Exploring the Implications of Bayesian Approach to Materials State Awareness

1. 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

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. Logic for Integrated, Automated Prognosis System Application Long term Advanced Material State Sensing Decision Capability for Legacy Engines Characterize Material Microstructures Mesomechanical Damage Models Lifing Algorithms Short term Analytical Stress Model Full-Authority Digital Engine Control (FADEC) Math Model Mission Simulation Installed Autonomous Sensors Long term Ready 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

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. Utopian View 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. Damage Progression Model Initial State Damaged State Failure Model Expected Lifetime Operational Environment Failure Criteria 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. 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 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. The reflection of sound at grain boundaries results in “noise” seen in UT inspections Incident sound pulse Grain boundary echoes Single crystal (“grain”) 100 mm Characterization of Grain Morphology 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. Nf Nf Nf Characterization of Fatigue Damage 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 observation 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 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 Likelihood of x  p(y/x) Prior distribution of x 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” 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 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 rejects false rejects false rejects false accepts false accepts false accepts 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 Stages 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 AFOSR Prognosis Workshop_February 2008