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Chanyoung Park Raphael T. Haftka Paper Helicopter Project.

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Presentation on theme: "Chanyoung Park Raphael T. Haftka Paper Helicopter Project."— Presentation transcript:

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2 Chanyoung Park Raphael T. Haftka Paper Helicopter Project

3 Structural & Multidisciplinary Optimization Group 2/25 Problem1: Conservative estimate of the fall time  Estimating the 5 th percentile of the fall time of one helicopter  Estimating the 5 th percentile to compensate the variability in the fall time (aleatory uncertainty)  The sampling error (epistemic uncertainty)  Estimating the sampling uncertainty in the mean and the STD  Obtaining a distribution of the 5 th percentile  Taking the 5 th percentile of the 5 th percentile distribution to compensate the sampling error Sampling  t,P  t,P tPtP t 0.05,P

4 Structural & Multidisciplinary Optimization Group 3/25  Estimating the 5 th percentile of the fall time of first helicopter (mean 3.78, std 0.37)  100,000 5 th percentiles of fall time  Helicopter 1 of the dataset 3  Height: 148.5 in  2.88 (sec) is the 5 th percentile of the histogram (a conservative estimate of the 5 th percentile of the fall time for 95% confidence) Problem1: Conservative estimate of the fall time

5 Structural & Multidisciplinary Optimization Group 4/25 Problem2: Predicted variability using prior  Calculating predicted variability in the fall time  We assume that the variability in the fall time is caused by the variability in the CD  The variability in the fall time is predicted using the computational model (quadratic model) and the distribution of the CD  The prior distribution represents our initial guess for the distribution of the CD Height at time t Steady state speed where

6 Structural & Multidisciplinary Optimization Group 5/25 Problem2: Comparing predicted variability and observed variability using prior  Area metric with the prior  Data set 3 0.290.340.49 CD from the fall time data Helicopter1Helicopter2Helicopter3 Sample mean of CDs 0.8960.8420.783 Sample STD of CDs 0.1900.1540.084 Prior Mean of CD = 1 / STD of CD = 0.28

7 Structural & Multidisciplinary Optimization Group 6/25 Problem3: Calibration: Posterior distribution of mean and standard deviation  Estimating parameters of the CD distribution  We assume that CD of each helicopter follows the normal distribution  The parameters, CD and σ test are estimated using 10 data  Posterior distribution is obtained based on 10 fall time data  Non informative distribution is used for the standard deviation After 1 updateAfter 5 updatesAfter 10 updates

8 Structural & Multidisciplinary Optimization Group 7/25 Problem3: Comparing predicted variability using posterior and observed variability  Comparing MLE and sampling statistics  MCMC sampling  10,000 pairs of the CD and the STD of CD are generated using Metropolis-Hastings algorithm  An independent bivariate normal distribution is used as a proposal distribution  MLE of the posterior distribution is used as a starting point CD from the fall time data Helicopter1Helicopter2Helicopter3 Sample mean of CDs 0.8960.8420.783 Sample STD of CDs 0.1900.1540.084 MLE of the CD mean 0.8960.8430.783 MLE of the CD STD 0.1820.1460.081

9 Structural & Multidisciplinary Optimization Group 8/25 Problem3: Comparing predicted variability using posterior and observed variability  Handling the epistemic uncertainty due to finite sample  How to handle epistemic uncertainty in the CD and the test standard deviation estimates?  Comparing the posterior predictive distribution of the fall time and the empirical CDF of tests (combining epistemic and aleatory uncertainties)  Using p-box with 95% confidence interval of epistemic uncertainty (separating epistemic and alreatory uncertainties)

10 Structural & Multidisciplinary Optimization Group 9/25 Problem3: Comparing predicted variability using posterior and observed variability  Area metric of the posterior predictive distribution of CD 0.150.110.07 CD from the fall time data Helicopter1Helicopter2Helicopter3 148.5 in 2 clips (ref) Sample mean of CDs 0.8960.8420.783 Sample STD of CDs 0.1900.1540.084

11 Structural & Multidisciplinary Optimization Group 10/25 Problem3: Comparing predicted variability using posterior and observed variability  Area metric of the posterior predictive distribution of CD 0.010.00 CD from the fall time data Helicopter1Helicopter2Helicopter3 148.5 in 2 clips (ref) Sample mean of CDs 0.8960.8420.783 Sample STD of CDs 0.1900.1540.084

12 Structural & Multidisciplinary Optimization Group 11/25 Problem4: Predictive validation for the same height and different weight  Area metric of the posterior predictive distribution of CD 0.190.340.38 CD from the fall time data Helicopter1Helicopter2Helicopter3 148.5 in 1 clips Sample mean of CDs 0.9160.9920.968 Sample STD of CDs 0.1470.0690.049 148.5 in 2 clips (ref) Sample mean of CDs 0.8960.8420.783 Sample STD of CDs 0.1900.1540.084

13 Structural & Multidisciplinary Optimization Group 12/25 Problem4: Predictive validation for the same height and different weight  Area metric of the distribution of CD with p-box 0.010.110.22 CD from the fall time data Helicopter1Helicopter2Helicopter3 148.5 in 1 clips Sample mean of CDs 0.9160.9920.968 Sample STD of CDs 0.1470.0690.049 148.5 in 2 clips (ref) Sample mean of CDs 0.8960.8420.783 Sample STD of CDs 0.1900.1540.084

14 Structural & Multidisciplinary Optimization Group 13/25 Problem6: Predictive validation for different height and the same weight  Area metric of the posterior predictive distribution of CD 0.340.190.09 CD from the fall time data Helicopter1Helicopter2Helicopter3 181.25 in 2 clips Sample mean of CDs 0.8860.8660.816 Sample STD of CDs 0.0790.1190.066 148.5 in 2 clips (ref) Sample mean of CDs 0.8960.8420.783 Sample STD of CDs 0.1900.1540.084

15 Structural & Multidisciplinary Optimization Group 14/25 Problem6: Predictive validation for different height and the same weight  Area metric of the distribution of CD with p-box 0.050.000.01 CD from the fall time data Helicopter1Helicopter2Helicopter3 181.25 in 2 clips Sample mean of CDs 0.8860.8660.816 Sample STD of CDs 0.0790.1190.066 148.5 in 2 clips (ref) Sample mean of CDs 0.8960.8420.783 Sample STD of CDs 0.1900.1540.084

16 Structural & Multidisciplinary Optimization Group 15/25 Problem5: Linear model  Area metric with the prior  Data set 3 0.650.680.81 CD from the fall time data Helicopter1Helicopter2Helicopter3 Sample mean of CDs 0.9780.9490.916 Sample STD of CDs 0.1020.0910.050 Prior Mean of CD = 1 / STD of CD = 0.28

17 Structural & Multidisciplinary Optimization Group 16/25 Comparison to predictive validation  Area metric of the posterior predictive distribution of CD  The predictive validation with the linear model is not as successful as that with the quadratic model  Area metric with p-box  Area metric with p-box tries to capture the extreme discrepancy between the predicted variability and the observed variability Helicopter1Helicopter2Helicopter3 148.5 in / 1 clips 0.130.060.02 148.5 in / 2 clips (ref) 0.010.00 Helicopter1Helicopter2Helicopter3 148.5 in / 1 clips 0.440.300.14 148.5 in / 2 clips (ref) 0.130.110.07

18 Structural & Multidisciplinary Optimization Group 17/25 Problem5: Linear model  Area metric of the posterior predictive distribution of CD 0.130.110.07 CD from the fall time data Helicopter1Helicopter2Helicopter3 148.5 in 2 clips (ref) Sample mean of CDs 0.9780.9490.916 Sample STD of CDs 0.1020.0910.050

19 Structural & Multidisciplinary Optimization Group 18/25 Problem5: Linear model  Area metric of the distribution of CD with p-box 0.010.00 CD from the fall time data Helicopter1Helicopter2Helicopter3 148.5 in 2 clips (ref) Sample mean of CDs 0.9780.9490.916 Sample STD of CDs 0.1020.0910.050

20 Structural & Multidisciplinary Optimization Group 19/25 Problem5: Linear model with one clip  Area metric of the posterior predictive distribution of CD 0.440.300.14 CD from the fall time data Helicopter1Helicopter2Helicopter3 148.5 in 1 clips Sample mean of CDs 0.8820.9230.911 Sample STD of CDs 0.0780.0330.025 148.5 in 2 clips (ref) Sample mean of CDs 0.9780.9490.916 Sample STD of CDs 0.1020.0910.050

21 Structural & Multidisciplinary Optimization Group 20/25 Problem5: Linear model with one clip  Area metric of the distribution of CD with p-box 0.130.060.02 CD from the fall time data Helicopter1Helicopter2Helicopter3 148.5 in 1 clips Sample mean of CDs 0.8820.9230.911 Sample STD of CDs 0.0780.0330.025 148.5 in 2 clips (ref) Sample mean of CDs 0.9780.9490.916 Sample STD of CDs 0.1020.0910.050

22 Structural & Multidisciplinary Optimization Group 21/25 Comparison between quadratic and linear models  Area metric of the posterior predictive distribution of CD  Area metric of the distribution of CD with p-box Helicopter 1 Helicopter 2 Helicopter 3 148.5 in / 1 clipLinear 0.190.340.38 Quadratic 0.440.300.14 148.5 in / 2 clips (ref) Linear 0.130.110.07 Quadratic 0.150.110.07 Helicopter 1 Helicopter 2 Helicopter 3 148.5 in / 1 clipLinear 0.130.060.02 Quadratic 0.010.110.22 148.5 in / 2 clips (ref) Linear 0.010.00 Quadratic 0.010.00

23 Structural & Multidisciplinary Optimization Group 22/25 Concluding remarks  Predictive validation for both quadratic and linear models  The predictive validation for different mass is a partially success  The predictive validation for different height is a success but the assumption of constant CD is not clearly proven  Comparison between models  Cannot conclude  Overall  Reason for the differences in the area metric is not clear  The effect of the manufacturing uncertainty is significant (i.e. very different area metrics for the same test condition)

24 Paper Clips HeightCalibrated?Model Area Metric Set 1 Model 1Set 1 Model 2Set 1 Model 3Set 3 Model 1Set 3 Model 2Set 3 Model 3 21noquadratic0.53760.83970.57170.30040.37060.5299 21yesquadratic0.1430.4420.190.3050.2670.0946 12yesquadratic0.62720.8330.65030.39270.5930.5442 21nolinear0.96391.02021.00620.63850.57370.7388 21yeslinear0.10730.50030.25870.35710.31040.1398 11yeslinear1.32161.53981.34570.2130.2070.168 22yesquadratic0.1690.2140.2840.4610.4240.269 Kaitlin Harris, VVUQ Fall 2013 Comments: Chose to maintain uniform distribution for calibration parameter based on histogram results (vs normal) Conclusions: Best models: calibrated quadratic at both heights and calibrated linear with 2 paper clips Worst models: un- calibrated linear with 2 paper clips and un- calibrated linear with 1 clip for data set 1

25 Validation of analytical model used to predict fall time for Paper Helicopter By Nikhil Londhe *Calibrated Analytical Model is validated to represent experimental data *Quadratic dependence is valid assumption between drag force and speed *For given difference in fall height, Cd can be assumed to be constant Comparison of Analytical Cdf and Empirical Cdf Data Set 5, H=18.832, No. of Pins=2Helicopter 1Helicopter 2Helicopter 3 Validation Area Metric Before Calibration0.73090.69320.6169 After Calibration0.28090.09060.142 Calibration Results Helicopter 1Helicopter 2Helicopter 3 Maximum Likelihood Estimate of Cd0.78890.89650.9111 Standard deviation in Posterior pdf of Cd0.0460.01910.0238 Predictive Validation for Data Set 5, No. of Pins =1 H=18.832ft.Helicopter 1Helicopter 2Helicopter 3 Validation Area Metric0.18690.24530.192 Validation Area Metric for Linear Dependence Model Data Set 5Helicopter 1Helicopter 2Helicopter 3 Validation Area Metric0.44350.26060.3506 Validation of Cd is constant at different height, h=11.482ft Data Set 5, Pins = 2Helicopter 1Helicopter 2Helicopter 3 Validation Area Metric0.41860.15920.1513

26 Quadratic Dependence 2 clips Different height. Quadratic Dependence 1 clip Predictive Validation. Linear Dependence 2 clips & 1clip Prior VS. Posterior Dist. Quadratic Dependence 2 clips Prior VS. Posterior Dist. Course Project: Validation of Drag Coefficient -Yiming Zhang Validation based on 1 set of data: Validation based on 3 set of data: Summary: (1) Quadratic dependence seems accurate using one set; Quadratic and linear dependence both don’t match well using 3 sets; Prior Area Metric:0.4042 Post Area Metric:0.1695 Prior Area Metric:0.4920 Post Area Metric: 0.2 Validation Area Metric:0.1923 Validation Area Metric:0.1267  2 clips:  1 clip: Posterior Area Metric:0.1729 Validation Area Metric:0.1960  Summary: Two confi. interval of Cd don’t coincide. Linear dependence seems inaccurate.  2 clips: Posterior Area Metric:0.1987  1 clip: Validation Area Metric: 0.1253  Summary: Seems reasonable, but not accurate Validation Area Metric: 0.0970 Validation Area Metric: 0.1394 (2) SRQ is required to be fall time. If use Cd as SRQ, comparison could be more consistent and clear; (3) 0.8 seems a reasonable estimation of Cd. This estimation would be more accurate while introducing more sets of data.

27 Backup Slides

28 Structural & Multidisciplinary Optimization Group 27/25 Problems  Problem1: Conservative estimate of the fall time  Problem2: Comparing predicted variability and observed variability using prior  Problem3: Comparing predicted variability and observed variability using posterior  Problem4: Predictive validation for the same height and different weight  Problem5: Comparing the quadratic and linear models  Problem6: Predictive validation for different height and the same weight (proving the assumption of constant CD)

29 Structural & Multidisciplinary Optimization Group 28/25 Problem1: Conservative estimate of the fall time  Estimating the 5 th percentile of the fall time of one helicopter  Since fall time follows a normal distribution, estimating the 5 th percentile is based on estimating the mean and standard deviation (STD) of the fall time distribution  The mean and STD are estimated based on 10 samples  There is epistemic uncertainty in the estimated mean and STD due to a finite number of samples  To compensate the epistemic uncertainty, a conservative measure to compensate the epistemic uncertainty is required  Estimating the 5 th percentile with 95% confidence level

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