4. Dresdner Probabilistik-Workshop. Probabilistic Tutorial. From the measured part to the probabilistic analysis. Part I

Transcription

1 4. Dresdner Probabilistik-Workshop Probabilistic Tutorial From the measured part to the probabilistic analysis Part I Matthias Voigt (TU Dresden)

2 Outline Part I 1. Introduction 2. Basic Statistics 3. Probabilistic methods (MCS) Part II 4. Optical measurement (3D scanning) 5. Identification of geometrical Parameter and statistical analysis of scanned blades 6. Probabilistic Model - Meshmorphing 7. Generation of probabilistic samples 8. Analysis of probabilistic FE simulation Part III 9. Analysis of cold-hot transformation 10. Deterministic CFD Modell 11. Analysis of probabilistic CFD simulation 12. Conclusion Slide 2

3 Outline Part I 1. Introduction 2. Basic Statistics 3. Probabilistic methods (MCS) Part II 4. Optical measurement (3D scanning) 5. Identification of geometrical Parameter and statistical analysis of scanned blades 6. Probabilistic Model - Meshmorphing 7. Generation of probabilistic samples 8. Analysis of probabilistic FE simulation Part III 9. Analysis of cold-hot transformation 10. Deterministic CFD Modell 11. Analysis of probabilistic CFD simulation 12. Conclusion Slide 3

4 Introduction Two turbine blades from same engine Massachusetts Institute of Technology, Prof. David L. Darmofal but with clearly different life. Slide 4

5 Introduction Classic simulation-based approach used in design Design Intent Simulation Real behavior of physical system Design Intent Reality Slide 5

6 Outline Part I 1. Introduction 2. Basic Statistics 3. Probabilistic methods (MCS) Part II 4. Optical measurement (3D scanning) 5. Identification of geometrical Parameter and statistical analysis of scanned blades 6. Probabilistic Model - Meshmorphing 7. Generation of probabilistic samples 8. Analysis of probabilistic FE simulation Part III 9. Analysis of cold-hot transformation 10. Deterministic CFD Modell 11. Analysis of probabilistic CFD simulation 12. Conclusion Slide 6

7 Boundary condition for probabilistic simulations probabilistic analysis of systems deterministic model Slide 7

8 Example of deterministic model beam with a semibeam input values: beam height H beam width W E-Modulus point load P point load P position S result values: deflections w m, w i Slide 8

9 Boundary condition for probabilistic simulations probabilistic analysis of systems deterministic model distribution type, distribution parameters, correlationen between input values Slide 9

10 Reason for scatter Slide 10

11 Statistical measures 1 arithmetic mean: [1] non-robust estimator, outlier sensitive median: is that value of the point that divides the area below the probability density function into two pieces of equal size a robust measure of central tendency modal value or mode: is that value of the data set that occurs with the greatest frequency, it is not necessarily unique. Slide 11

12 Statistical measures 2 Slide 12

13 Statistical measures 3 standard deviation: [1] coefficient of variation: Slide 13

14 Statistical distribution Zusammenhänge von Verteilungen [1] Slide 14

15 Kolmogorow-Smirnow-Test Slide 15

16 Anderson-Darling-Test the Anderson-Darling-Test is a modification of the Kolmogorow- Smirnow-Test this gives more weight to the tails than the K-S test does [2] is the cumulative distribution function of the test data tables of critical values for e.g. are available in [4] for different distributions Slide 16

17 Boundary condition for probabilistic simulations probabilistic analysis of systems deterministic model distribution type, distribution parameters, correlationen between input values Slide 17

18 Correlationen Pearson's correlation coefficient [1] range: [-1,1] Slide 18

19 Correlationen Spearman's Rank correlation coefficient [1] range: [-1,1] Slide 19

20 Correlationen Slide 20

21 Boundary condition for probabilistic simulations probabilistic analysis of systems deterministic model distribution type, distribution parameters, correlationen between input values probabilistic method Slide 21

22 Outline Part I 1. Introduction 2. Basic Statistics 3. Probabilistic methods (MCS) Part II 4. Optical measurement (3D scanning) 5. Identification of geometrical Parameter and statistical analysis of scanned blades 6. Probabilistic Model - Meshmorphing 7. Generation of probabilistic samples 8. Analysis of probabilistic FE simulation Part III 9. Analysis of cold-hot transformation 10. Deterministic CFD Modell 11. Analysis of probabilistic CFD simulation 12. Conclusion Slide 22

23 Monte-Carlo-Simulation (MCS) Slide 23

24 Results of MCS Application of Monte-Carlo methods for probabilistic investigations using optimized LHS under consideration of input parameter correlation Result of probabilistic Simulation pdf of input variables - roughly known (as in industry) Sensitivities Robustness Optimization Probability of failure - precisely known (rarely) required number of deterministic runs output - one single MC Simulation provides all result quantities - almost independent to no. of parameters - common: n sim = minimum: n sim = no. of par better confidence intervals for higher n sim Slide 24

25 Correlationen Slide 25

26 Correlationen [3] third-order response surface first-order response surface third-order response surface first-order response surface Slide 26

27 Correlationen Robustness 2D Ant Hill Plot Slide 27

28 Meta model R 2 = third-order response surfaces with mixed terms and Box-Cox-transformation Slide 28

29 Meta model R 2 = Slide 29

30 Probability of failure density (LHS, DS) (SRS) MCS RSM importance sampling, SORM, FORM Slide 30

31 Conclusions problems advantage in the application of probabilistic methods sufficient database for stochastic variables required increased engineering time - parametric model - validation of results - interpretation of results high computational cost due to multiple analysis probability to failure (no successive conservatism in the assessment) robustness of design sensitivity of result values w.r.t. stochastic variables cheaper design: - increased tolerances if possible - decreased tolerances if necessary Slide 31

32 Outline Part I 1. Introduction 2. Basic Statistics 3. Probabilistic methods (MCS) Part II 4. Optical measurement (3D scanning) 5. Identification of geometrical Parameter and statistical analysis of scanned blades 6. Probabilistic Model - Meshmorphing 7. Generation of probabilistic samples 8. Analysis of probabilistic FE simulation Part III 9. Analysis of cold-hot transformation 10. Deterministic CFD Modell 11. Analysis of probabilistic CFD simulation 12. Conclusion Slide 32

33 references [1] [2] [3] [4] Matthias Voigt, TU Dresden 3. Dresdner Probabilistik-Workshop Slide 33