Exploring the Implications of Bayesian Approach to Materials State Awareness R. Bruce Thompson...

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


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).


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


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).


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


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


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


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


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


ime



ime



Outline




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


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


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


Time Domain Waveforms


Characterization of Grain Structure

Grain noise inhomogeneity provides information about microstructure



Ultrasonic backscattering controlled by grain size

Theoretical base exists to quantify relationship (single scattering assumption)



Determining grain size and shape from single sided backscattering measurements


Characterization of Grain Structure Results obtained on rolled and extruded

aluminum


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


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


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


Outline




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)


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)


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


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


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


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


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


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


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


Outline




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)


Microstructural Fatigue Model


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


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


Outline




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

Exploring the Implications of Bayesian Approach to Materials State Awareness R. Bruce Thompson...

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