Application of the Generalized Conditional Expectation
Method for Enhancing a Probabilistic Design Fatigue
Code
Faiyazmehadi N. Momin1, Harry R. Millwater 2
Ronald W. Osborn3
Department of Mechanical Engineering and Biomechanics, University of Texas at San Antonio, San
Antonio, TX, 78249
Michael P. Enright 4
Reliability and Materials Integrity, Southwest Research Institute, San Antonio, TX 78238
Abstract
Traditionally, probabilistic design codes are developed for a particular field of
application such as aerospace, nuclear and off-shore structures. These codes contain
specific mechanics models and random variables. Also, the probabilistic methods
implemented are specific and highly optimized for the particular problem. Over a
period of time these codes may need to be enhanced and certain variables that were
once considered deterministic need to be made random. In addition, the source code
may not be available. In such a scenario, the generalized conditional expectation
method is implemented to enhance the probabilistic design code and add new
random variables without changing the source code. This methodology can also be
used to assist the developer in evaluating the enhancement of code, before changing
the source code. In this paper, the generalized conditional expectation methodology
is used to enhance a probabilistic fatigue code by adding more random variables to
the existing code without changing the source code. Also, new features such as
estimating the sensitivities of the probability of failure to the parameters of random
variables are demonstrated. A numerical example is solved to demonstrate the
methodology.
Nomenclature
CPOF
Pf
Pfi
g( )
A
r
X
X˜
1
!
= Conditional probability-of-failure
= Probability-of-failure
= Conditional probability-of-failure for the ith realization of conditional variables
= Limit state function
= Conditional variable
= Vector of random variables
= Conditional variable
Master of Science, Department of Mechanical Engineering, 6900 N. Loop 1604 West, San Antonio,
Texas, 78249, [email protected].
2
Assistant Professor, Department of Mechanical Engineering and Biomechanics, 6900 N. Loop 1604 West,
San Antonio, Texas, 78249, AIAA Member, harry. [email protected].
3
Graduate student, Department of Mechanical Engineering, 6900 N. Loop 1604 West, San Antonio, Texas,
78249, [email protected].
4
Principal Engineer, Reliability and Materials Integrity, Southwest Research Institute, San Antonio, TX,
78338, AIAA Member, [email protected]
Xˆ
N
!
!
!
"˜
"ˆ
ω
Po
ri
= Control variable
= Number of samples
= Distributional parameter of a conditional variable
= Distributional parameter of a control variable
= Rotational speed
= External pressure
= Inner radius
Introduction
Probabilistic design codes are generally developed for particular applications areas such as aerospace,
nuclear and offshore structures. Some prominent probabilistic codes in industry are PROF1, VISA2,
PRAISE3 and DARWIN®4. These codes contain specific mechanics models and random variables and are
highly optimized for specific types of problems. As the applications of these codes become more prevalent,
the capability to consider more random variables becomes important.
Over a period of time some of the variables that were considered deterministic, need to be made
random. Also, a user who without access to the source code wishes to consider additional random variables,
depending on problem specifications. In such a scenario, the generalized conditional expectation method
can be used to consider additional random variables without modifying the source code. The sensitivities
the probability of failure to the parameter of the random variables are also computed to assess the relative
importance of the random variables. This may be helpful for the developer, who is considering the
expansion of the code and must discern which random variables to consider before resorting to code
modification.
Conditional Expectation (CE)
The concept of conditional expectation is evolved from the concepts of conditional probability on one
hand and mathematical expectation on the other5. The knowledge of occurrence of certain events modifies
the possible outcomes, thereby affecting the probability. This results in the introduction of the concept of
conditional probability. By averaging the conditioned probability, the expected value is obtained called
conditional expectation (CE).
In reliability engineering, the probability of failure can be obtained by solving the integral
Pf =
$ """ $
f (x1, x 2 ,...x n )dx1,dx 2 ," " ",dx n
(1)
r
g( X )#0
where n is the number of random variables and g is the limit state defined such that
indicates failure, and f denotes the joint probability density function.
For a given variable A, the conditional probability of failure is then computed as
!
where
r
Pf ( X | A) =
#
r
f Xr ( X | A)dx1,...dx n
!
r
g( X )"0
r
g( X ) " 0
(2)
r
r
X is a vector of random variables x1,x2,,,xn and f Xr ( X | A) is the conditional probability
density function.
r
The variable
! A is called a “conditional” variable and variables X are called “control” variables. In the
above expression, the value of conditional variable A is deterministic. However, if A is made random, then
!
the probability of failure will also be a function of
! the conditional variable called conditional failure
probability. The conditional variable is generated randomly and for each realization of the conditional
variable, the conditional probability of failure (CPOF) !
is evaluated as a function of the control variables.
By taking the expected valued of the conditional failure probability, the total probability of failure is
obtained as
r
Pf = E[Pfi ( X | Ai )]
!
(3)
For Conditional Expectation there is one control variable, denoted Xˆ , and any number of conditional
variables, denoted X˜ . The conditional probability of failure can often be determined using the realizations
of the conditional variables and the cumulative distribution function of the control variable as
Pf = E X˜ FXˆ [g( X˜ 1, X˜ 2 ,..., X˜ M , Xˆ !
) " 0]
[
!
]
(4)
where X˜ denotes the conditional variables, m is the number of conditional variables,
variable, and
FXˆ is the CDF of the control variable Xˆ .
Xˆ is the control
!
Equation 4 can be approximated as
!
!
1 N
Pf = " Pfi!
N i=1
!
(5)
where N is the number of samples and
Pfi is the conditional probability of failure determined for the ith
realization of the conditional variables computed using any probabilistic method. Equation 5 can be
obtained by sampling methods, e.g., Monte Carlo sampling, Latin Hypercube etc.
! and coefficient of variation is then determined using the formulas 6,7,8, given by
The variance
!
N
# (P
fi
var(Pf ) =
" Pf ) 2
i=1
N(N "1)
COV (Pf ) =
var(Pf )
N
(6)
Generalized Conditional Expectation (GCE)
! reduction technique where there are more than
!
The generalized
conditional expectation is a variance
one control variables7 and the probability of failure is determined using
Pf = E x˜ [Pf ( X˜ , Xˆ )]
(7)
where Xˆ is now a set of control variables, and conditional probability of failure is estimated for each
realization of conditional variables using any probabilistic design method. The overall probability of failure
can be estimated as in equation 5.
!
!
Implementation
Traditionally the control variables are selected as those with the higher coefficient of variation, the
effect of which is to reduce or remove the variance of the control variables from the analysis. However, in
our application, the control variables are selected as those that are already in the probabilistic design code
hereafter called, “internal” variables. The random variables to be considered for addition, treated as
conditional random variables, are called, “external” variables. The external variables are generated
randomly and the generated values are substituted in the probabilistic design code and the conditional
probability of failure is determined for each set of external variables as a function of the internal variables.
The overall probability of failure is determined by taking the expected value according to equation (5). In
this way the probability of failure as a function of both internal and external variables is obtained. In the
current research, the application of conditional expectation to enhance the probabilistic design code is
illustrated using the probabilistic fatigue code DARWIN.
DARWIN is a computer code that integrates finite element stress analysis, fracture mechanics analysis,
non-destructive inspection schedules and probabilistic analysis to access the risk of rotor fracture9. It
computes probability-of-fracture as the function of flight cycles considering the random variables initial
defect size, life scatter, stress scatter, probability of detection and inspection schedule. DARWIN employs
Monte Carlo sampling and important sampling10 to compute the probability-of-fracture. Also included in
DARWIN is a fracture mechanics module called Flight_life to compute the fatigue crack growth.
Uncertainty due to loading, geometry and other variables not random within DARWIN is taken into
consideration using the generalized conditional expectation method.
The procedure to enhance the probabilistic fatigue code DARWIN using the generalized conditional
expectation methodology is illustrated by the flow chart shown in Figure 1.
Figure 1. Schematic for considering additional random variables in DARWIN
The main ingredients for this procedure include control software, a finite element (FE) solver,
translators for converting FE output to a format for use by the probabilistic design code and the
probabilistic design code itself. The FE model is first built parametrically in terms of external random
variables. Realizations for the external variables are generated randomly according to their distributions
using the control software. The finite element solver solves for stresses (.UOF) and geometry (.UIF) for the
realized external variables. RESULT2NEU is a translator that translates the finite element results to a
format usable by DARWIN. The DARWIN input file is modified based upon the realizations of the
external random variables by the control software. Using the DARWIN input file and the UIF and UOF
files, DARWIN computes conditional probability-of-fracture as a function of the internal variables. The
results are then retrieved to the control software for further computations. The process is continued until the
number of samples is complete. Then the expected value of the probability-of-fracture is computed to give
the overall probability-of-fracture.
Response Surface
In the GCE method, any computational method can be used to compute the expected value of the
conditional probability of failure and Monte Carlo sampling is the most common method for this purpose;
however, it may require excessive amount of computation due to a large number of samples. Hence, the
response surface method is implemented to reduce the computations. In the example problem, the central
composite design (CCD)11 is used to generate the design points and the number of runs required to build the
response surface for three random variables is 15. The response surface (RS) relating the external random
variables and the CPOF is generated and regression analysis is performed using the method of least squares
to obtain the relation between the CPOF and the external variables, e.g.,
CPOF( X˜ 1, X˜ 2 ) = A0 + A1 X˜ 1 + A2 X˜ 2 + A3 X˜ 12 + A4 X˜ 22 + A5 X˜ 1 X˜ 2
(8)
Monte Carlo sampling is then performed on the response surface and the expected value of CPOF is
obtained. Hence, only a small number of DARWIN runs are performed when using RSM compared to, for
example,
! 1000 in Monte Carlo sampling.
In the regression analysis, the coefficients of the regression terms are determined on the basis of
minimization of sum of the error terms, but the model fitted is not an exact representation of the actual
model. There will usually be some discrepancies in the model predicted. Therefore, there are several
criteria and statistical tools to evaluate the discrepancies called goodness-of-fit measures. The goodness-offit measures express how accurately a response surface represents the data points. Some of the goodnessof-fit measures includes12 error sum of squares (SSE), coefficient of determination (R2) and maximum
absolute residual. In addition various transformations may be applied to the data before computing the
response surface to predict the response surface that best represents the actual model data. Some of the
transformations that can be used to improve the response surface are Exponential, Logarithmic, Square
Root, Box-Cox etc11. Since there are many transformations that can be used, the optimal transformation that
gives the best fit for the response surface can be obtained using the maximum log-likelihood method12.
Sensitivities
Sensitivity analysis is important in that it helps to quantify the relative importance of the random
variables and define their role in the reliability analysis. The sensitivities of the POF with respect to
parameters of the internal and external variables are computed by taking the derivative of the expected
value of the POF with respect to distributional parameters13. Sensitivities for control variables, however,
are dependent on the sensitivities being intrinsic to the probabilistic design code; otherwise they must be
determined using a numerical differentiation approach for each realization of the external variables. The
sensitivities of control and conditional variables are given by
"Pfs
"
= ˜ E[Pfi ( x˜ , xˆ )]
˜
"# j "# j
where
"Pfs
"
=
E[Pfi ( x˜ , xˆ )]
"#ˆ j "#ˆ j
(9)
"˜ j represents a parameter of the external variable’s probability density function and "ˆ j
represents a parameter of the internal variable’s probability density function.
Sensitivity measures will be particularly helpful for the developer in assessing the importance of the
! variables before resorting to code modifications. !
external
!
Numerical Example
!
A numerical example to demonstrate the methodology is presented. The model consists of the titanium
ring outlined by the advisory circular AC-33.14-114. The geometry cross section is a square titanium ring
under rotation with an external load. The rotation alternates from zero to 6800 RPM. An external pressure
of 50 Mpa (7.25 ksi) is applied on the outer diameter to simulate blade loading. A surface crack with 1-1
aspect ratio is located on the inner bore. Since the model is axisymmetric, only the cross section of the ring
is considered for computation. The finite element model of the problem is shown in Figure 2. The model is
meshed using quadrilateral “plane42” elements. Fracture is defined as the crack the stress intensity factor
exceeding the fracture toughness. The POF is computed in this example is a conditioned on a crack being
present. The probability of having a defect was removed from the analysis for clarity only. The internal
random variable that is considered is the initial crack size. The external random variables are loading
variables - rotational speed (ω) and applied load (Po) and a geometric variable - inner radius (ri). The
random variables with their statistical characteristics are shown in Table 1. Any distribution can be used to
define the external random variables but for simplicity normal distributions are selected. Therefore, their
distributional parameters are defined by their mean and coefficient of variation (COV) as shown in Table 1.
The initial crack size follows an industry defined exceedance curve15 and amin and amax in Table 1 denote the
upper and lower bounds of the exceedance curve.
Variable
Type
External
External
External
Variable
Type
Internal
Name
Type
Omega
Pressure
Inner Radius Ri
Normal
Normal
Normal
Name
Type
Initial crack size
Exceedance curve
Parameter 1
(Mean)
2
712.35 rad/sec
7250 psi
11.81 inch
Parameter 1
(COV)
0.05
0.1
0.02
Parameter 1
(amin)
3.5236 mils2
Parameter 1
(amax)
111060.0 mils2
Table 1 Statistical characteristic of random variables
The ANSYS probabilistic design system (PDS) is used as the control software and ANSYS is used as
the finite element solver. The translator ANS2NEU is used to create neutral files from finite element
geometry and stresses and DARWIN is used to compute the conditional probability-of-fracture.
Po
Element type-Plane42
14400 elements
Speed !
R2
6800
rpm
t
Surface Crack
R1
L
x
!
Figure 2. Geometry of the Advisory Circular AC-33.14-1
Results and Discussion
Using the statistical characteristics given, the probability-of-fracture is computed using the following
methods: 1a) Standard Monte Carlo (no GCE) with DARWIN and 1b) Standard Monte Carlo (MC) with an
analytical MATLAB™ solution that mimics a DARWIN computation, 2) GCE and MC sampling using
ANSYS and DARWIN, and 3) GCE and response surface method using ANSYS™ and DARWIN.
First, the probability-of-fracture is obtained by Monte Carlo sampling using DARWIN considering
only the initial crack size as random. The external variables pressure, rotational speed and inner radius are
considered deterministic. Secondly, the POF is then obtained using Monte Carlo sampling using an
analytical code developed in MATLAB that mimics the DARWIN computations. The results are tabulated
in Table 2. From the table, we see that the expected POF from both methods is in good agreement. This
analysis verifies the analytical solution, which is used as a benchmark for further computations.
Method
Monte Carlo
(DARWIN)
Monte Carlo
(MATLAB)
Random Variables
No. of Samples
Expected POF
Initial Crack size (ai)
10,000
0.1703
Initial Crack size (ai)
10,000
0.1702
Table 2. Results for the POF with the initial crack size as a random variable
Next, the additional random variables are considered and results are obtained using three different
methods. First the total POF is obtained using direct Monte Carlo sampling with the analytical code
developed in MATLAB, which serves as a benchmark. Second, the POF is obtained by Generalized
Conditional Expectation using Monte Carlo sampling by interfacing ANSYS and DARWIN. Third, the
POF is obtained by the Generalized Conditional Expectation method using the response surface (RS)
method by interfacing ANSYS and DARWIN.
The results for all three methods are shown in Table 3. From the table we can conclude that the
expected POFs for each of the three different methods are in good agreement. Comparing Table 2 and 3,
we observe that there is an increase in the POF in Table 3. This can be attributed to the additional random
variables i.e. the external variables. Hence we can say that, the external random variables have significant
effect on the POF and using the GCE to consider additional random variables is useful. The significance of
external variables can be quantified using the sensitivity measures.
Method
Monte Carlo (analytical)
GCE (Ansys (MC) and
Darwin)
GCE (Ansys (RS) and
Darwin)
Random Variables
ai
Omega
Pressure
Inner Radius
ai
Omega
Pressure
Inner Radius
ai
Omega
Pressure
Inner Radius
No of Samples (N)
Expected
POF
1000
0.21300
1000 FE & DARWIN
0.20743
15 FE & DARWIN
with 100,000 MC of
the RS Simulations
0.20787
Table 3. Expected value of POF and variance considering additional random variables
The response surface results require that only 15 ANSYS-DARWIN runs are needed. Since the
response surface method is used to estimate the expected POF, different transformations are used to
determine the expected POF as shown in Table 4. Also, since the response surface is an approximation
method, therefore there will always be some discrepancy between the actual model and fitted RS model. To
account for these errors in the model, various goodness-of-fit measures are performed. Some of the
goodness-of-fit measures includes error sum of squares (SSE), coefficient of determination (R2) and
maximum absolute residual. Table 5 shows goodness for fit measures for different transformations.
Using the equations developed for sensitivities of probability of failure with respect to the parameters
of the random variable probability density functions, the sensitivities for the internal and external variables
of the POF with respect to the distributional parameters are computed. The sensitivities of both internal and
external variables are determined using four methods. First, the sensitivities are determined by direct Monte
Carlo method using MATLAB and serves as a benchmark. Second, the sensitivities are computed using the
equations developed for generalized conditional expectation using Monte Carlo sampling. Third, the
sensitivities are computed using the equations developed for generalized conditional expectation using the
response surface method. Fourth, the sensitivities are obtained using the numerical finite difference
method. Since the random variables pressure, speed and inner radius are normally distributed, the
sensitivity of POF with respect to their mean and standard deviation are determined. The random variable
initial crack size is defined using the exceedance curve15 whose distributional parameters are minimum and
maximum crack size. Hence, the sensitivities are determined with respect to the minimum and maximum
crack size of the exceedance curve. The results of the sensitivities of the POF are tabulated in Table 6.
From the table, we observe that the sensitivities computed using different methods are in good agreement
with one another. This validates the equation developed for sensitivities of the POF with respect to internal
and external variable parameters.
Parameter
Monte Carlo
Sampling
(1000 MATLAB
runs)
GCE
ANSYSDARWIN (1000
DARWIN runs)
GCE
ANSYS-DARWIN
(15 RS and
100,000 MC)
Finite difference
ANSYSDARWIN
(15 DARWIN &
100,000 MC)
0.00316
0.00334
0.00340
0.00351
0.00008
0.000074
0.000090
0.000087
0.1325
0.1156
0.1318
0.1618
"Pf
"µ!
(sec/rad)
!Pf
!µ Po
(1/psi)
!Pf
!µ ri
(1/inch)
Table 6.a Sensitivities of POF with respect to the mean of external random variables.
Parameter
Monte Carlo
Sampling
(1000 MATLAB
runs)
GCE
ANSYS-MCSDARWIN (1000
DARWIN runs)
GCE
ANSYS-RSDARWIN (15
DARWIN and
100,000 MC)
Finite difference
ANSYS-RSDARWIN (15
DARWIN runs
100,000 MC)
0.123E-2
0.143E-2
0.140E-2
0.124E-2
0.500E-4
0.694E-4
0.521E-4
0.551E-4
0.0287
0.0244
0.0268
0.0267
#Pf
#" !
(sec/rad)
"Pf
"! Po
(1/psi)
"Pf
"! ri
(1/inch)
Table 6.b Sensitivity of the POF with respect to the standard deviation of the external
random variables
Monte Carlo
Sampling
(1000 MATLAB
runs)
GCE
ANSYS-MCSDARWIN (1000
DARWIN runs)
GCE
ANSYS-RSDARWIN (15
DARWIN and
100,000 MC)
Finite difference
ANSYS-RSDARWIN (15
DARWIN runs
100,000 MC)
(mils-2)
0.0349
0.0372
0.0371
0.0340
(mils-2)
-6.977E-11
-7.441E-11
-7.398E-11
***
Parameter
!Pf
!amin
!Pf
!amax
*** The sensitivity of the POF with respect to amax is too small to be computed using the finite difference
method
Table 6.c Sensitivity of the POF with respect to the parameters of internal random
variables
Summary and Conclusions
A generalized conditional expectation methodology is presented and demonstrated that can be used to
consider the effects of additional random variables on a probabilistic design code without modifying the
source code. The random variables are partitioned into two categories: “internal” – already contained
within the probabilistic design code, and “external” – additional random variables under consideration. The
methodology involves computing the expected value of the conditional probability-of-failure using
realizations of the external random variables.
Computation of the expected value can be accomplished using Monte Carlo sampling or other similar
methods. In order to reduce the computational effort, the response surface method is used to generate a
relationship, typically second order, between the conditional probability of failure and the external random
variables. The expected value can then be computed using the response surface approximation. Goodnessof-fit tests can be used to confirm the validity of the response surface approximation.
Sensitivity equations have been developed that provide the derivative of the probability of failure with
respect to the parameters of the probability distributions of the external and internal variables.
Numerical examples have been presented in the context of the probability of fracture of a gas turbine
disk. The probability of fracture of the disk and the sensitivities were computed using GCE, with and
without the response surface method and verified against an independent analysis.
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