probabilistic design introduction an example motivation features benefits probabilistic methods...

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



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


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


– 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


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


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


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:


Mean Value Sample Plot:


Standard Deviation Sample Plot:


Histogram Plot:


Histogram for random input variables

Histogram for random output parameters

Cumulative Distribution Function:


Show probabilities asempirical cumulative distribution function

Cumulative Distribution Function:


Show probabilities as:- normal plot- log-normal plot- Weibull plot

Sensitivities:


Show sensitivities as:• Spearman rank order sensitivity plot

• Linear correlation sensitivity plot

Scatter Plot:


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!)• ...

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

probabilistic design introduction an example motivation features benefits probabilistic methods...

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