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

3. Purpose of a Probabilistic Design System (PDS) Introduction InputInput ANSYSANSYS OutputOutput  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!!

4. Introduction ANSYS PDS 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)? Purpose of Probabilistic Design System (PDS)

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

6. Motivation Influence of Young’s Modulus and Thermal Expansion Coefficient on thermal stresses:  thermal = E ·  ·  T Deterministic Approach: E mean 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)

7. Scatter in material properties and loads PropertySD/Mean % Metallic materiales, yield15 Carbon fiber composites, rupture17 Metallic shells, buckling strength14 Junction by screws, rivet, welding8 Bond insert, axial load12 Honeycomb, tension16 Honeycomb, shear, compression10 Honeycomb, face wrinkling8 Launch vehicle, thrust5 Transient loads50 Thermal loads7.5 Deployment shock10 Acoustic loads40 Vibration loads20 Source: Klein, Schueller et.al. Probabilistic Approach to Structural Factors of Safety in Aerospace. Proc. CNES Spacecraft Structures and Mechanical Testing Conf., Paris 1994

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

9. 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)

10. 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

11. PDS Benefits Deterministic Analysis Probabilistic Analysis Illustration of the Benefits of Probabilistic Analysis over Deterministic Analysis

12. 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

13. 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 Features of the ANSYS/Probabilistic Design System

14. 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

15. 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 Probabilistic Methods

16. Monte Carlo Simulation Method Scheme: ANSYSANSYS Simulation of input parameters at random locations Statistical analysis of output parameters X3X2X1 Repetitions = Simulations Probabilistic Methods

17. 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 Finite Element Runs for Monte Carlo Probabilistic Methods

18. – 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 Probabilistic Methods

19. Response Surface Methods Scheme: ANSYSANSYS Simulation of input parameters at specific locations Statistical analysis of output parameters Response Surface Fit Monte Carlo Simulations on Response Surface Evaluate input parameter values X3X2X1 Repetitions = Simulations Probabilistic Methods DOE

20. For Response Surface Methods the number of simulations depends on the number of random input variables only : No. of randomCoefficients CentralBox- input variablesof equationCompos.Behnken 13 269 3101513 4152525 5212741 6284549 7367957 8458165 955147121 1066149161... Finite Element Runs for Response Surface Probabilistic Methods

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

22. Direct (Crude) Monte Carlo Simulation: Probabilistic Methods Method Principles & Properties: Values for input parameters are sampled randomly User specified distribution function used for sampling Sampling process has no “memory” (clustering possible) = cluster

23. Latin Hypercube Monte Carlo Simulation: Probabilistic Methods Method Principles & Properties: Values for input parameters are sampled randomly User specified distribution function used for sampling Sampling process does have a “memory” (avoids clustering) No. of simulations does not depend on no. of input parameters

24. User Specified Monte Carlo Simulation: User provides the random samples (set of input values for each run) in a user- specified file. The samples will be simply executed - the user is responsible that the samples are correct. Benefits: –Flexibility: Use other third party code for sampling, rerun old analyses –Correlation: The result of different analysis type using the same FE model are correlated. E.g. a temperature variation will influence both the LCF and creep lifetime in the same direction. Generating the sample set in the first analysis and then “feeding” the same random input sample set into the second one, will cover that correlation correctly. –No programming, compiling and linking of USER-subroutines required. Probabilistic Methods

25. Central Composite Design (CCD): Probabilistic Methods X3 X1 X2 (0,0,0) (0,-a,0) (1,1,1) (1,1,-1) (1,-1,-1) (1,-1,1) (-1,-1,1) (-1,1,1) (-1,1,-1) (-1,-1,-1) (0,0,-a) (a,0,0) (-a,0,0) (0,0,a) (0,a,0) Observation Points: Center Point(1) Axis Points(2n) Corner Points(2 n ) Method Principles & Properties: Values for input parameters sampled at deterministic points User specified distribution function used for locations No. of simulations depends on no. of input parameters (n)

26. Box-Behnken Matrix (BBM): Probabilistic Methods X3 X1 X2 (1,0,-1) (0,1,-1) (1,0,-1) (0,-1,1) (-1,1,0) (-1,0,-1) Observation Points: Center Point(1) Mid-side Points(n*2 (n-1) ) Method Principles & Properties: Values for input parameters sampled at deterministic points User specified distribution function used for locations No. of simulations depends on no. of input parameters (n) (1,1,0) (0,1,1) (1,0,1)(-1,0,1) (0,-1,1) (-1,0,-1) (0,0,0)

27. User Specified Response Surface Method: User provides the observation samples (set of input values for each run) in user- specified file. The samples will be simply executed - the user is responsible that the samples are correct. Benefit: –Flexibility: –Use other third party code for Design of Experiments (DOE). –No programming, compiling and linking of USER-subroutines required. Probabilistic Methods

28. Main Menu PDS Tight Integration into ANSYS 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

29. 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? 2What is the probability that output parameters do not fulfil design criteria (failure probability)? 3How much does the scatter of the input parameters contribute to the scatter of the output? Probabilistic Results  Plot: Statistics (sigma), Histogram, Sample Diagrams  Plot: Cumulative Distribution Function, Probabilities  Plot: Sensitivities, Scatter Diagram, Response Surface

30. Simulation Value Sample Plot: Probabilistic Results

31. Mean Value Sample Plot: Probabilistic Results

32. Standard Deviation Sample Plot: Probabilistic Results

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

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

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

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

37. Scatter Plot: Probabilistic Results

38. Response Surface Plot: Probabilistic Results 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!)...

39. HTML Report: Probabilistic Results Sharing 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

40. 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 Summary