Materials State Awareness
R. Bruce Thompson
Director, Center for Nondestructive Evaluation
Professor, Materials Science &
Aerospace Engineering,
Iowa State University
Outline





Interpretation of Current Status of and Future
Needs for Prognosis
Microstructural Characterization Sensors
Integration within Bayesian Framework
A Conceptual Illustration
Conclusions
AFOSR Prognosis Workshop_February 2008
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L. Christodoulou and J. M. Larsen, “Using Materials Prognosis to Maximize the Utilization
Potential of Complex Mechanical Systems,” Materials Damage Prognosis, J. M. Larsen,
L. Christodoulou, J. R. Calcaterra, M. L. Dent, M. M. Derriso, J. W. Jones, ad S. M. Russ, Eds.
(TMS, 2005).
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Logic for Integrated, Automated
Prognosis System
Application
Long term
Characterize
Material
Microstructures
Advanced Material
State Sensing
Decision Capability for
Legacy Engines
Mesomechanical
Damage Models
Lifing Algorithms
Short term
Full-Authority Digital
Engine Control (FADEC)
Ready
Math Model
Mission Simulation
Long term
Analytical Stress Model
Installed Autonomous
Sensors
L. Christodoulou and J. M. Larsen, “Materials Damage Prognosis: A Revolution in Asset
Management,” Materials Damage Prognosis, J. M. Larsen, L. Christodoulou, J. R. Calcaterra,
M. L. Dent, M. M. Derriso, J. W. Jones, ad S. M. Russ, Eds. (TMS, 2005). (adapted from Cruse)
AFOSR Prognosis Workshop_February 2008
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New Ingredients
“In many ways, materials damage prognosis is
analogous to other damage tolerance
approaches, with the addition of
in-situ local damage and global state
awareness capability and much improved
damage predictive models”
L. Christodoulou and J. M. Larsen, “Materials Damage Prognosis: A Revolution in Asset
Management,” Materials Damage Prognosis, J. M. Larsen, L. Christodoulou, J. R. Calcaterra,
M. L. Dent, M. M. Derriso, J. W. Jones, ad S. M. Russ, Eds. (TMS, 2005).
AFOSR Prognosis Workshop_February 2008
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Utopian View
In principle, we simply need to execute the following
strategy
Initial
State
Damage
Progression
Model
Damaged
State
Operational
Environment
Failure
Model
Expected
Lifetime
Failure
Criteria
This would be a “done deal” if the input data were
correct/complete and models were of sufficient
accuracy and computationally efficient.
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Barriers to Reaching Nirvana

Missing information



Do not currently determine the initial state of individual
components/structures/systems with high precision
Have not traditionally monitored the operating environment
of individual components
Damage progression models have traditionally been
empirical (e.g., Paris Law)


Uncertainty


Difficult to incorporate the missing information if it were
available
There will always be uncertainty in the input data
Variability

Even if we eliminate uncertainty, we would have to take
variability into account
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Examples of Research Underway
and Gaps

Operational environment


State sensing data
 Global



Structures: strain, displacement, acceleration
Propulsion: vibration analysis
Local




Temperature, strain and chemical sensors under development
Guided waves to sense structural changes
Moisture
Ultrasonic, eddy current, … to sense microstructure
Damage models
 Under refinement in many programs
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Long Term Microstructural Sensor
Needs

Improved sensor and data interpretation
procedures to monitor evolution of
microstructure during damage



A key will be a well-developed, quantitative
understanding of relationship of sensor response
to microstructural changes
Physics-based models of the sensing process
Must work subject to practical constraints



Access
Survivability
Simplicity of implementation
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Long Term Integration Needs

Systems perspective to integrate all of the NDE state data with
damage model predictions
 Depot, field, on board sensors
 Global, local sensors
 Measurements of initial state, damage state
Must recognize fundamental difference in data structure for
traditional (depot and field) and on board NDE measurements
On board sensors provide information as a
function of time at discrete locations
Traditional NDE provides
information as a function
of position at discrete times
Time

Space
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Outline





Interpretation of Current Status of and Future
Needs for Prognosis
Microstructural Characterization Sensors
Integration within Bayesian Framework
A Conceptual Illustration
Conclusions
AFOSR Prognosis Workshop_February 2008
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Detailed Understanding of Microstructure must be a Key
Ingredient in Development of State Awareness Strategies

An idealized scenario

Generally, each link has it challenges




Non-uniqueness
Inadequate sensitivity to key parameters
Limitations of the theory base
Force a stochastic approach
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Need for Microstructural Characterization
Tools as Well as Flaw Detection Tools

Need to be able to assess the progression of
damage before cracks form



Quantification of initial state
Check of evolution of damage when possible
Validation of prognostic calls
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Characterization of Grain Morphology

The reflection of sound
at grain boundaries
results in “noise” seen
in UT inspections
Incident
sound
pulse
100 mm
Grain
boundary
echoes
Single
crystal
(“grain”)
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Time Domain Waveforms
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Characterization of Grain Structure

Grain noise inhomogeneity provides
information about microstructure
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Characterization of Grain Structure


Ultrasonic backscattering controlled by grain
size
Theoretical base exists to quantify
relationship (single scattering assumption)
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Characterization of Grain Structure

Determining grain size and shape from single
sided backscattering measurements
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Characterization of Grain Structure

Results obtained on rolled and extruded
aluminum
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Characterization of Fatigue Damage
Normalized
Harmonic Ratio -vs- Low Cycle Fatigue Life
Normalized Harmonic Ratio -vs- Percent Low Cycle Fatigue Life*
Ni-based
Engine
Alloy
Ni-based Aero
Aero Engine
Alloy
1.81.8
N3 - 51 ksi - N =180k
N4 - 47 ksi - fN =302k
N7 - 47 ksi - fN =290k
Normalized Harmonic Ratio
(A2/A12)/(A2/A12)unfatigued
1.61.6
1.41.4
f
1.2
1.2
1
1
0.8
0.8
0.6
0.6
N4
0.4
N3
0.4
0.2
N7
00.2
00
0
Serie
s4 20
Serie
20
s5
* 100 % is last data point prior to first detection of surface crack
* 100% is defined as the last data point prior to first appearance
of a surface crack found through visual or penetrant inspections
40
60
80
(Percent)
40Fatigue Life60
80
100
100
120
120
Fatigue Life (Percent)*
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The Way Forward

Significant benefits can be obtained from
further developing nondestructive
microstructural characterization tools


Best developed if seek relationship to
microstructure rather than properties
Need physics-based, rather than empirical
understanding

Needs collaboration of measurement and materials
experts
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Some Open Questions



Role of precipitates and grain boundary
decorations in ultrasonic and backscattering
measurements
Role of dislocations in attenuation
measurements
Relative roles of dislocations and microcracks
in harmonic generation
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Outline





Interpretation of Current Status of and Future
Needs for Prognosis
Microstructural Characterization Sensors
Integration within Bayesian Framework
A Conceptual Illustration
Conclusions
AFOSR Prognosis Workshop_February 2008
23
The Bayesian Approach

The essence of the Bayesian approach is to provide a mathematical
rule explaining how you should




From an intuitive perspective, we can consider the “utopian view” that we
discussed previously as existing knowledge
The new data are the results of NDE measurements about initial state,
operational environment, or the state of damage evolution
This approach addresses the non-uniqueness problem that plagues
the interpretation of many NDE measurements


combine new data with existing knowledge or expertise
A framework for data inversion
Enabling technologies are


Physics-based models of the NDE measurement process
High speed computational capability that makes implementation practical
(not the case a decade ago)
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Traditional Data Inversion

Consider a model relating input parameters (state of material
or flaw) x, to experimental observations, y, where y and x are
vectors
y = m x 
observation
material state parameters (e.g., flaw size)


One way to “invert” data is to adjust x to maximize the
pdf, p(y/x)



In principle, y might be a global or local variable
One seeks parameter values that maximize the probability of
the observed data
We do this all at the time in making least square fits to data
Need more observations than unknown parameters in order
for this to work
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Likelihood: Direct Use in Inversion

In the language of the likelihood approach,
 p  y x  is proportional to the likelihood function
 Sometimes written L  x  or L  x;y 



We seek to choose the values of x such that the likelihood
is maximized
These values are considered best estimates of x
In special cases, this approach is equivalent to the
more familiar least squares fitting procedures



y normally distributed about mean values
No systematic errors in models (model predicts mean
values)
No truncated or censored data
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Limitations of this Approach to
Inversion


This approach (including least squares fitting) breaks down if
 Data is not sufficient to determine parameters without auxiliary
information or assumption (i.e., solutions of inverse problem
would not be unique)
 One wishes to incorporate knowledge from past experience in a
systematic way
 One wishes to estimate probability of parameter values (not just
most likely values)
Bayes Theorem provides a path forward
 Allows direct incorporation of physical understanding of
processes (e.g., as incorporated in physics-based simulation
tools)
 Significant computations may be required

“Computational plenty” is reducing this objection
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Bayes Theorem for Continuous
Variables
Likelihood of x
 p(y/x)
Prior distribution of x
f ( y / x )f ( x )
f (x / y ) 
 f ( y / s )f (s )ds
Posterior pdf
Normalization
Note: Physical understanding of the measurement,
ideally as captured by a physics-based model, enters
through the likelihood  p(y/x). “How likely was the
observed state data for possible states in the prior
distribution”
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Summary of Bayesian Approach

Advantages

Framework to utilize “prior” knowledge




Update beliefs about probability of state in light of new
evidence, the measurement results y
Provides “posterior” (probability distribution of state),
not just most likely state
Depends in a simple way on the “likelihood”,
something that can be computed from forward models
Issues


Significant computations
Dependence on the prior


Posterior may not be highly sensitive to this
Sensitivity studies needed
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An Intuitive Description

The prior contains our knowledge about the
materials state that is expected to be present

In one way or the other, we often make such assumptions
in a less formalized way


We use the measurement results to determine
which of those possible states are most consistent
with the data


“If the defect were a crack, it would have the following size”
In essence, ruling out the portions of the prior distribution
that are inconsistent with the observations
The posterior is the sharpened distribution of states
that emerges
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Generalization to Failure Prediction

Probabilistic model for P(x,y,c)

x: state of defect

y: measured data

c: 1 if piece survives under specified conditions
0 if piece fails under specified conditions
From this model, want to infer the probability of failure (c) given the NDE data


P (c / y )   P ( c / x )
P( x / y ) dx

failure model


NDE data inversion
Note: P(x/y) will depend on the accept/reject criterion
Richardson
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Effects of Randomness and
Completeness
false accepts
false rejects
false rejects
false rejects
One measurement
One measurement
Complete measurement
 Failure uncertainty
 Failure perfect
 Failure uncertainty
 Measurement uncertainty  Measurement perfect
 Measurement perfect
false accepts
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false accepts
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Outline





Interpretation of Current Status of and Future
Needs for Prognosis
Microstructural Characterization Sensors
Integration within Bayesian Framework
A Conceptual Illustration
Conclusions
AFOSR Prognosis Workshop_February 2008
33
Waspalloy Disk
“The scatter in material behavior is attributed to the inhomogeneous
microstructure elements with metals.”
L. Nasser and R. Tryon, “Prognostic System for
Microstuctural-Based Reliability”, DARPA Prognostics web site
(with reference to work at Cowles, P&W)
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Microstructural Fatigue Model
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Potential Sensor Assistance at
Various Stages
Stage of
Fatigue
Potential Measurement
Status of Scientific
Foundation
Crack
nucleation
Grain size determination by
UT backscatter after
manufacturing

Short crack
growth
Ultrasonic harmonic
generation
Long crack
growth
Deploying tradition NDE
in-situ
Implementation
Issues
Well established for
single phase materials
Effects of precipitates
and grain boundary
decorations under study
No major “show
stoppers”

Mechanisms for
engineering materials
under study
(dislocations vs.
microcracks as sources)
Very challenging
measurement on
wing

Broad foundations in
place
Effects of morphology
e.g., closure, subject of
ongoing study
Challenging
measurement of
wing


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At the End of the Day
(In this or other applications)

When we balance






Our improving but incomplete understanding of failure
processes
The ideal characterization procedures based on
understanding of the measurement physics
The measurement possibilities as constrained by practical
constraints
We will be making prognoses based on incomplete
information
Exact data inversion will not be possible
Suggest use of Bayesian statistics to eliminate
possible outcomes inconsistent with sensor data
AFOSR Prognosis Workshop_February 2008
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Outline





Interpretation of Current Status of and Future
Needs for Prognosis
Microstructural Characterization Sensors
Integration within Bayesian Framework
A Conceptual Illustration
Conclusions
AFOSR Prognosis Workshop_February 2008
38
Conclusions

Realizing a full Materials State Awareness capability
will require a wide range of inputs



Mesoscopic damage models
Sensing of operational parameters of individual components
Advanced material state sensing



Bayesian statistics provides an attractive framework
for integrating these disparate inputs


Needs physics-based understanding of relationship to
microstructure
Constrain by access, survivability, need for simplicity
Enabled by physics-based models of the measurement
process
A conceptual example based on aircraft engine disks
was provided
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