“from the measured part to the probabilistic analysis“ · probabilistic tutorial “from the...
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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]