Probabilistic Design Introduction An Example Motivation Features Benefits Probabilistic Methods...
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Probabilistic Design
• Introduction • An Example• Motivation• Features• Benefits• Probabilistic Methods• Probabilistic Results/Interpretation• Summary
Purpose of a Probabilistic Design System (PDS)
Introduction
InputInputInputInput ANSYSANSYSANSYSANSYS OutputOutputOutputOutput
Material PropertiesGeometryBoundary Conditions
DeformationStresses / StrainsFatigue, Creep,...
It’s a reality that input parameters are subjected to scatter => automatically the
output parameters are uncertain as well!!
Introduction
ANSYS PDSANSYS PDSANSYS PDSANSYS 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)
An Example
Example: Lifetime of Components !!!
Random input
variablesRandom output
parametersFinite-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 !
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)
Scatter in material properties and loads
Property SD/Mean %
Metallic materiales, yield 15
Carbon fiber composites, rupture 17
Metallic shells, buckling strength 14
Junction by screws, rivet, welding 8
Bond insert, axial load 12
Honeycomb, tension 16
Honeycomb, shear, compression 10
Honeycomb, face wrinkling 8
Launch vehicle , thrust 5
Transient loads 50
Thermal loads 7.5
Deployment shock 10
Acoustic loads 40
Vibration loads 20
Source: Klein, Schueller et.al. Probabilistic Approach to Structural Factors of Safety in Aerospace.
Proc. CNES Spacecraft Structures and Mechanical Testing Conf., Paris 1994
Motivation
CFD
FEMCAD
FEMGeometry
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
PDS Benefits
Deterministic Analysis: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: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)
PDS Benefits
Deterministic Analysis: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: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
PDS Benefits
Deterministic AnalysisDeterministic Analysis
Probabilistic AnalysisProbabilistic Analysis
Illustration of the Benefits ofIllustration of the Benefits of
Probabilistic Analysis over Deterministic AnalysisProbabilistic Analysis over Deterministic Analysis
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
• 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
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
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) Fu
lly P
aral
lel
Probabilistic Methods
Monte Carlo Simulation Method Scheme:
ANSYSANSYSANSYSANSYS
Simulation of input parameters at
random locations
Statistical analysis of output parameters
X3X2X1
Repetitions = Simulations
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 functionNsim 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
– 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) F
ully
Par
alle
l
Probabilistic Methods
Response Surface Methods Scheme:
ANSYSANSYSANSYSANSYS
Simulation of input parameters at specific locations
Statistical analysis of output parameters
Response Surface FitResponse Surface FitResponse Surface FitResponse Surface Fit
Monte Carlo Simulations Monte Carlo Simulations on Response Surfaceon Response Surface
Monte Carlo Simulations Monte Carlo Simulations on Response Surfaceon Response Surface
Evaluate input Evaluate input parameter valuesparameter values
Evaluate input Evaluate input parameter valuesparameter values
X3X2X1
Repetitions = Simulations
Probabilistic Methods
DOE
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 32 6 93 10 15 134 15 25 255 21 27 416 28 45 497 36 79 578 45 81 659 55 147 121
10 66 149 161...
Finite Element Runs for Response Surface
Probabilistic Methods
Parallel Distributed Processing
Build the ModelIdentify MachinesClick “Run…”Post-process Results
Run analysis 1,4, …
Run analysis 2,5,6, ...
Run analysis 3,7
Model file+ Input
variables
Resultoutput
parameters
Client Server 1
Server 2
Server 3
PC to PCPC to UNIXUNIX to PCUNIX to UNIX
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
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?
Probabilistic Results
Plot: Statistics (sigma), Histogram, Sample Diagrams
Plot: Cumulative Distribution Function, Probabilities
Plot: Sensitivities, Scatter Diagram, Response Surface
Simulation Value Sample Plot:
Probabilistic Results
Mean Value Sample Plot:
Probabilistic Results
Standard Deviation Sample Plot:
Probabilistic Results
Histogram Plot:
Probabilistic Results
Histogram for random input variables
Histogram for random output parameters
Cumulative Distribution Function:
Probabilistic Results
Show probabilities asempirical cumulative distribution function
Cumulative Distribution Function:
Probabilistic Results
Show probabilities as:- normal plot- log-normal plot- Weibull plot
Sensitivities:
Probabilistic Results
Show sensitivities as:• Spearman rank order sensitivity plot
• Linear correlation sensitivity plot
Scatter Plot:
Probabilistic Results
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!)• ...
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
• 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