Probabilistic Design

1. Probabilistic Design • Introduction • An Example • Motivation • Features • Benefits • Probabilistic Methods • Probabilistic Results/Interpretation • Summary

2. Introduction Purpose of a Probabilistic Design System (PDS) Input Output ANSYS • Material Properties • Geometry • Boundary Conditions • Deformation • Stresses / Strains • Fatigue, Creep,... It’s a reality that input parameters are subjected to scatter => automatically the output parameters are uncertain as well!! Uncertain ! Scatter !

3. Introduction Purpose of Probabilistic Design System (PDS) ANSYS PDS Scatter ! Uncertain ! • Questions answered with probabilistic design: • How large is the scatter of the output parameters? • What is the probability that output parameters do not fulfill design criteria (failure probability)? • How much does the scatter of the input parameters contribute to the scatter of the output (sensitivities)?

4. An Example • Material • Strength • Material Properties • Loads • Thermal • Structural Evaluate reliability of products ! Geometry/ Tolerances Evaluate quality of products ! Evaluate warranty costs ! • Boundary Conditions • Gaps • Fixity To evaluate is the first step to improvement ! Example: Lifetime of Components !!! Random input variables Finite-Element Model Random output parameters • LCF lifetime • Creep lifetime • Corrosion lifetime • Fracture mechanical lifetime • …

5. Motivation Influence of Young’s Modulus and Thermal Expansion Coefficient on thermal stresses: thermal = E ·  ·T Deterministic Approach: Emean and mean => evaluate expected value: expect Probabilistic Approach: P(thermal > 1.05 expect) P(thermal > 1.10 expect) ‘E’ scatters ±5% 16% (~1 out of 6) 2.3% (~1 out of 40) ‘E’ and ‘ ‘ scatter ±5% 22% (~1 out of 5) 8% (~1 out of 12)

6. Scatter in material properties and loads Source: Klein, Schueller et.al. Probabilistic Approach to Structural Factors of Safety in Aerospace. Proc. CNES Spacecraft Structures and Mechanical Testing Conf., Paris 1994

7. Motivation Materials, Bound.- Cond., Loads, ... ±5-100% Thermal Analysis CAD FEM CFD LCF ±??% Geometry FEM ± 0.1-10% Materials, Bound.- Cond., ... Structural Analysis Materials Materials, Bound.- Cond., Loads, ... ±30-60% ±5-50% ±5-100%

8. PDS Benefits • Deterministic Analysis: • Only provides a YES/NO answer • Safety margins are piled up “blindly” (worst material, maximum load, … worst case) • 1 worst case assumption =10-2 • 2 worst case assumptions =10-4 • 3 worst case assumptions =10-6 • 4 worst case assumptions =10-8 • ... • => Leads to costly over-design • Only “as planned“, “as is” or the worst design • Probabilistic Analysis: • Provides a probability and reliability (design for reliability) • Takes uncertainties into account in a realistic fashion => This is closer to reality => Over-design is avoided • “Tolerance stack-up” is included (design for manufacturability)

9. PDS Benefits • Deterministic Analysis: • Sensitivities do not take range of scatter or possibilities into account • Sensitivities do not take interactions between input variables into account (second order cross terms) • Quality is “indirectly” affected • Probabilistic Analysis: • Range/width of scatter is “built-in” into probabilistic sensitivities • Interactions between input variables are inherently taken care of • Quality becomes a measurable, quantifiable and controllable quantity

10. PDS Benefits • Illustration of the Benefits of • Probabilistic Analysis over Deterministic Analysis Probabilistic Analysis • Deterministic Analysis

11. Features of the ANSYS/Probabilistic Design System • Free for ANSYS users (included in ANSYS since rel. 5.7) • Works with any ANSYS analysis model • Static, dynamic, linear, non-linear, thermal, structural, electro-magnetic, CFD .. • Allows large number random input and output parameters • 10 statistical distributions for random input parameters • Random input parameters can be correlated • Probabilistic methods: • Monte Carlo - Direct & Latin Hypercube Sampling • Response Surface - Central Composite & Box-Behnken Designs

12. Features of the ANSYS/Probabilistic Design System • Use of distributed, parallel computing techniques for drastically reduced wall clock time • Comprehensive probabilistic results • Convergence plots, histogram, probabilities, scatter plots, sensitivities, ... • State-of-the art statistical procedures to address the accuracy of the output data • Confidence intervals

13. Features of the ANSYS/Probabilistic Design System • ANSYS Customer Base • All “Top 10” Fortune 100 Industrial companies • 73 of the Fortune 100 Industrial companies • Over 5,700 commercial companies • Over 40,000 commercial customer seats • Over 100,000 university licenses • Probabilistic Design • Available since ANSYS 5.7 and after • Used by well over 100 companies in production

14. Probabilistic Methods Monte Carlo Simulation: Perform numerous analysis runs based on sets of random samples, and then evaluate statistics of derived responses. • Direct (Crude) Sampling Monte Carlo (DIR) • Latin Hypercube Sampling Monte Carlo (LHS) • User defined(USR) Fully Parallel

15. Probabilistic Methods Monte Carlo Simulation Method Scheme: Simulation of input parameters at random locations Statistical analysis of output parameters X1 X2 X3 Repetitions = Simulations ANSYS Scatter ! Uncertain !

16. Finite Element Runs for Monte Carlo Probabilistic Methods • For Monte Carlo Simulation the number of simulations does not depend on the number of random input variables, but on the probabilistic result you are looking for: • For assessment of the statistics of output parameters (Mean, sigma) • Nsim  30 … 100 • For histogram and cumulative distribution function • Nsim  50 … 200 • For assessment of low probabilities P (tails of the distribution) • Nsim  30/P … 100/P

17. Probabilistic Methods • Response Surface Methods: Select specific observation points for each random variable, run analyses, establish response surface for each response parameter, perform Monte Carlo Analysis on Response Surface. • Central Composite Design (CCD) • Box-Behnken Matrix (BBM) • User defined (USR) Fully Parallel

18. Probabilistic Methods Uncertain ! Response Surface Methods Scheme: Simulation of input parameters at specific locations Statistical analysis of output parameters X1 X2 X3 Evaluate input parameter values Monte Carlo Simulations on Response Surface Repetitions = Simulations DOE Response Surface Fit ANSYS Scatter !

19. Finite Element Runs for Response Surface Probabilistic Methods For Response Surface Methods the number of simulations depends on the number of random input variables only : No. of random Coefficients Central Box- input variables of equation Compos. Behnken 1 3 2 6 9 3 10 15 13 4 15 25 25 5 21 27 41 6 28 45 49 7 36 79 57 8 45 81 65 9 55 147 121 10 66 149 161 ...

20. Parallel Distributed Processing Model file + Input variables Server 1 Client Run analysis 1,4, … Result output parameters Server 2 Build the Model Identify Machines Click “Run…” Post-process Results Run analysis 2,5,6, ... PC to PC PC to UNIX UNIX to PC UNIX to UNIX Server 3 Run analysis 3,7

21. PDS Tight Integration into ANSYS Main Menu • Enter the PDS module from ANSYS Main Menu • Generate a loop file representing any type of analysis • Pre-processing • Define Methods and Run options • Fit Response Surfaces • Post-processing • Database handling

22. Probabilistic Results Post-processing of simulations results: The results should be displayed such that the user can graphically and intuitively answer the questions: 1 How large is the scatter of the output parameters? 2 What is the probability that output parameters do not fulfil design criteria (failure probability)? 3 How much does the scatter of the input parameters contribute to the scatter of the output? Plot: Statistics (sigma), Histogram, Sample Diagrams Plot: Cumulative Distribution Function, Probabilities Plot: Sensitivities, Scatter Diagram, Response Surface

23. Probabilistic Results Simulation Value Sample Plot:

24. Probabilistic Results Mean Value Sample Plot:

25. Probabilistic Results Standard Deviation Sample Plot:

26. Probabilistic Results Histogram Plot: Histogram for random input variables Histogram for random output parameters

27. Probabilistic Results Cumulative Distribution Function: Show probabilities as empirical cumulative distribution function

28. Probabilistic Results Cumulative Distribution Function: Show probabilities as: - normal plot - log-normal plot - Weibull plot

29. Probabilistic Results Sensitivities: • Show sensitivities as: • Spearman rank order sensitivity plot • Linear correlation sensitivity plot

30. Probabilistic Results Scatter Plot:

31. Probabilistic Results Response Surface Plot: • Response Surface Types: • Linear • Quadratic w/o X-terms • Quadratic with X-terms • Regression Analysis: • Full Regression • Forward-Stepwise- Regression • Transformations: • Logarithmic Y*=log(Y) • Square root Y*=sqrt(Y) • Power Y*=Y^a • Box-Cox (automatic!) • ...

32. Probabilistic Results Sharing HTML Report: • Note: • Report is automatically generated (push-button) • It includes all pictures according to user preferences/options • It includes explanations as text Click to see Report

33. Summary • Deterministic engineering design practices have matured and do not yield significant performance gains. • Future design improvements will require accounting for variations. • Probabilistic approach enables Design for Quality, Reliability and Robustness • Reduced warranty costs • Better resale value • Increased market size, market share, and margin on sales • Distributed computing allows faster simulation turn-around