46th SIAA Structures, Dynamics and Materials Conference Austin, TX, April 18-21, 2005 Application of the Generalized Conditional Expectation Method for.

46th SIAA Structures, Dynamics and Materials ConferenceAustin, TX, April 18-21, 2005

Application of the Generalized Conditional Expectation Method for Enhancing a Probabilistic Design Fatigue Code

Faiyazmehadi Momin, Harry Millwater, R. Wes Osborn

Department of Mechanical EngineeringUniversity of Texas at San Antonio

Michael P. EnrightSouthwest Research Institute

University of Texas at San Antonio

Motivation

Probabilistic design codes are specialized for particular application Highly optimized for particular application

Specific mechanics modelSpecific random variablesSpecific probabilistic methods

Prominent codes in industry include PROF, DARWIN, VISA

Codes may need to be enhanced Add more random variables Source code may not be available


Objective

Present the methodology of GCE to enhance a probabilistic design code by considering additional random variables

Compute the sensitivities of the probability-of-failure to ALL random variables

Demonstrate the methodology using a probabilistic fatigue code DARWIN


Approach

Basic idea - discrete distribution

40% 60%

Speed()

CPOF1

CPOF2POFTOTAL = CPOF1 * 0.4 +

CPOF2 * 0.6 = E[CPOFi]

1 2CPOF - Conditional

Probability of Failure


Approach

Generalized Conditional Expectation

€

POFTotal = CPOF(xInternal | xExternal ) fX (External )

dx∫

= E[CPOF(xInternal )]

≅CPOF(xInternal )∑

N

Multiple runs of the probabilistic design

code needed to compute expected

value


Methodology

Generalized Conditional Expectation (GCE) methodology is implemented without modifying the source code Random variables are partitioned as “internal” and

“external” variablesInternal - random variables already considered in

probabilistic design code (called control variables in GCE vernacular)

External - additional random variables to be considered (called conditional variables in GCE vernacular)


Variance Reduction Approach

Traditional use for GCE is variance reduction with sampling methods - reduce the sampling variance by eliminating the variance due to the control variables

Conditional (called “external” here)Control (called “internal” here)

Ayyub, B. M., Haldar, A., “Practical Structural Reliability Techniques,” Journal of Structural Engineering, Vol. 110, No. 8, August 1984, pp. 1707-1725.

Ayyub, B. M., Chia, “ Generalized Conditional Expectation for Structural Reliability Assessment,” Structural Safety, Vol.11, 1992, pp. 131-146


Generalized Conditional Expectation

The conditional expected value can be approximated by

Pfi is the conditional POF of ith realization is conditional variables

The variance and coefficient of variation are given by

∑=

≈N

ifif P

NP

1

1

)1(

)()var( 1

2

−

−=∑=

NN

PPP

N

i

ffi

ff

ff

P

PPCOV

)var()( =


Implementation

1. Partition random variables into two categories: • Internal - random variables within the probabilistic design code• External - additional random variables to be considered not

within the probabilistic design code

2. Generate a realization of external variables using Monte Carlo sampling

3. Determine the conditional probability-of-fracture (CPOFi) given this realization of external random variables

4. Compute the expected value (average) of the CPOF results using MC sampling

5. Compute the sensitivities of the POF to the parameters of the internal and external random variables


Implementation - Response Surface Option

1. Partition random variables into two categories: • Internal - random variables within the probabilistic design code• External - additional random variables to be considered not

within the probabilistic design code

2. Generate a realization of external variables using response surface designs

3. Determine the conditional probability-of-fracture (CPOFi) given this realization of external random variables

4. Build a response surface relating the external variables to the CPOF results.

5. Compute the expected value (average) of the CPOF results using MC sampling of the Response Surface

6. Compute the sensitivities of the POF to the parameters of the internal and external random variables


Response Surface Option

Build a response surface representing the relationship between the conditional POF and the external random variables

Use classical design of experiments and goodness of fit tests

€

CPOF( ˜ X 1, ˜ X 2 ) = A0 + A1˜ X 1 + A2

˜ X 2 + A3˜ X 1

2 + A4˜ X 2

2 + A5˜ X 1 ˜ X 2

The response surface is not use to approximate the limit state


Sensitivities

Methodology developed to compute the sensitivities of the POF to the parameters of the internal and external random variables Compare the effects of internal and external variables

€

dP fd ˜ θ i

= E Pf i( ˜ X , ˆ X )

∂fX j ( ˜ x )

∂ ˜ θ j

1

fX j ( ˜ x )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

dP fd ˆ θ i

= E∂Pfi( ˜ X , ˆ X )

∂ ˆ θ j

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Internal

External

No additional limit state analyses needed


DARWIN®


Implementation with DARWIN

1. Partition random variables into two sets • Internal - DARWIN variables (crack size, life scatter, stress

scatter)• External - non-DARWIN (geometry, loading, structural and

thermal material properties, etc.)2. Generate a realization of external variables using Monte Carlo

sampling3. Run the finite element solver to obtain updated stresses 4. Execute DARWIN given this realization of external random

variables and associated stresses to determine CPOFi 5. Compute the expected value (average) of the DARWIN CPOF

results using Monte Carlo sampling6. Compute the sensitivities of the POF to the parameters of the

internal and external random variables


Implementation - Response Surface Option

1. Partition random variables into two sets • Internal - DARWIN variables (crack size, life scatter, stress

scatter)• External - non-DARWIN (geometry, loading, structural and

thermal material properties, etc.)2. Generate a realization of external variables using response

surface design points3. Run the finite element solver to obtain updated stresses 4. Execute DARWIN given this realization of external random

variables and associated stresses to determine CPOFi 5. Build a response surface relating conditional variables to

DARWIN CPOF6. Compute the expected value (average) of the DARWIN CPOF

results using Monte Carlo sampling of the response surface 7. Compute the sensitivities of the POF to the parameters of the

internal and external random variables


FLOW CHARTstart

Parametric Deterministic Model

Enter RV

Results2NEU .UIF/.UOF

DARWINInput file

Darwin ResultsDarwin CPOF

i = K

Expected CPOFSensitivities

No Yes

Results file

Control Software

Finite Element Solver

Generate samples, Build RS,

Compute Expected CPOF,

Sensitivities

Design Point Loop


Implementation

Ansys probabilistic design system used to control analysis

Ansys FE solver used to compute stressesANS2NEU used to extract stresses for DARWINDARWIN used to compute the CPOFText utility used to extract DARWIN CPOF results and

return to AnsysSensitivity equations programmed within AnsysPOF determined by computing the expected value of the

CPOF using Monte Carlo or Monte Carlo with Response Surface


UTSA FLOW CHARTstart

Parametric Deterministic Model

Enter RV

ANS2NEU .UIF .UOF file

DARWINInput file

DARWIN ResultsDARWIN POF

i = K

Expected POFSensitivities

No Yes

Results file

Ansys PDS

Ansys Solver

Design Point Loop

Generate samples, Build RS,

Compute Expected CPOF,

Sensitivities


Application Example

FA Advisory Circular 33.14 test case Internal variables: initial crack size(a)External variables: rotational speed(RPM), external

pressure(Po), inner radius(Ri)Surface crack on inner boreConsider POF (assuming a defect is present) at 20,000

cycles

Solve using GCE with Darwin and AnsysCompare to independent Monte Carlo solution


FEM MODEL

R2

R1L

6800 rpm

t

Speed

Por

x

Surface Crack

Element type - Plane42

1444 elements


Initial Crack Size

aMIN aMAX

Exceedance Curve


Probabilistic Model

Procedure not limited to Normal distributions

No Name TypeParameter1

(mean)

Parameter2

(COV)

2 Pressure Normal 7250 psi 0.1

3 Speed Normal 712.35 rad/sec 0.05

4 Inner radius Normal 11.81 inches 0.02

No. Name Type

Parameter 1

amin (mils2 )

Parameter 2

amax (mils2 )

1Initial crack

sizeExceedance

Curve3.5236 111060.0

Inte

rnal

Ext

ern

al


Independent Benchmark Solution Developed

MethodRandom variables

No. of Samples

POF

Monte Carlo (Benchmark)

Initial crack size (ai)

10,000 0.1702

Monte Carlo

(DARWIN)Initial crack size

(ai)10,000 0.1703

Approximate analytical fatigue algorithm developed for verification Uses standard Monte Carlo sampling (no GCE) of all variables


Probabilistic Results Using GCE Method

MethodRandom variables

No. of Samples

POF

Monte Carlo

(Benchmark)

ai

Omega

Pressure

Radius

1000 0.2040

GCE

(Ansys (MC) and Darwin)

ai

Omega

Pressure

Radius

1000

DARWIN0.2074

GCE

(Ansys (RS) and Darwin)

ai

Omega

Pressure

Radius

15 DARWIN &100,000

RS0.2079

MC-Monte Carlo

RS-Response Surface simulations


Effects of Response Surface Transformations

Transformation No. of SamplesExpected

POF

Monte Carlo

(Comparison)1000 Darwin 0.2040

Linear15 Darwin &

100,0000.2832

Quadratic

with cross-terms15 Darwin &

100,0000.2077

Exponential15 Darwin &

100,0000.2073

Logarithmic15 Darwin &

100,0000.2119

Power15 Darwin &

100,0000.2077

Box-Cox15 Darwin &

100,0000.2077

All quadratic


Goodness of Fit

TransformationError sum of

Squares

(Close to Zero)

Coefficient of Determination (R2)

(Close to one)

Maximum Absolute Residual

(Close to Zero)

Linear 8.634E-2 0.8908 0.14078

Non-linear quadratic

2.450E-3 0.9969 0.01784

Exponential 2.474E-3 0.9910 0.01813

Logarithmic 7.840E-3 0.9393 0.06510

Power 1.014E-4 0.9996 0.00514

Box-Cox 9.196E-5 0.9996 0.00464


Response Surface Implementation

Note: Response Surface is only used to compute the expected value of a function

This is completely different from the traditional use of RS in probabilistic analysis, i.e., to approximate the limit state and estimate an often very small probability

Curse-of-Dimensionality is still present if a quadratic model is used; however, only the external random variables enter the equation


Sensitivity Results (Mean)

Parameter Monte Carlo(Comparison)

GCE

ANSYS-DARWIN

GCE

Ansys-Darwin(Finite difference)

Rotational speed

0.00316 0.00334 0.00351

External pressure

0.00008 0.000074 0.000087

Inner radius 0.1325 0.1156 0.1618

Sensitivity of POF with respect to mean value€

∂Pf

∂μ

Sensitivities have units




GCE

ANSYS-DARWIN

GCE


Rotational speed

10.8 11.5 12.1

External pressure

2.79 2.59 3.04

Inner radius 7.54 6.58 9.21

Sensitivity of POF with respect to mean value€

∂Pf

∂μ i

*μ i

Pf


Sensitivity Results (Std Dev)


GCE

ANSYS-DARWINGCE


Rotational speed

0.123E-2 0.143E-2 0.124E-2

External pressure

0.500E-4 0.609E-4 0.551E-4

Inner radius 0.0287 0.0244 0.0258

Sensitivity of POF with respect to standard deviations€

∂Pf

∂σ





GCE

ANSYS-DARWINGCE


Rotational speed

0.21 0.25 0.21

External pressure

0.17 0.21 0.19

Inner radius 0.03 0.03 0.03

Sensitivity of POF with respect to standard deviations€

∂Pf

∂σ i

*σ i

Pf


Sensitivities of Internal RV

Parameter

Monte Carlo sampling

(1000 Monte Carlo MATLAB runs)

GCE ANSYS-DARWIN (1000 DARWIN runs)

GCE

ANSYS-DARWIN (15 RS and 100,000

MC)

Finite Difference

ANSYS-DARWIN (15 DARWIN & 100,000

MC)

(mils-2) 0.03494 0.03726 0.0371 0.0340

(mils-2) 6.977E-11 7.441E-11 7.398E-011 ***

mina

Pf

∂∂

maxa

Pf

∂∂

*** The sensitivity of CPOF with respect to amax is too small to be computed using finite difference method



Sensitivities of Internal RV

Parameter Monte Carlo sampling

(1000 Monte Carlo MATLAB runs)

GCE ANSYS-DARWIN (1000 DARWIN runs)

GCE

ANSYS-DARWIN (15 RS and 100,000

MC)

Finite Difference


MC)

(mils-2) 0.59 0.63 0.63 0.58

(mils-2) 4E-5 4E-5 4E-5 ***

mina

Pf

∂∂

maxa

Pf

∂∂

€

∂Pf

∂amin/ max

*amin/ max

Pf

Non-dimensionalized sensitivities are significantly smaller than other random variables

Application Problem

Zone 14

Zone 13

Zone 12

Zone 11

Zone 10

Zone 9 Zone 2

Zone 3

Zone 4

Zone 5

Zone 6

Zone 7

Zone 1Zone 8


POF per Flight results

Method Random Variables No. of SamplesExpected POF per

Flight

Monte Carlo (within DARWIN)

Initial crack size 10,000 / Zone 1.330E-9

GCE (ANSYS MC and DARWIN)

Initial crack size ai

Pressure Po

Rotational Speed Inner Radius ri

1000 DARWIN runs

1.917E-9

GCE ANSYS RS and DARWIN

(Power Transformation)

Initial crack size ai

Pressure Po

Rotational Speed Inner Radius ri

15 DARWIN and 100,000 RS

1.935E-9


Response Surface Transformations

Transformations No. of Samples Mean POF

Monte-Carlo 1000 DARWIN 1.957E-9

None - Linear15 DARWIN & 100,000 MC

2.637E-9

None - Quadratic

with Cross Terms15 DARWIN & 100,000 MC

1.945E-9

Logarithmic15 DARWIN & 100,000 MC

1.932E-9

Square Root15 DARWIN & 100,000 MC

1.930E-9

Power (0.45)15 DARWIN & 100,000 MC

1.935E-9

Box-Cox (0.45)15 DARWIN & 100,000 MC

1.966E-9

All quadratic


Goodness of Fit Measures

TransformationError sum of

Squares

(Close to Zero)

Coefficient of Determination

(Close to Zero)

Maximum Absolute Residual

(Close to Zero)

Linear 3.077E-17 0.7184 3.083E-9

Quadratic

with Cross Terms

4.322E-19 0.9565 6.238E-10

Logarithmic 3.991E-19 0.9598 1.162E-9

Square root 2.090E-20 0.9978 2.006E-10

Power (0.45) 7.716E-20 0.9922 4.354E-10


Sensitivity Results (Mean) -

ParameterGCE

ANSYS-DARWIN (1000 DARWIN runs)

GCE

ANSYS-DARWIN (15 RS and 100,000 MC)

Finite Difference


MC)

Rotational speed

4.092E-11 4.479E-11 4.727E-11

External pressure

9.974E-13 1.079E-12 1.035E-12

Inner radius

1.609E-09 1.562E-09 1.660E-09

€

∂Pf

∂μ i




ParameterGCE


GCE


Finite Difference


MC)

Rotational speed

15.2 16.7 17.6

External pressure

3.77 4.06 3.92

Inner radius

9.91 9.61 10.3

€

∂Pf

∂μ i

*μ i

Pf



ParameterGCE


GCE


Finite DifferenceANSYS-DARWIN (15 DARWIN & 100,000

MC)

Rotational speed

2.930E-11 5.515E-11 3.4665E-11

External pressure

9.659E-13 5.172E-13 3.0758E-13

Inner radius

1.634E-09 1.684E-09 6.740E-10


€

∂Pf

∂σ i


Sensitivity Results (Std Dev) -

ParameterGCE


GCE


Finite DifferenceANSYS-DARWIN (15 DARWIN & 100,000

MC)

Rotational speed

0.55 1.03 0.65

External pressure

0.36 0.19 0.12

Inner radius

0.201 0.207 0.083

€

∂Pf

∂σ i

*σ i

Pf


Conclusions

Methodology to consider affects of additional random variables is developed and demonstrated on a probabilistic fatigue analysis

POF from MC sampling and GCE method are in good agreement

Response surface method used to reduce signicantly the computational time with good accuracy

The sensitivities obtained from MC simulations, GCE formulae and finite difference method are in good agreement and indicate the importance of internal and external random variables on the POF


Conclusions

Enables the user to consider additional random variables without modifying the source code

Enables the developer to consider the importance of implementing additional random variables

46th SIAA Structures, Dynamics and Materials Conference Austin, TX, April 18-21, 2005 Application of the Generalized Conditional Expectation Method for.

Documents

conditional variables

control variables conditional

random variables source

source code random variables

partition random variables

san antonio methodology

conditional pof

san antonio michael