a u.s. army center of excellence for modeling and...

REC 2014; Chicago, IL 1

Recent Advances in Reliability Estimation of Time-Dependent Problems Using the Concept of

Composite Limit State

Automotive Research CenterA U.S. Army Center of Excellence for Modeling and Simulation of Ground Vehicles

Zissimos P. Mourelatos

Monica Majcher

Vijitashwa Pandey

Mechanical Engineering Department

Oakland University


Background information

Definition of time-dependent Pf

Out-crossing rate approach

Proposed approach to estimate time-dependent Pf using

Composite Limit State (CLS)

Identification of CLS

Calculation of time-dependent Pf

Implementation points

Summary and future work

Overview


SystemInput Output

Uncertainty

(Quantified)Uncertainty

(Calculated)

Propagation

Design

• Random Variables (Time-Independent)

• Random Processes (Time-Dependent)

Challenges:

• Quantification of a Random Processes

• Estimation of time-dependent reliability

Background


Time-Dependent Probability of Failure

2t Ttn 3t01 t

timet

n

ii

ti

tgPTfP

1

0,,,0 YX

0,,:,0,0 ttgTtPTf

P YX

Series System Reliability Problem


Time-Dependent Probability of Failure

T

ttif

PTf

P0

dexp011,0

Failure rateOut-crossing rate approach

0),(,0),(,

0lim

ttgttgPt

YXYX

tt

Up-crossing rate

Accurate ONLY if up-crossings are

statistically independent


K-L expansion is used to represent random process Y(t) as :

: mean of random process Y(t)

: eigenvalue and eigenvector of covariance matrix of Y(t)

: standard normal random variable

tY

tΦ,

Z

Y(t) is assumed Gaussian

p

i

iTiiY ZtttY

1

Φ

Characterization of Input Random Process


The Composite Limit State defines a convex

domain representing the intersection of safe

regions of all instantaneous limit states

Concept of Composite Limit State is used

Calculation of Time-Dependent

Probability of Failure (Pf)

0,,:,0,0 ttgTtPTf

P YX


Identification of Composite Limit State

z1

z2

Composite limit state

*iz

*kz

cosik

Correlation

Coefficient

Two-step approach to identify composite limit state



z1

z2


*iz

*kz

Step 1: Delete highly correlated

instantaneous limit states (almost parallel) 99.0ij



z1

z2


*iz

*kz

Step 2:

Delete instantaneous limit

states that are not part of

the composite

Check by solving a series of LPs

jii

itg

jtg

,1

0,,0,: ZZZIf set is null,

the jth limit state is deleted


Calculation of Time-Dependent Pf

z1

z2


*iz

*kz

dzztt

ttP

ij

ji

jif

ij

0

;,

i

Bivariate standard normal vector



Our approach calculates Pf exactly as

by eliminating ALL other terms using the convex

polyhedron of the safe domain

This is a substantial contribution in

both Time-Dependent Reliability and

System Reliability



G

F

H

I

z1

z2

g3


g6 g4 g

5

B

B A

C

D

E

z2*

z3*

z1

*

0,,*** kjikkjj zzz

*2zIf is a positive linear

combination of and ,

( )

can be eliminated

enlarging the safe domain

(SD) so that :

*1z

2g

*3z

fABGGCDEFGABCDEFA PPPFP 11

Original SDEnlarged

SD

ffffABG PPPgggPP 1323122321 0,0,0



G

F

H

I

z1

z2

g3


g6 g4 g

5

B

B A

C

D

E

z2*

z3*

z1

*

f

FHIf

DHEf

ABGGCIf

DHEf

ABG

GCHFGf

ABGGCDEFGABCDEFA

PPPPPP

PPPPFP

1

111

fffGCI PPPP 1341341431

Finally:

This is an EXACT

calculation of Pf

involving the convex

polyhedron of the SD


Example : Hydrokinetic Turbine Blade under Time-dependent

River Flow Loading*

EI

tCtv

EI

tMtg m

allow

flap

allow2

1

21

: Gaussian random process with

autocorrelation coefficient function

tv

1221 2cos, ttttv

*Hu, Z. and Du, X., (2012), “Time-dependent Reliability Analysis by a Sampling Approach

to Extreme Values of Stochastic Processes,” Proceedings of the ASME 2012 IDETC/CIE

is calculated from 0 to 12 monthsFP

Illustration of Composite Limit State

: Allowable strain

: Random variables

: ConstantsECm ,,

allowItallow ,, 1


t = 8.2 months


Instantaneous Limit State

Safe

Region



Time, months

Rel

iab

ilit

y I

nd

ex t

t = 8.2 months


0.2 month discretization


t = 5.2 months


Instantaneous Limit State

Safe

Region



Time, months

Rel

iab

ilit

y I

nd

ex t

t = 5.2 months






-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

z1

z2


Probability of Failure Calculation


Implementation Points

Observations:

• Proposed approach requires a time-

independent analysis (beta and MPP) at ALL

time steps.

• Only low-beta limit states contribute to

composite limit state.

To avoid calculating beta and MPP at ALL time

steps:

• Build a surrogate of beta curve using Kriging

• Build composite limit state (CLS) progressively

starting with times where beta is low.

• Stop if CLS does not change further.


Reliability Index Estimation using Kriging



Only 14 Evaluations


β(t

)

Time t

5Δt 10Δt

15Δt

20Δt

Progressive Estimation of Composite


Total Probability Theorem

Proposed approach is based on FORM.

FORM’s accuracy deteriorates if

• Limit state is nonlinear

• Random variables are non-normal and / or

correlated.

The Total Probability Theorem can increase

accuracy

wwWW

dfFPFPTfP

,0

Mean value of conditional probability


Total Probability Theorem

032,, 15

2

2124

23

221

31 tYXtYXtXtXXXXttg YX

tYtY 21 , : Gaussian Processes

542 ,, XXX : Non-normal R.V.s

Example :

tYxtyxtxtXxxxtg 15

2

2124

23

221

31 32,

tYXXXX 25421W W

FPIf , then is calculated using

Realization of X1Normal R.V. Function of

Normal R.V.s

Linear function of Normal R.V.s


Key Points of Composite Limit State

Method

Efficient estimation of reliability index

Kriging metamodel using prediction variance estimation

Identification of Composite Limit State using

Convex domain formed by instantaneous limit states with

“smallest” betas

Calculation of using Composite Limit State

Exact calculation using the convex polyhedron of the safe

domain

fP

t

t


Developed a new time-dependent reliability

estimation method using the concept of Composite

Limit State (CLS). It

Identifies the CLS automatically

Calculates time-dependent probability of

failure exactly for convex linear safe sets

Summary and Future Work

Improve computational efficiency (e.g. use Kriging

to estimate MPP locus)

Apply method to estimating remaining life due to

fatigue failure

Future Work


Thank you for your

attention


A novel MC-based method to calculate the

time-dependent reliability (cumulative

probability of failure) based on :

short-duration data and an

exponential extrapolation using MCS

or Importance Sampling (Infant

Mortality)

Poisson’s assumption (Useful Life)

Our Approach


btet 0)(ˆ

00

ˆ1

tdt

db

MCS

Exponential

Extrapolation

0

Poisson’s Assumptiontint

Efficient MC Simulation Approach

],[,))(1(1

],0[,1)(

int

)(

int

int

ˆ

int

0

f

ttc

T

dtt

c

T

tttetF

ttetF

m

t


Objectives and Scientific Contributions

• Develop a methodology to estimate reliability and remaining life of a vehicle system using time-dependent reliability / durability principles.

• Use the methodology to improve existing accelerated Life Testing (ALT) methods by

– Shortening testing time, and

– Using realistic testing conditions

• Implement all developments in TARDEC’s Physical Simulation Lab

Objectives for Project 5.3

• Developed advanced statistical methods to calibrate a math model using a limited number of tests.

• Developed a novel time-dependent reliability method using the concept of composite limit state. The method also advances state-of-the-art in system reliability

• Developed a new paradigm for Accelerated Life Testing

Fundamental Scientific Contributions


Relates reliability measured under high

stress conditions to expected reliability

under normal conditions

Advantage:

Shortens testing time

Disadvantage:

Uses unrealistic testing conditions

Our Goal: Shorten

testing time and use

realistic testing

conditions

Accelerated Life Testing (ALT)


Vehicle speed : 20 mph; Mission distance : 100 miles

Simulation can be practically performed for a

short-duration time

0 200 400 600 800 1000-3

-2

-1

0

1

2

3

4

Roa

d H

eigh

t, in

Longitudinal Distance, ft

20 40 60 80 100

-2

-1

0

1

2

3

Time

Ver

tica

l A

ccel

erat

ion

in

G, S

(d,X

,t)

Durability/Performance

Measures in Time

Terrain, Engine

Load, etc.

Random Variables

Problem Description


0 200 400 600 800 1000-3

-2

-1

0

1

2

3

4

Roa

d H

eigh

t, in

Longitudinal Distance, ft

20 40 60 80 100

-2

-1

0

1

2

3

Time

Ver

tica

l A

ccel

erat

ion

in

G, S

(d,X

,t)

Durability/Performance

Measures in Time

Terrain, Engine

Load, etc.

Random Variables

Model input random processes (terrain, engine load)

Develop detailed, and yet simple and accurate vehicle

math models

Run math models for long time

Major Challenges:

Observations / Challenges


Model input random processes (terrain, engine load)

Time series and spectral decomposition methods

Develop detailed, and yet simple and accurate vehicle math

models

Use available math models

Calibrate math models using tests to improve their

accuracy

(Model V&V approach)

Run math models for long time

Run calibrated math models for a short duration

Our Approach to Address Challenges


Random process

characterized by

time series

Available math model

calibrated using tests

years

tC

Degraded vehicle

parameter (e.g. or )sk sb

t years

Response

years

Proposed Approach for ALT

REC 2014; Chicago, IL 48 t years

Response

months

Reliability at time t is the probability that the system

has not failed before time t.

Calculate reliability through time

Must calculate

time-dependent

probability of

failure

Main Task in Proposed ALT

REC 2014; Chicago, IL 497/23/2014

Definitions / Observations

0 Lt time

L

c

T tF

Time-Variant

Reliability0 Lt time

L

i

T tF

Time-Invariant Reliability

Reliability: Ability of a system to carry out a function in a time period

[0, tL]

Cumulative Prob.

of Failure 0,,,0 ttgthatsuchttPtF LL

c

T X

Instantaneous Prob. of Failure 0, LLL

i

T ttgPtF X

L

c

TL

c

f tFttPp Prob. of Time to Failure

REC 2014; Chicago, IL 507/23/2014

Definitions / Observations

0 Lt time

L

c

T tF

Time-Variant Reliability

0 Lt time

L

i

T tF

Time-Invariant Reliability

Reliability: Ability of a system to carry out a function in a time period [0, tL]

Cumulative

Prob. of Failure 0,,,0 ttgthatsuchttPtF LL

c

T X


i

T ttgPtF X

L

c

TL

c

f tFttPp Prob. of Time to Failure

REC 2014; Chicago, IL 517/23/2014

Calculation of Cumulative

Probability of Failure

1t Ft2Kt 1Kt2t0

timet

tKtF

• State-of-the-art Approaches

PHI2 method (Andrieu-Renaud, et al., RESS, 2004)

Set-Based approach (Son and Savage, Quality & Rel.

Engin., 2007)

• State-of-the-art approaches are in general, inaccurate due to:

Choice of

Not including contribution of all discrete times

t


What is Reliability?Cumulative Probability of Failure

Cumulative

Prob. of Failure 0,,,0 ttgthatsuchttPtF LL

c

T X


i

T ttgPtF X

Reliability at time t is the probability that the system

has not failed before time t.

Maximum Response Method

Niching GA & Lazy Learning Local Metamodeling

MCS / Importance sampling

Analytical

Simulation-based

Calculation Methods for tF c

T

t

c

T dtttF0

]exp[1


HMMWV Control arm / Spring

STM Table

Control

Arm

Actuation

point for

road input

Output : Stress or

Strain at different

locations on control

arm

Test Fixture at TARDEC

REC 2014; Chicago, IL 54MotionView Math Model Physical Model

HMMWV Lower Control Arm Fixture


Characterization of random processes

Calculation of time-dependent

probability of failure (or reliability)

Model validation through calibration

Time series and Spectral

Decomposition


Approach

Companion

project

Prediction bias Zero mean random error

,,,,, xxcxx mt yy

Key Points of Proposed Approach

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