1. On the formulation and application of design rules Barna Szabó Washington University in St. Louis Engineering Software Research and Development, Inc. St. Louis, Missouri USA The University of Texas at Austin – March 22, 2016

2. Collaboration Prof. Ivo Babuška* Prof. Raul Tempone*, Marco Scavino and Zaid Sawlan Dr. Ricardo Actis* Mr. David Rusk* 3/22/20162 * Participants in the MITACS Workshop on Methodology of Validation and Verification Banff, Alberta, Canada, April 27-May 1, 2008

3. FEA timeline 3/22/20163

4. Dangerous trends 3/22/20164 Numerical simulation is being confused with finite element modeling. Recipe for generating mis- information on a very large scale.

5. Outline Formulation and application of design rules A new class of predictors for high cycle fatigue Calibration The design curve: Ranking The case for simulation governance 3/22/20162

6. Design rules 3/22/20166

7. Application of design rules Designers are obligated to consider the worst case scenario: Numerical errors penalize design Certification of design is not possible without knowing the numerical error 3/22/20167 F max ≤ F all | F max - F num | ≤  F max F num = (1 –  ) F max F num ≤ (1 –  ) F all

8. Design rules for high cycle fatigue of metals Goal: Establish design rules, given S-N data from coupon tests Account for arbitrary cycle ratios Formulate statistical models of S-N data Define predictors of damage accumulation for the generalization of S-N data Estimate the value of each predictor for the objects of interest Calibrate the predictors Rank the predictors (order of predictive performance) Update the design rules: Simulation governance 3/22/20168

9. Design/certification methodologies Safe life design Damage tolerant design Flaw-tolerant safe life design Barely detectable flaws will not initiate a propagating crack within the service life of a component. Clearly detectable flaws will not initiate a propagating crack between inspection intervals. 3/22/20169

10. Stress We will consider macro-mechanical stress: Average stress over a representative volume element (RVE) Metal fatigue is a highly nonlinear process: Irreversible changes (dislocations) accumulate over time, leading to crack initiation Phenomenological models are used for correlating failure events with macro-mechanical stress cycles. 3/22/201610

11. Classical machine design 3/22/201611

12. Peterson’s notch sensitivity factor 3/22/201612 Source: W. D. Pilkey, Peterson’s Stress Concentration Factors, 2nd Edition, John Wiley & Sons, New York (1997) p. 39.

13. SAE shaft – bending 3/22/201613

14. Formulation 3/22/201614

15. Generalization 3/22/201615

16. Formulation of predictors 3/22/201616

17. The effect of notches 3/22/201617 Dogbone specimen Double edge notched tensile specimen

18. NACA/NASA fatigue data 3/22/201618

19. Specimen types #1, #8, #9 3/22/201619

20. NASA test specimen type #9 (K t =4.0) 3/22/201620

21. NACA test specimen type #6 (K t =4.0) 3/22/201621 Fillet radius: 0.0195 ± 0.0005 inches Specimen used for calibration

22. Statistical models 3/22/201622

23. Calibration curve for Model 2a 233/22/2016

24. Profile likelihoods for A 3 3/22/201624

25. Calibration for  3/22/201625

26. Calibration for  3/22/201626

27. Empirical and predicted CDFs (  = 0.5) 3/22/201627

28. Specimen type #9: Which  ? 3/22/201628

29. Which  ? 3/22/201629

30. Random fatigue limit models 3/22/201630

31. Design curves for RFL models Design curves are contours of the CDF with the contour level selected such that failure prior to the end of design life is unlikely (p = 0.001). 3/22/201631

32. Design curves RFL - normal 3/22/201632

33. Design curves RFL - SEV 3/22/201633

34. Validation or ranking? Why reject a model if a better one has not been identified? It is not possible to claim that “I have a validated model” (there is no such thing) The last word will never be spoken 3/22/201634

35. Ranking design curves Statistical models are calibrated on the basis of highly probable data Design curves are extrapolations to low probability events (p < 1/1000). Few (if any) data points are available Interpretation of data from full-scale fatigue tests and tear-down inspection is complicated by numerical and modeling errors (finite element modeling practices, aka variational crimes) 3/22/201635

36. Bayes factors for RFL models 3/22/201636 BF = Prob(Data|M 1 )/Prob(Data|M 2 ) M 1 : The log of RFL is normal M 2 : The log of RFL is sev

37. Dependence on statistical model Design curves strongly depend on the choice of statistical model. For example: 3/22/201637 Allowable stress (ksi) for N = 10 5 probnormalsev 10 -3 23.221.0 10 -4 21.417.5 10 -5 20.014.5

38. The case for simulation governance Simulation governance is the exercise of command and control over all aspects of numerical simulation If the mission is application of design rules: Selection and adoption of the best available numerical simulation technology Procedures for data and solution verification Standardization of routine tasks Economic benefits: Productivity, reliability, repeatability 3/22/201638

39. The case for simulation governance If the mission is formulation of design rules: Formulation of mathematical models Collection, maintenance and documentation of experimental data Management of solution and data verification procedures Revision and updating of mathematical models in the light of new information collected from physical experiments and field observations Economic benefits: Substantial savings through optimization of design and maintenance procedures 3/22/201639

40. References Babuška I, Sawlan Z, Scavino M, Szabó B and Tempone R. Bayesian inference and model comparison for metallic fatigue data. ArXiv: 1512.01779v1 [stat.CO] 6 Dec. 2015. To appear in: Comput. Methods Appl. Mech. Engng. (2016), http://dx.doi.org/j.cma.2016.02.013 http://dx.doi.org/j.cma.2016.02.013 Szabó B. and Actis R. Simulation governance: Technical requirements for mechanical design. Comput. Methods Appl. Mech. Engng. 249 (2012) 158-168 3/22/201640