Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

4. Dresdner Probabilistik-Workshop

Probabilistic Tutorial

“From the measured part to the probabilistic analysis“

Part IMatthias Voigt (TU Dresden)

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 2

Part I1. Introduction2. Basic Statistics3. Probabilistic methods (MCS)

Part II4. Optical measurement (3D scanning)5. Identification of geometrical Parameter and statistical analysis of

scanned blades6. Probabilistic Model - Meshmorphing7. Generation of probabilistic samples8. Analysis of probabilistic FE simulation

Part III9. Analysis of cold-hot transformation10.Deterministic CFD Modell11.Analysis of probabilistic CFD simulation12.Conclusion

Outline

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 3

Part I1. Introduction2. Basic Statistics3. Probabilistic methods (MCS)

Part II4. Optical measurement (3D scanning)5. Identification of geometrical Parameter and statistical analysis of

scanned blades6. Probabilistic Model - Meshmorphing7. Generation of probabilistic samples8. Analysis of probabilistic FE simulation

Part III9. Analysis of cold-hot transformation10.Deterministic CFD Modell11.Analysis of probabilistic CFD simulation12.Conclusion

Outline

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 4

Massachusetts Institute of Technology, Prof. David L. Darmofal

Two turbine blades from same engine…

… but with clearly different life.

Introduction

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 5

• Classic simulation-based approach used in design

• Real behavior of physical system

Reality

Introduction

Simulation

Design Intent

Design Intent

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 6

Part I1. Introduction2. Basic Statistics3. Probabilistic methods (MCS)

Part II4. Optical measurement (3D scanning)5. Identification of geometrical Parameter and statistical analysis of

scanned blades6. Probabilistic Model - Meshmorphing7. Generation of probabilistic samples8. Analysis of probabilistic FE simulation

Part III9. Analysis of cold-hot transformation10.Deterministic CFD Modell11.Analysis of probabilistic CFD simulation12.Conclusion

Outline

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 7

Boundary condition for probabilistic simulations

probabilistic analysis of systems

deterministic model

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 8

Example of deterministic model

beam with a semibeam

result values: • deflections wm, wi

input values:• beam height H• beam width W• E-Modulus• point load P• point load P position S

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 9

Boundary condition forprobabilistic simulations

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

probabilistic analysis of systems

deterministic model

distribution type, distribution parameters, correlationen between

input values

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 10

Reason for scatter

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 11

• arithmetic mean:

− 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.

Statistical measures 1

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

[1]

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 12Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Statistical measures 2

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 13

• standard deviation:

• coefficient of variation:

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

[1]

Statistical measures 3

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 14

Zusammenhänge von Verteilungen

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

[1]

Statistical distribution

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 15

Kolmogorow-Smirnow-Test

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 16

• 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

• is the cumulative distribution function of the test data• tables of critical values for e.g. are available in [4] for

different distributions

Anderson-Darling-Test

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

[2]

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 17

Boundary condition forprobabilistic simulations

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

probabilistic analysis of systems

deterministic model

distribution type, distribution parameters, correlationen between

input values

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 18

Pearson's correlation coefficient

range: [-1,1]

Correlationen

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

[1]

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 19

Spearman's Rank correlation coefficient

range: [-1,1]

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

[1]

Correlationen

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 20Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Correlationen

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 21

probabilistic analysis of systems

deterministic model

Boundary condition for probabilistic simulations

distribution type, distribution parameters, correlationen between

input values

probabilistic method

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 22

Part I1. Introduction2. Basic Statistics3. Probabilistic methods (MCS)

Part II4. Optical measurement (3D scanning)5. Identification of geometrical Parameter and statistical analysis of

scanned blades6. Probabilistic Model - Meshmorphing7. Generation of probabilistic samples8. Analysis of probabilistic FE simulation

Part III9. Analysis of cold-hot transformation10.Deterministic CFD Modell11.Analysis of probabilistic CFD simulation12.Conclusion

Outline

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 23

Monte-Carlo-Simulation (MCS)

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 24

Application of Monte-Carlo methods for probabilistic investigations using optimized LHS under consideration of input parameter correlation

Result of probabilistic Simulation

Sensitivities Robustness Optimization Probability of failure

pdf of input variables- roughly known (as in industry)

- precisely known (rarely)

required number of deterministic runs

- almost independent to no. of parameters- common: nsim = 50…100- minimum: nsim = no. of par + 10…20 - better confidence intervals for higher nsim

output- one single MC Simulation provides all result quantities

Results of MCS

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Result of probabilistic Simulation

Sensitivities Robustness Optimization Probability of failure

pdf of input variables- roughly known (as in industry)

- precisely known (rarely)

required number of deterministic runs

- almost independent to no. of parameters- common: nsim = 50…100- minimum: nsim = no. of par + 10…20 - better confidence intervals for higher nsim

output- one single MC Simulation provides all result quantities

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 25

Correlationen

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 26Probabilistic Tutorial “From the measured part to the probabilistic analysis”

[3]

third-order response surface

first-order response surface

Correlationen

first-order response surface

third-order response surface

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 27

CorrelationenRobustness

2D Ant Hill PlotProbabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 28

third-order response surfaces with mixed terms and Box-Cox-transformation

Meta model

R2=0.997

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 29

Meta model

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

R2=0.997

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 30

(LHS, DS) (SRS) MCS

Probability of failure

RSM

importance sampling, SORM, FORM

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

dens

ity

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 31

Conclusions

problems advantage

• 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

in the application of probabilistic methods

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 32

Part I1. Introduction2. Basic Statistics3. Probabilistic methods (MCS)

Part II4. Optical measurement (3D scanning)5. Identification of geometrical Parameter and statistical analysis of

scanned blades6. Probabilistic Model - Meshmorphing7. Generation of probabilistic samples8. Analysis of probabilistic FE simulation

Part III9. Analysis of cold-hot transformation10.Deterministic CFD Modell11.Analysis of probabilistic CFD simulation12.Conclusion

Outline

Probabilistic Tutorial “From the measured part to the probabilistic analysis”

Faculty of Mechanical Engineering · Institute of Fluid Mechanics · Chair of Turbomachinery and Jet Propulsion

Matthias Voigt, TU Dresden Slide 33

references

[1]

[3]

[2]

3. Dresdner Probabilistik-Workshop

[4]