uncertainty quantification in time- dependent...

Uncertainty Quantification in Time-Dependent Reliability Analysis

1

Zhen Hua, Sankaran Mahadevana, and Xiaoping Dub

a. Department of Civil and Environmental Engineering

Vanderbilt University, Nashville, TN, USA b. Department of Mechanical and Aerospace Engineering

Missouri University of Science and Technology, Rolla, MO, USA

Outline

2

• Background - Time-dependent reliability analysis - Epistemic uncertainty

• Uncertainty quantification in time-dependent reliability analysis - Modeling of epistemic uncertainty in random variables - Modeling of epistemic uncertainty in stochastic processes - Uncertainty quantification of time-dependent reliability

• Numerical examples • Conclusions and future work

Time-dependent reliability analysis

3

0 0( , ) Pr{ ( , ( ), ) 0, [ , ]}e eR t t g t t t t t= ≤ ∀ ∈X Y

( , ( ), ) 0g t t >X YFailure:

Reliability:

Random variables

response

Stochastic processes

0 0

( , ) Pr{ ( ) ( , ( ), ) 0, [ , ]}f e e

p t t G g t tX Y

• The ability that a system performs its intended function over a time period of interest.

• The probability that the first time to failure (FTTF) is larger than a time period of interest.

• The longer the time period, the lower the reliability.

An example

Time-dependent reliability analysis

4

State of the art • Composite limit-state function method (Singh and Mourelatos, 2010, Du, 2013) • Importance sampling approach (Mori and Ellingwood, 1993, Dey and Mahadevan, 2000, Singh and Mourelatos, 2011) • Upcrossing rate-based methods (Madsen, 1990, Zhang and Du, 2011, Hu and Du, 2013) • Extreme value response method (Wang and Wang, 2012, Hu and Du, 2015) • Other sampling-based methods (Hu and Du, 2014, Jiang, et al., 2014, Wang and Mourelatos, 2014) Only aleatory uncertainty

Challenges

• Representation of epistemic uncertainty in time-dependent reliability analysis • Time-dependent reliability analysis including epistemic uncertainty

Epistemic uncertainty

5

Epistemic uncertainty

Data uncertainty Model uncertainty Surrogate model uncertainty

Distribution parameter uncertainty

Distribution type uncertainty

Model parameter uncertainty

Model discrepancy

Method error

Hyper-parameter uncertainty

Prediction uncertainty

Linearization error

• Epistemic uncertainty in random variables • Epistemic uncertainty in stochastic processes • How to efficiently perform uncertainty quantification in time-dependent reliability

analysis?

Modeling of Epistemic Uncertainty in Random Variables

6

θ θθ

θ θ θ

( ) ( )( )

( ) ( )

Lp

L d

xx

x

θ θ θ( ) ( ) ( )p L x x

Data uncertainty Distribution type uncertainty Distribution parameter uncertainty

Bayes’ theorem

Posterior distribution

58 60 62 64 66 68 70 72 74 76 780

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

50 60 70 800

2

4

6

8

X2

Freq

uenc

y

Distribution parameters Mean Standard deviation

Prior distribution

An example

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Modeling of Epistemic Uncertainty in Stochastic Processes

Modeling of stochastic processes

• Karhunen-Loeve (KL) expansion method • Polynomial Chaos Expansion (PCE) • Linear Expansion (LE) method • Orthogonal Series Expansion (OSE) method • Expansion Optimal Linear Estimation (EOLE) method • ARMA or ARIMA model

(0) (1) (2) ( ) (1) ( )1 2 1

( ) ( ) ( ) ( ) ( ) ( ) ( )p qj i j j j i j j i j j i p i j i j i q

Y t Y t Y t Y t t t tϕ ϕ ϕ ϕ ε ω ε ω ε− − − − −= + + + + − − −

ARMA (p, q)

(0)

(1) ( )1j

j

Y pj j

(1) ( )1

(1) ( ) 21

( ) ( 1) ( ) 21

(1 ) , for 0

( ) , for 1, ,

0 for 1

pk j k j k p

qj j q

k k qj j j q k

k

k q

k q

(1) (2) ( )( ) ( 1) ( 2) ( )j j j j

pY j Y j Y j Y

k k k k p

Mean

Auto-covariance

Auto-correlation

Required in time-dependent reliability analysis

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Bayesian ARMA model

For given time series coefficients , the standard deviation in the noise term: φ ω,k k

2 2

1 1

1 ˆ[ ( ) ( )]( 1)

ts tn nj

k i k ij i pts

Y t Y tn n p

Observations Estimations of the time series

φ ω( , )k k k

L DThe likelihood function

Number of trajectories Number of time instants

φ ω φ ω1

( , ) ( , )tsn

ik k k k k ki

L L

D D

φ ω φ ω1 2

( , ) ( ( ), ( ), , ( ) , )i i i ik k k k k k nt k k

L L Y t Y t Y tD

φ ω φ ω φ ω

1 2( ( ), ( ), , ( ) , ) ( ( ) , , ) ( , )

nt nt

i i i ik k k nt k k k nt k k t t k k

L Y t Y t Y t L Y t LY Y

All trajectories

One trajectory

Time instants

φ ω φ ω φ ω( , ) ( , ) ( ) ( )k k k k k k k k

p L D D

9


0 100 200 300 400 500-10

-5

0

5

10

15

t

Y(t)

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30

1

2

3

4

5

6

7

8

9

φ1

PD

F

Prior DistributionPosterior Distribution

-0.4 -0.2 0 0.2 0.40

2

4

6

8

10

12

φ2

PDF

Prior Distribution

Posterior Distribution

-0.02 0 0.02 0.04 0.06 0.080

10

20

30

40

50

φ3

PDF

Prior DistributionPosterior Distribution

21 2 3

( ) 0.8231 ( ) 0.1256 ( ) 0.0812 ( ) (0, 2 )i i i i

Y t Y t Y t Y t N

An example

• Model the time series model based on collected data

• MCMC is used for calibration • Data from MCMC is recorded to

preserve the correlation between posterior distributions

MPP-based time-dependent reliability analysis

10

MPP searches

Upcrossing rate

Random field modeling

Importance sampling

Time-dependent reliability

• The effects of epistemic uncertainty on MPP-based time-dependent reliability analysis • Problems with stationary stochastic processes • Two key elements: and


*Yu

Affects failure threshold in U space and correlation over time

Affects Correlation over time

ρρ

ρ

0 0

0

* *T * *T02

( , ) ( , )

( , )

11 ( ( , ) )

T TL

T T T T

t t t t

t t

t t

X X Y Y

X X Y Y Y Y Y Y

Y Y Y Yu u u u


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Quantify the uncertainty in the time-dependent reliability analysis result

A straightforward way

• Dimensionality of the surrogate model may be high: m(1+p+q)+n×nr

• Computationally expensive

Number of stochastic processes

Number of random variables


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New approach

• Group one: -- random variables and stochastic processes that are exactly modeled • Group two: , -- random variables and stochastic processes with epistemic uncertainty

Classification of random variables and stochastic processes:

0 0( ) , Pr{ ( ) ( , , ) }a

fp t G t g e x y X x y

For given realization of variables with epistemic uncertainty:

aXX φ ω( , )

( )tY

For given value of epistemic parameters and :

0 0( ) ( ( ) , ) ( ) ( )

f fp t p t f f d d x y x y x y

( )f x ( )f y

( , )0

( , )0

2 2 20 ( )

( )

min ( )

( ( ), ( ), ( ))

a

a

t

t

t

g T T T e

ϕ ω

ϕ ω

XX Y

XX Y

u u u

u u u

Main task: how to efficiently obtain the key elements and for given value of the epistemic uncertainty

*Yu


13

Introducing an auxiliary variable:

0( ) Pr{ } ( ) ,

f fp a p fu U u p t x y

φ ω( , )10 0 0

( ) Pr{ ( ( ) , ( )) 0}f a f

p t U p t t X Y

φ ω

φ ω

( , )0

( , )0

0

( )

0 ( )

min ( )

[ , , ]

( ) ( ) ( ), ( )

a t

a f t

t

u

u p t T T

u

X Y

X Y

u

u u u

u u

MPP search

New approach (Cont.)

φ ω

φ ω

( , )0

( , )0

0

( )

( )

min ( )

[ , , ]

( ), ( )

a t

Ca t

t

u

u T T

u

X Y

X Y

u

u u u

u u

Conditional reliability index

• Independent from the distribution of variables with epistemic uncertainty

• Dependent on value of the variables with epistemic uncertainty

• Still applicable when the distribution changes

• Dimension: m(1+p+q)+n×nr m+n

Surrogate model of conditional reliability index

Implementation procedure

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New approach (Cont.)

Numerical Examples

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1 2 1( ) ( )g t X X Y t

A mathematical example

0 1 2 1 0

( , ) Pr{ ( ) ( ) 0, [ , ]}f e e

p t t g X X Y t t

0 20 40 60 80 100140

150

160

170

180

Cycles

Y(t)

50 60 70 800

2

4

6

8

X2

Freq

uenc

y

is exactly modeled , and are modeled based on data. 1

X2

X1( )Y t

Experimental data

Numerical Examples

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0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

[0, t] cycles

Tim

e-de

pend

ent p

roba

bilit

y of

failu

re

0 10 20 300

0.1

0.2

0.3

0.4

PDF

pf(0

, t)

"True" pf curve

95% Prediction Interval

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

5

10

15

Time-dependent probability of failure

Prob

abili

ty D

ensit

y Fu

nctio

n (P

DF)

Aleatory+Epistemic"True" pfOnly aleatory

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

5

10

15

20

25

30

35

40

45

Time-dependent probability of failure

Prob

abili

ty D

ensit

y Fu

nctio

n (P

DF)

100 Cycles200 Cycles500 Cycles"True" pf

A mathematical example (Cont.)

UQ results of time-dependent reliability analysis

Last time instant

More data

Numerical Examples

17

A beam example

220 0

0 0

( ) 1( , ( ))

4 8 4b st b

u

F t L a b Lg t a bX Y

Numerical Examples

18

0 5 10 15 20 25 30 35 40 45 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

[0, t] cycles

Tim

e-de

pend

ent p

roba

bilit

y of

failu

re

0 100 2000.02

0.04

0.06

PDF

pf(0

, t)

"True" pf curve

95% Prediction Interval

0.025 0.03 0.035 0.04 0.045 0.05 0.0550

20

40

60

80

100

120

140

pf (0, 50)

PDF

Aleatory+Epistemic"True" pfOnly aleatory

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.0550

20

40

60

80

100

120

pf(0, 50)

PDF

1000 Cycles

"True" pf

200 Cycles

2000 Cycles

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.0650

50

100

150

200

250

pf (0, 50)

PDF

1000 Cycles

2000 Cycles200 Cycles

2000 Cycles and 2000 Samples"True" pf

A beam example (Cont.)

• Uncertainty in the reliability analysis result is more sensitive to the epistemic uncertainty in the random variable

More stochastic load data More random variable data

Conclusions

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• A framework is developed for time-dependent reliability analysis in the presence of data uncertainty

• A new method is proposed to improve the efficiency of uncertainty quantification in time-dependent reliability analysis

• Numerical examples demonstrated the effectiveness of the proposed method

Future work

• Inclusion of other sources of epistemic uncertainty in time-dependent reliability analysis (i.e. model uncertainty)

• Time-dependent sensitivity analysis to quantify contributions of epistemic uncertainty sources

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• Air Force Office of Scientific Research (Grant No. FA9550-15-1-0018, Technical Monitor: Dr. David Stargel)

• National Science Foundation through grant CMMI 1234855

• The Intelligent Systems Center at the Missouri University of Science and Technology.

Acknowledgement

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Thank You

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