Application of the Generalized Conditional
Expectation Method for Enhancing a
Probabilistic Design Fatigue Code
Faiyazmehadi Momin, Harry Millwater,
R. Wes Osborn
Department of Mechanical Engineering
University of Texas at San Antonio
Michael P. Enright
Southwest Research Institute
46th SIAA Structures, Dynamics and Materials Conference
Austin, TX, April 18-21, 2005
Motivation
 Probabilistic design codes are specialized for particular
application
 Highly optimized for particular application
Specific mechanics model
Specific random variables
Specific 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
University of Texas at San Antonio
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
University of Texas at San Antonio
Approach
Basic idea - discrete distribution
POFTOTAL = CPOF1 * 0.4 +
CPOF2 * 0.6 = E[CPOFi]
CPOF2
CPOF1
40%
CPOF - Conditional
Probability of Failure
60%
1
2
Speed()
University of Texas at San Antonio
Approach
Generalized Conditional Expectation
POFTotal 
 CPOF(x
 E[CPOF(x Internal)]
CPOF(x


N
Internal
)
Internal
| x External) f X (External)dx
Multiple runs of the
probabilistic design
code needed to
compute expected
value
University of Texas at San Antonio
Methodology
 Generalized Conditional Expectation (GCE) methodology
is implemented without modifying the source code
 Random variables are partitioned as “internal” and
“external” variables
Internal - 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)
University of Texas at San Antonio
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
University of Texas at San Antonio
Generalized Conditional Expectation
 The conditional expected value can be approximated by
1
Pf 
N
N
P
i 1
fi
 Pfi is the conditional POF of ith realization is conditional
variables
 The variance and coefficient of variation are given by
N
var( P f ) 
2
(
P

P
)
f
 fi
i 1
N ( N  1)
COV ( P f ) 
var( P f )
Pf
University of Texas at San Antonio
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
University of Texas at San Antonio
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
University of Texas at San Antonio
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  A1X˜1  A2 X˜ 2  A3 X˜12  A4 X˜ 22  A5 X˜1X˜ 2

The response surface is not use to
approximate the limit state
University of Texas at San Antonio
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

˜ ) 1 
f
(
x
dPf
X
j
˜ , Xˆ )



E
P
(
X
f
i
˜
˜
di
j f X j ( x˜ ) 



P ( X˜ , Xˆ ) 
dPf

 E  fi
ˆ
ˆ
di

 j 

External
Internal
No additional limit state analyses needed

University of Texas at San Antonio
DARWIN®
University of Texas at San Antonio
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
sampling
3. 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 sampling
6. Compute the sensitivities of the POF to the parameters of the
internal and external random variables
University of Texas at San Antonio
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 points
3. 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 CPOF
6. 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
University of Texas at San Antonio
FLOW CHART
start
Parametric Deterministic Model
Enter RV
Control Software
Finite Element Solver
Results file
Input file
Darwin CPOF
Design Point Loop
Results2NEU
Generate samples,
Build RS,
Compute Expected
CPOF,
Sensitivities
.UIF/.UOF
DARWIN
Darwin Results
No
i=K
Yes
Expected CPOF
University
of Texas at San Antonio
Sensitivities
Implementation
 Ansys probabilistic design system used to control
analysis
 Ansys FE solver used to compute stresses
 ANS2NEU used to extract stresses for DARWIN
 DARWIN used to compute the CPOF
 Text utility used to extract DARWIN CPOF results and
return to Ansys
 Sensitivity equations programmed within Ansys
 POF determined by computing the expected value of the
CPOF using Monte Carlo or Monte Carlo with Response
Surface
University of Texas at San Antonio
UTSA FLOW CHART
start
Parametric Deterministic Model
Enter RV
Ansys PDS
Ansys Solver
Results file
ANS2NEU
Input file
DARWIN
DARWIN POF
DARWIN Results
Design Point Loop
No
i=K
Generate samples,
Build RS,
Compute Expected
CPOF,
Sensitivities
.UIF .UOF file
Yes
Expected POF
University
of Texas at San Antonio
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 bore
 Consider POF (assuming a defect is present) at 20,000
cycles
 Solve using GCE with Darwin and Ansys
 Compare to independent Monte Carlo solution
University of Texas at San Antonio
FEM MODEL
r
Po
Element type - Plane42
1444 elements
Speed 
R2
6800
rpm
t
Surface Crack
R1
L
x

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Initial Crack Size
Exceedance
Curve
aMIN
aMAX
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External
Internal
Probabilistic Model
Parameter 1
Parameter 2
No.
Name
Type
amin (mils2 )
amax (mils2 )
1
Initial crack
size
Exceedance
Curve
3.5236
111060.0
No
Name
Type
Parameter1
(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
Procedure not limited to Normal distributions
University of Texas at San Antonio
Independent Benchmark
Solution Developed
 Approximate analytical fatigue algorithm developed for
verification
 Uses standard Monte Carlo sampling (no GCE) of all
variables
Method
Random
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
University of Texas at San Antonio
Probabilistic Results Using GCE Method
Method
Random
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
DARWIN
0.2074
GCE
(Ansys (RS)
and Darwin)
ai
Omega
Pressure
Radius
15 DARWIN
&100,000
RS
MC-Monte Carlo
RS-Response Surface
simulations
0.2079
University of Texas at San Antonio
Effects of Response Surface
Transformations
Transformation
No. of Samples
Expected
POF
Monte Carlo
(Comparison)
1000 Darwin
0.2040
Linear
15 Darwin &
100,000
0.2832
Quadratic
with cross-terms
15 Darwin &
100,000
0.2077
Exponential
15 Darwin &
100,000
0.2073
Logarithmic
15 Darwin &
100,000
0.2119
Power
15 Darwin &
100,000
0.2077
Box-Cox
15 Darwin &
100,000
0.2077
University of Texas at San Antonio
All
quadratic
Goodness of Fit
Error sum of
Squares
Transformation
(Close to Zero)
Coefficient of
Maximum Absolute
Determination (R2)
Residual
(Close to one)
(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
University of Texas at San Antonio
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
University of Texas at San Antonio
Sensitivity Results (Mean)
Pf

Sensitivity of POF with respect to mean value

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
Sensitivities have units
University of Texas at San Antonio
Sensitivity Results (Mean)
Pf i
*
i Pf
Sensitivity of POF with respect to mean value

Parameter
Monte Carlo
(Comparison)
GCE
ANSYS-DARWIN
GCE
Ansys-Darwin
(Finite difference)
Rotational
speed
10.8
11.5
12.1
External
pressure
2.79
2.59
3.04
Inner radius
7.54
6.58
9.21
University of Texas at San Antonio
Sensitivity Results (Std Dev)
Pf

Sensitivity of POF with respect to standard deviations

Parameter
Monte Carlo
(Comparison)
GCE
ANSYS-DARWIN
GCE
Ansys-Darwin
(Finite difference)
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
Sensitivities have units
University of Texas at San Antonio
Sensitivity Results (Std Dev)
Pf  i
*
 i Pf
Sensitivity of POF with respect to standard deviations

Parameter
Monte Carlo
(Comparison)
GCE
ANSYS-DARWIN
GCE
Ansys-Darwin
(Finite difference)
Rotational
speed
0.21
0.25
0.21
External
pressure
0.17
0.21
0.19
Inner radius
0.03
0.03
0.03
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Sensitivities of Internal RV
Parameter
GCE
ANSYS-DARWIN
Finite Difference
ANSYS-DARWIN
(1000 DARWIN runs)
(15 RS and 100,000
MC)
(15 DARWIN & 100,000
0.03494
0.03726
0.0371
0.0340
6.977E-11
7.441E-11
7.398E-011
***
Monte Carlo
sampling
(1000 Monte Carlo
MATLAB runs)
GCE
ANSYS-DARWIN
MC)
Pf
amin
(mils-2)
Pf
amax
(mils-2)
*** The sensitivity of CPOF with respect to amax is too small to be computed using finite difference method
Sensitivities have units
University of Texas at San Antonio
Sensitivities of Internal RV
Parameter
Pf
amin/ max
GCE

ANSYS-DARWIN
*
amin/ max
Pf
Monte Carlo
sampling
GCE
ANSYS-DARWIN
(1000 Monte Carlo
MATLAB runs)
(1000 DARWIN runs)
0.59
0.63
0.63
0.58
4E-5
4E-5
4E-5
***
(15 RS and 100,000
MC)
Finite Difference
ANSYS-DARWIN
(15 DARWIN & 100,000
MC)
Pf
amin
(mils-2)
Pf
amax
(mils-2)
Non-dimensionalized sensitivities are
significantly smaller than other random variables
University of Texas at San Antonio
Application Problem
Zone 8
Zone 1
Zone 9
Zone 2
Zone 10
Zone 3
Zone 11
Zone 4
Zone 12
Zone 13
Zone 14
Zone 5
Zone 6
Zone 7
POF per Flight results
Method
Random Variables
No. of Samples
Expected 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
Initial crack size ai
Pressure Po
Rotational Speed 
Inner Radius ri
15 DARWIN and
100,000 RS
1.935E-9
GCE ANSYS RS
and DARWIN
(Power
Transformation)
University of Texas at San Antonio
Response Surface Transformations
Transformations
No. of Samples
Mean POF
Monte-Carlo
1000 DARWIN
1.957E-9
None - Linear
15 DARWIN &
100,000 MC
2.637E-9
None - Quadratic
with Cross Terms
15 DARWIN &
100,000 MC
1.945E-9
Logarithmic
15 DARWIN &
100,000 MC
1.932E-9
Square Root
15 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
University of Texas at San Antonio
All
quadratic
Goodness of Fit Measures
Transformation
Error 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
University of Texas at San Antonio
Sensitivity Results (Mean) -
Pf
i
GCE
ANSYS-DARWIN
GCE
ANSYS-DARWIN
Finite Difference

ANSYS-DARWIN
(15
(1000 DARWIN runs)
(15 RS and 100,000 MC)
DARWIN & 100,000 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
Parameter
Sensitivities have units
University of Texas at San Antonio
Sensitivity Results (Mean)
Pf i
*
i Pf

Finite Difference
GCE
ANSYS-DARWIN
GCE
ANSYS-DARWIN
(1000 DARWIN runs)
(15 RS and 100,000 MC)
DARWIN & 100,000 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
Parameter
ANSYS-DARWIN
University of Texas at San Antonio
(15
Sensitivity Results (Std Dev)
Pf
 i
Finite Difference

ANSYS-DARWIN
GCE
ANSYS-DARWIN
GCE
ANSYS-DARWIN
(1000 DARWIN runs)
(15 RS and 100,000 MC)
(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
Parameter
Sensitivities have units
University of Texas at San Antonio
Sensitivity Results (Std Dev) -
Pf  i
*
 i Pf
Finite Difference

GCE
ANSYS-DARWIN
GCE
ANSYS-DARWIN
(1000 DARWIN runs)
(15 RS and 100,000 MC)
(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
Parameter
ANSYS-DARWIN
University of Texas at San Antonio
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
University of Texas at San Antonio
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
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