1. Conference on Systems Engineering Research Use of Akaike’s Information Criterion to Assess the Quality of the First Mode Shape of a Flat Plate 14th Annual Conference on Systems Engineering Research (CSER 2016) March 22-24, 2016 Huntsville, Alabama John H. Doty, Ph. D. Doty Consulting Services CSER 2016, Doty Consulting Services

2. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 2 Purpose: Perform structural mode analysis of flat plate Determine mode shapes for 1 st 3 modes Extract mass-weighted effective independence (MWEI) retained degrees of freedom (DOFs) for 6 methods (referenced AIAA paper) Determine if AIC methodology produces consistent results to MWEI for similar mode shapes Determine if AIC methodology produces acceptably-better results MWEI for similar mode shapes CSER 2016, Doty Consulting Services

3. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 3 Flat Plate Results: 1 ft x 1 ft square flat plate, aluminum, 0.1 in thick, clamped at all edges Frequencies and 1 st 3 mode shapes determined Extracted mass-weighted effective independence (MWEI) retained degrees of freedom (DOFs) for 6 methods Applied corrected Akaike Information Criterion (AICc) to results & compare to MWEI CSER 2016, Doty Consulting Services

4. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 4 Results: Grid for FEA (400 elements for 1ft x 1 ft) 441 DOFs total, 361 DOFs ‘free’ to move, 80 DOFs removed via clamped Boundary Conditions CSER 2016, Doty Consulting Services

5. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 5 Results: 1 st Mode Shape 361 DOFs ‘free’ to move, All DOFs used 80 DOFs are clamped Boundary Conditions CSER 2016, Doty Consulting Services

6. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 6 Results: 2 nd Mode Shape 361 DOFs ‘free’ to move, All DOFs used CSER 2016, Doty Consulting Services

7. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 7 Preliminary Methodology: Given that mode shapes for 1 st 3 modes resemble bivariate cubic: –Perform cubic spline extrapolant as statistical model of surfaces Extrapolates well beyond region of fit –Also enables model to be ‘fit’ to arbitrary grid locations, not just nodal points With fitted equation  f(x,y) grid locations: –Perform leave-one-out cross validation, understand quality of model for prediction –Perform predictions of ALL un-constrained mode shapes –Assess quality of fit: residuals, mean-squared error (MSE)  Akaike Information CSER 2016, Doty Consulting Services

8. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 8 Theory Overview: –Akaike Information Criterion (AIC) performs estimate of most-likely ‘fit’ of an unknown function that generated a data set –Corrected Akaike Information Criterion (AICc) provides ‘small’ sample correction for asymptotically-equivalent bias correction as for large samples –Select model with lowest AICc as ‘best’ of those considered Small Sample Correction Bias Bias Correction Additive Constant CSER 2016, Doty Consulting Services

9. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 9 Theory Overview (cont’d): –Mass-Weighted-Effective-Independence paper reference: –Notation: * Reference: The Effect of Mass-Weighting on the Effective Independence of Mode Shapes, Jeffrey A. Lollock and Thomas R. Cole, 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, 18 - 21 April 2005, Austin, Texas CSER 2016, Doty Consulting Services

10. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 10 CSER 2016, Doty Consulting Services

11. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 11 Theory Overview (cont’d): Mass-Weighted Effective Independence (MWEI) –Model 4: Square Root of the Diagonal Mass Matrix –Model 5: Guyan Reduced Mass Matrix –Model 6: Identity Matrix CSER 2016, Doty Consulting Services

12. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 12 Results: Degrees of Freedom (DOFs) kept, All Methods CSER 2016, Doty Consulting Services

13. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 13 Results: DOFs kept, Method 1 DOF positions in red circles: 92 98 134 136 138 140 152 154 162 164 178 180 196 198 202 204 238 240 244 246 262 264 278 280 288 290 302 304 306 308 344 350 CSER 2016, Doty Consulting Services

14. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 14 Results: DOFs kept, Method 1 Overlaid on mode shapes From Paper: From my FEA analysis and overlay of Method 1 DOFs: Red circle: Method 1 DOFs kept Black circle: DOFS removed due to clamped BC CSER 2016, Doty Consulting Services

15. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 15 Which dofs (1  # dofs, or combinations) important? Which model is best (least loss of information relative to mode shapes (MS) as ‘truth’)? Proposed Models Data DOF # Mode Shape Value 1 441 MS(1) MS(441) Develop AICc methodology to relate quality of model relative to DOFs retained Use saturated model as reference (all DOFs except BC’s retained (361) ) DOF #Response 10 20 30 …… 200 210 220 230.0213 240.1303 250.3172 …… 4410 CSER 2016, Doty Consulting Services

16. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 16 Numerical results (decomposed) corrected Akaike Information Criterion (AICc) allocation of information SourceModel 1 Model 2 Model 3 Model 4 Model 5 Model 6 n dofs32 30 76 36 28 116 K33 31 77 37 29 117 SS error 1428.37 1461.59 1428.91 857.73 1762.84 1092.14 Bias606.56 616.70 606.72 381.65 699.34 488.20 2K66 62 154 74 58 234 AIC672.56 678.70 760.72 455.65 757.34 722.20 2nd order correction5.51 4.85 33.09 6.98 4.23 85.49 AICc678.07 683.55 793.81 462.63 761.57 807.68 Bias Bias Correction Small Sample Correction Model 4 Best by AICc Model 5 Best by MWEI Notable difference in information value CSER 2016, Doty Consulting Services

17. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 17 Results for 1 st Mode Shape Comparison of Methods 4 (AICc best) and 5 (MWEI best) Projected onto 1 st Mode Shape CSER 2016, Doty Consulting Services

18. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 18 Using results from paper (Method 5 as ‘best’): Perform perturbation analyses using AICc relative to the ‘best’ Method 5 in paper Determine if there is an improved solution (fewer dofs with ~ same AICc) Initial Method 5 DOFs Initial Method 5 DOFs, Remove 4 near center Initial Method 5 DOFs, Remove 4 near center, add center CSER 2016, Doty Consulting Services

19. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 19 Method 5 as ‘best’ Reduce by 4 –Arguably ~similar 4 Fewer dofs than method 5 Reduce by 4, add CP –Arguably no worse 3 Fewer dofs than method 5 corrected Akaike Information Criterion (AICc) allocation of information: Method 5 SourceModel 5 Model 5 Reduce 4 dofs Model 5 Reduce 4 dofs add CP n dofs28 24 25 K29 25 26 SS error 1762.84 1779.80 Bias699.34 703.60 2K58 50 52 AIC757.34 753.60 755.60 2nd order correction4.23 3.10 3.40 AICc761.57 756.70 758.90 CSER 2016, Doty Consulting Services

20. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 20 Summary: An alternative methodology to MWEI is offered based upon information theory –Akaike Information Criterion, corrected version (AICc) Results for plate 1 st mode shape indicate that AICc methodology represents the mode shape information –But, different result than MWEI Potential reduction in DOFs for ~same information value CSER 2016, Doty Consulting Services

21. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 21 Extensions and applications Compare experiments & simulations  residuals –Use logistic function to characterize & quantify differences –Use AICc to determine value of information –Goal: understand where and by how much models offer ‘trustability’ Fewer sensors with same level of risk in decision making Where to put sensors to provide same quality of information (e.g. frequenciers, bending modes, etc.) –Reduced power, weight, volume required for sensors CSER 2016, Doty Consulting Services

22. Flat Plate FEA Analysis and Akaike Information Criterion (AIC) Slide - 22 Acknowledgements This effort is supported by a grant from NASA through the University of Alabama in Huntsville System Engineering Consortium and is greatly appreciated. The author specifically thanks Dr. Michael D. Watson from the NASA Marshall Space Flight Center and Professor Phillip A. Farrington from The University of Alabama in Huntsville, Industrial & Systems Engineering & Engineering Management Department, for their valuable feedback and motivation for this effort. CSER 2016, Doty Consulting Services