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Exploring the Implications of Bayesian Approach to Materials State Awareness
R. Bruce ThompsonDirector, Center for Nondestructive Evaluation
Professor, Materials Science & Aerospace Engineering,
Iowa State University
AFOSR Prognosis Workshop_February 2008 2
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 3
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).
AFOSR Prognosis Workshop_February 2008 4
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)
Long term Advanced MaterialState Sensing
MesomechanicalDamage Models
CharacterizeMaterial
Microstructures
Full-Authority DigitalEngine Control (FADEC)
Math Model Mission Simulation
Ready
Short term
Application
Decision Capability forLegacy Engines
Lifing Algorithms
Analytical Stress Model
Installed AutonomousSensorsLong term
Logic for Integrated, Automated Prognosis System
AFOSR Prognosis Workshop_February 2008 5
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 6
In principle, we simply need to execute the following strategy
This would be a “done deal” if the input data were correct/complete and models were of sufficient accuracy and computationally efficient.
Utopian View
DamageProgression
Model
DamagedState
InitialState
OperationalEnvironment
FailureModel
ExpectedLifetime
FailureCriteria
AFOSR Prognosis Workshop_February 2008 7
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) Difficult to incorporate the missing information if it were
available Uncertainty
There will always be uncertainty in the input data Variability
Even if we eliminate uncertainty, we would have to take variability into account
AFOSR Prognosis Workshop_February 2008 8
Examples of Research Underway and Gaps Operational environment
Temperature, strain and chemical sensors under development State sensing data
Global Structures: strain, displacement, acceleration Propulsion: vibration analysis
Local Guided waves to sense structural changes Moisture Ultrasonic, eddy current, … to sense microstructure
Damage models Under refinement in many programs
AFOSR Prognosis Workshop_February 2008 9
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
AFOSR Prognosis Workshop_February 2008 10
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
Long Term Integration Needs
Space
Traditional NDE providesinformation as a functionof position at discrete timesT
ime
On board sensors provide information as a function of time at discrete locations
Space
Traditional NDE providesinformation as a functionof position at discrete timesT
ime
On board sensors provide information as a function of time at discrete locations
Traditional NDE providesinformation as a functionof position at discrete timesT
ime
On board sensors provide information as a function of time at discrete locations
AFOSR Prognosis Workshop_February 2008 11
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 12
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
AFOSR Prognosis Workshop_February 2008 13
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
AFOSR Prognosis Workshop_February 2008 14
Incidentsoundpulse
100 m
Single crystal
(“grain”)
Grain boundary echoes
Characterization of Grain Morphology
The reflection of sound at grain boundaries results in “noise” seen in UT inspections
AFOSR Prognosis Workshop_February 2008 15
Time Domain Waveforms
AFOSR Prognosis Workshop_February 2008 16
Characterization of Grain Structure
Grain noise inhomogeneity provides information about microstructure
AFOSR Prognosis Workshop_February 2008 17
Characterization of Grain Structure
Ultrasonic backscattering controlled by grain size
Theoretical base exists to quantify relationship (single scattering assumption)
AFOSR Prognosis Workshop_February 2008 18
Characterization of Grain Structure
Determining grain size and shape from single sided backscattering measurements
AFOSR Prognosis Workshop_February 2008 19
Characterization of Grain Structure Results obtained on rolled and extruded
aluminum
AFOSR Prognosis Workshop_February 2008 20
Characterization of Fatigue Damage Normalized Harmonic Ratio -vs- Percent Low Cycle Fatigue Life*
Ni-based Aero Engine Alloy
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 20 40 60 80 100 120Fatigue Life (Percent)
N3 - 51 ksi - =180kN4 - 47 ksi - =302kN7 - 47 ksi - =290k
* 100 % is last data point prior to first detection of surface crack
N
fN
fN
f
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 20 40 60 80 100 120
Nor
mal
ized
Har
mon
ic R
atio
(A2/
A12 )/
(A2/
A12 ) u
nfa
tigu
ed
AFOSR Prognosis Workshop_February 2008 21
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
AFOSR Prognosis Workshop_February 2008 22
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
AFOSR Prognosis Workshop_February 2008 23
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 24
The Bayesian Approach
The essence of the Bayesian approach is to provide a mathematical rule explaining how you should combine new data with existing knowledge or expertise 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 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)
AFOSR Prognosis Workshop_February 2008 25
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
In principle, y might be a global or local variable One way to “invert” data is to adjust x to maximize the
pdf, p(y/x) 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
y = m xobservation
material state parameters (e.g., flaw size)
AFOSR Prognosis Workshop_February 2008 26
Likelihood: Direct Use in Inversion
In the language of the likelihood approach, is proportional to the likelihood function
Sometimes written 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
p y x x ;yL or L x
AFOSR Prognosis Workshop_February 2008 27
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
AFOSR Prognosis Workshop_February 2008 28
Bayes Theorem for Continuous Variables
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”
( / ) ( )( / )
( / ) ( )
f y x f xf x y
f y s f s ds
Likelihood of x p(y/x) Prior distribution of x
Posterior pdf Normalization
AFOSR Prognosis Workshop_February 2008 29
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
AFOSR Prognosis Workshop_February 2008 30
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 “If the defect were a crack, it would have the following size”
We use the measurement results to determine which of those possible states are most consistent with the data 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
AFOSR Prognosis Workshop_February 2008 31
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
failure model NDE data inversion
Note: P(x/y) will depend on the accept/reject criterion
( / ) ( / ) ( / )P c y P c x P x y dx
Richardson
AFOSR Prognosis Workshop_February 2008 32
Effects of Randomness and Completeness
One measurement Failure uncertainty Measurement uncertainty
One measurement Failure perfect Measurement perfect
Complete measurement Failure uncertainty Measurement perfect
false accepts false accepts false accepts
fals
e re
ject
s
fals
e re
ject
s
fals
e re
ject
s
AFOSR Prognosis Workshop_February 2008 33
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 34
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)
AFOSR Prognosis Workshop_February 2008 35
Microstructural Fatigue Model
AFOSR Prognosis Workshop_February 2008 36
Potential Sensor Assistance at Various StagesStage of Fatigue
Potential Measurement Status of Scientific Foundation
Implementation Issues
Crack nucleation
Grain size determination by UT backscatter after manufacturing
Well established for single phase materials
Effects of precipitates and grain boundary decorations under study
No major “show stoppers”
Short crack growth
Ultrasonic harmonic generation
Mechanisms for engineering materials under study (dislocations vs. microcracks as sources)
Very challenging measurement on wing
Long crack growth
Deploying tradition NDE in-situ
Broad foundations in place
Effects of morphology e.g., closure, subject of ongoing study
Challenging measurement of wing
AFOSR Prognosis Workshop_February 2008 37
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 38
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 39
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
Needs physics-based understanding of relationship to microstructure
Constrain by access, survivability, need for simplicity Bayesian statistics provides an attractive framework
for integrating these disparate inputs Enabled by physics-based models of the measurement
process A conceptual example based on aircraft engine disks
was provided